QUANTUM DOTS: RESEARCH, TECHNOLOGY AND APPLICATIONS
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QUANTUM DOTS: RESEARCH, TECHNOLOGY AND APPLICATIONS
RANDOLF W. KNOSS EDITOR
Nova Science Publishers, Inc. New York
Copyright © 2009 by Nova Science Publishers, Inc.
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CONTENTS Preface
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Chapter 1
Few-Electron Semiconductor Quantum Dots in Magnetic Field: Theory and Methods Orion Ciftja
1
Chapter 2
Investigations of Electronic States in Self-assembled InAs/GaAs Quantum-Dot Structures Shiwei Lin and Aimin Song
47
Chapter 3
Chemically Deposited Thin Films of Close Packed Cadmium Selenide Quantum Dots: Photophysics, Optical and Electrical Properties Biljana Pejova
109
Chapter 4
Numerical Modelling of Semiconductor Quantum Dot Light Emitters for Fiber Optic Communication and Sensing Mariangela Gioannini
169
Chapter 5
Quantum Dot Technology for Semiconductor Broadband Light Sources C.Y. Ngo, S.F. Yoon and S.J. Chua
203
Chapter 6
Quantum Dots in Medicinal Chemistry and Drug Development Ian D. Tomlinson, Michael R. Warnement and Sandra J. Rosenthal
243
Chapter 7
Strain Relief and Nucleation Mechanisms of InN Quantum Dots J.G. Lozano, A.M. Sánchez, R. García, S. Ruffenach, O. Briot and D. González
267
Chapter 8
Electronic Structure and Physical Properties of Semiconductor Quantum Dots Xiu-Wen Zhang, Yuan-Hui Zhu and Jian-Bai Xia
299
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Chapter 9
Ge Nanoclusters in GeO2 Films: Synthesis, Structural Research and Optical Properties V.A. Volodin and E.B. Gorokhov
331
Chapter 10
Model for the Coherent Optical Manipulation of a Single Spin State in a Charged Quantum Dot Gabriela M. Slavcheva
371
Chapter 11
Sub-diffraction Quantum Dot Waveguides Chia-Jean Wang and Lih Y. Lin
393
Chapter 12
Three-Dimensional Imagings of the Intracellular Localization of mRNA and Its Transcript Using Nanocrystal (Quantum Dot) and Confocal Laser Scanning Microscopy Techniques Akira Matsuno, Akiko Mizutani, Susumu Takekoshi, R. Yoshiyuki Osamura, Johbu Itoh, Fuyuaki Ide, Satoru Miyawaki, Takeshi Uno, Shuichiro Asano, Junichi Tanaka, Hiroshi Nakaguchi, Mitsuyoshi Sasaki, Mineko Murakami and Hiroko Okinaga
413
Chapter 13
Unified Description of Resonance and Decay Phenomena in Quantum Dots Ingrid Rotter and Almas F. Sadreev
427
Chapter 14
Theoretical Study on Quantum Dots Using Effective-Mass Envelope Function Theory Shu-Shen Li and Jian-Bai Xia
493
Chapter 15
Transmission through Quantum Dots with Variable Shape: Bound States in the Continuum Almas F. Sadreev, Evgeny N. Bulgakov, Konstantin N. Pichugin, Ingrid Rotter, and Tatyana V. Babushkina
545
Chapter 16
Optical Properties of Quantum Dots: Possible Control of the Impurity Absorption Spectra and Factor of Geometric Form V.D. Krevchik, M.B. Semenov and R.V. Zaitsev
577
Chapter 17
Post-growth Energy Bandgap Tuning of InAs/InGaAs/InP Quantum Dot Structures: Intermixing of Quantum Dot Structures Tang Xiaohong and Yin Zongyou
623
Chapter 18
Application of Quantum Dots in Organic Memory Devices: A Brief Overview Kaushik Mallick and Michael .J Witcomb
651
Index
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PREFACE Since first developed in the early sixties, silicon chip technology has made vast leaps forward. From a rudimentary circuit with a mere handful of transistors, the chip has evolved into a technological wonder, packing millions of bits of information on a surface no larger than a human thumbnail. And most experts predict that in the near future, we will see chips with over a billion bits. Quantum dots are small devices that contain a tiny droplet of free electrons. They are fabricated in semiconductor materials and have typical dimensions ranging from nanometres to a few microns. The size and shape of these structures and therefore the number of electrons they contain can be precisely controlled; a quantum dot can have anything from a single electron to a collection of several thousands. The physics of quantum dots shows many parallels with the behavior of naturally occurring quantum systems in atomic and nuclear physics. As in an atom, the energy levels in a quantum dot become quantized due to the confinement of electrons. Unlike atoms however, quantum dots can be easily connected to electrodes and are therefore excellent tools for studying atomic-like properties. This new book presents the latest research developments in the world. Semiconductor quantum dots represent nanoscale systems with few electrons confined in a semiconductor host crystal. The importance of semiconductor quantum dots lies primarily in their tunability and sensitivity to external parameters as electrons are confined in all dimensions. The bulk of semiconductor quantum dots are fabricated by applying a lateral confinement potential to a two-dimensional electron gas. Quantum confinement profoundly affects the way electrons interact with each other and with external parameters, such as a magnetic field. Quantum confinement of electrons is just one of several ways quantum mechanics reveals itself. Another pure quantum phenomena associated with electrons is their spin. An external magnetic field affects both orbital and spin motion of electrons. External control of the full quantum wave function in a semiconductor quantum dot may lead to novel technological application involving both charge and spin. From a theoretical point of view, semiconductor quantum dots represent a unique opportunity to study fundamental quantum theories in a tunable atomic like set-up. In Chapter 1, the author reviews some of the theoretical approaches used to study two-dimensional few-electron semiconductor quantum dots. The main emphasis is to clarify the relations between different theories and methods for few-electron semiconductor quantum dots in an external parameter, a perpendicular magnetic field. Properties of few-electron semiconductor quantum dots in the weak magnetic regime are explained well through single-electron theory concepts. However, challenges do exist when considering stronger external magnetic fields. A strong magnetic field, when applied
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perpendicular to the quantum dot, changes the quantum nature of the electronic correlations and spin-polarizes the electrons. As the strength of the external magnetic field increases, the confined electrons start to manifest collective quantum behavior as seen in the integer and fractional quantum Hall effect regime. Theoretical and computational challenges to studies of semiconductor quantum dots as the magnetic field changes from weak to strong are reviewed. Specific examples are introduced to illustrate the transformation of the quantum wave function into a Laughlin-like one as the magnetic field increases. Space-charge techniques, such as capacitance-voltage (CV) spectroscopy and deep-level transient spectroscopy (DLTS), are used to examine the electronic states of ensembles of selfassembled InAs quantum dots (QDs), embedded in a GaAs matrix and grown by the 3D Stranski-Krastanow growth mode. In Chapter 2 the authors present direct experimental evidence of the coexistence of deep levels in the same epitaxial layer of optically active quantum dots. The InAs quantum dots show very good optical properties, as evidenced by the strong photoluminescence (PL) at room temperature at ~1.3 μm. The reverse-bias dependence of the DLTS signal together with results from the reference samples, containing thin InAs layers but no quantum dots, confirms that the deep levels coexist in the dot layer and are most likely caused during the lattice-mismatched growth process. Laplace deep-level transient spectroscopy (LDLTS) is a technique developed primarily to study the point defects in semiconductors, which has also recently been applied to the semiconductor quantum-dot structures. The newly developed technique can provide orders of magnitude better resolution than the conventional DLTS method. By applying the LDLTS technique, the authors are able to study the electronic fine structure of the deep-level states coexisting in the dot layer. As a way of tuning the electronic properties, postgrowth rapid thermal annealing (RTA) has been applied to the semiconductor quantum dots, and the induced optical and electrical changes are studied using PL and DLTS. These combined optical and electrical experiments also confirm our findings of the coexistence of the deep levels with the QDs. By a comparison of the DLTS data with the PL spectra, the authors find that the effects of RTA on the optical spectra are closely linked with the alternations of the electronic structures, and that a new deep level (0.62 eV) is created in the structure, which dominates the whole spectra at certain annealing temperatures. Furthermore, by combining the CV, conventional and Laplace DLTS techniques, the authors systematically and quantitatively investigate the underlying emission mechanisms in the QD single-level two-electron system. Electron emissions from the singly and doubly occupied QD s states can be resolved by the LDLTS technique. The emission processes are investigated in detail by the pulse-bias dependency. The electron distribution profile in quantum dots is identified by applying an appropriate set of voltage pulses across the Schottky diode structure. A recently developed chemical method for synthesis of close packed cadmium selenide quantum dots (QDs) in thin film form is reviewed in Chapter 3. By controlling the chemical composition of reaction solution and post-deposition treatment, the presented method permits optoelectrical properties of CdSe QD thin films to be designed. Synthesized CdSe QDs crystallize in cubic crystalline system and are characterized with high chemical and crystallographic purity. Such properties of synthesized CdSe QD thin films are quite distinct in comparison with QDs of the same material synthesized by other methods reported in the literature. The average crystal radius of as-deposited CdSe QDs, calculated by the Debye-
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Scherrer approach, is 2.6 nm. Upon annealing at 300 oC, this value increases to 12 nm. Optical band gap energies of as-deposited and thermally treated CdSe QD thin films are 2.08 and 1.77 eV correspondingly. The notable blue shift of band gap energy of 0.34 eV for asdeposited thin films with respect to the bulk value is due to the pronounced quantum size effects. Upon thermal treatment, the absorption edge of CdSe thin films is red shifted and the band gap energy tends to approach the bulk value. The experimental blue shifts of the band gap energies of as-deposited and annealed CdSe QD thin films (with respect to the corresponding bulk value) were compared with the theoretical ones, predicted by the effective mass approximation model. Electrical and photoelectrical properties of the synthesized cubic CdSe QDs in thin film form (including the relaxation dynamics of photocarriers) were investigated as well. On the basis of measured temperature dependence of dark electrical resistance of annealed CdSe QD thin films in the intrinsic conductivity region, thermal band gap energy value of 1.85 eV was calculated (corresponding to 0 K). In lower-temperature region, the conductivity of QD thin films was shown to be a two-channel temperatureactivated process with activation energies of 0.74 and 0.43 eV. The possible physical interpretations of these values are discussed. Time-resolved studies of photoconductivity relaxation dynamics showed that within a very short starting time interval immediately upon light excitation switch off (< 0.2 ms) the photocarriers are relaxed according to the quadratic relaxation mechanism. After about 0.2 ms the non-equilibrium charge carriers recombine according to the linear relaxation mechanism with a relatively high relaxation time value of 0.4 ms. This indicates a potential applicability of the synthesized QD thin films in solar cells engineering. In Chapter 4 the author presents a review of her research work on the modelling of the optical properties of light emitting devices having a semiconductor quantum dot material as active region. The gain region is obtained by a Strasky-Krastanov growth of several layers of quantum dots that are not uniform in size. This causes an inhomogeneous broadening of the gain spectrum that is a peculiar characteristics of these light emitters. The numerical model is based on a multi-population rate equation model used for describing the dynamics of electrons and holes in an inhomogeneous material and in the several energy states confined in the dots. The rate equations of the carriers are also coupled with the rate equations of the photons generated by spontaneous and/or stimulated emission. In this review the author provides several examples of simulation results of the optical characteristics of InAs/GaAs quantum dot semiconductor lasers and superluminescent diodes emitting in the near infrared with application in optical communications, sensing and optical coherent tomography. In particular, the author shows how the inhomogeneous gain broadening and the presence of more than one confined energy state in the dots can influence the laser properties such as the shape of the emitted spectrum, the maximum modulation bandwidth and the frequency fluctuations (chirp) under large signal modulation. The results of this analysis gives useful insights on the meaning, in the quantum dot case, of various parameters (linewidth enhancement factor, differential gain) that are routinely measured in the lab with the standard characterization techniques for semiconductor quantum well or bulk lasers. The author also provides some examples of calculated emission characteristics (light versus current curves and output spectra) of quantum dot superluminescent diodes to highlight the relevant differences respect to the laser case. The author also shows how the inhomogeneous broadening of the gain, the quantum dot layer composition, and the device geometry can be
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engineered to get bright sources with broad spectrum useful for medical and sensing applications. Semiconductor broadband light sources (e.g., superluminescent diodes) are important light sources for fiber optic gyroscopes and sensors, optical fiber communications, and biomedical imaging. To minimize undesired interference effects in these applications, low temporal coherence of the light sources is desired. Since the wider the emission spectrum, the lower the temporal coherence; there have been great efforts to increase the spectral bandwidth of the emission spectrum so as to improve the performance of the abovementioned applications. Quantum dots (QDs) have recently been proposed as the perfect material for broadband light sources since the inherited inhomogeneity of the self-assembly QD growth mode is an intrinsic advantage for wideband emission. In Chapter 5, both the broadband light sources (together with its applications) and the justification for the use of QDs (as compared to higher dimensional systems) were mentioned. Existing methods to increase the spectral bandwidth were discussed. In particular, the authors’ theoretical and experimental approaches to optimization of the InAs QD layers for high areal density and wideband emission were presented. The origins of the high radiative efficiency and wideband emission of the optimized QD sample were also determined. Lastly, the potential challenges associated with the use of QDs were highlighted with the solutions proposed. Quantum dots have increasingly been incorporated into a wide variety of biological assays as improved fluorescent probes. Their photophysical properties permit the investigation of cellular processes and biological phenomena with unprecedented spatial resolution and temporal longevity. Consequently, quantum dots are poised to facilitate advances in future drug development applications. Multiplexed detection in whole cell assay format may ultimately provide added insight into the extremely complex biochemical mechanisms involved in drug receptor interactions. Chapter 6 provides a detailed discussion of biological applications which have incorporated quantum dot detection, with a particular emphasis on their possible integration into drug discovery and medicinal chemistry applications. As presented in Chapter 7, in recent years, Indium nitride (InN) based nanostructures are the focus of special and increasing attention. The combination of the intrinsic properties of InN –best theoretical electronic properties among the III-nitrides and a recently established bandgap of 0.7 eV suitable for the telecommunications field– with those related to quantum confinement phenomena promises interesting applications. Here the authors present a complete characterization by transmission electron microscopy (TEM) of uncapped and GaN capped InN quantum dots grown on sapphire/GaN substrates by MOVPE. Morphological aspects such as height, area or roundess of the QDs, as well as the effect of the GaN capping layer on them will be discussed. The nucleation mechanisms of the InN QDs will be studied, showing that they preferentially nucleate on top of pure edge type threading dislocations located in the GaN and that do not propagate into the QDs. This mechanism of InN QDs nucleation on GaN has never been reported before, and has to differ notably of the more classical ones found in the literature, like the Burton-Cabrera-Frank mechanism, since the Burgers analysis showed that these dislocations present a pure edge character; or the StranskiKrastanov model, since the relaxation occurs by the formation of the misfit dislocations network instead of by surface islanding. Finally, the strain state of the QDs will be also reported, showing that they are almost fully relaxed due to the introduction of a misfit dislocations (MD) network in the interface QD/GaN. Strain maps at atomic scale in plan-view
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orientation allow a complete characterization of this network, consisting of three sets of misfit dislocations lying along the <11-20> directions without interaction between them or generation of threading dislocations. One of the most important challenges in order to achieve functional InN QDs based devices is the obtention of a good crystalline quality capping layer due to the difficulties associated to its growth, namely the low temperature neccesary to avoid the previously InN deposited decomposition. In this case, it was succesfully achieved and unexpectedly, the introduction of this GaN capping layer is shown to induce a rearrangement of these MDs, lowering the degree of plastic relaxation of the heterostructures. Along with growing of synthesizing methods of semiconductor quantum dots, they are widely investigated experimentally and theoretically. The electronic structure and optical, magnetic property of colloidal quantum dots are investigated in the framework of effective mass envelope function theory by expanding the envelope function in spherical Bessel functions and spherical harmonic functions. On the basis of calculating the energy levels and envelope functions the various physical properties of semiconductor quantum dots are investigated. Chapter 8 will be organized as following: 1. Effective-mass envelope function theory for quantum dots. 2. Polarization properties of emission, including: strong linear polarization along the c-axis of wurtzite quantum ellipsoids, circular polarized property of wurtzite quantum dots ensemble in the magnetic field. 3. Electon g factors, including: electron g factors as functions of size and shape of dots, direction of magnetic field, and electric field tunable electron g factor of quantum dots. 4. Highly anisotropic Stark effect of quantum ellipsoids. 5. Giant Zeeman splitting, including: Zeeman splitting energies as functions of radius of dots, Mn ion concentration, magnetic field, highly anisotropic Zeeman splitting in wurtzite quantum dots, and radius sensitive Zeeman splitting of zero-gap quantum dots. 6. Curie temperature of DMS quantum dots, including: definition of Curie temperature in quantum dots, effect of hole number on the Curie temperature, room temperature ferromagnetism of (Zn,Mn)O quantum dot, electric field tunable ferromagnetism of quantum dots, and highly anisotropic ferromagnetism in oblate quantum dots. Semiconductor nanostructures, namely, quantum dots and quantum well wires, attract a lot of interest due to its new electronic and optical properties that can be modified artificially. The quantum size effects in semiconductor quantum dots lead to possibility of application of semiconductors with indirect band structure (Si and Ge) in optoelectronics. The germanium has several advantages comparing with silicon (relatively low temperature of processing, bigger Bohr radius, bigger electrical permittivity). In Chapter 9, germanium nanoclusters in GeO2 films have been obtained with the use of two methods. The first method of Ge nanocluster formation is a film deposition from supersaturated GeO vapor with subsequent dissociation of meta-stable (in solid phase) GeO on hetero-phase system Ge:GeO2. The second method is growth of anomalous thick native germanium oxide layers with chemical composition GeOx(H2O) during catalytically enhanced Ge oxidation, x~1. The obtained films were studied with the use of photoluminescence, Raman scattering spectroscopy, IR-spectroscopy, ellipsometry, high-resolution electron microscopy. Strong photoluminescence signals were detected in GeO2 films with Ge nanocrystals at room temperature. “Blue-shift” of the photoluminescence maximum was observed with reducing of Ge nanocrystal size in anomalous thick native germanium oxide films. So, the correlation between reducing of the Ge nanocrystal sizes (estimated from position of Raman peaks) and photoluminescence “blue-shift” was observed. The Ge nanocrystals presence was confirmed by high-resolution electron microscopy data. The
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optical gap in Ge nanocrystals was calculated with taking into account quantum size effects and compared with the position of the experimental photoluminescence peaks. It can be concluded that a Ge nanocrystal in GeO2 matrix is a quantum dot of type I. It was shown that “band gap engineering” approaches can lead to creation of Ge:GeO2 heterostructures with required properties. The possibility of relatively low-temperature crystallization of dielectric GeO2 based film was demonstrated, the crystallized films have device quality. This heterostructures can be perspective for using in opto-electronics, for creation of elements of quasi-nonvolatile MOS memory, etc. In Chapter 10, the optically-driven coherent dynamics associated with the single-shot initialization and readout of a localized spin in a charged semiconductor quantum dot embedded in a realistic structure is studied theoretically using a new Maxwell-pseudospin model. Generalized pseudospin master equation is derived for description of the time evolution of spin coherences and spin populations in terms of the real state pseudospin (coherence) vector including dissipation in the system through spin relaxation processes. The equation is solved in the time domain self-consistently with the vector Maxwell equations for the optical wave propagation coupled to it via macroscopic medium polarization. Using the model the long-lived electron spin coherence left behind a single resonant ultrashort optical excitation of the electron-trion transition in a charged QD is simulated in the low- and highintensity Rabi oscillations regime. Signatures of the polarized photoluminescence (PPL) resulting from the numerical simulations, such as the appearance of a second echo pulse following the excitation and a characteristic non-monotonic PPL trace shape, specific for initial spin-up orientation, are discussed for realization of high-fidelity schemes for coherent readout of a single spin polarization state. Quantum dots (QD) have been popularized in biological tagging applications and low threshold lasers. However, the unique 3D confinement, size and surface chemistry properties may also be employed for high component density photonic circuit applications. With conventional dielectric waveguides subject to the diffraction limit, the authors proposed the QD cascade array, which operates on the principle of stimulated emission of a signal light given a pump excitation source. The device is designed to guide light within several hundreds of nanometers or smaller. In Chapter 11, the authors focus on the modeling, fabrication and experimental results, which together form a comprehensive discussion. In particular, simulation of the gain, inter-dot coupling and overall transmission behavior provide theoretical insight. Furthermore, two different fabrication processes are outlined, implemented and compared. Finally, a presentation of the measured loss and crosstalk characteristics under a near field optical test setup reveals that the QD nanophotonic waveguide is a technique with high potential for sub-diffraction guiding and opens up an opportunity to create wavelength specific, nanoscale optical logic structures. Confocal laser scanning microscopy (CLSM) combined with computed imaging analysis enables observation of subcellular organelles, mRNA and protein, three-dimensionally, in routinely processed light microscopic specimens. Meanwhile, recently developed semiconductor nanocrystals (Quantum dots, Qdots), which do not fade upon exposure to light, enables generation of multicolor images of molecules due to a narrow emission peak that can be excited via a single wavelength of light. Qdots have recently been used in biological research, and they are utilized to detect signals of immunohistochemistry and fluorescence in situ hybridization (FISH). Recently, the authors successfully applied the above-mentioned advantages of Qdots and CLSM to three-dimensional imagings of the
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intracellular localization of mRNA and protein. In Chapter 12, the authors describe their new technique of three-dimensional imaging using Qdots and CLSM and discuss the advantages of this method. In situ hybridization and immunohistochemistry using Qdots combined with CLSM can optimally illustrate the relationship between protein and mRNA simultaneously in three dimensions. Such an approach enables visualization of functional images of proteins in relation with mRNA synthesis and localization. The authors exploit the analogy between light nuclei and quantum dots (QDs) for applying the Feshbach projection operator (FPO) formalism onto the description of the transmission through QDs with a small number of states. In the first part of Chapter 13, the exact solutions of the formalism as well as the S matrix are derived. The spectroscopic information on the system is contained in the complex eigenvalues and eigenfunctions of a non-Hermitian Hamilton operator that describes the localized part of the system. It depends explicitly on energy. The eigenfunctions are biorthogonal. The eigenvalues give the positions as well as the decay widths of the resonance states. The unitarity of the S matrix is guaranteed at all parameter values (including energy). Very often, it is achieved by the parameter dependence of the eigenvalues, above all of their imaginary parts. The properties of branch points (exceptional points) in the complex plane are considered and their role for physical processes is discussed. Avoided level crossings lead to level repulsion at small coupling strength between system and environment and to widths bifurcation at larger coupling strength. They cause an internal impurity of an open quantum system which quantitatively can be expressed by the phase rigidity of the wave function that varies between 1 and 0. It does not vanish at zero temperature. Due to the widths bifurcation, bound states in the continuum (BICs) may appear. They do not decay although they lie above particle decay thresholds and their decay is not forbidden by any selection rule. In the second part of the review, the FPO formalism is applied to the description of QDs. By means of analytical and numerical studies, it is shown that the generic properties of open quantum systems can be seen also in QDs. The topology of the branch points is compared to that of diabolic points. The geometrical phase of a branch point is half of the Berry phase. The role of the branch points for the spectroscopic properties of different QDs is discussed. They cause avoided level crossings of resonance as well as of discrete states. In double QDs, resonance states with vanishing widths (BICs) appear when the system is symmetrical, and with almost vanishing widths when the symmetry is somewhat disturbed. The branch points govern, generally, the crossover from standing to traveling modes in the transmission. Here the phase rigidity is reduced and the transmission probability is enhanced. Some results obtained in experimental studies of high accuracy, which cannot be explained in the framework of the standard theory, are qualitatively discussed. In Chapter 14, the authors will review the results of their theoretical research on quantum dots. Based on the effective-mass envelope function theory, their investigation primarily covers single quantum dots, coupled quantum dots, and N quantum dot molecule. For single quantum dots, the authors mainly present their study on the InAs/GaAs single quantum dots and the hierarchical self-assembly of GaAs/AlxGa1-xAs quantum dots. They discuss the electronic states, valance band structures, quantum-confined Stark effects, properties in magnetic field, and application as single-electron dot qubit of InAs/GaAs quantum dots. Then they will turn their attention to the electronic and optical properties, and the asymmetric quantum-confined Stark effects of the hierarchical self-assembly of GaAs/AlxGa1-xAs
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quantum dots. As to the coupled quantum dots, the authors focus on their research into InAs/GaAs strained coupled quantum dots, and the properties of coupled quantum dots arranged as superlattice. Finally, the authors discuss the electronic structures of N quantum dot molecule. In Chapter 15 the authors consider open quantum dots (QD) whose spectra can be varied continuously by variation of gate voltage. The authors show that bound states in the continuum (BICs) may occur for discrete values of the voltage and energy of incident electrons. They are localized inside the QD and superposed by the transport solution. However superposition coefficient depends on the way the BIC point is approached. For integrable QD this phenomenon occurs, if the QD spectrum is degenerated incidentally. However a BIC might occur for irregular shape of QD. Both types of QDs are considered analytically in the simplest case of a two level QD and are complemented by numerical calculations for the realistic QB. Although each eigen state of QD is coupled to waveguide, the coupling of BIC with propagating mode of the waveguide turns to zero because of interference with other resonances. As a result, resonance width tends to zero for approaching to the BIC point. In order to find explicitly BICs, the authors look for the complex eigenvalues of the effective non hermitian Hamiltonian which respond for positions and widths of the resonance states. In particular the authors show that BIC is an eigenstate of the effective Hamiltonian with real eigenvalue. The authors present a few numerical examples of BICs in realistic QDs and in systems of double QDs coupled by a wire with variable spectrum. In the framework of the impurity Anderson model the authors took into account Coulomb effects. Such an approach allows one to find the Green function of the closed QD exactly. Further, the solution of the Dyson equation for full Green function describes the open QD. The authors show that the Coulomb repulsion does not eliminate the BIC, but on the contrary, replicates BICs as two-electron BICs. As explained in Chapter 16, research of the magnetic freezing effect for D(-) – states in quasi-zero-dimensional structure with parabolic confinement potential has been fulfilled in frames of common theoretical approach, which is based on the zero-range potential method. It has been shown that the D(-) – state binding energy for quantum dot (QD) in magnetic field can exceed by many times its “bulk” value, because of hybrid quantization. The magnetooptical impurity absorption spectra in quasi-zero-dimensional structure with D(-) – centers has been also calculated. It has been shown that for such structures there is absorption dichroism, which is connected with change in selection rules under optical transitions of electron from the D(-) – center ground state to hybrid-quantizing states of quasi-zero-dimensional structure. It has been demonstrated that possible control of the magneto-optical impurity absorption spectra has been provided by the spectrum parameters dependence from characteristic frequencies: the confinement potential frequency, cyclotron and hybrid frequencies. The light impurity absorption features, which are connected with the geometric form change for quasi-zero-dimensional structures of two types: QD with the ellipsoid of revolution shape, and the disk-shaped QD, have been theoretically investigated. The dispersion equation for electron, which is localized on D(0) – center in QD with the ellipsoid of rotation shape with parabolic confinement potential, has been obtained in the zero-range potential model. It has been shown, that character of the binding energy spatial anisotropy for D(-) – state is comparable with case of D(-) – state in the sphere-shaped QD under influence of external magnetic field. The optical impurity absorption coefficient for quasi-zero-
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dimensional structure with the ellipsoid of rotation – shaped QD has been calculated in dipole approximation. It has been demonstrated that for quasi-zero-dimensional structure with nonspherical QD there is the impurity absorption dichroism, which is connected with the selection rules change for magnetic quantum number in radial direction and for oscillator quantum number in z-direction of QD. Under this situation, spectral dependence of the impurity absorption coefficient has oscillating character with the oscillation period, which is determined by corresponding characteristic frequencies of the confinement potential. The light impurity absorption in quasi-zero-dimensional structures with the disk-shaped QD has been also theoretically investigated. Theoretical approach is based on the D(-) – state energy spectrum investigation in model of the zero-range potential with account of the logarithmic divergence in the one-electron Green function. For simulation of the quantum disk confinement potential in radial direction the potential of “rigid wall” has been used; and in z-direction – potential of the one-dimensional harmonic oscillator. It has been shown that there is spatial anisotropy for the D(-) – state binding energy in quantum disk, that is due to feature of the quantum disk geometric shape. Calculation of the optical impurity absorption coefficient has been maid in dipole approximation for quasi-zero-dimensional structure with the disk-shaped QD with account of their characteristic sizes dispersion. It has been shown that in the case of transversal light polarization (in relation to the quantum disk axis) optical transitions are possible only to the dimensionally-quantizing states of quantum disk with even values of the oscillator quantum numbers and with values of magnetic quantum number ± 1. It has been also demonstrated that spatial dependence for the absorption coefficient has oscillating character with pronounced peaks, position of which is determined by characteristic sizes of quantum disk and by amplitude of confinement potential in z-direction. It has been revealed that factor of the QD geometric form essentially influence as on coordinate dependence of the D(-) – state binding energy, as also on the optical properties of structures with QD. It is very important, because the non-uniform broadening for energy levels in the QD set can be connected with factor of the QD-nonidentity; and the QD-set can be used as active environment for laser structures. Semiconductor quantum dots (QDs) have become a topic of intensive research due to much interest in the fundamental physics of three dimensional (3D) quantum confinement, together with the novel device functionality that they can provide. For example, a QDs based semiconductor laser shows much lower threshold current density and lower temperature sensitivity of the threshold current, etc. Post-growth energy bandgap tuning of semiconductor QD structures is very important for monolithic photonic integration of QDs based passive and active optoelectronic devices. In Chapter 17, post-growth thermal annealing intermixing of InAs/InGaAs/InP quantum dots has been investigated in detail. The energy bandgap tuning of InAs/InGaAs/InP QD structures through the thermal annealing intermixing under a wide temperature range is studied. To increase the energy bandgap tuning, argon (Ar) plasma exposure enhanced intermixing of the InAs/InGaAs QD structure has been investigated. The energy bandgap blue shift of InAs/InGaAs/InP QD structure through the Ar plasma enhanced intermixing achieves 159 meV. By using a SiO2 mask layer, selective intermixing of an InAs/InGaAs/InP QD structures has been studied. The largest intermixing selectivity of the same wafer reaches 77 meV. Three different energy bandgap tuning across an InAs/InGaAs/InP QD wafer has been achieved using the post-growth selective Ar plasma enhanced intermixing. This large post-
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growth selective bandgap tuning of the QD structures paves a way for monolithic integration of QDs based passive and active devices. Quantum dots (QDs) are nanosized regions capable of restricting a single electron, or a few electrons, to the region in three dimensions and in which the electrons no longer occupy band-like energy states, but rather discrete energy states just as they would in an atom. Quantum mechanical phenomena result from this, hence the term quantum confinement. Originally, QDs were grown from semiconductors such as cadmium selenide or cadmium telluride. Since then, however, the synthesis of QDs from nearly every semiconductor and from many metals and insulators has been reported. Quantum dots of semiconductors and metals are currently the focus of intense research. Their electrical, optical, and magnetic properties are different from those of the bulk systems being more like those from molecular-like clusters in which a large number of atoms are on or near the surface. Apart from unique physical properties, QDs also exhibit interesting applications. With their advantage of size, they are ideal for data storage or memory applications to provide high-density memory elements. Potential applications of nonvolatile flash memory devices utilizing QDs have resulted in extensive efforts being made to form QDs, acting as both charging and discharging islands, by a variety of methods. Semiconductor or metallic QDs incorporated within organic or polymeric materials have demonstrated a memory effect when subjected to an electrical bias voltage. Memory phenomenon in QDs arise from their electrical bistability, which is triggered by charge confinement via a suitable voltage pulse. These materials have shown potential applications in digital information storage because of their good stability, flexibility and fast response speed. Organic electrical bistable materials are those that exhibit two kinds of different stable conductive states by applying appropriate voltages. The materials can be switched from low conductive state (“0” or OFF state) to high conductive state (“1” or ON states) by applying an activation voltage. This process is called ‘write’. The high conductive state can remain stable without a bias voltage, and can be read back at a lower voltage. The reverse process is realized by applying a reverse bias when the conductive status changes from a high conductive state to a low conductive state, this being termed ‘erase’. Materials functionalized with ‘erase’ and ‘write’ can be used as RAM (random access memory) and Flash memory. Some materials are write-once-read-many times (WORM), which can be used as ROM (readonly-memory) devices. For commercial use of data storage, devices should satisfy a number of requirements, such as, room temperature operation, low activation voltage to save energy, high ON/OFF ratio, short response time, long retention time and durability. Chapter 18 reviews the recent progress of memory devices exhibiting electric bistability, such devices being based on composites containing quantum dots of semiconductors or metals embedded in organic macromolecular materials.
In: Quantum Dots: Research, Technology and Applications ISBN 978-1-60456-930-8 c 2008 Nova Science Publishers, Inc. Editor: Randolf W. Knoss, pp. 1-46
Chapter 1
F EW-E LECTRON S EMICONDUCTOR Q UANTUM D OTS IN M AGNETIC F IELD : T HEORY AND M ETHODS Orion Ciftja Department of Physics, Prairie View A&M University, Texas 77070, USA
Abstract Semiconductor quantum dots represent nanoscale systems with few electrons confined in a semiconductor host crystal. The importance of semiconductor quantum dots lies primarily in their tunability and sensitivity to external parameters as electrons are confined in all dimensions. The bulk of semiconductor quantum dots are fabricated by applying a lateral confinement potential to a two-dimensional electron gas. Quantum confinement profoundly affects the way electrons interact with each other and with external parameters, such as a magnetic field. Quantum confinement of electrons is just one of several ways quantum mechanics reveals itself. Another pure quantum phenomena associated with electrons is their spin. An external magnetic field affects both orbital and spin motion of electrons. External control of the full quantum wave function in a semiconductor quantum dot may lead to novel technological application involving both charge and spin. From a theoretical point of view, semiconductor quantum dots represent a unique opportunity to study fundamental quantum theories in a tunable atomic like set-up. In this work, we review some of the theoretical approaches used to study two-dimensional few-electron semiconductor quantum dots. The main emphasis is to clarify the relations between different theories and methods for few-electron semiconductor quantum dots in an external parameter, a perpendicular magnetic field. Properties of few-electron semiconductor quantum dots in the weak magnetic regime are explained well through single-electron theory concepts. However, challenges do exist when considering stronger external magnetic fields. A strong magnetic field, when applied perpendicular to the quantum dot, changes the quantum nature of the electronic correlations and spin-polarizes the electrons. As the strength of the external magnetic field increases, the confined electrons start to manifest collective quantum behavior as seen in the integer and fractional quantum Hall effect regime. Theoretical and computational challenges to studies of semiconductor quantum dots as the magnetic field changes from weak to strong are reviewed. Specific examples are introduced to illustrate the transformation of the quantum wave function into a Laughlin-like one as the magnetic field increases.
2
1.
Orion Ciftja
Introduction
Nanoscience is the study of novel phenomena and properties of materials that occur at extremely small length scales, typically on the nanoscale that is the size of atoms and molecules [1]. Nanotechnology is the application of nanoscience and engineering to produce novel materials and devices [2]. Among the many advances in the field of nanotechnology, invention of sophisticated experimental tools has made possible the fabrication of various nanoscale semiconductor structures in a precise and controlled way. In such semiconductor devices, electron’s quantum mechanical nature dominates. The payoff of this behavior is that electronic devices built on nanoscale not only can pack more densely on a chip, but also can operate far faster than conventional transistors. With the shrinking size of these devices, electrons manifest pronounced quantum behavior and their motion becomes confined in one, two, or three dimensions. The properties of confined two-dimensional (2D) electronic systems in semiconductor materials, that we refer to as 2D semiconductor quantum dots are a topic of intensive ongoing research [3–7]. They consitute a whole new class of semiconductor devices representing one of the most promising avenues for meeting the new technological challenges of the 21-st century. 2D semiconductor quantum dots are expected to provide the basis for future generations of device technologies such as threshold-less lasers and ultra-dense memories. 2D semiconductor quantum dot structures generally hold only a few electrons in contrast to standard bulk semiconductor devices therefore, they represent the ultimate limit of the semiconductor device scaling. As the device sizes are reduced the number of carriers involved in the operation of a single device is reduced as well. In fact, state-of-theart semiconductor structures will soon be plagued by dopant fluctuation and particle noise problems. Quantum dot device concepts utilize the discreteness of the electron charge and they offer a possible breakthrough in device and circuit technology. 2D semiconductor quantum dots are fabricated semiconductor nanostructures in which charge carriers, such as electrons, are confined in a small 2D region of space [8–14]. The size and shape of these structures and therefore the number of electrons they contain can be precisely controlled. A 2D semiconductor quantum dot can have anything from a single electron to a collection of several thousands. The physics of 2D semiconductor quantum dots shows many parallels with the behavior of naturally occurring quantum systems in atomic and nuclear physics. In atomic systems, electrons are confined by the attraction of the positivily charged nucleus. In 2D semiconductor quantum dots the confinement of electrons is instead due to an artificially created potential, formed by the electrodes connected to layers of semiconductor. Because of analogies to real atoms, semiconductor quantum dots are frequently referred to as artificial atoms. As in an atom, the energy levels in a 2D semiconductor quantum dot become quantized due to the confinement of electrons. Unlike atoms however, 2D semiconductor quantum dots can be easily connected to electrodes and are therefore excellent tools to study atomic-like properties in a controllable way. There is a wealth of interesting phenomena that have been seen in 2D semiconductor quantum dot devices over the past decade. Modern microfabrication technology can fabricate 2D semiconductor quantum dots that are sufficiently small that they contain only a small number of mobile electrons. Initial studies were focused on parabolically confined 2D semiconductor quantum dots and elucidated their atomic-like properties at low magnetic fields [15–17].
Few-Electron Semiconductor Quantum Dots in Magnetic Field
3
In particular, capacitance spectroscopy studies of 2D semiconductor quantum dots in the low magnetic field regime indicated that the ground states of parabolic 2D semiconductor quantum dots exhibit shell structure and obey Hund’s first rule [15]. The shell structure is particularly evident in measurements of the change in electrochemical potential due to the addition of one extra electron. In this regime, parabolic 2D semiconductor quantum dots exhibit pronounced shell structure manifesting their atomic-like nature. However, with the strengthening of the magnetic field, the structure of single-electron levels changes to highly degenerate Landau levels and electron correlations start playing a major role. Transport properties of 2D semiconductor quantum dots in this regime are highly interesting because of possible applications of single-electron tunneling in electronic devices [18]. The application of new and extraordinary experimental tools to nanosystems in general and 2D semiconductor quantum dots in particular has created an urgent need for a quantitative understanding of new physical phenomena at nanoscale lengths. New models and robust tools for the quantitative description of properties at the nanoscale are urgently needed in order to capitalize on the important scientific opportunities in nanoscience. With each new experimental finding in nanoscience comes new opportunities to introduce new theories and appraches as well as test the theoretical foundations of standard theories and models.
2.
2D Semiconductor Quantum Dots
In a 2D semiconductor quantum dot, electrons move in a plane in a lateral confinement potential. The relative strength of the electron-electron interaction and electron confinement energy can be experimentally tuned over a wide range of parameters. As a result, 2D semiconductor quantum dots have highly tunable physical properties. 2D semiconductor quantum dots can contain anything from a single electron to a collection of thousands of electrons and many of the parameters that describe them can be precisely controlled by standard nanofabrication methods. A standard theoretical model for 2D semiconductor quantum dots involves a number of approximations. The most common approximations regard: i) the motion of electrons which is considered to be exactly 2D, ii) the confining potential which is simplified, and iii) the interaction potential between electrons which is considered to have a Coulomb form. The basic technological motivation to study semiconductor quantum dots is that smaller components should be faster, dissipate less heat, and quantum mechanical effects are so relevant in such systems that devices with fundamentally new properties can be obtained. For instance, semiconductor quantum dots could be used in single electron transistor devices, for computer memory storage of huge capacity if dense packing of quantum dot matrices becomes possible, or in new quantum information devices [19]. Apart from their potential use as novel devices, 2D semiconductor quantum dots are also interesting from a fundamental point of view, since modern microfabrication technologies allow us to control their number, shape, size, as well as energy structure. From the theoretical point of view, 2D semiconductor quantum dots constitute a unique system where various quantum theories and methods can be directly compared to experiment. Spin effects are also pronounced and they do occur at ordinary magnetic fields. In this regard, 2D semiconductor quantum dots are an ideal laboratory to investigate the interplay between confinement, magnetic field and
4
Orion Ciftja
electronic correlations effects. The main effect of the magnetic field, as it becomes stronger, is to change the single-electron levels from 2D harmonic oscillator to Landau levels where states with different angular momentum become degenerate and electron correlations play a very important role, similar to the role they play in the fractional quantum Hall effect (FQHE) for bulk 2D electron systems (2DES) in high magnetic fields [20, 21]. The properties of N -electron 2D semiconductor quantum dots subject to a magnetic ~ = (0, 0, Bz ), are generally calculated by field perpendicular to the quantum dot plane, B considering the following Hamiltonian: ˆ = H
N X 1 hˆ i=1
2m
i2
~ ri) p~i + e A(~
+ V (~ri) +
N X 1 e2 + g µB Bz Sz , 4 π 0 r i>j |~ri − ~rj |
(1)
where the first term is a one-electron term, the second term is the Colomb potential energy, the last term is the Zeeman energy and V (~r) is the one-electron confinement potential. In a symmetric gauge, the magnetic vector potential is: ~ r) = Bz (−y, x, 0) , A(~ (2) 2 where ~r = (x, y) is the 2D position vector, −e (e > 0) is electron’s charge, m is electron’s mass, g is electron’s g-factor, µB is Bohr’s magneton, r is the dielectric constant and Sz is the z-component of the total spin. To obtain the many-electron energy spectrum and wave functions one must solve the stationary Schr¨odinger equation for the Hamiltonian above: ˆ Ψ(~r1 , . . ., ~rN ) = E Ψ(~r1, . . . , ~rN ) . H
(3)
Clearly, this is a formidable task and this quantum problem cannot be solved exactly even for 2D semiconductor quantum dot systems having as few as N = 2 electrons. The case of few-electron 2D semiconductor quantum dots [22–25] is of particular interest, since single-electron confinement energy, the cyclotron energy for ordinary magnetic fields and electron-electron correlation are all of the same order of magnitude. As a result a rich physics and a variety of complicated quantum phenomena are manifested.
3.
Parabolic Confinement Potential
An isotropic parabolic confinement potential of the form: m 2 2 ω r , (4) V (r) = 2 0 is the most common choice to describe electron’s confinement in a 2D semiconductor quantum dot [26–28]. The parabolic confinement model explains reasonably well some of the main features associated with most common 2D semiconductor quantum dots. If we start with a system of parabolically confined non-interacting electrons, neglecting the Coulomb interaction, the Hamiltonian Eq.(1) (without the Zeeman term) is reduced to a sum of single particle Hamiltonians, H0 (~r) each of the same form: ˆ = H
N X i=1
ˆ 0 (~ri) . H
(5)
Few-Electron Semiconductor Quantum Dots in Magnetic Field
5
For a parabolic confinement potential the single-particle quantum problem can be solved exactly. Therefore, Eq.(5) admits an exact analytic solution for the case of non-interacting electrons. This fact provides a strong justification for the wide use of parabolic confinement potentials when studying confined electrons in a 2D semiconductor quantum dot. The study of single-particle features in a perpendicular magnetic field is not only of theoretical interest, but is also very useful to explain important experimental features observed in 2D semiconductor quantum dots, for instance the existence of shell structures. ˆ 0(~r) (without the Zeeman term) for a single electron in a 2D parabolic The Hamiltonian, H confinement potential subject to a uniform perpendicular magnetic field is: h i2 ~ r) + m ω 2 r2 , ˆ 0 (~r) = 1 ˆ (6) p + e A(~ ~ H 2m 2 0 where ¯ h ω0 is the strength of the parabolic confinement potential and we are using the symmetric gauge vector potential in Eq.(2). Such problem was first investigated by Fock [29] and Darwin [30] in the context of diamagnetism and the eigenstates are routinely called the Fock-Darwin (FD) states:
r2 Ψnmz (r, ϕ) = Nnmz exp − 2 4 lΩ
!
r lΩ
where
|mz |
Ln|mz |
r2 2 2 lΩ
!
×
exp (−i mz ϕ) √ , 2π
(7)
s
h ¯ , (8) 2mΩ is an effective magnetic length, n = 0, 1, . . . is the radial quantum number, mz = 0, ±1, . . . lΩ =
|mz |
is the z-angular momentum quantum number and Ln
r2 2 l2Ω
are associated Laguerre’s
polynomials [31]. The frequency, Ω appearing in the expression for the effective magnetic length is: ω2 (9) Ω2 = ω02 + c , 4 where ωc = e Bz /m is the cyclotron frequency. The normalization constant, Nnmz is: Nnmz =
s 2 2|mz | lΩ
n! . (n + |mz |)!
(10)
The energies (without the Zeeman energy) are: h ωc ¯ mz , (11) 2 and depend on the magnetic field through the ωc dependence. In Figure 1 we plot the h ω0 ) as a function of energy spectrum of FD states in dimensionless units, = En mz /(¯ the dimensionless magnetic field parameter, ωc /ω0 . The expression in Eq.(11) predicts that energy levels with positive mz shift downward and levels with negative mz shift upward as magnetic field increases. When ω0 = 0 or in the limit of very large magnetic fields (ωc ω0 ), the effective magnetic length, lΩ becomes the electronic magnetic length, l0: h Ω (2 n + 1 + |mz |) − En m z = ¯
lΩ →
s
¯ h = m ωc
s
¯ h = l0 e Bz
;
ωc →∞. ω0
(12)
6
Orion Ciftja
10 Energy ( ε )
8 6 4 2 0 0.5
1
1.5
2 2.5 ωc / ω0
3
3.5
4
Figure 1. Single-particle energy levels, = En mz /(¯ h ω0 ) as a function of the dimensionless magnetic field parameter, ωc /ω0 .
If we assume ω0 6= 0, the effective magnetic length, lΩ can also be written as: 1 2 2 = 2α lΩ
s
1 1+ 4
where the parameter
ωc ω0
2
;
ω0 6= 0 ,
(13)
r
m ω0 , (14) h ¯ is the inverse oscillator length. This way it is easy to recover the 2D harmonic oscillator states from the FD states in the limit of zero magnetic field ( ωc /ω0 → 0). As seen in Figure 1, the full energy spectrum of FD states is rather complex. However, we note that the ground state wave function has always zero angular momentum, n = 0 and mz = 0 and is not degenerate for any value of the magnetic field. α=
4.
Other Confinement Potentials
The most common theoretical model used to study 2D semiconductor quantum dots considers an isotropic parabolic confinement. The assumption of a parabolic confinement potential model explains reasonably well some main features associated with common semiconductor quantum dots. However, such model is limited in its applicability and is somehow inadequate because of the implied infinite range and height of the confinement potential, an assumption that is unphysical in a real experiment. Anisotropic non-circular parabolic confinement potentials of the form: m 2 2 ωx x + ωy2 y 2 , (15) V (x, y) = 2 where ωx and ωy are confinement frequencies in respective x, y directions have also been investigated [32, 33]. Despite their wide use, both isotropic and anisotropic parabolic confinement models have limitations and are somehow inadequate because of the infinite range
Few-Electron Semiconductor Quantum Dots in Magnetic Field
7
and infinite height of potential at large distances. While the parabolic model is certainly appropriate at low energies, for instance 2D semiconductor quantum dots with few electrons, it is unsuitable to study larger systems where a substantial portion of electrons have their energies close to the energy continuum threshold. Close to the energy continuum threshold, the shape of the confinement potential felt by electrons is no longer parabolic therefore an infinite range parabolic potential is no longer justified. Consideration of 2D confinement potentials with finite depth solves such defficiencies. A very common choice would be a finite 2D cylindrical well potential: V (r) =
−V0 ; 0 ≤ r ≤ R
,
0 ; R
(16)
where R is the radius of the semiconductor quantum dot and V0 > 0 is the depth of the potential well. A similarly common choice would be a finite 2D square well potential: V (x, y) =
Ly Ly L L −V0 ; − 2x ≤ x ≤ + 2x ; − 2 ≤ y ≤ + 2
,
(17)
0 ; otherwise
where Lx × Ly is the area occupied by the confined electrons. These two potentials are quantum mechanics textbook favorite examples. In addition to having a finite depth, they also represent a specific class of confinement potentials with sharp boundaries/edges. Few other potentials with finite depth, but with smooth boundaries that do not become infinite at large distances have been proposed. One of them is a Gaussian confining poten0.2 0
V(r) / V0
-0.2 -0.4 -0.6 -0.8 -1 -1.2 0
0.5
1
1.5
2 r/R
2.5
3
3.5
4
Figure 2. Plot of V (r)/V 0 as2a function or r/R for a Gaussian confining potential of the form: V (r) = −V0 exp − 2rR2 , where V0 > 0 is the depth of the potential well and R is the range. tial [34] shown in Figure 2 which has the form: r2 V (r) = −V0 exp − 2 R2
!
,
(18)
Orion Ciftja
V(r) / V(R)
8
1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 0
0.5
1 r/R
1.5
2
Figure 3. A truncated parabolic confinement potential with a cut-off range, R.
where V0 > 0 is the depth of the potential well and R is the range of confinement potential. Note that the Gausian potential has no sharp edges and smoothly goes to zero in the r → ∞ limit. Another confining potential with smooth boundaries [35] that goes smoothly to zero in the r → ∞ limit can be written as: V (r) = −
V0 1+
r2 R2
2 ,
(19)
where V0 > 0 and R is an adjustable parameter. At small distances the above confining potentials are parabolic, while, at large distances they tend to a finite (zero) asymptotic value. While the above potentials do not have some of the shortcomings of the infinite range and height parabolic potential, their physical origin and justification is not completely clear. As previously pointed out the “ideal” parabolic confining potential has limitations and in certain regimes is unphysical due to its infinite range and infinite height at large distances. Close to the energy continuum threshold, the shape of the real confinement potential felt by electrons is no longer parabolic therefore the use of an infinite range and height parabolic potential cannot be justified. In all experimental situations there is a “finiteness” of the domain in which electrons are confined. In particular, there are systems such as laterally coupled semiconductor quantum dots (for instance double dots) where an infinite range and height parabolic confinement model cannot be used since it does not allow unbinding of states from individual harmonic wells. Perhaps, the simplest way to improve the “ideal” parabolic confining potential and make it more realistic is to substitute it with a “truncated” parabolic potential shown in Figure 3. Such potential is parabolic up to a finite cut-off distance R and is finite (constant) at distances larger than R:
V (r) =
m 2 2 2 ω0 r ; 0 ≤ r ≤ R
m ω 2 R2 ; R < r < ∞ 2 0
.
(20)
Few-Electron Semiconductor Quantum Dots in Magnetic Field
9
A detailed study of the properties of such confinement potential (number of bound states, discrete energies, effect of electronic interactions on the discrete energy spectrum, etc) might shed light on various properties of 2D semiconductor quantum dots in those regimes where parabolicity is no longer guaranteed. While the “truncated” parabolic confining potential may look too simple, we remark that it captures some of the main properties of a “realistic” semiconductor quantum dot potential: it is parabolic in the center, it has finite range/height and allows unbinding of states from individual semiconductor quantum dots. We now describe in more detail another confinement potential for electrons in a 2D semiconductor quantum dot that appears to be more physically motivated and better experimentally justified than the commonly used infinite range parabolic potential or few other choices. The motivation for the choice comes from the consideration of the specific experimental setup in a 2D semiconductor quantum dot. Such setup involves application of gate potentials who cause electron depletion in the area near the gates. The area depleted from electrons acts as a positively charged region which most simply can be modeled as a uniformly charged 2D disk with a positive background charge. In this experimental setup, individual electrons feel a confinement potential originating from the uniformly positively charged 2D background disk. Differently from the infinitely high parabolic confinement potential, the resulting 2D charged disk potential has a finite depth. The resulting 2D charged disk potential has a form that can be reasonably approximated as a parabolic potential in the central region of the semiconductor quantum dot (for states with low energy), however outside that region is no longer parabolic but behaves more like a Coulomb potential that obviously has a finite (zero) asymptotic value at large distances. To derive the exact form of the 2D charged disk electrostatic confining potential, we first consider the interaction potential between an electron with charge, −q0 (q0 > 0) and a uniformly charged finite 2D disk (the depleted region) with total positive charge Q and radius R. Both parameters, Q and R can be tuned experimentally. It has been calculated [36] that the resulting electrostatic confining potential energy between an electron and a 2D charged disk can be written as: V (r) = −q0 V0 F (r, R) ; F (r, R) =
Z ∞ dz 0
z
J0
r z J1 (z) , R
(21)
where F (r, R) is a function that depends only p on the ratio r/R (given in integral form), Jn (z) are n−th order Bessel functions, r = x2 + y 2 is the distance of the electron from the center of the disk, V0 = (2 k Q)/R is the electrostatic potential created by the disk at its center (r = 0) and k = 1/(4 π 0 ) is Coulomb’s electric constant. While the integral presentation of F (r, R) is rather convenient, the integration in Eq.(21) can also be carried out analytically resulting in an expression involving complete elliptic integrals of the second kind and hypergeometric functions [36]. Two special values of this function are: F (r = 0, R) = 1 ;
F (r = R, R) =
2 . π
(22)
In Figure 4 we show V (r)/(q0V0) = −F (r, R) as a function of r/R, where r is the distance of the electron from the center of the disk. One immediately notices that the function, −F (r, R) is approximately parabolic for the range 0 ≤ r ≤ R and becomes zero asymptotically at large distances. The confining potential under consideration, V (r) originates
10
Orion Ciftja 0.2 0
V(r) / (q0 V0)
-0.2 -0.4 -0.6 -0.8 -1 -1.2 0
0.5
1
1.5
2 r/R
2.5
3
3.5
4
Figure 4. Electrostatic confinement potential, V (r) between an electron of charge −q0 (q0 > 0) and the the uniformly charged 2D disk with radius R and positive charge Q. The quantity V0 = 2 k Q/R is the electrostatic potential at the center of the disk (r = 0) and k is the electric Coulomb’s constant. from a setup in which electrons are embedded in a single positively charged circular 2D disk layer. Thus the positive background and electrons all belong to the same plane and are not vertically separated. Given that the 2D charged disk potential closely resembles a parabolic function inside the disk let us find a reasonable parabolic approximation to V (r) that we denote Vp(r) and parametrize it. The simplest approach is to impose the constrains: Vp(r = 0) = V (r = 0) and Vp(r = R) = V (r = R) which give a simple parabolic function approximation of the form: Vp(r) = −q0 V0 + c q0 V0
r R
2
,
(23)
where c = 1 − 2/π > 0 is a constant number. Obviously, Vp(r) is a good approximation to V (r) only in the range 0 ≤ r ≤ R but not for r > R. For simplicity we can map: m 2 c q0 V0 ω , = R2 2 0
(24)
and write Eq.(23) as:
m 2 2 ω r , (25) 2 0 ˆ the Hamiltonian for an electron in a where ω0 is given in Eq.(24). Let us denote by H realistic 2D charged disk confinement potential: Vp(r) = −q0 V0 +
2 2 ˆ = pˆ + V (r) = pˆ − q0 V0 F (r, R) . H 2m 2m
(26)
ˆ p the Hamiltonian for the parabolic confinement potential Vp(r): If one denotes H 2 2 ˆ p = pˆ + Vp(r) = −q0 V0 + pˆ + m ω 2 r2 , H 2m 2m 2 0
(27)
Few-Electron Semiconductor Quantum Dots in Magnetic Field
11
one can immediately see that: where
ˆ , ˆ =H ˆp + W H
(28)
ˆ = q0 V0 [1 − F (r, R)] − m ω02 r2 , W 2
(29)
ˆ may be treated represents the departure from the parabolic model. In perturbation theory, W as a perturbing term. To simplify notation, we denote: ˆp + W ˆ , ˆ =H ˆ + q0 V0 = ∆H ∆H
(30)
where
2 ˆ p + q0 V0 = pˆ + m ω 2 r2 , ˆp = H (31) ∆H 2m 2 0 is a pure 2D parabolic potential. Although one cannot overlook the role of electron correlations, the first step in understanding the electronic properties of 2D semiconductor quantum dots is to study the single-particle aspects of the behavior of electrons under confinement. To calculate the energy spectrum of an electron in a 2D charged disk confinement potenˆ (therefore H). ˆ To tial we need to solve the Schr¨odinger equation for the Hamiltonian ∆H achieve this task, we resort to the exact numerical diagonalization method [37]. A detailed description of the exact numerical diagonalization method will be given in the following pages. For now it suffices to say that within such method one tries to solve the stationary ˆ Ψ = E Ψ, by expanding the (unknown) function Ψ as a linear Schr¨odinger equation: ∆H combination of basis functions. Such basis functions are for instance the exact orthonormalized eigenfunctions, |n mz i = Φnmz (r, ϕ) of the 2D harmonic oscillator with energy eigenvalues: h ω0 (2 n + |mz | + 1) , (32) Enmz = ¯
where n = 0, 1, . . . is the radial quantum number and mz = 0, ±1, ±2, . . . is the z-angular momentum quantum number. In order to set up the Hamiltonian matrix that needs to be ˆ in the basis of the eigendiagonalized, one must first calculate the matrix elements of ∆H functions |n mz i of the 2D harmonic oscillator. Non-diagonal terms arise only from the ˆ which is diagonal with respect to mz but not n. For any given value of the operator, W angular momentum, mz we have: ˆ mz i hn0 mz |∆H|n = (2 n + |mz | + 1) δn0 n + hn0 n , h ω0 ¯ where
(33)
ˆ |n mz i hn0 mz |W . (34) h ω0 ¯ Note that energies are measured in units of ¯ h ω0 . One can calculate numerically all the desired quantities, hn0 n . For chosen z−angular momentum values, mz = 0, ±1, . . . we then build sufficiently large Hamiltonian matrices and then solve the matrix eigenvalue problem by means of standard diagonalization tools. The resulting eigenvalues of the diagonalized matrix represent the allowed energies of an electron in a 2D charged disk confinement potential. The smallest of the energy eigenvalues represents the ground state energy corresponding to any given radius of the 2D disk. It is convenient to express the 2D charged disk hn 0 n =
12
Orion Ciftja 16
α R=3.0
14 12 10
ε
8 6 4 2 0 -2 -2
0
2
4
6
8
10
12
|mz|
ˆ h ω0 ) for the electron with charge −q0 Figure 5. Bound energy spectrum, = h∆Hi/(¯ (q0 > 0) in the 2D charged disk confinement potential, V (r) for a radius of disk, α R = 3. The solid circles represent the bound energies corresponding to each z-angular momentum h ω0 ) of quantum numbers, |mz | = 0, . . . , 10. The solid line represents the depth, q0 V0/(¯ the confining well above which we have unbound (scattering) states.
p
h is the standard harmonic oscillator radius in dimensionless units, α R where α = m ω0 /¯ parameter that has the dimensionality of an inverse length. Bound states of electrons in the 2D charged disk confinement potential are of major interest, therefore we consider a specific value, α R = 3 for the disk radius. This choice h ω0 ) = −(α R)2/(2 c) which roughly results in a quantum well with a depth: −q0 V0/(¯ guarantees to accomodate more than ten electrons. In Figure 5 we show the resulting bound energy spectrum for an electron in dimensionless harmonic oscillator energy units and meah ω0 ) for selected z-angular momensured with respect to the bottom of the well, −q0 V0/(¯ tum values, |mz | = 0, . . . , 10 and for α R = 3. The corresponding enegy spectrum for an electron in a 2D parabolic confinement potential is shown in Figure 6. Since the accuracy of results from expansions in a finite basis is usually best for the lowest energy states and deteriorates with increasing energy, the energy spectrum close to the energy continuum threshold is less accurate than the spectrum close to the bottom of the finite quantum well. As clearly seen from Figure 5 and Figure 6 the general effect of the 2D charged disk potential on the energy spectrum of electrons as compared to the 2D parabolic potential is a lowering of the corresponding harmonic oscillator energies. Such effect becomes more pronounced for states closer to the energy continuum threshold. Closer to the energy continuum threshold the available energies in a 2D charged disk potential are also more closely packed than their parabolic counterparts resulting in a higher density of states. A comparison of ground state and excited states corresponding to the 2D charged disk confinement potential with those for a 2D parabolic confinement potential clearly indicates similarities at low energy states with pronounced differences at higher energy states. The two ground state wave functions are very similar (for low energy states the 2D charged disk potential is approximately parabolic). However, this is not the case for excited higher energy
Few-Electron Semiconductor Quantum Dots in Magnetic Field
13
16 14 12 10
ε
8 6 4 2 0 -2 -2
0
2
4
6
8
10
12
|mz|
Figure 6. The corresponding energy spectrum for a particle in a 2D parabolic confinement h ω0 ) of the 2D charged disk confinepotential. The solid line represents the depth, q0 V0 /(¯ ment potential.
states. Closer to the energy continuum threshold the excited state wave functions for the 2D charged disk potential spread out much more than the parabolic counterparts thus allowing leaking of electrons away. For relatively large semiconductor quantum dots, a sizeable number of bound electrons have their energies close to the energy continuum threshold therefore in this scenario the parabolic model is a rather poor representation. In particular, for the case of laterally coupled quantum dots, such as double quantum dots [38–41], an infinite range parabolic confinement model cannot be used since it does not allow unbounding of states from individual harmonic wells. On the other hand, the 2D charged disk confining potential introduced here decays smoothly with distance. Because the 2D charged disk potential decays smoothly with distance (unlike the parabolic potential that increases quadratically), it naturally allows “leaking” of electrons from the interior of the quantum dot therefore it is ideally suited to model systems of laterally coupled quantum dots without resorting to additional artificial assumptions about the form of the confinement. For a given system of laterally coupled quantum dots the overall confinement potential would be a combination of individual 2D charged disk potentials. The net result is a confinement potential with multi-minima that allows inter-dot tunneling and does not become infinite far away from the central confinement regions.
5.
Theory
From the theoretical perspective, 2D semiconductor quantum dots are an ideal ground to study novel quantum phenomena. In particular, the study of the strong magnetic field regime with all electrons fully spin polarized is relevant because this is the crossover regime between microscopic 2D semiconductor quantum dots and macroscopic 2DES of FQHE type [42–52]. The weak (or zero) magnetic field regime is also very interesting [53–57], with the main focus on the Fermi liquid-Wigner solid crossover regime, a problem closely
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related to the nature of metal-insulator transition in 2D [58]. The transitional regime of intermediate (weak) magnetic fields is expected to have prominent spin effects and is not well investigated. The richness and complexity of the phenomena associated with the behavior of 2D semiconductor quantum dot systems in a perpendicular magnetic field makes them fascinating and at the same time challenging from the theoretical and modeling point of view. In the following we specifically consider 2D semiconductor quantoms dots with parabolic confinement potential and try to classify the various quantum regimes that influence the properties of the confined electrons. There are two main dimensionless parameters that determine the behavior of 2D semiconductor quantum dots in a perpendicular magnetic field. One is the dimensionless Coulomb correlation parameter: λ=
e2 α 1 , 4 π 0 r ¯ h ω0
and the other one is the dimensionless magnetic field parameter: ωc . γ= ω0
(35)
(36)
The parameter λ gauges the strength of the Coulomb correlation energy relative to the confinement energy, while the parameter γ gauges the strength of the magnetic field p relative to = h/(2 m Ω) ¯ the confinement energy. The characteristic length scale of the system is: l Ω q
where Ω = ω02 + ωc2/4 is a characteristic frequency, ωc = e Bz /m is the cyclotron frequency and α is the harmonic (parabolic) oscillator inverse length. Another parameter related to λ which characterizes the interaction strength of the electrons is the Wigner-Seitz dimensionless density parameter, rs . In homogeneous 2D systems it is defined by the radius of the circle that every electron occupies effectively in units of the Bohr radius: aB =
4 π 0 r ¯ h2 , m e2
(37)
n0 =
1 . π (rs aB )2
(38)
so that the homogeneous density is:
In the low-density (strong interaction) limit, rs → ∞, classical considerations suggest the stabilization of a Wigner molecule [59] or a Wigner crystal-like phase [60]. In contrast, at high densities (weak interaction) limit, rs → 0, a Fermi liquid-like description is expected to be valid. The phase diagram of 2D semiconductor quantum dots in a strong perpendicular magnetic field is intricate. The overwhelming majority of studies to date have considered only the stability between competing liquid and Wigner crystal phases. We speculate that it is likely that other more exotic intermediate phases, that have not been yet considered so far, can stabilize in systems of confined electrons in a 2D semiconductor quantum dot. The most conventional of such quantum phases would be the equivalent of the charge density wave [61–63] stripe state seen in quantum Hall samples in weak magnetic field. The possibility of other intermediate quantum phases with liquid crystalline order and broken rotational symmetry [64–72] cannot be excluded. Naive arguments may even suggest that, for
Few-Electron Semiconductor Quantum Dots in Magnetic Field
15
a careful choice of parameters (temperature, magnetic field, density, etc) because of confinement, some intermediate exotic liquid crystalline phases [73–75] might be energetically more robust than their anisotropic quantum Hall counterparts [76]. The following physical constrains need to be incorporated at any microscopic theory that applies to 2D semiconductor quantum dot systems in a perpendicular magnetic field: • In the low-density limit and at arbitrary magnetic fields, the sys• In the hightem should resemble a Wigner molecule or a Wigner crystal state . density limit and at strong magnetic fields the system should resemble a strongly correlated fully spin-polarized Laughlin-like FQHE-type liquid state . • In the highdensity limit and at zero magnetic field the system should resemble a correlated spin-unpolarized Fermi liquid system that can be suitably described by a Jastrow-Slater wave function. • In the high-density limit and for intermediate magnetic fields, the system should resemble a correlated partially spin-polarized Fermi liquid state with pronounced spin-dependent effects. In short, all these important physical constrains should be carefully incorporated in any description or microscopic theory dealing with confined electrons in a 2D semiconductor quantum dot in a perpendicular magnetic field. If one wants to start with a microscopic wave function, a trial wave function for N electrons should have, at the very least, the following form: Ψγ =
N Y i<j
J(rij )
N Y
(zi − zj )np D↑(φ)D↓(φ) χ(S)
(39)
i<j
where J(rij ) are Jastrow factors, zj is the 2D position of electrons in complex notation, rij = |~ri − ~rj | is the distance between a pair of electrons, D↑ ↓ (φ) are Slater determinants of single-particle orbitals, φ corresponding to N↑ spin-up and N↓ = N − N↑ spin-down electrons, χ(S) is the spin function, S = {s1 , s2, . . . , sN } represents all spin coordinates and np = 0, 1, . . . is an integer quantum number that determines the parity of the Laughlin Q np polynomials, N i<j (zi − zj ) . The single-particle orbitals entering the Slater determinants may be chosen to be FD orbitals when describing a liquid state (high density) or localized Gaussian orbitals when describing a Wigner molecule/crystal state (low density):
φ=
|mz | |mz | |z|2 |z|2 z φ ∝ exp − L , n lΩ FD 4 l2Ω 2 l2Ω φ ∝ exp −c |~ ~2 , r − R| ~ R
(40)
where n = 0, 1, . . . is the radial quantum number, mz = 0, ±1, . . . is the angular momen~ are the lattice centers of vibration of the electrons. Antisymmetrizatum and the vectors, R tion of the product of Gaussian orbitals might not be crucial. In the low-density limit we expect the system to manifest its crystalline nature (Wigner molecule/crystal). If we start with a crystalline wave function, it is expected that the presence of a perpendicular magnetic field will further stabilize the crystalline phases. On the other hand, in the high-density limit , more exotic liquid phases may emerge and become energetically favorable because of the presence of a perpendicular magnetic field.
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Therefore, we focus our attention in the high-density limit trying to anticipate possible quantum phases and the form of a suitable microscopic wave function that might describe such phases. In the strong magnetic field limit ( Bz → ∞), which corresponds to γ → ∞, we expect the wave function to become similar to a Laughlin-like wave function which describes fully spin-polarized strongly correlated electrons in the FQHE regime. In this limit, the Jastrow factor, J(rij ) of the microscopic wave function above should become a slight correction to the Laughlin-like wave function if we assume that a FQHE-type phase stabilizes. In the weak (or zero) magnetic field limit (Bz → 0), which corresponds to γ → 0, the wave function is expected to become a standard Jastrow-Slater wave function representing a spin-unpolarized state. In this limit, the FD orbitals transform into 2D harmonic oscillator Q np states. Pauli’s principle imposes the choice: np = 0 for the polynomial N i<j (zi − zj ) , although in a more general setting one can consider any even value, np = 2, 4, . . . and treat np as a variational parameter. For weak (or zero) confinement and for intermediate values of the magnetic field (when a fraction of electrons is spin-reversed) the general wave function in Eq.(39) should represent a partially spin-polarized state where spin effects take prominence.
6.
General Methods
Let us now mention briefly some important theoretical methods that have been employed to study 2D semiconductor quantum dot systems, principally aiming to elucidate the electronic structure and transport properties of such systems. Obviously this brief review is not meant to be complete in any sense. There are very few analytical results that apply to semiconductor quantum dots with parabolic confinement potential and Coulomb interaction between electrons. Exact results that have been obtained so far are relevant only for the two-body problem at some special parabolic confinement frequencies and magnetic fields [77–80]. Thus these calculations, while interesting, do not represent a general exact solution of the problem even in the twobody sense. Ab-initio calculations for semiconductor quantum dots which incorporate the detailed form of the confinement potential and many-electron interactions are extremely challenging. They rely on a combinition of Schr¨odinger-Poisson equations to obtain realistic abinitio confining potentiald and then calculate the electronic structure numerically [81–83]. Exact numerical diagonalization studies [84–87] are accurate for a very small number of electrons. In practice they have been used for up to N = 8 electrons [88], but the accuracy is very limited for all cases except those at very small N . The computational cost grows very rapidly with the increase on the electron number because the number of basis functions needed to obtain a given accuracy becomes too large. Oftenly, one has to impose other constrains to make the computation feasible such as truncating the basis set, freezing some degrees of freedom, etc in a way that does not compromise the overall accuracy. Overall, the difficulty of the problem sets a serious limit to the application of the exact numerical diagonalization method for larger systems of electrons. Hartree [89–91], restricted Hartree-Fock [92] and unrestricted Hartree-Fock [93–95] methods applied to semiconductor quantum dots have given results that are satisfactory for a qualitative understanding of some systematic properties at zero and non-zero magnetic
Few-Electron Semiconductor Quantum Dots in Magnetic Field
17
field. However, all Hartree and/or Hartree-Fock methods suffer from sizeable systematic errors as indicated from comparisons with exact numerical diagonalization results. Density functional theory methods with various flavours (local spin-density, current density functional theory, etc) represent another class of methods used to study semiconductor quantum dots [96–99]. At last we mention the quantum Monte Carlo (QMC) methods [100–103] which have been succesfully applied to study semiconductor quantum dot systems with or without a magnetic field. The advantage of QMC methods with respect to other approaches is that they are not limited by the number of electrons and in principle can be extended to very large systems of electrons with relative ease. Despite various numerical challenges, QMC methods tend to be very accurate and reliable. For this reason, we dedicate more time to describe some basic principles involving QMC methods with focus on two of them: the variational Monte Carlo (VMC) and the diffusion Monte Carlo (DMC) method.
7.
Quantum Monte Carlo methods
QMC methods have been succesfully applied to a variety of classical and quantum problems. The core idea of QMC methods is to model any problem by involving stochastic variables. The presence of electron-electron interactions in strongly correlated electronic systems makes the many-electron problem one of the most difficult problems to solve in quantum mechanics. In almost all cases the many-electron stationary Schr¨odinger equation cannot be solved exactly. Thus one has to resort to various computational methods. The QMC method is a numerical method which can solve the many-electron quantum problem almost exactly. While facing numerical challenges too, the QMC methods have the greatest advantage of all since they can be extended to large systems with many electrons in a straightforward manner. Differently from the exact numerical diagonalization method that is applicable only for few electrons, QMC methods can deal with systems containing hundreds of electrons. The attention given to QMC methods is justified by the expectation that such methods will definitily become the most accurate and reliable methods to calculate the properties of various nanoscale electronic systems including the 2D semiconductor quantum dots. Common QMC methods which are used to study confined electrons in a 2D semiconductor quantum dot are the variational Monte Carlo method and the more sophisticated diffusion Monte Carlo method. In the following, we focus on some basic principles involving these two simulation methods, without considering other QMC methods/techniques in circulation. The brief description of these two methods also serves the purpose of pointing out key numerical challenges faced while trying to simulate confined systems of interacting electrons in 2D semiconductor quantum dots.
7.1.
Variational Monte Carlo Method
Variational Monte Carlo (VMC) methods [104–111] are robust and yield very accurate results at a computational cost that grows relatively modestly with the number of electrons and the statistical error can be made very small. The VMC method is based on a direct application of the variational principle for finding ground state energies of a quantum system. One starts with an “intelligent guess” of a trial trial wave function that depends on
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several adjustable parameters. This trial wave function is used as an approximation to the true many-body ground state wave function. The variational principle states that the expectation value of the Hamiltonian with respect to a trial wave function, ΨT (~r1, . . . , ~rN ) will be greater than or equal to the (unknown) exact ground state energy: R
~ ΨT (R) ~ ∗H ~ ˆ ΨT (R) dR ≥ E0 , E= R ~ ΨT (R) ~ ∗ ΨT (R) ~ dR
(41)
~ = {~r1, ~r2, . . . , ~rN } represents all coordinates, E0 is the exact ground state energy where R and the asterisk indicates complex conjugation. The variational method then consists of optimizing the adjustable parameters of the trial wave function so as to minimize the trial energy, E, thus providing an upper bound to the exact ground state energy. One can rewrite the above equation in an importance sampled form as: Z
ˆ ~ ∗ ~ ~ ~ p(R) ~ EL(R) ~ ; p(R) ~ = R ΨT (R) ΨT (R) ~ = H ΨT (R) , ; EL (R) dR ~ ΨT (R) ~ ∗ ΨT (R) ~ ~ dR ΨT (R) (42) ~ ~ where p(R)R is the probability density function and EL(R) is the so-called local energy. ~ p(R) ~ = 1. The Metropolis algorithm samples the configuration, R ~ from Obviously, dR ~ The variational energy is obtained by averaging the local the probability distribution p(R). ~ ~ energy, EL(R) over the set of configurations R: E=
E=
M 1 X ~ i) , EL(R M i=1
(43)
where M is the number of Monte Carlo configurations (steps) used in the averaging process. ~ be the instantaneous configuration of the system. The Metropolis algorithm will Let R ~ 2, . . . , R ~ M which, upon reaching equilibrium, will sam~ 1, R then produce a random walk R ~ ple configurations with probability p(Ri), provided that the ratio of the acceptance probabilities satisfies the condition of detailed balance. The following is an outline of the algorithm. Firstly, select a particle i at position ~riold and attempt to move it at a new position ~rinew using: (44) ~rinew = ~riold + ∆ (RN D − 0.5) , where RN D is a random vector uniformly distributed in [0, 1] and ∆ is a length scale for the random displacement. The amplitude of ∆ determines the average acceptance probability of the move. In most implementations, particles are displaced one at a time. Let us denote ~ new the new configuration with a particle moved to position ~ old the old configuration and R R new ~ri . We then calculate the probability ratio: p=
~ new )∗ Ψ(R ~ new ) ~ new ) Ψ(R p(R = , ~ old) ~ old )∗ Ψ(R ~ old ) p(R Ψ(R
(45)
and if p > RN D we accept the move, otherwise we reject it. In this way, each particle i is moved one at a time, in a cube of dimension ∆, centered at ~riold with an acceptance probability min(1, p). The closer the trial wave function is to the exact ground state wave function the more accurate is the variational upper bound of energy.
Few-Electron Semiconductor Quantum Dots in Magnetic Field
19
Obviously, one uses the VMC method and the Metropolis algorithm to evaluate numerically multi-dimensional integrals. It √can be shown that the Monte Carlo integration yields an error which decreases with 1/ M independent of the number of dimensions of the integral. This makes Monte Carlo integration the preferred method for integrals in high dimensions that always come up while solving quantum problems.
7.2.
Diffusion Monte Carlo Method
The diffusion Monte Carlo (DMC) method starts with the idea that the time-dependent Schr¨odinger equation looks like a diffusion/branching equation when written in imaginary time [112]. For a quantum system of N particles (for now we consider a zero magnetic field), the time-dependent Schr¨odinger equation is written as: i¯ h
∂ ~ t) = H ˆ Ψ(R, ~ t) , Ψ(R, ∂t
(46)
ˆ = Tˆ + Vˆ is the many-body Hamiltonian, Tˆ is the total kinetic energy operator, where H ˆ ~ = {~r1, . . . , ~rN } denotes the set of all N V is the total interaction energy operator and R particles coordinates, let’s say in three dimensions. By introducing the imaginary time, τ = i t/¯ h we can rewrite the many-body time-dependent Schr¨odinger equation as: −
∂ ~ τ) = H ˆ Ψ(R, ~ τ) . Ψ(R, ∂τ
(47)
To understand better how this equation is related to a diffusion/branching equation, let us specifically write it as: −
h2 2 ~ ∂ ~ τ) = − ¯ ~ Ψ(R, ~ τ) , Ψ(R, ∇ Ψ(R, τ ) + V (R) ∂τ 2m
(48)
where ∇2 is the Laplacian (for all coordinates). The above expression can be rewritten as: ∂ ~ τ ) = D ∇2 Ψ(R, ~ τ ) − V (R) ~ Ψ(R, ~ τ) , Ψ(R, ∂τ
(49)
where D = ¯ h2 /(2 m) is a constant. The first term in the right hand side of Eq.(49) represents a diffusion term. If we ignore this term, the result would be a first order differential equation that describes a branching process such as an exponential birth/death process in a population (depending on the sign of V ). Thus, the equation above can be interpreted as a diffusion/branching equation. To make the analogy clear, compare Eq.(49) to an or~ t) dinary diffusion/branching differential equation for a system with particle density, ρ(R, ~ and source potential, S(R): ∂ ~ t) = D ∇2 ρ(R, ~ t) − S(R) ~ ρ(R, ~ t) . ρ(R, ∂t
(50)
~ τ ) is the analogous On can see that the many-body wave function in imaginary time, Ψ(R, ~ t). The particles in the case of the Schr¨odinger’s equation in of the particle density, ρ(R, imaginary time are the so-called walkers that diffuse through the phase space. At a point
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~ the number of walkers increases or decreases depending on the sign of V (R). ~ The R, walkers do not interact with each other. Note that this probabilistic interpretation requires ~ τ ) be non-negative, since the particle density ρ(R, ~ t) is non-negative. A formal that Ψ(R, solution of Eq.(47) is: ~ 0) , ~ τ ) = e−τ Hˆ Ψ(R, (51) Ψ(R, ~ 0) is the initial time wave function. If we expand Ψ(R, ~ 0) in the basis set of the where Ψ(R, ˆ (unknown) exact eigenstates, {Φn } of the Hamiltonian and act with e−τ H we would have:
~ τ ) = e−τ E0 c0 Φ0 + Ψ(R,
∞ X n6=0
cn e−τ (En −E0 ) Φn .
(52)
Assuming a non-degenerate ground state (En − E0 > 0 ; n 6= 0) , the above wave function becomes proportional to the exact ground state wave function in the limit of infinite imaginary time: ~ τ → ∞) = c0 Φ0 e−τ E0 . (53) Ψ(R, ~ τ ) and hence the population of walkers will eventually deHowever, we also see that Ψ(R, cay to zero unless E0 = 0. This problem can be avoided by measuring E0 from an arbitrary reference energy ET , which can be adjusted so that an approximate steady population of walkers is obtained. With this modification we have: h i ∂ ~ τ) = − H ~ τ) . ˆ − ET Ψ(R, Ψ(R, ∂τ
(54)
Accordingly Eq.(49) can be conveniently written as: h i ∂ ~ τ ) = D ∇2 Ψ(R, ~ τ ) − V (R) ~ − ET Ψ(R, ~ τ) , Ψ(R, ∂τ
(55)
~ τ → ∞) = c0 Φ0 e−τ [E0 −ET ] . Ψ(R,
(56)
and h
i
~ − ET = 0, then Eq.(55) would be a standard diffusion equation with diffusion If V (R) constant D, hence h i easy to simulate by performing random walks. Alternatively, if only ~ the V (R) − ET term would have been present, Eq.(55) becomes a standard branching equation describing a branching process such as an exponential birth/death process in a given population. As a result, the imaginary time Schr¨odinger equation can be simulated using a combination of diffusion/branching processes acting on a population of walkers. ~ increases or decreases depending upon The number of diffusing walkers at a given point R h i ~ − ET . The adjustable energy shift ET the density of walkers there and the value of V (R) serves to keep the total population of walkers approximately constant. The use of an energy shift ET does not alter the set of eigenstates of the total Hamiltonian thus is fully justified. It can be shown that the ground state energy E0 can be expressed as: E0 =
R
~ Ψ(R, ~ τ ) V (R) ~ dR . R ~ Ψ(R, ~ τ) dR
(57)
Few-Electron Semiconductor Quantum Dots in Magnetic Field
21
~ over the Thus the ground state energy, E0 can be obtained by averaging the potential V (R) population (number) of the walkers: E0 =
P
~ i) (R , i Ni
i Ni V
P
(58)
~ i at time τ . A simple estimate for E0 can be obtained where Ni is the number of walkers at R by averaging the sum in Eq.(58) for several values of τ once a steady population of walkers has been reached. An improvement upon this simple random walk algorithm can be obtained by using the formalism of Green’s functions. With this formalism, the imaginary time Schr¨odinger equation modelled by diffusion/branching processes can be rewritten as: ~ 0, τ ) = Ψ(R
Z
~ G(R, ~ R ~ 0, τ ) Ψ(R, ~ 0) , dR
where
ˆ
~ 0 |e−τ H |Ri ~ , ~ R ~ 0, τ ) = hR G(R,
(59)
(60)
is the Green’s function. One can give a probabilistic interpretation to the Green’s function in ~ to R ~ 0 given the walker Eq.(59) by presenting it as the probability of transition from a state R ~ is Ψ(R, ~ 0). One can use the so-called Green’s function Monte Carlo method density at R ~ R ~ 0, τ ) is not to solve the integral in Eq.(59) but the difficulty is that the exact form of G(R, known. What is known is that the Green’s function satisfies the following equation: h i ∂ ˆ − ET G(τ ) G(τ ) = − H ∂τ
with a formal solution:
ˆ
G(τ ) = e−τ (H−ET ) .
(61)
(62)
One difficulty with Eq.(62) is that the kinetic energy and potential energy operators which ˆ = Tˆ + Vˆ do not commute with each other. If one wants to factor Eq.(62) one are part of H needs an approximation for the exponential. Such approximations are generally valid only for short time τ . To first order in τ , we have: G(τ ) ≈ Gb (τ ) Gd(τ ) Gb(τ ) , where
ˆ
Gd (τ ) = e−τ T ,
(63) (64)
represents diffusion and τ
ˆ
Gb (τ ) = e− 2 [V −ET ] ,
(65)
corresponds to branching. The factorized Green’s functions satisfy the following differential equations (in operatorial form):
and
∂ Gd (τ ) = −Tˆ Gd(τ ) = D ∇2 Gd (τ ) , ∂τ
(66)
i 1h ∂ Gb (τ ) = − Vˆ − ET Gb(τ ) . ∂τ 2
(67)
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The solution of the diffusion equation is: ~ 2 /(4 D τ ) ~ R ~ 0, τ ) ∝ e−(R~ 0 −R) . Gd (R,
(68)
The solution of the branching equation can be written as: ~ Vˆ (R ~ 0 )]+τ ET ~ R ~ 0, τ ) = e− τ2 [Vˆ (R)+ . Gb (R,
(69)
This solution assumes that half the branching occurs before the diffusion and half after as implied by Eq.(63). The method described above has an important limitation. The branching rate of walkers ~ and this can lead to large fluctuations in the walker is proportional to the potential V (R) population especially when fluctuations of the potential are very large as is the case of a Coulomb potential. As a result the method becomes numerically very inefficient. This problem might be solved by using an importance sampling method [113]. The idea is to ~ to guide the walkers towards the more important use a known trial wave function, ΨT (R) ~ regions of V (R). A suitable choice for a trial wave function is one used in a VMC simu~ τ ) given lation. To implement the idea, we introduce a new probability distribution f (R, by: ~ τ) . ~ τ ) = Ψ(R, ~ τ ) ΨT (R, (70) f (R, ~ τ ) satisfies the following differential equaWith help from Eq.(49) one can show that f (R, tion: h i h i ∂ ~ τ ) = D ∇2 f (R, ~ τ) − ∇ ~ f (R, ~ τ) F ~ (R) ~ − EL (R) ~ − ET f (R, ~ τ) , f (R, ∂τ h
(71)
i
~ = H ~ /ΨT (R) ~ is the local energy and F ~ (R) ~ is given by: ˆ ΨT (R) where EL (R) ~ ~ ~ = 2 D ∇ΨT (R) . F~ (R) ~ ΨT (R)
(72)
Compare Eq.(71) with the standard diffusion/branching/drift equation for a system of particles: h i ∂ ~ t) = D ∇2 ρ(R, ~ t) − ∇ ~ ρ(R, ~ t) ~v(R) ~ − S(R) ~ ρ(R, ~ t) . ρ(R, (73) ∂t ~ in Eq.(71) corresponds to a drift of walkers away from regions The term containing F~ (R) 2 ~ ~ (R) ~ in Eq.(71) is the counterpart of the particle drift velocity where |ΨT (R)| is small and F ~ ~ in Eq.(73). Since now the branching rate is controlled by the fluctuations of EL (R) ~v (R) ~ rather than V (R) the algorithm is much more stable. The integral equation equivalent to Eq.(59) looks exactly the same: ~ 0, τ ) = f (R
Z
~ G(R, ~ R ~ 0, τ ) f (R, ~ 0) , dR
(74)
except that now the Green’s function is modified by a drift term. With inclusion of impor~ τ ) which must be non-negative everywhere tance sampling it is the mixed distribution f (R, since a negative density of walkers makes no sense. For a Bose system of particles this is
Few-Electron Semiconductor Quantum Dots in Magnetic Field
23
always true. However, for Fermi systems such as confined electrons in a semiconductor quantum dot, the ground state wave function should be antisymmetric. This means that the ~ τ ) for electrons can be positive and negative. Positivity of f (R, ~ τ ) can wave function, Ψ(R, ~ everywhere. ~ τ ) to have the same sign as ΨT (R) be guaranteed only if we constrain Ψ(R, The best that we can do for Fermi systems of particles is to find the lowest energy state wave function with the same nodal surface as the trial wave function. This is the core idea ~ are the same as the nodes of of the fixed node (FN) approximation. If the nodes of ΨT (R) ~ then in the τ → ∞ limit one has: the ground state, Φ0 (R) ~ Φ0(R) ~ . ~ τ → ∞) = ΨT (R) f (R,
(75)
~ ΨF N (R) ~ , ~ τ → ∞) = ΨT (R) f (R,
(76)
Otherwise: ~ is the lowest energy state with the given nodal surface. Within the FN where ΨF N (R) approximation, one can calculate the ground state energy from the equation: EDM C =
Z
~ ρDM C (R) ~ EL(R) ~ , dR
(77)
where the FN-DMC algorithm samples the following phase space probability distribution: ~ ~ ~ = R ΨT (R) ΨF N (R) . ρDM C (R) ~ ΨT (R) ~ ΨF N (R) ~ dR
(78)
The DMC method attempts to solve exactly the Schr¨odinger equation for a given Hamiltonian. This can be done exactly for a Bose system. However, for Fermi systems such as confined electrons in zero magnetic field, an implementation of the DMC method involves the so-called FN approximation [114]. The FN-DMC method, in which the nodes of the ground state wave function are approximated with those of a good variational function is very accurate for electrons and provides a ”quasi-exact” solution of the quantum problem. It is imperative though that FN-DMC be guided by the best possible variational wave function at disposal. For this reason we stress the importance of developing the variational method first even in presence of ”quasi-exact” simulation methods like FN-DMC for systems of fermions. The FN ground state energy usually is a very accurate variational upper bound to the exact ground state energy. Despite the high accuracy of the FN-DMC method, one can further improve it by using the released-node method [115] where walkers are allowed to cross the nodes under special circumstances. Other more advanced quadratic and fourth-order DMC algorithms to solve atomic and quantum many-problems have been devised [116, 117]. Presence of a magnetic field, for instance when confined electrons in a semiconductor quantum dot are subjected to a perpendicular magnetic field affects the way the DMC calculations are performed. Presence of a magnetic field makes the Hamiltonian complex and explicitly breaks the time reversal invariance of the Hamiltonian. Complex-valued wave functions (such as those for confined electrons in a magnetic field) are dealt by the fixed-phase DMC (FP-DMC) method [118] rather than the FN-DMC method. Within the FP-DMC approach one first makes a choice for the phase of the complex many body wave
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functions and then solves exactly the Bose problem for the magnitude (absolute value) of the wave function using the DMC method. The FP-DMC method provides a variational bound for the energy and, for a given trial phase, it provides the lowest energy consistent with that phase. The accuracy of the FP-DMC method can be assessed and further improved through the use of the released-phase method, which handles the fixed-phase bias [119]. Obviously, many other technical details are left out in this brief description. For further insight on other QMC methods, their applications to physics and chemistry and various algorithms many excellent sources are avalaible [120–124].
8.
The Simplest 2D Semiconductor Quantum Dot (N=2)
The simplest, yet non-trivial, 2D semiconductor quantum dot system consists of two electrons (N = 2) confined in a parabolic confinement potential. If the system is subjected to a perpendicular magnetic field, the quantum Hamiltonian is written as: ˆ r1, ~r2) = H(~
N =2 X i=1
(
)
ωc ˆ m 1 pˆ2i e2 + Liz + Ω2 ri2 + + g µB Bz Sz , (79) 2m 2 2 4 π 0 r |~r1 − ~r2|
ˆ iz is the z-component angular where Ω2 = ω02 + ωc2/4 is a characteristic frequency, L momentum operator for the i-th electron and ωc = eBz /m > 0 is the cyclotron frequency. Despite its small number of electrons, this 2D semiconductor quantum dot system shows many characteristic features which persist to larger structures. In particular the ground state energetics as a function of the magnetic field is highly complex [125]. To study this system we will use the exact numerical diagonalization method and a microscopic wave function written as a product of a Laughlin-type wave function with a Jastrow factor: |mz |
Ψ(~r1 , ~r2) = J(r12) × (z1 − z2 )
r2 + r2 exp − 1 2 2 4 lΩ
!
.
(80)
The Jastrow factor, J(r12) is chosen to have a displaced Gaussian form: !
b2 2 + c b r12 , J(r12) = exp − r12 2
(81)
where zj = xj − iyj is the 2D position in complex notation [this choice is consistent with Eq.(7)], r12 = |~r1 − ~r2| is the inter-electron distance, mz = |mz | = 0, 1, . . . is the angular momentum number, with b and c being two non-negative variational parameters to be optimized. The parity of the space wave function depends on the value of angular momentum |mz |. For even values, |mz | = 0, 2, 4, . . . the space ground state wave function is symmetric and the spin function corresponds to a spin-singlet state ( S = 0). For odd values, |mz | = 1, 3, 5, . . . the space ground state wave function is antisymmetric, thus the spin function becomes a spin-triplet ( S = 1). The rationale behind the choice of a displaced Gaussian correlation factor is better understood if one considers a system of two electrons in zero magnetic field. In absence of
Few-Electron Semiconductor Quantum Dots in Magnetic Field
25
an electronic repulsion between electrons, the relative coordinate ground state wave function with be a Gaussian centered at relative coordinate, r12 = 0, will have zero angular momentum and will correspond to a spin singlet state. When the Coulomb repulsion is taken under consideration, the ground state will still have zero angular momentum [101] therefore it is plausible to expect that the main effect of the Coulomb repulsion is simply to separate further the electrons. This will result in a new relative coordinate ground state wave function which will resemble a Gaussian centered at r12 6= 0 values. The choice of the Jastrow factor in Eq.(81) mimicks this physical effect. Despite the usual controversies involved on the selection of a Jastrow correlation factor it appears that the displaced Gaussian pair correlation factor has all the attributes to capture effectively most of the electronic correlations present on the ground state of such system. Some indication of the eventual high quality of the trial wave function also comes from a previous study [126] where the same Jastrow pair correlation factor was used. Motivated by these arguments, we embarked on a complete study of this trial wave function Eq.(80) for arbitrary values of the magnetic field ranging from very weak (or zero) to very strong [127]. To gauge the accuracy of the microscopic wave function under consideration, we first solve the problem exactly (in numerical sense) by using the exact numerical diagonalization method. Therefore, we start first with a brief general description of the exact numerical diagonalization method and then use it to solve the problem at hand. After achieving this task, we turn our attention to the VMC simulation method and study the properties of the trial wave function under consideration. We calculate various quantities at arbitrary values of the perpendicular magnetic field and then compare variational and exact diagonalization results to each other.
8.1.
Exact Numerical Diagonalization
Let us give a very brief description of how the exact numerical diagonalization method would be applied to solve a quantum eigenvalue equation: ˆ |ψi = E |ψi , H
(82)
ˆ is some Hamiltonian that does not allow an exact analytic solution. To search for where H a solution of the above problem, let’s express the (unknown) function |ψi as a linear combination of a certain number Nmax of linearly independent functions, |φ1 i, |φ2 i, . . ., |φN i: |ψi =
NX max
ci |φii ,
(83)
i=1
where the constants, ci are unknown. For simplicity we can consider the functions |φi i to be orthonormalized, for example exact orthonormalized eigenstates for a simpler Hamiltonian ˆ 0 so that hφi |φj i = δij . H Substituting Eq.(83) into Eq.(82) and performing the inner product of the resulting equation with bra-s: hφ1|, hφ2|, . . . , hφNmax | gives a system of N linear equations relative to the variables c1 , c2, . . . , cNmax . Such system of equations can be written in matrix
26
Orion Ciftja
form as:
H11 H21 ... ... HNmax 1
H12 H22 ... ... ...
... ... ... ... ...
... ... ... ... ...
H1Nmax ... ... ... HNmax Nmax
c1 c2 c3 ... cNmax
= E
c1 c2 c3 ... cNmax
,
(84)
where ˆ ji , Hij = hφi |H|φ
(85)
denotes the so-called Hamiltonian matrix. This equation represents a matrix eigenvalue equation where the column of ci -s is the matrix eigenstate to be determined and E-s are the matrix eigenvalues. Thus, the whole problem of finding the exact energy eigenvalues of the stationary Schr¨odinger equation is reduced to the numerical diagonalization of the resulting Hamiltonian matrix. The matrix to be diagonalized has Nmax × Nmax elements where Nmax is the dimensionality of the truncated (finite) basis set. Diagonalization of the Hamiltonian matrix produces a set of energy eigenvalues and expansion coefficients (from where the eigenfunctions can be obtained). The energy eigenvalues obtained for a series of finite values of Nmax are extrapolated to the Nmax → ∞ limit. When large basis sets are considered, a linear fit (in 1/Nmax) is generally a good fit to the data. The values extrapolated in the Nmax → ∞ limit represent the exact numerical solution of the energy eigenvalue problem (ground and excited state energies). Fort the 2D semiconductor quantum dot (N = 2) under consideration, the exact numerical diagonalization technique simplifies if we separate the Hamiltonian into two parts, one representing the center-of-mass (CM) motion and the other one representing the relative motion. The CM and relative coordinates/momenta are defined as follows: ˆ p~ − ˆ p~2 ~ =ˆ ~ = ~r1 + ~r2 ; Pˆ p~1 + pˆ . ~2 ; ~r = ~r1 − ~r2 ; ˆ p= 1 ~ R 2 2
(86)
Accordingly, the total mass of the system is M = 2 m and the reduced mass is µ = m/2. In this representation the Hamiltonian of the system decouples and can be written as: ~ +H ˆ r (~r) , ˆ R, ~ ~r) = H ˆ R (R) H(
(87)
~ and H ˆ r (~r) are respectively the CM and relative motion Hamiltonians given ˆ R(R) where H by: ˆ2 ˆ z + M Ω2 R2 , ~ = P + ωc L ˆ R (R) (88) H 2M 2 2 and 2 1 e2 ˆ r (~r) = pˆ + ωc lˆz + µ Ω2 r2 + , (89) H 2µ 2 2 4 π 0 r r ˆz , ˆ lz are respectively the CM and relative angular momentum operators. The CM where L wave function is wellknown and the CM eigenenergies are: h Ω (2 nCM + 1 + |mCM |) − ECM = ¯
h ωc ¯ mCM , 2
(90)
Few-Electron Semiconductor Quantum Dots in Magnetic Field
27
where nCM = 0, 1, . . . and mCM = 0, ±1, . . . When looking for the ground state energy, the CM motion will be restricted to the lowest energy level, ¯ h Ω that corresponds to nCM = 0 and mCM = 0. If the electrons were not affected by the Coulomb interaction, the relative wave function would be of the same form as Eq.(7) with the ~r the relative coordinate, mass replaced by the reduced mass, µ and lΩ modified accordingly. As in any standard numerical diagonalization technique we need to calculate the matrix elements of the Hamiltonian in the basis: |nCM mCM ; n mz i. The only non-diagonal terms arise from the Coulomb potential, which is diagonal with respect to nCM , mCM and mz , but not n. As a result the most general non-zero Hamiltonian matrix elements have the form: ˆ CM mCM ; n mz i = hnCM mCM |H ˆ R|nCM mCM iδn0 n+hn0 mz |H ˆ r |n mz i . hnCM mCM ; n0 mz |H|n (91)
For a given nCM , mCM and mz , we have: ˆ R|nCM mCM i Ω 1 ωc hnCM mCM |H = (2 nCM + 1 + |mCM |) − mCM . h ω0 ¯ ω0 2 ω0
(92)
ˆ r |n mz i/(¯ h ω0 ) = hn0 n is given by: The other term, hn0 mz |H hn0 n
r s 1 ωc λ Ω n0 ! n! Ω In0 n |mz | , = (2 n + |mz | + 1) − mz δn n0 + √ ω0 2 ω0 2 ω0 (n0 + |mz |)! (n + |mz |)! (93)
where In0 n |mz | represents the integral: In0 n |mz | =
Z ∞ 0
1 |m | dt t|mz | e−t Ln0 z (t) Ln|mz | (t) √ . t
(94)
We apply the exact diagonalization technique to solve the problem for various values of dimensionless parameters, λ and γ that have been already defined in Eq.(35) and Eq.(36). One cam immediately see that λ = l/aB , where l = 1/α is the harmonic oscillator length h2 /(m e2) is the effective Bohr radius. For any value of the quantum and aB = 4 π 0 r ¯ number, mz = 0, ±1, . . . we build sufficiently large matrices with elements, hn0 n and diagonalize them by using standard numerical methods. For a given λ and γ the smallest of the eigenvalues represents the ground state energy for the relative Hamiltonian. With the addition of the CM energy to the ground state energy of the relative motion we obtain the exact numerical diagonalization value of the ground state energy of this system at an arbitrary perpendicular magnetic field. In Table 1 we show the exact numerical diagonalization ground state energies, = E/(¯ h ω0 ) for the 2D semiconductor quantum dot system (N = 2) in a perpendicular magh ω0 ) = 0, 1, . . ., and netic field, for values of Coulomb correlation, λ = e2 α/(4 π 0 r ¯ values of magnetic field, γ = ωc /ω0 = 0, 1, . . . , 6. The ground state angular momentum, mz is also specified. In Figure 7 we plot the exact numerical ground state energy, = E/(¯ h ω0 ) as calculated from the diagonalization procedure in terms of the dimensionless magnetic field parameter, γ for given values of λ. As expected, the ground state energy always shifts upward as magnetic field increases while the confinement energy strength, h ω0 is kept constant. ¯
28
Orion Ciftja
Table 1. Exact numerical diagonalization ground state energies, = E/(¯ h ω0 ) for a 2D semiconductor quantum dot (N = 2) subject to a perpendicular magnetic field as a function of the dimensionless Coulomb coupling parameter, λ = e2α/(4π0 r ¯ h ω0 ) = 0, 1, . . ., 8 and values of magnetic field, γ = ωc /ω0 = 0, 1, . . ., 6. Thepangular momentum, mz of the ground state is also h has the dimensionality of an inverse length. specified. The parameter α = m ω0/¯
λ=0 mz λ=1 mz λ=2 mz λ=3 mz λ=4 mz λ=5 mz λ=6 mz λ=7 mz λ=8 mz
γ=0 2.00000 0 3.00097 0 3.72143 0 4.31872 0 4.84780 0 5.33224 0 5.78429 0 6.21129 0 6.61804 0
γ=1 2.23607 0 3.30508 0 4.06684 1 4.60594 1 5.11165 1 5.58995 1 6.04534 1 6.48130 1 6.90054 1
γ=2 2.82843 0 3.95732 1 4.61879 1 5.23689 1 5.73642 2 6.21499 2 6.67999 2 7.09645 3 7.49604 3
γ=3 3.60555 0 4.71894 1 5.43123 2 6.01256 2 6.53522 3 7.01716 3 7.46782 4 7.89020 4 8.30327 5
γ=4 4.47214 0 5.61430 1 6.30766 2 6.89002 3 7.41600 4 7.90109 4 8.34530 5 8.77568 6 9.17695 6
γ=5 5.38516 0 6.53067 2 7.22681 3 7.81384 4 8.33874 5 8.82281 6 9.27057 6 9.69413 7 10.10134 8
γ=6 6.32456 0 7.47400 2 8.17960 4 8.76220 5 9.28511 6 9.76657 7 10.21735 8 10.64382 8 11.04743 9
The value of |mz | determines the value of the total spin too. Wave functions with even |mz | are even functions of r12 and denote a singlet spin state with total spin, S = 0. Those with odd |mz | are odd functions of r12 and represent a triplet spin state with total spin, S = 1. Because the ground state spin can be either S = 0 or S = 1, it can serve as gate of a quantum computer [128].
8.2.
Variational Theory
Now we apply variational theory to study the same system. We use the microscopic wave function of Eq.(80) with a Jastrow pair correlation factor that has a displaced Gaussian form given by Eq.(81). We will show that, after optimization, the proposed trial wave function is an excellent representation of the true ground state at any value of the magnetic field and compares very favorably to the exact numerical diagonalization results. Since the parity of space wave function is determined by the value of the angular momentum, |mz |, the ground state angular momentum value determines whether the ground state corresponds to a spinsinglet or spin-triplet state. Therefore, a stringent test of quality for this trial wave function
Few-Electron Semiconductor Quantum Dots in Magnetic Field
29
14 12 10
ε
8 6 4 2 0 0
1
2
3
γ
4
5
6
Figure 7. Exact numerical diagonalization ground state energy, = E/(¯ h ω0 ) for a 2D semiconductor quantum dot (N = 2) as a function of the dimensionless magnetic field Coulomb coupling parameter, λ = parameter, γ = ωc /ω0 for values of the dimensionlessp 2 hω0 ) = 0, 1, . . ., 10. The parameter α = m ω0 /¯ h has the dimensionality of e α/(4π0r ¯ an inverse length. The lowest curve corresponds to λ = 0 and, in ascending order, the top curve corresponds to λ = 10. The solid lines are a guide to the eye.
is to check whether the lowest variational energy state has always the correct angular momentum number for different combinations of Coulomb correlation, confinement strength and perpendicular magnetic field. In a general situation where both Coulomb correlation and magnetic field are present, there is no way to anticipate the correct value of the ground state angular momentum number. An exception are the simple cases of: (i) absence of Coulomb correlation, or (ii) absence of a perpendicular magnetic field. In both cases the ground state is expected to have zero angular momentum. Obviously, such behavior should be reflected by the microscopic wave function under investigation. With no Coulomb correlation ( λ = 0), the ground state has zero angular momemtum (|mz | = 0) both in presence or in absence of the perpendicular magnetic field. For these conditions, the corresponding trial wave function becomes that of Eq.(80) with Jastrow correlation J(r12) = 1 and angular momentum, |mz | = 0 in agreement with the expected result. In absence of a perpendicular magnetic field, the groundstate still has zero angular momentum, (|mz | = 0) with or without the Coulomb correlation, therefore a trial wave function with J(r12) 6= 1 and an angular momentum, |mz | = 0 is again consistent with the expected result. However, when both Coulomb correlation and perpendicular magnetic field are present, the situation changes drastically. As the magnetic field increases, a groundstate with nonzero angular momentums (|mz | 6= 0) may arise, therefore in addition to the Jastrow pair correlation factor also the Laughlin factor starts contributing to optimize the separation between the two electrons. We express the variational ground state energy in dimensionless units, = E/(¯ h ω0 ) ˆ where E = hΨ|H|Ψi/hΨ|Ψi is the expectation value of the Hamiltonian with respect to the
30
Orion Ciftja 18
γ=0 γ=3 γ=6
16 14 12
ε
10 8 6 4 2 0 0
2
4
λ
6
8
10
Figure 8. Variational ground state energy, = E/(¯ h ω0 ) for a 2D semiconductor quantum dot (N = 2) in a perpendicular magnetic field, as a function of the dimensionless Coulomb coupling parameter, λ for values of magnetic field corresponding to γ = 0, 3 and 6.
trial wave funtion in Eq.(80). We use VMC simulations to accurately calculate the ground state energy as a function of λ, γ and variational parameters, B = b/α and c. Note that the Zeeman energy term is not specifically included in the expression for the variational energy. The calculation of the best value of the adjustable variational parameters, B and c is straightforward. Given the values of Coulomb and magnetic field parameters, λ and γ we calculate the lowest energies for a set of integer values of mz and minimize it with respect to B and c through standard numerical procedures. In Table 2 we show the variational ground state energies and optimal values of parameters, B and c at different values of λ and γ. We also verified that the lowest variational energy is always reached for an angular momentum that corresponds exactly to the exact numerical diagonalization values. In Figure 8 we plot the variational energy, = E/(¯ h ω0 ) as a function of the dimensionless Coulomb correlation parameter, λ for various values of magnetic field parameter, γ = ωc /ω0 . The results in Table 1 and Table 2 indicate excellent agreement between variational and numerical diagonalization energies for the whole range of Coulomb correlations and magnetic fields considered here. In the strong magnetic field limit, the variational energies are practically identical (within the range of very small statistical errors) to the exact numerical diagonalization values. We also calculated the mean square distance, h|~r1 − ~r2 |2i between two electrons. In Figure 9 we plot α2 h|~r1 − ~r2|2 i as a function of λ for magnetic field values that correspond to γ = 0, 3 and 6. In absence of Coulomb correlations, the increase of the magnetic field brings electrons closer to each other resulting in a reduced mean square distance. However,
Few-Electron Semiconductor Quantum Dots in Magnetic Field
31
Table 2. Variational ground state energies, = E/(¯ h ω0 ), values of optimal variational parameters, B = b/α and c and ground state angular momentum, mz for a 2D semiconductor quantum dot (N = 2) subject to a perpendicular magnetic field as a function of the dimensionless Coulomb coupling parameter, λ = 0, 1, . . ., 8 and magnetic field values, γ = 0, 1, . . ., 6. Numerical uncertainty is in the last digit. λ=0 B c mz λ=1 B c mz λ=2 B c mz λ=3 B c mz λ=4 B c mz λ=5 B c mz λ=6 B c mz λ=7 B c mz λ=8 B c mz
γ=0 2.00000 0 0 0 3.00174 0.401849 1.67676 0 3.72565 0.497908 2.21655 0 4.32576 0.542433 2.57492 0 4.85637 0.566026 2.85052 0 5.34141 0.579761 3.07820 0 5.79354 0.588185 3.27473 0 6.22032 0.593558 3.44916 0 6.62674 0.597084 3.60699 0
γ =1 2.23607 0 0 0 3.30578 0.416269 1.63743 0 4.06704 0.290392 1.64297 1 4.60635 0.341374 1.98559 1 5.11233 0.379338 2.26315 1 5.59088 0.408618 2.49976 1 6.04652 0.431917 2.70721 1 6.48271 0.450862 2.89276 1 6.90215 0.466771 3.06083 1
γ=2 2.82843 0 0 0 3.95737 0.228093 1.11252 1 4.61899 0.308128 1.55874 1 5.23732 0.366692 1.88097 1 5.73655 0.30103 1.88523 2 6.21518 0.330052 2.09758 2 6.68025 0.355122 2.28554 2 7.09656 0.306556 2.26553 3 7.49618 0.325128 2.41150 3
γ =3 3.60555 0 0 0 4.71899 0.242474 1.05000 1 5.43127 0.240407 1.26233 2 6.01263 0.283543 1.54922 2 6.53525 0.257704 1.62705 3 7.01722 0.284468 1.81530 3 7.46785 0.262571 1.85449 4 7.89024 0.280511 2.00118 4 8.30329 0.258563 2.03383 5
γ=4 4.47214 0 0 0 5.61435 0.255996 0.996915 1 6.30769 0.251836 1.20167 2 6.89004 0.237944 1.34215 3 7.41601 0.225906 1.45132 4 7.90111 0.25575 1.60855 4 8.34532 0.243209 1.66757 5 8.77569 0.229349 1.72987 6 9.17696 0.244089 1.84758 6
γ =5 5.38516 0 0 0 6.53068 0.188161 0.816413 2 7.22683 0.205845 1.04900 3 7.81385 0.212876 1.18987 4 8.33875 0.208064 1.30691 5 8.82282 0.208553 1.38742 6 9.27058 0.227768 1.52332 6 9.69414 0.221357 1.58141 7 10.10140 0.187824 1.71598 8
γ=6 6.32456 0 0 0 7.47400 0.198808 0.782671 2 8.17961 0.182162 0.937437 4 8.76220 0.184837 1.09835 5 9.28511 0.199867 1.18835 6 9.76658 0.168675 1.36590 7 10.21740 0.193236 1.37076 8 10.64380 0.210169 1.47576 8 11.04740 0.199625 1.54589 9
in presence of Coulomb correlations there are values of λ and γ where electrons find energetically favorable to jump to outer orbits (increasing the angular momentum and the mean square distance, as well) despite the effect of the magnetic field. A manifestation of this
32
Orion Ciftja 12
γ=0 γ=3 γ=6
α2 <|r1-r2|2>
10 8 6 4 2 0 -2
0
2
4
6
8
10
12
λ
Figure 9. Variational mean square distance between two electrons, α2 h|~r1 − ~r2|2i for a 2D semiconductor quantum dot (N = 2) in a perpendicular magnetic field as a function of the dimensionless Coulomb coupling parameter, λ for magnetic field values corresponding to γ = 0, 3 and 6. The line joining the data points for γ = 0 serves as a guide to the eye. Note the non-monotonic jumps for λ = 3, 5 and 10.
behavior comes in the form of non-monotonic jumps of α2 h|~r1 − ~r2|2 i as seen in Figure 9 when comparing larger magnetic fields, γ relative to smaller values of γ.
9.
Quantum Hall Limit
Beyond the ground-state energetics and ground-state angular momenta, we test the accuracy of the trial wave function under consideration in the quantum Hall regime limit which corresponds to an infinite magnetic field (γ → ∞). We expect that in this limit, the variational ground state energy should approach the ground state energy of a Laughlin-like wave function for N = 2 electrons: ΨL (~r1, ~r2) = (z1 − z2)|mz |
r2 + r2 exp − 1 2 2 4 lΩ
!
.
(95)
Note that, ΨL (~r1, ~r2) differs from the authentic Laughlin wave functionpfor oddh/(e Bz ) denominator-filled FQHE states [21] since the electronic magnetic length, l0 = ¯ of the original Laughlin wave function is replaced with lΩ in Eq.(95). To check the quantum Hall limiting behavior, we use the Laughlin wave function, ΨL (~r1, ~r2) to calculate the ground state energies of the system at the same values of parameters λ and γ used in the earlier calculations. We then compare the results directly. The ground state energies obtained with a Laughlin-like wave function are displayed in Table 3. The angular momentum, mz for which the lowest energy is obtained is also specified. One immediately observes that the Laughlin-like wave function is not at all a good description of the system at weak (and zero) magnetic fields. By looking at the data in Table 3
Few-Electron Semiconductor Quantum Dots in Magnetic Field
33
Table 3. Ground state energies, = E/(¯ h ω0 ) corresponding to a Laughlin-like wave function for a 2D semiconductor quantum dot (N = 2) system subject to a perpendicular magnetic field for given values of dimensionless parameters λ and γ. The ground state angular momentum, mz is also specified.
λ=0 mz λ=1 mz λ=2 mz λ=3 mz λ=4 mz λ=5 mz λ=6 mz λ=7 mz λ=8 mz
γ=0 2.00000 0 3.25331 0 4.25331 1 4.87997 1 5.50663 1 6.13329 1 6.75994 1 7.28995 2 7.75994 2
γ=1 2.23607 0 3.51671 1 4.17932 1 4.84193 1 5.45996 2 5.95692 2 6.45388 2 6.95083 2 7.40322 3
γ=2 2.82843 0 3.98787 1 4.73309 1 5.33361 2 5.89253 2 6.39990 3 6.86566 3 7.33143 3 7.74564 4
γ=3 3.60555 0 4.74972 1 5.47320 2 6.09150 3 6.61737 3 7.11735 4 7.57749 4 8.01830 5 8.43243 5
γ=4 4.47214 0 5.64527 1 6.34988 2 6.93735 3 7.46625 4 7.95855 5 8.41976 5 8.84800 6 9.26526 7
γ=5 5.38516 0 6.54155 2 7.24827 3 7.84253 4 8.37252 5 8.86033 6 9.31802 7 9.74881 7 10.15680 8
γ=6 6.32456 0 7.48489 2 8.19251 4 8.78138 5 9.30931 6 9.79480 7 10.24890 8 10.67860 9 11.08890 10
one sees that there are several occasions in which the Laughlin ground state has the wrong angular momentum, such as in the cases γ = 0 and λ = 2, 3, . . ., etc. However, a comparison of the energies in Table 2 and Table 3 indicates that, in the limit of strong magnetic fields (increasing γ-s), the ground state energy of the displaced Gaussian variational wave function quickly approaches that of the Laughlin-like wave function. This is clearly seen in Figure 10 where we plot the ground state energy corresponding to the displaced Gaussian variational wave function and the Laughlin wave function for two values of λ as a function of the dimensionless magnetic field parameter, γ. The convergence of variational and Laughlin ground state energies in the quantum Hall regime limit ( γ → ∞) is quite general and happens at any arbitrary value of the Coulomb correlation strength, λ. For clarity of the plot, we do not display curves corresponding to other values of λ. We also studied the classical limit of the system. The lowest classical energy, Ec for N = 2 point charges in a harmonic potential (for zero kinetic energy) was calculated to be: c =
3 Ec = (2 λ)2/3 . (¯ h ω0 ) 4
(96)
We note that the classical energy is not affected by the presence or absence of a magnetic ~ does not change the kinetic energy of a field since a Lorentz magnetic force, F~L = q ~v × B
34
Orion Ciftja 12
Var λ=2 Laughlin λ=2 Var λ=4 Laughlin λ=4
10
ε
8
6
4
2 0
1
2
3
4
γ
5
6
Figure 10. Ground state energy, = E/(¯ h ω0 ) as a function of the dimensionless magnetic field parameter, γ for the case of displaced Gaussian variational wave function (Var) and Laughlin’s wave function (Laughlin) for two values of the Coulomb correlation strength: λ = 2 and 4. The solid lines joining the data points serve as a guide to the eye. charge, q. The classical ground state configuration for N = 2 electrons is one in which the respective positions of the particles are exactly opposite to each other at an optimal distance, ~r1 = −~r2 6= 0. Naturally, one cannot immediately compare the quantum variational energy, to its classical counterpart, c since without Coulomb correlations ( λ = 0) the lowest classical energy, c is simply zero while the the lowest quantum energy, is nonzero. The energy difference, − c represents the quantum zero-point energy (at λ = 0) which in dimensionless units is given by: E0 =2 ∆0 = h ω0 ¯
s
1+
γ2 . 4
(97)
If we adjust the classical energy by the quantity ∆0 above, then we can compare the quantum energy, with the “adjusted” classical energy, (c + ∆0 ). Figure 11 shows the variational ground state energy, of a 2D semiconductor quantum dot system (N = 2) in a perpendicular magnetic field, and the “adjusted” classical energy, c + ∆0 (solid lines) as a function of the dimensionless Coulomb correlation parameter, λ for selected values of the dimensionless magnetic field parameter, γ. Quite surprisingly there is very good agreement between the quantum ground state energy and the “adjusted” classical ground state energy at all magnetic fields considered including weak magnetic fields. As the magnetic field grows (increasing values of γ) the agreement only improves as we can clearly see from the plot. We also verified that in the infinite magnetic field limit ( γ → ∞) both (variational)−∆0 and (diag) − ∆0 tend to c (classical). Generally, the physics of 2D semiconductor quantum dots in high magnetic field has been described with the use of the composite fermion (CF) wave functions [129–137] which represent a generalized version of the Laughlin wave function. However, recently, a novel class of wave functions dubbed rotating Wigner molecule (RWM) wave functions [138,139]
Few-Electron Semiconductor Quantum Dots in Magnetic Field 14
35
γ=0 γ=2 γ=4
12 10
ε
8 6 4 2 0 0
2
4
λ
6
8
10
Figure 11. Variational quantum ground state energy for a 2D semiconductor quantum dot system (N = 2) in a perpendicular magnetic field, = E/(¯ h ω0 ) as a function of the dimensionless Coulomb coupling parameter, λ for values of magnetic field corresponding to γ = 0, 2 and 4. The solid lines represents the ”adjusted” classical energy, c + ∆0 as a function of λ calculated at the selected values of γ. has emerged as a serious competitor to the CF description.
10. Generalized Description of Few-Electron Semiconductor Quantum Dots in an Arbitrary Perpendicular Magnetic Field Despite the great theoretical progress, a common unified description of 2D semiconductor quantum dots for the whole range of perpendicular magnetic fields is a very challenging problem far from being fully resolved. A large number of proposed theoretical models treat separately the weak (or zero) magnetic field regime [140] from the strong magnetic field regime [101] and generally have a limited domain of applicability. Routinely, a microscopic description that works well in the strong magnetic field regime, does not apply to the weak (or zero) magnetic field limit and vice versa. Thus, it is highly desirable to have a generalized unifying framework for the study of 2D semiconductor quantum dots at any arbitrary magnetic field. A microscopic description would start with a general ground state wave function that applies to both regimes of weak (or zero) and strong magnetic field. The most reasonable physical requirements to be imposed upon would be: (i) The microscopic wave function should be Laughlin-like in the strong magnetic field limit, (ii) The microscopic wave function should have a Jastrow-Slater form in the zero magnetic field limit (where depending on localization/correlation strength parameters, a Fermi liquid or a Wigner molecule/crystal state may emerge). One way to go is to generalize the microscopic wave function introduced earlier to 2D semiconductor quantum dot systems with more electrons. In such case the liquid or solid (crystalline)
36
Orion Ciftja
character of larger 2D semiconductor quantum dot systems in a magnetic field is rather non-trivial and crucially depends on the values of γ, λ as well as density of the system under consideration. To describe Wigner molecules/crystal phases, at the minimum one needs to employ a Jastrow-Slater crystalline wave function where the Slater determinant should ~ as indicated in Eq.(40): contain Gaussian functions centered at sites of a given lattice, {R}
~ 2 , φR~ ∝ exp −c |~r − R|
(98)
where c is an adjustable parameter. To describe liquid-like phases, the following variational wave function may serve as a generalization for 2D semiconductor quantum dot systems containing any number of electrons both at zero and nonzero magnetic fields: Ψγ =
N Y
J(rij )
i<j
N Y
(zi − zj )np D↑(φF D )D↓(φF D ) χ(S) ,
(99)
i<j
where np = 0, 1, . . . is a non-negative integer, S = {s1 , s2, . . . , sN } represents all spin coordinates and the space wave function corresponds to a spin function, χ(S) = χ(s1 , s2, . . . sN ) in which the first N↑ electrons are spin-up and the remaining electrons, N↓ = N − N↑ are spin-down. The Slater determinants for spin-up, D↑(φF D ) and spindown, D↓(φF D ) electrons are built out of FD states of the form: φF D ∝
z lΩ
|mz |
|z|2 exp − 2 4 lΩ
!
|z|2 2 2 lΩ
Ln|mz |
!
,
(100)
where z is 2D position in complex notation, n = 0, 1, . . ., and mz = 0, ±1, . . .. The parity of the space wave function crucially depends on the value of the integer number, np = 0, 1, 2, . . ., which is even/odd in accordance with Pauli’s principle. In the strong magnetic field limit (Bz → ∞), which corresponds to γ → ∞, the wave function, Ψγ reduces to a Laughlin-like wave function: Ψγ→∞ =
N Y
J(rij )
i<j
N Y
np +1
(zi − zj )
i<j
exp −
N X |zi |2 i=1
!
2 4 lΩ
χ(S) ,
(101)
where the spin function, χ(S) represents a fully spin-polarized state. The overall antisymmetry of the wave function requires np to be even with J(rij ) ' 1 becoming a minor correction in this limit. This result is explained by noting that for a fully spin-polarized system ( N↑ = N ), only D↑(φF D ) exists. The other Slater determinant, D↓(φF D ) can be formally set to 1 and in this limit FD states transform into Landau states where D↑(φF D ) =
QN
i<j (zi
− zj ) exp −
PN
|zi |2 i=1 4 l2 Ω
and lΩ → l0.
In the weak (or zero) magnetic field limit (Bz → 0), which corresponds to γ → 0, the variational wave function, Ψγ reduces to a standard Jastrow-Slater wave function of the form: Ψγ→0 =
N Y i<j
J(rij ) D↑(φHO ) D↓(φHO ) χ(S) ,
(102)
Few-Electron Semiconductor Quantum Dots in Magnetic Field
37
where the spin function, χ(S) is expected to be antisymmetric (like a spin-unpolarized state) and the FD states become 2D harmonic oscillator (HO) states: φHO ∝ (α r)|mz |
α2 r2 exp − 2
!
Ln|mz | α2 r2 exp(i mz ϕ) .
(103)
Q
np Pauli’s principle imposes the choice: np = 0 for the polynomial N i<j (zi − zj ) , although any even value, np = 2, 4, . . . would also work in a more general setting. The general wave function in Eq.(99) should apply to any 2D semiconductor quantum dot having N electrons in a zero or nonzero perpendicular magnetic field. Clearly we can see that, for N = 2, it reduces to the wave function in Eq.(80) where np = |mz |. For weak (or zero) confinement and for intermediate values of the magnetic field (when a fraction of electrons is spin-reversed) the wave function becomes:
Ψγ
=
N Y
J(rij )
i<j
N↑ Y
(zi − zj )np +1
i<j
× exp −
N↓ Y
(zi − zj )np +1
i<j
N↑ N ↓ Y Y
i=1 j=1
N↑ N↓ X X ri2 ri2 exp − χ(S) . i=1
2 4 lΩ
i=1
2 4 lΩ
(zi − zj )np
(104)
This is reminiscent of the Halperin’s spin-reversed wave function [141] for FQHE-like states in moderate magnetic fields where the J(rij ) factors above play the role of additional correction factors.
11. Conclusion A number of experimental discoveries in the field of 2D semiconductor quantum dots call for development of new theoretical concepts, models and methods not only to study the properties of available systems, but also to anticipate theoretically the properties of many other nanoscale structures that will be manufactured in the future. The case of 2D semiconductor quantum dots with an arbitrary number of electrons poses not only challenging theoretical problems, but also represents a tremendous computational/simulation task for accurate numerical studies. The transitional properties of 2D semiconductor quantum dots between mesoscopic and bulk regimes are hard to predict. By its very nature, the study of nanoscale 2D semiconductor quantum dots involves multiple lengths and scales as well as the combination of theories that have been traditionally studied separately. This means that fundamental theoretical methods that were developed in separate contexts will have to be combined and eventually new ones invented in order to describe and model such nanoscale systems. In this work we try to give an introductory theoretical review to the many-particle properties of 2D semiconductor quantum dot systems with a few number of electrons in presence of a perpendicular magnetic field. For brevity of treatment, we ommit from our treatment many other interesting phenomena that occur in a tilted magnetic field. Our objective is not to give a comprehensive review of the vast experimental and theoretical literature published over the last decade. The main emphasis is to clarify some novel physical properties of 2D
38
Orion Ciftja
semiconductor quantum dots in comparison with extended bulk materials that arise because of the interplay between quantum confinement, correlation effects and perpendicular magnetic field. To this effect we try to point out general features of confined electronic structures in a 2D semiconductor quantum dot and their origins. We also give a brief overview of different computational methods applied to the field. The focus is mainly on the exact numerical diagonalization method and quantum Monte Carlo methods. Therefore we give a lengthier description of these techniques, while being very brief when describing other methods. Among the variety of quantum Monte Carlo methods, we specifically focus our attention on two of them, variational Monte Carlo (VMC) and diffusion Monte Carlo (DMC). Such methods now provide a nearly exact description of the electronic structure of a variety of systems ranging from atoms/molecules to strongly correlated electronic systems with hundred and thousands of particles. The VMC method is robust and yields very accurate results at a computational cost that grows relatively modestly with the number of electrons and the statistical error can be made very small. The variational approach gives an insight into the physical processes related to the behavior of 2D semiconductor quantum dots in a magnetic field, provides very accurate results for a number of fundamental properties and can be applied to systems with many electrons. A good variational wave function improves the accuracy of the more accurate (but also more computationally demanding) DMC simulations who would provide a ”quasi-exact” solution of the quantum problem (within the fixed-node, released-node, fixed-phase and/or released-phase approximations). Among the many theoretical and computational challenges that have hindered the theoretical understanding of 2D semiconductor quantum dot devices in a perpependicular magnetic field a few ones that we identify are: • bridge electronic and spin properties from microscopic to macroscopic lengths. • understand the nature of the electronic liquid-solid phase transition from weak to strong interaction limit in an arbitrary magnetic field. • understand the evolution of Wigner molecules (i.e. the small finite-size counterpart to the Wigner crystal) from a low-density (strong interaction) regime to a high-density (weak interaction) regime. • device novel theoretical and simulation approaches to study electronic and spin properties of strongly correlated confined electronic systems. • simulate with reasonable accuracy the electronic and spin properties at an arbitrary magnetic field. • shed light on spin-dependent properties especially at an intermediate (weak) magnetic field. In this work we analyze theoretically the electronic properties of few-electron 2D semiconductor quantum dots in a perpendicular magnetic field. We develop the theoretical background for the analysis of quantum confinement effects and its relation to the single particle spectrum, introduce the different quantum regimes that apply to such systems and give an
Few-Electron Semiconductor Quantum Dots in Magnetic Field
39
overview of various calculation methods. We then attempt to illustrate key theoretical properties of 2D semiconductor quantum dots in a pedagogical way by focusing on a simple 2D quantum dot system that serves as a good benchmark and illustrates well the challenges of the field. Despite many studies our understanding of 2D semiconductor quantum dot systems in a perpendicular magnetic field is far from being complete. So far, the following distinct magnetic field regimes have been identified: 1) Strong magnetic field, where all electrons are fully spin polarized. This is the crossover regime between microscopic 2D quantum dots and macroscopic 2D electronic systems. 2) Weak or zero magnetic field which corresponds to the crossover region between a Fermi liquid and a Wigner solid. 3) Intermediate (weak) magnetic field where prominent spin effects are expected to occur. A unified description of 2D semiconductor quantum dots for the whole range of magnetic fields is hard to achieve and many current models treat separately the weak (or zero) magnetic field regime from the strong magnetic field regime. Such models have a limited domain of applicability and fail to describe consistently the behavior of 2D semiconductor quantum dots at an arbitrary magnetic field. Thus, we raise the question of what would be a good microscopic description of a 2D semiconductor quantum dot system for an rabitrary value of the magnetic field which incorporates weak, intermediate and strong magnetic field regimes. This guarantees a common theoretical framework to study the properties of 2D semiconductor quantum dots from weak to strong interactions as the size (number of electrons) and the magnetic field is varied. To this effect we introduce some elements of a good microscopic ground state wave function that would apply to both regimes of weak (or zero) and strong magnetic field. A key novel element of this description is a Jastrow pair correlation factor that has a displaced Gaussian form and contains two adjustable variational parameters. A Jastrow pair correlation factor, such as the displaced Gaussian factor introduced in this work, is essential to provide an accurate description of the system at any value of the magnetic field not limited to high magnetic fields only. It also provide quite an accurate description of the system and compare favorably with exact diagonalization results. When compared to other Jastrow correlation factors, the displaced Gaussian pair correlation factor is the most intuitive and the simplest physical choice that guarantees a consistent and excellent description of the system under consideration at all magnetic fields ranging from weak (zero) to infinity. We demonstrate the quality of this wave function for the special case of few-electron 2D semiconductor quantum dot systems (N = 2) and discuss general outcomes for zero, intermediate and strong magnetic fields. The variational energies are in excellent agreement with exact numerical diagonalization calculations at any value of the perpendicular magnetic field including weak (and zero) or strong fields. For a given Coulomb correlation strength and in the limit of infinite magnetic field, the energies of the displaced Gaussian trial wave function agree very well with the energies obtained from a Laughlin-like wave function, though we note that Laughlin’s wave function is a poor description of the system for weak (zero) and intermediate magnetic fields. For weak (zero) and intermediate magnetic fields a Jastrow pair correlation factor of the nature studied in this work should be included in the total wave function in conjuction with the Laughlin component, which is effective only in high magnetic fields. A generalization of this trial wave function for N -electron quantum dots in a perpendicular magnetic field is also discussed. Most of the
40
Orion Ciftja
results shown in Figures and Tables are our own calculations. A lot of work has been done over the last decade and obviously many phenomena shown here have been studied by other authors. While we have tried to give credit to earlier research, certainly and inadvertently we might have overlooked other valuable published work on the topic. For brevity of treatment and space limitation, we were forced to leave out of consideration several other aspects of the physics of 2D semiconductor quantum dots. In this work we have attempted to present and introductory review of the physics of few-electron 2D semiconductor quantum dots in a perpendicular magnetic field, with particular focus on confinement effects, energy spectra and key simulation methods. We also introduced some ideas about key elements useful for a generalized microscopic description of 2D semiconductor quantum dots at all magnetic field regimes. By further developing this initial conceptual understanding, we hope that in the future we will have methods to determine directly the magnitude, size, shape, and orientation of 2D semiconductor quantum dot structures that ultimately will lead us to determine accurately their electronic, optical and vibrational properties. A complete theoretical understanding of the interplay between all these factors will put us in position to attack a number of important fundamental problems that have already been identified, and no doubt some that have not yet been recognized. It will also help us to develop methods that would directly determine the conditions under which a given 2D semiconductor quantum dot can operate with the desired high selectivity and sensitivity. While we have attempted to illustrate some relevant phenomena present in 2D semiconductor quantum dots in a perpendicular magnetic field, the intricate interplay between confinement, electronic correlations, magnetic field, spin effects, and low dimensionality certainly provides fertile ground for many other surprises that we could not discuss in this work.
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[128] G. Burkard, D. Loss, and D. P. DiVincenzo, Phys. Rev. B 59, 2070 (1999). [129] O. Heinonen (Ed.), Composite Fermions, World Scientific, New York (1998). [130] J.K. Jain, Phys. Rev. Lett. 63, 199 (1989). [131] J. K. Jain, Phys. Rev. B 41, 7653 (1990) [132] J. K. Jain and T. Kawamura, Europhys. Lett. 29, 321 (1995). [133] O. Ciftja and C. Wexler, Solid State Commun. 122/7-8, 401 (2002). [134] O. Ciftja and C. Wexler, Eur. Phys. J. B 23, 437 (2001). [135] O. Ciftja, Physica E 9, 226 (2001). [136] O. Ciftja, Eur. Phys. J. B 13, 671 (2000). [137] A. D. G¨ucl¨u, G. S. Jeon, C. J. Umrigar, and J. K. Jain, Phys. Rev. B 72, 205327 (2005). [138] C. Yannouleas and U. Landman, Phys. Rev. B 68, 035326 (2003). [139] C. Yannouleas and U. Landman, Phys. Rev. B 69, 113306 (2004). [140] F. Pederiva, C. J. Umrigar, and E. Lipparini, Phys. Rev. B 62, 8120 (2000) ; Phys. Rev. B 68, 089901(E) (2003). [141] B. I. Halperin, Helv. Phys. Acta 56, 75 (1983).
In: Quantum Dots: Research, Technology and Applications ISBN: 978-1-60456-930-8 Editor: Randolf W. Knoss, pp. 47-107 © 2008 Nova Science Publishers, Inc.
Chapter 2
INVESTIGATIONS OF ELECTRONIC STATES IN SELF-ASSEMBLED INAS/GAAS QUANTUM-DOT STRUCTURES Shiwei Lin a,b and Aimin Song a a
School of Electrical & Electronic Engineering, University of Manchester, Manchester M60 1QD, United Kingdom b College of Materials Science and Chemical Engineering, Hainan University, Haikou 570228, People’s Republic of China
Abstract Space-charge techniques, such as capacitance-voltage (CV) spectroscopy and deep-level transient spectroscopy (DLTS), are used to examine the electronic states of ensembles of selfassembled InAs quantum dots (QDs), embedded in a GaAs matrix and grown by the 3D Stranski-Krastanow growth mode. We present direct experimental evidence of the coexistence of deep levels in the same epitaxial layer of optically active quantum dots. The InAs quantum dots show very good optical properties, as evidenced by the strong photoluminescence (PL) at room temperature at ~1.3 μm. The reverse-bias dependence of the DLTS signal together with results from the reference samples, containing thin InAs layers but no quantum dots, confirms that the deep levels coexist in the dot layer and are most likely caused during the latticemismatched growth process. Laplace deep-level transient spectroscopy (LDLTS) is a technique developed primarily to study the point defects in semiconductors, which has also recently been applied to the semiconductor quantum-dot structures. The newly developed technique can provide orders of magnitude better resolution than the conventional DLTS method. By applying the LDLTS technique, we are able to study the electronic fine structure of the deep-level states coexisting in the dot layer. As a way of tuning the electronic properties, postgrowth rapid thermal annealing (RTA) has been applied to the semiconductor quantum dots, and the induced optical and electrical changes are studied using PL and DLTS. These combined optical and electrical experiments also confirm our findings of the coexistence of the deep levels with the QDs. By a comparison of the DLTS data with the PL spectra, we find that the effects of RTA on the optical spectra are closely linked with the alternations of the electronic structures, and that a new deep level
48
Shiwei Lin and Aimin Song (0.62 eV) is created in the structure, which dominates the whole spectra at certain annealing temperatures. Furthermore, by combining the CV, conventional and Laplace DLTS techniques, we systematically and quantitatively investigate the underlying emission mechanisms in the QD single-level two-electron system. Electron emissions from the singly and doubly occupied QD s states can be resolved by the LDLTS technique. The emission processes are investigated in detail by the pulse-bias dependency. The electron distribution profile in quantum dots is identified by applying an appropriate set of voltage pulses across the Schottky diode structure.
1. Introduction The first semiconductor low-dimensional heterostructures, known as quantum wells (QWs), were developed in the early 1970s. They form the basis of many optoelectronic devices available today and their importance was recognized by the award of the 2000 Nobel Prize in Physics to Zhores Alferov and Herbert Kroemer. Such structures are often referred to as two dimensional (2D), because the charge carriers (electrons and holes) are confined in a 2D plane. This has two advantages. First, the optical properties of QWs can be tuned simply by changing their structural parameters, typically thickness and composition (so-called band-gap engineering). Secondly, the reduced dimensionality leads to improved optical performance, especially by increasing the probability of electron-hole radiative recombination. Following the developments of 2D systems, scientists also investigated the possibility of reducing the dimensionality further to create 1D (quantum-wire) and 0D (quantum-dot) structures. In the last decades, considerable interest has been focused on the semiconductor self-assembled quantum dots (SAQDs).[1-6] The QDs are self-assembled by the StranskiKrastanow (SK) growth mode (3D clusters on a wetting layer).[7] This growth mode occurs in heteroepitaxial systems with significant lattice mismatch, such as InAs on GaAs (7.2%). When the amount of deposited InAs material exceeds a critical thickness, 3D coherent islands are formed to relieve the built-up elastic strain. The 3D islands are then made into QDs by covering them with another larger bandgap semiconductor layer, such as GaAs. The SAQDs are characterized by a strong three-dimensional quantum confinement of charge carriers, leading to the discrete density of states in the dots. For this reason, the SAQDs are often described as artificial atoms. This property makes the quantum dots important in device application and fundamental physics. Self-assembled dots are considered as coherent (i.e., defect-free) 3D islands with intriguing quantum nature until island coalescence and defect incorporation occur as the dot-layer thickness is larger than a critical thickness [typically ~ 3 monolayers (ML) for InAs/GaAs QDs].[8-10] Such a coherent structure opens up applications in electronics and optoelectronics,[11-15] e.g., low-threshold semiconductor lasers, infrared single-phonon sources/detectors, and flash memories. In a practical QD-based device, the precise information of the electronic level structure and thorough understanding of the charge carrier exchange between the QDs and their host material are of considerable importance. So far, extensive experimental and theoretical investigations have been reported. On the theoretical side, the multiband k•p model[16-18] and empirical pseudopotential theory[19] have enabled a deeper insight into the electronic structure of strained QDs. However, these model calculations rely strongly on the detailed structural properties of the dots and the input materials parameters. Experimentally, efforts have been devoted to the investigation of QD electronic structures, either by electrical or
Investigations of Electronic States in Self-assembled InAs/GaAs Quantum-Dot…
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optical techniques. The optical experiments, such as photoluminescence (PL) spectroscopy,[20,21] photoluminescence excitation (PLE) spectroscopy,[22] are successful in elucidating the excitonic properties of QDs, but cannot access the energy levels and carrier dynamics of electrons and holes separately. Space-charge techniques (electrical techniques), such as capacitance-voltage (CV) spectroscopy,[23-28] admittance spectroscopy[28,29] and deep-level transient spectroscopy (DLTS),[30-36] have been recently applied to QD structures. They allow quantitative determination of the QD levels and examine the carrier capture and escape dynamics for the electrons and holes separately. As such, they provide complementary information to optical techniques. More recently a newly developed Laplacetransform isothermal deep-level transient spectroscopy (LDLTS) gives about an order of magnitude better energy resolution than conventional DLTS.[37] It thus enables us to reveal the fine electronic state structures in the carrier emission process. Up to now, the QD structures have been widely investigated by both static and timeresolved optical experiments. Plenty of valuable information was obtained to allow systematic description of the excitonic properties not only in quantum-dot ensembles but also in individual dots. However, investigations on the self-assembled QD structures by electrical techniques are relatively limited. More importantly, in recent years, an increasing number of PL measurements have suggested the possibility that electronic deep levels might exist around or inside QDs, in contrast to the early belief that SAQDs are coherent and hence defect free.[38-40] Due to their generally nonradiative nature, PL experiments could not provide direct proof or disproof of possible deep levels. However, in the case of deep centers, the electrical space-charge technique, DLTS, has long been proven to be a powerful tool in investigating their electronic properties, and their carrier capture and emission dynamics. As such, it is of great importance to apply such technique to investigate the electronic states in the QD structures, which can not only reliably determine the spatial and energy positions of the intrinsic QD states as well as the nonradiative deep levels, but also provide useful guidance to the optimizations of QD electronic and optical devices. In this chapter, we combine both optical and electrical techniques, i.e., PL, CV, conventional and Laplace DLTS, to investigate the electronic states in the self-assembled InAs/GaAs quantum-dot structures, and study the electron interaction between the QD electronic states and the host semiconductor. In Section 2 the space-charge techniques that we used are introduced. Section 3 focuses on the coexistence of the deep levels with optically active quantum dots. The quantum confinement of the carriers in QDs has investigated using CV spectroscopy, and its dependence on the temperature and measurement frequency is presented. In this section, we present the direct evidence of the coexistence of deep levels with optically active QDs, where the complete DLTS spectra of both the deep levels and InAs QD intrinsic energy states are determined. Furthermore, the high-resolution LDLTS has been applied to resolve the fine structure of the deep levels. In Section 4, we investigate the effects of postgrowth rapid thermal annealing (RTA) on the intrinsic states and deep levels in the QD structures. Finally, in section 5, the electron emission from intrinsic QD energy states has been studied by applying both the electrical and optical techniques. The LDLTS technique allows us to resolve the electron emissions from the singly and doubly occupied QD s states.
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Shiwei Lin and Aimin Song
2. Space-Charge Techniques 2.1. Capacitance-Voltage Spectroscopy Valuable information on the electronic structure can be provided by capacitance-voltage spectroscopy on dot layers incorporated in metal-insulator-semiconductor (MIS) devices or in Schottky diodes. Figure 1 presents the schematic measurement diagram. In capacitance measurements on quantum dots in Schottky diode the interaction between charges in the dot and the reservoir is significant and clearly observed. In pure capacitance spectroscopy the measurement frequency is sufficiently low so that the electrons in the dots are always in equilibrium with the reservoir. When a small AC voltage Vmod is added to the applied gate voltage Vg, the resulting shift of charge back and forth between the back contact and the dots manifests itself by an increased capacitive signal in the external circuit, which results in a plateau-like structure in the CV trace. Once an electron is injected from the contact into a dot, further charging is inhibited by the so-called Coulomb blockade as long as the contact potential is not readjusted. While capacitance spectra yield information on the level separation and the Coulomb charging energy, they do not provide direct information on the absolute binding energies of the dot states.
Figure 1. Quantum-dot field-effect device used to probe the electronic states of self-assembled QDs.
By applying Poisson’s equation and Gauss’ theorem, the exact expression for the capacitance in terms of the total voltage across the depletion region for uniformly doped materials is given as following[41]: 1 2
kT ⎡ ⎛ εε N ⎞ ⎧ ⎛ − eV C = A⎜ 0 d ⎟ ⎨V − 1 − exp⎜ ⎢ e ⎣ ⎝ kT ⎝ 2 ⎠ ⎩
⎞⎤ ⎫ ⎟⎥ ⎬ ⎠⎦ ⎭
−
1 2
⎡ ⎛ − eV ⋅ ⎢1 − exp⎜ ⎝ kT ⎣
⎞⎤ ⎟⎥ ⎠⎦
(1)
where A is the Schottky contact area, Nd doping concentration, V= Va + Vb the sum of the built-in voltage Vb and the applied bias Va, and T the measured temperature. For practical
Investigations of Electronic States in Self-assembled InAs/GaAs Quantum-Dot…
51
purposes it can be simplified by noting that at room temperature (kT/e) ≈ 0.026 V and in reverse bias V > Vb ≈ 0.7 V for common semiconductors, so that the exponential terms can be neglected to give 1
kT ⎞ ⎛ εε eN ⎞ 2 ⎛ C = A⎜ 0 d ⎟ ⎜V − ⎟ e ⎠ ⎝ 2 ⎠ ⎝
−
1 2
.
(2)
A plot of C-2 versus reverse applied bias Vr in uniform material is linear and has a slope proportional to Nd-1 and intercept of (Vb - kT/e). With the assumption V >> kT/e and the total band bending voltage is
V =
eN d 2 xd . 2εε 0
(3)
We can obtain the important (but approximate) result
C=
εε 0 A xd
.
(4)
This is identical to the expression for the capacitance of two parallel plates of area A containing a dielectric of relative permittivity ε and spaced apart xd a distance equal to the depletion layer width: the depletion layer appears to behave as a parallel plate condenser. This is surprising because unlike a conventional dielectric the depletion region of a semiconductor contains distributed charges so the electric field is not uniform but increases in magnitude linearly with distance from the depletion layer edge, as can be shown below:
F ( x) = −
eN d
εε 0
( x d − x) .
(5)
With the depletion approximation, it is also easy to derive the capacitance-voltage behaviour for an arbitrary space-charge density distribution ρ(x). The capacitance is the same as given by the parallel-plate expression (Equation (4)) irrespective of the charge distribution in the depletion region. In uniformly doped material a plot of C-2 versus Vr is linear with a slope determined by the doping intensity Nd. However, a linear plot is not obtained for nonuniform material. But as demonstrated by P. Blood,[41] it is also possible to calculate the local doping intensity Nd(x) from the local slope of the CV curve, which can be written in the form:
− C 3 ⎛ ΔC ⋅⎜ N ( xd ) = eεε 0 A ⎜⎝ ΔVr
⎞ ⎟⎟ ⎠
−1
(6)
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Shiwei Lin and Aimin Song
where we allow for the possibility that the semiconductor is compensated by Na acceptors, in which case N(xd)=Nd(xd)-Na(xd). For heterobarriers, the redistribution of free carriers across the barrier leads to an accumulation and depletion of free carriers on opposite sides of the barrier, which shows the spike in the vicinity of the barrier in the CV profile. When the applied bias is not too large, free carriers remain trapped at the band edge discontinuity. The CV profiling process lumps two components of the charge increment together: one is the localized component at the band discontinuity, and the other is the smear-out component by the Debye tail at the depletionlayer edge. Such carrier concentration profile is not due to real variation in Nd(x) but to the artifacts associated with the redistribution of free charge across the barrier. Similarly, the quantum confinement of the QDs can also be schematically shown by the apparent carrier concentration profile of Nd as a function of depletion depth xd.
2.2. Deep-Level Transient Spectroscopy Deep-level transient spectroscopy has been very successfully applied for decades to investigate deep traps in semiconductors and more recently to self-assembled QDs.[30-35] The deep traps or QDs must be located in the depletion region of a Schottky diode or a p-n junction. By observing the capacitance transient associated with the return to thermal equilibrium occupation of the level following a non-equilibrium condition, one can determine the energy level of the traps or QDs. This can be done by measuring the time constant of the transient as a function of temperature and obtaining the activation energy for the level from Arrhenius plot. In the DLTS measurement, the QDs are charged by a voltage pulse applied across the Schottky diode that pushes the boundary of the depletion zone through the QD layer.[36] The value of the filling-pulse bias Vf controls the charging state of the dots. After the filling pulse the gate voltage is reduced to a so-called reverse bias value Vr, which lifts the dot levels above the Fermi energy of the contact. Since the charged dots are not in an equilibrium state, thermal or tunneling emission will take place and the occupation of the dots decreases. Accordingly, the width of the depletion zone in the Schottky diode will slowly decrease, which is monitored by the time-resolved capacitance measurement. The capacitance transient C(t) recorded after the end of the filling pulse thus reflects the emission of the charge trapped in the dots. While the filling-pulse bias controls the occupation state of the dot at the beginning of the emission process, the reverse-bias value determines the electric field across the dots during the emission process and whether the new equilibrium state corresponds to partially filled or empty dots. The essential feature of the DLTS technique is the ability to set an emission rate window such that the measurement equipment only responds when it detects a transient within this window. The emission rates are thermally activated and can be expressed as[41,42]:
⎛ E ⎞ en (T ) = γT 2σ ∞ exp⎜ − a ⎟ ⎝ kT ⎠
(7)
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53
where Ea is the activation energy, σ∞ the capture cross section for T = ∞, k the Boltzmann constant, and γ a temperature-independent constant. In the simple case of emission by thermal excitation from the QD ground state to the continuum Ea corresponds to the binding energy of the ground state. In the case of more complex scenarios Ea may deviate from this energy. Indirect emission processes have been proposed that involve a thermal excitation to a bound dot level and a subsequent tunnel process.[28,29,43] Neglecting the tunnel time, above analysis can still hold if Ea now is identified with the separation between the ground and excited states. The use of a rate window is illustrated in Fig. 2. The left-hand figure shows the capacitance transient at various temperatures, while the right-hand side shows the corresponding DLTS signal which is the difference between the capacitance at time t1 and the capacitance at t2 as a function of temperature. It is clear from Fig. 2 that C(t1)-C(t2) goes through a maximum when τ, the inverse of the transient constant, is of the order of (t2-t1). It is assumed that, at temperatures close to the DLTS peaks, the capacitance transients are dominated by the emission from one level and thus can be described by a single emission rate en = τ-1:
⎛ t⎞ C (t ) = C ∞ − ΔC 0 exp⎜ − ⎟ . ⎝ τ⎠
(8)
We can define a normalized DLTS signal as
S (T ) =
[C (t1 ) − C (t 2 )] . ΔC 0
Figure 2. Illustration of how to define a rate window.
(9)
54
Shiwei Lin and Aimin Song By differentiating S(T) we can find the maximum value of τ:
τ max =
t2 − t1 . ln(t2 / t1 )
(10)
The emission rate corresponding to the maximum of a peak observed in a DLTS thermal scan is thus a precisely defined quantity and can be used together with the corresponding temperature to give one point on the Arrhenius plot. By changing the window it is possible to obtain a complete semilog activation energy plot. Beginning from Poisson’s equation in a Schottky diode, the capacitance transient from a QD electronic level measured with respect to the steady state C(∞) at t = ∞ can be derived as:
ΔC (t ) = ΔC 0 exp(−en t )
(11)
with the amplitude of the DLTS signal
xQD N QD ΔC 0 =− 2 C xd N d
(12)
where ΔC is the DLTS peak amplitude, C the steady state capacitance at detection bias, NQD the QD state density, Nd doping concentration, xQD the distance between the QD layer and the gate contact, and xd the depletion depth at the applied bias. Note that this equation also describes the relative DLTS-peak amplitude of deep levels located on a plane. In this case, xQD is replaced by xT the distance between the top gate contact and the plane where the deep levels are located while NQD is replace by NT the sheet density of the deep levels.
2.3. Laplace-Transform Deep-Level Transient Spectroscopy The time-constant resolution of standard DLTS is too poor for studying electronic fine structures in the emission process. Among the numerous reasons for this is the fundamental way that the spectrum is obtained, i.e., even a perfect defect, with no complicating factors, produces a broad line on the DLTS spectrum, shown as Fig. 2. Any variation of time constant present in the defect emission results in an additional broadening of the peak, so closely spaced energy states are practically impossible to resolve unless the time constants are well separated. Numerous authors have tried to overcome this limitation, usually by applying sophisticated peak deconvolution methods. However, the problem of extracting multiple closely spaced decaying exponentials is fundamentally ill posed. Recently, the Laplacetransform deep-level transient spectroscopy has been developed and can give us more than one order of magnitude better resolution than the conventional DLTS method.[37,44-46] A brief description of the Laplace DLTS method is given as follows. For a detailed description, please refer to Ref. [37]. The task of separating multiple, closely spaced, decaying exponential components in measured data recurs throughout science. In DLTS there have been two broad categories of
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approach which can be classed (perhaps somewhat simplistically) as analogue and digital signal processing. All analogue signal processing is undertaken in real time as the sample temperature is ramped, picking out only one or two decay components at a time. Fixed filters produce an output proportional to the amount of signal that they see within a particular time constant range. This is done by multiplying the capacitance meter output signal by a timedependent weighting function. Many weighting function waveforms have been investigated, e.g., double boxcar, exponential and multiple boxcar. In summary, it appears that the most elaborate weighting function waveforms are unable to provide a very significant selectivity improvement over Lang's original scheme. Digital schemes digitise the analogue transient output of the capacitance meter, typically with a sample held at a fixed temperature and averaging many digitised transients to reduce the noise level. All of the accessible decay time constants are then picked out of the acquired waveform by software. The problem of what algorithm to use is difficult. A common approach to the quantitative description of non-exponentiality observed in the capacitance transients is to assume that they are characterised by a spectrum of emission rates, ∞
f (t ) = ∫ F ( s )e − st ds
(13)
0
where f(t) is the recorded transient and F(s) is the spectral density function. For simplicity, this spectrum is sometimes represented by a Gaussian distribution overlaying the logarithmic emission-rate scale. In this way it was possible to describe the nonexponentiality of the transient in terms of broadening of the activation energy for emission. The possibility that the spectrum contains fine structure is ignored. A mathematical representation of the capacitance transients given by Equation (13) is the Laplace transform of the true spectral function F(s). Thus, to find a real spectrum of the emission rates present in the transient it is necessary to use a mathematical algorithm that effectively performs an inverse Laplace transform for the function f(t). The result of such a procedure is a spectrum of delta-like peaks for multi-, mono-exponential transients, or a broad spectrum with no fine structure for continuous distribution. In this method it is not necessary to make any a priori assumptions about the functional shape of the spectrum, except that all decays are exponential in the same direction. Despite the fact that the problem is defined in a very general way, one has to remember that Equation (13) has not a general solution for any given function f(t). As a result, an approximate spectral function can be obtained only from complex numerical approximation methods. The Tikhonov regularization method is very effective for the LDLTS case. In general, in this method an oscillatory character of the spectral function, which could be a result of a simple least-square fitting procedure when the number of peaks (mono-exponential components) is not constrained, is suppressed by an additional constraint imposed on the second derivative of the spectral function. In order to determine how much this second derivative has to be suppressed it is necessary to use a numerical method based on a statistical analysis of the magnitude and spectral distribution of the noise within the experimental data. Additionally all numerical methods employed in the LDLTS system attempt to find a spectral function with the least possible number of peaks, which is consistent with the data and experimental noise; a procedure referred to as the principle of parsimony. This approach has
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Shiwei Lin and Aimin Song
the consequence that the computed spectra obtained are “noise free” in a sense that peaks having amplitudes around the noise level are removed from the spectra by the numerical procedures. The actual algorithm employed is more involved than a simple Laplace transformation. However, the end product of a time constant versus spectral density plot justifies describing this as Laplace DLTS (just as a spectral density-frequency plot is described as a Fourier plot). In our experimental LDLTS system three different software procedures can be used for the numerical calculations. All of them are based on the Tikhonov regularization method, however they differ in the way the criteria for finding the regularization parameters are defined. The first one (CONTIN) is in the public domain and has been obtained from a software library and modified in order to integrate it with our system. The outline code of the second one (FTIKREG) is distributed by the same library, but it has been substantially modified by the original authors for operation within the LDLTS system. The last one (FLOG) has been specifically developed for the system. The parallel use of three different software packages substantially increases the level of confidence in the spectra obtained. Additionally, for preliminary data analysis a discrete (multi-exponential) deconvolution method can be used. This method is based on a simple integration procedure. Besides a variety of tests performed on the software used for solving Equation (13), a further test of the method has been undertaken through a long series of measurements on different point and extended defects in different semiconductors. The LDLTS spectra have been investigated starting from a simple point defect in an elemental semiconductor, such as the platinum-related centre in silicon, to very complicated centers, such as the DX defects in AlGaAs. This work has demonstrated that the LDLTS technique can provide at least one order of magnitude better energy resolution than the conventional DLTS technique. The evolution of this technique over the last few years has enabled us, by using an isothermal method, to achieve the theoretical limit of resolution of DLTS. For relatively shallow states that emit at low temperature, the reduction in line-width is remarkable and can give us even more than two orders of magnitude better resolution than Lang's technique. Similar to the conventional DLTS method, LDLTS also allows us to obtain the activation energy for the emission process from the spectra taken at different temperatures, and to determine the corresponding concentration for each component in the spectrum by Equation (12). Note that due to the Laplace method used to calculate the spectrum, it is the area under a peak, rather than the peak value, that determines the magnitude of the charge exchange associated with each peak.
3. Coexistence of Deep Levels with Optically Active Quantum Dots Self-assembled QDs are grown by the Stranski-Krastanow mode (3D clusters on a wetting layer), which occurs in heteroepitaxial systems with significant lattice mismatch, such as InAs on GaAs (7.2%). When InAs coverage increases to about 1.8 ML, InAs strained film will be partially relieved and the growth mode changes from two-dimensional (2D) to threedimensional (3D). The 2D to 3D transition during the initial stage of growth will not introduce any misfit dislocation, and self-assembled dots are considered as coherent (i.e., defect-free) 3D islands. But as the InAs coverage increases to about 3 ML, island coalescence
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57
and defect incorporation will occur.[8-10] Generally, the lattice defects in semiconductors samples are a very serious problem, as they degrade the electronic properties. Due to the small size of the a QD, one expects that even a single lattice defect makes the QD nonluminescent, i.e.—in a more general term—optically inactive, since it may trap charges resulting in distortion of the electrostatic potential. The defect-containing dots are called incoherent opposite to the perfect coherent ones. In recent years, however, an increasing number of experiments have suggested the possibility that electronic deep levels might exist around or in what were regarded as coherent QDs that exhibit strong PL signals. For example, the presence of defects was suggested as a possible reason for the absence of the so-called phonon-bottleneck effect.[47,48] Other optical experiments, such as the quenching of PL signals at high temperatures, also suggested possible existence of deep-level defects around the QDs.[40,49,50] Dai et al. pointed out that the defect-related centers existed at the InAs/GaAs interface and played an important role in the PL quenching process.[38] They also indicated that the energy of the interface defects depended on the size of the quantum dots. By performing time-resolved optical characterizations of InAs QDs in GaAs, Fiore et al. speculated on the existence of nonradiative traps in the (In)GaAs matrix in the close vicinity of the QDs, which would capture the carriers before they relaxed into the QDs.[39] Despite the evidence of the existence of unknown deep levels, the optical experiments could not provide confirmation and direct information, such as energy levels and concentrations, of possible deep levels due to their generally nonradiative nature. The electrical space-charge technique, deep-level transient spectroscopy, has recently been used to characterize QD structures, which can supply complementary information to PL measurements. More importantly, the DLTS technique allows reliable determination of both the spatial and energy positions of the QD intrinsic states as well as the nonradiative deep levels. However, to this day, no study has reported clear detection of the coexistence of the deep level with optically active QD intrinsic energy states. J. S. Wang et al. observed several deep traps in the GaAs cap layer or near the QDs due to relaxation-induced defects, which occurs while the InAs thickness increases to 3.4 ML.[10] Using the capacitive transient spectroscopy and transmission electron microscopy (TEM), C. Walther et al. detected two deep levels which are energetically too deep to be the intrinsic electron levels of the quantum dots, and proposed the levels are due to point defects in or near the QDs.[34] Krispin et al. also reported a series of defect states in their InAs/GaAs structures.[51] Since no intrinsic quantum dot states were observed in these experiments, however, it was not clear whether these deep levels were actually present in optically active QD structures, and if so, whether they were spatially localized and whether they coexisted in the QD layers. Such information is important for physical studies and optimizations of both electronic and optical devices based on self-assembled QDs. It is also useful to determine if the deep levels result from the lattice mismatch induced strain, or the deep levels are actually created only during the Stranski-Krastanow island formation process. The experimental samples in our work have different structures and were grown for different purpose. The samples described in this section and Section 4 were grown to study the electronic structures and to identify the coexistence of deep levels with the quantum-dot intrinsic states using PL, CV, conventional and Laplace DLTS. In Section 5, different samples were grown to investigate the electron emission from QD intrinsic states. These
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Shiwei Lin and Aimin Song
samples have a lower background doping concentration and can increase the resolution of the DLTS measurement and reduce the tunneling rates. Samples with either InAs QDs or only a pseudomorphic InAs wetting layer were studied. The InAs layers (both QDs and pseudomorphic) were sandwiched by two 0.4-μm-thick Sidoped GaAs layers grown by molecular beam epitaxy (MBE) on (100) n+-GaAs substrates as shown in Fig. 3. The growth was performed in an Oxford Instrument VGSemicon V90H system under the conditions that were tailored to yield large InAs QDs at a growth temperature of 480 °C. The GaAs matrix was grown at 580 °C. The formation of QDs with 2.8 ML coverage was monitored in situ by reflection high-energy electron diffraction (RHEED), and the QD nucleation was observed via the change of the RHEED pattern from steady [two dimensional (2D) growth] to spotty (3D growth). The resulting quantum-dot density was approximately 3×109 cm-2, as determined by imaging uncapped samples using an atomic force microscope (AFM), and is of the same order of magnitude as the density estimated from the DLTS signals. AFM analysis also shows that the QDs have a base width of about 20 nm and a height of about 2 nm. The Ohmic contact was fabricated by alloying Au-Ge-Ni/Au on the backside of the structure; the top Schottky contacts, 1 mm in diameter, were defined by evaporating Al on the top surface.
Figure 3. Schematics of the material and device structures: (a) a QD sample with 2.8 ML InAs coverage and (b) two reference samples with 1.2 and 1.5 ML InAs wetting layers, respectively.
PL measurements were performed at room temperature with laser excitation at a wavelength of 532 nm. The signal was dispersed by a monochromator and collected using an InGaAs detector for the wavelength range between 1000 and 1500 nm, while using a charge-
Investigations of Electronic States in Self-assembled InAs/GaAs Quantum-Dot…
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coupled device (CCD) detector for shorter wavelengths. The large size of the dots correlates with a strong PL emission at about 1.26 μm shown in Fig. 4 (according to Gaussian fitting as Ref. [24], E = 0.99 eV, standard deviation ΔE = 0.042 eV), which corresponds to the interband transition from the QD ground electronic state to the ground hole state.[24,27] In addition, the PL spectrum of the QD sample exhibits another peak at about 1.18 μm (E = 1.05eV, ΔE = 0.090 eV) which is due to the recombination of first excited state electrons with holes in the first excited states.[27] The strong PL at room temperature shows that the QDs are optically active.
2.5 Q D sam ple PL intensity [a.u.]
2.0 1.5 1.0 0.5
R ef.A
0.0 1000
1100
1200
1300
1400
W avelength [nm ] Figure 4. PL spectra at room temperature for the QD sample and the Ref. A sample with a 1.2 ML InAs wetting layer.
For comparison, two reference samples with 1.2 and 1.5 ML InAs wetting layers (WL), respectively, were grown with exactly the same layer structures and under the same conditions as the QD sample (Referred to as Ref. A and Ref. B, respectively). Figure 3(b) shows their material and device structures. The PL signals from the GaAs matrix and InAs wetting layer in the reference samples are observed at around 890 and 910 nm, respectively, which are the same as those in the QD sample. The absence of QDs in the reference samples is confirmed by the disappearance of QD PL signals at room temperature as shown in Fig. 4. Since sample Ref. B shows very similar experimental results to sample Ref. A, in the following we concentrate on the results that have been obtained on the QD and Ref. A samples.
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3.1. Characterization of Electronic Structure by CV Spectroscopy The CV characteristics of the devices were measured over a frequency range from 10 KHz to 1 MHz and a temperature range from 30 K to room temperature. The amplitude of the measurement signal (Vosc) was 10 mV. Figure 5 shows the typical CV characteristics of the QD and Ref. A samples measured at a temperature of 120 K. The pronounced plateau-like structure of the QD sample occurring between the reverse biases Vr = –3.0 V and –1.0 V, as compared with the almost smooth curve of the reference sample, suggests that certain electronic states exist in the QD layer that capture and emit electrons.[24,52] The capacitance value at the plateau agrees well with the distance between the QD layer and the Al top Schottky contact. The width of the plateau is determined by the energy spread of the electronic states.[35,53] Such states may be the intrinsic QD energy levels, deep levels due to defects, or both. The DLTS experiments applied later identify that the pronounced plateau in the CV trace is a result of capture and emission of electrons in the QDs in pace with the measurement frequency. The CV result reveals that at Vr = –1.0 V the electronic states are lifted close to the bulk Fermi level, and electrons begin to escape from the states. At Vr = –3.0 V, all the electronic states involved are raised above the Fermi level and are therefore fully discharged.
Figure 5. CV characteristics taken at 120 K and 1 MHz for the QD and Ref. A samples.
3.1.1. Influence of Temperature on the CV Characteristics Figure 6 shows the CV characteristics of the QD sample at different temperatures at a measurement frequency of 1 MHz. Applying the depletion approximation to the CV characteristic, we calculate the apparent concentration profile N(xd), shown in Fig. 6(b) using Equation (6), which can clearly show the electron confinement in the QD layer. When the temperature is lowered (from 296 to 115 K), the peak intensity increases and the peak width narrows, suggesting quantum confinement. The plateau-like structure in the CV characteristic
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in Fig. 6(a) is related to the charging and discharging of QDs.[53] The width of a plateau in the CV characteristic depends on the steady-state occupation of the electron levels in the QDs. This, in turn, is determined at a given temperature by the sheet concentration NQD of QDs and the Fermi-Dirac function determining the relative positions of the electron level in the QDs (EQD) and the bulk Fermi level (EF) in the GaAs matrix.[24]
17
-3
C oncentration [cm ]
1.0x10
16
8.0x10
16
6.0x10
30K 50K 78K 115K 196K 296K
16
4.0x10
16
2.0x10
(b) 0.0 0.2
0.3
0.4
0.5
0.6
D epletion depth [μm ] Figure 6. CV curves (a) and the corresponding apparent carrier concentration (b) for the InAs/GaAs QD sample at a measurement frequency of 1 MHz.
Upon plotting the same capacitance spectroscopy data as a concentration profile in Fig. 6, a strong peak reflecting the carrier accumulation in the QD layer, and depletion regions at both sides of the peak, become apparent. The peak positions appear shifted towards somewhat
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Shiwei Lin and Aimin Song
larger depth instead of 400 nm as shown by the thickness of the capping layer in Fig. 3. This can be expected since the electrons in a QD energy level below the conduction band can only be detected for a somewhat higher reverse bias when the level is lifted close to the bulk Fermi level, corresponding to a larger depth in the concentration profile.[43] The temperature influence on the CV profile can be discussed in two temperature range -from 115 to 296 K and from 30 to 115 K. In the range from 115 to 296 K, Figure 6(b) shows the broad CV peak at high temperature, which is similar to the CV profiles reported on the compositional quantum well (CQW).[54] The full width at half maximum (FWHM) of the apparent carrier distribution peak in CQW is mainly affected by the Debye averaging process at high temperatures and the change in the position expectation value of the 2D electrons at low temperatures. At high temperatures, the FWHM is large, attributed to the Debye averaging process between 2D and 3D electrons. As the temperature is decreased, the 3D electron contribution becomes negligible and the apparent carrier distribution is mainly determined by a change of the position expectation value of 2D electrons under the sweeping CV measurement. The small width of the CV peak is the result of a very small change of the position expectation value of 2D electrons at low temperatures. Similarly, when the capacitance is measured in our QD system by superimposing a small oscillation signal Vosc at a frequency f on the applied DC reverse bias Vr, Vosc modulates the charge both at the edge of the space charge region (ΔQ3D) and at the point where the Fermi level crosses the electron level in the QDs (ΔQQD). As the temperature decreases, the Debye tail becomes short and the electron distribution on the QD levels becomes narrow, both of which result in ΔQQD dominating in the sweeping CV measurement. This leads to the narrower and more apparent CV plateau in Fig. 6(a), and the sharper and stronger peak in Fig. 6(b) with decreasing temperature. On the other hand, as the temperature decreases from 296 to 115 K, the thermal emission rate decreases and a higher reverse bias is required to lower the emission barrier to discharge the electrons from the QDs, which leads to the shift of the concentration peak towards a larger depth, shown in Fig. 6(b). The CV characteristic of the QD structure in Fig. 7 indicates that, in the region of the capacitance plateau from –1 to –3 V, as temperature decreases, the change in the space-charge width Δxd due to the increment of the reverse bias ΔVr becomes so small that ΔQQD is larger than ΔQ3D, i.e. the capacitance part contributed from QDs, CQD, is higher than that from the modulation of the edge of the depletion region, C3D. The small Δxd change corresponds to the sharp and strong peak in the concentration profile, which is associated with the obvious step in CV curve (Fig. 6(a)) and in great agreement with the discussion above in the range from 115 to 296 K. As the temperature is lowered from 115 to 30 K, the quantum part of capacitance CQD decreases (Fig. 6(a)). At T = 30 K, CQD almost disappears (Fig. 6(a)), and the Δxd/ ΔVr increases (Fig. 7). Considering that escape of electrons from the QDs is a slower process than capture, at 30 K the thermionic emission rate of electrons (en) from QDs is much lower than the angular measurement frequency 2πf (f = 1 MHz), i.e. freezing of electrons in the QD levels takes place.[55] To remove electrons from QDs at T = 30 K, a higher electric field is required so that the electrons leave the dots by tunneling through a narrow triangular potential. This process gives rise to a weak plateau in the CV characteristic at high reverse biases between –2 and –3.5 V (Fig. 6(a)), leading to the second peak in the concentration profile at xd ≈ 0.5 µm (Fig. 6(b)). Thus, the freeze-out of electrons in the QD levels is responsible for the decrease of accumulation peaks at around 400 nm and the introduction of the second
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accumulation peaks at around 500 nm. On the other hand, since the electrons are frozen and saturated in the QD electronic levels, the positions of concentration peaks at around 400 nm are fixed in the temperature range from 30 to 115 K, while the second peaks at around 500 nm shift to larger depth due to fewer electrons freezing at relatively high temperature.
296.3K 115.5K 30K
-3
C oncentration [cm ]
16
8.0x10
0 -1
16
6.0x10
-2
16
4.0x10
-3 16
2.0x10
R everse voltage [V]
17
1.0x10
-4 0.0 0.2
0.3
0.4
0.5
0.6
D epletion depth [μm ] Figure 7. Illumination of the change of the space-charge width Δxd with the reverse voltage at different temperatures.
3.1.2. Influence of Frequency on the CV Characteristics Figure 8(a) presents the CV characteristics of the QD sample as a function of measurement frequencies, while Fig. 8(b) the corresponding apparent carrier concentration profile. The influence of frequency on the CV trace is quite similar to the results presented by P. N. Brounkov et al.[55] The thermionic emission rate depends exponentially both on the temperature and the energy of the QD electronic states according to Equation (7). There are two paths for electrons to leave the QDs: by thermionic emission over the barrier (when en >> 2πf) and by tunneling through the triangular barrier (when en << 2πf). The relationship between these two components depends on the relation between the thermionic emission rate en and the angular measurement frequency 2πf. At a given temperature, decreasing the measurement frequency tends to increase the number of QDs from which electrons can thermally escape, which results in the broad plateau (Fig. 8(a)) and the strong concentration peak at about 400 nm (Fig. 8(b)). The electrons which could not thermally emit out from QDs at lower measurement frequency required higher reverse bias to tunnel through the triangular barrier, shown as the second peak at the larger depletion depth in Fig. 8(b).
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Shiwei Lin and Aimin Song
16
-3
C oncentration [cm ]
8.0x10
16
6.0x10
1M H z 100KH z 10KH z
16
4.0x10
16
2.0x10
(b) 0.0 0.2
0.3
0.4
0.5
0.6
D epletion depth [μm ] Figure 8. CV characteristic (a) and apparent concentration profile (b) of the structure with InAs QDs measured at 30 K.
3.1.3. Indication of Deep Levels in the CV Profiles The corresponding apparent carrier distributions of the QD and reference samples in Fig. 9 can be obtained from the local slope of the CV curves in Fig. 5,[41] and indicate electron accumulation in the QD/WL plane at around 400 nm under the Schottky contact, in good agreement with the location of the InAs layer from the growth parameters. There is another accumulation peak at around 530 nm in the Ref. A concentration profile, suggesting the electronic deep levels in the structure, which is confirmed by the DLTS data in the following section. The charge on the deep levels can only be detected by applying sufficiently high
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65
reverse voltage at which the deep levels can be lifted above the bulk Fermi level. No similar peak is found in the electron concentration profile of the QD sample probably because of the strong electrostatic field in the depletion regions on both sides of the pronounced QD accumulation peak and lower densities of the deep levels in the QD structure.
Figure 9. Apparent carrier concentration profiles taken at 120 K and 1 MHz for both the QD and Ref. A samples.
According to the DLTS results later, there are deep levels in the QD sample similar to those in the reference samples. Besides the reasons given above, the absence of the electronaccumulating peak related to deep levels in the QD sample might also be due to the following reasons. Firstly, there exist the unintentional fluctuations in the doping concentrations in the two materials, the different surface Fermi-level pinning caused by slight variations in the sample fabrications, etc. Even on the same sample such fluctuations would appear on different devices. These could cause different electrostatic potentials around the deep levels. Secondly, the spatial resolution of CV profile shows strong temperature dependence.[54] This strong temperature dependence is due to the temperature dependence of the Debye screening length that is given by 1/ 2
⎛ εε kT ⎞ LD = ⎜ 20 ⎟ ⎝ e Nd ⎠
(14)
where ε is the dielectric permittivity, k the Boltzmann constant, and Nd the doping concentration. That is, low temperatures give good resolution, but quite low carrier emission rate from deep levels. Thus it is difficult to detect the accumulation peak due to the deep levels in the QD sample in the CV profile.
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Figure 10. CV traces (a) and apparent carrier concentration profiles (b) of the QD sample at 115 and 247 K with the measurement frequency of 1 MHz.
Only one among twenty devices shows a clear accumulation peak probably due to the deep levels in the QD sample, shown as in Fig. 10. For clarity, only two typical CV curves and the corresponding concentration profiles at 115 and 247 K are presented. The reverse voltage here increases to –6 V, which gives rise to a larger space charge region to study whether or not some deep traps exist in the QD structure. And the oscillating voltage Vosc is fixed at 50 mV to lower the vibration of the concentration profile. Unlike the electron concentration profiles reported on the QD structures in previous works,[10,43] a second accumulation peak appears at above –4 V. The second accumulation peak in Fig. 10(b) is different from the second peak in Fig. 6 due to the freeze-out of electrons in the QD levels because the measurement temperature (115 and 247 K) is much higher than the low freezing-
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temperature range from 30 to 78K, which suggests an electronic deep level in the structure. The electrons accumulated in the deep state are far less than those in the QDs so that it cannot be presented in the CV trace shown in Fig. 10(a). Also, from Fig. 10(b) the deep level can only be detected by applying high reverse bias above –4 V at which the deep level can be lifted above Fermi level of the bulk. Furthermore, a key fact is that the location of such deep levels, either in the QDs layer or in the GaAs matrix material, cannot be directly determined from Fig. 10 because of large peak shift in the concentration profile. Further analysis by conventional DLTS and Laplace DLTS will be made to localize this peak position and identify its origin in detail. Just mention here that the accumulation peak due the deep levels in the QD sample observed in Fig. 10 is associated with the coexisted deep levels in the QD layer, which is identified to be the 115 K peak in the conventional DLTS spectra.
3.2. DLTS Characterization of Electronic Structure in Quantum-Dot Structures An alternative method to probe the electronic level structure in QD structures is deep-level transient spectroscopy, which has been very successfully applied for decades to investigate deep traps in semiconductors and more recently to self-assembled QDs.[30-36,42,53] Here, similar to ionization experiments on atoms, the emission of electrons from the quantum dots is probed. In contrast to the CV measurements, the electron reservoir, from which the dots have been charged, is generally assumed not to play a role in the thermal electron emission from the dots. Changing the bias in this experiment can not only charge or discharge the dots but also control the electric field at the dots. It thus shows that DLTS is a powerful tool in investigating electronic properties and carrier capture and emission dynamics in the QD structures.
3.2.1. Coexistence of Deep Levels with Optically Active QDs In Fig. 10, the second accumulation peak suggests the electronic deep levels in the structure. In this section, we demonstrate that deep levels coexist with the QD intrinsic states by applying the transient capacitance spectroscopy. To identify the nature of the electronic states that result in the plateau in the CV characteristics, DLTS experiments were carried out in the QD and reference samples at different reverse biases. All the DLTS measurements were performed from 30 to 320 K in a wide range of rate windows from 0.8 to 5000 s-1 under dark condition. Before each scan, the sample was cooled down under zero bias and the measurements were made during the warm-up cycle. Figure 11 shows the DLTS spectra of the QD and Ref. A samples with different reverse voltages, which induce different electric fields in the structures. The rate window is set as 200 s-1 and the filling pulse bias is –0.5 V. The reverse bias is increased in a step of 0.5 V from –2.5 to –5.0 V and from –3.0 to –5.5 V for the QD sample and the Ref. A sample, respectively.
68
Shiwei Lin and Aimin Song 1.8 D LTS SignalΔC [pF]
(a)Q D sam ple -2.5 V -3.0 V -3.5 V -4.0 V -4.5 V -5.0 V
Q D intrinsic states
1.6 1.4
P 1.2 1.0 0.8
250K peak
0.6 115K peak
0.4 0.2
Trap 1
0.0 0
D LTS SignalΔC [pF]
1.4
50
100 150 200 250 Tem perature [K]
-3.0 V -3.5 V 115K peak -4.0 V -4.5 V -5.0 V -5.5 V
1.2 1.0 0.8
Trap 2 300
350
(b)R ef.A
250K peak
0.6 0.4 0.2 0.0 0
50
100
150
200
250
300
350
Tem perature [K] Figure 11. DLTS spectra for the QD sample (a) and the Ref. A sample (b). Referring to the DLTS peak positions, the two pronounced peaks observed in both samples are named as 115 K peak and 250 K peak, respectively. Trap 1 and Trap 2 denote bulk defects in the QD sample. Arrow P in (a) shows the point at T = 35 K and Vr = –3.0 V where the tunneling rate directly from the QD ground state to the GaAs conduction band is approximately equal to the thermal excitation rate from the ground state to the excited state. (Reprinted figure with permission from Ref. [53]. Copyright 2005 by the American Physical Society.)
Five peaks are observed in the DLTS spectra of the QD sample as shown in Fig. 11(a), where the reverse bias Vr varies between –2.5 and –5.0 V. The two peaks, marked as trap 1 and trap 2, can still be detected at much lower reverse biases Vr > –1.5 V in Fig. 12, when the intrinsic electron states in the QDs as well as any possible deep levels in the QD layer are well below the bulk Fermi energy. We therefore conclude that they are due to bulk electron
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traps. This is confirmed in other GaAs samples grown under the same conditions but without an InAs layer. In the reference samples, these two traps are also observed despite their weaker densities in Fig. 11(b). By comparison with the findings in the literature, we conclude that trap 1 and trap 2 are the M3 and E4 levels respectively, which are commonly detected in GaAs materials grown by MBE.[56] We are not interested in the bulk electron traps (trap 1 and trap 2). In the following, we discuss in detail the data of the other three peaks, which are related to the QD intrinsic energy states and the deep levels.
D LTS signalΔC [pF]
0.020 Trap 1
0.015
Trap 2
0.010 0.005 0.000 V f= -0.5 V V r = -1.5 V -1 200 s
-0.005 -0.010 0
50
100
150
200
250
300
350
Tem perature [K] Figure 12. DLTS spectrum at quite a low reverse bias of –1.5 V.
The pronounced peak at about 50 K at Vr = –3.0 V in Fig. 11(a) is attributed to the electron emission from the intrinsic states in the quantum dots since there is no such DLTS signature at a similar temperature in the DLTS spectra of the reference samples, shown in Fig. 11(b). The post-growth rapid thermal annealing (RTA) of the QD sample also confirms such an identification (detailed RTA results to be in Section 4). We hence conclude that the pronounced plateau in the CV trace in Fig. 5 is a result of capture and emission of electrons in the QDs in pace with the measurement frequency. The activation energy of 76 meV (σ∞ = 1.27×10-14 cm2), obtained from the Arrhenius plot in Fig. 13, is similar to the values reported by Kapteyn et al.[43,57] Using the same theory, we interpret the emission process as electrons in QDs being firstly thermally activated from the ground state into the excited states and then fast tunneling out from the excited states. The calculated thermal activation energy of 76 meV therefore corresponds to the energy difference between the QD ground and excited states. As shown in Fig. 11(a) at different reverse biases, such QD thermal excited DLTS peaks are well resolved only at certain bias voltages. This is because when Vr < –3.0 V the emission is dominated by tunneling, which is rather temperature independent and therefore less visible in the DLTS spectra. The flat region in the DLTS spectrum at Vr = –3.0 V below 35 K is related to electron tunneling directly from
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Shiwei Lin and Aimin Song
the QD ground state into the GaAs conduction band. Assuming a triangular barrier the tunneling rate can be written as
et =
⎛ −4 2m* ΔE 3/ 2 ⎞ t exp ⎜ ⎟, * ⎜ ⎟ 3 e = F 4 2m ΔEt ⎝ ⎠ eF
(15)
where F is the electric field, e the electron charge, ћ the reduced Planck constant, m* the GaAs effective electron mass, and ΔEt the tunneling barrier height.[58,59] In order to estimate ΔEt, we assume that the tunneling rate is approximately equal to the thermal excitation rate from the QD ground state to the excited state at T = 35 K, marked by the arrow P in Fig. 11(a). The obtained barrier height for the QD ground state is 171 meV, similar to the values reported by others.[32,33,36,43,57,60]
76m eV
171m eV
2
-1
ln(en/T )[ln((K s) )]
-2
-4
2
160m eV
Q D intrinsic states 115K peak in Q D sam ple 250K peak in Q D sam ple 115K peak in R ef.A 250K peak in R ef.A
484m eV
-6 493 m eV
-8 5
10
15
20
-1
1000/T [K ] Figure 13. Arrhenius plots of the emission rates determined from the maximum positions of the DLTS spectra at different rate windows. The activation energies are determined from the slopes of a linear fit to the data (straight lines). The apparent electron capture cross sections are determined from the intercepts of the linear fit on the vertical axis.
Apart from the QD signal in the DLTS spectra in Fig. 11(a), two pronounced peaks are located at about 115 K and 250 K. In the DLTS spectra of the Ref. A sample in Fig. 11(b), there are two DLTS peaks at similar temperature positions, suggesting that these 115 and 250 K peaks in the QD sample are not related to the QD intrinsic states but some type of deep levels in the structure. We can also conclude that these deep levels could be created by the strain field induced by the lattice mismatch alone, not necessarily due to the three-dimension QD formation process.
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1.0
ΔC /ΔC m ax
0.8 0.6 0.4 0.2 0.0
115K peak 250K peak (a)Q D sam ple -5.5 -5.0 -4.5 -4.0 -3.5 -3.0 -2.5 -2.0 -1.5 R everse Voltage [V]
1.0
ΔC /ΔC m ax
0.8 0.6 0.4 0.2 0.0
115K peak 250K peak (b)R ef.A -5.5
-5.0
-4.5
-4.0
-3.5
-3.0
-2.5
R everse Voltage [V] Figure 14. Bias dependence of normalized DLTS peak height, ΔC/ΔCmax, at 115 K (squares) and 250 K (dots) in the DLTS spectra of the QD sample (a) and Ref. A (b). The filling pulse bias is fixed at –0.5 V. (Reprinted figure with permission from Ref. [53]. Copyright 2005 by the American Physical Society.)
From the Arrhenius plots in Fig. 13, the excitation energies of these two deep levels in the QD sample are determined to be about 160 and 484 meV, respectively. The bias dependence of the relative capacitance change, ΔC/ΔCmax, shown in Fig. 14(a), indicates that the deep levels are not bulk defects, but have a rather narrow spatial distribution. Both peaks do not appear until the reverse biases reach their thresholds, at which their electronic energy states are raised up to the bulk Fermi energy. The 115 K peak starts to appear at Vr = –2.0 V and reaches its maximum at Vr = –3.1 V, whereas the 250 K peak starts at Vr = –3.5 V and becomes maximal at Vr = –4.5 V. As shown in Fig. 14(a), the relative height of the 250 K peak can be determined reliably only when the bias is beyond –3.0 V because of the presence of traps 1 and 2. Beyond the maxima, the DLTS signals gradually decrease because the
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Shiwei Lin and Aimin Song
relative change in the capacitance becomes smaller due to the increased depletion depth. The behaviour follows evidently from Equation (16), which is the same as Equation (12) in Section 2 and describes the relative DLTS peak amplitude for deep levels located on a plane.[43,61]
ΔC nT xT = 2 . C w Nd
(16)
Here, ΔC is the DLTS peak amplitude, C the steady state capacitance at the reverse bias, nT the sheet density of the deep levels, xT the distance between the top Schottky contact and the plane where the deep levels are located, w the depletion depth at the reverse bias, and Nd the doping concentration. Note that since the deep levels usually have a finite distribution of energy, nT in Equation (16) is a function of reverse bias. Below the threshold bias of the deep levels, nT is equal to zero because all the electronic states are below the bulk Fermi level; when the reverse bias is larger than the threshold value, nT increases with increasing reverse bias, and then fixes at the maximal value when all the electronic states are raised above the Fermi level. To determine the locations of the deep levels in the QD sample, we model the QD structure as an ideal Schottky diode and neglect the effect of charging and discharging of QDs on the electric field. By solving Poisson’s equation, the potential profile of the QD structure is given by
V ( x) = −
eN d ( w − x)2 , 2ε 0ε r
(17)
where ε0 is the permittivity of free space, εr the GaAs dielectric constant, and x the distance from the top Schottky contact in the depletion region. Equation (16) shows that the reverse bias to reach the maximal relative capacitance change, ΔC/ΔCmax in Fig. 14(a), neither exactly corresponds to the moment when the bulk Fermi level passes the full energy distribution of the deep levels, nor when the Fermi level aligns with the energy corresponding to the distribution peak. We therefore use the threshold biases, at which the deep levels start to be detected, to estimate their locations by applying Equation (17). From the reverse-bias threshold of the 115 K peak, Vr = –2.0 V, shown in Fig. 14(a), we obtain a location of 395 nm below the surface, in very good agreement with the designed QD location in the material (400 nm). From the threshold reverse bias of –3.5 V of the 250 K peak, the location is calculated to be about 391 nm below the surface, also corresponding well to the depth of the QD layer. Furthermore, when we etched off the capping layer and the QD layer from the QD sample, we found no DLTS signals at around 50 K, 115 K and 250 K in Fig. 15, further supporting that the deep levels coexist in the dot layer. The drastic increase of the DLTS signal after 250 K is due to plenty of deep traps located close to the highly doped GaAs layer, which is much stronger than the signal from trap 2 at around 300 K. Thus, to the best of our knowledge, this is the first direct observation and strong confirmation of the coexistence of deep levels with the intrinsic quantum-dot states.
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0.8
D LTS SignalΔC [pF]
0.6 0.5 0.4 0.3
D LTS signalΔC [pF]
0.04
0.7
0.03 Trap 1 0.02 0.01 0.00 -0.01 100 120 140 160 180 200 220 240 Tem perature [K]
0.2 0.1
V f= -0.5 V V r = -3.0 V 200/s
0.0 -0.1 0
50
100
150
200
250
300
350
Tem perature [K] Figure 15. DLTS spectrum of the sample in which the QD layer and the GaAs capping layer were etched away. The inset zooms in the DLTS spectrum between 100 and 250 K.
Table 1. Data about the deep levels in the QD and reference samples, calculated from the DLTS results
115K peak Ea a (meV) QD sample Ref. A Ref. B
160 171 166
σb (cm2) 9.5×10-16 1.0×10-15 9.3×10-16
250K peak Sheet densityc (cm-2) 2.5×109 1.1×1010 1.4×1010
Ea (meV) 484 493 495
σ (cm2) 1.9×10-13 2.1×10-13 2.4×10-13
Sheet density (cm-2) 5.0×109 1.1×1010 1.4×1010
a
Thermal excitation energy obtained from the slope of the Arrhenius plot. The apparent electron capture cross section calculated from the intercept of the Arrhenius plot on the vertical axis. c Sheet density calculated from the maximal amplitude of the DLTS spectra by Equation (16). b
Experiments were also performed on the Ref. A and B samples. Both reference samples show very similar PL, DLTS spectra and bias dependence of the relative capacitance change. Here, we only show the results of Ref. A to compare with those of the QD sample. Figure 11(b) shows no QD DLTS peak at around 50K in Ref. A, in agreement with the roomtemperature PL in Fig. 4. This is as expected since the InAs layer is well below the critical thickness of 1.8 ML for QD formation. However, two strong DLTS peaks appear at about 115 and 250 K, the same positions as those of the deep levels in the QD sample, shown in Fig. 11(a). The Arrhenius plots shown in Fig. 13 allow us to calculate their thermal activation energies and the corresponding apparent electron capture cross sections, which are listed in
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Shiwei Lin and Aimin Song
Table 1 for comparison with the results in the QD sample. For both the 115 and 250 K peaks, the activation energies and capture cross sections in the reference samples are quite close to those in the QD sample. In addition, as shown in Figs. 14(a) and (b), the bias dependences of the DLTS spectra in the QD and Ref. A samples are very similar, with virtually the same threshold biases. This also confirms that they are spatially localized in the growth direction. A close look at Fig. 14 reveals a difference in the threshold biases of the peaks between the QD and reference samples. This is expected because of the unintentional fluctuations in the doping concentrations in the two materials, the different surface Fermi-level pinning caused by slight variations in the sample fabrications, etc. By determining the threshold biases and applying the same calculation model as above, the 115 and 250 K peaks in the reference samples have almost identical locations, also confirming that these two types of deep levels coexist in the same layer. Furthermore, the calculated depth is very close to the growth parameter. All the evidence above from the QD sample and the two reference samples thus suggests that the deep levels related to the 115 K peak among the three samples are the same type of defects, and so are the deep levels related to the 250 K peak in the three samples.
3.2.2. Origins of the Coexisting Deep Levels in the QD Structures We now discuss the origins of such coexisting deep levels in the InAs layer. Two categories of defects are usually observed in semiconductors: one is extended defects such as dislocations, and the other is point defects, such as vacancies, interstitials, or chemical impurities in the lattice. The analysis of our experimental results below rules out the possibility of being dislocation-related extended defects. Instead, the observed deep levels might be some type of point defects introduced by the lattice-mismatch strain during the growth process. The previous work by Wang et al. on the relaxation-induced defects in QD structures indicated that beyond the critical coverage of about three monolayers of InAs, threading dislocations and stacking faults could appear in the GaAs cap layer, whereas misfit dislocation and cross-hatched stacking faults might appear near the QDs boundary.[10] In comparison with the experimental results by Wang et al., the coexisting deep levels in our samples have different activation energies and apparent electron capture cross sections. Furthermore, the thickness of our InAs QD layer is 2.8 ML, less than the reported critical thickness of three monolayers. More importantly, we observe strong photoluminescence signals even at room temperature and apparent quantum confinement in the CV and DLTS measurements. We therefore conclude that the deep levels observed in our experiments are not dislocation-related traps. M-series defects, reported by Lang,[62] are commonly-observed bulk defects in MBEgrown GaAs, which are attributed to defect-impurity complexes with growth temperaturedependent concentrations. M1 and M4 have similar DLTS signatures and thermal activation energies to the deep levels observed in our QD and reference samples. However, the concentrations of M1 and M4 are typically very low, around 1012 cm-3 at a growth temperature of 580 ºC.[56] In order to compare the densities of the deep levels with those bulk defects commonly presented in MBE-grown GaAs, the volume densities of deep levels in our samples can be simply calculated from
Investigations of Electronic States in Self-assembled InAs/GaAs Quantum-Dot…
N v = 2(ΔC / C ) N d ,
75 (18)
a formula used to evaluate the densities of bulk defects in GaAs.[42] The densities of the deep levels related to the 115 and 250 K peaks are thus calculated to be 5.7×1013 and 1.3×1014 cm-3, respectively, in the QD sample, and 2.8×1014 and 3.2×1014 cm-3, respectively, in the Ref. A sample. The densities are therefore more than one order of magnitude higher than those of M1 and M4 in MBE-grown GaAs under the similar growth conditions. Furthermore, the observed deep levels in our samples are localized in space and coexist in the InAs layer. There hence is a possibility that the strain field, induced by the lattice mismatch between the InAs and GaAs, could enhance the creation of M1 and M4, and localize them by inducing migration of M1 and M4 from neighbouring layers. Since the strain around the dots is quite different from that between the dots, the DLTS peak shapes of the deep levels in the QD sample should be different from the symmetric DLTS signature of the ideal bulk defects. This might be reflected by the observation that the 115 and 250 K peaks in our QD sample are broadened and extended on the low-temperature sides. Another possibility is that the coexisting deep levels are native point defects caused by the strain during the InAs/GaAs growth process. From the DLTS amplitudes, the sheet densities of these deep levels can be calculated (see Table 1), and are found to be comparable with the QD density determined by AFM scans. Note that in both reference samples the sheet density ratio between the 115 and 250 K deep levels is 1:1, strongly suggesting that these two deep levels have the same origin. The lattice mismatch between InAs and GaAs is 7.2%. The strain is built up by growing InAs on GaAs, until the growth mode changes from 2D to 3D and QDs form to relax the strain. The point defects related to the 115 and 250 K DLTS peaks, such as vacancies and interstitials, might be introduced by the strain during the growth process and accumulated in the pseudomorphic wetting layer in the reference samples. As shown in Table 1, the sheet densities of the deep levels in Ref. B are larger than those in Ref. A, which could be because the strain field in Ref. B with a 1.5 ML InAs layer is much stronger than in Ref. A with a 1.2 ML InAs layer. In the QD sample, much of the strain is relaxed by the formation of QDs, which can explain the lower sheet densities of the deep levels. The partial strain relaxation in the QD sample may also cause a broadening in the energy distribution of the deep levels, which may be the reason for the wider DLTS peaks shown in Fig. 11(a). Interestingly, the sheet density ratio between the 115 and 250 K deep levels is 1:2, rather than 1:1 in the reference samples. The reason is unknown and is subject to future studies. Moreover, although the formation of QDs relaxes some of the strain, the remaining strain still exists in and around QDs. It has also been shown that strain field can extend from the QDs into the surrounding GaAs matrix over a typical scale of ≥10 nm.[2,63,64] The extended strain field might cause migrations of native defects, such as vacancies and interstitials, to the vicinity of the quantum dots. During our sample growth, because of the relatively thick upper GaAs confining layers, the dots were subjected to an anneal of about 20 minutes at 580 ºC, which could enhance the migration of the defects. Such accumulation of defects is in agreement with the previous observation by Walther et al., where the clustering of the trap states occurred in regions of randomly higher quantum-dot concentration.[34]
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3.2.3. Effects of the Coexisting Deep Levels on the Properties of QD Structures Having identified the deep levels coexisting in the same layer of QDs and their sheet densities, it is important to gain further insight into the effects of the deep levels on the properties of the QD structure, which may allow improvements of the optical and electrical properties of the QD materials. Sercel proposed that the presence of interface states or point defects formed during the growth process and correlated with the QDs could provide an efficient relaxation path for electrons through multiphonon-assisted tunneling.[48] Such a lack of the phonon bottleneck effect was found in recent time-resolved PL measurements, which was in disagreement with the theoretical prediction for an ideal quantum dot.[65,66] The binding energy of the QD ground state in our QD sample has been estimated to be 171 meV, close to the 160 meV of the coexisting deep level related to the 115 K DLTS peak. If the deep states lie close to the QDs, they may strongly couple with the dots and provide a rapid energyrelaxation channel through which electrons thermalize to enhance the luminescence efficiency.[48] For electronic device applications, the existence of states that are much deeper than the intrinsic QD states could be utilized to design memory devices. We recently demonstrated a fully electrically controlled, room-temperature memory device where InAs QDs were embedded in a modulation-doped high electron-mobility transistor structure. The coexistence of similar deep levels in the QD layer was confirmed by the experiments using different gate biases to control the transfer of electrons between the deep levels and a twodimensional electron gas.[67] In such devices, the slow charging of the deep levels, rather than the emission or discharging process, provided a long memory time even at room temperature.
3.3. Fine Structures of the Deep Levels Probed with LDLTS The LDLTS technique provides about an order of magnitude better energy resolution than the conventional DLTS method. With a real spectrum of the emission rate, the LDLTS technique can reveal the fine structure in the thermal emission process. In this section, we employ the LDLTS technique to investigate the fine electronic structure of the deep levels, related to the 250 K peak in the conventional DLTS spectra in Fig. 11. Such study could provide insight into the nature of the deep levels and guide the improvements of the optical and electrical properties of the QD materials. The Laplace DLTS system records the carrier thermal emission as a function of time at a fixed temperature, after excitation by a filling pulse. A large number (typically over 1000) of such transients are averaged. The data are processed by implementing the inverse Laplace transform, which provides an emission-rate spectrum of the deep levels.[37] Figure 16(a) shows the LDLTS spectrum in the QD sample at a fixed temperature of 250 K. Three well-resolved peaks were observed in contrast to the broad peak observed at around 250 K in the conventional DLTS experiments in Fig. 11(a). The corresponding Arrhenius plots are shown in Fig. 16(b), which are produced by analysing the results at different temperatures. Due to the method used to calculate the spectrum, it is the area under a peak, rather than the peak value, that determines the magnitude of the charge exchange associated with each peak. The magnitude of peak A is calculated to be 0.20 pF, which corresponds well to the peak height of trap 2 at Vr = –5.0 V in Fig. 11(a). This peak is due to one of the
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commonly-observed bulk traps in MBE-grown GaAs materials,[53] and will be ignored in the following discussions. The activation energies of 485 and 468 meV for peaks B and C, respectively, are in good agreement with that of the 250 K peak (484 meV) in Fig. 11(a), which has been identified by the conventional DLTS analysis. The energy difference between these two peaks is only 17 meV, which is smaller than the thermal energy of about 22 meV at 250 K, demonstrating the exceptional capability of the LDLTS technique in resolving the fine structures of the deep-level states. 0.025 (a)
B
T= 250 K
Am plitude [a.u.]
0.020 0.015 A 0.010
C
0.005 0.000 100 1000 -1 Em ission rate [s ]
(b)
-5
2
2
-1
ln(e/T )[ln((K s) )]
-4
C :468 m eV 2 2.11E-13 cm B:485 m eV 2 2.03E-13 cm
-6 -7 -8
A:491 m eV 2 3.96E-14 cm
-9 4.0
4.1
4.2
4.3
4.4
4.5
-1
1000/T [(m K) ] Figure 16. (a) LDLTS spectrum at 250 K and at a reverse bias of –5.0 V, and (b) Arrhenius plots of the emission rates for the three peaks in (a).
The Arrhenius plots in Fig. 16(b) were also used to calculate the apparent electron capture cross sections, which are summarized in Table 2 together with the activation energies and the sheet densities. Peaks B and C have virtually identical capture cross sections.
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Furthermore, the sum of the magnitudes of peak B (~0.32 pF) and peak C (~0.11 pF) agrees very well with that of the deep levels (~0.46 pF) in the conventional DLTS spectra in Fig. 11(a). Therefore we conclude that peaks B and C represent the fine structure of the deep levels observed as the 250 K peak in the conventional DLTS spectra. The peak broadening in the LDLTS spectra is attributed to the corresponding energy-state distribution, so we determine that the energy distributions of peaks A, B and C are 29.1 meV, 22.9 meV, and 13.7 meV, respectively. From the LDLTS spectrum, the sheet densities of peaks B and C are found to be 3.6×109 and 1.2×109 cm-2, respectively, the sum of which is in good agreement with the value of 5.0×109 cm-2 for the 250 K peak in the conventional DLTS spectra in the QD sample. Table 2. Data related to the fine structure of the deep levels in the QD and reference samples, calculated from the LDLTS results. For the purpose of comparison, the data of the M4 defect in the GaAs control sample are also shown.
QD sample Ref. A Control sample (M4)
Peak B (strong peak) Sheet Ea σ density 2 (meV) (cm ) (cm-2) -13 485 2.03×10 3.6×109 -13 499 3.46×10 1.3×1010 479 1.12×10-13 4.25×1012 cm-3
Peak C (weak peak) Sheet σ Ea density 2 (meV) (cm ) (cm-2) -13 468 2.11×10 1.2×109 -13 478 2.88×10 5.2×108 Note: M4 is the bulk defect in the MBE-grown GaAs.
The fact that the ratio between the densities of deep levels B and C is an integer of three may be significant. This not only seemingly further supports that deep levels B and C are of the same origin and represent the fine structure of the deep levels, but also provides an important clue to identify the nature of the deep levels. Recently, under the application of uniaxial <111> stress, point defects with a trigonal symmetry, such as bond-center hydrogen in Silicon or Germanium, were studied in LDLTS experiments. A single emission peak was observed to split into two components with the amplitude ratio 3:1 under the stress.[37,68] Since strain also exists in our QD structures due to the lattice mismatch, our experimental results may suggest that peaks B and C in Fig. 16(a) could similarly be due to the lifting of the degeneracy of the coexisting deep levels by the lattice-mismatched strain. To prove or disprove the above picture, experiments were also performed on the Ref. A sample and on a control sample that was used to investigate the M4 defect. The control sample is an MBE-grown GaAs sample with the uniform Si-doping concentration of 1.25×1016 cm-3. The LDLTS spectra of the QD, Ref. A and control samples are shown in Fig. 17, while the calculated data are listed in Table 2. In the Ref. A sample, the Laplace DLTS spectrum shows much sharper peaks than those in the QD sample. The ratio between the magnitudes of peaks B and C becomes much larger than three, and the peak at the higher emission rate nearly vanishes. The apparent electron capture cross sections slightly increase but remain in the same order of magnitude. The slightly different thermal excitation energies and capture cross sections may be due to the influence of different strains on the deep levels in the QD and reference samples. However, in the control sample, only a single peak appears at the similar position of Peak B in the QD sample.
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B
242K
V r=-5.0V,V f=-0.5V,L=1m s
A
Q D sam ple
C
D LTS signal[a.u.]
79
B V r=-5.0V,V f=-0.5V,L=1m s R ef.A sam ple C
V r=-3.0V,V f=-0.5V,L=1m s C ontrolsam ple 10
100
1000
10000
-1
Em ission rate [s ] Figure 17. LDLTS spectra of the QD sample, the Ref. A sample and the control sample at 242 K.
As mentioned by the conventional DLTS analysis, the deep levels related to the 250 K peaks in the QD and Ref. A samples exist in the InAs layer. They have the same origin and are most likely caused by the strain induced by the lattice mismatch during the growth process. These deep levels are either point defects due to the creation and accumulation of the M4 levels, which are commonly observed bulk defects in MBE-grown GaAs, or native defects induced by the strain during the InAs/GaAs growth process. Both peaks B and C in Fig. 17 show the fine structure of the deep levels, related to the 250 K peaks in the conventional DLTS spectra in the QD and Ref. A samples as shown in Figs. 11(a) and (b). However, the dramatic change of the magnitude ratio from peak B to peak C in the QD and Ref. A samples seems to rule out the possibility of the split of point-defect levels with a trigonal symmetry. To look for other possible pictures, it is noted that the strain around the dots is quite different from that between the dots. Peak B and peak C might hence correspond to the same type of deep states but are energy-shifted by the different strain strengths depending on their spatial locations in the wetting layer.[69] Such a picture seems to explain the fact that the deep levels of the 250 K DLTS peak in the reference sample has only one dominant peak in the LDLTS spectrum because of the relatively homogeneous strain field at the InAs/GaAs interface. The difference in the peak broadening can also be explained, since the large variation of the strain strength in the QD sample should result in more variations in the defect energy. Also, the single Laplace DLTS peak in the control sample supports the explanation above. We expect that further experimental and theoretical investigations will provide important implications to the physical nature of the deep levels, and optimization of the QD growth for both electrical and optical applications.
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4. Effects of Rapid Thermal Annealing on QD Structures 4.1. Postgrowth Rapid Thermal Annealing Self-assembled quantum dots are expected to enable great improvements in device performance because of the delta-function-like QD density of states,[70] such as larger differential gains and lower threshold current densities of QD lasers than those of QW lasers.[5,6] However, the nonuniformity of QD size distribution as well as the deep levels in the QD structures, such as the coexisting deep levels mentioned in Section 3, have hampered these advantages and constituted a technical barrier for the development of QD-based devices. As a way of tuning the device properties, postgrowth rapid thermal annealing has been applied to the semiconductor quantum dots, and the induced structural and optical changes were studied by using transmission electron microscopy (TEM) and photoluminescence.[7177] Commonly observed effects of the annealing on self-assembled QD structures are the blueshift and enhancement of the PL emissions in the temperature range between 600 and 800 ºC, and the decrease of PL intensity at higher annealing temperatures due to the degradation in the material quality.[73-75] However, recent experiments also showed a monotonic reduction of PL intensity by postgrowth annealing beginning at a temperature of about 600 ºC.[76,77] There has not been a broad consensus on the differences largely because of the lack of quantitative information about the energies and densities of intrinsic QD states and defect states in the studied structures. Such information is difficult to obtain in optical experiments alone. The mechanisms underlying the variation of the PL spectra are therefore yet to be clarified. The deep levels observed in our QD structures most likely originate from point defects, which are difficult to observe directly by microscopies such as TEM. Due to their generally nonradiative nature, static PL experiments could provide neither confirmation nor direct information of possible deep levels, such as energy levels and concentrations. On the other hand, the change of such coexisting deep levels at different annealing temperatures is likely to strongly influence the PL spectra.[78,79] Despite much effort on RTA, the effects of RTA on deep levels have not been studied systematically and quantitatively to provide information on the changes in the defect concentrations, any introduction of nonradiative defects, and the variations of the intrinsic QD states during RTA processes, etc. Such information is important for physical studies and optimizations of electronic and optical devices based on selfassembled QDs. Therefore, in this section, we combine optical and electrical experiments to study the influence of RTA on a self-assembled InAs/GaAs QD structure using the PL and DLTS techniques. At a series of annealing temperatures, the effects of RTA on the optical spectra are found to be closely linked with the variations of both the intrinsic QD and deep-level states. Compared with the PL spectra, the DLTS data are more complex, revealing a number of electronic states varying with the annealing temperature and particularly the formation of a new deep level (0.62 eV). The new deep level becomes dominant in the whole spectra at certain annealing temperatures.
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4.2. Effects of Annealing on PL Spectra The annealing experiments were performed on the QD and Ref. A samples which have been studied in Section 3. The samples were proximity capped and RTA treated in nitrogen ambient for 60 seconds at temperatures ranging from 500 to 800 ºC in a step of 50 ºC. Although there was most likely some As loss from the surface after the proximity annealing, no obvious change in the sample surface was observed. Furthermore, the CV measurements of the annealed samples showed similar doping profiles to that of the as-grown sample, indicating that the effect of As loss due to annealing on the Schottky contact and electrical properties may not play an important role here. The PL measurements were performed at room temperature (RT) with laser excitation at a wavelength of 532 nm as mentioned in Section 3. For the purpose of comparison, low-temperature PL measurements were also performed in a helium cryostat with laser excitation at 633 nm. The signal was detected by a N2-cooled Ge detector and amplified using the conventional lock-in techniques. The QD PL signal at a low temperature of 12 K in Fig. 18(b) is almost the same as that at room temperature in Fig. 18(a) except a small blueshift of peak positions due to the increase of the band gap and a less pronounced second peak. Actually, we observed only about 30% of reduction in PL peak intensity as the device was cooled to 12 K, which suggests that the quantum dots are indeed very optically active. The low-temperature PL spectra of the annealed QD samples are not included in the following discussion because they similarly show little difference from the room-temperature results.
(a)R T
PL intensity [a.u.]
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1300
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Figure 18. Room-temperature (a) and low-temperature (b) photoluminescence spectra of the as-grown QD sample. The dash lines in (a) show the deconvolution of the QD PL peak by fitting with two Gaussians.
Figure 19(a) shows the room-temperature QD PL spectra from the samples annealed at different temperatures. In order to quantitatively study the influence of RTA, the PL spectra have been fitted with Gaussian curves as sketched in Fig. 18(a). The PL peak position and intensity are plotted as a function of annealing temperature in Fig. 20(a). The QD PL intensity
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decreases with increasing annealing temperature up to 600 ºC and then stabilizes up to 700 ºC, suggesting an introduction of nonradiative paths in the structure which will be confirmed by the DLTS measurements later. The interdiffusion of In and Ga atoms at the interface between the QDs and GaAs barrier occurs only when the annealing temperature rises beyond 700 ºC, as revealed by the onset of blueshift of the QD and wetting-layer PL peaks in Fig. 20.[72,,74,75]
as-grow n
(a)Q D s
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o
500 C o 550 C o
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550 C o
650 C
800
850
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950
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W avelength [nm ] Figure 19. Room-temperature PL spectra of the as-grown and annealed samples from (a) QDs, and (b) InAs wetting layer and GaAs matrix. (Reprinted figure with permission from Ref. [79]. Copyright 2006 by the American Institute of Physics.)
The interdiffusion also causes a reduction of the confinement potential in the QDs, and therefore more photo-generated carriers in QDs to be thermally emitted into the wetting layer
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and/or GaAs continuum.[21,80] This results in the decrease of the QD PL intensity and the increase of the PL signals from the wetting layer and GaAs matrix at temperatures higher than 700 ºC, as clearly shown in Figs. 20(a) and (b). Because two different detectors were used, an InGaAs detector for the QD PL signal in the wavelength range between 1000 and 1500 nm and a CCD detector for the PL signal from the wetting layer and GaAs matrix in the shorter wavelength range, it is however difficult to determine whether the integrated PL intensity increased or decreased.
Peak position [eV]
1.20
Peak 1 position Peak 2 position Peak 1 intensity Peak 2 intensity
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o
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1.344 1.342 1.340 1.338 500
550
600
650
700
750
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o
Annealing tem perature [C ] Figure 20. The position and intensity of the (a) QD and (b) InAs wetting-layer PL peaks as a function of the annealing temperature. (Reprinted figure with permission from Ref. [79]. Copyright 2006 by the American Institute of Physics.)
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Typically, the PL peak due to the InAs wetting layer is stronger than that due to the GaAs matrix because of the better carrier confinement in the wetting layer.[72] In Fig. 19(b), however, the wetting-layer PL intensity is much weaker than that of the GaAs matrix. Such apparent contradiction can be accounted for by the coexistence of deep levels with QDs in the InAs wetting layer, which has been demonstrated in Section 3 and is revealed by the DLTS spectra in Fig. 21(a) for the purpose of comparison. The nonradiative nature of the deep levels may play an important role in reducing the PL efficiency in the wetting layer.
4.3. Effects of Annealing on DLTS Spectra When the applied reverse bias Vr is much beyond –3.0 V, the electron emission from the QDs is more dominated by tunneling, which is rather temperature independent and hardly visible in the DLTS spectra. The dependence of the DLTS peak of the QD intrinsic states on the annealing temperature is therefore determined at a reverse bias of Vr = –3.0 V as shown in Fig. 21(b).
D LTS SignalΔC [pF]
1.4 1.2
(a)
Q D intrinsic states
Vr= -3.0 V Vr= -4.0 V Vr= -5.0 V
1.0 0.8 0.6
250K peak
0.4
115K peak
0.2 0.0 0
Trap 1 50
100
150
200
Trap 2
250
300
Tem perature [K]
D LTS SignalΔC [pF]
(b)
Q D intrinsic states
1.4
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1.2 as grown 1.0
o
600 C
0.8 o
0.6
700 C 115K peak
0.4 0.2 0.0 0
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40
60
80
100 120 140
Tem perature [K]
Figure 21. Continued on next page.
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0.8 D LTS signalΔC [pF]
(c) 250K peak
0.6
Vr= -4.5V
0.4 115K peak 0.2
o
0.0 50
550 C as-grown o 650 C o 600 C o 750 C 100 150 200 250 300 350 Tem perature [K]
Figure 21. (a) DLTS spectra of the as-grown sample at various reverse biases. (b) and (c) temperature dependence of DLTS spectra at Vr = –3.0 and –4.5 V, respectively.
The result in Fig. 21(b) shows that the DLTS peak at around 50 K decreases as the annealing temperature increases, and eventually vanishes after annealing at temperature higher than 700 ºC. Up to 700 ºC, the PL peak does not show an obvious shift, suggesting that no interdiffusion has occurred. The decrease of DLTS amplitude might be due to the introduction of newly created defects at 600 ºC, which will be confirmed later and shown in Fig. 21(c). When the annealing temperature increases beyond 700 ºC, the interdiffusion of In and Ga atoms at the interface between the QDs and GaAs barrier occurs, as evidenced in the PL spectra in Fig. 19(a). The interdiffusion lowers the QD confinement barrier, which results in more direct electron tunneling from the QD quantum states to the GaAs continuum. Annealing at even higher temperatures (> 800 ºC) may dissolve the QDs and significantly modify the material composition of the wetting layer.[75] This is indeed supported by the observations of the disappearance of the QD DLTS peak and quenching of the roomtemperature QD PL. Therefore, the overall behaviour of the PL spectra at different annealing temperatures seems to be explained and in good agreement with the variations of the electronic states revealed in the DLTS spectra. To study the effect of RTA on other electronic structures, particularly the coexisting deep levels as well as any new defects created during the RTA processes, which are not optically active and hence not revealed in the PL spectra, we perform the DLTS measurements at a reverse bias of –4.5 V in Fig. 21(c). Under this condition, the DLTS peak related to the QD intrinsic states disappears because electrons in the dots directly tunnel out to the GaAs conduction band, at a rate much higher than the thermal emission in the high electric field. As the annealing temperature increases, new defects are introduced in the structure as shown by the remarkable increase of the DLTS signal at around 275 K. These new defects do not exist in the as-grown sample, and become fully developed after annealing at 600 ºC. Both the sharp annealing-temperature dependence and the large densities of the defects are remarkable.
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8
2
10
6
(a) o 750 C Vr= -4.5V
-4.8 -5.0 -5.2 620 m eV
2
D LTS signalΔC [pF]
-1
ln(en/T )[ln((K s) )]
12
-5.4 -5.6 2.96 3.00 3.04 -1 1000/T [K ]
4 2 0 240
280
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Tem perature [K]
D LTS peak am plitude [a.u.]
40
(b)
35 30 25 20 15 10 5 0 -5 -6 10
-5
10
-4
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-3
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Pulse length [s] Figure 22. Characteristics of the new defects created by annealing. (a) A full DLTS spectrum after annealing at 750 ºC at a rate window of 200 s-1. The inset shows the Arrhenius plot of the main DLTS peak. (b) The DLTS peak amplitude as a function of the filling-pulse length. (Reprinted figure with permission from Ref. [79]. Copyright 2006 by the American Institute of Physics.)
Figure 22(a) shows a full DLTS spectrum of the new defects in the sample after annealing at 750 ºC. The strong DLTS peak appears at around 340 K and a much weaker one on the side. The Arrhenius plot, shown in the inset of Fig. 22(a), yields a thermal excitation energy of 620 meV for the main DLTS peak. Moreover, the peak amplitude is found to have a logarithmic dependence on the duration of the filling pulse in the range from 10-6 to 10-3 s, as shown in Fig. 22(b). The logarithmic dependence is characteristic of extended defects, such as dislocation-related traps, and is attributed to the coulombic repulsion of the carriers captured at the traps.[10,81] We speculate that the formation of the extended defects occurs along with a redistribution, merge, and/or elimination of point defects.[82] The introduction of the new
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D LTS signalΔC [pF]
defects and the probable redistribution of the coexisting point defects into the QD islands could explain the observed decrease in the PL emissions from the intrinsic QD states and InAs wetting layer after annealing at temperatures up to 700 ºC, which is shown in Figs. 20(a) and (b). Furthermore, the introduction of defects into the QDs might be responsible for the decrease of the DLTS signal from the intrinsic QD states below 700 ºC in Fig. 21(b) when no interdiffusion occurs. To identify the origin of the new deep levels at 620 meV, we also perform DLTS measurements to study the influence of annealing on a reference sample (Ref. A), which contains a 1.2 ML InAs wetting layer but no quantum dots. Figure 23 shows the annealingtemperature dependence of the DLTS spectra of the reference sample at a reverse bias of –5.0 V. The two pronounced peaks have been previously studied in Section 3 and identified as deep levels located in the InAs wetting layer.[53] They have the same origins as the corresponding peaks in the QD sample in Fig. 21(a), and are most likely caused by the strain field during the lattice-mismatched growth process. However, unlike the DLTS spectra of the QD sample in Fig. 21(c), no obvious increase of the DLTS signal at around 275 K is observed in the reference sample at annealing temperatures of both 700 and 800 ºC. This indicates that the new deep levels are associated with the QDs. The DLTS signals of the annealed reference samples become slightly negative on both sides of the 240 K peak in Fig. 23. The reason is not clear but may be due to acceptor levels introduced by the annealing process.
1.0 R eference sam ple as-grow n o 0.8 700 C o 800 C 0.6 0.4 0.2 0.0 0
50
100
150
200
250
300
350
Tem perature [K] Figure 23. DLTS spectra of the as-grown and annealed reference samples using a rate window of 80 s-1.
This monotonic decrease in the QD PL intensity with increasing annealing temperature is in agreement with the observations by Lee et al.[76] and Raghavan et al.[77], and is attributed to the formation of nonradiative defects in the structure. The DLTS measurement here provides more direct information about the characteristics of the newly created defects than previous optical experiments. The thermal activation energy can be accurately determined in the Arrhenius plot, and the corresponding density can be calculated from the DLTS peak amplitude. However, an increase of the intrinsic QD state PL signal with the
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3.24
0.20
2.70
0.16
2.16
9
D LTS signalΔC [pF]
-2
0.24
Sheetdensity [x10 cm ]
annealing temperature was also reported.[73-75] Since no DLTS experiments were carried out in these experiments, the detailed information about defect types and concentrations in these QD structures is not clear. The inconsistency is most likely caused by differences in detailed growth conditions. The amplitude of the DLTS 115 K peak is proportional to the density of the related deep levels in the InAs wetting layer as described in Equation (16). Figure 24 thus shows that the density of the coexisting deep level starts to decrease when the annealing temperature reaches 650 ºC, which is slightly lower than the onset annealing temperature, 700 ºC, of the PL blueshift observed in Fig. 20(a). For the PL to show obvious blueshift, a rather significant interdiffusion into the QD centers might be necessary. At 650 ºC, the interdiffusion may have just started at the interface between the QDs and the wetting-layer material, which is detectable in the DLTS spectra but not clearly revealed in the PL results. At 700 ºC, however, the interdiffusion seems to have occurred inside the QDs as suggested by the blueshift. The drop of the DLTS signal at 115 K suggests that when the interdiffusion starts, the number of defects is reduced in the InAs wetting layer. Furthermore, the interdiffusion causes the potential depth of the QDs to reduce, allowing more carriers to populate the wetting layer. Both factors may explain the observation of the enhancement of the PL signal from the wetting layer shown in Fig. 20(b).
115 K peak 0.12
1.62 550
600
650
700
750
800
o
Annealing tem perature [C ] Figure 24. The amplitude of the 115 K peak at Vr = –4.5 V as a function of annealing temperature. (Reprinted figure with permission from Ref. [79]. Copyright 2006 by the American Institute of Physics.)
Unlike the 115 K DLTS peak, it is difficult to quantify the density of the deep levels corresponding to the 250 K peak because of the mixture of trap 2 and the influence of the RTA-induced extended defects.[79] For the 340 K peak, a close look at the Fig. 21(c) reveals that the disappearance of trap 2 and the formation of the extended defects occur at almost the same time, making an accurate determination of the defect density difficult. Furthermore, there is a weaker DLTS peak on the side of the 340 K peak. The detailed evolution of these
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different types of defects during thermal annealing processes seems to be rather subtle and complex, which requires further experimental as well as theoretical investigations.
5. Electron Emissions from QD Intrinsic States 5.1. Preliminary Investigation of Carrier Emission from the Electronic States of Self-assembled InAs QDs by LDLTS Although there has been a spasmodic effort over a long period of time to study the electrical properties of quantum wells[83,84] and quantum dots[30,36,85], the interpretation of the data is problematic. Perhaps the most fruitful work in the last few years has been effected by growing a plane of self-assembled quantum dots in a semiconductor doped to a level whereby a depletion region formed underneath a Schottky barrier could be modulated so that its edge passed through the plane of quantum dots, according to the applied bias. The presence of charge exchange with the dots is observed in capacitance-voltage measurements, admittance spectroscopy, and perhaps most fruitfully, deep-level transient spectroscopy. In the latter technique the depletion region is made narrow by the application of a zero or low applied reverse bias, under which circumstances the dots have the opportunity to capture carriers and then in the second phase of the experiment the depletion region is extended by the application of a larger reverse bias, so the dots are contained in the depleted region. In an attempt to return to equilibrium charge is released from the dots driven by thermal energy. By conducting the experiment over a range of temperatures and observing the emission rate, an Arrhenius plot can be constructed, the slope of which represents an energy. The technique of DLTS provides a fundamentally different approach to the study of confined states compared to photoluminescence. DLTS opens up the possibility of studying wells in a one-carrier system, so that at least in principle dipole effects are under control. However, the emission process from which the DLTS spectra are derived is rather complex. Figure 25(i) provides a schematic illustration of the conduction band in the vicinity of the quantum dot. At this stage in the discussion it should be pointed out that the detail of the system is dramatically affected by the materials and size of the dots and so as a starting point we will consider InAs dots in a GaAs matrix, with dots approximately 10 nm high and 20 nm wide. Such a system is likely to possess 6 quantized states - 2 with s-like character and 4 with p-like character.[86] When the dot captures a carrier (in this case an electron), the bands bend, effecting a Coulomb blockade which makes capture of a second carrier more difficult (see Fig. 25(ii)). In the experimental case which we are considering, capture takes place in a neutral region of the semiconductor as represented by (ii) in the diagram, whereas emission takes place in a high field region (within the depletion region) as illustrated in (iii). It is evident that there are a number of mechanisms for release that can take place. The most obvious (and the one which dominates the thinking of early papers) is a simple thermal excitation from the s state to the band shown as process (c) in Fig. 25(iii). Conceptually this will result in two peaks in the DLTS spectrum with activation energies which relate to the energetic positions of the two s states. However, another alternative is that the electrons are excited to the p states, from which they are then thermally emitted to the conduction band or, as has been mentioned previously, electrons are excited thermally from the s states to the p states and from there they tunnel through the barrier into the band; this is shown as process
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(b). Process (a) must also be considered and that is the possibility of tunneling from the s states directly into the band. In all cases the electrical techniques can only detect charge exchange between the confined states and the band (bound to free transitions) so that the rearrangement of charge among the quantized states does not result in any detectable signal using these techniques.
Figure 25. Schematics of the conduction band in the vicinity of the quantum dot. (i) A neutral region. (ii) One electron captured in the dot. (iii) In a high electric field. (a), (b) and (c) show different electron emission paths from the dot.
All the above processes could occur but the relative probabilities will depend on the temperature, the electric field (which is a function of the applied voltage and the carrier concentration of the matrix), and on the matrix and quantum-dot materials. In principle it is possible to separate the processes using carefully designed thermal emission experiments. However, there are two major obstacles in the experimental realization of such measurements. The first is that all such measurements to date have been undertaken on ensembles of quantum dots. Unfortunately the Stranski-Krastanow self-assemble technique, which relies on the relief of strain in the lattice-mismatched systems, produces a distribution of dot sizes and hence within the ensemble there is a distribution of energies of the quantized states. In luminescence it is technically possible to examine a single dot and a number of publications have resulted from such work.[87,88] Similarly, the absorption
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spectra of a single dot have also been determined by using refined optics[89] or photoconductivity[90]. At the present time the measurement of individual dots has not been achieved using the DLTS methods. The second issue is specific to DLTS. Because the technique is a thermally driven process the energy resolution will be limited to a value of the order of kT, where T is the measurement temperature. However, this is unlikely to be a problem as the quantized states are separated by a significantly greater energy than kT in the system under consideration. Unfortunately in practice the resolution of DLTS is limited to a much worse value which revolves around the way in which the measurement is done and is essentially instrumentation limited.[42] The end result is that the conventional technique is unable to provide a clear separation of emission from the quantized states and some deconvolution of the DLTS spectrum is necessary in order to interpret the result. LDLTS described here eliminates the instrumental broadening of DLTS,[37] increasing its resolution by approximately an order of magnitude over the conventional technique and therefore, in principle, providing adequate resolution of not only the s states but also the p states. In Section 3, the QD sample was studied by the conventional DLTS technique and presented the DLTS spectra with both the deep levels and QD intrinsic states in Fig. 11(a). The peak at around 50 K is attributed to the electron emission from the intrinsic states in the quantum dots. The rapid thermal annealing results in Section 4 also confirm such identification. However, there is no fine structure discernible in the conventional DLTS spectra. In this section, we apply the high-resolution LDLTS technique to examine this peak in order to study the electron emission processes from the QD electronic states.
D LTS signal[a.u.]
0.015
0.010
0.005
T= 60K V r=-3.0V,L=1m s Filling-pulse bias -2.80V -2.75V -2.70V -2.65V -2.60V
0.000 10
100
1000 -1
Em ission rate [s ] Figure 26. LDLTS spectra of the quantum-dot states at 60 K obtained by using different filling-pulse biases.
In Fig. 11(a) the peak at around 50 K at Vr = –3.0 V has a broad shoulder extending to lower temperature, which suggests that it results from either a non-exponential transient or a number of exponential components. Figure 26 shows the LDLTS spectra taken at 60 K; this
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D LTS signal[a.u.]
provides an insight into the low-temperature DLTS peak. The reverse bias is set Vr = –3.0 V, but a range of filling voltages is used so as to selectively fill the confined states. LDLTS always represents the discrete emission rates referenced to perfect exponentials as almost delta functions. It is very evident in Fig. 26 that there is a range of emission rates, which cannot be separated unambiguously. The system is designed to use the principle of parsimony and so in such circumstances where there is a continuum of emission rates or emission rates that are very close together (about a factor of 2), such broad spectra are generated.
0.005 V r= -3.5V,V f= -2.8V,L= 1m s 55 K 50 K 0.004 45 K 40 K 0.003 0.002 0.001 0.000 10
100
1000 -1
Em ission rate [s ] Figure 27. LDLTS spectra of the quantum-dot states at a larger reverse bias than in Fig. 26 and at lower temperatures in which it would be expected that tunneling is more significant than in Fig. 26.
If the temperature is reduced and the reverse bias increased, a detailed structure starts to emerge in the LDLTS spectrum as shown in Fig. 27. Quite obviously, reducing the temperature reduces the probability of thermal emission while increasing the reverse bias increases the probability of tunneling. In the temperature range between 50 and 55 K, the positions of the peaks shift, i.e., they activate with temperature. However, below 50 K there is no meaningful change in the position of the peaks with temperature, suggesting an athermal process. However if the peaks are examined as a function of the applied reversed bias, then the emission rate increases with increasing reverse bias. We conclude from this observation that tunnelling dominates the spectrum. Moreover this provides strong evidence that there are a number of discrete tunnelling rates operating in parallel within our sample. Although LDLTS provides clear and reproducible data on these quantum-dot samples, its interpretation is unclear apart from the fact that at low temperatures tunnelling processes appear to dominate, as shown by processes (a) and (b) in Fig. 25(iii). Some additional information can be obtained from quantifying the charge exchange. From the DLTS measurements we can quantify, with considerable precision, the charge released to the band. This can be computed from the area enclosed within the peak. In an ideal situation and exercising care in the measurement of all the parameters concerned, this figure can be determined to within a few percent. If the dot concentration and its position are precisely
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known within the depletion region, we can then calculate how many electrons per dot the peak represents. Unfortunately in this case we do not have a precise measure of the number of dots involved in the samples we studied. The data are derived from AFM measurements, where by necessity we are measuring a similar sample but without the gallium arsenide capping layer. However the comprehensive study of Ledentsov et al, in the same material system with very similar growth conditions indicates that the dot concentrations will not change dramatically during the capping process.[91] In consequence if we take the AFM observed value and consider the peak at 50 K in Fig. 11(a) we are removing at least four electrons from each quantum dot. Considering the Laplace spectra taken at 60 K in Fig. 26, the total spectrum represents approximately two electrons per dot. As the filling conditions are set so as to only fill the s states, this is consistent with expectations of a combination of thermal excitation and tunnelling. In Fig. 27 the fact that the four Laplace peaks represent a sequence of discrete electron transfer rates due to tunnelling is quite remarkable. It seems highly unlikely that this has anything to do with the energy levels of the s or the p states in the dots. The energetic separation of s1 and s2 is generally calculated to be rather small as is the separation of individual p states. The energy difference between the two groups in this system is expected to be about 80 meV. The emission rates are independent of temperature but dependent on the field. As it seems likely that there is a size distribution of the dots, the mechanism is probably more associated with the amount of charge on the dots rather than on the energies of the quantized states. A possible hypothesis is that due to the band bending the tunnelling rate is dependent on the charge state of the dot, so that if the dot is highly charged and the band bending severe, the tunnelling rate is high and in consequence represents the loss of the first electron.[92] When the electron is lost from any particular dot, the band bending changes and the tunnelling rate decreases. This would explain the sequence of the discrete tunnelling rates, and, as the coulombic field extends well beyond the physical size of the dot, would be independent of the dot size.
5.2. Electron Emission from QD Intrinsic States In Section 5.1 above, the electron emission from the QD intrinsic states has just been preliminarily investigated in the InAs quantum-dot structures, and the results show that LDLTS is a valuable technique for examining carrier emission processes with a better resolution than the convention DLTS technique. However, the previous QD sample shows one strong DLTS peak at about 50 K related to the QD intrinsic states in Fig. 11(a). The lack of the fine structure inside this peak makes it difficult to resolve carrier thermal emissions from the different electron shells (the s and p shells) and from the two different s-electron configurations. For this sample it is impossible to provide definitive identification of the peaks in the LDLTS spectra. For device application and fundamental physics, the capture and emission of charge carriers in the selected QD intrinsic states are of considerable importance; this requires a new structure design and optimal sample fabrication.
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Figure 28. Schematics of the layer structure of the QD sample with a 0.4 µm capping layer, named as SE40.
In this section, we study the QD sample grown with low-doping background, which can increase the resolution of the DLTS measurement and reduce tunneling rates. The InAs QD plane is embedded in the lightly Si-doped (1x1016 cm-3) GaAs Schottky diode grown by MBE on (001)-oriented n+-GaAs substrates. The epitaxial layer sequence together with the growth temperatures is depicted in Fig. 28. Self-assembled QDs were formed by the StranskiKrastanow effect using the growth-interruption technique. The 3 ML thick InAs layer was grown at 510 ºC under a repeated sequence of 0.1 ML InAs followed by a two-second-growth interruption under a continuous flux of As2. The formation of QDs was monitored in situ by RHEED, and the QD nucleation was observed via the 2D to 3D transformation. The QDs obtained have a base/height size of 20/10 nm and with a concentration of 1-2×1010 cm-2 defined by the AFM image. For the purpose of comparison, a reference sample with a 0.40 µm capping layer was also grown. In the reference sample (referred to as RE40), only an InAs wetting layer was deposited. Gold was evaporated on the capping layer to form a Schottky barrier with 1 mm diameter and on the highly doped substrate back-side as an Ohmic contact. Low-temperature PL measurements were performed on the QD and reference samples in a helium cryostat with laser excitation at 633 nm. The signal was detected by a N2-cooled Ge detector and amplified using the conventional lock-in techniques. In Fig. 29 the PL spectrum of the QD sample exhibits a strong peak at 1169 nm (E = 1.063 eV, ΔE = 35 meV) which corresponds to the interband transition from the QD ground electronic state to the ground hole state.[27] The absence of QDs in the reference sample is confirmed by the disappearance of
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the QD PL signal in the spectrum, and the broad PL peak at 961 nm is from the pseudomorphic InAs wetting layer.
PL intensity [a.u.]
500
7.7K SE40 R E40
400 300 200 100 0 800
900
1000
1100
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1300
W avelength [nm ] Figure 29. PL spectra at 7.7 K from the QD sample (SE40) and the reference sample (RE40).
In the following sections, the electronic properties of the InAs/GaAs QDs are investigated firstly by means of capacitance-voltage measurements. Next the thermal emission of electrons from the selected s-shell configurations in InAs QDs is studied using the conventional DLTS technique. Finally, Laplace DLTS is applied to study the detailed mechanisms of electron emission from the QD intrinsic states.
5.2.1. CV Characteristics of the InAs/GaAs QD Structures The base length of the QDs in SE40 is about 20 nm, with a variation of about ±5 nm, while the height is about 11 nm with a variation of ±2 nm as determined in an earlier work.[35,93] QDs of these dimensions normally have two clear electron shells, one of s character capturing two electrons, with an energy position at about 0.15 eV, and another of p character with four electrons, at about 0.1 eV from the conduction band edge.[86] Figure 30 shows a CV characteristic of one of the Schottky diodes of the QD sample with a 0.40 µm capping layer. The CV trace was recorded at 80 K in order to ensure sufficiently fast charge exchange between the QDs and electron reservoir at the modulation frequency of 10 KHz. The plateau-like structure occurring between –0.32 and –1.3 V reflects the charging and discharging QDs in pace with the signal frequency of the capacitance meter. This increases the capacitance when the bulk Fermi level is close to the energy levels of the QDs. The capacitance value of the plateau corresponds well to the calculated value of 226 pF corresponding to the distance between the dot layer and the front gate of d = 0.4 µm and the gate area of 7.85×10-3 cm2. The width of the plateau is determined by the spread in QD energy levels and of the energy distance between the s and p electrons. At closer inspection, a substructure is discernible in the plateau which we associate to the s-shell and the p-shell
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filling of the dots.[36] Correspondingly, we expect that the s and p levels start to become occupied at Vr = –1.3 and –1 V, respectively. The arrows in Fig. 30(a) denote the gate voltage intervals at which two and four electrons are loaded into the dots, respectively. The ratio of the interval lengths from s to p is about 1:2, which confirms our interpretation of the substructure since we expect the QDs to accommodate two and four electrons in the s shell and the p shell, respectively. 180 SE40
150 p 120 90
200
60
160
o
s
240
Phase angle []
C apacitance [pF]
280
(a)
30 120 0 -5
-4
-3
-2
-1
0
R everse bias [V]
-3
C arrierconcentration [cm ]
0 (b)
p
s
-2 -3
1E16
R everse bias [V]
-1
1E17
-4 -5 0.4
0.5
0.6
0.7
D epletion depth [μm ] Figure 30. CV characteristic (a) and the corresponding carrier concentration profile (b) of the QD sample (SE40) at a measurement frequency of 10 KHz and at 80 K.
The corresponding carrier concentration profile is shown in Fig. 30(b). The two separate accumulation peaks might or might not be related to the electrons loaded in the s and p shells. The reason for the difficulty to make the conclusion is that the CV profiling process lumps
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two components of the charge increment together: one is the localized component in the QD/QW layer, and the other is the smear-out component by the Debye tail at the depletionlayer edge.[41] Therefore the distance extracted from capacitance is the mean position of the charge displaced by the oscillatory test signal. Since the s and p shells are in the same depth in the sample, it is hard to directly relate them to the accumulation peaks at different depths in Fig. 30(b). However, the CV profile in Fig. 30(b) is indeed different from that in Fig. 9 in the previous QD sample with the similar structure. The conventional DLTS spectra later shows the sample measured here has much better resolution in the energy spacing between the s1 and s2 states than that in the previous QD sample. Although there is not indication on the CV characteristics, it is also found that deep levels exist in this InAs/GaAs quantum-dot structure.[94,95] Three deep-level traps have been found to be grown-in defects and induced by the strain present in the QD sample.[94] DLTS depth profiling procedures indicated that the deep-level related defects were localized in GaAs in the vicinity of the QD plane. In the following, we only concentrate on the carrier emission from QD intrinsic states.
5.2.2. Carrier Emission from s States Studied by Conventional DLTS By selecting appropriate voltage intervals in pulsing sequences of the Schottky diodes, Engström et al.[35] and Schulz et al.[36] separated and measured the activation energies for the thermal emission of electrons from the two s-shell configurations in InAs/GaAs QDs. In this section, we use the similar method to resolve the thermal emission of electrons from the s1 and s2 states, and determine their activation energies respectively. The results obtained by using the conventional DLTS technique is of importance to determine the electron capture and emission mechanisms in QD intrinsic states, and to be used to identify the QD signals in the LDLTS spectra in the next section. In Fig. 30, the plateau-like structure, corresponding to the capturing and emptying electrons in the dots, occurs in the CV trace between –0.32 and –1.3 V. The observable substructure in the plateau reflects the capturing and emptying electrons from the s and p shells. One can thus conclude that the filling-pulse bias of Vf ≥ –0.32 V can fully charge the QDs, and for reverse voltages above the ledge value at –1.3 V, all electrons are emitted from the QDs. Due to the detected limit of the convention DLTS technique, the reverse biases used to observe the electron emission from the QD states are larger than those estimated by the CV measurements. By using appropriate voltage pulse sequences in the DLTS measurement, the two configurations, with one and two electrons captured in the s shell, can be separated as illustrated in Figs. 31 and 32. Due to the variation in QD size, the electron energy levels will have a certain distribution. Biasing the Schottky diode to a voltage of 0 V will bring the bulk Fermi level into a position above all these energy levels, which will fill the s shell with two electrons. When the reverse voltage is decreased to –1.4 V, the upper part of the energy-level distribution reaches above the bulk Fermi level and electrons start to be emitted to the GaAs conduction band. As a result, for lower emission voltages, only one electron in the majority of QDs will leave the s shell, so that the configuration goes from one with two electrons (s2) to one with a single electron captured (s1). Correspondingly, starting at a reverse voltage of –2.0 V, the whole energy-level distribution falls above the bulk Fermi level; all QDs are emptied of electrons, and are then neutral in charge. Increasing the voltage to –1.6 V, the lower part of the energy-level distribution will “dip” below the Fermi level and be occupied by the first
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electron. Switching the Schottky barrier between –1.6 and –2.0 V, therefore, makes the electron configuration change from s1 to an empty dot, and the s1 configuration can be studied separately. 3.0 C apture/Em ission
D LTS signalΔC [pF]
SE40 2.5
0.0V/-2.0V 0.0V/-1.8V 0.0V/-1.6V 0.0V/-1.4V 0.0V/-1.2V
2.0 1.5 s2
1.0 0.5 0.0 20
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50
60
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80
90 100
Tem perature [K] Figure 31. DLTS spectra for a constant capture voltage and a varying emission voltage in the QD sample (SE40) at a rate window of 200 s-1.
Figure 31 shows a series of DLTS spectra with the filling-pulse bias kept at 0 V while varying the reverse pulse bias between –1.2 and –2.0 V, separating the s2 configuration. As the reverse bias increases from –2.0 to –1.4 V, the peak shifts to low temperature and fixes at Vr > –1.4 V, reflecting the emission of the electrons leaving the s2 state. Emission from the p shell occurs at temperatures below 40 K. Because of the limitation of the hardware, the peak related to electron emission from the p levels cannot be detected. For higher emission voltages, this emission is dominated by tunneling, which causes a temperature-independent background at low temperature (T < 35 K). In Ref. [36], the electron emission from the p shell can be clearly identified in the conventional DLTS spectra. This may be expected because of the larger QD size and the lower background-doping concentration in their QD samples. In Fig. 32, the reverse-pulse bias is kept at a value of –2.0 V, high enough for both s electrons to be emitted, while the filling-pulse bias is varied from 0 to –1.8 V. As the filling bias decreases from 0 to –1.8 V, the peak shifts to high temperature. For the capture voltage at –1.6 V, only the s1 state will capture and emit electrons at the majority of QDs and the s1 state is separated. Determining the activation energies of the s1 and s2 emissions by standard procedure, we obtain the Arrhenius plots shown in Fig. 33. The activation energies are found to be 102 meV for the s1 emission and 70 meV for the s2 emission. However, the thermal activation energies for the singly and doubly occupied s states in our experiments are less than those simulated by Ref. [86] (about 150 meV) and those reported in the similar structure by Ref. [35] (160 and 120 meV for the s1 and s2 states, respectively). Further measurements on the activation
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energies at different emission voltages by the conventional Arrhenius analysis demonstrate that the activation energies show a very strong reverse-bias dependence. From the electricfield dependence, Schulz et al. proposed that the electrons escape from the dots by phononassisted tunneling processes and revealed that the charge state of the dots could strongly influence the emission process.[36] Therefore, the different activation energies determined in our experiments from those values reported in the similar structures may be explained by the different selections of the voltage pulses to capture and emit electrons from the selected s states. 2.5
C apture/Em ission -1.8V/-2.0V -1.6V/-2.0V -1.4V/-2.0V -1.2V/-2.0V -1.0V/-2.0V -0.8V/-2.0V s1 0.0V/-2.0V
D LTS signalΔC [pF]
SE40 2.0 1.5 1.0 0.5 0.0 20
30
40
50
60
70
80
90 100
Tem perature [K] Figure 32. DLTS spectra for a constant high emission voltage and a varying capture voltage in the QD sample (SE40) at a rate window of 200 s-1.
C apture/Em ission 0V/-1.6V -1.6V/2.0V
-2 s1 102m eV
2
-1
ln(en/T )[ln((K s) )]
-1
2
-3 s2 70m eV
-4
-5 14
SE40 15
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17
18
19
20
21
22
23
-1
1000/T [(m K) ] Figure 33. Arrhenius plots obtained from the separated DLTS curves for the s2 and s1 states.
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Due to the identical orbital wave functions of the s electrons, they occupy the same energy level with different spins within the QDs. The different activation energies determined from the conventional Arrhenius analysis of the emission rate data for the s1 and s2 states may be due to the variation of the QD size,[35] the different electric fields in the barrier determined from the reverse bias and the QD charge states.[94] Also, the calculated activation energies depend on the controlling emission mechanism during the electron emitting processes, which may be the thermal emission, the direct tunneling, or some other emission mechanisms.
5.2.3. Carrier Emission from s States Studied by LDLTS Using appropriate voltage-pulse sequences in the conventional DLTS measurements has separated the two configurations, with one and two electrons captured in the s shell. With a real spectrum of the emission rate, the LDLTS technique gives better resolution than the conventional DLTS technique, and enables us to reveal the fine structure in the electron emission process from the QD intrinsic states. LDLTS can separate the maxima associated to the emissions from the singly and doubly occupied s shell instead of a broad peak by the conventional DLTS method. As illustrated in Figs. 34 and 35, the electron emissions from the s1 and s2 configurations can be identified by the pulse-bias dependency similar to the conventional DLTS analysis in Section 5.2.2. 0.10 s1 0.08
D LTS signal[a.u.]
s2
C apture/Em ission -1.0V/-2.0V -1.1V/-2.0V -1.2V/-2.0V -1.3V/-2.0V -1.4V/-2.0V -1.5V/-2.0V -1.6V/-2.0V -1.7V/-2.0V -1.8V/-2.0V
SE40 60K
0.06 0.04 0.02 0.00 1
10
100
1000
10000
-1
Em ission rate [s ] Figure 34. LDLTS spectra for a constant high emission voltage and a varying capture voltage in the QD sample (SE40) at 60 K.
The sheet densities of the electrons escaping from the s1 and s2 states can be determined from the peak amplitude of the LDLTS spectra. At the bias condition of the capture/emission voltage at –1.0 V/–2.0 V in Fig. 34, the peak amplitudes for the electron emissions from the s1 and s2 states are 2.69 and 2.79 pF, respectively. According to Equation (12), the sheet densities for the s1 and s2 electrons are 1.49×1010 and 1.55×1010 cm-2, respectively. The
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calculated values are in good agreement with the value of 1-2×1010 cm-2 estimated from the AFM image, which confirms the identification of the two maxima in the LDLTS spectra with the s1 and s2 emissions.
0.12 0.10
D LTS signal[a.u.]
s2
SE40 60K
C apture/Em ission -0.5V/-1.0V -0.5V/-1.1V -0.5V/-1.2V -0.5V/-1.3V -0.5V/-1.4V -0.5V/-1.5V -0.5V/-1.6V -0.5V/-1.7V -0.5V/-1.8V -0.5V/-1.9V -0.5V/-2.0V
s1
0.08 0.06 0.04 0.02 0.00 1
10
100
1000
10000
-1
Em ission rate [s ] Figure 35. LDLTS spectra for a constant capture voltage and a varying emission voltage in the QD sample (SE40) at 60 K.
D LTS signal[a.u.]
0.020
SE40 60K
C apture/Em ission -0.8V/-1.0V -1.0V/-1.2V -1.2V/-1.4V -1.4V/-1.6V -1.6V/-1.8V -1.8V/-2.0V -2.0V/-2.2V
0.015 0.010 0.005 0.000 1
10
100
1000
10000
-1
Em ission rate [s ] Figure 36. LDLTS spectra for the varying capture/emission voltages with a constant small voltage interval of –0.2 V in SE40 at 60 K.
In order to resolve the electron distribution in the QDs, the approach of the DLTS depth profile is duplicated here to gradually change the QD intrinsic energy levels relative to the bulk Fermi level by using different filling-pulse bias and emission reverse bias with a
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Shiwei Lin and Aimin Song
constant small bias interval of –0.2 V.[51] The result is shown in Fig. 36. It is a distinct feature of interfacial energy states that their DLTS response has a maximum, when, at a certain bias, the Fermi level crosses the energy level at the interface.[51] Figure 36 shows two groups of LDLTS peaks, corresponding to the electron emission from the s1 and s2 configurations. For clarity, the dependence of the peak position on the emission reverse bias is depicted in Fig. 37. The emission rate saturates at the low and high quiescent reverse biases, respectively. This indicates that the electron emission from the same s shell can be divided into two groups, which are associated with the emissions from the s1 and s2 states, respectively. Since the electron emission processes are quite complex and depend on the measuring conditions, i.e., the reverse bias and the temperature, further experimental and theoretical investigation is require to clarify whether the electrons escape from the QD intrinsic states by the thermal emission, direct tunneling, or their combined processes.[92]
-1
Em ission rate [s ]
2000
SE40 60K
1500
s2
1000 500
s1
0 -2.2
-2.0
-1.8
-1.6
-1.4
-1.2
R everse bias [V] Figure 37 LDLTS peak positions as a function of the emission reverse bias in SE40 at 60 K from Fig. 36.
6. Conclusion Electronic states in self-assembled InAs/GaAs quantum-dot structures have been systematically investigated by applying both optical and electrical techniques, such as PL, CV, conventional and Laplace DLTS. We present the direct evidence of coexistence of deep levels with optically active QDs. The complete DLTS spectra have been demonstrated to present both the deep levels and InAs QD intrinsic states. The reverse-bias dependence of the DLTS signal together with experimental results from the reference samples, containing thin InAs layers but no quantum dots, confirms that the deep levels coexist in the same layer as the InAs dots. The deep levels are most likely caused by the strain field during the latticemismatched growth process. Physical parameters of the intrinsic and deep-level states are
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quantitatively determined in the DLTS experiments. These include the thermal excitation energies, densities, and their location in the QD structures. Furthermore, we investigate the effects of postgrowth rapid thermal annealing on the intrinsic states and deep levels in the QD structures. At a series of annealing temperatures, the effects of annealing on the optical spectra are found to be closely linked with the variations of both the intrinsic QD and deeplevel states. Compared with the PL spectra, the DLTS data are more complex, revealing a number of electronic states varying with the annealing temperature and particularly the formation of a new deep level. The coexistence of deep levels with quantum dots plays an important role both in fundamental physics and device applications. If the deep states lie close to the QDs, they may strongly couple with the dots and provide a rapid energy-relaxation channel through which electrons thermalize to enhance the luminescence efficiency, causing the lack of phonon bottleneck effect. For electronic device applications, the existence of states that are much deeper than the intrinsic QD states could be utilized to design memory devices. Further experimental as well as theoretical studies on the deep levels could not only provide important insight into the physics of electronic states but also have implications to the optimization of the QD growth and device performance. The electron emission from QD intrinsic states is complicated, including a number of mechanisms, such as thermal excitation, direct tunneling, and even both. All these processes are physically feasible but their relative probabilities depend on the temperature, the electric field and the QD and host materials. Using the CV spectroscopy, the conventional DLTS and LDLTS techniques, we have studied the electron emission from the QD intrinsic states. By applying an appropriate set of voltage pulses across the Schottky diode, the two different selectron configurations have been separately using the conventional DLTS and LDLTS techniques. In contrast to one broad peak on the conventional DLTS spectrum, two wellseparate maxima are determined on the LDLTS spectrum, which are related to the electron emission from the QD singly and doubly occupied s states. All the measurements indicate that LDLTS is valuable for examining carrier emission process from QDs and can provide information that is difficult or perhaps impossible to obtain by other methods.
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In: Quantum Dots: Research, Technology and Applications ISBN: 978-1-60456-930-8 Editor: Randolf W. Knoss, pp. 109-168 © 2008 Nova Science Publishers, Inc.
Chapter 3
CHEMICALLY DEPOSITED THIN FILMS OF CLOSE PACKED CADMIUM SELENIDE QUANTUM DOTS: PHOTOPHYSICS, OPTICAL AND ELECTRICAL PROPERTIES Biljana Pejova* Institute of Chemistry, Faculty of Natural Sciences and Mathematics, Sts. Cyril and Methodius University, POBox 162, 1001 Skopje, Macedonia
Abstract A recently developed chemical method for synthesis of close packed cadmium selenide quantum dots (QDs) in thin film form is reviewed. By controlling the chemical composition of reaction solution and post-deposition treatment, the presented method permits optoelectrical properties of CdSe QD thin films to be designed. Synthesized CdSe QDs crystallize in cubic crystalline system and are characterized with high chemical and crystallographic purity. Such properties of synthesized CdSe QD thin films are quite distinct in comparison with QDs of the same material synthesized by other methods reported in the literature. The average crystal radius of as-deposited CdSe QDs, calculated by the Debye-Scherrer approach, is 2.6 nm. Upon annealing at 300 oC, this value increases to 12 nm. Optical band gap energies of asdeposited and thermally treated CdSe QD thin films are 2.08 and 1.77 eV correspondingly. The notable blue shift of band gap energy of 0.34 eV for as-deposited thin films with respect to the bulk value is due to the pronounced quantum size effects. Upon thermal treatment, the absorption edge of CdSe thin films is red shifted and the band gap energy tends to approach the bulk value. The experimental blue shifts of the band gap energies of as-deposited and annealed CdSe QD thin films (with respect to the corresponding bulk value) were compared with the theoretical ones, predicted by the effective mass approximation model. Electrical and photoelectrical properties of the synthesized cubic CdSe QDs in thin film form (including the relaxation dynamics of photocarriers) were investigated as well. On the basis of measured temperature dependence of dark electrical resistance of annealed CdSe QD thin films in the intrinsic conductivity region, thermal band gap energy value of 1.85 eV was calculated (corresponding to 0 K). In lower-temperature region, the conductivity of QD thin films was shown to be a two-channel temperature-activated process with activation energies of 0.74 and *
E-mail address:
[email protected]
110
Biljana Pejova 0.43 eV. The possible physical interpretations of these values are discussed. Time-resolved studies of photoconductivity relaxation dynamics showed that within a very short starting time interval immediately upon light excitation switch off (< 0.2 ms) the photocarriers are relaxed according to the quadratic relaxation mechanism. After about 0.2 ms the non-equilibrium charge carriers recombine according to the linear relaxation mechanism with a relatively high relaxation time value of 0.4 ms. This indicates a potential applicability of the synthesized QD thin films in solar cells engineering.
1. Introduction The remarkably rapid development of contemporary technologies, which is evident in the last decades, has lead to a necessity for miniaturization of electronic components and devices. Reduction of the size up to the micrometer range, aside from technological problems related to its particular realization, does not lead to any substantially new properties of the materials used for device or device component production. However, when the nanometer range is entered, further size reduction has been clearly shown to lead to a wide class of quite novel and unusual phenomena (at least in comparison to the corresponding macrocrystalline analogues of the investigated materials) [1-8]. These new phenomena, which arise when obtained particles reach some size limit in the production of nanosize materials, are quite understandable and in fact expected from the viewpoint of quantum mechanics. Nanosize materials are, in their properties and behavior, much more similar to atomic and molecular systems than to macrocrystalline (bulk) counterparts. Therefore, in the last period numerous studies have been devoted to a thorough understanding of the size-evolution of materials’ properties [1-8 and references therein]. A standard model which has often been used to analyze the properties of a bulk (macrocrystalline) solid state material involves the concept of a perfect crystal. This entity is often thought of as an ideal three-dimensional pattern (i.e. crystal lattice) convoluted with a particular structural motif, free from any irregularities, including vacancies, interstitial atoms or impurity atoms. The periodicity in three spatial dimensions of such perfect crystal has certain consequences on the electronic properties of the solid state material in question [9-17]. Most of these consequences may be derived from the properties of one-electron wavefunctions for a potential periodic in three spatial dimensions (3D). The potential experienced by an electron in a perfect crystal with lattice translational vector (in real space)
G R is a periodic function of the form:
G G G V r + R = V (r )
(
)
(1)
G
where r is the radius vector of the electron and V is the potential it experiences. It can be shown that the solutions of the electronic Schrödinger equation with potential of the form (1) have the following form [9-17]:
G
G
( G)
G
φ kG , j (r ) = exp ik r u kG , j (r )
(2)
Chemically Deposited Thin Films…
111
G
G
In (2), u kG , j (r ) is a function with a periodicity of the lattice, while k is the electron’s wavevector. Wavefunctions of the form (2), which have the periodicity of the crystal lattice, are known as Bloch functions. As evident from (2), the indexing of one-electron
G
wavefunctions is carried out by the electron’s wavevector k . It may be shown that the electronic energy, on the other hand, is a periodic function in the reciprocal space (also called
G
Fourier space or k - space [12]):
G G G E (k ) = E (k + 2πbg )
(3)
G In (3) bg is a vector of the reciprocal lattice, defined with the following equation:
G
(b , aG ) = 2πδ g
j
(4)
gj
G
where a j is the unit vector in the real space, while δgj is the Cronecker symbol. Such
G
periodicity in k - space implies that all sufficient information concerning the electronic energy structure of the solid should be contained in the primitive unit cell of the reciprocal lattice. This unit cell is called the first Brillouin zone [12]. In a one-dimensional model of a perfect crystal, the first Brillouin zone actually corresponds to the interval [-π/a, π/a] of the
G
reciprocal space. Other intervals from the reciprocal ( k ) space also appear to be significant
G
for certain applications of the solid state theory. Thus, the interval k ∈ [-2π/a, -π/a] ∪ [π/a, 2π/a] corresponds to the second Brillouin zone, etc. From these properties of oneelectron wavefunctions, the concepts of energy bands (allowed and forbidden) emerge. For a free electron with mass m the dispersion relation may be expressed by the following simple formula [14]:
G G =2k 2 E (k ) = 2m
(5)
G
()
In the case of an electron experiencing a periodic potential, on the other hand, E k is a complex discontinual function. The discontinuities appear at the points corresponding to the Brillouin zones borders in reciprocal space (see Fig. 1 for a schematic presentation). For each Bloch wavefunction of the form (2) the corresponding energy eigenvalues E kG , j obtained by
G
variation of k describe an “energy band” for each value of the other index j [11]. In cases where there is no overlap in energy between bands with different indices j, energy ranges appear where there are no stationary values of E kG , j . These correspond to the “energy gaps” (i.e. “forbidden bands”). It is the magnitude of the energy gap between the last “filled” band and the first “unfilled” band that is usually referred to as the “bad gap energy”. The value of this parameter actually classifies a given solid state material as a metal, semiconductor or
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Biljana Pejova
isolator, although the limiting value discriminating between the last two classes of materials is somewhat arbitrary. To describe the behavior of a charge carrier in a perfect crystal under the influence of an external electric or magnetic field, it is intuitively clear that it has to be considered as a wavepacket which could be basically represented as a superposition of Bloch wavefunctions. The superposition is obtained by summing (or integrating) the Bloch functions within an
G
G
interval of k values around given value of k 0 [2,13]: Wavepackets of the form (6) are known in the literature as Wannier functions [2,13].
G
Under the influence of an external force F the wavepacket energy changes as:
G G GG dE (k ) = Fds = F vg dt
(7)
G
where v g is the group velocity of the wavepacket. The last quantity is defined with the following expression:
G G 1 vg = grad k E (k ) = G
(8)
G
φ k (r ) = ∑ a k e ik r u k (r ) o
k
a) Figure 1. Continued on next page.
(6)
Chemically Deposited Thin Films…
113
b) Figure 1. a) The dispersion relation for an electron moving in a periodic potential in the extended-zone scheme (the case corresponding to the free electron is represented by a dashed curve); b) Dispersion relation in the reduced-zone scheme, where all bands are shown within the first Brillouin zone.
For a one-dimensional case, the expression (8) could be rewritten in a more simple form:
vg =
1 dE = dk
(9)
On the other hand, combining the expressions (7) and (9), having also in mind that:
G G G G dE ( k ) G G dk = =v g dk dE ( k ) = dk
(10)
one easily arrives at the following formulation of the second Newton’s law:
G G dpG dk = =F= dt dt
(11)
G
In the last equation, p is the quasi-momentum of the “free” charge carrier. Under the
G
influence of an external force, thus, the wavepacket moves with acceleration a given by:
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Biljana Pejova
G G G d v g 1 ∂ 2 E 1 ∂ 2 E dk 1 ∂2E G F = a= = = dt = ∂k ∂t = ∂k 2 dt = 2 ∂k 2
(12)
Comparing the last equation with a standard formulation of the second Newton’s law:
G 1 G a= F m
(13)
it follows that the free charge carrier, under the influence of an external force, moves in the periodic potential imposed by the crystal lattice with an effective mass m* defined as:
1 1 ∂2E 1 ∂2E ; i , j = x, y , z = = = 2 ∂k 2 = 2 ∂k i ∂k j m∗
(14)
The concept of effective mass is of prime importance in semiconductor physics. This quantity, in a sense enables us to consider a charge carrier (electron or hole) in a crystal as a free particle with a modified mass (equal to the effective mass) which moves under the influence of an external force. The overall effect of the interaction of the charge carrier with the lattice periodic potential is therefore accounted for through this quantity. As obvious from the definition equation (14), the effective mass is a tensorial quantity, i.e. it depends on the direction in which the charge carrier moves.
Eg,1
Eg,2
w1 w2 a) Figure 2. Continued on next page.
Chemically Deposited Thin Films…
115
w w b)
w
w w
c) Figure 2. A schematic presentations of some low-dimensional structures: a) multiple quantum well (system with a confinement in one dimension); b) quantum wire (confinement in two dimensions); c) quantum dot (confinement in three dimensions); w is a characteristic linear dimension of the lowdimensional structures (w ≈ 2 – 20 nm); Eg is the band gap energy.
If one or more dimensions are “left out” from the concept of ideal periodic systems in 3D, the charge carrier motion will be confined in one or more spatial dimensions [1-5]. Confinement in one spatial dimension is realized in systems often referred to as quantum films. Depending on the particular experimental design (the specific number, thickness and configuration), quantum films may be realized as single quantum wells, multiple quantum wells (schematically depicted in Fig. 2 a) or superlattices. If the charge carrier motion is confined in two spatial dimensions, one deals with quantum wires (Fig. 2 b). Finally, confinement of the charge carriers’ motions in all three spatial dimensions, leads to quantum dots (Fig. 2 c). As this review paper will be mostly devoted on semiconductor materials, we will further focus on physics and chemistry of low-dimensional semiconductors. Most differences in the electronic properties of bulk and confined (low-dimensional) semiconductors are manifested as a result of the differences in the corresponding densities of states (DOS – N(E)). Qualitative trends in the functions N(E) starting from a bulk semiconductor and further passing to the low-dimensional counterparts are given in Fig. 3. In a bulk semiconductor, N(E) is a continuous functions of the form N(E) ∝ E1/2 (Fig. 3 a). In the case of quantum films, N(E) is a step function (Fig. 3 b), while for quantum dots N(E) consists of a series of discrete states (Fig. 3 d), i.e. it may be regarded as a superposition
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of Dirac-delta-like functions. The N(E) function in the case of quantum wires, as can be seen from Fig. 3 c, has a form which is intermediate between that characteristic of quantum films and quantum dots [3]. The central importance of low-dimensional semiconducting materials in contemporary science and technology is mostly due to the remarkable size-dependent optical and electronic structure properties that they exhibit [4,5]. Beyond any doubt, the most notable among these is the size-induced band gap variation. This property is directly related to the enabled spectral tunability of light absorption and emission as well as the oscillator strength induced due to quantum confinement effects. Such exciting new physics, aside from its fundamental importance, allows for a wide application of these nanosize materials in optoelectronics and non-linear optical applications. A particularly interesting topic within this research area is related to the optical, electrical and photophysical properties of semiconducting quantum dots (QDs) deposited as continual thin films. An individual QD nanostructure is actually a zerodimensional analogue of the two-dimensional quantum well (QW), as implied before. Such nanostructure is characterized by discretized energy level structure and discrete electronic transitions that shift to higher energies upon decrease of the QD linear dimensions (i.e. radius).
a)
b)
c)
d)
Figure 3. Plots of the density of states vs. energy in the case of: a) bulk semiconductor; b) quantum well; c) quantum wire; d) quantum dot.
Chemically Deposited Thin Films…
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In case when one deals with individual QDs which are close-packed (forming e.g. threedimensional assembly or array of QDs), further new opportunities are opened and fundamentally new aspects may be explored. In three-dimensional arrays of QDs the collective physical phenomena that develop upon interaction of the proximal QDs may be explored. At the same time, certain properties, which are characteristic of individual QDs, are retained [18-25 and references therein]. In some recent publications devoted to this subject [18-25], the terms “QD solid” or “colloidal crystal” have been coined and extensively used referring to 3D QD arrays. The new terms were in a sense required in order to emphasis that one actually deals with a new form of organization of matter. Such unique properties characteristic of an individual QD on one hand, and the cooperative effects in QD solids on the other hand, make these novel types of superstructures very convenient media with great potential for application in optical and electronic devices. Having in mind the possibilities for their application and the fundamental questions with respect to solid state science that arise from studies of these materials, the continuously increasing interest in this area is selfunderstood. In this context, it is worth mentioning that even purely fundamental considerations concerning physicochemical properties of low-dimensional systems may be of crucial importance for development of new technological breakthroughs. Actually, when one considers QDs of spherical shape, there is a certain similarity between these low-dimensional structures and isolated atoms. With the language of quantum mechanics, the mentioned case corresponds to the confining potential of spherical symmetry. In many cases which could be experimentally realizable, the charge carriers’ motions are strongly confined within the nanocrystal (the QD). These systems are quite well described by an infinitely deep potential well, i.e. with a potential which is actually zero within the quantum dot with a radius R (i.e. inside the dot), and rises abruptly to infinity at the borders [10]. Such potential may be quantitatively expressed with:
G ⎧ V (r ) = 0 ⎨ G ⎩V (r ) = ∞
if r < R otherwise
(15)
The Schrödinger equation for a problem with spherical symmetry may be conveniently written in spherical coordinates. In the present case, it takes the form [10]:
⎡ =2 ⎛ 1 ∂ ⎛ 2 ∂ ⎞ ⎜ ⎢− ⎟− ⎜r 2 ⎜ ⎢ 2m * r ∂r ⎝ ∂r ⎠ ⎝ ⎣
Gˆ L2 r2
⎤ ⎞ ⎟ + V (rG )⎥ Ψ (rG ) = EΨ (rG ) ⎟ ⎥ ⎠ ⎦
(16)
Gˆ
In (16), L is the orbital angular momentum operator. The solutions to Schrödinger equation (16) with the confining potential (15) have the form [10]:
⎛ α nl r ⎞ ⎟ Υ lm (θ , ϕ ) ⎝ R ⎠
Ψ (r ,θ , ϕ ) = A jl ⎜
(17)
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Biljana Pejova
where A is a normalizing constant, jl is spherical Bessel function, n is a positive integer, while l is the orbital angular momentum quantum number. Coefficients αnl are zeros of the spherical Bessel functions (their labeling with an integer is according to the increasing energy). The energy eigenvalues, corresponding to solutions (17) are given by:
= 2 ⎛ α nl ⎞ ⎜ ⎟ ; 2m * ⎝ R ⎠ 2
E nl =
n = 1,2,3…; l = 0,1,2…
(18)
Obviously, quantum dots have a discrete energy spectrum, just like isolated atomic systems (this property is independent on the actual dot shape). For that purpose, quantum dots are usually described as a sort of “artificial atoms”. Energy levels of a spherical QD can be labeled with an atom-like notation. For example, the state denoted as 1s is characterized by l = 0 and n = 1. The analogy between QDs and isolated atoms is, however, not quite complete. Thus, the degeneracy of levels in not the same in QDs as in atoms, since in the QD case there is no any restriction of the values that the quantum number l could have for a given n. In the present review, we recapitulate the most important results from our studies concerning chemical synthesis, optical absorption and photoelectrical properties of closepacked variable-sized cubic CdSe quantum dots in thin film form. We put a special emphasis on the consequences of the spin-orbit (SO) splitting of the valence band in the title compound and the characteristics of the higher excited electronic states which arise from this effect. To the best of our knowledge, systematic experimental studies of the size-dependence of SO coupling related phenomena in thin films constituted of low-dimensional semiconductor QDs have not been performed. The Section 2 of this review is devoted to the developed synthetic route to CdSe quantum dots in thin film form. In Section 3, details are presented concerning the identification and characterization of the structural properties of the synthesized close packed CdSe QDs in thin film form. In Section 4, optical properties of the synthesized materials are considered in details, including an in-depth analysis of the fundamental and higher-order electronic transitions, with an emphasis on the effects related to SO splitting of the valence band in macrocrystalline analogues and the size quantization effects. In Section 5 the emphasis is put on the studies of charge-carrier transport properties in CdSe QD thin films, including the analysis of temperature-dependent thin film resistivity data. Finally, in Section 6, the photophysical properties and photoconductivity relaxation dynamics phenomena in photoexcited CdSe QD thin films are analyzed.
2. The Chemical Synthetic Route to Nanostructured CdSe in Thin Film Form We have recently developed a novel, original chemical synthetic route to nanostructured cadmium selenide in thin film form [26]. As discussed further in this review paper, our method has certain advantages over other methods which have been presented in the literature for synthesis of this material [27-33]. Firstly, it enables deposition of CdSe quantum dots as both thin films and bulk precipitates that belong to the cubic modification of this semiconductor. As the parallel deposition of the hexagonal modification of CdSe is
Chemically Deposited Thin Films…
119
completely avoided by our synthetic approach, this means that our method enables synthesis of cubic CdSe with high crystallographic purity. This aim seemed to be practically unachievable by other chemical methods which have been previously reported in the literature. Besides that, as discussed later in more details, the method developed by our group also enables presence of Cd(OH)2 in the as-deposited material to be practically completely avoided. This component, if present in significant concentrations in the deposited films, could potentially significantly degrade its photoelectrical properties. Thin films of cadmium selenide quantum dots were deposited onto glass and also on polyester substrates with dimensions of a standard microscope glass. The adhesion with the substrate is of significant importance for the quality of the deposited thin films and it is also of crucial importance for further thermoelectrical and photoelectrical investigations of this semiconductor QDs deposited in thin film form. Therefore, in order to improve the film adhesion, the used substrates were immersed in a diluted solution of tin(II) chloride prior to the deposition process and afterwards thermally treated at 200 oC. As a result of this treatment, small crystals of tin(II) oxide with stochastic distribution are formed onto the substrate surface which, during the deposition process, initiate heterogeneous nucleation [34]. Nanostructured cadmium selenide thin films were prepared by the method of chemical deposition. As precursor of selenide ions, we have used the sodium selenosulfate. For control of cadmium(II) ion concentration and alkalinity of the reaction system an ammonia buffer solution (with pH = 9) was used. The experimental conditions of chemical deposition were optimized (in a classical way) with respect to the thin film’s photoelectrical performances. We found that the pH value of the reaction system is most important for chemical deposition of nanostructured cadmium selenide thin films (i.e. thin films composed of 3D arrays of QDs of this semiconductor) which manifest photoconductivity. According to our experimental results, the photoelectrical properties can be modified by controlling the volume of the used buffer solution, prepared from NH3 and NH4Cl, with c0(NH3) = 1 mol/dm3 and c0(NH4Cl) = 1 mol/dm3. The precursor of selenide ions, sodium selenosulfate, was used in the form of solution, which has been obtained by adding gray selenium to a hot solution of sodium sulfite, stirring this mixture for 1 hour at 90 ºC and filtering the excess of gray selenium. The solution of sodium selenosulfate is relatively unstable and therefore it must be freshly prepared prior to the thin film deposition process. The sulfite ions (due to their reduction properties; E°(SO42-/SO32-) = -0.90 V [35]) present in excess in the selenosulfate solution, are of primary importance for its stability. In fact, selenide ions in the reactor may be relatively easily oxidized to elemental selenium by the oxygen dissolved in the reaction solution [36,37]:
Se2–(aq) + H2O(l) +1/2O2(g) ® Se(s) + 2OH–(aq)
(19)
Occurrence of this process is prevented in the selenosulfate solution by the sulfite ions, due to the reducing properties of these species, which are enhanced in alkaline medium. This follows from analysis of the Nernst equation for the corresponding redox system [35]: SO42–(aq) + H2O(l) +2e ª SO32–(aq) + 2OH–(aq)
D ESO = – 0.90 V 2− /SO 2 − 4
3
(20)
120
Biljana Pejova D − ESO 2 − /SO 2 − = ESO 2− /SO 2 − 4
3
4
3
a(SO 32− ) 0.059V log + 0.059V (14 − pH) 2 a(SO 24− )
(21)
Also, the oxidation properties of the dissolved oxygen become less pronounced in alkaline medium, as implied by the Nernst equation for the corresponding redox pair:
E OD 2 /H 2O = +1.23 V
(22)
0.059V log a (O 2 ) − 0.059V pH 4
(23)
O2(g) + 4H+(aq) + 4e ª 2H2O(aq)
E O 2 /H 2O = E OD 2 /H 2O +
It therefore follows that, from the aspects of stability of selenide ions generator, alkaline medium is very suitable for synthesis of metal selenide thin films. The optimal chemical composition of the reaction system for preparation of photoconductive cadmium selenide QD thin films was obtained by mixing the following solutions: 10 cm3 CdNO3 (c(CdNO3) = 0.1 mol/dm3), 70 cm3 ammonia buffer solution, 15 cm3 Na2SeSO3 (c(Na2SeSO3) = 1 mol/dm3) and distillated water to a total volume of 100 cm3. According to the obtained experimental results, the optimal temperature for chemical deposition of cadmium selenide thin films is 60 oC [26]. As mentioned before, our developed synthetic route to thin films constituted by 3D arrays of cadmium selenide QDs was based on usage of the sodium selenosulfate as a precursor of selenide ions. In principle, the method is based on hydrolysis of selenosulfate ions in alkaline solution of complexed cadmium(II) ions. Several equilibria exist in the reaction system which can be presented in the following way: NH3 + H2O ª NH4+ + OH–
(24)
Cd2+ + 4NH3 ª [Cd(NH3)4]2+
(25)
SeSO32– + H2O ª H2Se + SO42 –
(26)
H2Se + OH– ª H2O + HSe–
(27)
HSe– + OH– ª H2O + Se2–
(28)
Cd2+ + Se2– ª CdSe
(29)
The selenosulfate ions hydrolyze to hydrogenselenide acid which is characterized by acidity constant (Ka) of 1,3·10-15 mol2dm-6 at 25 oC [35]. As a result of the low acidity constant value, the concentration of selenide anions in the reaction system is very small. The concentration of (hydrated) cadmium cations in the solution is determined by thermodynamic stability of tetraamminecadmium ions. The deposition of cadmium selenide starts when the
Chemically Deposited Thin Films…
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product of cadmium and selenide ion equilibrium concentrations is at least equal to the solubility product (Ksp) of cadmium selenide: (30) Cd2+(aq) + Se2– (aq) ª CdSe(aq) ª CdSe(s) Ksp(CdSe) ≤ c(Cd2+)c(Se2–
(31)
In the matter of photoelectrical performances of thin films of cadmium selenide, the control of pH of the reaction system is of prime importance. Namely, the presence of CdO and Cd(OH)2 impurities in relatively large quantities reflects negatively on the photoelectrical properties of cadmium selenide. On the other side, in acidic medium selenide ions oxidize to red selenium. For these reasons, the chemical deposition of nanostructured thin films of cadmium selenide was carried out in the presence of ammonia buffer solution with pH = 9. As it was mentioned before, the ammonia buffer has a double function. It controls the alkalinity of the reaction system and, on the other side, the ammonia forms tetraamminecadmium ions with stability constant of 1,32·107 mol-4dm12 [35] and thus determines the concentration of cadmium ions. To the best of our knowledge, usage of buffer solution with double role in chemical deposition of cadmium selenide (or metal chalcogenides in general) thin films has not been presented up to now in the literature. Having in mind that during the chemical deposition of cadmium selenide thin films, the concentrations of relevant ions in the reaction system should be kept at low values, the small concentration of Cd2+ ions was controlled indirectly using ammonia as a complexing reagent, whereas in the case of selenide ions, it was achived by slow dissociation of hydrogenselenide ions. The thin film deposition involves two processes: process of nucleation and process of crystal growth [38]. The crystal size of synthesized cadmium selenide thin films is closely related with the ratio of rates of these two processes. The rate of nucleation depends on the relative supersaturation of solution [38-40]. Therefore, low relative supersaturation of the reaction solution favors heterogeneous nucleation. By increasing the solution supersaturation, on the other hand, the rate of nucleation increases exponentially and the homogeneous nucleation dominates. In principle, the crystal growth process can be based on two types of mechanisms: ion by ion mechanism and cluster or colloidal mechanism [41]. According to the ion by ion mechanism, the crystal growth is a result of the reaction between cadmium(II) and selenide ions which are present in low concentration in the growth solution: (CdSe)n + Cd2+ + Se2- ª (CdSe)n+1
(32)
(CdSe)n+1+ Cd2+ + Se2- ª (CdSe)n+2
(33)
where n is the number of formula units which form a stable nucleus. On the other side, the low equilibrium ionic concentrations lead to practically only heterogeneous nucleation and absence of precipitation in the growth solution. Cluster mechanism of crystal growth is based on adsorption and coagulation (which involves coalescention and aggregation) of colloidal particles: (CdSe)l + (CdSe)k ª (CdSe)l+k
(34)
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Biljana Pejova
where l,k > n. According to the cluster mechanism, besides heterogeneous nucleation, the homogeneous nucleation also precedes to the crystal growth process. The colloidal particles, formed in the solution, migrate to the substrate surface. Regarding the chemical composition of colloidal particles, they can be cadmium selenide, cadmium hydroxide and hydrated cadmium oxide, which spontaneously convert to cadmium selenide as a result of the lower solubility product of CdSe in comparison to cadmium hydroxide and cadmium oxide. The mechanism of cadmium selenide crystal growth (in thin film form) was investigated using a light scattering experiment [41]. Thus, if the crystal growth process is based on the cluster mechanism, the colloidal particles, which are present in the solution will scatter the used monochromatic light. In the opposite case, the crystal growth is based on ion by ion mechanism. According to the experimental results, the cluster mechanism was dominating in deposition of cadmium selenide thin films in our case.
3. Structural Characterization of Close Packed CdSe QDs in Thin Film Form 3.1. Identification and Estimation of the Average Crystal Size of the Nanostructured CdSe Thin Films Cadmium selenide appears in two polymorphic modifications: cubic and hexagonal [42]. The differences in the crystal structures between the two polymorphs of this semiconductor are, of course, reflected in the differences in the corresponding optoelectrical and photophysical properties, which are relevant to potential applicability of this material in contemporary microelectronics. The cubic modification of CdSe is of sphalerite structural type, while the hexagonal modification is of wurtzite structural type. The phase transition temperature at which the cubic modification transforms to the hexagonal one is in the interval from 350 to 400 ºC [43]. Despite the fact that the overall number of publications in the scientific literature concerning CdSe is rather large, this material in thin form (especially the cubic modification) is one of the least studied semiconductors of the type AIIBVI. It has been found that the structure of synthesized CdSe in thin film form depends critically on the experimental deposition conditions [27-33]. Synthesis of the cubic polymorphic modification of this semiconductor by chemical route has been shown to be a rather difficult task. A proof of this statement is the small number of papers devoted to this problem that have appeared in the literature [27,28]. According to Kainthla and co-workers [29,30], thin films of CdSe synthesized by the chemical deposition method usually contain a mixture of the cubic and hexagonal phases of this material. If the synthesis is carried out in strongly alkaline media, saturated with Cd(OH)2, the hexagonal modification is a predominant one. On the other hand, if the synthesis is carried out at lower pH values, it is the cubic modification that predominates. However, it is a very difficult task to synthesize thin films of cubic CdSe with high crystallographic (phase) purity. To synthesize thin films constituted by the cubic polymorph of CdSe, often more sophisticated techniques have been employed, such as molecular beam epitaxy, electrodeposition, thermal evaporation methods etc [31-33].
Chemically Deposited Thin Films…
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The X-ray powder diffraction method was used for identification of the chemically deposited materials as both thin films and bulk precipitates. X-ray diffraction patterns were recorded on a Philips PW 1710 diffractometer, using monochromatic Cu-Kα radiation. Also, the recorded XRD patterns were employed for estimation of the average crystal size. This was done with two variants of the Debye-Scherrer’s method [9,44]. Within the simple DebyeScherrer approach, considering the deposited crystallites as spheres, their average diameter
d is given by:
d =
4 0.9 ⋅ λ 3 β ⋅ cosθ
(35)
β is the full width at half maximum
where λ is the wavelength of used X-ray radiation,
intensity of the peak and θ is the angle which corresponds to diffraction maximum [44]. Within the more elaborate Debye-Scherrer’s approach, to calculate the average crystal size in the nanostructured CdSe films on the basis of intrinsic broadening of the diffraction maxima, the following working equation was implemented [9,44]:
d =
4 3
0.9 λ
(36)
β − β s2 cosθ 2 m
where βm is the raw value of the full width at half maximum intensity (FWHMI) of the peak (which contains both instrumental and intrinsic broadening factors), while βs is a standard FWHMI value referring to a correspondent peak in the case of macrocrystalline and strainfree Al2O3 sample. Obviously, equation (35) may be easily derived from (36) assuming that βs << βm (i.e. assuming a negligible contribution of instrumental factors to β). In the case of the presently studied CdSe nanostructured films, it was found that neglect of the instrumental broadening in (36) leads to relative errors in the calculated average crystal radii of less than 12 percents. To determine correctly all of the parameters in eq. (36), the diffraction peaks were interpolated with linear combinations of Gaussian and Cauchy (Lorentzian) functions by the least-squares based methodology. Each peak was thus interpolated by a function of the form:
I (2θ ) = γ
[
2 2 1 + 4 (2θ ) Hπ
]
−1
+ (1 − γ )
4 ln 2 H π
[
exp − 4 ln 2 (2θ )
2
]
(37)
where the parameters H and γ were determined by least-squares techniques. Note that the function of the form (37) is frequently called a "pseudo-Voigt" profile in the context of Rietveld analysis. According to the results of X-ray diffraction analysis (Figs. 4 and 5) the thin films and corresponding precipitates of CdSe, obtained under identical experimental conditions, are polycrystalline. The prepared CdSe thin films belong to cubic crystal system (with unit cell parameter of 6.077 Å [45]) and are characterized with sphalerite type of structure. The results
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Biljana Pejova
about the crystal structure of the obtained thin films of cadmium selenide are in correlation with the conclusions of Kainthla and coworkers [29,30]. The broad diffraction peak in the diffractograms of CdSe thin films (which corresponds to 2 θ values of 20 – 40 o) is due to the amorphous structure of the glass substrates. 3500
c
3000
Intensity
2500
b
2000
1500
a 1000
500
0
10
20
30
40
2Θ /
50
60
70
o
Figure 4. The XRD patterns of: a) standard, b) as-deposited thin film constituted of cubic CdSe QDs c) annealed thin film of CdSe (at 300 oC). 3000
2500
c
Intensity
2000
b
1500
1000
a 500
0
10
20
30
40
2Θ /
50 o
60
70
Chemically Deposited Thin Films…
125
Figure 5. The XRD patterns of: a) standard, b) as-deposited precipitate constituted of cubic CdSe QDs c) annealed precipitate of CdSe (at 300 oC).
Using the equation of Debye and Scherrer, on the basis of the full width at half maximum of more intensive diffraction peaks, the average crystal sizes of unannealed and annealed (at 300 oC in air atmosphere) cadmium selenide (in form of thin film and corresponding precipitate) were estimated. The average crystal diameter of unannealed cadmium selenide (thin film and precipitate) is 5.3 nm. In the case of annealed cadmium selenide at 300 oC, as thin film and precipitate as well, the average crystal diameter of 24 nm was estimated. The identical values of crystal sizes of CdSe thin films and of the corresponding precipitates give a further support for the process of crystal growth according to the cluster mechanism [46].
4. Electronic Transitions and Optical Properties of the Synthesized Thin Films Composed by 3D Arrays of CdSe Quantum Dots 4.1. Band Structure Considerations for Macrocrystalline Cubic CdSe Since the focus of this review is put on the cubic phase of CdSe, in order to be able to understand the optical properties of synthesized 3D arrays of cubic CdSe QDs more thoroughly, the basic concepts concerning the band structure of macrocrystalline cubic CdSe will be firstly reviewed. Later on, the effect of low-dimensionality of the samples on the electronic transitions will be considered. As it was mentioned in the Introduction section of this review, the band structure of semiconducting materials is a function of a three-
G
dimensional wave vector ( k ) within the Brillouin zone which depends on the crystal structure and corresponds to the Wigner-Zeits-type primitive unit cell of the reciprocal lattice [2,47]. First Brillouin zone in the case of a cubic semiconductor with sphalerite (zinc-blende) type of structure is schematically presented in Fig. 6. In this figure, also the notation of more characteristic points and lines in the reciprocal (Fourier) space is presented. Actually, the band structure of crystals with cubic zinc-blende (sphalerite) lattice is very similar to that of the crystals with diamond structural type [13,48,49]. In the case of zinc-blende semiconductors, the HOMO bands originate primarily from the anionic p-orbitals, while LUMO bands originate from the cationic s-orbitals. In semiconductors of this simple structural type, the direct gap is located at the center of
G
the first Brillouin zone (i.e. k = {0,0,0} - the Γ point of the first Brillouin zone). The
conduction band is nondegenerate (neglecting the spin) and nearly isotropic, while the valence band has a three-fold degenerate maximum. Inclusion of spin, however, affects significantly the band structure in qualitative as well as in quantitative sense. In fact, even in the case of structurally simple semiconductors, such as sphalerite polymorphs of CdSe (and e.g. ZnSe), valence bands often show remarkable complexities [13,48,50]. In the case of mentioned semiconductor compounds, these complexities in hole dispersion relations are mainly due to the fact that the valence band originates from anionic porbitals. This band is therefore six-fold degenerate (counting the spin). The effects of coupling of the spin and orbital angular momentum are here very significant, as discussed in
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more quantitative details below in this review. At this point, having in mind the previously
Figure 6. The first Brillouin zone in a cubic semiconductor with sphalerite (zinc-blende) type of structure, along with the notation of more characteristic points and lines in the reciprocal (Fourier) space.
outlined arguments, it should be pointed out that the purely quantum property of chargecarriers – spin, enters the semiconductor physics as a substantially important “new” variable. The spin physics of semiconductors is, actually, a rapidly expanding field of contemporary science [51-55]. A particularly explosive development has been noted in relation to spinrelated optical and transport properties in low-dimensional semiconductor structures. The spin-orbit (SO) interaction has attracted considerable attention for a number of reasons, due to which it is worth studying in details. First of all, it enables optical spin orientation and detection. It could exhibit significant influence on the mesoscopic transport phenomena and quantum Hall effect. Besides that, in most cases the SO coupling is responsible for spin relaxation phenomena, and in addition, it introduces an interdependency between the transport and spin phenomena. A whole new field of microelectronics, usually called spinelectronics or spintronics has been developed and it has already offered unique opportunities for construction of a new generation of multifunctional devices. Such devices would be based on addition of the spin degree of freedom to the conventional (charge-based) microelectronics. Aside from the relevance of the SO coupling – related phenomena from a purely applicative point of view, research in this field is also of certain fundamental significance. Despite the quantitative disagreements between theory and experiments in certain cases, the basic physics which governs the size-quantization effects on optical properties of low-dimensional semiconductor nanostructures seems to be rather well understood in a qualitative sense. On the other hand, the mechanisms governing the SO coupling as well as its strength in mentioned systems has not been understood in details yet and it is therefore still a very active area of research. In case of compounds constituted only by light elements the electronic spin (S) and angular momentum (L) are both described by good quantum numbers, as the magnetic field generated by orbiting electron is too weak. Therefore, it does not induce any significant coupling between the electron spin (i.e. its intrinsic angular momentum) and its orbital angular momentum. In semiconductor compounds containing heavier elements, on the other hand, the nearly relativistic electronic velocities lead to sufficiently large magnetic fields. In such cases, L and S are coupled and give a total angular momentum J = L + S. Now, J is the
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quantity which is described by good quantum number (in contrast to L and S separately, as in the previous example). Generally speaking, the most important effect of the spin-orbit interaction is the inducement of a coupling of the electron dynamics in ordinary and spin spaces. This effect, on the other hand, reduces the overall symmetry of the Hamiltonian. In semiconductors of zinc-blende structural type, the spin-orbit interaction splits the valence (HOMO) band, which is six-fold degenerate counting the spin, into an upper and lower component (Fig. 7). As can be seen from Fig. 7, the upper (Γ8) component is four-fold degenerate. It is characterized by the following combinations of quantum numbers characterizing the overall angular momentum and its projection on the z-axis: (J, MJ) = (3/2, ±3/2) and (3/2, ±1/2). The lower (Γ7) component is two-fold degenerate, characterized by (J, MJ) = (1/2, ±1/2).
G Ek
()
G k
hh lh
so
J = 3/2
Γ8v
ΔSO J = 1/2
Γ 7v
Figure 7. The valence band in semiconductors of zinc-blende structural type. ΔSO – spin-orbit splitting energy, hh – heavy hole band, lh – light hole band, so – split-off band, J = L + S.
4.2. The Influence of Size Quantization. A Simple Picture In this review Chapter, a concise overview of the available results in the literature published by our and other research groups will be made, concerning the evolution of electronic transitions (not just the fundamental interband transition, but also the higher order ones) upon increase of the average crystal size from strongly quantized to practically non-quantized case of CdSe QD thin films. To be able to understand the experimentally observed trends, as well as to assign correctly the experimentally detected electronic transitions in the nanostructured films, it is important to have an idea about the influence of size quantization on the spin-orbit split valence band in semiconductor compounds with zinc-blende structural type. This issue was actually analyzed some time ago for semiconductor slabs [56,57] and in a more general sense [49,58-60]. It was concluded that the split-off valence band component Γ7 upon quantization gives rise to one series of separate hole states, while the degenerate Γ8 component is split and quantized due to symmetry lowering (Fig. 8). Two series of quantized hole states are therefore expected to arise from the valence band components upon
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nanocrystal size decrease, which may be labeled as “heavy holes” and “light holes”. The effective masses of the holes are given by:
mh =
m0 γ 1 − 2γ 2
(38)
ml =
m0 γ 1 + 2γ 2
(39)
In (38) and (39), m0 is the free electron mass while γ1 and γ2 are the Luttinger parameters which characterize the band shape [13,48,50]. The outlined results are in line with those obtained by Luttinger and Kohn for bulk specimen [61]. They have originally introduced three parameters γ1, γ2 and γ3 besides the spin-orbit splitting energy to describe the corresponding bulk band structure. Description of the valence band in the manner of Luttinger and Kohn leads to a small warping of this band which is proportional to (γ2 – γ3). This small anisotropy is, however, usually neglected. In fact, setting γ = γ2 = γ3, the usual description of the heavy-hole and light-hole band is obtained (four-fold degenerate at Γ point, including spin – with the effective masses given by (38) and (39)) and a split-off band with an (isotropic) effective mass given by:
m s = mi = E
m0
γ1
(40)
1D 1P 1S
bulk conduction band bulk band gap HOMO-LUMO gap 0.0
bulk valence band 1S 1P 1SΔ
1D
1PΔ 1DΔ
Figure 8. The discrete hole and electron states arising from valence and conduction bands in the case of semiconductor nanoclusters of zinc-blende structural type according to the simplest model that does not account for the mixing between hole states (according to Ref. [58]).
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We will later on point out in this review that such a simple picture of size quantization influence on the hole and electron states does not allow for a proper interpretation of the experimental optical spectra of nanostructured CdSe thin films. However, this simple physical picture enables a basis for a better in-depth understanding of the more complex approaches which will be used later.
4.3. Experimental Electronic Spectra of Nanostructured CdSe Thin Films. The Fundamental “Band-to-Band” Electronic Transitions The optical spectra of the thin films constituted by close packed cadmium selenide QDs were recorded in the spectral range from 190 to 1100 nm. On the basis of the experimental data for the spectral dependence of transmission coefficient, the absorption coefficient (for each wavelength) was calculated using the following equation:
α (λ ) =
1 I 0 (λ ) ln d I (λ )
(41)
where d is the thickness of the investigated thin film, while I0 and I are the intensity of incident and transmitted light at a given incident photon wavelength λ, correspondingly. The ratio I / I0 defines the transmission coefficient, i.e. the transmittance for each given wavelength. The spectral dependence of absorption coefficient for nanostructured as-deposited cadmium selenide thin film is presented on Fig. 9. As can be seen, the magnitude of absorption coefficient in the investigated spectral range is of the order of 104 cm-1 or even higher. This fact clearly indicates that the explored spectral region is the region of intrinsic absorption of cadmium selenide and the corresponding electronic transitions are of direct dipole-allowed type. The direct type of electronic transitions implies that identical electronic wave vector corresponds to the absolute minimum of the conduction band and the absolute
G
maximum of valence band in the k – space and the electronic transitions do not involve phonons. Values of absorption coefficient larger than 104 cm-1 correspond to electronic transitions from valence band to higher energy levels of the conduction band. According to the literature data, the excited electrons in this way are thermalized to the lower edge of conduction band within 10-12 to 10-13 s [3]. Our experimental data have unequivocally revealed that the optical properties of nanostructured cadmium selenide thin films change irreversibly upon thermal treatment. The most prominent visual indication of such changes is the observed irreversible change of color from yellowish-orange to brown induced upon thermal treatment of the films. Dependence of absorption coefficient vs. photon energy in the case of annealed CdSe thin film (in air atmosphere at 300 oC) is presented on Fig. 10. It can be clearly seen from this figure that upon thermal treatment the absorption edge shifts towards lower photon energies (i.e. undergoes a “red shift”).
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20 18 16 -1
12
4
α /(10 cm )
14
10 8 6 4 2 0 1.5
1.7
1.9
2.1
2.3
2.5
2.7
2.9
3.1
3.3
3.5
3.7
hv /eV Figure 9. The dependence of absorption coefficient of as-deposited CdSe QD thin film on incident photon energy.
20 18
4
-1
α /(10 cm )
16 14 12 10 8 6 4 2 0 1.2
1.4
1.6
1.8
2
2.2
2.4
2.6
2.8
3
3.2
3.4
3.6
3.8
hv /eV Figure 10. The dependence of absorption coefficient of annealed CdSe QD thin film on incident photon energy.
By a thorough analysis of the experimental optical spectroscopy data, first of all, we have calculated the optical band gap energy corresponding to the lowest-energy “band-to-band” electronic transitions in the case of nanostructured cadmium selenide thin films. This was
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done within the framework of parabolic approximation for dispersion relation, on the basis of equations which arise from the Fermi’s golden rule for electronic transitions from the valence to conduction band. Namely, the semiconductor’s band structure determines the functional dependence of the absorption coefficient (α) on the photon energy. The Fermi’s golden rule for fundamental “band to band” electronic transitions could be formulated in a general case with the following expression relating the experimental optical spectroscopy data to the band structure parameters:
αhv = C ∑ Pvc g ( E )δ ( E c − E v − hv) G 2
(42)
k
2
In the last equation, Pvc is the transition dipole moment, g(E) is the density of energy levels (density of states), while hv is the incident photon energy. Ec and Ev, on the other hand, correspond to the minimum of conduction band and the maximum of valence band correspondingly [16]. Further mathematical elaborations based on (42) lead to the following equation which is the basis of optical band gap energy calculations:
(αhv) n = A(hv − E g )
(43)
(see, for example Reference [62]). The index n in (43) depends on the type of electronic transitions. In the case of direct dipole-allowed and dipole-forbidden transitions the values of n are 2 and 2/3 correspondingly, whereas for indirect type of transitions (allowed and forbidden) n has a value of 1/2 and 1/3 correspondingly. Experimental optical absorption spectra of the nanostructured CdSe thin films were mathematically transformed according to the expressions arising from Fermi’s golden role. The constructed dependences of the type αhv = f(hv) were fitted according to the equation (43). Linear dependences of (αhv)n vs. hv were obtained only in the cases when n = 2. This finding is in agreement with the previously derived conclusions, based on the magnitude of the absorption coefficients, that near the absorption edge the electronic transitions are direct (dipole-allowed) and do not involve phonons. A typical dependence of (αhv)2 vs. hv in the case of as-deposited nanostructured CdSe thin film is presented on Fig. 11. The deviation from the linearity in the dependence of (αhv)2 vs. hv in the lower-energy region is a consequence of deviations from parabolic approximation for the dispersion relation. By linear correlation analysis of experimental data within the region of linear dependence of (αhv)2 vs. hv the optical band gap energy was calculated. As explained in details in our published papers [26,34,62-74], this analysis was carefully carried out by successively including or eliminating of a number of neighboring points in the correlation ranges and parallel monitoring the R2 value. After determination of the sets of relevant points which belong to the linear (αhv)2 vs. hv dependence, we have extrapolated (αhv)2 vs. hv dependences to αhv = 0 and calculated the corresponding transition energies on the basis of previously derived correlation equations. According to this analysis the optical band gap energy of as-deposited nanostructured thin films of cadmium selenide (at room temperature) is 2.08 eV.
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The band gap value of as-deposited nanostructured cadmium selenide thin films is higher by 0.34 eV in comparison with the value corresponding to a bulk specimen of this semiconductor compound (1.74 eV [75]).
Figure 11. The dependence of (αhv)2 on hv in the case of as-deposited CdSe QDs in thin film form.
This notable blue-shift of Eg is due to the pronounced three-dimensional charge carrier confinement effects (so called quantum size effects), which is discussed in somewhat more details in the next section of this review. At this point, we just mention that all of these conclusions are in correlation with the results of X-ray diffraction analysis. On the basis of the constructed (αhv)2 vs. hv dependence, implementing the described linear correlation analysis, we have calculated the optical band gap energy value in the case of annealed cadmium selenide, which is 1.77 eV at room temperature (Fig. 12). The last value has practically converged to the previously mentioned bulk value for this semiconductor of 1.74 eV [75].
Figure 12. The dependence of (αhv)2 on hv in the case of annealed CdSe QDs in thin film form.
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The pronounced red shift of the absorption onset upon thermal annealing of the films, besides on the basis of analysis of the semiconductor absorption functions, is evident also upon comparison of the optical absorption spectra (Fig 13).
12
4 -1 α /(10 cm )
10 8 6
b
4
a
2 0 1.2
1.5
1.8
2.1
2.4
2.7
3.0
hv /eV Figure 13. The spectral dependence of the absorption coefficient for as-deposited (a) and thermally treated CdSe quantum dot thin films at 300 ºC (b).
4.4. Size Evolution of the Fundamental “Interband” Electronic Transitions: Analysis of the Three-Dimensional Charge Carrier Confinement Effects (Quantum Size Effects) As it has been already mentioned before, the strongly blue-shifted absorption onset of the asdeposited nanostructured CdSe thin films (compared to a bulk specimen of the same material), along with the red shift induced upon thermal treatment (which is accompanied by average crystal size decrease), are strong indications of the so-called size quantization effects. These effects occur due to the confined charge carriers’ motions in three spatial dimensions in the case of a quantum dot. It is certainly beyond the scope of the present review to make a thorough retrospective of the existing literature models aiming to offer a clear physical picture of these effects, and also to allow for quantitative predictions of the size evolution of the absorption onsets in size-quantized semiconductors. Besides the existing rather famous excellent review papers written by the most eminent experts in this area, a modest contribution to the subject has been reviewed in somewhat more details in a previous publication of the author of this review [62]. For the purpose of this review, we will focus on the analysis of size quantization effects in the studied nanostructured CdSe thin films by one of the most-quoted literature models – the so-called Brus’ model, also known as the effective mass approximation model [76-79]. The 3D charge carriers’ quantum confinement effects should become observable if the quantum dot radius becomes less than (or comparable to) the Bohr excitonic radius rB for the
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corresponding material. This extraordinarily important quantity in the physics and chemistry of low-dimensional systems is given by the following relation [76-79]:
rB =
4πε 0 ε r = 2 e 2 m0
⎛ 1 1 ⎞ ⎜ * + *⎟ ⎜m ⎟ ⎝ e mh ⎠
(44)
In (44), = = h /( 2π ) − h being the Planck’s constant, m0 is the electron mass, while me* and mh* are the electron and hole relative effective masses, respectively (i.e. the effective masses expressed in units of electron mass m0), e is the electron charge, ε0 the permitivity of vacuum and εr is the relative dielectric constant of the semiconductor. It has been generally proven that the size of a bulk exciton (i.e. the excitonic radius of a bulk semiconductor crystal) is a natural intrinsic measure of linear dimensions for which crystal (quantum) size effects will predominate and affect to a significant degree the electrical and optical properties of the semiconductor in question. Depending on the exact ratio between the quantum dot (nanocrystal) size (R) and Bohr’s radius, several size quantization regimes could be defined. In case when R >> rB, one speaks of the weak confinement regime [4,5]. Within this regime, while the character of the exciton as a quasiparticle is preserved, its translational degrees of freedom are modified because of the size quantization effects, leading to just a slight increase of the exciton energy. When R << rB, the strong confinement regime is entered, within which both the electrons and holes are separately quantized. The spatial correlation between the charge carriers is very small, as well as the Coulomb interaction between them. Between these two extreme cases is the regime of medium confinement. The estimated literature value for the excitonic Bohr’s radius in the case of macrocrystalline (bulk) CdSe is 5.6 nm. As it has been outlined in more details in Section 3 of this review, the average crystal radius in the case of unannealed (as-deposited) CdSe thin films is 2.65 nm. Upon post-deposition thermal annealing treatment, this value increases to 12.0 nm. Note once again that the post-deposition annealing of the films is accompanied only by average crystal size increase, and not by any changes in the chemical composition of the films. Obviously, therefore, our newly developed synthetic method for deposition of nanostructured CdSe QD thin films allows a specifically fine tuning of the semiconductor properties. An especially good advantage of the method is that it enables synthesis of CdSe QDs in thin film form with an average crystal radius of the as-deposited material which is lower than rB. The post-deposition annealing, on the other hand, induces coalescence and crystal growth, so that the final average crystal size after the annealing is more than twice larger than rB, so that a switch between two different confinement regimes is allowed. The famous model of Brus is essentially based on the concepts of charge carriers’ effective masses, more specifically on the particle in a box model. It considers the confined motion of charge carriers in a spherically symmetrical potential. According to this model, the band gap energy of a quantum confined nanocrystal is blue-shifted in comparison to the bulk specimen by a value given by the following expression (the Brus formula) [76-79]:
Chemically Deposited Thin Films…
ΔE g (R ) =
h2 8m0 R 2
⎛ 1 4π 2 e 4 m0 1 ⎞ 1.8e 2 ⎜⎜ * + * ⎟⎟ − − 0.248 ⎛ 1 1 ⎞ 2 ⎝ me mh ⎠ 4πε 0 ε r R 2(4πε 0 ε r ) h 2 ⎜⎜ * + * ⎟⎟ ⎝ me m h ⎠
135
(45)
In (45), R is the QD radius, while all other quantities have their usual meanings. Asides from its simplicity, the main reason for the extensive usage of the Brus’ model in analyzing the quantum confinement effects is the clear physical picture that it provides for an explanation of the phenomena in questions, which are rather complex. For example, it is relatively easy to assign an exact physical meaning to each of the terms on the right-hand side of equation (45). The first term is called the quantum localization term and arises solely due to the kinetic energy of the charge carriers. This term shifts Eg to higher energies proportionally to R-2. The second term on the right-hand side of (45) is, as may be intuitively expected, due to the Coulomb interaction energy between electron and hole (screened by the value of dielectric constant for the investigated semiconductor). This term induces a shift of Eg(R) to lower energies as R-1. Finally, the third term in (45), which is independent on the QD size, is due to spatial correlation effects. It has been often found that this term is negligibly small in comparison to the other two. This is so in the sense of its absolute value and also in the sense of its contribution to the overall ΔEg(R). For that purpose, therefore, this term has often been either neglected or simply omitted from the calculations in the literature. However, one has to keep in mind that for some semiconductor samples (especially those with a small value of εr), it may become significant in both senses implied before, especially with respect to its contribution to the overall band gap blue shift ΔEg(R) under conditions when the opposite trends induced by the quantum localization and Coulomb terms tend to cancel. For the purpose of analyzing the experimental results in the case of our chemically deposited nanostructured CdSe films, we have calculated the predicted band gap energy blue shifts by the equation (45) for both as-deposited an annealed films. In the case of as-deposited films, the experimental band gap energy blue shift (ΔE(R) = Eg(R) – Eg,bulk) is 0.34 eV. The effective mass approximation model of Brus, on the other hand, predicts a value of 0.53 eV for ΔE(R), on the basis of available experimental data for parameters appearing in (45), characteristic for CdSe (for R = 2.65 nm). Upon thermal annealing of the films (R = 12.0 nm), the experimental ΔE(R) value is 0.03 eV. In this case, the model of Brus predicts a value which is in excellent agreement with the experimental one (exactly 0.03 eV). It could be therefore concluded that the model of Brus seems to give very good agreement with the experimental data in the case of CdSe QDs in the region of R values spanning the weak confinement regime, while the agreement is significantly poorer when the regime of stronger confinement is entered. As discussed before by our group [62,67], the general reasons for the inadequacy of the Brus model to quantitatively predict the ΔEg(R) values could be grouped in several categories: (i) the breakdown of the effective mass approximation upon crystal size decrease (i.e. the apparent non-parabolicity of energy bands), (ii) the implied possibility for a reduced dielectric screening in small-sized semiconductor nanocrystals reflecting in a smaller value of εr (arising due to the inability of the lattice polarization to follow the more rapid electron and hole motions associated with a smaller crystal radius) and, last but not least, (iii) the existence of a finite barrier height at the nanocrystal boundary (allowing charge carrier leakage outside the quantum dot). For example, in the case of nanostructured ZnSe QD thin
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films we have found that the most possible reasons for the disagreement between the Brus’ model and the experimental data for this particular semiconductor are the differences in the εr values between the QD solid and the macrocrystalline one, and also the finiteness of barrier heights at the nanocrystal boundaries [67]. In the presently studied case of CdSe QD thin films, however, we attribute the predicted too high values of ΔE(R) for as-deposited films to the breakdown of the effective mass concept for ultrasmall nanocrystals. In fact, the used values for charge carriers’ effective masses were those characteristic of a macrocrystalline CdSe. Actually, the applicability of the concept of effective mass to a charge carrier which is confined within a QD with radius of 2-3 nanometers is questionable. Upon significant enlargement of the QD radii upon annealing (from 2.65 to 12.0 nm), the agreement between theory and experiments becomes excellent.
4.5. A Simple Explanation of Electronic Transitions Accounting for the SpinOrbit Splitting of the Valence Band Besides the fundamental band-to-band electronic transitions in size-quantized CdSe QD thin films, which were discussed up to this point in the current review, even a visual inspection of the thin films’ optical absorption spectra enables detection of additional inflection points in the spectral dependencies of α. This certainly implies the possibility to detect also some of the higher-order transitions in the films. The higher excited electronic states in ZnSe and CdSe (as well as in ZnS) nanoclusters, with an emphasis on the SO coupling were studied by Chestnoy, Hull and Brus [80]. However, these authors have focused their attention on colloidal systems of the title semiconducting materials, synthesized by arrested precipitation colloidal technique. Velumani et al. have studied the optical properties of hot wall deposited CdSe thin films obtained under various experimental conditions, and in this context they also presented the energies of the higher-order electronic transitions due to SO splitting of the valence band in this semiconductor [81]. The main emphasis in this paper was, however, put on the influence of deposition conditions on the optical transitions, because the average crystal diameters were all close to 30 nm and size quantization effects were not exhibited by the studied films. Nĕmec et al. [82] have utilized light-controlled chemical deposition technique for synthesis of nanocrystalline CdSe thin films. On the basis of optical absorption and photoluminescence data these authors have calculated the spin-orbit splitting energy in series of thin films with varying average crystal size deposited under different experimental conditions. Series of profound works by Norris, Bawendi [83-86], as well as Alivisatos, Brus [87], Efros [88], Xia [89] and Hodes [90,91] have been devoted to correct assignments of the higher energy transitions detected in the photoluminescence and optical spectra of CdSe quantum dots in various size regimes (from strong to weak confinement). On the other hand, the groups of Artemyev and Woggon [21-23,92-96] have devoted particular attention to the phenomena which arise in ensembles of close-packed QD solids, specifically addressing the semiconducting CdSe QDs. The question of excitons in CdSe quantum dots has been theoretically addressed by Laheld and Einevoll [97]. The relation of interdot interactions to optical properties of CdSe nanocrystal arrays has been investigated in series of profound works [24,98,99]. A particularly interesting study related to excitonic levels of CdSe QDs embedded in an amorphous GeS2 thin film matrix has been published by Raptis, Nesheva et
Chemically Deposited Thin Films…
137
al. [100]. These authors have observed resonant Raman effects which have been related to resonant light absorption in several excitonic transitions of the CdSe QDs. To characterize in a quantitative manner such higher-order transitions, the semiconductor absorption functions arising from the Fermi’s golden rule were constructed for the studied direct band gap semiconductor. The plots of (αhv)2 vs. hv dependencies for the as-deposited and thermally-treated CdSe QD thin films in a wider energy range are given in Figs. 14 and 15. Obviously, two linear trends could be clearly observed in both cases, the extrapolations of which to αhv = 0 gives energies of 2.08 and 2.33 eV in the case of as-deposited, and 1.77 and 2.01 eV for annealed films.
Figure 14. The plot of (αhν)2 versus hν in a wider energy range for as-deposited CdSe quantum dot thin film, together with the extrapolations of linear segments to αhν = 0.
Figure 15. The plot of (αhν)2 versus hν in a wider energy range for thermally treated CdSe quantum dot thin film (at 300 ºC), together with the corresponding extrapolations of linear segments to αhν = 0.
While the first two energy values for both cases correspond to the band gap energies of as-deposited and annealed films (already discussed before in this review), it is the interpretation of the second transitions that we pay special attention to in this context.
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To be able to make a correct interpretation of all experimentally detected electronic transitions (not just the lowest energy ones) in the optical spectra of nanostructured CdSe films, we first consider the simplest model of semiconductor cluster electronic structure, following the works of Brus et al. [58,80,87]. Within the model proposed by these authors, the quantum-confined states for electron-hole pair can be calculated employing the effective mass approximation. Simply speaking, this means that the bulk solid energy bands are transformed into discrete sets of cluster molecular orbitals, which are well separated. Such energy levels, corresponding to the resulting particle-in-a-sphere wavefunctions (if a spherical shape of the semiconductor cluster is assumed), are labeled according to their symmetry. The conduction band of bulk CdSe is nondegenerate (ignoring spin) and almost isotropic near its
G
absolute minimum at k = {0,0,0}. This means that the corresponding effective mass is a
scalar, and the lowest few discrete particle-in-a-sphere levels are characterized by radial (N) and angular (L) quantum numbers. One should be aware in this context that there isn’t an apparent analogy of the present case of spherical nanometer-sized semiconductor clusters with the hydrogen atom situation. This is so since the potential in this case is not Coulombic. The situation gets much more complicated when one wishes to consider the energy levels near the absolute maximum of the valence band (similarly as in the case of the bulk semiconductor). Basically, it could be stated that the spin-orbit split valence band transforms
G
upon size-quantization into two sets of occupied states (Fig. 8). Except at k = {0,0,0}, these
bands can be described by a 6×6 tensor Hamiltonian, which, as shown by Brus et al. following the approach by Baldereshi and Lipari [50,80], in a spherical harmonic basis may be written in the form:
Hˆ h = Hˆ S + Hˆ D
(46)
Hˆ S is a diagonal matrix containing S-type hole momentum operators and the spin-orbit splitting energy Δ (see the matrix representation of this operator given in references [50,80]). In this operator, only the isotropic hole mass defined before by (40) is actually included, while the d-like operators are contained in Hˆ D . Of course, the last term in (46) is nondiagonal. Ignoring this term in (46) leads to appearance of two series of discrete hole states, which are offset by a spin-orbit splitting energy Δ - Fig. 8. The spin, N, L, and the total angular momentum J (J = L + S) are individually all good quantum numbers. Under such conditions, the single isotropic hole mass governs the dependence of the positions of the mentioned levels upon particle size change. According to this discussion, the lowest-lying occupied hole states are the 1S and 1SΔ levels (i.e. the Γ8 1S and the Γ7 1S level). If one assumes absence of any mixing between these levels, both would be expected to remain relatively pure, regardless on the semiconductor nanocrystal size. When the particle size decreases, both of these states will therefore shift together (simultaneously) to higher energy via the isotropic hole mass. The lowest “band to band” transitions in the very small nanoclusters, as well as in the bulk-like clusters will thus correspond to 1S → 1S and 1SΔ → 1S. The discussed two transitions should, therefore, be split by a value Δ. The last quantity, on
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the other hand, is expected to be equal (or, at least, very similar) to the value of the spin-orbit splitting energy characteristic for the corresponding bulk material. Analyzing the semiconductor absorption functions, constructed on the basis of the UVVIS spectra for the films constituted by 3D arrays of variable-size CdSe quantum dots, we have indeed clearly observed two lowest-lying electronic transitions. In the case of asdeposited nanostructured CdSe films, these two transitions occur at 2.08 and 2.33 eV, while upon thermal treatment both exhibit significant red shifts to 1.77 and 2.01 eV. On the basis of previously outlined arguments, these two transitions may be assigned as the 1S → 1S and 1SΔ → 1S ones (denoted by arrows in Fig. 8). Although there is an overall red shift of both transition energies upon average crystal size increase, the energy difference between them remains unchanged. An overall general conclusion could therefore be derived that both energetically lowest-lying electronic transitions in the case of CdSe quantum dots in thin film form are blue-shifted with respect to the expected transitions in the case of bulk material, but the splitting between them remains practically unchanged (0.25 eV in the as-deposited films and 0.24 eV in the annealed ones). According to the elaborated simple model, the observed splitting could be attributed solely due to the spin-orbit interaction (i.e. it corresponds to the spin-orbit splitting energy). The last quantity is, thus, independent on the average nanocrystal size. If the assignment outlined before was correct, this energy would correspond to the spinorbit splitting energy of the valence band in CdSe. The bulk (macrocrystalline) value for this quantity, on the other hand, is expected to be similar to that in the case of ZnSe (about 0.40 eV [80]). This is so since the spin-orbit splitting of the valence band is a phenomenon governed solely by the selenide anion, i.e. it is in a sense of an “atomic” and not crystalline nature. It is worth noting in this context, however, that our experimental data for the energy splitting between the first two “band-to-band” transitions in the case of CdSe QD thin films are in very good agreement with other recently published data in the literature [18,19,2123,25,81-83,85,94,95]. The current trends are very similar to the situation encountered in the case of thin films constituted of close-packed ZnSe quantum dots, studied also by our [66,67] and other groups. In the last case, however, the energy difference between the first two electronic transitions is almost twice larger than in CdSe. It could be therefore concluded that although the outlined simple reasoning explains quite nicely the independence of the Δ value on the average nanocrystal size, the previous seeming inconsistency of Δ for the two studied semiconductors clearly shows that it is impossible to rely solely on such simple theory if one wants to make a more profound analysis of the optical spectroscopic data for QD thin films. Let us also mention at this point that, up to now, we haven’t discussed one extremely important issue concerning the overall appearance of the optical absorption spectra of the studied QD thin films. If we consider electronic transitions between molecular-like energy levels in QDs, then the absorption spectra should contain absorption bands. Instead, in the present case, as can be seen from Figs. 9 and 10, the spectral appearances closely resemble the case of bulk-like materials (except for the blue-shifts of optical absorption onsets), i.e. they are essentially structureless. This observation will be explained in the following section.
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4.6. The More Profound Physical Picture of Electronic Transitions, Accounting for the Hole Energy Levels Mixing To point out at the weak points of the simple model aiming to explain the experimental observations, outlined in the previous section, we consider in more details some aspects concerning a more rigorous approach to the size-quantization influence on the electron and hole energy spectra in ultrasmall quantum dots. If the QD size enters the strong confinement regime of electrons and holes, the confinement energy is much larger than the energy of Coulomb interaction between the charge carriers. Therefore, these particles may be treated independently. In other words, in the strong confinement regime, the electron and hole wavefunctions may be represented by a product of unit cell basis function and an envelope function (which satisfies the spherical boundary conditions). It is physically obvious to assume that the unit-cell component is identical to the case of a bulk material, and to focus the actual effort in explanation of the QD energy level structure on determination of the envelope function [83,85,88,89,95]. Assuming a simple two-band isotropic effective mass model as an approximation to the bulk valence and conduction bands, and a confinement of charge carrier motions by an infinite potential barrier at QD boundaries, the electrons and holes may be described by “particle in a sphere” envelope functions [83,85,88,89,95]. These envelope functions are labeled by the radial quantum numbers (Nh i.e. Ne) and the angular momenta (Lh i.e. Le). The total QD wavefunction is, therefore, a simple product between the individual electron and hole components. For example, the first excited state may be denoted as (1Sh, 1Se) with both the hole and electron in their S-like envelope functions. Both radial quantum numbers, in such case, are equal to 1. However, accounting for the complexities in the band structures of the studied zinc-blende semiconductor, as described by the Luttinger-Kohn approach, the following additional important conclusions could be derived. When the Luttinger-Kohn Hamiltonian is combined with a spherical potential in the spherical band approximation, certain mixing between bulk valence bands occurs. Such mixing of states becomes particularly important when the particle size decreases (i.e. going from bulk materials to QDs). Accounting explicitly for this energy level mixing phenomena, one arrives at the following results. In this case, the only good quantum numbers for the envelope wavefunction are the parity and total hole angular momentum F = Lh + J. J is the Bloch bandedge angular momentum (3/2 for the light and heavy hole bands and 1/2 for the split-off band), while L is the envelope angular momentum. The hole states within a QD contain certain contributions from spherical harmonics described by Lh and Lh + 2. This observation is usually referred to in the literature as the “S-D mixing”. Also, a given hole QD energy level contains certain contributions from the heavy-hole, light-hole and split-off valence subbands too. Within this advanced treatment, the QD hole states are labeled as NhLF, where LF denotes the combination of L and L + 2 spherical harmonics having a total angular momentum F. The electronic envelope functions, on the other hand, are not affected by the valence band complexities. They are still labeled as NeLe. Within this approach, the first excited state in a QD nanocrystal is denoted as (1S3/2, 1Se). Due to the previously mentioned mixing between hole states, this level contains contributions from the following three hole components: (F = 3/2, J = 3/2, Lh = 0), (F = 3/2, J = 3/2, Lh = 2) and (F = 3/2, J = 1/2, Lh = 2). Advanced quantitative theoretical treatments of the mentioned aspects have been published in the series of works of Bawendi et al. [18,19,83-86]. In their papers, even the nonparabolicity of the bands as well as the finiteness of the barrier height at QD edge were taken into account. These
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treatments are in fact analogous to the early works of Brus et al. [80] concerning ZnSe QDs in which the Hˆ D term in (46), causing certain mixing of the QD energy levels, has been taken explicitly into account. Having in mind the previous discussion concerning the actual existence of a series of states split from the first excited state (1S3/2, 1Se) by the influence of spin-orbit coupling and size quantization effects on the band structure of zinc-blende type CdSe semiconductor, on the basis of quantitative results outlined in Refs. [83-97], we make the following assignments of the observed electronic transitions in our case. The lowest-energy electronic transition in quantized films could be attributed, without any doubt, to the ground state - (1S3/2, 1Se) transition in the nanocrystals. This transition is strongly blue-shifted with respect to the bulk value, which is, as discussed in the previous section, due to three-dimensional quantum confinement effects in CdSe QD thin films. A further confirmation of the last statement is the fact that upon particle size increase due to high-temperature annealing (which is not accompanied by changes in chemical composition of the QD films) the ground state - (1S3/2, 1Se) transition energy exhibits a continual red shift. It finally converges to the bulk band gap energy corresponding to the Γ8 → Γ6 interband electronic transition in essentially nonv
c
quantized samples. As can be inferred from the reported transition energy value, the confinement energies (ΔE = E(1S3/2, 1Se) – Eg,bulk) in the presently studied QD film samples are large. In other words, the films are strongly quantized. One may therefore expect that the optical absorption spectra should exhibit patterns characteristic of discretized energy levels in individual QDs (i.e. clearly visible excitonic peaks in the absorption spectra). Such patterns are, however, not present in the optical absorption spectra of CdSe QD thin films (Figs. 9 and 10). We attribute the observance of a structureless absorption spectrum in the studied case to the formation of collective electronic states in an ensemble of quantum dots, which are delocalized within a finite number of nanocrystals. Such states are formed as a result of the interdot electronic coupling – a phenomenon that seems to determine the complete optical properties of close-packed nanocrystallites (e.g. in thin films form). More detailed and seriously founded studies of interdot couplings occurring in close-packed quantum dot solids have been carried out only recently [18-25]. According to these recently accumulated data [18-25], the interdot electronic coupling occurs when very small quantum dots are close packed. The basic mechanism of the coupling seems to be the tunneling one, due to the charge carrier leakage outside each nanoparticle constituting the QD ensemble. From the viewpoint of the optical properties of quantum dot thin films, this interdot coupling is manifested through broadening of the exciton peak as compared to the case of diluted ensembles of quantum dots. An excellent pictorial explanation of the interdot coupling phenomena has been presented in the series of works by Woggon et al. [21-23,92-96]. Concerning the correct assignment of higher-energy electronic transitions in quantized CdSe films, it appears to be a much more complex task. According to Refs. [83-97], the second excited state in zinc-blende semiconducting QDs is the (2S3/2, 1Se) one. The expected energy differences between the ground state - (1S3/2, 1Se) and ground state - (2S3/2, 1Se) electronic transitions are, however, less than 0.1 eV (for ground state - (1S3/2, 1Se) transition energies of about 2.08 eV). Also, according to [83-97], the ground state - (2S3/2, 1Se) transition energy is expected to be a linear function of the ΔE(ground state - (1S3/2, 1Se)). We thus rule out the possibility that the second clearly differentiated feature of the absorption
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spectra of CdSe QD thin films corresponds to the ground state - (2S3/2, 1Se) electronic transition. It is, however, possible that this transition feature is masked by the much stronger ground state - (1S3/2, 1Se) one. By carefully reanalyzing the (αhv)2 vs. hv dependence in the energy region just above the absorption onset in the case of CdSe thin films of closely packed QDs, we have indeed observed an additional inflection point. It is, however, much less clearly differentiated than the other two – Fig. 16. Comparison of the energy difference between this and the fundamental ground state (1S3/2, 1Se) transition, implies the possibility that it corresponds to the ground state - (2S3/2, 1Se) electronic transition. According to energy ordering [83-97], the third excited state, should be (1S1/2, 1Se). For thin films of close packed CdSe QDs, the ground state - (1S1/2, 1Se) transition energy should be much less size-dependent than the ground state - (2S3/2, 1Se) one. 350
2
(αhv ) /(10 cm eV )
300
-2
250
8
200
2
150 100 50 0 1.70
1.90
2.10
2.30
2.50
2.70
hv /eV
Figure 16. The (αhv)2 vs. hv dependence in the energy region just above the absorption onset in the case of CdSe thin films of closely packed QDs. The additional, much less clearly differentiated inflection point corresponds to the ground state - (2S3/2, 1Se) electronic transition.
The agreement of our optical spectroscopy data with the predictions given in Refs. [8397] is excellent, and we could thus assign the second feature in the absorption spectra of our CdSe thin films exactly to the ground state - (1S1/2, 1Se) electronic transition. However, Woggon and collaborators have given a rather important note in Ref. [95], which is worth mentioning in this context. On the basis of their investigations of optical transitions in CdSe quantum dots which employed photoluminescence, pump-probe and photoluminescence excitation spectroscopies, these authors have concluded that in the case of quantum dots with average sizes smaller than Bohr’s excitonic radius, the (2P3/2, 1Pe) state could possibly appear much closer in energy to (2S3/2, 1Se) than (1S1/2, 1Se). Such arguments seem to be also supported by the fact that the previously mentioned energy ordering of the electron-hole excited states in CdSe QDs are based on theoretical approaches which are of questionable validity for P states. Therefore, in the case of very small nanocrystals the situation concerning the correct assignments of electronic transitions may be further complicated. The detected “energy splittings” between two lowest-lying electronic transitions in the studied CdSe QD thin films could, at first sight, seem to correspond to the spin orbit splitting energy of the zinc-blende valence bands. As it has been already mentioned before, if this
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energy difference corresponded to the spin-orbit splitting energy, then the actual value for this quantity in the case of CdSe would be drastically different than in the case of ZnSe. Since the spin-orbit splitting is an atomic phenomenon, no differences between the measured values are expected between CdSe and ZnSe films. If one accounts, however, for the apparent mixing of hole energy levels within the CdSe (and also ZnSe) QDs, it is clear that the observed “splittings” do not correspond exactly to spin-orbit splitting energies in both semiconductors, but are rather complex functions of the last quantity. The exact position of each hole energy level depends, besides on the “spin-orbit splitting energy”, also on other materialcharacteristic parameters, allowing the seeming discrepancy between expectations and experimental data to be eliminated in the light of more profound analyses of the QD hole energy spectra. The ground state - (1S1/2, 1Se) transition energy also exhibits a red shift upon high temperature annealing treatment of the films. It would be expected that this particular transition should converge to the Γ7 → Γ6 interband electronic transition in bulk zincv
c
blende semiconductors, which should occur at Eg + Δ0 (where Δ0 is the spin-orbit splitting energy of the valence band in the case of bulk semiconductor specimen). In the case of ZnSe QD thin films, studied previously by our group [66,67], the energy difference between the ground state - (1S1/2, 1Se) and the ground state - (1S3/2, 1Se) transitions was found to be 0.40 eV (in very good agreement with the literature value for the spin-orbit splitting energy of a bulk specimen of zinc-blende polymorph modification of this semiconductor). The energy difference between the ground state - (1S1/2, 1Se) and the ground state - (1S3/2, 1Se) transitions in the case of essentially non-quantized CdSe films found in the present study, on the other hand, is somewhat smaller than the literature value for the spin-orbit splitting for the hexagonal (wurtzite) polymorph of this semiconductor. Since the spin-orbit coupling is, as mentioned before, essentially an intraatomic phenomenon, it is expected that the magnitude of the spin-orbit splitting of the valence band in this case should not depend drastically on the crystal structure of the compound in question. This expectation has been confirmed by a number of other studies as well. For example, Janowitz et al. have estimated the spin-orbit splitting energy of the valence band in the case of cubic modification of CdSe, on the basis of their analysis of the pseudodielectric function of this material [101]. They obtained a value of 0.41 eV, which is quite close to Δ for wurtzite sample, in line with the previous discussion. The analysis that these authors have performed was, however, based on the second derivative spectra calculated for MBE-grown CdSe films on GaAs substrates. The presence of interference fringes due to the finite thickness and low absorption coefficient of CdSe films in the energy range below 2.5 eV has complicated the analysis of the pseudodielectric function, which also included contributions from GaAs substrate. Other authors, such as Velumani et al. have reported [81] much smaller, thickness-dependent values for the spin-orbit splitting energy in the case of CdSe films deposited on ITO substrates at various temperatures, ranging from 0.03 to 0.13 eV. Even the retrospective of such a brief literature data survey certainly indicates that the issue concerning the exact magnitude of the valence band spin-orbit splitting energy in this semiconductor has not been unequivocally solved yet. Alternatively, this quantity could also depend on the exact route employed to synthesize this semiconducting compound. In our study, we deal with thin films of close-packed CdSe QDs. In the weak confinement regime, characteristic for films with average crystal radii larger than 15 nm, although the quantum confinement effects on the absorption onset are absent, the hole
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energy level mixing may still be very significant. Therefore, the observed difference between the ground state - (1S1/2, 1Se) and the ground state - (1S3/2, 1Se) transition energies may still not correspond to the exact spin-orbit splitting energy characteristic for this semiconductor. It could be therefore concluded that the hole energy level mixing in the case of ZnSe QDs is not as significant as in CdSe, i.e. the states are purer in the former case. It is not clear, at least not in a quantitative manner, how should the close packing of the quantum dots affect the exact positions of the hole and electron energy levels in 3D assemblies of QDs deposited as thin films. Generally speaking, when nanocrystals are assembled and form a macroscopic colloidal crystal (a quantum dot solid) one deals with a kind of condensed matter with spatial organization on a length scale comparable to the electron de Broglie wavelength. Considering the dense quantum dot ensembles as analogous structures to multiple quantum wells, actually the QD solid may be viewed as a threedimensional superlattice formed by QDs as basic building blocks. As discussed in a series of works by Artemyev et al. [18,19,21-25,96], the ensembles of densely packed QDs deposited as thin films are expected to resemble the energy band structures of “conventional” solids. This implies an existence of energy bands in a perfect three-dimensional superlattice and coexistence of both localized and delocalized states in a lattice with some degree of disorder. These concepts are illustrated in Fig. 17, in which one-dimensional multiple quantum well structure is depicted, consisting of a random arrangement of different wells. The real nanocrystal assembly that we deal with is a three-dimensional analogue of Fig. 17. In comparison to an assembly of isolated QDs, it is expected that such a structure should exhibit a slight red shift of optical absorption onset and a structureless appearance of the optical spectra.
Figure 17. Schematic depiction of one-dimensional multiple quantum well structure consisting of a random arrangement of different wells.
Even if the distribution function characterizing the random arrangement of different wells was known, it would be a very difficult task to predict quantitatively the influence of such ordering on the issues related to spin-orbit splitting discussed before. Much more involved theoretical insights are required for such a purpose.
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5. Charge-Carrier Transport Properties in Close Packed CdSe Quantum Dots Deposited as Thin Films Under Equilibrium Conditions 5.1. Electrical Contact with the Nanostructured CdSe QD Thin Films
I / μA
Silver paste electrodes were used in the present study to provide ohmic (non-rectifying) contact with nanostructured CdSe QD films. The measured current-voltage characteristics of Ag-nanostructured cadmium selenide contact are presented in Fig. 18.
2 1.5 1 0.5 0
-6
-5
-4
-3
-2
-1
-0.5
0
1
2
3
4
5
6
V /V -1 -1.5 -2
Figure 18. The measured current-voltage characteristics of silver-cadmium selenide QD thin film contact.
As can be seen, the observed dependence of I on V is linear within the studied voltage
range. The derivative (∂I / ∂V )
−1
is constant and practically equal to the dark electrical
resistance of investigated thin film, which proves the non-rectifying character of the contact. This indicates a negligible value of the contact resistance [102,103]. At atomic level, ohmic heterocontact provides a negligible barrier to charge carrier transport at the boundary between the two substances. This, on the other hand, enables two-directional barrierless charge carrier transfer in and out of the studied semiconductor. It is of particular importance to perform the electrical and photoelectrical measurements in the voltage range in which the electrical conductivity is ohmic, in order to be able to focus on the processes of thermal and photoexcitation of charge carriers inherent to the nanocrystalline films, instead of the processes of large current injection directly from the electrodes (e.g. the space-charge limited current – SCLC [104-106]).
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5.2. Determination of Type of Electrical Conductivity in Thin Films Composed of Close Packed Cadmium Selenide QDs On the basis of the obtained sign of the thermoelectric force of CdSe-metal thermocouple (employing the hot point probe method), the dominant charge carriers are electrons. This is in agreement with the theoretical predictions, since the effective electron mass is smaller than ∗
∗
effective hole mass ( me < mh ) [75] for this semiconductor.
5.3. Temperature Dependence of the Equilibrium Conductivity of the Nanostructured CdSe QD Thin Films The dark electrical resistance of thin films composed by close packed cadmium selenide QDs was measured using the two-point and four-point probe methods, as well as the constant field method. In the case of two-point probe method, the silver electrodes were applied 1 cm in length and 1 cm apart (i.e. the length and width of thin films were 1 cm) and the values of thin film resistance were reported in “ohms per square” units. In our particular implementation of the four-point probe (van der Pauw) method fourpoint silver probe (with spacing between point contacts much larger compared to the film thickness (d)) was applied on the film’s surface as shown in Fig. 19. Two sets of currentvoltage measurements were performed: (i) the point probes A and B were connected to the d.c. source and the voltage drop was detected between points C and D (VCD); (ii) the contacts B and C were connected to the d.c. source and the A and D points detected the voltage drop (VDA).
Figure 19. The experimental setup for dark electrical resistance measurements using the four point probe method (van der Pauw method).
On the basis of the two sets of measurements, the dark electrical resistivity of investigated samples was calculated numerically solving the following equation [13]:
Chemically Deposited Thin Films…
exp(-π
d VCD d VDA ) + exp(-π ) =1 ρ I AB ρ I BC
147
(47)
In (47), d is film’s thickness, ρ is the resistivity of semiconducting material, VCD and VDA are the voltage drops between points (C and D) and (D and A) correspondingly, while IAB and IBC are the corresponding electric currents passed between (A and B) and (B and C) respectively. The constant field method for measurement of film resistance is based on usage of electrical circuit presented on Fig. 20 a). This circuit contains serially connected d. c. source to a standard resistor R (with resistance Rs) and to the investigated thin film sample F (with dark electrical resistance Rf).
V
RS
B
F
a)
b) Figure 20. The experimental setup for measurements of dark electrical resistance of the thin films by the constant field method (a); the setup for measurement of temperature dependence of the thin film dark electrical resistance (b).
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The dependence of thin film dark resistance on temperature was measured in inert (argon) atmosphere at P = 80 kPa using a set-up (presented in Fig. 20 b). It contains a special furnace (FH), a thermocouple with voltmeter (TC), variable transformer (T), a d.c. source and a standard resistor (Rs). The investigated thin film sample, characterized with surface area of 1 cm2, was placed in the furnace, which was subsequently slowly heated by the variable transformer (T). Thin film’s temperature was registered directly on its surface on the basis of temperature dependence of thermoelectromotive force of the used Fe-Constantane thermocouple. By mathematical analysis of electrical circuit, which was applied in film resistance measurements using the constant field method, the following equation could be easily obtained:
Rf =
Vb Rs − Rs Vs
(48)
where Vb is the applied voltage and Vs is the voltage drop at the ends of the resistor (Rs). In the case when Rs<
Rf =
Vb Rs Vs
(49)
The measured dark electrical resistances of thin film samples using the two-point, fourpoint probe methods, as well as the constant field method were found to be in excellent agreement. On the basis of our experimental data, it could be concluded that the dark electrical resistance of CdSe QD thin films depends on experimental conditions of the deposition process, initial concentrations of reactants and post-deposition treatment. Asdeposited cadmium selenide thin films composed by close packed QDs are characterized with high dark electrical resistance of the order of GΩs, regardless on chemical composition of reaction system. Such high value for the dark electrical resistance of as-deposited thin films could be attributed to the electrical isolation between CdSe quantum dots (i.e. localization of charge carriers to the nanocrystal). Physically, on an atomic level, the electrical isolation between CdSe nanocrystals implies existence of a large potential barrier between them [90]. As discussed before in this review, according to our investigations, the quantum size effects are strongly manifested in the case of chemically deposited CdSe thin films [26]. Thermal treatment of the nanostructured cadmium selenide thin films was found to lead to processes of coalescence of CdSe nanocrystals. All of these processes leaded to increase of connectivity between nanocrystals and, eventually, to crystal growth. An overall consequence of these processes is electrical connection between nanoparticles, which resulted in irreversible loss of confinement effects, as discussed previously. According to the results of our electrical measurements, upon thermal treatments of the studied films, their dark electrical resistance decreases (with respect to the unannealed thin films) and eventually becomes of the order from several GΩs to several MΩs depending on the chemical composition of growth solution. The decrease of dark electrical resistance of CdSe thin films upon annealing processes is, as
Chemically Deposited Thin Films…
149
implied before, due to electrical contact between the QDs. On the other hand, the chemical composition of reaction system influences the stoichiometry of obtained cadmium selenide. Small changes in the stoichiometry, which are under detection limit of XRD analysis, could act as impurities with chemical character. Their presence in band structure is manifested through discrete impurity levels, in the forbidden band, which are almost completely ionized at room temperature (300 K). As a result of ionization of impurity levels, the equilibrium concentrations of charge carriers increase, leading to a decrease of the dark electrical resistance. On the other hand, cadmium selenide thin films composed by close packed QDs, which manifest maximal photoconductivity, correspond to a practically stoichiometrical material, which acts as intrinsic semiconductor (i.e. it is characterized with a relatively high value of the dark electrical resistance). We have further investigated the temperature dependence of equilibrium electrical conductivity of the nanostructured CdSe QD films. A typical temperature dependence of dark electrical resistance (measured in inert argon atmosphere) for one of the investigated samples is presented in Fig. 21. Obviously, the apparent exponential decrease of electrical resistance upon temperature increase is related to the semiconducting nature of investigated system i.e. with increase of concentration of free charge carriers (holes and electrons). Two mechanisms are involved in generation of the free charge carriers: (i) interband electronic transitions between valence and conduction band; (ii) ionization of impurity levels present in forbidden band (the band gap states). 90 80 70
R / kΩ
60 50 40 30 20 10 0 250
300
350
400
450
500
550
T /K Figure 21. The measured dependence of R vs. T in temperature interval from 373 to 543 K for a CdSe QD thin film.
Experimentally measured data from temperature dependence of dark electrical resistance were interpreted in the framework of solid state theory for semiconducting materials. When
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one deals with macrocrystalline semiconductors, the charge carrier transport in these systems may occur through delocalized conduction band states, either upon their excitation from the valence band or from each of the n distinct discrete band-gap states. In such systems, the temperature dependence of dark (equilibrium) conductivity - σ is described with the following function [2,9,13,15,16,102,103,107]:
σ (T ) = σ 0 ' e
n
+ ∑ σ 0' ,i e −ΔEi / k BT
− Eg / 2 k BT
(50)
i =1
where σ0’ and σ0,i’ are constants, ΔEi is ionization energy of i-th impurity level from forbidden gap, while other symbols have their usual meaning. A function of the form (50) which describes the temperature dependence of R implies an (n + 1)-channel conduction mechanism. Generally, the electrical conductivity depends on two parameters: concentration of free charge carriers and their mobility as well. Both of these quantities are temperature depended, but usually the temperature dependence of mobility is much less pronounced. Therefore, the considered equation (50) has been actually derived neglecting the fact that the charge carriers’ mobility is a temperature-dependent quantity. However, in the case of semiconductors with an appreciable degree of disorder, the situation is somewhat more complicated, as discussed further below.
8 7
α1
6
ln(R /Ω)
5 4
α2
3 2
Eg
α3
1
tgα3 = Eg/2k
0 1.7
1.8
1.9
2
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
-1 3
-1
-1
10 T /K
Figure 22. The dependence of lnR vs. 1/T in temperature region from 373 to 543 K for a CdSe QD thin film.
From the experimental data related to temperature dependence of dark electrical resistance, the lnR = f(T –1) dependence for investigated samples was constructed (Fig. 22).
Chemically Deposited Thin Films…
151
As could be seen from Fig. 22, three linear trends in the explored temperature region (from 373 to 543 K) appear. The two linear trends from these dependence appearing in lowertemperature region (from 373 – 403 K and from 403 – 478 K) are close related with activation of extrinsic conductivity mechanism, whereas in higher-temperature one (from 478 to 543 K) it is due to intrinsic type of conduction. Since the linear trend from higher temperature region could be attributed to interband electronic transitions, the corresponding experimental data were used for calculation of band gap energy (Eg) of investigated semiconductor. In this region, the contributions of other terms in (50) to the overall conductivity may be neglected, and the expression (50) takes the form:
σ (T ) = σ 0 ' e
− Eg / 2 k BT
(51)
Switching from conductivities to electrical resistance, the experimental data from the highest temperature region were least-squares fitted by the following function:
ln R(T ) =
Eg 1 + ln R0 ' 2k B T
(52)
On the basis of the slope obtained by linear interpolation of the experimental data for ln(R) vs. 1/T dependence, a value of 1.85 eV for the band gap energy of the studied material was calculated. The calculated value of band gap energy corresponds to temperature of 0 K [66,68,69,71], as discussed in details in our recent review paper [62] and in several other papers. It is in excellent agreement with the literature values (1.85 eV and 1.90 eV [75]), and also with the optical band gap value calculated on the basis of optical spectra and on the spectral dependence of photoconductivity. Concerning the interpretation of the results from thermoelectrical measurements in the lower temperature region, first of all, one has to bear in mind that in the present case we deal with nanocrystalline thin films. Therefore, one can expect quite a high density of localized states in the forbidden gap. The situation is actually close to that in amorphous semiconductors, with an existence of band tails accompanied by a number of other donor-like and acceptor-like defect states in the band gap. These states are usually not expected to be discrete in highly disordered materials, but to be characterized by some distribution function (e.g. normal, i.e. Gaussian). The dark conductivity of polycrystalline n-semiconductor is proportional to the concentration of the dominant charge carriers n:
σ = enμ
(53)
where e is the electron charge, while μ is the drift mobility. The concentration of the dominant charge carriers is a temperature dependent quantity, i.e.:
⎛ E ⎞ n ∝ exp⎜⎜ − d ⎟⎟ ⎝ k BT ⎠
(54)
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The quantity Ed, appearing in the last equation, is the sum of grain boundary barrier height and the Fermi level еnergy EF. (i.e., more precisely, the energy distance between the Fermi level and conduction band bottom) [74]:
E d = Eb + E F
(55)
Ed = Eb + EF. Following Kazmerski [108], two processes are essential in thermally activated conductivity in lower temperature region. The first one includes thermally activated transitions of charge carriers from the Fermi level to the conduction band in the “bulk-like” part of the nanocrystals. The second one, on the other hand, is the thermally activated transport over the nanocrystal boundary barriers. Having this discussion in mind, the experimental data corresponding to the second linear trend in the overall R = f(T) dependence (at temperatures just below the range in which the intrinsic conduction mechanism is activated, i.e. in the temperature range from 403 – 478 K) were interpolated with a function of the form:
ln R(T ) =
Eb + E F 1 + ln R0,1 ' kB T
(56)
10
8
ln(R /Ω)
6
4
2
0 1.5
1.6
1.7
1.8
1.9
2
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
3
-2
103T -1/K-1
Figure 23. The dependence of lnR vs. 1/T during a single heating-cooling cycle for a CdSe QD thin film.
From the slope of the interpolating linear function (in the least-squares sense), a value of 0.74 eV was obtained for the sum of the Fermi energy and the barrier height. At still lower temperatures, due to the existence of considerable band tail in the studied systems (analogously as in amorphous materials), the dark current activation energy is often ascribed
Chemically Deposited Thin Films…
153
to the energy distance between the Fermi level and the lowest energy of the corresponding band tail. On the basis of linear interpolation of our experimental data in temperature region from 373 – 403 K, a value of 0.43 eV was obtained for the mentioned energy difference. The measurements of temperature dependence of dark electrical resistance are reproducible. In Fig. 23 the dependence of lnR vs. 1/T during a single heating-cooling cycle is presented.
6. Photophysical Properties and Relaxation Dynamics in Photoexcited CdSe Quantum Dots in Thin Film Form Both the stationary and time-dependent non-equilibrium conductivity (photoconductivity) of the synthesized CdSe quantum dots in thin film form were investigated. These investigations were carried out in order to test the photophysical/photoelectrical performances of the nanostructured films, and also to derive more in-depth knowledge concerning the band structure in the synthesized low-dimensional material.
6.1. The Spectral Dependence of Stationary Photoconductivity in Nanostructured CdSe QD Thin Films Spectral dependence of stationary photoconductivity of cadmium selenide QDs deposited as continual thin films was measured on the basis of the constant field method, using the experimental set-up shown in Fig. 20 a). The set-up contains d.c. source serially connected to a standard resistor (Rs) and investigated CdSe thin film sample (F), which is illuminated by incident light within the wavelength range from 400 to 1200 nm. As a source of incident radiation, a monochromator from spectrophotometer Beckman DU-2 was used. The thin films samples, which were used in photoelectrical investigations, were previously appropriately prepared. They are characterized with length of 4 mm and width of 3 mm. The silver electrodes were applied 4 mm apart. The applied voltage and electrical resistance of standard resistor were 30 V and 1 kΩ, correspondingly. The interaction of semiconducting materials with electromagnetic radiation from the intrinsic region is followed by generation and recombination of non-equilibrium charge carriers. Under stationary conditions the rates of these two processes are equal and the photoconductivity (or non-equilibrium conductivity) is constant [103]. It is exactly this physical quantity that we determine in our stationary measurements. Experimental measurements of spectral dependence of stationary photoconductivity are actually based on registration of the voltage drop (v) at the ends of the resistor (Rs) during interaction between the investigated sample and monochromatic radiation. The voltage drop at the ends of the resistor (with resistance R) as a result of interaction with monochromatic radiation is determined with the following equation:
v = (I l − I d ) ⋅ R
(57)
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Biljana Pejova
where Id and Il are dark current and current under light interaction correspondingly. On the other hand, Id and Il are determined in the following manner:
V R + r0
(58)
V R + r0 − Δr
(59)
Id =
Il =
where V is the applied voltage, r0 is the dark electrical resistance and r0-Δr is the electrical resistance during the interaction of the investigated sample with monochromatic radiation. On the basis of previous equations (58) and (59), the following expression for v could be obtained:
⎛ V V v = ⎜⎜ − ⎝ R + r0 − Δr R + r0
⎞ V ⋅ Δr ⋅ R ⎟⎟ R = (R + r0 − Δr ) ⋅ (R + r0 ) ⎠
(60)
Series of simple algebraic transformations lead to the following formula for Δr:
v ⋅ (R + r0 ) v ⋅ (R + r0 ) + V ⋅ R 2
Δr =
(61)
From (61), it follows that the stationary photoconductivity is given by:
Δσ st. = σ l − σ d =
1 1 Δr − = r0 − Δr r0 r0 (r0 − Δr )
(62)
From (61) and (62), it follows that the stationary photoconductivity depends on dark electrical resistance of the investigated sample and resistance of the standard resistor according to the following equation:
Δσ st. =
v ⋅ (R + r0 )
2
(63)
r0 VR − vr0 R ⋅ (r0 + R ) 2
In the case where R << r0 (constant field method), the stationary photoconductivity is proportional to the voltage drop at the ends of the resistor, i.e. is to a good approximation given by:
Δσ st. =
v ⋅ r0
r0 VR − vr R 2
v ⋅ r0
2 2 0
=
2
r0 R ⋅ (V − v ) 2
=
v R ⋅ (V − v )
(64)
Chemically Deposited Thin Films…
155
Finally, since when R << r0 also V >> v:
Δσ st. =
v R ⋅V
(65)
Our experimental studies have shown that the as-deposited CdSe QD thin films, regardless on the chemical composition of the reaction system, are non-photoconductive. The absence of photoconductivity and the previously discussed very high dark electrical resistance of as-deposited cadmium selenide QD thin films are due to the localization of electrons and holes in a confined space (quantum dot). Upon thermal treatment, which induces coalescence and crystal growth phenomena, the nanocrystals constituting the films are electrically connected and the confinement effects disappear irreversibly. The annealed thin films, which are characterized with dark electrical resistance of the order of several GΩs, manifest high photoconductivity. For example, upon interaction with white light (from an overhead projector), the electrical resistance of the thin film decreases to 18 MΩ. After light switch-off the resistance momentarily increases to dark value. This finding indicates that cadmium selenide thin films exhibit fast photoconductivity. By changing the reactant concentrations (within a narrow range) in the growth solution, thin films characterized with dark electrical resistance of the order of MΩs could be synthesized. These films were found to manifest small or negligible photoconductivity. 1.2
Δσ st./Δσ st.max
1 0.8 0.6 0.4
0.2 0 300
400
500
600
700
λ / nm a) Figure 24. Continued on next page.
800
900
1000
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Biljana Pejova
1.2
Δσst. max
Δσ st ./Δσ st.max
1 0.8
0,5·Δσst. max
0.6 0.4 0.2 0 1
1.2
1.4 1.6
1.8
2
2.2
2.4 2.6
2.8
3
3.2 3.4
hv / eV b) Figure 24. The measured spectral dependence of photoconductivity for a CdSe thin film composed by QDs of this semiconductor: a) Δσ / Δσmax. vs. wavelength; b) Δσ / Δσmax. vs. incident photon energy.
A typical spectral dependence of stationary photoconductivity for a cadmium selenide QD thin film (which is characterized with dark electrical resistance of 2 GΩ) is presented in Fig. 24. For a better insight into the photoelectrical properties of the studied QD thin films, this function has been plotted using both the incident photon energy and wavelength scale on the x-axis. As in the case of any other semiconductor, the spectral dependence of photoconductivity of CdSe QDs deposited in thin film form is determined from their absorption spectra. The manifested photoconductivity is mainly a consequence of the internal photoelectric effect i.e. it is due to the fundamental (band to band) electronic transitions and generation of non-equilibrium charge carriers. Transitions from various band-gap states to the conduction band edge (or to band tail states) add a certain contribution to the internal photoeffect, especially in the sub-band gap region. As can be seen from Figure 24, the absorption edge, i.e. the “red limit” of internal photoelectric effect is about 1.60 eV. Maximal stationary photoconductivity of cadmium selenide QD thin films is achieved upon interaction with photons with energy of 1.94 eV. On the side of larger photon energies, the stationary photoconductivity decreases as a result of surface relaxation processes [103]. The interaction of CdSe with photons with energies larger than the band gap energy (1.77 eV [26]) is followed by intense surface absorption and increase of non-equilibrium charge carrier concentrations. On the other hand, the high concentration of photocarriers favors the annihilation processes and decrease of photocarrier’s lifetime. Since the spectral dependence of stationary photoconductivity of semiconducting materials is determined from their absorption spectra (i.e. from their band structure), several approaches for determination of optical band gap energy, on the basis of spectral dependence
Chemically Deposited Thin Films…
157
of stationary photoconductivity, have been proposed in the literature [109-111]. According to the so-called Moss’ rule [109-111], the band gap energy is actually equal to the photon energy which corresponds to 0.5·Δσst,max. Some other methodologies, on the other hand, relate the band gap energy with the maximal stationary photoconductivity or to the red limit of internal photoelectric effect. In our case, the determined optical band gap energy of photoconductive CdSe thin film, on the basis of photon energy which corresponds to 0.5·Δσst,max, is 1.75 eV (Fig. 24 b). This is in excellent agreement with the value of 1.77 eV [26] calculated on the basis of absorption spectrum in the framework of parabolic approximation for dispersion relation using Fermi’s golden rule for electronic transitions from valence to conduction band.
6.2. Relaxation Dynamics of Non-equilibrium Charge Carriers in Photoexcited CdSe Quantum Dots Deposited in Thin Film Form Kinetics of charge-carrier recombination in the studied CdSe QDs deposited in thin film form was investigated by the oscilloscopic method, which is essentially based on studies of the photoconductivity relaxation dynamics. On the basis of experimentally obtained relaxation curves, the mechanism of relaxation and relaxation time of non-equilibrium charge carriers are determined. The used set-up contains serially connected d.c. source, standard resistor (Rs) and investigated sample (F) (Fig. 25). OSC
RS
F
B
FL
Figure 25. The experimental setup used for investigation of the relaxation dynamics of non-equilibrium charge carriers in CdSe QD thin films by the oscilloscopic method.
A sample of CdSe thin film, prepared in an equal manner as in the case of measurement of spectral dependence of stationary photoconductivity, was placed in a dark chamber. To achieve generation of non-equilibrium charge carriers the film was illuminated with white light impulse from a flash lamp (FL). In principle, the measurement of time dependence of non-equilibrium charge carriers’ concentration, after impulse interaction with white light from flash lamp, is based on registration of voltage drop at the ends of standard resistor using oscilloscope with previously calibrated time and voltage axes. The electrical resistance of investigated sample is calculated according to the formulae arising from the constant field method.
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Biljana Pejova
Classification of the recombination phenomena of non-equilibrium charge carriers in semiconductor quantum dots may be carried out according to various criteria. Concerning the type of recombination, they can be classified as direct (band to band recombination) and indirect (via bandgap recombination centers) processes. The later are characteristic of the wide-band gap semiconductors. The released energy that results from the recombination process can be emitted as a photon or dissipated to the lattice in the form of heat. If the charge carrier recombination leads to emission of a photon, the process is called radiative recombination. Otherwise, one deals with nonradiative recombination. The main criterion concerning the type of relaxation processes is related to their kinetics. From kinetic aspect, the non-equilibrium conductivity may be relaxed according to the linear or quadratic mechanism [103]. In Fig. 26, several typical oscillograms for a CdSe thin film constituted by close packed QDs of this material (with dark conductivity of 5·10-10 Ω-1) are shown. Using the equation (65), on the basis of time dependence of voltage drop at the end of the resistor Rs, the relaxation curve (i.e. the time dependence of the photoconductivity) was constructed. To derive conclusions concerning the charge carriers’ recombination kinetics, the experimentally obtained data for the dependences Δσ / Δσ max = f(t) were fitted with functions which determine the time dependence of photocarrier’s concentration in the case of linear and quadratic recombination processes. On the basis of the obtained results it could be concluded that within the short starting time interval (< 0.2 ms) the photocarriers are relaxed according to the quadratic relaxation mechanism, while after this relatively short initial time period, relaxation continues in accordance with the linear kinetics law. Considering photoexcitation with Heaviside - type rectangular light impulses, the overall dynamics of photoexcited non-equilibrium charge-carriers is governed by the following differential equation [112]:
d(ΔN ) = vgen. − v rec. dt
(66)
where ΔN denotes the excess number concentration of charge carriers with respect to the equilibrium value (i.e. the concentration of photo-generated charge carriers), vgen. denotes the rate of charge-carriers generation, while vrec. is the rate of their recombination. Second-order kinetics of the relaxation process implies that vrec. is proportional to concentrations of both type charge carriers, i.e.:
v rec . = γ ΔN e ΔN h
(67)
where ΔNe and ΔNh denote the number concentrations of photogenerated electrons and holes respectively, while γ is the recombination coefficient. If the condition ΔNe = ΔNh = ΔN is fulfilled, it follows that:
v rec . = γ (ΔN ) 2
(68)
Chemically Deposited Thin Films…
159
In such case, the general kinetic equation (66) takes the form:
d(ΔN ) = β α I − γ (ΔN ) 2 dt
(69)
a)
b)
c) Figure 26. The recorded oscillograms for a photoexcited thin film composed by CdSe QDs with various t - axes (a – 100 μs/cm, b – 200 μs/cm, c – 400 μs/cm) after the interaction with electromagnetic radiation is switched off.
160
Biljana Pejova
When both the processes of photogeneration and relaxation are in progress in the QD thin films, under initial condition Δn(t = 0) = Δn0 (= τβαI, where α, β, τ and I are absorption coefficient, quantum yield, the average lifetime of photogenerated charge carriers and the intensity of incident radiation, correspondingly), the solution of (69) gets the form:
ΔN (t ) =
αβI tgh (t α β γ I ) γ
(70)
When only the relaxation processes occur (i.e. after the light impulse has been switched off), the time dependence of ΔN is governed by:
αβI 1 γ t α β γ I +1
ΔN (t ) =
(71)
On the basis of the previous analysis, it follows that in the case of quadratic relaxation processes, 1/Δσ is expected to depend linearly on t. Analyzing our experimental results, we found out that within the short starting time interval (< 0.2 ms), the function:
1 = f (t ) Δσ
(72)
is indeed linear (Fig. 27) which is in agreement with the previously implied second-order recombination kinetics (quadratic recombination mechanism) in the beginning of relaxation process. In the first time interval the concentration of non-equilibrium charge carriers in photoexcited CdSe quantum dots is very high and the electrons from conduction band are annihilated with the holes from valence band. The rate of recombination, in this case, depends on the concentrations of both charge carriers [103,112]. After about 0.2 ms the photocarrier’s concentrations decrease and conditions are fulfilled under which the non-equilibrium charge carriers could be relaxed via bandgap recombination centers according to the linear relaxation mechanism. For linear recombination mechanism, the rate of recombination is proportional to the concentration of only one type of charge carriers, i.e.:
v rec . =
ΔN
τ
(73)
As the rate of generation is given by:
vgen. = β α I
(74)
d(ΔN ) ΔN = βα I − dt τ
(75)
the kinetic equation (66) takes the form:
Chemically Deposited Thin Films…
161
180000 160000
1/(Δσ / Ω-1)
140000 120000 100000 80000 60000 40000 20000 0 0.03
0.05
0.07
0.09
0.11
0.13
0.15
0.17
0.19
t / ms Figure 27. The dependence of 1/Δσ vs. t in the staring time interval of relaxation process of nonequilibrium charge carriers in nanostructured CdSe QD thin film.
The solution of this ordinary linear differential equation, with initial condition Δn(t = 0) = 0 is of the form:
ΔN (t ) = τ β α I (1 − e − t / τ )
(76)
The product before the parentheses in (76) is usually denoted as ΔNst. (the stationary value of the photogenerated charge carriers concentration), and the function ΔN(t) is written in the form:
ΔN (t ) = ΔN st. (1 − e − t / τ )
(77)
Under conditions when only the recombination processes remain active in the QD samples (i.e. when the light impulse is switched off and vgen. = 0), eq. (75) gets the form:
d(ΔN ) ΔN =− dt τ
(78)
ΔN (t ) = ΔN st. e −t / τ
(79)
The solution of (78) is:
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Biljana Pejova
Accounting for the proportionality of Δσ and ΔN, first-order relaxation kinetics leads to exponential decay function Δσ (t):
Δσ (t ) = Δσ st. e − t / τ
(80)
In other words, when relaxation occurs according to the linear mechanism, the dependence of ln(Δσ) on t is linear. The function ln(Δσ) = f(t), constructed on the basis of our oscilloscopic data for t > 0.2 ms is given in Fig. 28. The data from this dependence were fitted with a linear function. On the basis of the slope of ln(Δσ) as a function of t the average lifetime (i.e. relaxation time) of non-equilibrium charge carriers in photoexcited CdSe quantum dots in thin film form was calculated. The corresponding value is 0.4 ms, which is relatively high on an absolute scale. This finding implies that the synthesized CdSe QD thin films have a potential application in fabrication of solar cells. As can be seen from the oscillogram presented in Fig. 29, registered at extended t-axis of the oscilloscope (1 ms/cm), the obtained CdSe thin films do not manifest residual photoconductivity.
-10 0
0.2
0.4
0.6
0.8
1
1.2
-10.5 -11
-1
ln(Δσ /Ω )
-11.5 -12 -12.5 -13 -13.5 -14 -14.5
t / ms Figure 28. The dependence of ln(Δσ/Ω-1) vs. t within time interval which corresponds to the linear relaxation process of photocarriers in CdSe QD thin film.
Chemically Deposited Thin Films…
163
Figure 29. The recorded oscillogram for a photoexcited thin film composed by CdSe QDs at extended taxis of the oscilloscope (1 ms/cm) after the interaction with electromagnetic radiation is switched off.
6.3. Lux-Ampere Characteristics of the Photoconductive Thin Films. Having in mind that photoconductive CdSe thin films are characterized with maximal photoconductivity upon interaction with light with wavelength of 640 nm, the dependence of non-equilibrium conductivity on light intensity is investigated using monochromatic radiation with wavelength of 640 nm. Constant field circuit-based method was implemented for measurements of the non-equilibrium conductivity, while the incident light intensity (i.e. its flux and consequently the sample illumination) was modulated by changing the slit width of the Beckman DU-2 spectrometer light source. Sample illumination was measured by a LUXUV-IR METER 666 230. It is well known that the photoconductivity (Δσ) can be expressed as a power function of the light intensity (I) [112]:
Δσ ~ I k
(81)
where k is 1 or 1/2, depending on the type of relaxation mechanism (linear or quadratic, correspondingly). This may be proved by analyzing the kinetic equation (66) under stationary conditions, i.e.:
d(ΔN ) =0 dt
(82)
Under such circumstances, it follows from (75) that:
ΔN st. = β ⋅ α ⋅ I ⋅ τ st.
(83)
Δσ st. = const. ⋅ I
(84)
and therefore:
164
Biljana Pejova In case of the quadratic recombination mechanism, imposing the condition (82) leads to:
β ⋅ α ⋅ I = γ (ΔN st. ) 2
(85)
i.e.:
β ⋅α ⋅ I γ
(86)
Δσ st. = const. I
(87)
ΔN st. = Therefore:
On the basis of dependence of ln(Δσ) on lnΦ (presented in Fig. 30) the calculated value of k is approximately 1. According to these findings, in cumulative relaxation processes the linear mechanism has a dominant role.
-17.5 -18
3
4
5
6
7
8
ln(Δσ /Δσ max)
-18.5 -19 -19.5 -20 -20.5 -21 -21.5 -22
ln(F / lux) Figure 30. The dependence of ln(Δσ) on lnΦ in the case of nanostructured CdSe QD thin film.
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In: Quantum Dots: Research, Technology and Applications ISBN 978-1-60456-930-8 c 2008 Nova Science Publishers, Inc. Editor: Randolf W. Knoss, pp. 169-201
Chapter 4
N UMERICAL M ODELLING OF S EMICONDUCTOR Q UANTUM D OT L IGHT E MITTERS FOR F IBER O PTIC C OMMUNICATION AND S ENSING Mariangela Gioannini Dipartimento di Elettronica, Politecnico di Torino, Torino, Italy
Abstract We present a review of our research work on the modelling of the optical properties of light emitting devices having a semiconductor quantum dot material as active region. The gain region is obtained by a Strasky-Krastanov growth of several layers of quantum dots that are not uniform in size. This causes an inhomogeneous broadening of the gain spectrum that is a peculiar characteristics of these light emitters. The numerical model is based on a multi-population rate equation model used for describing the dynamics of electrons and holes in an inhomogeneous material and in the several energy states confined in the dots. The rate equations of the carriers are also coupled with the rate equations of the photons generated by spontaneous and/or stimulated emission. In this review we provide several examples of simulation results of the optical characteristics of InAs/GaAs quantum dot semiconductor lasers and superluminescent diodes emitting in the near infrared with application in optical communications, sensing and optical coherent tomography. In particular, we show how the inhomogeneous gain broadening and the presence of more than one confined energy state in the dots can influence the laser properties such as the shape of the emitted spectrum, the maximum modulation bandwidth and the frequency fluctuations (chirp) under large signal modulation. The results of this analysis gives useful insights on the meaning, in the quantum dot case, of various parameters (linewidth enhancement factor, differential gain ) that are routinely measured in the lab with the standard characterization techniques for semiconductor quantum well or bulk lasers. We also provide some examples of calculated emission characteristics (light versus current curves and output spectra) of quantum dot superluminescent diodes to highlight the relevant differences respect to the laser case. We also show how the inhomogeneous broadening of the gain, the quantum dot layer composition, and the device geometry can be engineered to get bright sources with broad spectrum useful for medical and sensing applications.
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Introduction
In the last ten years there has been an increasing interests in the applications arising by the possibility of growing semiconductor quantum dots (QDs) using the well established MBE or MOCVD techniques and a Stransky-Krastanov growth process [1]. According to this technique the large lattice constant mismatch between a few monolayer of deposited material and the substrate give the natural formation of islands of size of few nanometers [1]. Starting from the first growths of QDs on GaAs [2] or on InP [3] and the demonstration of their optical and electrical properties [4], [5], [6], several research groups around the world have focused their interests in the study of light emitting sources realized using semiconductor QDs as gain medium. The emission wavelength is in the range 980-1300 nm for QDs grown on GaAs and around 1550 nm and beyond for QDs grown on InP. In the recent years we have assisted to the realization of edge-emitting lasers with a record low threshold current density [7], of QD VCSELS [8], of QD lasers in modelocking regime [9], of single photon emitters [10] and of super-luminescent light emitting diodes [11], [12]. It was indeed predicted that the trapping of injected carriers in a three dimensional box would have lead to lasers with high gain and reduced threshold current , infinite T0, high differential gain and high modulation speed, zero linewidth enhancement factor (α-parameter [13]) and chirp-free modulation as well as insensitivity to optical feedback [14]. Unfortunately the practical realizations of the devices have shown that the theoretical predictions could not be realized all together in the same device due to the intrinsic nature of the QD materials grown using a self-assembled process [2], but some of them where reached separately and were proved to superior respect to the quantum well case. Despite of this, the semiconductor QD material still remains of high interest for the scientific community thanks to: 1)the large freedom of tailoring the optical emission properties through the engineering of the QD size and composition [15], 2) the possibility of realizing broad spectrum laser sources [16], 3) the possibility of actually increasing the modulation speed of lasers using alternative injection schemes [17], 4)the possibility of generating record short and high peak power pulses [9], 5) the promising use of the QD material in single-photon emitters as well as in devices for the control [18] and the amplification of light [19]. The peculiar property of the semiconductor QDs grown by self-assembling are: 1) the strong compressive strain inside the QDs which leads to a significant deformation of the potentials respect to the bulk case [20], 2) the ihomomogeneous distribution of the QD size [21], [22] and 3) the presence of more than one confined energy state inside the QDs [23]. In Fig. 1 we show how this properties can be observed in the experiments. An example of QD size dispersion is visible in the SEM image shown in Fig.1a, whereas an example of shape and composition of a semiconductor QD is shown in Fig. 1b. The inhomogeneous broadening of the emission line from an ensemble of dots and the presence of more than one confined state can be revealed by the measured electroluminescent, EL, spectra shown in Fig. 1c. The presence of two emission peaks in the EL spectrum indicates that the radiative recombination between electron and holes can occur from the lowest energy confined states in the QDs (the ground state, GS, indicated with E0 in Fig. 1c) or from a second, higher energy, confined state (the excited state, ES, indicated with E1 in Fig. 1c). The fitting of the EL spectrum with two gaussian functions can give a quantitative estimation of the
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Figure 1. (a) Scanning electron microscope (SEM) image of InAs QDs grown in GaAs. The QD islands are the bright spots; the different size of the spots revels the non uniform distribution of the QD size. (b) Transmission electron microscope (TEM) image of an InAs QD. The QD has a lens shape with thickness of about 5 nm and base diameter of about 25 nm. (c) Electroluminescent spectra and photocurrent spectra of a InAs QD ensable. (Images courtesy provided by the Department of Electronic and Electrical Engineering of the University of Sheffield ,UK.) width of the inhomogeneous distributions of the recombination energies and therefore of the spreading of the QD size. Further radiative recombination energies due to several other states confined in the QDs (indicated with E2 and E3 in Fig. 1c) can also be revealed as absorption peaks in the spectra of the measured photo-current. A strongest absorption peak at high energy is attributed to excitation of carriers in the two dimensional carrier reservoir, generally called the wetting layer. The optical and electrical properties of the semiconductor QD lasers and other devices are significantly affected by this peculiar characteristics of the QD material. For a correct theoretical analysis of the device characteristics it is therefore essential to account of these complex properties. The carrier and photon dynamics can not be approximated, as in the bulk or quantum well case, with a simple standard rate equation system (i.e. one equation for carriers and one for photons [24]) but it is necessary to introduce a model based on several rate equations (multi-population rate equation MPRE [25]) for representing the carrier dynamics in the various QD confined states. Various models have been proposed in the literature and have been applied to the study of various devices such as lasers [23, 26, 27], amplifiers [28] and SLDs [11]. Most of them have been used for the analysis of the static
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characteristics of QD lasers [23, 26, 27] and few were used to investigate their dynamic properties [29–31]. A comprehensive model for the simulation of the gain and refractive index dynamics in QDs has to include the presence of a barrier state where carriers are injected [26], a wetting layer state (WL) that acts as a common carrier reservoir [23, 26, 27], and the dot states coupled to the WL. The dot states should also be able to represent the inhomogeneous broadening of the gain due to the dot size distribution [26, 27]. In this chapter we present a review of the research work on the theoretical modelling of QD laser diodes (QD-LD) and QD superluminescent diodes (QD-SLD) based on the MPRE model. This work was carried on at the Dipartimento di Elettronica of Politecnico di Torino (Torino, Italy) in the framework of several projects supported by the EU and devoted to the realizations of QD lasers and SLDs on both GaAs [32] and InP [33]. This chapter is organized as follows: in section 3 the numerical model is described with particular emphasis to the various input parameters required. In section 4 we provide several examples of simulation results of the static and dynamic characteristic of QD lasers. The purpose is to demonstrate how the MPRE model can reproduce several of the static and dynamic characteristics measured in the experiments and can be used to understand and interpret the experimental results. The results of this analysis will also give useful insights on the significance, for the quantum dot case, of various parameters (linewidth enhancement factor, differential gain ) that are routinely measured in the lab with the standard characterization techniques of bulk or quantum well semiconductor lasers. In section 5 we will give examples of calculated emission characteristics (light versus current curves and output spectra) of quantum dot superluminescent diodes to highlight the relevant differences with respect to the laser case. We will also show how the inhomogeneous broadening of the gain, the QD layer composition and the device geometry can be engineered to get wide and bright sources useful for optical coherent tomography and other sensing or medical applications. Section 6 is devoted to a critical analysis of the model proposed; in section 7 we draw the conclusions. In the recent years we have seen an ever increasing number of papers and conferences dealing with theory and experiments with QD devices. This has produced a huge amount of information and sometimes contradictory results. As a consequence it can be difficult for new starters finding the right way for getting a basic understanding of the physics of QD devices. In this chapter we have therefore introduced several references to those papers we judged to be fundamental for understanding the basic physics of QD lasers with the hope of providing a useful guideline for those who start adventuring in this field.
2. 2.1.
Numerical Model Description and Definitions
Fig.2a shows an example of a semiconductor edge emitting laser grown on GaAs. When the waveguide is tilted and/or anti-reflection coatings are deposited at the facets the same structure can work as a SLD or an amplifier. The active waveguide is a separate confinement heterostructure (SCH) with the cladding layer in AlGaAs; the core and the active region is constituted of several InAs QD layers embedded in a InGaAs well and separated by a GaAs barrier [12]. The InAs QDs have generally a lens or a truncated pyramid shape, with an
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underlaying thin 2D substrate in InAs which is due to the initial stage of the self-assembled growth process [1]. Even if this thin substrate is usually refereed in the literature as the wetting layer (WL), we prefer defining the WL in a broader sense. In our model we define the WL as the energy reservoir which accounts for all of the continuum of the states which are not confined in the QDs. This continuum of states has a two dimensional density of states and origins from the higher energy states of the different dots which can be strongly quantum mechanically coupled together via the thin substrate under the QDs [34].
Figure 2. (a) Schematic structure of a ridge semiconductor laser with QD active material; in the example the QD layer is repeated 6 times. (b) Conduction band energy diagram of an InAs QD along the z-axis indicated in the inset. The inset in (b) shows the electron wavefunction for the lowest energy WL state. In Fig.2b the calculated conduction band energy diagram for the InAs QD is shown. The height of the potential barrier, formed by the energy band-offset between the strained InAs and the surrounding materials, the QD size and shape determines the number of electron confined states in the QDs and their corresponding energies. The energy band-offset is significantly influenced by the strain formed in the QDs [20]. In our QD model the elastic deformation is calculated using a continuous medium model and the strain is obtained by minimizing the total elastic energy. We used for this purpose the simulation software, nextnano3 [35], which provides the conduction band and valence band energy diagram, accounting for the band structure modification due to strain. The energy states confined in the QDs were calculated in conduction band and solving the single-particle Schrodinger equation [20]. As suggested in [36] the effective mass of the electrons was modified with respect to the bulk case for accounting of the strain modification. As shown in Fig.2b, we
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have a GS, a first ES (ES1) with double degeneracy and a second ES (ES2) with degeneracy of 3. The figure also shows evidence of the energy position of the WL. As shown in the inset of Fig.2c, at this energy level the electron wavefunction is poorly confined in the QD and almost spread in the thin InAs substrate. These electrons can therefore be coupled with adjacent dots (separated by about 20-10 nm when the dot density ranges between 5 − 10 · 1010cm−2 ) and can be shared by all of the ensemble. In this example (InAs QDs grown on GaAs for emission at 1.3µm) the energy separation between the GS and the GaAs barrier is quite big (about 280 meV). This means that when a carrier is captured from the GaAs barrier with a time constant of about 25 ps, it can escape back to the barrier with a time constant of about 800 ps. Such a large difference between these time constants have several effects: 1) the carrier thermalization out of the QD is reduced and therefore the characteristics of the device should be less sensitive to temperature effect [37], 2) the carrier diffusion lengths is reduced because the only possible mechanism for diffusion is the thermal escape out of the dot with the consequent diffusion in the WL (for carrier diffusion in one layer) or the 3D GaAs barrier (for carrier diffusion between different layers) [38]. As it will be shown in section 6 this slow escape time can have a negative effect on the uniform carrier filling of the different layers when a multi-layer structure is used. To study the carrier dynamics with our MPRE model we consider only the electron dynamics in the hypothesis that the holes can follow the electrons (exciton model [23]). We also neglect the presence of the second ES (ES2 in Fig.2b) because for the injection current used in practical laser applications the ES2 is always empty. Only recently it has been shown that this second excited state can be used to increase the output power and the emission bandwidth of QD SLDs but at the expense of a higher injection current. We are, however, aware that a more detailed model should include the presence of ES2 as well as equations for modelling separately the hole dynamics (see section 6). Our simplified picture has the advantage of representing quite well the physics of the inhomogeneous material with a reasonable number of equations that can be solved in an acceptable computation time. To include the effect of the size fluctuation of the QDs, we have divided the QD ensemble in several sub-groups [25], each characterized by an average energy of the GS, EGSn and of the ES, EESn respectively. We assume that the QD size distribution is gaussian with a consequent gaussian distribution of the recombination energies from the GS and the ES. In Fig.3 we present a schematization of the carrier dynamics in the QDs. The carriers are injected in the SCH barrier, relax in the WL state with a rate 1/τs or escape back in the barrier with the rate 1/τqe ; from the WL they can be captured in the dots of different size. Therefore the WL state acts as a common reservoir from which the carriers are captured in the ES of the n−th subgroup with a rate 1/τcn and from the ES to the GS with rate 1/τdn . The carriers escape also from the GS back in the ES with rate 1/τeGSn or from the ES back in the WL with rate 1/τeESn . It is generally assumed that carriers can fill the GS only via a relaxation process from the ES [23]. Very recently it has been shown that in QDs grown on InP a direct capture from the WL to the GS may also be possible [39]; in this case an additional capture channel has been added to the model with rate τd2 as shown in dashed line in Fig.3. Carriers can also recombine with radiative and non radiative processes from the SCH, from the WL and from the various confined states (GS,ES). The rates 1/τsr , 1/τqr , 1/τr in Fig. 3 indicate the time constants of the radiative recombination from the SCH, the WL, and the QD respectively. We assumed that the stimulated emission can take
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place only due to recombination between the electron and hole in the ES and in the GS. The rate of photons emitted out of the cavity is Sm /τpm with τpm the photon lifetime. Sm represents the photon density of each m-th longitudinal mode of the laser (see section 3.2) or the photon density in the m − th energy slice of the SLD (see section 3.3).
Figure 3. Schematic of the carrier dynamics in a QD material.
2.2.
Multi-population Rate Equations for QD Lasers
We use the following rate equations to describe the carrier dynamics in the various QD states and to describe the photon dynamics in the several cavity longitudinal modes. dNs dt dNq dt dNESn dt
= = = −
dNGSn dt
dSm dt
I Ns Ns Nq − − + e τs τsr τqe X Ns NESn Nq Nq Nq + − − − τs τeESn τqr τqe τc n
(1) (2)
Nq Gn NGSn (1 − PESn ) NESn NESn + − RspESn − RAugerESn − − + τcn τeGSn τeESn τdn cΓ X gmnES Sm , n = 0, 1, ... , N −1 (3) nr m
NESn NGSn (1 − PESn ) − −RspGSn − RAugerGSn − + τdn τeGSn cΓ X gmnGS Sm , n = 0, 1, ... , N −1 − nr m Sm Nm cΓ X = βsp + (gmnES + gmnGS )Sm − , τr nr n τpm =
(4)
(5)
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In the system of equations (1)-(5) we have one rate-equation for the total number of carriers in the SCH (Ns ), one rate-equation for the total carriers in the WL (Nq ) and N rate-equations for the carriers in the various QD subgroups ( NESn and NGSn ). These equations are then coupled with the lasing photon rate equation (5). In equation (5) the first ) represents the rate of photons emitted by spontaneous term on the right end side (βsp Nτm r emission coupled into the lasing mode. In equation (5) the stimulated emission rate of photons is proportional to the gain at the energy Em . The gain is calculated as the sum of the contributions from the various states (GS,ES) and the various QD sub-groups. The terms gmnGS and gmnES in equations (3), (4) and (5) are calculated as: gmnGS = µGS Cg ND
σ 2 |PGS | (2PGSn − 1) Gn Bcv (Em − EGSn ) , EGSn
(6)
σ |2 |PES (2PESn − 1) Gn Bcv (Em − EESn ) . (7) EESn The expressions (6) and (7) represent the gain at Em due the GS (gmnGS ) and of the ES (gmnES ) of the n-th sub-group of dots. e µGS = 2 and µES = 4 are the degeneracy of the GS and ES levels including the spin; Cg is a constant [40], ND is the dot density, Γ is σ 2 | is the density matrix momentum the optical confinement factor in the active region, ,|PGS [26], PGSn and PESn are the carrier filling probability of the GS and ES calculated as:
gmnES = µES Cg ND
PGSn =
NGSn NDtotGn
and
(8)
NESn (9) NDtotGn with NDtot the total number of dot in the device and Gn is the existence probability of a the n−th QD sub-group. Gn is calculated assuming a Gaussian distribution of the QD size. Bcv (E − EGS,ESn ) in equations (6) and (7) is the homogeneous broadening function with width ~Γhom [26] and accounts for exciton dephasing time [41]. The shape of the homogenous linewidth is assumed to be Lorentzian, even if more complicate shapes have been measured [41]. The full width half maximum of the Lorentzian is generally taken from experiments; ~Γhom depends significantly on the temperature, on the current injection and can be different for emission from the GS and ES [41]. From the stimulated emission term in equations (3) and (4) we see that the photons Sm with energy Em can deplete all the carrier populations NESn ,GSn inside the homogeneous broadening linewidth Bcv (Em − EESn ,GSn ). As a consequence the various QD sub-groups are coupled together not only via the common WL reservoir but also by the homogeneous broadening of the emission linewidth. This effect becomes significant when the inhomogeneous broadening is comparable with the homogeneous linewidth [25]. The application of the MPRE to the study of QD-LD has also shown that homogeneous linewidth influences: PESn =
• the shape of the laser emission spectrum [25] • the turn-on delay during the switch-on transient [6] • the frequency chirp of the directly modulated laser [42]
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The terms RspGSn,ESn in equations (3) and (4) represent the spontaneous emission rate from each state. They are calculated as the integral, over all of the energy emission range, of the spontaneous emission rate from the GS or the ES of the n−th sub-group. The spectrum of the spontaneous emission rate (rsponGSn,ESn ) is given by: σ rsponGSn,ESn (E) = µGS,ES nL ND Csponρopt Gn|PGS,ES |2 Bcv (EGSn,ESn − E) in eV−1 s−1 m−2 (10)
and RspGSn,ESn = wL
Z
∞
rsponGSn,ESn (E)dE
(11)
0
where nL is the number of QD layers, w is the waveguide width, L is the device lengths, ρopt is the density of the optical modes and Cspon is a constant similar to Cg . The terms RAugerGSn,ESn represent the Auger non radiative recombination rate from the QD confined state. One electron of the GS/ES recombines with one hole of GS/ES and release the energy to another GS/ES electron of the same dot. This second electron is scattered to higher energies in the SCH. This rates are calculated according to the model recently proposed in [Blood]; in our MPRE model the Auger non radiative recombination rates are written as:
RAugerESn =
4 X k=2
PESn µES ! k! k µES −k P (1−PESn ) Gn ND (12) (k − 2)!2! τAuger (µES − k)!k! ESn
and RAugerGSn =
2 PESn Gn ND τAuger
(13)
where τAuger is a time constant, characteristic of this non-radiative recombination, and in the range 0.2-0.5 ns [43]. The size inhomogeneity of the dots is also included in the expression of the capture and c0 escape times; in equations (2)-(4) these time constants are given by: τcn = (1−PτESn )Gn , P τd0 −1 −1 τdn = 1−PGSn and τ c = n τc0 (1 − PESn )Gn ; where τc0 and τd0 are the average capture time from the WL to the ES and from the ES to the GS when the final state is empty. τ c is the average capture time from the WL into the QD ensemble and is weighted by the existence probability Gn of each nth subgroup and by the probability of having an empty place in the ES of the n-th subgroup (1−PESn ) [26]. At room temperature and without stimulated emission the system must converge to a quasi-thermal equilibrium characterized by a Fermi distribution of the carriers in all the states. To ensure this convergence we impose an equilibrium condition: in absence of photons we must have equilibrium between the ES and the N G N (1−P ) = GSnτe ESn ). We calcuWL ( τqc n = τNeESn ) and between the GS and the ES ( NτESn dn n ESn GSn late then the escape times as function of τcn and τdn forcing Nq , NESn and NGSn to follow a Fermi distribution with a common quasi-Fermi level. The carrier escape times are theree fore given by: τeGSn = τd0 µµGS ES
EESn −EGSn kB T
and τeESn =
µES ND VA ρW Lef f VW L
τc0 e
EW L −EESn KB T
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with n = 0 , 1 , . . . , N − 1. In the above expressions ρW Lef f is the effective density of states in the WL, VA and VW L are the volume of the QD active region and of the WL respectively. From the expression of τeESn we see that the carrier escape time out of the QD is dependent on the ratio between the number of available states in the QDs and in the WL and also on the energy separation between ES and WL. The values of the capture, relaxation and escape times are important parameters determining several key properties of the QD lasers. In particular it can be shown that: 1. the value of τc0 and τd0 influence the slope of the light-current characteristics as well as the separation between the GS and the ES threshold currents (see [44], [23] and section 4.1) 2. τc0 and τd0 influence the laser response during the laser switch-on (see [45] and section 4.3) and determine the small signal modulation bandwidth of the laser 3. a big escape time out of the QDs (obtained for example when the energy separation EW L − EESn is high) reduces the number of carriers in the WL and therefore the carriers which can diffuse in the following layers (see section 6) 4. the dependence of the escape time on the ratio between the ES and WL density of states influence significantly the laser dynamics and limit the modulation bandwidth [29, 30] Typical values for τc0 and τd0 have been measured of few picoseconds for the electron dynamics [46] and less than one picosecond for the hole dynamics [47]. The hole dynamics is characterized by a fast capture in the QDs and a fast thermalization out of the QDs [47]. The measurements have also shown that these time constants are inversely proportional to the carrier density accumulated in the WL [39], indicating that the Auger carrier-carrier scattering fasten the capture and the relaxation in the QDs [48]. For this reason the dependence on NW L should also be included in the expressions of τc0 and τd0 as: τc0 =
AW
1 + CW nW L
(14)
and 1 (15) AE + CE nW L where nW L is the WL carrier density per unit volume. The values of the coefficients AW,E and CW,E where experimentally estimated of about AW = 1.35 · 1010s−1 , CW = 5 · 10−15m3 s−1 , AE = 1.5 · 1010s−1 , CE = 5 · 10−14 m3s−1 for InAs/InP QDs [49] and AW = 1 · 1012s−1 , CW = 1 · 10−14m3s−1 , AE = 1 · 1011s−1 , and CE = 7 · 10−12m3 s−1 for InAs/GaAs QDs [50]. In our model we include also the calculation of the refractive index variation due to changes of the carrier density in the QDs, in the WL and in the SCH [40], [51]. The variation of the refractive index at the lasing mode energy Em is given by the sum of two contributions. The first contribution, ∆nQD in equation 16, is the term linked, through the Kramers-Kroning relation, to the gain variation in the active layer. The second contribution τd0 =
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is due to the plasma effect (∆nplasma in equation 18) and is caused by the free-carrier accumulation in the two dimensional WL and the three dimensional SCH [52]. The expressions of these two contributions are calculated as follows:
∆nQD (Em) = Γ
~c Cg ND 2E
X
X
k=GS,ES
n
µk
|Pkσ |2 (2Pkn − 1) Gn Ncv (Em − Ekn ) . Ek
(16) where the index k stays for the GS or ES transition; Ncv (E) is the homogenous broadening function of the refractive index spectrum, it is given by: (Em − Ekn )/π (Em − Ekn )2 + (~Γhom )2
Ncv (Em − Ekn ) =
(17)
The free-carrier contribution is given by: ∆nplasma (Em) = ΓSCH Kn
Ns Nq + ΓW L Kn 2 2 Em Em
where ΓSCH,W L is the optical confinement factor in the SCH and WL and Kn = The frequency fluctuation (chirp) of the lasing mode is then given by: ν(t, Em ) = −
Em ∆nef f (t, Em) 2π~ng
(18) ~2 e2 2ε0 nr m∗e .
(19)
where ∆nef f (t, Em) is the total effective refractive index variation given by the sum of the terms in (16) and in (18). For studying the frequency dynamics of the lasing mode it is important to note that a variation of the carrier density in the state Ekn causes in turn a variation of the refractive index at the lasing energy Em due to the Ncv (Ekn − Em) broadening function. For clarity sake the broadening functions Bcv (E − Em ) and Ncv (E − Ekn ) are shown in Fig. 4; we observe that the width of the homogenous broadening function Ncv in Fig. 4b is much larger that the width of Bcv . As shown later in section 4.3.2 this has important consequences in the chirp of QD-LD.
2.3.
Multi-population Rate Equations for QD SLDs
The output power of the SLD is obtained by the amplification, along the device, of the spontaneous emitted photons coupled with the waveguide mode. The SLD is modelled using a travelling-photons approach [53]. As shown in Fig.5a the device is divided in several sections indicated with zh and with sampling step ∆z. The photon density in each node zh is obtained from the propagation of the forward and backward photons of the two nearest sections . Given the importance of the SLD emission spectra in practical applications, the model is also based on a spectral slicing approach. In each node zh the model calculates the spectrum of the spontaneous emission given by the sum of contributions in equation (10) from each QD sub-group: X rsponk n (E, zh) (20) rspon (E, zh) = k=ES,GS
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Figure 4. Homogeneous broadening function of (a) the gain and (b) the refractive index variation. As shown in Fig. 5b the spontaneous emission rate is then sliced in several energy intervals, each with energy Em and with sampling step ∆Em. The total spontaneous emission in the spectral slice ∆Em is given by: Z rspon (E, zh)dE (21) Rspon (Em , zh ) = ∆Em
Figure 5. (a) Schematic of the longitudinal discretization of the QD-SLD and (b) spectrum of spontaneous emission rate calculated in the zh slice. The photon density in each node zh is given by the sum of the forward and backward contributions, propagated from the adjacent slices, plus the spontaneous emission in the zh section: Sm (zh ) =
X f,r
f,r Sm (zh ∓∆z) exp (Γ
X
gmnk (zh ) − αi )∆z+βsp Rspon (Em , zh )∆tw∆z
k=GS,ES
(22) where the apices f, r stands for forward and reverse direction respectively. The time step ∆t is related to ∆z through the group velocity vg : ∆t = ∆z/vg . βsp is the amount of
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spontaneous emission coupled with the guided mode. The photon density Sm in expression (22) is used in the stimulated emission terms of equations (3) and (4). In the first and last longitudinal section of the device (ie: zh = 0 and zh = L) we set the boundary condition to account for the field reflected back in the cavity due the non zero facet reflectivity caused for example by non ideal AR coatings. Solving the MPRE model in each zh section self-consistently with equations (20)-(22) we get the output power and output spectrum of the SLD as well as the spatial distribution of photons along z. We can therefore calculate how much the carrier population in the GS and ES are depleted by the photons of the various sections (spatial hole burning) and how much the different QD sub-groups are depleted by the photon density in the different spectral slices (spectral hole burning).
3.
Numerical Results: QD Lasers
In this section we analyze the static and dynamic characteristics of QD-LD using the MPRE model. All of the characteristics presented are at room temperature, because the model is not suitable for low temperature simulations where the hypothesis of thermal distribution of carrier is not satisfied. At low temperature the master equation model can indeed provide more correct results [54] and can predict the negative T0 measured at low temperature in several lasers [55]. The QD-LD characteristics are presented in this section following the characterization sequence generally followed in the lab: first static characteristics with light versus current curves (L-I) and opticla spectra, then small signal analysis to extract the parameters useful to quantify the dynamic properties and finally large signal analysis.
3.1.
Static Characteristics of QD Lasers
QD semiconductor lasers present unique properties respect to quantum well and bulk lasers. Due to the large inhomogeneous broadening of the gain and the slow capture and relaxation times, the carrier density is no more clamped at threshold but it continue to increase with current in those QD-subgroups which are not resonant with the lasing modes [23]. This can lead to very broad emission spectra above threshold and in some cases to the turn-on of the ES lasing, simultaneous with the GS lasing [23]. In Fig.6a we show the typical power versus current characteristic and in Fig. 6b the laser spectrum calculated applying the MPRE model presented in section 3.2 to the analysis of a Fabry-Perot laser. The GS threshold is at 25 mA; at 100 mA we observe in the spectrum also the ES emission indicating also the ES is lasing. From the spectra we see that, when the emitted power saturates at one wavelengths (ie. 1315 nm at 50 mA), the spectrum continues to grow and broaden on the shorter wavelengths side. This is because, with increasing current, the QD sub-groups at lower energies continue to collect carriers and eventually the gain can reach the threshold value. These very broad spectra obtained by the inhomogeneous gain broadening plus the double lasing has been used for the realization of broad band lasers useful for low cost CWDM in silicon photonic technologies [16]. In [44] it has been shown that the double-lasing emission is significantly dependent on the electron dynamics through the time constants τd0. In Fig. 7 we compare various power
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Figure 6. Power versus current characteristic (L-I) of a QD laser and (b) corresponding laser spectra calculated applying the MPRE model to the analysis of a QD Fabry-Perot laser. versus current characteristics and lasing spectra obtained changing the capture and relaxation times. We see in Fig. 7a that the variation of τc0 has only effect on the slope of the L-I curve but does not cause the turn-on of the ES. The corresponding spectra are shown in Fig. 7b and are calculated for an output power of 20 mW. We observe that increasing the capture time (slower carrier injection in the QD) the spectrum broadens significantly towards higher energy. This spectral broadening is caused by the lasing of more and more QD-subgroups, because, for slower capture, the central lasing QD sub-group can not collect carriers at a rate fast enough to compensate the loss due to stimulated emission . Adjacent sub-groups have therefore time enough to collect carriers in competition with the central sub-group and reach also the threshold. The broadening of the spectrum is indeed accompanied by the formation of an hole at the center of the spectrum which is the sign of a strong spectral hole burning effect. The trends are different changing the relaxation time τd0 in Fig. 7c and 7d. In this case an increasing of τd0 from 7 ps to 20 ps changes the threshold,the slope of the L-I curve (Fig. 7c) and the width of the lasing spectrum (Fig. 7d). A further increase of τd0 to 40 ps causes also the turn-on of the ES emission. When the ES turns on the with of the GS emission remains quite narrow, because all of the injected carriers are now used to maintain the lasing of the ES and do not have time enough to relax to the GS to make more GS sub-groups participate to the lasing.
3.2.
Small Signal Analysis
The dynamic properties of QD lasers have been at the center of an intense debate generated by the fact that the measured modulation bandwidth of QD lasers is generally comparable or even less than the one obtained in quantum well lasers [2], [56]. Simple theoretical models had indeed predicted that the increase of the carrier confinement from a 2D quantum well to a 3D box should have led to an increase of the modulation bandwidth. Broad modulation bandwidth is indeed required for fast optical communication systems and QD lasers could be promising used for this application. More sophisticate models [29], [57], which includes the realistic properties of the QD material, have shown that the modulation bandwidth is limited by: • the inhomogeneous broadening of the gain [57]
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Figure 7. (a) L-I curve and (b) output spectra calculated for different values of capture time c. (c) L-I curve and (d) output spectra calculated for different values of relaxation time d. • the carrier escape to energy states with energy higher than the resonant QD state (for example carrier accumulation in the WL) [29] • the hole thermalization among the several confined states in valence band [58] These factors all together bring to a reduction of the differential gain and to an increase of the gain compression factor [2], [57], [58]. 3.2.1. IM Response of QD Lasers The intensity modulation (IM) and frequency modulation (FM) [24] response of the laser can be numerically calculated from the model presented in section 3.2 simulating the laser response to a very small current step given around a bias point [30]. From this analysis we have calculated the -3 dB modulation bandwidth of a single mode laser (SML) with lasing wavelength centered at the GS peak (GS-SML) or at the ES peak (ES-SML) wavelengths respectively. The result reported in Fig. 8 shows that the modulation bandwidth is significantly limited by the inhomogeneous broadening of the gain spectrum and is higher operating at the ES wavelengths [31]. On the ES we have indeed an increased differential gain thanks to the double degeneracy of the ES. The limit coming out from the inhomogeneous broadening of the gain can be overcome only forcing the injection of carriers in the resonant QDs as done for example with the tunnel injection of carriers from a quantum well adjacent to the QD layer [17], [31].
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Figure 8. Calculated -3dB modulation bandwidth as function of the ratio between bias and threshold current for different single mode QD lasers emitting from the GS (GS-SML) or from the ES (ES-SML) and with different values of the inhomogeneous broadening of the recombination energy ( ∆Einhom ). An important parameter usually used to quantify the small signal properties of semiconductor laser is the differential gain. For the QD laser this parameter has been studied in detail in [57], [58]. In the lab it is usually extracted from a measurement of the resonance frequency of the IM responce as function of the bias output power. We repeated with simulations the same kind of experiments done in the lab in order to extract an equivalent differential gain above threshold and compare it with the one obtained at threshold [40]. In Fig. 9a we report the square of the resonance frequency fR as function of the bias photon 0 ) as: density Np. From the values of fR we can extract the equivalent differential gain (geq 0 = geq
(2πfR)2 τpm . vg Np
(23)
where τpm is the photon lifetime at the lasing energy Em . In Fig. 9b we report the extracted 0 as function of Np; as expected it is higher without inhomogeneous broadening but in all geq of the cases it reduces with increasing Np. The reduction of the differential gain is caused by the strong compression factor of QD lasers, much higher respect to quantum well and bulk lasers [2]. To quantify the gain compression in the QD case and compare it with the 0 as function values reported in the literature for the quantum well or bulk case we fitted geq of Np with the following expression: 0 (Np) = geq
geq (Np = 0) 1 + ε1 Np + ε2 Np2
(24)
where we define ε1 as the gain compression factor [24] and ε2 as a second order compression factor necessary for the fitting. For the laser with inhomogeneous gain broadening (∆Einhom = 40 meV ) we got: ε1 = 3.3 · 10−16cm3 and ε2 = 1.4 · 10−30cm6 for the GS-SML; ε1 = 0.9 · 10−16 cm3 and ε2 = 0.1 · 10−30cm6 for the ES-SML. These are about
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one order of magnitude higher than the compression factors obtained in InGaAs/InGaAlAs quantum well lasers and are consistent with those measured in the experiments. For example in [2] it is reported a measured gain compression coefficient of 4 · 10−16cm3 for a InAs/GaAs QD laser emitting from the GS. The strong compression factor in QD lasers is caused by two effects: • the slow capture/relaxation rate of carriers which can not fill the lasing state at a rate fast enough to compensate the carrier loss due to stimulated emission (spectral hole burning). • the inhomogeneous distribution of the QDs which causes states that can capture carriers in competition with the lasing states
Figure 9. (a) Resonance frequency versus photon density ( Np) of a GS-SML and an ESSML with and without inhomogeneous gain broadening due to QD size dispersion. (b) Equivalent differential gain extracted from the curves in (a). Figures reproduced from [40] with kind permission of Springer Science and Business Media.
3.2.2. FM Response and ”α”-Parameter of QD Lasers An important parameter characterizing the small signal FM response of the laser is the linewidth enhancement factor (LEF) or α-parameter [13]. It was predicted that QD laser with a gain spectrum perfectly symmetric around the gain peak should lead to zero linewidth enhancement and absence of frequency chirp [2] which is the ideal condition for long haul transmission systems. Several measurements have shown that the LEF in QD lasers can be low [59], but it can never be zero. It is also very dependent on the working conditions and on the measurement procedure [40], [51], [60]. We have used the MPRE model to calculate the LEF of the laser below threshold and reproduce some of the experiments done in the lab to extract the LEF above threshold [40]. In Fig. 10a we report the spectra of the LEF calculated at the threshold current of the GS-SML and the ES-SML. At threshold the LEF dnef f has been calculated applying the definition α = − 4π λ · dgef f , where dnef f and dgef f are the variation of the effective refractive index and of the gain caused by a small current change around the threshold point. The variation of refractive index and gain where obtained from expressions (6), (7), (16) and (18). The LEF above threshold can be measured using various techniques such as the linewidth method, the FM/IM method and the optical feedback method [13]. For the QD
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laser the LEF above threshold have been extracted measuring the FM/IM response [61]. According to this technique, the LEF is given by the asymptotic value of the FM/IM ratio evaluated well above the resonance frequency. The calculated FM/IM is shown in Fig. 10b. The asymptotic value at high frequencies gives αeq = 0.31 for the GS-SML and αeq = 0.13 for the ES-SML. We obtain a discrepancy of about 30% respect to the threshold value for the GS emission and about 60% for the ES emission. We have also observed that the differences between the LEF static value (at threshold) and the dynamic values (from FM/IM) is strongly dependent on the operating wavelength and on the bias current as confirmed also by the experiments [62]. This discrepancy is caused by the following reasons: • the carrier density is not clamped at threshold and therefore changing the bias point we get a different number of carriers in the non-resonant states. Carriers accumulated in the non-resonant state can change the refractive index of the lasing mode (see section 4.3.2 and [42]). • The carriers in the non-resonant states can respond in a different way (slower or faster) to a small signal current modulation. The type of the response depends on the position of the state respect to the WL and on the existing filling of the state. • The carrier accumulation in the WL and in the SCH increases with increasing bias and causes an additional variation of the refractive index due to plasma-effect [51]
Figure 10. (a) spectra of the LEF calculated at the threshold current of the GS-SML (dashed line) and of the ES-SML (solid line). The arrows indicate the operating point of the two lasers. (b) FM/IM ratio versus the modulation frequency. The asymptotic value at high frequencies gives the LEF above threshold. Figures reproduced from [40] with kind permission of Springer Science and Business Media.
3.3.
Large Signal Analysis
In this section we analyze the laser response to a large signal current modulation; in particulary we study the dynamics of the photons and of the refractive index when the laser is modulated with a large signal current step or with a large signal sinusoidal modulation.
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3.3.1. Laser Switch-on The laser switch-on (ie: response to a current step from below to above threshold) has been studied in [6], [30], [45], [63]. In [6] it was shown that the width of the homogenous broadening linewidth influences delay of the turn-on response and the number of modes that switch-on during the transient. In [45] it was evidenced that during the turn on we can have both GS and ES emission. In [63] it was shown that the damping of the relaxation oscillations, present during the transient, is also dependent on the Coulomb interaction between electrons and holes. We have used the MPRE model to analyze the effects of the capture and relaxation times on the characteristics of the laser turn-on. In Fig. 11 we analyze the laser response to two different current steps: from the threshold Ith to 2Ith and from Ith to 4.2Ith . The laser analyzed in this case has a slow capture time in the QD ( τc = 30 ps) and a faster relaxation time from ES to GS (τd = 7 ps). From Fig. 11 we see that the response is dominated by the GS dynamics. The ES emission appears, only during the transient, when a larger current step is applied (Fig. 11b). With the time constant of these simulations, the GS of the resonant QDs can indeed be filled by carriers at a rate fast enough to compensate the photon loss due to fast stimulated emission rate; as a consequence carriers never accumulate on the ES to allow it to reach threshold. The ES emission is only possible when the current step is large enough to cause an initial high injection rate in the ES such that the ES accumulates carriers at the very beginning of the transient and lases as far as the carrier do not relax down to the GS and make the GS lasing. In Fig. 12 we analyze the opposite case of a fast capture time (τc0 = 7 ps) and a slower relaxation time (τd0 = 17.5 ps). As shown in Fig. 12, the type of the response changes when the time constants are inverted (fast capture, slow relaxation). We have pure GS emission only for quite small current step (Fig. 12a). For a bit larger current step (Fig. 12b ) we have both GS and ES emission with an ES turn-on fast and strongly over-damped and a GS turn-on significantly slow and under-damped . With the time constant of these simulations, the ES is indeed filled at a rate fast enough to cause a fast accumulation of carriers in the ES and make it lasing before the carrier can relax to the GS at a much slower rate. The filling of the GS is therefore delayed not only by the slower relaxation rate but also by the stimulated emission on the ES. This is shown in Fig 12c where we observe first the lasing of the ES only (from 6.5 ns to 7.0,ns), and then the turn-on of the GS at 7.2 ns. When the GS turns on, the ES starts losing power because some of the injected carriers relax to the GS and no more sustain the ES emission. 3.3.2. Chirp Analysis Using our MPRE model, we have studied in detail the chirp properties of modulated QDLD [42]. We have shown that QD lasers can give non zero frequency chirp even in the most promising condition when the LEF is zero at the laser threshold. Applying equation s(16) and (19), we see that the homogeneous broadening of the refractive index couples the carrier variation of several states with the refractive index variation at the lasing frequency. This concept is shown in Fig. 13. We have two kinds of coupling: 1. as in Fig. 13a the coupling is via the homogeneous broadening of the gain: the lasing
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Figure 11. Turn-on dynamics of a QD laser with τc = 30 ps and τd = 7 ps under the injection of a current step (a) from Ith to 2Ith and (b) from Ith to 4.2Ith. Solid line represents power emitted from the GS, the dashed line power emitted form the ES.
Figure 12. Turn-on dynamics of a QD laser with c=7 ps and d=17.5 ps under the injection of a current step (a) from Ith to 1.5Ith , (b) from Ith to 2.1Ith and (c) from Ith to 2.5Ith . Solid line represents power emitted from the GS, the dashed line power emitted from the ES. photons with energy Em depletes the population with energy Ek n . A variation of carriers in Ek n causes in turn a variation of refractive index in Em 2. as in Fig. 13 b the coupling is only via the homogeneous broadening of the refractive index: carriers in states Ek n , far away from the lasing energy, can fluctuate (for example due to current modulation) and can cause a refractive index variation at Em due to the broader Ncv function. To quantify and compare the chirp under large signal modulation and in various operating conditions (change in bias current, modulation frequency, modulation depth ...) we have defined in [42] an equivalent ”α-parameter” that quantifies the transient contribution to the chirp [24]. We have shown that the equivalent ”α-parameter” is significantly dependent on the working conditions (GS or ES lasing, bias current, modulation depth and frequency ...).
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Figure 13. Schematic of the two mechanisms responsible for the refractive index variation at the lasing energy Em caused by the carrier variation inside the QD. A third mechanism, not shown here, is the carrier accumulation in the WL and SCH. The definition of an equivalent ”α-parameter” has also been useful to study the physical origin of the non-zero chirp. The non-zero frequency chirp is given by the combination of several contributions to the refractive index variation. These several contributions are due to carrier accumulated in the following states: • in QD states resonant with the lasing mode (lasing contribution, coupling as in Fig. 16a with Em = Ek n ) • in states non-resonant with the lasing mode and with energy higher/smaller than the lasing frequency (blue/red contributions, coupling as in Fig. 16a Em 6= Ek n ) • in the ES when the lasing is from the GS or carriers in the GS when lasing is from the ES (ES/GS contributions, coupling as in Fig. 16b)
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• in the WL and SCH , defined as plasma contribution according to equation (18) As an example we report in Fig. 14a the calculated output power and the frequency chirp of a single mode laser emitting from the GS and with LEF null at threshold. The laser has been modulated with 2.5 GHz sinusoidal modulation with bias current of 25 mA and peak to peak amplitude of the modulation of 44 mA. In Fig. 14b we plot the chirp in the power-frequency plane (fish-diagram [64]). This representation is useful to identify the adiabatic contribution to the chirp proportional to the power (straight line, in Fig. 14b) and the transient contribution as the deviation from the straight line. The same analysis has been repeated for a single mode laser emitting from the ES with LEF null at threshold (Fig. 14c). The comparison of the fish diagrams of Fig. 14b and Fig. 14c shows that we get higher chirp when we operate on the GS. Our analysis in [42] has shown that in both cases the lasing contribution is always null because the refractive index variation due to carrier variation in resonant QDs is always zero. Blue and red contributions have always opposite signs (even symmetry of Ncv ) and tend to minimize the chirp. For the GS emitting, laser the ES contribution is always the dominant contribution to the chirp; in this case a current modulation produces indeed a large carrier variation in the ES which causes a large refractive index variation on the GS. On the contrary, when the laser is emitting from the ES, the major contribution to the chirp are always due to the non exact compensation of blue and red contributions, whereas the lasing and GS contributions are null. The plasma contribution is significantly dependent of the separation between ES and WL and was practically null in our analysis. It can play a significant role if the variation between ES and WL reduces or when the ES is absent as in [51].
4.
Numerical Results: QD SLD
QD SLD has gained an increasing interests in the last few years as promising broad spectrum devices for sensors and medical applications that require optical sources with broad emission spectrum and high output power. The QD material seems indeed a good candidate for generating broad amplified spontaneous emission thanks to the large inhomogeneous broadening of the gain spectrum [65] as well the presence of more than one state (ie: GS plus ES) in the QDs [11], [12]. For obtaining a broad output spectrum at relatively small injection currents, it is indeed important not only a broad and flat optical gain spectrum, obtainable also with asymmetric quantum wells, but also an independent carrier filling of the states with different energies. This means that the thermalization of carriers at room temperature and in presence of photons should be avoided as much as possible, such that states with different energies (for example GS and ES) could be filled at the same level. This is not possible in quantum wells since the very fast intra-band relaxation times always lead to a quasi-Fermi distribution of carriers. On the contrary, in the QD case, the slower relaxation time from the ES to the GS, causes an accumulation of carriers in the ES even when we have fast stimulated emission from the GS. In this way the ES can give also a contribution to the amplified spontaneous emission and to the broadening of the spectrum. The same broad spectrum could be obtained with quantum wells (exploiting emission from a second confined state in the well) but at the expense of an higher injection current. In this section we will present results on QD SLD output spectra and light versus current
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Figure 14. (a) Output power and frequency chirp of the GS-SML. (b) and (c) Fish diagram of the GS-SML and the ES-SML. In (a),(b) and (c) the laser is modulated at 2.5 GHz with sinusoidal current with bias of 25 mA and peak-to-peak amplitude of 44 mA. The thick solid line in (b) and (c) indicates the leading edge (LE) of the pulse, the dashed line the trailing edge (TE); the arrows the time evolution. The thin line indicates the adiabatic contribution to the chirp, directly proportional to the power. In (b) ∆νmax LE,T E indicate the maximum of the transient chirp, obtained as maximum deviation from the adiabatic contribution.
curves calculated using our MPRE model. In Fig. 15a we report an examples of the typical output spectra and in Fig. 15b of the output power versus current characteristics of two QD SLDs. The simulated device is 6 mm long and we compare the cases of SLDs with 6 and 20 QD layers respectively. The waveguide structure and material composition is similar to the one reported in [12]. At low current and low power the spectrum is dominated by the GS contribution as shown in Fig. 15a. Increasing current the ES starts to be filled and a second peak appears on the spectrum (Fig. 15a). At high currents and high output power the spectrum is dominated by the ES contribution because the ES gain is nearly doubled respect to the GS gain thanks to the doubled degeneracy of the ES (Fig. 15a). Fig. 15b evidences also two important parameters used to evaluate the performance of the SLDs: the maximum -3 dB bandwidth (obtained when the power from the GS equals the power from the ES) and the total output power corresponding to this bandwidth (equal power point, EP). From Fig. 18b it is clear that the EP can be increased increasing the number of QD layers but at the expense of a higher injection current and at the risk of a non-uniform carrier filling of the various
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layers (see section 6 of this chapter). Alternative approaches for increasing the EP power consist in the use of p-type modulation doping to increase the number of holes per dot and therefore the gain [11], in the optimization of the waveguide design to increase the optical confinement factor in the QDs, in the increasing the dot density per layer.
Figure 15. (a) Output power spectrum and (b) L-I curve of a QD-SLD. In (b) we compare two SLDs with 6 (dashed line) and 20 (solid line) QD layers respectively. The insets in (b) show the output spectra and -3dB bandwidth in the EP point (when power from ES equals the power from GS). As shown in the insets of Fig. 15b the spectrum in the equal power condition has a significant dip at the center of the -3 dB bandwidth due to the separation between the GS and ES emission. The flatness of the spectrum can be improved increasing the inhomogeneous broadening of the QD material. This is possible in several ways: increasing the sensitivity of the emission wavelengths to the QD size fluctuations [65], growing QD layers with different emission wavelengths obtained for example changing the QD size [15], or changing the composition or the thickness of the surrounding well (chirping [66]). In Fig. 16a we show the GS and ES emission wavelengths calculated changing the thickness of the InGaAs cap layer covering the InAs QD. This chirping scheme has then been used to simulate a SLD with three different layers (cap thickness 2 nm, 4 nm and 6 nm as shown in Fig. 16a) repeated five times. The calculated output spectra, shown in Fig ˙,16b, demonstrates that the chirping has been successful in improving the flatness of the spectrum.
5.
Limitations of the Model
As shown in this chapter the MPRE model is a useful tool for understanding the basic physics behind QD lasers and SLDs. The model suffers however of two major limitations: it does not account for the hole dynamics (being based on the hypothesis that the holes follow the electrons) and it does not consider the effect of the finite diffusion time of the carriers in the SCH. In this section we will show that the inclusion of separate equations accounting for the fast hole dynamics (non-exciton model) and the simulation of the carrier transport along the SCH have a significant impact on the optical properties of the QD material. As a consequence they could significantly affect also the performance of the lasers and SLDs.
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Figure 16. (a) GS and ES emission wavelength from QDs with different thickness of the InGaAs cap layer and (b) calculated output spectrum of an SLD with three different QD layers (cap thickness of 2 nm,4nm and 6 nm) repeated five times.
In [47] it was recently proved that the hole dynamics is characterized by a fast capture time in the QDs (less than 1 ps) and a fast escape time out of the QDs (about 1 ps). We have therefore modified our model including additional equations for the holes. As done in [47] we assume that the holes are captured from the WL in one equivalent state inside the dot; this equivalent state has a degeneracy equal to the total number of hole confined states in valence band. The hole states are indeed many more than the electron states because holes have an higher effective mass than electrons. The hole states have also an energy separation around the thermal energy or even less and therefore we assume that the holes in the QD are always in thermal equilibrium. In Fig. 17 we show a comparison of the gain versus current curves (Fig. 17a) and of the gain and the absorption spectra (Fig. 17b) calculated using an exciton and a non-exciton model. The figure shows that the non-exciton model predicts a lower gain respect to the exciton analysis. The lower gain is caused by the thermal spreading of holes in several states and by the fast escape of holes out of the dot with their consequent accumulation in the WL. This lower gain can lead to an increase of the threshold current and an increase of the laser T0 as well as a reduction of the differential gain [58]. In the SLD case it can lead to a reduction of the output power and an increase of the current necessary to reach the EP point. To study the effect of the carrier transport along the SCH we have very recently adapted the model presented in [67] to the case of the QD material. The QD material has been represented with our exciton MPRE model in the ideal case without inhomogeneous broadening. According to this model the injected carriers can diffuse along the SCH and can be captured in the first QD layers; from this layer they can recombine or escape out to diffuse to the next layer. We have seen that the very slow escape rate out of the QDs to the SCH state causes a very non uniform distribution of carriers among the layers with a consequent reduction of the gain as function of current. This is shown in the example reported in Fig. 18 where we plot the GS occupation probability in the different layers of an active waveguide having 6 (Fig. 18a) and 15 (Fig. 18b) layers respectively. The two figures show that in the case of 6 layers all the layers are uniformly filled and can reach saturation ( PGS = 1) at high injec-
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Figure 17. Comparison between exciton (thin lines) and non-exciton (thick lines) models: (a) gain peak as function of current density and (b) absorption and gain spectra.
tion current. On the contrary, when the number of layers increases, those layers which are further from the n-side are filled less and never reach saturation even if the current injection increases. As a consequence, the total modal gain per layer is reduced and the total modal gain does not increase proportionally to the increase of the number of layers stacked in the waveguide.
6.
Conclusion
We have presented a review of our research work on the modelling of QD lasers and superluminescent diodes. The modelling is carried on with a MPRE model to represent the carrier dynamics in an inhomogeneously broadened gain material characterized by the presence of several different energy states that can capture the injected carriers. We have shown that the model can be used to understand the basic properties of QD-LD and SLDs an in particular to study the static characteristics of QD-LD and SLDs as well as the dynamic properties (small and large signal responce) of QD-LD.
Acknowledgements The author wishes to acknowledge Prof. Ivo Montrosset from Politecnico di Torino for his assistance in reading and reviewing this chapter and supervising this research work. The author also thanks the students that during the years have been working on this subject; in particular, Alberto Sevega, George The, Mattia Rossetti and Paolo Bardella. This work was partially supported by the European Project Nano-UB Sources, FP6, contract 017128.
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Figure 18. Calculation of the GS occupation probability versus current in the different layers of an active waveguide with (a) 6 and (b) 15 layers. (c) Comparison of GS and ES gain per layer as function of the current per layer.
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In: Quantum Dots: Research, Technology and Applications ISBN: 978-1-60456-930-8 Editor: Randolf W. Knoss, pp. 203-242 © 2008 Nova Science Publishers, Inc.
Chapter 5
QUANTUM DOT TECHNOLOGY FOR SEMICONDUCTOR BROADBAND LIGHT SOURCES C.Y. Ngo and S.F. Yoon Compound Semiconductor and Quantum Information Group Nanyang Technological University, Singapore
S.J. Chua Institute of Materials and Research Engineering Agency for Science, Technology and Research (A*STAR), Singapore
Abstract Semiconductor broadband light sources (e.g., superluminescent diodes) are important light sources for fiber optic gyroscopes and sensors, optical fiber communications, and biomedical imaging. To minimize undesired interference effects in these applications, low temporal coherence of the light sources is desired. Since the wider the emission spectrum, the lower the temporal coherence; there have been great efforts to increase the spectral bandwidth of the emission spectrum so as to improve the performance of the abovementioned applications. Quantum dots (QDs) have recently been proposed as the perfect material for broadband light sources since the inherited inhomogeneity of the self-assembly QD growth mode is an intrinsic advantage for wideband emission. In this chapter, both the broadband light sources (together with its applications) and the justification for the use of QDs (as compared to higher dimensional systems) were mentioned. Existing methods to increase the spectral bandwidth were discussed. In particular, our theoretical and experimental approaches to optimization of the InAs QD layers for high areal density and wideband emission were presented. The origins of the high radiative efficiency and wideband emission of the optimized QD sample were also determined. Lastly, the potential challenges associated with the use of QDs were highlighted with the solutions proposed.
1. Introduction This section starts with an introduction to the common applications that require broadband light sources. In particular, it describes briefly the operating principles of the applications and
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the requirements needed from the light source for each application. This is followed by an introduction to the common types of broadband light sources available. The spectrum characteristics of the light sources are mentioned, which eventually leads to one, i.e. the semiconductor superluminescent diode (SLD), being the most popular light source for the mentioned applications. The main factor determining the broadband emission lies in the p-n junction, i.e. the active layer. The last portion of this section thus proposes why, as compared to higher dimensional systems, the quantum dot system will be a better material system for ultra-broadband (>100 nm) SLDs.
1.1. Applications of Broadband Light Sources Broadband light sources have been a topic of continuing research because of their wide range of applications, which include (i) fiber-optic gyroscopes [1], (ii) fiber-optic sensors [2], (iii) optical coherence tomography [3,4], and (iv) wavelength-division multiplexing transmission [5,6]. Fiber-optic gyroscopes (FOGs) are devices which measure rotation. They are generally used for surveying [7], positioning [8], and inertial navigation tasks [9]. A simplified FOG setup is shown in Fig. 1. The working principle is based on the measurement of the phase shift of the counter-propagating light waves after traveling though the coil of optical fiber that is subjected to rotation. Due to the Sagnac effect [10,11], light wave traveling along the rotation will take a longer time interval than the one traveling against the rotation. When the light waves recombine, they will no longer be in phase, thus resulting in a phase shift. This amount of phase shift, calculated from the decrease in the intensity, is proportional to the rotation rate of the rotating system. The output power of the light source is not very critical for FOG applications, and 10 mW is considered acceptable [12]. On the other hand, wide emission spectrum of the light source is critical for FOG applications since broadband light sources are useful for reducing the fluctuation of the Rayleigh backscattered light [13,14], thus resulting in the reduction of excess noise. This will increase the sensitivity for measuring systems with very low rotation rates.
Light source
Photodetector
Coupler
Optical fiber coil
: Incoming light wave (from source) : Outgoing light wave (to detector) : Clockwise light wave (in fiber coil) : Anti-clockwise light wave (in fiber coil)
Figure 1. Simplified illustration of the fiber-optic gyroscope (FOG) setup.
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Fiber-optic sensors (FOSs) can be used to detect changes in pressure [15], strain [16], temperature [17], current [18], etc. Compared to electronic-based sensors, FOSs are preferred since they are immune to radio frequency (RF) and electromagnetic (EM) interference, and they can be easily embedded within the structures to be measured. A simplified FOS setup is shown in Fig. 2, with the fiber Bragg grating (FBG) acting as a strain sensor. The gratings, written by high power ultraviolet (UV) laser, are actually periodic variation in the refractive index on the core of the optical fiber. When a broadband light source is coupled to the FBG, light with a wavelength (λ) that corresponds to the grating spacing will be reflected while the remaining spectrum will be transmitted. Changes to the strain of the FBG sensor will result in changes to the grating spacing. This, in turn, results in a new reflected wavelength (λ’) from the broadband light source that propagates through the FBG sensor. The amount of strain can then be obtained from the amount of wavelength shift, Δλ, where Δλ=│λ’–λ│. The output power of the light source depends on the applications and will be higher (tens of mW) if multiple FBG sensors are deployed over an extended areas. While a broad emission spectrum (> 40 nm) is desired, the profile of the light source is more important as it is necessary to have a flat emission spectrum for interrogation of the FBG sensors.
Fiber Bragg grating (FBG) sensor
Light source
Photodetector
Coupler
: Incoming light wave (from source) : Outgoing light wave (to detector) : Transmitted light wave (through FBG) : Reflected light wave (from FBG)
Figure 2. Simplified illustration of the fiber-optic sensor (FOS) setup.
Optical coherence tomography (OCT) is an advanced imaging technique for non-contact, non-invasive, high-resolution, cross-sectional imaging in both biomedical [19] and nonbiomedical applications [20]. Compared with other imaging techniques such as magnetic resonance imaging (MRI), and ultrasound scanning (USS), OCT can provide higher resolution of a few μm or even sub-μm depending on the light source used. A simplified OCT setup is shown in Fig. 3. The working principle is based on optical low-coherence interferometry, hence the need for the broadband light source. The light source is being split (by the beam splitter) into two, with one propagating towards the sample and the other one towards the movable mirror. The reflected light from the sample interfaces (or discontinuities) will combine with the reflected light from the mirror, thus creating an interference pattern. Due to the low coherence of the broadband light source, interference
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pattern only occurs within a narrow width of a few μm [21]. Therefore, as shown in Fig. 3, the reflectivity profiles of the three interfaces can be easily resolved. Consequently, locations of the discontinuities within the sample can also be determined. At this point, it is obvious that wide emission spectrum from the broadband light source is important for high resolution imaging since wider spectrum will give a narrower width for the interference pattern, thus improving the ability to resolve two interfaces that are very close together. However, in addition to the requirement of a broad spectral width, a Gaussian profile of the light source is also preferred for OCT applications since the spectral profile is equally important in determining the image quality [22].
Movable mirror
Sample with two layers
Light source
Beam splitter
Photodetector : Incoming light wave (from source) : Outgoing light wave (to detector) : Reflected light wave (from mirror) : Reflected light wave (from sample)
Figure 3. Simplified illustration of the optical coherence tomography (OCT) setup.
Wavelength division multiplexing (WDM) transmission in passive optical subscriber loop architectures can be achieved with the use of broadband light sources through the spectrum-slicing technique [23]. A simplified spectrum-sliced WDM setup is shown in Fig. 4. All the light sources (i.e. transmitters) can have identical spectral range since each port of the multiplexer has a unique spectral passband. Therefore, only the light that falls within the spectral passband of the respective input port will be able to reach the output port. Consequently, each input port will take a different “slice” of the broadband light source. Compared with WDM transmission using numerous diode lasers with different wavelengths, spectrum-slicing technique allows all users to use identical transmitters [24]. This greatly simplifies the installation, provisioning, and maintenance of the network. At this point, it is evident that wide emission spectrum from the broadband light source is beneficial for passive optical networks (PONs) since wider spectrum implies the ability to have more “slices”, thus
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more users. However, a drawback of the spectrum-slicing technique is the utilization of only a small fraction of power from each broadband light source. As such, this tight power budget restricts the per-channel bit rates compared to those achieved by diode lasers [5]. Hence, in addition to the requirement of a wide emission spectrum, high output power (hundreds of mW) from the light source is also preferred.
Light source
Light source
Light source
Port #1
Port #2
De-MUX
Port #3 Multiplexer (MUX)
Figure 4. Simplified illustration of the spectrum-sliced wavelength-division multiplexing (WDM) setup.
1.2. Types of Broadband Light Sources In general, wide emission spectrum can be obtained from (i) tungsten-halogen light sources, (ii) optically pumped rare-earth-doped fiber-based amplified spontaneous emission sources, (iii) optically pumped crystal lasers, and (iv) semiconductor superluminescent diodes. Typical values of the abovementioned light sources are depicted in Table 1. Tungsten-halogen lamps consist of tungsten filaments sealed in glass bulbs that are filled with halogen gas. This allows higher operating temperature of the tungsten filaments, thus providing higher luminous efficiency compared to incandescent lamps. As tungsten-halogen lamps behave like blackbody radiators, they exhibit very wide emission spectrum with the properties of the spectrum and the tungsten temperature related according to Planck’s radiation law and Wien’s displacement law [25]. However, the low power (in μW range) renders this light source unsuitable for many applications, since the signal-to-noise ratio and thus sensitivity of the system will be very poor.
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C.Y. Ngo, S.F. Yoon and S.J. Chua Table 1. Typical values of various broadband light sources.
Light source Tungsten-halogen [26] Rare-earth-doped fiber sources [27] Crystal lasers [28] Super-luminescent diodes [29]
Center wavelength (nm) 880 1050 1548 800 1250 1310 1310 1550 1550
Spectral width (nm) 320 50 45 210 150 38 83 40 95
Power (mW) 0.0002 25 50 500 250 50 15 35 0.2
Typical dimensions (mm) 120 x 90 x 15 940 x 280 x 200 30 x 13 x 9
Rare-earth-doped fiber sources operate by optically pumping the rare-earth-doped fiber (gain medium) with another laser, normally the 980 nm diode laser. The type of rare-earth dopant used depends on the wavelength range required. In particular, dopants like ytterbium (Yb), praesodymium (Pr), and erbium (Er) are used for wavelength coverage of 1020–1100 nm, 1280–1330 nm, and 1530–1565 nm, respectively. Recently, wide emission spectrum in the 1150–1300 nm range was demonstrated with bismuth (Bi) dopant [30]. This suggests the possibility of obtaining wide emission spectrum (> 100 nm) from rare-earth-doped fiber sources. However, because the amplified spontaneous emission (ASE) relies on an indirect light generation process through high power laser pumping, the quantum efficiency of rareearth-doped fiber sources is inherently low [31]. Furthermore, variation in the power density of the emission spectrum is about 2–10 dB/nm [27]. This will be detrimental for applications like optical coherence imaging where spectral shape of the source is of prime importance in determining the image quality [22]. Crystal lasers operate by optically pumping the doped crystal (gain medium) with another laser, normally the 514.5 nm argon-ion (Ar+) laser. The type of doped crystal used depends on the center wavelength. In particular, titanium-sapphire (Ti:sapphire), chromium-forsterite (Cr:forsterite), and chromium-yttrium aluminium garnet (Cr:YAG) are required for center wavelengths of 800 nm, 1250 nm, and 1500 nm, respectively. From Table 1, it seems that this is the best light source since it is able to provide wide spectrum width (hundreds of nm) and high output power (hundreds of mW). However, while high probe power can increase the signal, it might also increase the excess noise of the system, resulting in no improvement in the sensitivity. Furthermore, high power is not preferred for biomedical applications like ophthalmology since the eyes cannot withstand high probe power. Therefore, it is more important to have a variety of center wavelengths from a given light source since proper choice of the center wavelength can reduce scattering in samples and increase the probing depth [4]. Superluminescent diodes (SLDs) generate amplified spontaneous emission (ASE) by electrically pumping a semiconductor pn-junction (active layer). The emitted wavelength depends on the material composition of the active layer – this implies the ease of tuning the center wavelength to that required by the application. Its optical characteristics are intermediate between edge-emitting laser diodes (LDs) and light emitting diodes (LEDs). A
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comparison between LED, SLD and LD is given in Table 2. On one hand, the light emission from LED is mainly spontaneous with incoherent radiation. The spectral width (Δλ) is 30–120 nm, and the approximate value will depend on the emission wavelength (λpeak) since Δλ = 1.45λ peak kT , where k is the Boltzmann constant [32]. The LED output power 2
(Po) increases linearly with the injected current and is usually low. On the other hand, the light emission from LD is mainly stimulated with highly coherent radiation. The spectral width is the narrowest and the increase in the LD output power as a function of injected current can be divided into three regions, i.e. linear increase below the threshold current (Ith), superlinear increase above Ith, and power saturation at higher current. The output power of the LD depends on applications, and can range from 5 mW (for laser pointers) to 1 kW (for direct material processing). Table 2. Comparison between light emitting diode (LED), superluminescent diode (SLD) and laser diode (LD).
Output power Spectral width Nature of emission Optical feedback
LED
SLD
LD
Low
Medium
High
Wide
Medium
Narrow
Spontaneous
Spontaneous
Stimulated
No
No
Yes
Po
Po
Po
Light-current (L-I) curves
“knee”
I
“knee”
I
Ith
I
The characteristics of the SLD lie between that of LED and LD. In terms of geometry, SLD has similar structure to an edge-emitting LD. However, while high-reflection coatings are applied on the LD facets to promote stimulated emission so as to achieve lasing, the facets of the SLD are intentionally antireflection-coated to suppress the optical feedback mechanism. Compared to LD, SLD has higher gain, higher current density, and larger nonuniformity of carrier and photon density distribution [33,34]. The light-current (L-I) characteristics of the SLD and LD look similar except that there is no obvious “knee” at the transition from the linear region to the superlinear region for the SLD – ideally, there should be no “knee”. Most importantly, the SLD output power and spatial coherence are similar to LD, while the emission spectrum and temporal coherence are similar to LEDs. High spatial coherence of SLD implies that the beam emitted has a small divergence angle. This is beneficial for coupling into fiber pigtails since high coupling efficiency can be obtained. Poor
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temporal coherence of SLD implies that the coherence length is very short. In the spectral domain, this implies a broadband spectrum. Furthermore, as seen from Table 1, the size of the SLD can be very small. Therefore, good beam and spectrum quality, together with compactness, makes SLD the popular light source for many applications.
1.3. Quantum dots (QDs) vs. Higher Dimensional Systems Although the SLD structures and operating conditions can be varied to improve the spectrum characteristics [35,36], the main factor determining the broadband emission lies in the p-n junction, i.e. the active layer. The spectrum characteristics thus depend on the material systems employed for the active layer. An illustration of the commonly used material systems are shown in Fig. 5, namely the bulk, quantum well (QW) [37] and quantum dot (QD) [38] material systems. Bulk z
y
Quantum well (QW)
Quantum dot (QD)
z
z
x
y x
y x
H H
H
No carrier confinement (zero confinement)
Carrier confinement in zdirection (1-D confinement)
H ~ 80–200 nm [39]
H ~ 5–10 nm [39]
B
Carrier confinement in x-, y-, and z-directions (3-D confinement) H ~ 2–8 nm B ~ 20–40 nm [40]
Figure 5. Illustration of the commonly used material systems.
As pointed out by Chen et al. [41], QW structures will have higher current densities than bulk double-heterojunction (DH) structures for a given amount of current injection. This is due to it being a much smaller dimensional system. As depicted in Fig. 6, the increase in the current density thus resulted in the deeper penetration of the quasi-Fermi levels into the conduction and valence bands. As a result, the gain spectrum increases since the gain condition [42,43] is satisfied over a wider spectral range. Given that the working mechanism of SLDs is based on amplified spontaneous emission, the emission spectrum is directly related to the gain spectrum. Therefore, an increase in the gain spectrum implies an increase in the emission spectrum. As expected, a wider emission spectrum of 54 nm is obtained from SLDs with quantum well (QW) structures [41] as compared to an emission spectrum of 20 nm from SLDs with bulk double heterojunction (DH) structures [44].
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Quantum Well (QW)
Ez
Ez EqFc
EqFc Eg,eff
Eg,eff EqFv
Ez: Energy in the z-direction EqFc: quasi-Fermi level, conduction band
EqFv
Eg,eff: Effective bandgap EqFv: quasi-Fermi level, valence band
Gain condition[42,43]: Gain exists if the energy (E’) is such that Eg,eff < E’ < EqFc – EqFv Figure 6. Illustration of the energy-momentum (E-k) diagrams for bulk and quantum well systems (top), and definition of the gain condition (bottom).
Based on the same argument regarding the reduced dimensional system that leads to larger gain and emission spectra, Sun et al. [45] proposed that the three-dimensionally confined quantum dot (QD) systems will be able to achieve even wider emission spectrum than that of the QW systems. This is further supported by the fact that the self-assembled quantum dots (SAQDs) will introduce a size inhomogeneity of approximately 10% [46]. As shown in Fig. 7, the actual emission spectrum from SAQDs will deviate significantly from the delta-like spectrum of an ideally uniform QD ensemble. As a result, wider emission spectrum of 60 nm can easily be obtained from SLDs utilizing the QD structures [47]. Hence, although the inherited inhomogeneity issue is undesirable for narrowband devices like lasers, it is an intrinsic advantage for broadband devices like SLDs. Interestingly, Zhao et al. [48] conducted an experiment recently to investigate the effect of proton radiation on QW-SLDs and DH-SLDs (i.e. SLDs with active layers of different dimensional systems) and the results show that QW systems are less radiation sensitive than bulk systems. Even though the QD systems are not investigated in this case, we can expect higher radiation tolerance from the QD systems. This is due to strong exciton localization in the QDs, therefore resulting in a lower probability of carrier nonradiative recombination with the radiation-induced defect centers outside the QDs [49]. With the development of the aerospace industry, these findings suggest that QD-SLDs might also find applications in the future space systems.
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QD
QD
QD
PL intensity
Ideal QDs
λ
QD
QD QD
PL intensity
Actual QDs
λ
Figure 7. Illustration of the emission spectrum for ideal QDs and actual QDs. For the energy band structures, only the ground state transitions are represented.
2. Methods to Increase the Spectral Bandwidth Being a broadband light source, the spectral width of SLD is definitely one important performance parameter. Therefore, to achieve as large spectral bandwidth as possible, various techniques have been employed. It must be highlighted that, even though the device structures and operating conditions can also be varied to improve the SLD spectrum characteristics [35, 36], this section will only focus on techniques that are applied to the QD active layers for extension of the spectrum bandwidth. In general, these techniques can be broadly classified into (i) utilization of ground and excited QD transitions, (ii) QD intermixing, (iii) bandgap engineering, and (v) active layer optimization.
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2.1. Utilization of Ground and Excited QD Transitions This is the easiest technique to extend the emission spectrum. As illustrated in Fig. 8, the emission spectrum can be easily widened by employing the transitions from both the ground and excited states. In practice, this can be achieved by increasing the current density of the SLDs such that the gain from the ground state saturates and the gain from the excited states starts to set in. Ideally, the widest emission spectrum should be obtained when the excited state gain is equivalent to that of the ground state. However, as seen from the typical photoluminescence spectra in Fig. 8, the emission spectrum is normally irregular in shape and is very sensitive to the excitation power or current injection. Therefore, this method is seldom applied independently. Instead, it is commonly employed to the below-mentioned techniques as a mean to further extend the emission spectrum towards the shorter wavelength (higher energy) range due to the contribution of the excited QD states. Valence band
Conduction band GaAs
InAs QDs
GaAs
Figure 8. Illustration of the QD energy band diagram, showing the transitions from ground state and excited states (top); and typical photoluminescence (PL) spectra as function of PL excitation power (bottom).
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2.2. Quantum Dot (QD) Intermixing This is a post-growth technique where spectral bandwidth broadening is performed only after the growth is completed. As shown in Fig. 9, the working principle is based on induced intermixing of constituent atoms though the InAs/GaAs heterointerface. As a result, the intermixed InxGa1-xAs region will have a larger effective bandgap than the non-intermixed InAs region, i.e. Eg2 > Eg1. Non-intermixed
Atom arrangement
GaAs
InAs
GaAs
GaAs
Ga atom In atom
GaAs
EC
Eg1 EV
InxGa1-xAs
Ga atom In atom
EC
Band structure
Intermixed
Eg2 EV
Figure 9. Illustration of the constituent atoms arrangement and band structure before and after In/Ga intermixing. For the energy band structures, only the ground state transitions are presented.
Based on the same working principle, QD intermixing can be induced by many ways with the commonly employed methods being impurity free vacancy disordering (IFVD) [50], ion implantation followed by thermal annealing [51], and laser irradiation [52]. As depicted in Fig. 10, by selectively covering the QD-SLD surface with a dielectric material or reflective layer before applying the abovementioned methods, only a portion of the QD region will be subjected to intermixing. The intermixed QDs will then emit at a shorter wavelength as compared to the non-intermixed QDs. Ideally, contributions from both regions should result in broader emission spectrum. However, since the intermixed QDs will emit at a slightly higher intensity [53] and narrower spectral width [54,55], care must be taken when employing the intermixing technique. Otherwise, the emission profile will follow that of the intermixed QDs, resulting in narrower linewidth as shown in Fig. 10 for the non-optimized case. In general, even though ultrawide emission width can be obtained from the emission of both intermixed and non-intermixed QDs, the output power is low. This is because there is a tradeoff between the spectral width and the output power for QDs employing the intermixing
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technique, and is due to the finite number of QDs that can contribute to optical emission. Therefore, for a fixed QD density, “stretching” the spectrum width means that the number of QDs that can contribute to each wavelength will be smaller, thus implying a smaller power density. One way to overcome the low output power issue is to increase the QD density since this will imply a larger number of QDs to start with. In general, this can be achieved by increasing the QD areal density [56] or stacking up of multiple QD layers [57]. To date, the widest emission width of 360 nm is obtained by utilizing laser annealing on three QD layers [52]. However, it must be highlighted that since the laser annealing process consists of melting and subsequent fast re-crystallization of the material, high concentration of nonequilibrium point defects will be created. Consequently, this will result in strong degradation in the luminescence efficiency of the laser annealed region, with the drop in the intensity being as much as 20 times as compared to the non-intermixed region [58]. Therefore, this might be the reason why the output power obtained from Ref. [52] is less than 4 mW.
Ion implantation or laser irradiation
Selectively covered surface for QD intermixing process
Dielectric material or reflective layer
non-intermixed QDs
intermixed QDs
: non-intermixed : intermixed : resultant
PL intensity
Emission spectrum contributed by nonintermixed and intermixed QDs (non-optimized)
λ
: non-intermixed : intermixed : resultant
PL intensity
Emission spectrum contributed by nonintermixed and intermixed QDs (optimized)
λ
Figure 10. Illustration of the QD intermixing process and the emission spectra for both non-optimized and optimized cases.
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2.3. Bandgap Engineering This is a structural design technique where multiple QD layers are required. However, the spectrum range and profile emitted by each QD layer can be very different. In general, this can be achieved by varying the composition of the matrix surrounding the QDs [59], varying Structure
DCMWELL structure
GaAs
Energy band diagram Valance band
Conduction band
Valance band
Conduction band
Valance band
Conduction band
InAs QDs in InGaAs QW InAs QDs in GaAs QW InAs QDs in AlGaAs QW GaAs
GaAs InGaAs on InAs QDs
Chirped QD structures
InGaAs on InAs QDs InGaAs on InAs QDs GaAs
GaAs InGaAs on InAs QDs GaAs on InAs QDs AlGaAs on InAs QDs GaAs
Figure 11. Illustration of the DCMWELL and chirped QD structures. For the energy band structures, only the ground state transitions are presented.
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the amount of QD monolayer coverage [60], or inserting overgrown layer of varying composition on top of the QDs before growing the matrix material [61,62]. The first one is also known as dots in compositionally modulated well (DCMWELL) structures [59], while the remaining two are commonly known as chirped QD structures [60-62]. A simplified illustration of the DCMWELL and chirped QD structures is shown in Fig. 11. For clarity, only the ground-state transitions are presented. InAs, InGaAs, GaAs, and AlGaAs, which are in increasing bandgap energies, are used in the illustration. As a result, the QD bandgap can be engineered with each layer emitting at a different wavelength range, thus providing a wide emission spectrum. To date, the emission spectrum obtained by this technique is still not as wide as that obtained by the intermixing technique; however, much higher output power had been demonstrated. In particular, Yoo et al. [60] had demonstrated an emission spectrum of 98 nm at output power of 32 mW from a six-layer chirped structure utilizing three different amount of InAs QD monolayer coverage.
2.4. Active Layer Optimization This technique is a bottom-up approach, with an aim of optimizing the most important layer of the QD-SLD, i.e. the QD active layer. As pointed out by Sun et al. [45], size distribution is the most important factor for achieving wide emission spectrum from QD-SLD. In addition, unlike QD lasers, emission from QD-SLDs is contributed by all QDs of various sizes. Consequently, to obtain QD-SLDs with high power and broadband emission, large QD size inhomogeneity and high QD areal density (i.e. number of QDs per layer) are required from the optimized QD layer. An optimized QD layer with high areal density is beneficial for the QD-SLD structures since lesser QD layers will be required for a given QD density. This is because self-assembled QDs growth is a strain-driven process, and larger number of QD layers will imply larger strain in the system. Consequently, this might lead to dislocations [63] and degradation in the optical and electrical properties [64]. Therefore, to attain an overall QD density of 2x1011 cm2 , two QD layers with areal density of 1x1011 cm-2 will be more favorable than five QD layers with areal density of 4x1010 cm-2. More importantly, the optimized QD layers can be implemented with the abovementioned techniques, thus suggesting the possibility of obtaining QD-SLDs with ultra-wide emission spectrum and reasonably high output power. Despite numerous publications on QD-SLDs, most of them focus on utilizing the previous three techniques for the extension of QD spectral width. Only a handful [65-,,68] recognizes the importance of the bottom-up approach and the benefits that can be reaped by implementing the optimized QD layers together with the abovementioned techniques. In particular, our group had obtained a record high InAs QD areal density of 1.5x1011 cm-2 and the broadest spectral width of 136nm without any forms of bandgap engineering [68]. The next two sections will focus on the theoretical and experimental approaches taken by us in optimizing the InAs QD layers for high areal density and wideband emission. The origins of high radiative efficiency and wideband emission from the optimized QD sample will also be determined.
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3. Theoretical Calculation In this section, both analytical and numerical calculations were performed with an attempt to understand the effect of the QD size on the energy states. It is well-known that the energy states of QDs depend strongly on the size parameters, i.e. QD height, base length, and volume. However, to quantify the energy dependencies for each size parameter, theoretical calculations will be required. In particular, the size parameter with the strongest dependency will be identified, i.e. the one that gives the largest change in the energy states. Although the material considered in this work is the InAs/GaAs QD system, we believe that the calculated energy trends as function of QD sizes and shapes should be similar for another material system.
3.1. Analytical Derivation – QD with Infinite Potential Barriers The expressions derived in this subsection are based on the square-based cuboid QD confined in infinite potential barriers, as shown in Fig. 12.
8
8 E1,1,2
h
QD
E1,2,1 & E2,1,1 ΔE1
E1,1,1
b b
Figure 12. Illustration of a square-based cuboid QD, and the corresponding energy well with infinite potential barriers. The dimensions of the cuboid and four lowest energy states of the energy well are also indicated.
With the same terminology as that described in Fig. 12, the confinement energy can be written as [69]
Enx ,ny ,nz = and the energy spacing as
π 2 = 2 ⎛ nx2
nz2 ⎞ + + ⎜ ⎟ 2m* ⎜⎝ b 2 b 2 h 2 ⎟⎠ n y2
(1)
Quantum Dot Technology for Semiconductor Broadband Light Sources
ΔE =
π 2 = 2 ⎛ nx' 2 − nx2 ⎜ 2m* ⎜⎝
b2
+
n'y 2 − ny2 b2
nz' 2 − nz2 ⎞ + ⎟ h 2 ⎟⎠
219
(2)
Therefore, the ground-state energy (i.e. nx=ny=nz=1) can be written as
E1,1,1 =
π 2=2 ⎛ 2
1 ⎞ ⎜ 2+ 2⎟ 2m ⎝ b h ⎠ *
(3)
and the energy spacing between the ground and first excited state can be written as
ΔE1 =
π 2=2 ⎛ 3 ⎞ ⎜ ⎟ 2m* ⎝ b 2 ⎠
(4)
To consider the effect of a specific size parameter (e.g. height, h) on the ground-state energy, the remaining parameters (i.e. base length, b) can be fixed as constants. Therefore, the ground-state energy dependency due to the QD height (h) for a given base length (b0) is h 1,1,1
E
=
π 2=2 ⎛ 2
1 ⎞ + ⎜ ⎟ 2m* ⎝ b02 h 2 ⎠
(5)
and the ground-state energy dependency due to the QD base length (b) for a given height (h0) is
b 1,1,1
E
=
π 2=2 ⎛ 2
1 ⎞ ⎜ 2+ 2⎟ 2m ⎝ b h0 ⎠ *
(6)
Similarly, we can also consider the effect of the volume on the ground-state energy by assuming a constant aspect ratio (r), where r =
h . As such, the ground-state energy b
dependency due to the volume (vol) for a given aspect ratio (r0) is vol 1,1,1
E
π 2=2 ⎛
⎞ −32 = ⎜ 2r + r ⎟ vol 2 m* ⎝ ⎠ 2 3 0
−4 3 0
(7)
The analytical results thus provide an insight on the effect of size variation on the QD energy states. The main results can be summarized below:
ΔE1 ∝ b −2
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The ground-to-excited-state energy spacing depends only on the QD base length. Therefore, for wider emission spectrum with the utilization of multiple QD states as discussed in subsection 2.1, the QD base length should be wider. This will thus prevent undesirable intensity dips, as depicted in Fig. 8. h 1,1,1
E
−2
∝h , E
b 1,1,1
−2
∝ b , and E
vol 1,1,1
∝ vol
−2 3
The ground-state energy dependencies of the QD height, base length and volume were obtained. As seen from the relations, the ground-state energies show a stronger dependency on the QD base length and height, as compared to the QD volume. Therefore, for effective bandgap tuning with the technique discussed in subsection 2.3, experimentalists should focus on varying the QD base length or height, rather than the QD volume.
3.2. Numerical Formulation – QD with Finite Potential Barriers To understand qualitatively the effect of QD size on the energy states, the above expressions derived based on an ideal case of a cuboid QD with infinite potential barriers will be sufficient. However, to determine quantitatively the effects of QD size and its fluctuation on the energy range, numerical calculations will be required. Consequently, a strained-modified, single-band, constant potential three-dimensional model is formulated to study the effect of QD size on the ground-state energy [69]. In particular, InAs QDs embedded in GaAs matrix (i.e. InAs/GaAs QDs) will be considered. It must be highlighted that, despite greater computational complexity as compared to single-band model, the commonly used eight-band k.p model does not fare significantly better in terms of accuracy for calculating the energy states [70]. Consequently, simplicity in the calculation and computational efforts thus justified the use of the single-band model for the numerical calculation. In the framework of the envelope function and effective mass theory, the Hamiltonian can be written as
H=
1 −=2 ∇ * ∇ + V ( x, y , z ) 2 m ( x, y , z )
Where m* (x,y,z) =
m*InAs m*GaAs
, in QD , otherwise
V (x,y,z) =
0 ΔEV or ΔEC
, in QD , otherwise
m*InAs and m*GaAs are the carrier effective masses of InAs and GaAs, respectively; ΔEV and ΔEC are the valence band and conduction band discontinuities, respectively.
(8)
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The Schrödinger’s equation (H.ψ = E.ψ) can be written as
−= 2 1 ∇[ .∇ψ ( x, y, z )] + V ( x, y, z ).ψ ( x, y, z ) = E.ψ ( x, y, z ) 2 m *( x, y, z )
(9)
and the wave-functions are expanded in terms of normalized plane-waves, i.e.
ψ ( x, y , z ) =
1 Lx .Ly .Lz
∑
anx ,ny ,nz .e
i ( knx . x + kny . y + knz . z )
(10)
nx , ny , nz
where Lx, Ly, Lz are lengths of unit cell along x-, y- and z-directions; nx, ny, nz are the number of plane-waves along x-, y- and z-directions; knx = kx + nxKx , Kx = 2π/Lx; kny = ky + nyKy , Ky = 2π/Ly; knz = kz + nzKz , Kz = 2π/Lz. At this point, it must be pointed out that the attraction of this normalized plane-waves approach is that there is no need to explicitly match the wave functions across the boundary of the barrier and QD materials. Hence, this method is easily applicable to an arbitrary confining potential [71]. Multiplying ψ* to the left side of Eq. (9) and integrating over the unit cell, one obtains the matrix equation
( Anx ',ny ',nz ',nx ,ny ,nz − Eδ nx ',nxδ ny ',nyδ nz ',nz ).anx ,ny ,nz = 0
(11)
where
Anx ',ny ',nz ',nx ,ny ,nz =
−=2 1 ∫ψ *nx ' ny ' nz ' ∇[ .∇ψ nxnynz ]dxdydz + ∫ψ *nx ' ny ' nz ' .V .ψ nxnynz dxdydz m* 2 (12)
Performing partial differentiation to the first integral term on the right-hand-side of Eq. (12)
1 .∇ψ nxnynz ]dxdydz m* 1 1 ).(∇ψ nxnynz )dxdydz + ∫ψ *nx ' ny ' nz ' .∇ (∇ψ nxnynz )dxdydz = ∫ψ *nx ' ny ' nz ' (∇ m* m* ∫ψ *nx ' ny ' nz ' ∇[
(13) Performing integration by parts to the second integral term on the right-hand-side of Eq. (13) and assuming that the wave functions are negligible at the boundaries of the unit cell
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1 .∇(∇ψ nxnynz ) dxdydz m* 1 1 ).(∇ψ nxnynz )dxdydz = − ∫ ∇ψ *nx ' ny ' nz ' ( ).∇ψ nxnynz dxdydz − ∫ψ *nx ' ny ' nz ' (∇ m* m* ∫ψ *nx ' ny ' nz '
(14)
Substituting Eqs. (13) and (14) back to Eq. (12), one obtains the matrix elements
=2 1 ∫ ∇ψ *nx ' ny ' nz ' .∇ψ nxnynz dxdydz + ∫ψ *nx ' ny ' nz ' .V .ψ nxnynz dxdydz 2 m*
Anx ',ny ',nz ',nx ,ny ,nz =
(15) The electron and hole energy states are calculated using discretized Schrödinger’s equation technique. As such, the matrix elements can be calculated rapidly as there is no integration involved. Thus, this method is very useful for treating problems with complex geometry. However, the main drawback of this technique is with the errors involved in the large computation. As such, optimization steps had been taken to ensure that the errors involved in the calculation are as small as possible, typically less than 0.001 %. A basis of seven normalized plane-waves in each direction is used for solving the Hamiltonian matrix, i.e. with nx, ny, nz each ranging from -3 to 3 [69]. The strain-modified parameters [72-74] used in the calculation are listed in Table 3. *
While some publications [73,75] specify two values for the heavy-hole effective mass mhh , *
*
i.e. mhh , xy and mhh , z , only a single value is used here since the inclusion of two values for * mhh increase the computational effort significantly, yet only improve the computed results by
2.5 % [75]. Based on the same rationale, unnecessary complications that do not significantly improve the computed results will be ignored. As such, additional effects such as potential due to piezoelectricity, Coulombic interaction, and strain distribution within the InAs QD had been excluded [69]. Table 3. Parameters used in the numerical calculation. Parameters * e
GaAs
InAs
m
0.0665 m0
0.04 m0
* mhh
0.377 m0
0.590 m0
EG
1.424 eV
0.720 eV
ΔEC
380 meV
ΔEV
324 meV
Table 4 illustrates the QD shape and size variations considered in this study. In particular, the QD shapes considered are namely cuboid, cylindrical and lens-shaped rather than pyramidal and conical shapes, since the transmission electron microscopy (TEM) images
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reflect QDs with broad tips from our InAs QD samples (as shown in the inset of Fig. 20). The size variations considered are realistic since these are the commonly reported dimensions [40]. Care had been taken to study solely the effect of a single size parameter on the QD energy states by isolating the remaining parameters. As such, when varying the QD height, the base length is kept constant at 25 nm. Similarly, when varying the QD base length (volume), the height (aspect ratio) is kept constant at 5 nm (0.5). Table 4. QD shape and size variations considered in this study Illustration
QD shape
Size variation Given h/b = 0.5, vol = 102–104 nm3
Cuboid
h b b
h
Given b = 25 nm, h = 2–12 nm
Cylindrical b
h
Lens-shaped b
Given h = 2.5 nm, b = 5–40 nm
where b: QD base length h: QD height vol: QD volume
3.3. Results and Discussion In this subsection, the calculated results will be discussed. Note that the electron (heavy hole) ground-state energies are taken with respect to the InAs conduction (valence) band. The energies and dimensions are in unit of meV and nm, respectively. The calculated electron and heavy hole ground-state energies for cuboid, cylindrical and lens-shaped QDs as function of QD volume are depicted in Fig. 13. As expected, decrease in the energy states is observed for all three shapes considered, following increase in the QD volume. This conclusion can also be inferred from the analytical expression derived in Eq. (7) and the phenomenon can be understood by a reduction in the carrier confinement. In addition to the similarity between the energy trends of the three QD shapes, it was also found that the largest energy difference between the QD shapes is only approximately 10meV across the range of QD volume studied in our case. This implies that the effect of the QD shapes on the energy states is not very significant in this work. Hence, for clarity, only cuboid QD will be presented from now onwards. Nevertheless, it must be pointed out that QD shapes do affect the QD energy states. However, as shown in Ref. [69], the small energy difference is due to
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the fact that the QD shapes considered in this work are all QDs with broad tips. If QDs with narrow tips (e.g. pyramidal or conical) are considered across the same range of QD volume, the energy difference can be as large as 70 meV.
Figure 13. Plot of ground-state energy for electron (hollow symbols) and heavy hole (filled symbols) as function of QD volume. The QD shapes considered are cuboid ( ), cylindrical ( ) and lens-shaped ). The dotted lines are guides to the eyes. (
The calculated electron and heavy hole ground-state energies for cuboid QDs as function of QD base length are depicted in Fig. 14. The observed decrease in the energy states can be explained by the reduction in the carrier confinement along the horizontal direction. However, notice that significant changes to the energy states only occurs for QD base length less than 20 nm. From the growth optimization point of view, this is not useful since the QD base length is in general more than 25 nm [76]. As seen from Fig. 14, further increase in the QD base length will not result in significant changes to the QD energy states. Nevertheless, as inferred from Eq. (4), larger QD base length will result in smaller energy spacing between the ground and first-excited QD states. Hence, this might be beneficial for the technique discussed in subsection 2.1 since wider emission spectrum without intensity dips can be realized when utilizing multiple QD states. The calculated electron and heavy hole ground-state energies for cuboid QDs as function of QD height are depicted in Fig. 15. Again, decrease in the energy states is observed following increase in the QD height, and can be explained by the reduction in the carrier confinement along the vertical direction. Notice that for QD height ranging from 2–6 nm, significant decrease in the ground-state energy is observed, especially for the electron groundstate energy. Therefore, to optimize a single InAs QD layer for large energy range, large height inhomogeneity has to be intentionally introduced. In particular, InAs QD height ranging from 2–6 nm will be preferred.
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Figure 14. Plot of ground-state energy for electron (hollow symbols) and heavy hole (filled symbols) as function of cuboid QD base length. The dotted lines are guides to the eyes.
Figure 15. Plot of ground-state energy for electron (hollow symbols) and heavy hole (filled symbols) as function of cuboid QD height. The dotted lines are guides to the eyes.
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4. Optimization for High Quantum Dot Areal Density and Wideband Emission It is well-known that the surface morphology and optical properties of InAs QDs depends strongly on its growth temperature and monolayer coverage. As such, the effects of growth temperature and monolayer coverage were varied with an aim of obtaining both high areal density and wideband emission from the InAs QDs. In addition, the origins of high radiative efficiency and wideband emission from the optimized InAs QD sample were also discussed.
4.1. Experimental Details The layer structure, growth conditions, and sample description of the InAs QD samples under investigation were depicted in Fig. 16. As described, the InAs QD monolayer coverage (θC) investigated were 2.0, 2.5, and 3.0 ML, and the growth temperature (TG) investigated were 450, 480, and 510 oC. For ease of reference, they are labeled as samples A to E. The samples were grown using solid-source molecular beam epitaxy, and reflection high-energy electron diffraction (RHEED) was employed for in-situ monitoring of the QD formation. Layer structure InAs QDs GaAs In0.15Ga0.85As SRL InAs QDs GaAs In0.15Ga0.85As SRL InAs QDs GaAs buffer GaAs substrate
Growth conditions θC / TG 50 nm / 580 oC 5 nm / TG θC / TG 40 nm / 580 oC 5 nm / TG θC / TG 250 nm / 580 oC –
θC (ML)
TG (oC)
2.0
510 480
2.5 3.0
450
Sample ID A B C D E
Figure 16. Layer structure, growth conditions, and sample description of the InAs QD samples under investigation. The strain-reducing layer (SRL), monolayer coverage (θC), and growth temperature (TG) were also indicated.
Oxide desorption of the GaAs (001) substrate was performed at 580 oC before growing a 250 nm thick buffer layer. The temperature was then ramped down to TG before growing θC of InAs QDs. As shown in Fig. 17, the two- to three-dimensional transition of the InAs QDs was confirmed by the RHEED patterns changing from streaky to spotty. This was followed by the growth of 5 nm In0.15Ga0.85As strain-reducing layer (SRL) to tune the ground state emission to around 1.3 µm. The growth temperature was then ramped up to 580 oC for the growth of 40 nm GaAs spacer layer. The aim of the thick GaAs spacer layer is to decouple the electronic and strain effects of the first InAs QD layer from the subsequent InAs QD layer [77,78]. As such, the surface morphology and optical properties of the second InAs QD layer are identical to a structure with only a single InAs QD layer grown at the same TG and θC [79]. After that, the growth sequence for the InAs QDs, In0.15Ga0.85As SRL, and GaAs layer
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was repeated. Finally, a layer of uncapped InAs QDs with the same TG and θC as the previous two capped InAs QD layers was grown for atomic force microscopy (AFM) and scanning electron microscopy (SEM) characterization. The room temperature photoluminescence (PL) characteristics of the InAs QD samples were measured using the 514.5 nm line of an Ar+ laser. The PL signals were detected using a liquid nitrogen-cooled Ge-detector in conjunction with a standard lock-in technique.
Streaky RHEED patterns (2D growth)
Spotty RHEED patterns (3D growth)
Figure 17. Streaky and spotty RHEED patterns for two-dimensional (2D) and three-dimensional (3D) growth, respectively.
4.2. Effect of Growth Conditions on QD Surface Morphology and Optical Properties The AFM and SEM images of the uncapped InAs QDs as function of growth conditions are depicted in Fig. 18, while the room temperature PL spectra and the corresponding full-width half-maximum (FWHM) spectral range of the InAs QD samples as function of growth conditions are depicted in Fig. 19. In addition, a summary of the extracted values of the QD areal density, room temperature PL spectral width, and normalized integrated PL intensity for the InAs QD samples at different growth conditions are shown in Table 5. As mentioned in Ref. [68], the trend of variation can be classified into two regions, i.e. the variation of QD growth temperature at 2.0 ML, and the variation of QD monolayer coverage at 450 oC. On one hand, one could see from Fig. 18 that high QD areal density can be obtained at 450 oC, regardless of the monolayer coverage. As presented in Table 5, the lowest QD areal density of 7.6 x 1010 cm-2 obtained from TG = 450 oC is still more than four times higher than that grown at higher QD growth temperature. The observed increase in the QD areal density at low growth temperature is due to the reduction of the adatom migration length [80]. Therefore, impinging adatoms is more likely to form new nucleation sites instead of combining with existing InAs QDs. On the other hand, one can see from Fig. 19 that wide emission spectrum can be obtained at 2.0 ML, regardless of the QD growth temperature. As presented in Table 5 again, the smallest spectral width of 105 nm obtained from θC = 2.0 ML is almost two times wider than
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that grown at higher monolayer coverage. As reported [81], this is due to the fact that monolayer coverage of 2.0 ML does not provide sufficient material for the InAs QDs to reach its mature size. As such, there exists a large number of InAs QDs at various stages of size development. The large QD size variations thus result in wide emission spectrum from the 2.0 ML InAs QDs. Growth conditions
AFM images (1 µm x 1 µm)
SEM images (30,000 magnification)
2.0 ML 510 oC (Sample A)
2.0 ML 480 oC (Sample B)
2.0 ML 450 oC (Sample C)
2.5 ML 450 oC (Sample D)
Figure 18. Continued on next page.
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AFM images (1 µm x 1 µm)
229
SEM images (30,000 magnification)
3.0 ML 450 oC (Sample E)
Figure 18. Three-dimensional AFM images and the corresponding SEM images for the InAs QD samples at different growth conditions.
Most importantly, as summarized in Table 5, both high areal density (1.5 x 1011 cm-2) and wideband emission (136 nm) can be obtained from the optimized InAs QD sample, i.e. sample C with TG = 450 oC and θC = 2.0 ML. While the widest emission spectrum of 155 nm can be obtained from sample A (i.e. TG = 510 oC and θC = 2.0 ML), the high growth temperature increases the adatom migration length [80] and results in four times decrease in the areal density to 3.8 x 1010 cm-2. The observed 30 % drop in the integrated PL intensity of sample A as compared to sample C might thus be due to the significant decrease in the areal density. Interestingly, while the highest areal density of 1.6 x 1011 cm-2 can be obtained from sample E (i.e. TG = 450 oC and θC = 3.0 ML), the integrated PL intensity was found to be the lowest. As seen from the AFM and SEM images in Fig. 18, this is due to the large number of islands with width of approximately 100 nm for sample E. As reported by Joyce et al. [81], presence of dislocations within these large islands is observed from the TEM images. Therefore, these islands are nonradiative in nature and act as sinks for photogenerated carriers. This explains the significant (60 %) drop in the integrated PL intensity of sample E as compared to sample C, despite the highest QD areal density obtained. QD growth conditions
Room temperature PL spectra
2.0 ML 510 oC (Sample A)
FWHM Spectral range
155 nm 1185–1340 nm
Figure 19. Continued on next page.
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C.Y. Ngo, S.F. Yoon and S.J. Chua QD growth conditions
Room temperature PL spectra
FWHM Spectral range
2.0 ML 480 oC (Sample B)
105 nm 1230–1335 nm
2.0 ML 450 oC (Sample C)
136 nm 1195–1331 nm
2.5 ML 450 oC (Sample D)
60 nm 1266–1326 nm
3.0 ML 450 oC (Sample E)
55 nm 1275–1330 nm
Figure 19. Room temperature PL spectra and the corresponding FWHM spectral range for the InAs QD samples at different growth conditions.
Hence, by investigating the effect of growth temperature and monolayer coverage on the surface morphology and optical properties, high areal density and wideband emission can be simultaneously obtained from the InAs QD samples. In particular, this is obtained at a relatively low growth temperature of 450 oC for 2.0 ML coverage of InAs QDs. The obtained
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areal density of 1.5 x 1011 cm-2 is four times higher than those previously reported [82-84]. In addition, the emission spectrum of 136 nm is the broadest spectral width obtained without any forms of bandgap engineering [61,62]. Furthermore, the intensity profile is relatively smooth with no undesirable 3dB-dip in the PL intensity as compared to those reported [61,62,82,83]. These results will definitely contribute to an improvement in the performance of QD-SLDs. Table 5. Extracted values of the QD areal density, room temperature PL spectral width, and integrated PL intensity for the InAs QD samples at different growth conditions. The values of the integrated PL intensities are normalized with reference to the InAs QD sample with the highest integrated PL intensity. QD ML coverage 2.0 2.5 3.0
QD growth temperature (oC) 510 480 450
QD areal density (cm-2) 3.8 x 1010 1.8 x 1010 1.5 x 1011 7.6 x 1010 1.6 x 1011
PL spectral width (nm) 155 105 136 60 55
Integrated PL intensity 0.7 0.8 1.0 0.95 0.4
4.3. Origins of High Radiative Efficiency and Wideband Emission The previous section had investigated the growth conditions required for high areal density and wideband emission. In particular, low growth temperature and low monolayer coverage of the InAs QDs are required for high areal density and wideband emission, respectively [68]. However, knowing the growth conditions alone is insufficient. For that to be applicable to different growth techniques (e.g. metal organic chemical vapor deposition) or material systems (e.g. InAs/InP), the origins for high radiative efficiency and wideband emission from the optimized InAs QD sample must be understood [85]. An attempt to understand the radiative efficiency of the InAs QD samples is made by comparing the surface morphologies between samples C (i.e. the optimized InAs QD sample) and E. This is because the areal densities for both samples are similarly high, yet sample C has the highest integrated PL intensity while sample E has the lowest. As mentioned briefly in the previous subsection and depicted in Fig. 18, an obvious difference between them is the absence of large islands (~ 100 nm wide) in sample C. An enlarged plan-view SEM image of sample E, taken at 60,000 magnifications is depicted in Fig. 20. From the SEM image, one can see the significant presence of large islands among the high density of small QDs. Further investigation on the large islands is provided by cross-sectional TEM, and the image is depicted in the inset. As expected, a dislocation line underneath a large island is observed, and this is consistent with that reported by Joyce et al. [81]. This might be due to misfit dislocations at the interface of the InAs QDs and the underlying GaAs layer [86,87]. Therefore, it is confirmed that these islands act as sinks for the photogenerated carriers and the presence of them degrade the radiative efficiency of the InAs QD samples. Figure 21 presents the InAs QD height and base length histograms of samples A, C, and E, i.e. QD sample at high TG, the optimized QD sample, and QD sample with high θC, respectively. From the cross-sectional TEM image in the inset of Fig. 20, the size of the
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nonradiative large island is found to be approximately 60 nm wide and 16 nm high – this thus gives us an idea of the minimum dimensions of the nonradiative islands. Consequently, dashed lines that denote the critical height (16 nm) and base length (60 nm) are drawn in the InAs QD height and base length histograms in Fig. 21, and these lines define the limit for good optical properties of the InAs QDs. However, as seen from Fig. 21, no QD exceeding the critical height and base length was found in sample C. Therefore, as supported by the plan-view SEM and AFM images of Fig. 18 and the InAs QD size histograms of Fig. 21, high radiative efficiency (or integrated PL intensity) from sample C is due to the absence of the large nonradiative islands. Qualitatively, one can infer from eqns. (5)–(7) that the QD energy range depends on both its size and fluctuation: (i) for a given QD size, the energy range is proportional to its size fluctuation, and (ii) for a given size fluctuation, the energy range is inversely proportional to the QD size. However, to quantitatively determine the effects of the QD size and its fluctuation on the energy range of the InAs QD samples, one need to calculate the effects of the QD size and shape on the energy states. Recognizing this, theoretical calculations were performed to determine the combined effects of the InAs QD size and its fluctuation. While we had obtained good agreement between the theoretical and experimental results from the InAs QD samples at various growth conditions [85], only the combined effects of the size and its fluctuation on the energy range of sample C (i.e. the optimized InAs QD sample) will be discussed in details. From the QD size histograms in Fig. 21, the height and base length fluctuations obtained for sample C was found to be 2–3.5 nm and 24–34 nm, respectively. As seen from Fig. 14, energy range for the QD base length fluctuation of 24–34 nm is found to be negligible. This is expected since the low QD aspect ratio of ~0.1–0.3 [88] implies that the energy range is more sensitive to the QD height fluctuation. Hence, in the following analysis, only the effect of the QD height fluctuation on the energy range will be considered. Figure 22 depicts (a) the “zoom-in” view of Fig. 15, with InAs QD height ranging from 2–8 nm, and (b) the room temperature PL spectra of the optimized InAs QD sample at low excitation power. It must be mentioned that low excitation power was necessary to ensure that only the ground state emissions were allowed since the theoretical calculations were only performed for the QD ground-state energies. From Fig. 22(a), an energy range of 103 meV was obtained from the QD height fluctuation of 2–3.5 nm. This large energy range is contributed by both the electron energy variation (78 meV) and heavy-hole energy variation (25 meV). Most importantly, the theoretical result (103 meV) is in good agreement with the experimental result (102 meV) depicted in Fig. 22(b). Nevertheless, it must be pointed out that even though cross-sectional TEM images were obtained from the InAs QD samples, the QD dimensions were not extracted for the theoretical calculations. Instead, dimensions from the uncapped InAs QD layers were used for the theoretical calculations. This is due to the large lattice mismatch (7%) of the InAs/GaAs QD system, thus resulting in substantial strain-field contrast in the TEM image. Therefore, as the strain extends further out of the QD into the surrounding material, the QDs will appear larger. On the other hand, it was recently reported that InAs QDs capped with a strain-reducing layer (e.g. In0.15Ga0.85As) is capable of suppressing the QD decomposition and thus retain the QD size upon capping [89]. Hence, dimensions obtained from the uncapped InAs QDs will be able to reflect well those InAs QDs capped with In0.15Ga0.85As strain-reducing layer. It is worthwhile to note that, despite the fact that the calculation described in the previous section was based on InAs QDs with a GaAs cap layer, the current experimental results based on InAs QDs capped
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with In0.15Ga0.85As are still valid. This is due to the insignificant height dependency difference on the electronic states for InAs QDs capped with the two materials [69].
SEM image of large islands TEM image of large islands
Figure 20. SEM image of uncapped InAs QDs layer of sample E, i.e. TG = 450 oC and θC = 3.0 ML. Inset: TEM image with dotted white ring showing a dislocation line beneath a large island of approximately 60 nm wide and 16 nm high [85].
QD growth conditions
QD height histogram
QD base length histogram
2.0 ML 510 oC (Sample A)
2.0 ML 450 oC (Sample C)
Figure 21. Continued on next page.
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QD growth conditions
QD height histogram
QD base length histogram
3.0 ML 450 oC (Sample E)
Figure 21. InAs QD height and base length histograms as function of growth conditions. The dashed lines denote the critical height and base length that define the limit for good optical properties of the InAs QDs [85].
(a)
(b)
Figure 22. (a) “Zoom-in” view of Fig. 15, with InAs QD height ranging from 2–8 nm. The black and red dotted curves are fitting curves to the electron and heavy hole ground-state energies. The black and red dashed lines are guides to the eyes. (b) Room temperature PL spectra of the optimized InAs QD sample at low excitation power [85].
5. Potential Challenges As shown, having the self-assembled quantum dots (SAQDs) as the active material is beneficial for broadening the spectral width of the broadband light source since SAQDs have an inherited size inhomogeneity. However, SAQDs have an inherited problem, i.e. the output power differs significantly between the transverse-electric (TE) and transverse-magnetic (TM) polarization modes. (For clarity, the definitions of the TE and TM polarization modes in a waveguide configuration are illustrated in Fig. 23.) This is due to the fact that the QD shapes are flat (i.e. low QD aspect ratio) and are under biaxial compressive strain. Therefore, the degeneracy of the heavy-hole (HH) and light-hole (LH) is lifted, with the HH states above that of LH states. The ground-state energy transitions will then have electron-to-HH
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characteristics, which coupled to light polarized perpendicular to the growth direction, i.e. the TE polarization mode. While some applications might be able to take advantage of this feature, e.g. polarization-sensitive optical coherence tomography (PS-OCT) [90], other applications like the FOS, FOG and spectrum-sliced WDM would prefer polarization insensitive broadband light sources for higher sensitivity [91,92]. Otherwise, polarization maintaining fibers and other polarization control components will be required, which will eventually increase the cost and complexity of the systems [93].
TE Propagation QD active layer Substrate
direction TM
Figure 23. Illustration of the TE and TM polarization modes in a waveguide configuration.
Figure 24. The PL spectra of InAs/GaAs QDs with InxGa1-xAs capping layer for (a) x=0 and (b) x=0.13. The closed and open circles indicate TE- and TM-mode PL spectra, respectively [94].
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Figure 25. Cross-sectional TEM images of the InAs QDs for (a) GaAs and (b) In0.13Ga0.87As capping layer [94].
The anisotropy of the InAs QD polarization can be resolved either by covering the QDs with InxGa1-xAs capping layer of suitable Indium (In) composition [95] or controlling the number of stacking layers in the multi-stacked QDs, i.e. columnar QDs [96,97]. As depicted in Fig. 24, the former approach demonstrates that TE-to-TM polarization mode inversion can be achieved by covering the InAs QDs with In0.13Ga0.87As instead of GaAs capping layer. As seen from the cross-sectional TEM images in Fig. 25, the inversion is suggested to be due to an enhancement of the shape anisotropy of the InAs QDs, i.e. from QD aspect ratio of approximately 0.2 to 0.7. The result suggests that, upon proper choice of the InxGa1-xAs composition, equal TE and TM polarization PL intensities (i.e. zero degree of polarization) can be obtained from InAs QDs with InxGa1-xAs capping layer.
Figure 26. Control of gain polarization with close stacking of QDs, and polarization-resolved PL spectra measured with closely stacked InAs QDs on GaAs. Ref. [99] © 2007 IEEE.
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Figure 27. Cross-sectional TEM images of columnar QDs with (a) 16- and (b) 35-repetition cycles of InAs/GaAs superlattice [98].
Columnar QDs are multi-stacked QDs with thin spacer so that the QDs are aligned vertically and coupled electronically. As explained in Ref. [99] and depicted in Fig. 26, by increasing the number of stacking QD layers, vertical wavenumber k⊥ of the ground states can either be reduced to be similar to in-plane wavenumber k& for equal TE and TM polarization modes, or much less than k& for TM-dominant polarization mode. In addition, according to the calculated results based on elastic continuum theory for the QD strain distribution and eight-band k.p theory for the QD electronic structures, the increase in the TM component for the columnar QDs is also due to the enhanced HH-LH mixing as a result of a reduction of the biaxial strain in the center portion of the columnar QDs [100]. Furthermore, as pointed out by Li et al. and depicted in Fig. 27, QD aspect ratio as high as 4.1 can be obtained upon proper growth optimization of the columnar QDs [97,101]. Such QD structures are very promising for polarization-insensitive broadband light sources. While the columnar QD structure seems to be the solution towards polarizationinsensitive emission, there are still many challenges ahead. In particular, the areal density of the columnar QDs is only about 1.7 x 1010 cm-2 [101]. As shown in the previous section, the low QD areal density is detrimental for the radiative efficiency of the QDs. Additionally, as reported in Ref. [102], the lateral confinement potential of the columnar QDs was found to be very weak since the Indium compositions of the QDs and the surrounding matrix are approximately 30 % and 16 %, respectively. As such, only single electron-HH and electronLH transitions are identified. This implies that the utilization of both the ground and excited states to extend the spectral bandwidth will be impossible since there is no excited QD state being confined in the shallow confinement potential. Furthermore, the shallow confinement potential implies low thermal stability since the thermally excited carriers can escape from the shallow confinement potential easily. Therefore, one might want to consider lower growth
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temperature (~ 450 oC) so as to obtain higher areal density [68] and better thermal stability [103] for the columnar QDs. For higher lateral confining potential, the thin GaAs spacer material between the stacked QDs can be replaced by AlxGa1-xAs, i.e. a larger bandgap material. Consequently, QDs with high areal density and ultrawide polarization-insensitive emission spectrum can be realized.
6. Conclusions An introduction on the common applications that require broadband light sources, together with the common types of broadband light sources, was provided. The spectrum characteristics of the light sources are mentioned, which eventually leads to one, i.e. the semiconductor superluminescent diode (SLD), being the most popular light source for the mentioned applications. Justification for the use of QDs (as compared to higher dimension systems) as the active material of the SLDs was also provided. Existing methods to increase the spectral bandwidth of the light source were discussed, with emphasis being placed on our “bottom-up approach” of optimizing the QDs active layers. In addition, analytical derivations and numerical formulations were performed to determine the energy dependencies for each size parameters for the InAs/GaAs QD system. In particular, calculated results suggest that large energy range can be obtained from InAs QD height ranging from 2–6 nm. By investigating the effect of growth temperature and monolayer coverage on the surface morphology and optical properties, high areal density (1.5 x 1011 cm-2) and wideband emission (136 nm) can be simultaneously obtained from the InAs QD samples. As verified, the high radiative efficiency (or integrated PL intensity) from the optimized InAs QD sample is due to the absence of the large nonradiative islands, while the wideband emission was originated from InAs QDs with height fluctuation of 2–3.5 nm. Lastly, the potential challenges associated with achieving QDs with high areal density and ultrawide polarizationinsensitive emission spectrum were highlighted with the solutions proposed.
Acknowledgements The authors would like to express their acknowledgements to Dr. L. H. Li (University of Leeds), Dr. Z. Y. Zhang (University of Sheffield), and Dr. Z. Z. Sun (Nanyang Technological University) for fruitful discussions and valuable suggestions. They are also grateful to D. M. Y. Lai and S. Y. Chow for their help in the SEM and TEM measurements, respectively. One of the authors (C. Y. Ngo) would like to acknowledge the financial support from the A*STAR Graduate Scholarship program.
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In: Quantum Dots: Research, Technology and Applications ISBN: 978-1-60456-930-8 Editor: Randolf W. Knoss, pp. 243-265 © 2008 Nova Science Publishers, Inc.
Chapter 6
QUANTUM DOTS IN MEDICINAL CHEMISTRY AND DRUG DEVELOPMENT Ian D. Tomlinson, Michael R. Warnement and Sandra J. Rosenthal Dept. of Chemistry, Vanderbilt Univ., Nashville, TN
Abstract Quantum dots have increasingly been incorporated into a wide variety of biological assays as improved fluorescent probes. Their photophysical properties permit the investigation of cellular processes and biological phenomena with unprecedented spatial resolution and temporal longevity. Consequently, quantum dots are poised to facilitate advances in future drug development applications. Multiplexed detection in whole cell assay format may ultimately provide added insight into the extremely complex biochemical mechanisms involved in drug receptor interactions. This article provides a detailed discussion of biological applications which have incorporated quantum dot detection, with a particular emphasis on their possible integration into drug discovery and medicinal chemistry applications.
Introduction Traditionally drug discovery has focused on a single biological target in the development of new drugs and, while this methodology has been productive, there has been a steady decline in new drugs reaching the marketplace over the last two decades. The introduction of techniques, such as combinatorial chemistry and high throughput screening have increased the number of lead compounds, but these techniques address the issue of quantity rather than quality. Many of these compounds have subsequently failed due to toxicity or poor bioavailability. Critics have claimed that these screening techniques are not a true representation of the complex chain of events that occur when a drug binds to its target. This target based approach focuses on a single gene or the molecular mechanism of a key event in a disease process. Consequently, these approaches cannot probe the complex interplay between bioavailability, binding at the desired target and expression of other systems or the
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activation of alternate biochemical pathways. To overcome the limitations of the target based approach for drug discovery a number of researchers have proposed alternate approaches based on whole cell assays. These may be more realistic and enable the research community to gain valuable insights into the biochemical mechanisms of drug receptor interactions and, by using whole cells, biological complexity may be introduced into the drug discovery process. This should reveal both the intra- and intercellular processes occurring in a specific disease state. Improved fluorescent detection, incorporating instrumentation such as the FlexstationTM fluorescent plate reader, flow cytometry or laser scanning cytometry, has resulted in the development of assays which can ultimately be adapted to screen a wide variety of fluorescently tagged drug candidates in a whole cell assay format. Unfortunately, this approach has limited utility due to the properties of current organic fluorophores, which include photobleaching (many organic fluorophores bleach within seconds), chemical degradation, toxicity, broad emission spectra and low quantum yields. Consequently cellular imaging has remained a largely descriptive tool and, until recently, has only been amenable to small scale experimental samples. However this situation is likely to change with the development of new fluorophores such as quantum dots. Quantum dots are nanometer sized crystals that offer the promise to solve some of these issues in drug development, while simultaneously being useful for developing deeper insights into long term drug-receptor interactions and downstream biochemical changes. Their unique photophysical properties make them ideal for the development of new diagnostic tools with a much higher signal to noise ratio than can ever be achieved with a conventional fluorophore, and they may be utilized in biological screens that require longer periods of illumination. Since many drugs interact with unknown receptors or enzymes, fluorescent nanocrystals may find utility in the purification process to identify new drug receptors and enzymatic targets. In addition to drug development, cellular imaging, and diagnosis applications, quantum dots may also have the potential as nanoscale drug-delivery devices or nano vectors. Given the multivalent nature of their surfaces, this alternate approach has focused on incorporating both targeting molecules and therapeutic molecules on the quantum dot surface to permit selective treatment of diseased cells and tissue. Several unique photophysical properties make quantum dots useful as fluorescent markers in a wide range of biological applications, and their first reported use as imaging agents occurred in 1998.[1,2] Subsequently they have become a tool in many sandwich based assay platforms, cellular imaging and in vivo imaging protocols. A review of the literature shows that quantum dots have been utilized in a wide variety of biological applications. Of these, perhaps the most important from a medicinal chemistry and drug development perspective are antibody, peptide and small molecule conjugated quantum dots. To understand the importance of quantum dots in drug development it is necessary to understand both their physical properties as well as the current state of the art applications which employ them. As such, this review will initially provide an introduction to quantum dots and the physical underpinnings of their unique photophysical properties. We will then focus our discussion on surface modifications which have been employed to facilitate biological compatibility and ensure a specific interaction with an intended biological target. A review of current biological applications is then presented with a focus on antibody, peptide and small molecule quantum dot conjugates. We conclude by highlighting future
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directions for the incorporating quantum dots in medicinal chemistry and drug development applications.
An Introduction to Quantum Dots The photophysical properties of quantum dots arise from their small size, causing them to behave quite differently than bulk semiconductor material. A cursory examination of solid state physics indicates that the electrons in any semiconductor material are partitioned into either a ground state valence band or a higher energy conduction band. Absorbing a photon of sufficient energy can cause promotion of an electron in the valence band to the conduction band, leaving behind a ‘hole’ in the valence band. Due to numerous attractive and repulsive forces, this electron-hole pair, or exciton as it is commonly referred, seeks to be separated by a finite distance, known as the Bohr exciton radius. Nanoparticles, however, typically have radii on the order of 1-4 nm which is smaller than the typical Bhor exciton radius. Consequently, the exciton formed upon absorption of a photon cannot achieve its desired separation, resulting in an effect known as quantum confinement. The size-tunable properties of quantum dots are all due to this quantum mechanical quantum confinement effect, hence the name “quantum dots”. Smaller particles are increasingly confined and give rise to higher energy electronic transitions and, consequently, a blue-shifted fluorescent emission. Additionally, optimization of this fluorescent emission can be achieved by wrapping this initial nanoparticle in a shell of a different semiconductor material. A wide variety of quantum dots synthesized in this manner have been reported in the literature[3-7], while the most extensively studied of these are quantum dots that have a cadmium selenide or cadmium telluride core encapsulated in a cadmium doped zinc sulfide shell (this review will focus on these quantum dots). The shell acts to passivate potential trap states on the surface of the core, enabling the quantum confinement of an electron hole pair, resulting in an increased likelihood of fluorescent emission. Consequently, the fluorescent properties of these core shell quantum dots offer several improvements compared to organic fluorophores. Most notably, they are highly fluorescent and their quantum yields are significantly greater than fluorescent dyes (commercially available quantum dots have quantum yields in excess of 80-90% which is due in part to the cadmium dopant in the shell).[8-10] Additionally, the size dependant extinction coefficients of quantum dots are exceptionally large (on the order of 1x106) [11,12] which means imaging applications based on quantum dots can utilize low powered excitation sources. This combination of extremely high extinction coefficients and quantum yields leads to exceptional brightness. Fluorescent dyes are also susceptible to photobleaching and metabolic and chemical degradation, while quantum dots have demonstrated excellent photostability in biological environments. The absorption spectrum of quantum dots (Figure 1A) is continuous in nature, so a single excitation source may be used to excite a wide range of colors.[13] Additionally they have size tunable narrow emission spectra, commonly less than 30 nm full width at half maximum[14-17] (Figure 1B), which enables the development of multiplexing experiments where several distinct biological targets may be monitored simultaneously.
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Figure 1. Photophysical properties and characterization of quantum dots. (A) Representative absorption spectra for various sizes of CdSe quantum dot nanocrystals. This tunable absorption is the result of quantum confinement effects upon absorption of a photon. Since this absorption is continuous above the band gap for all nanocrystals, a single excitation source (i.e. 350 nm) can be used to excite all sizes of nanocrystals. (B) UV illumination illustrates that the fluorescent emission from cadmium selenide nanocrystals can be tuned across the visible spectrum, and can even be extended into the near-IR or UV by varying composition. (C) High resolution atomic number contrast scanning transmission electron (ZSTEM) micrograph illustrating the atomic structure of an individual CdSe/ZnSe/Cd quantum dot.[8] Adapted from Rosenthal et. al. Surface Science Reports.[9]
Quantum Dot Surface Chemistry Quantum dot synthesis generally utilizes a coordinating solvent such as trioctyl phosphine oxide (TOPO) to control growth rate, and are consequently coated with this ligand after synthesis.[18] TOPO coated dots are stable and highly fluorescent in organic solvents, but they have low solubility and quantum yields in aqueous solution. To make them compatible with an aqueous environment it is necessary to modify the surface chemistry.[19] This may be achieved by replacing the TOPO with a water soluble ligand such as a thiolated acid.[20] Examples of this approach include mercaptoacetic acid[2], mercaptopropionic acid[21], dihydrolipoic acid (DHLA)[22,23], and DL-cysteine.[24] As these thiols are not covalently bound to the surfaces of dots, an equilibrium exists in aqueous solution with dynamic ligand exchange. This may be problematic if there are other species present in the buffer that can displace the thiol. To overcome the limitations of thiolated ligands a number of alternative surface chemistries have been explored. These include displacing the TOPO with organic phosphine oligomer heterofunctional ligands[25], displacing the TOPO with peptides[26], encapsulating the TOPO coated quantum dots in a water soluble polymer[27,28] or cross linking the mecapto acids with the amino acid lysine.[29] Derivatives of TOPO that can be chemically modified to form a water soluble polymer after the quantum dot synthesis have
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also been reported.[30] To be useful for biological applications, additional surface modifications must facilitate improved colloidal stability, low nonspecific adsorption to cellular components and photostability in a wide range of buffers and pHs, regardless of salt concentration. Colloidal stability and photostability in a wide range of buffers can be achieved by a number of capping methodologies where the dot is coated in materials such as amphillic polymers[31-33], multifunctional polymers[34] dendrimers[35,36] or creating a silicon dioxide layer around the dot.[1,19,37-40] Low nonspecific adsorption may be achieved by adding inert polymers to the surface such as polyethylene glycol[41], sugars, polysacarides,[45] proteins,[42] or encapsulating the quantum dots in micells.[43-46] The photostability of quantum dots has been studied in numerous buffers, and it has been shown that quantum dots will degrade when stored in buffers at low pH (pH~5) for a period of several days. For this reason quantum dots are usually supplied and stored in borate buffer at pH 8.4 until required. This degradation has been shown to be size dependant and occurs quicker at higher dilutions.[47] Additionally, the photostability and colloidal stability of quantum dots is dependant upon their surface coatings. Quantum dots with thio glycerol on their surfaces had better colloidal and photostability in 5 M sodium chloride solution than mercaptoacetic acid coated quantum dots, which aggregate after 30 minutes.[48] Quantum dots coated in polyethylene glycol, however, have even greater colloidal stability in high salt buffers.[49] A reduction in quantum yield has been observed in commonly used buffers such as Tris and PBS, but this is a slow process, requiring a period of many hours or days before it becomes significant, which is not likely to be problematic during the time frame of the average experiment.
Bioactivation of Quantum Dots Initial attempts to introduce biological activity onto quantum dots were first reported in 1998. One involved conjugating the quantum dots to transferin which were subsequently used for cellular imaging[2]; while in the same issue of Science, Bruchez et al. used phalloidin, an actin binding molecule, to image cells.[1] Following this pioneering work a great deal of effort has been expended to make the surfaces of quantum dots compatible with biologically active molecules. There are now many different strategies to conjugate biologically active molecules to quantum dots. These include conventional methods such as covalently attaching a reactive functionality on the ligand to the surface coating of the dot via a reactive ester, or to maleimide. The biologically active ligand may also be bound to the surface of the dot via electrostatic interactions or acid base interaction such as a thiol interacting with the zinc sulfide coat. Quantum dots have been coated in proteins such as avidin and streptavidin and biotinylated ligands may be conjugated to these protein modified quantum dots.[50] Using these approaches a diverse range of biologically active molecules have been attached to quantum dots. Ligands that have been conjugated to quantum dots include antibodies[51], peptides[52], proteins[53-57], RNA[58], DNA[59-68], PNA[69], cytokines[70], viruses[71] and small molecules such as drugs and neurotransmitters.[72, 137] The biological properties of the quantum dot may be altered by changing the surface coating and enabling the development of cell penetrating quantum dots; a variety of different coatings have been reported including cationic cell penetrating peptides and nanogels.[2,73-78]
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Quantum dots have been utilized to label proteins and receptors in the cells membrane[79] and nuclear targeting peptides have also been attached to quantum dots to image cell nuclei.[80] Peptides that are cleaved by proteolytic enzymes have been attached to the surfaces of quantum dots enabling the development of fluorescent probes for the expression of protease enzymes.[81-83] One such quantum dot assay for enzymatic activity was developed by Xu et al. and uses a FRET based system for the detection of β-lactamase activity. In this assay, a βlactam was derivatized with biotin and the fluorescent dye Cy5, as illustrated in Figure 2. Cy5 acted as a fluorescence acceptor and quenched the fluorescent emission of the quantum dot. However, the presence of β-lactamase initiates cleavage of this lactam linker, permitting the Cy5 to dissociate from the dot’s surface and restore the fluorescent quantum dot emission.[84] HO3S
Quantum Dot
Biotin
H N
H N
S
HO
HO
O N
O
O
O
SO3 H
S
N
N+
O
S
H N
N H
O
Lactamase HO3 S
Quantum Dot
Biotin
H N HO
O
HO
H N
S
O N
HO N
+ O
SO3 H
O
SH S
N H
N+
O H N O
Figure 2. A FRET sensor for β-Lactamase activity.
In addition to in vitro applications, quantum dots have been used in a wide variety of deep tissue[85] and in vivo applications[86-91] including tracking metastatic tumor cell extravasation[92] and sentinel lymph node mapping during surgery.[93,94] Unlike fluorescent dyes, the multivalent nature of quantum dots enables multiple copies of the same ligand to be attached to the surface of the dot; alternately, several different ligands may be attached to the surface of the dot. For instance, specificity can be introduced through the conjugation of an appropriate ligand, while another surface modification may confer an additional characteristic to the quantum dot. An example of this may include the addition of paramagnetic coatings for use in MRI.[95,96] The multivalent nature of quantum dots may lead to the development of smart drug targeting nano conjugates with enhanced sensitivity and selectivity for desired biological targets, and nano particle based gene delivery systems.
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Antibody Conjugated Quantum Dots 1. Quantum Dot Based Fluorescent Assays One of the widest applications of quantum dots as biological labels to date has been to label biological targets with antibodies. Given the low signal to noise ratios of fluorescent dyes and their susceptibility to photobleaching, it is not surprising that a variety of sandwich based assay systems for many different cellular components and toxins have been developed to incorporate quantum dots. In these systems, the antibody may be attached to the surface of modified poly acrylamide (AMP) coated quantum dots via a biflunctional linker such as sulfo-SMCC (illustrated in Figure 3).[97] Alternately, the antibody may be attached to an adaptor protein[98] which has been conjugated to the dots surface. Na+ O
-
O
O O
S O
N
O
N O
O
O
sulf o-SMCC Figure 3. The structure of sulfo-SMCC.
These conjugates have been used for the detection of staphylococcal enterotoxin B (SEB) in plate based assays,[98] and in continuous flow immunoassays for the detection of the explosive 2,4,6-trinitrotoluene (TNT) in aqueous samples.[98] The narrow fluorescent emission spectra of quantum dots has been exploited by many groups and multiplex toxin assays have been developed with antibody conjugated quantum dots using four colors of quantum dots.[99] Another method of antibody conjugation uses biotinylated antibodies which are attached to quantum dots via an interaction with avidin or streptavidin. Avidin conjugated quantum dots have been used as an assay in conjunction with biotinylated antibodies to detect SEB and cholera toxin.[100] Pegilated streptavidin conjugates have been used in the development of tissue microarray sandwich assays for the detection of proteins expressed by malignant cells[101,102] as well as sufamethazine residue in chicken muscle tissue[103] and an immunosensor for the detection of prostate-specific antigen.[104] Antibody conjugated quantum dots have also been employed in one recent application which provides an improved detection methodology for respiratory syncytial virus (RSV) infections.[105] This study utilized viral fusion (F) and attachment (G) proteins, which are incorporated into the host cell membrane following viral infection, as antigenic markers of RSV infection. The enhanced sensitivity and improved signal to noise of quantum dot probes permits the detection of viral infection as early as one hour following infection, even at an exceedingly low multiplicity of infection (Figure 4). Traditional western blot or PCR methodologies, currently utilized for clinical detection, can require as many as four days of
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cell culture to verify infection, but antiviral treatment is only effective if administered early in the course of infection. Consequently, quantum dots are being pursued in a clinical diagnostic application and may ultimately provide a method for early, rapid detection of RSV infection.
Figure 4. Progression of RSV infection as monitored by antibody conjugated quantum dots. A series of images showing the detection of RSV fusion proteins with antibody conjugated quantum dots at various time points after infection. Traditional methodologies for RSV detection typically require at least four days of cell culture before detection is possible. Quantum dots, however, show remarkable sensitivity and are able to identify an RSV infection after only one hour. Adapted from Bentzen et. al.[105]
Quantum dots conjugated to antibodies may also be used in western blot detection systems, facilitating ultra sensitive and quantitative detection for multiplexed proteomic analysis.[106,107] Given their size-tunable fluorescent emission, it is possible to develop assays where an antibody is conjugated to one color quantum dot while an analyte is bound to a quantum dot of another color. The resulting antigen/antibody immunocomplex has a color that is a combination of these two colors. This principle has been demonstrated with quantum dots conjugated to BSA and the corresponding anti-BSA antibody (IgG).[108] A flouroimmuno assay as also been developed which relies upon Förster resonance energy transfer (FRET) to detect human estrogen receptor-β (ER-β). In this assay, quantum dots with a fluorescent emission at 565 nm were used as a FRET donor, while a polyclonal anti-ER-β antibody labeled with either an Alexa Fluor® 568 or an Alexa Fluor® 633 dye served as the acceptor.[109] Another application targeted Listeria monocytogenes, an important food born pathogen, by detection of Internalin A (In1A) and Internalin B (In1B), proteins which promote host cell invasion. Fluorescent microscopy on fixed cell samples was performed using a primary anti-In1A antibody, followed by a biotinylated secondary antibody and subsequent streptavidin quantum dot detection. [110] Formalin fixed paraffin embedded tonsil and lymphoid tissue has also been imaged using streptavidin conjugated quantum dots in conjunction with an appropriate primary antibody and multiple biotinylated secondary antibodies.[111] Finally, formalin fixed Purkinje cells in the glia have been labeled in
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cerebellum tissue sections with an antibody specific for the glial fibrillaray acidic protein (GFAP).[112]
2. Quantum Dot-Antibody Based Live Cell Assays In addition to sandwich assays, antibody conjugated quantum dots have been used to label live and fixed cells. Specific labeling of live cells with avidin conjugated quantum dots and biotinylated antibodies has been demonstrated for HeLa cells which were transfected to express the an extracellular loop of the multidrug transporter P-glycoprotein (pgp).[113] In an elegant experiment, Dahan showed that the glycine receptor in live cells could be labeled with primary antibody (mAb2b), a biotinylated anti-mouse Fab fragment and streptavidin conjugated quantum dots.[114] Cancer markers in live Her2 cells[115], TrkA receptor expression in live PC12 cells[116], and single bacterial pathogens such as E.coli DH5α[117] have been labeled using antibodies and streptavidin conjugated quantum dots. Additionally, SiHa cells have been labeled with quantum dots coated in streptaividin and conjugated to anti-EGFR. These conjugates have been utilized in a diagnostic test for use in early cervical cancer detection.[118] The narrow fluorescent emissions of quantum dots have also been used to expand the capabilities flow cytometry. In a recent article, Chattopadhyay et al. demonstrated the simultaneous detection of 17 different fluorescent emissions, each corresponding to a specific antigen expressed in T-cells, utilizing quantum dot fluorphores.[119] Streptavidin conjugated quantum dots and dots coated in an amphiphillic polymer have been used in conjunction with a biotinylated anti-CD33 antibody to label and track human leukemic, bone marrow and cord blood cells via flow cytometry.[120] Furthermore, a fluorescent assay using biotinylated antibodies and streptavidin conjugated quantum dots has been developed which permits the simultaneous detection of Escherichia coli 0157:H7 and Salmonella Typhimurium.[121] In addition to these cell based assays, antibodies have been conjugated to quantum dots and used to image tumors in vivo.[122]
Peptide Conjugated Quantum Dots Peptides may potentially be better suited to serve as targeting ligands for biological applications due to their considerably smaller size compared to antibodies. This size difference allows tens or even hundreds of peptides to be attached to the surface of a single quantum dot. Consequently, peptide conjugated quantum dots may exhibit stronger binding affinity and better targeting efficacy as a result of this polyvalent effect. A variety of peptides have been conjugated to quantum dots and used to image live cells. Some notable peptidebased quantum dot applications include angiotensin II binding to the angiotensin I receptor,[123] biotinylated epidermal growth factor (EGF) detecting the erbB/HER receptor,[124] and neuronal growth factor conjugated quantum dots binding TrKA receptors in PC12 cells.[125] Cai et al. have conjugated pegilated cadmium telluride quantum dots to the arginineglycine-aspartic acid (RGD) peptide, illustrated in Figure 5, for in vivo imaging of integrin αvβ3-positive tumor vasculature in a murine xenograft model.[126,127] Integrin αvβ3 plays a key role in tumor angiogenesis and metastasis and it is significantly upregulated in invasive
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tumor cells of certain cancer types (glioblastoma, melanoma, breast, ovarian, and in prostate cancer and in almost all tumor vasculature) but not in quiescent endothelium and normal tissue. Integrins expressed on endothelial cells modulate cell migration during angiogenesis, and inhibition of integrin αvβ3 has been shown to prevent tumor growth and cause tumor regression. Several integrin αvβ3 inhibitors are currently in clinical trials as therapeutics, and the ability to image this receptor in vivo would have potential applications in both cancer imaging and image guided surgery. Near IR quantum dots were functionalized by attaching a thiolated derivative of the cyclic peptide RGD to the pegilated surface of the cadmium telluride quantum dots via a SMCC linker. A thiolated derivative of RGD was synthesized by reacting the lysine ε-amino residue with S-acetylthioglycolic acid N-hydroxysuccinimide ester (SATA), followed by a thiol deprotection using hydroxylamine under neutral conditions. The resulting conjugates were injected into U87MG tumor-bearing mice and a rapid uptake in tumor tissue was observed relative to background fluorescence. The quantum dots were also observed to accumulate in the liver, spleen, bone marrow and lymph nodes. H N
O
NH NH 2
O HN
NH
HO
O OO
HN
NH
NH 2
O HO
RGD Figure 5. RGD a cyclic peptide that has high affinity for integrin αvβ3.
A recent cancer therapy application has utilized peptide conjugated quantum dots targeted to specific heat shock proteins as potential tumor imaging agents. Heat shock proteins are a set of chaperone proteins involved in many intra- and intercellular processes, including protein synthesis and folding, vesicular trafficking, and antigen presentation and processing. Glucose-regulated protein is a member of this family and is usually expressed and located in the edoplasmic reticulum. However stressful conditions, such as heat exposure, lead to an increase in expression of this protein and the presence of a surface membrane bound form of glucose-regulated protein. In addition, the over expression of this protein has been shown to occur in cancer. Consequently, this receptor has been identified an attractive target for drug delivery of chemotherapeutic agents for cancer. A number of different cyclic peptides have been identified that bind to this protein and are internalized by cancer cells. One of these, the peptide designated Pep42, was covalently attached to the surfaces of modified amphiphillic poly acrylamide (AMP) coated quantum dots using an EDC coupling. The binding and internalization of these conjugates in the highly metastatic human melanoma Me6652/4 or Me6652/56 cells was then studied by a FACS analysis.[128]
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COOH
O
O
H N
H N
O
N H O
N H
N H2 O
N H
N
NH
O
L LP1A (1) COOH
O
O
H N
H N
O
N H O
N H
N H2 O
N H
N
NH
O
L LP2A (2)
Figure 6. The chemical structures of LLP1A and LLP2A.
Phage-display peptide libraries are an established approach for identifying specific peptide sequences with a high affinity for cellular targets. In one such application, the multiple cell binding tetrameric peptide TP H1299.2 was identified and used in conjunction with streptavidin quantum dots to image live H1299 cells.[129] High affinity and specific targeting peptides for the α4β1 integrin receptor have also been identified using phage-display peptide libraries. The endogenous ligands for the α4β1 integrin receptor are the QIDS and ILDV motifs of vascular cell adhesion molecule-1 (VCAM-1) and fibronectin, respectively. This receptor is expressed in proliferating endothelial cells in angiogenesis during tumor development and therefore is also an attractive target for imaging and therapeutic agents for cancer. Using one compound on one bead combinatorial chemistry, several high affinity peptidomimetics have been identified; two of these compounds, LLP1A and LLP2A, are shown in figure 6. The binding affinities of LLP2A and LLP2B for the α4β1 integrin receptor were measured using an α4β1-mediated adhesion assay with Jurkat cells and an immobilized
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CS-1 peptide (a 25-amino acid linear peptide of fibronectin that interacts with the α4β1 integrin receptor). The IC50 of LLP2A and LLP1A were found to be 2 ± 1.4 pM and 22 ± 18 pM respectively. A biotinylated version of LLP2A was subsequently used to image Jurkat cells expressing the α4β1 intergrin receptor by incubating the cells in the presence of the biotinylated LLP2A followed by quantum dot 605 conjugates. In addition LLP2A biotin was also conjugated to streptavidin-quantum dots and used for in vivo imaging which showed strong binding to α4β1 integrin receptor expressing cells Molt-4 cells.[130] NH 2 HN NH
N NH O O
H2 N
O
O
H N H
O
H N
N H H
O
O
H H N
N H
H
O
H
O
H N
NH 2
O
O N H
OH
NDP OH O O H2 N H
H N H
H N O
H
O
H N H
O
H N
O N H
H
H H N O
O N H
H N
SH H
O (PEG) 2
O
N H
H N
R
O
HO H N
O O
O
O
O
O
H N
(PEG)2
Deltorphin-II Figure 7. NDP and Deltorphin-II ligands used to label G-protein coupled receptors in live cells.
Zhou et al. have recently reported the development of peptide labeled quantum dots for imaging G protein coupled receptors in live cells. In their system, they attached the peptides either via an EDC coupling to carboxylic acids on the quantum dot surface or via a SMCC cross linker. These dots were coated in a low molecular weight (~1200 Da) diblock copolymer which encompassed acrylic acids as hydrophilic segments and amino-octyl side chains as hydrophobic segments. Two different peptides were employed to introduce specificity (Figure 7). The NDP peptide is an α-MSH analog that has a high affinity for human melanocortin receptor, while the Deltrophin-II analog is specific for δ-opioid
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receptors. After conjugation, the quantum dots were used to label live either HEK cells expressing the NDP receptor or live CHO cells expressing the δ-opioid receptor.[131]
Small Molecule Conjugated Quantum Dots The first report of small molecule quantum dot conjugates employed in a biological imaging application was published by Rosenthal et al.[72] In this work, quantum dots were used to image the serotonin transporter (SERT) following surface modification with pegilated serotonin ligands[132], illustrated in Figure 8. Attachment directly to the surfaces of quantum dots was carried out via the thiol terminus. These conjugates antagonized the serotonin transporter protein (SERT) with an IC50 of 115 µM and facilitated fluorescence imaging of SERT expression in transfected HEK-293 cells. NH2 O
HS
O
O N H
Figure 8. The serotonin ligand conjugated to mercapto acetic acid coated quantum dots and used to image SERT expressing cells.
Following the early success with these serotonin ligand conjugates, we have since synthesized a variety of ligands that are targeted to SERT, the dopamine transporter (DAT) and the 5HT2A receptor. These were conjugated to quantum dots using a variety of methodologies including biotin-streptavidin interactions, EDC couplings to the surfaces of AMP dots and thiol exchange reactions to the surfaces of mercapto acetic acid conjugated NH2 O X O N O
S HN NH O N H R
(I) R = SH (II) R = PEG600NH2
(III) X = NH (IV) X = NHCOCH2PEG3400NH
Figure 9. Continued on next page.
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R
O H N
N
X
N
(V) R = H, X = C 11H22SH (VI) R = F, X = C11H22SH (VIII) R = F, X = Biotin (X) R = H, X = Biotin (XI) R = F, X = C11H22NHBiotin
Figure 9. Ligands that have been conjugated to quantum dots.
quantum dots. A selection of these compounds is shown in Figure 9. All of these ligands have been subsequently tested against their biological targets and shown to be biologically active.[133-137] HO N O HN O
N O HO
O
HN
HN O
NH PEG3400
O
PE G3
O
400
HN
OO OH O
N H O
O HN PEG3400 O HN
HO O
HN
O
0 40 G3 PE
PE G 34 00
NHHN O HN O N
O HN
OH
O NH
O HN
O N
O N
OH HO
Figure 10. Schematic illustration of AMP dots conjugated with a muscimol ligand.
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C
E
B
D
F
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Figure 11. Bright field and fluorescent images of oocytes incubated with a 34 nM solution of either muscimol conjugated dots or a 34 nM solution of AMP dots. Panels A and B are the fluorescent and bright field images of a GABAC expressing Oocyte that was incubated with a 34 nM solution of muscimol conjugated dots. Panels C and D shows a fluorescent and bright field image of a GABAC expressing Oocyte incubated with a 34 nM solution of AMP dots. Panels E and F show the fluorescent and bright field images of an untransfected Oocyte incubated with muscimol conjugated AMP dots.
Additionally, recent efforts have resulted in a pegilated derivative of muscimol designed to interact specifically with the GABAC receptor. Following conjugation to AMP quantum dots (Figure 10), Gussin et al. demonstrated that this pegilated muscimol derivative could be used to image GABAC expressed in Xenopus Laevis oocytes. Quantum dot conjugates specifically labeled GABAC expressing oocytes and AMP coated quantum dots gave very little nonspecific labeling with transfected and untransfected oocytes, as illustrated in Figure 11.[138] In addition to our work in the small molecule conjugates field, Clarke et al. have conjugated a dopamine derivative to the surface of quantum dots, via acid base conjugation chemistry, and have subsequently used these conjugates to target cells expressing the D2 receptor. [139] These conjugates were readily internalized by D2 expressing HEK293 and 3T3 cells, and could be blocked by a 10 fold excess of free dopamine. This internalization was shown to be specific as these conjugates did not bind to cells lacking the D2 receptor. This report also detailed the formation of a dopamine quinone complex and described its observed cellular toxicity. Dopamine can undergo a light insensitive oxidization to form a quinone, as shown schematically in Figure 12. This compound is toxic, causing oxidative damage to > 90% of the cells. However, the addition of mercapto ethanol, a reducing agent which inhibits quinone formation, results in enhanced labeling and normal cell viability. Furthermore, they have proposed an energy transfer process by which a molecule in close proximity to the quantum dot surface may act as a photosensitizer and lead to the generation of free radicals such as 1O2. Consequently, the ability of these dopamine conjugates to induce oxidative damage in cellular systems may ultimately permit quantum dot photodynamic therapy applications.[139]
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HO
O
O O
HO
N H O HO
N H O
Ox S NH
O
S NH
CdSe
CdSe O
HO
Figure 12. Oxidation of a dopamine ligand on the surfaces of quantum dots.
Future Applications of Quantum Dots in Drug Development and Medicinal Chemistry Cell based assays may be used as part of the drug discovery process, and it is likely that small molecule or peptide conjugated quantum dots will form the basis of such assays. The nature of the quantum dot surface is crucially important for the development of such probes, requiring excellent colloidal stability and photostability in a wide range of buffers with little or no nonspecific adsorption to a variety of cell types. In addition, the ligands must have high affinity, in the low nanomolar or sub nanomolar range, and selectivity for the desired receptor. The multiplexing afforded by quantum dots would enable adaptation to a high throughput format. Many different drug development fluorescent assay based platforms can be envisaged. For example transfected cells may be plated out in multi well plates and incubated with quantum dots conjugated to a specific antagonist or agonist. The resultant fluorescently labeled cells could be incubated with a wide range of test compounds for an allotted time period and subsequently washed with buffer. Any displacement of the quantum dot conjugates would result in a reduction of fluorescent intensity and would indicate that the test compound is biologically active. To be useful in the clinic for in vivo applications, it is apparent that quantum dots must first be capable of demonstrating little to no cellular toxicity. Currently, however, there is little information present in the literature regarding added toxicity as a result of quantum dot exposure. Both cadmium and selenium are known toxins and, additionally, cadmium is a suspected carcinogen. Cadmium has a half life of 15-20 years in humans and is systemically transported around the body, with the ability to cross the blood brain barrier, eventually accumulating primarily in the liver and kidneys. The possibility that cadmium may leak from quantum dots and have a deleterious effect on cellular physiology has been studied in the literature. These studies, however, were limited to cadmium selenide cores[140] lacking any zinc sulfide shell or the wide variety of capping ligands routinely used with quantum dots. Loric et al. found that CdTe quantum dots had variable toxicity in PC12 rat cytoma cells depending upon their surface modification, In this study, quantum dots coated with mecaptoacetic acid and cystine had an observed toxicity at concentrations of 10 µg/ml while uncoated quantum dots were cytotoxic at 1 µg/ml. Additionally, cytotoxicity was significantly greater for small quantum dots with a positive charge than larger quantum dots with a similar
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charge. The distribution inside the cell was also affected by the size of the quantum dot, as small quantum dots were capable of entering the nucleus while larger quantum dots remained in the cytosol.[141] Other groups have suggested that the quantum dot capping material may be responsible for added cytotoxicity. Notably, Hoshino et al. found that mercaptoundecanoic acid alone caused toxicity in murine T-cell Lymphoma EL-4 cells.[87] Several in vitro and in vivo studies have been cited in the literature as demonstrating a lack of evidence for quantum dot cytotoxicity including Ballou et al.[88], Dubertret et al.[44] and Jaiswal et al.[142]
Conclusion Quantum dots are increasingly finding a diverse range of applications in enzyme assays as well as fluoro immuno assay based applications. This review has highlighted numerous applications where the unique photophysical properties of quantum dot fluorophores have allowed unprecedented insight into biological processes. These properties enable long periods of illumination and high quantum yields permit detection at the sub nanomolar range. The narrow emission spectra of quantum dots facilitate their used in several multiplexed assay systems, and they have been used as fluorescence donors in many FRET based assay systems. Additionally, the multivalent nature of their surfaces may be useful for the development of nano vectors for drugs and gene therapy. As the size of quantum dots is larger than 3.5 nM, generally agreed to be the maximum particle size for renal clearance,[89] their application in some in vivo imaging systems may be limited, especially those where a low background fluorescence or a rapid clearance is required. The continued development of alternate surface modifications should facilitate improved biologically inert probes with enhanced colloid stability and reduced nonspecific cellular interactions. Incorporating quantum dots in whole cell assays capable of simultaneously screening a wide variety of drug candidates will move conventional drug development approaches beyond the current single target approach. Consequently, quantum dots are likely to be of great benefit in future drug discovery applications.
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In: Quantum Dots: Research, Technology and Applications ISBN: 978-1-60456-930-8 Editor: Randolf W. Knoss, pp. 267-298 © 2008 Nova Science Publishers, Inc.
Chapter 7
STRAIN RELIEF AND NUCLEATION MECHANISMS OF INN QUANTUM DOTS J.G. Lozano1, A.M. Sánchez1, R. García1, S. Ruffenach2, O. Briot2 and D. González1 1
Departamento de Ciencia de los Materiales e Ingeniería Metalúrgica y Química Inorgánica, Universidad de Cádiz, 11510, Puerto Real, Cádiz, Spain 2 Groupe d’Etudes des Semiconducteurs, UMR 5650 CNRS, Place Eugène Bataillon, Université Montpellier II, 34095 Montpellier, France.
Abstract In the last years, Indium nitride (InN) based nanostructures are focus of special and increasing attention. The combination of the intrinsic properties of InN –best theoretical electronic properties among the III-nitrides and a recently established bandgap of 0.7 eV suitable for the telecommunications field– with those related to quantum confinement phenomena promises interesting applications. Here we present a complete characterization by transmission electron microscopy (TEM) of uncapped and GaN capped InN quantum dots grown on sapphire/GaN substrates by MOVPE. Morphological aspects such as height, area or roundess of the QDs, as well as the effect of the GaN capping layer on them will be discussed. The nucleation mechanisms of the InN QDs will be studied, showing that they preferentially nucleate on top of pure edge type threading dislocations located in the GaN and that do not propagate into the QDs. This mechanism of InN QDs nucleation on GaN has never been reported before, and has to differ notably of the more classical ones found in the literature, like the Burton-Cabrera-Frank mechanism, since the Burgers analysis showed that these dislocations present a pure edge character; or the Stranski-Krastanov model, since the relaxation occurs by the formation of the misfit dislocations network instead of by surface islanding. Finally, the strain state of the QDs will be also reported, showing that they are almost fully relaxed due to the introduction of a misfit dislocations (MD) network in the interface QD/GaN. Strain maps at atomic scale in plan-view orientation allow a complete characterization of this network, consisting of three sets of misfit dislocations lying along the <11-20> directions without interaction between them or generation of threading dislocations. One of the most important challenges in order to achieve functional InN QDs based devices is the obtention of a good crystalline quality capping layer due to the difficulties associated to its growth, namely the low temperature neccesary to avoid the previously InN deposited decomposition. In this case, it was succesfully achieved and unexpectedly, the introduction of
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J.G. Lozano, A.M. Sánchez, R. García et al. this GaN capping layer is shown to induce a rearrangement of these MDs, lowering the degree of plastic relaxation of the heterostructures.
I. Introduction Among the III-V semiconducting compounds, those constituted by the Nitrides subgroup (IIIN) have become a focus of special attention during the last decade, mainly because of their attractive optoelectronic properties. Their use makes possible the enlargement of the emission and detection wavelength range significatively, and as a consequence, the application field of the nitrides-based devices. For instance, the achievement of shorter wavelengths (GaN, AlN) led to increase in one order of magnitude the density of information optically recordable, to fabricate high-power lasers or to use very short wavelength detectors for aerospace applications. In the same way, the use of AlInGaN alloys would lead to the performance of detectors or emitters covering the whole optical spectral range, from the infrared to the near ultraviolet. Thus, it is clear that the economic and social repercussion of the potential applications of this kind of materials justifies the huge effort carried out by the scientific community during the last years to achieve their optimum development. Nevertheless, Indium Nitride (InN) has been the element of this family historically less studied and developed. This was due to the fact that its band gap formerly assigned[1], 1.9 eV, was covered enough by other semiconductors well known and developed. This reason, together with the important difficulties associated to its growth, made of InN a material with relative low interest. However, in 2002 different groups demonstrated by photoluminescence measurements that the real bandgap of InN should be reestablished in a value close to 0.7 eV[2,3]. Therefore, we may assume this new value, InN would become a very suitable material for the fabrication of optoelectronic devices, especially those operating in the infrared range, with frequencies near the optical fibre transmission window[4]. Moreover, InN has the best theoretical electronic properties among the nitrides[5], with high carrier mobility (above 4400 cm2V-1s-1 at 300K), a low effective mass for the electrons and a high saturation velocity, that makes it undoubtly useful for the fabrication of high frequency transistors. It has also proven that InN presents an intrinsic surface charge accumulation[6], around 1013 cm-2, that could serve as a basis for the development of chemical or biological sensors. However, nowadays, the achievement of fully functional InN-based devices is still a challenge, mainly due to the difficulties associated to the growth of high crystalline quality structures of this material. One of the most important is the lack of suitable substrates: the high reticular mismatch and different thermal expansion coefficient with the traditional substrates, such as sapphire or Si(111), make the InN structures grow highly faulted and dislocated, with the consequent pernicious effect on the electronic properties. The other difficulties are related to the growth conditions: the low dissociation temperature for InN and the high vapour pressure of N2 needed implies a low growth temperature. This is a huge problem in MOVPE where, as in other similar techniques, a high temperature is needed to obtain an efficient decomposition of the NH3 precursor. In consequence, the first attempts to grow InN[7] resulted in structures with a very poor crystalline quality. The use of intermediate layers, typically AlN or GaN, to reduce the lattice mismatch and plasma assisted growth techniques to obtain a more efficient NH3 dissociation, considerably improved the
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quality of these layers, reducing the dislocations density and enhancing the electronic properties. Up to now, the best results reported in the literature correspond to samples with a carrier mobility of 2050 cm2V-1s-1 and a carrier density of 3.5·1017 cm-3 regarding the electronic properties[8], and a dislocation density of 109 cm-2 regarding the structural properties[9]. On the other hand, as a consequence of the interest in reducing much more the integration scale of the different components that constitute the optoelectronic devices, the relatively new Nanoscience and Nanotechnology disciplines have boomed during the last decade, where nanometric scaled materials are obtained and manipulated,. Simply as an interesting fact, more than 230 billion $ is the market demand forecast for nanotechnologic applications in 2010, half of it corresponding to the optoelectronics field[10]. These disciplines are based on the understanding and use of the phenomena associated to the quantum confinement of the matter[11], for optoelectronic devices, namely better efficiency, spectral gain and monocromaticity of the emitted radiation than those for the massive material, as Easki and Tsu predicted in 1969[12]. Typically, for the fabrication of these nanostructures, litographic or ion bombardment techniques are used to create regular patterns on the substrates, on top of which the structures nucleate[13,14]. Nevertheless, the mechanism known as “self-ordering”, where uniquely by controlling the growth parameters self assembled heterostructures with a high homogeneity are spontaneously obtained, has shown to be itself effective enough[15]. In light of all these arguments, it seems clear that the synergic combination of the properties of the material (InN in our case), and those derivated of quantum confinement phenomena, would be extremely interesting. However, the achievement of nanostructuresbased functional optoelectronic devices requires the combination of several factors still uncontrolled: on one side, a regular distribution of the nanomotives with a perfect homogeneity in sizes and shapes. Otherwise, a variation in the energetic levels would occur, with the subsequent difficulties for tuning the emission. On the other hand, the structures must grow free of defects, specially threading dislocations, as they behave as free carries traps[16], impoverishing their emission efficiency and their transport and electronic properties. Moreover, to achieve the functionality of the devices, the semiconducting nanostructures must suffer a capping process, typically with another material with different bandgap, and during this growth, new strain fields influencing the residual strain of the buried structures, or intermixing processes and morphological changes may occur. During the last few years, some works relating changes in shape and density of InN quantum dots (QDs) with growth condition parameters such as temperature[17] or In/N flux ratio[18], or with the use of different pseudosubstrates[19] (AlN, GaN or Si) have been reported. However, these characterizations were performed using Atomic Force Microscopy. Thus, to our knowledge, the study presented in this chapter is one of the first concerning characterization by Transmission Electron Microscopy (TEM) of InN quantum dots. The aim of this work is to develop a full characterization of the morphology of these nanostructures, their strain state and structural properties, and the effect on the former of an overgrown GaN capping layer. With this, design rules are intended to be established, in order to obtain in a near future fully functional high quality devices based on InN quantum dots.
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II. Experimental InN quantum dots were grown onto sapphire substrates using a thick (~1μm) GaN buffer layer by Metalorganic Vapor Phase Epitaxy[20]. The schematic of the final structure is shown in Figure 2.1. First, a buffer layer of GaN was grown using the classical two-step process on (0001) sapphire at a temperature close to 1000 ºC. The temperature was then lowered to 550º C, and InN QDs were deposited using a V/III ratio of 15000 and NH3 as introgen precursor. For the capped QDs, we used a process where GaN is first deposited at low temperature (550 ºC) above the InN dots, in order to cover them and to prevent their decomposition at a higher temperature. Once protected in such a way, the growth temperature is raised to 1050 ºC to recrystallize the low-temperature GaN previously deposited. This is necessary because the low-temperature GaN has poor crystallinity. Samples in plan-view and cross-sectional geometries were prepared by mechanical grinding and dimpling to a final thickness of 10 μm, followed by Ar+ milling in a Gatan PIPS. All TEM related techniques were carried out in a JEOL 1200EX, working at 120 kV, and a JEOL 2011 and JEOL 2010FEG, working at 200 kV.
Figure 2.1. Schematic of the studied samples structure.
III. Morphological Characterization As mentioned previously, knowing the morphology of the QDs is especially relevant in order to establish reciprocal relationships between different parameters such as area, height or: density of the QDs, and the optical emission properties of these nanostructures. In order to clarify some of these factors, a statistical study by conventional TEM, was developed in both types of samples. A first part of the study was carried out analyzing uncapped InN quantum dots samples prepared in plan-view orientation, determining that the density of QDs is relatively low, with an average value of (4±2)·108 cm-2, and that most of them present a well defined flat hexagonal shape. The projection of some of them through the [0001] zone axis is shown in Figure 3.1. The statistics were carried out sampling over a significant number of QDs, some of them displayed in Figure 3.2, showing a large variety of shapes and sizes.
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The average diameter found for the QDs was d=73±11 nm, the average area A=5300±1700 nm2 and the average perimeter, p=270±50 nm. Thus, if we define the parameter roundness (R) as:
R=
p2 4πA
it results R = 1.16±0.08, very close to the calculated roundness for a perfect hexagon (R=1.103), and further from the roundness for the perfect circle (R = 1). Nevertheless, as shown in the histogram corresponding to this parameter (Figure 3.3), the deviation in the distribution of R is large, and only a 33% of them are between the characteristic values 1 and 1.10.
Figure 3.1. Plan view micrograph of uncapped InN QDs.
50 nm Figure 3.2. Examples of quantum dots where the statistics were performed.
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The area distribution of the QDs is displayed in Figure 3.4, where the histogram represents the frequency of QDs over the projected area along the [0001] zone axis. The standard deviation σ = 1700±200, and it follows a normal distribution clearly adjustable to a gaussian curve.
12
10
Counts
8
6
4
2
0 1.0
1.1
1.2
1.3
1.4
1.5
Roundness
Figure 3.3. Histogram showing the distribution of R for uncapped QDs.
5
Counts
4 3 2 1 0 2000
4000
6000
8000 10000 12000 14000 16000 18000 20000 22000 24000
Uncapped quantum dots area, nm
2
Figure 3.4. Histogram showing the area distribution for uncapped QDs.
In order to estimate the height of the QDs, TEM samples were prepared in cross-sectional geometry. In this case, the sampling was performed over a significantly reduced number of QDs, since the number of them observable in the electron transparent area of the sample is less than in plan view orientation, as a consequence of the low density of QDs. The average height found was h = 12±2 nm, and thus, the aspect ratio h/d results 1/6. As can be observed in Figure 3.5, there exist two preferential facets, one of them forming an angle of 59º with the (0001) plane, and the other forming an angle of 10º. From the equation that determines the angle between two planes, θ, in the hexagonal system:
Strain Relief and Nucleation Mechanisms of InN Quantum Dots
cos θ 12 =
4 3a 2
273
⎡ ⎤ 1 3 a2 ( ) + + + + h h k k h k h k ll 1 2 1 2 2 1 ⎢ 1 2 2 1 2⎥ 2 4c ⎣ ⎦
where h,k and l represent the planes indexes, and a and c the lattice parameters. Substituting for h1=0, k1=0, l1=1, and the values for θ expressed above, we obtain that the planes corresponding to the facets of the truncated pyramid are of the type {10 1 1} (59º) and
10º 59º
Figure 3.5. Cross-sectional micrograph of an InN QD showing the preferentical facets.
{10 1 10} (10º). Putting together all the obtained values so far, we have constructed a three-dimensional model for an uncapped InN quantum dot, as displayed in Figure 3.6:
12 nm
} {10-110
{0001}
{10-11} 73 nm
Figure 3.6. 3D reconstruction of an InN QD.
IV. Nucleation Rather significant is the fact that all the QDs when observed in plan view orientation present a well defined set of three directional moiré fringes pattern. An example of this can be seen in Figure 4.1, where a small tilt of the sample with respect to the zone axis exists in order to
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excite one of the fringe families. These patterns arise from the interference between two overlapping materials with different lattice parameters, as occurs in the present
Figure 4.1. Plan view micrograph of an InN QD showing a set of moiré fringes. Indicated by an arrow, interruption of a fringe indicating the presence of a threading dislocation.
InN/GaN system. In our case, the fringes are produced by the overlapping of the different sets of { 11 00 } planes in the two materials. Diffraction patterns show a perfect alignment between the spots corresponding to InN and GaN crystals, and thus, the moiré fringes are of the translational type. Although the main utility of these patterns is the determination of the strain state of the heterostructures as will be shown in the next section of the chapter, they also provide information about the presence and localization of dislocations in the materials[21]. Indicated by an arrow in Figure 4.1, a interruption in the moiré fringes is clearly visible in the center of the area corresponding to the QD, a fact associated to the presence of a threading dislocation (TD) in that region. In order to clarify the location of the TD, i.e., whether is into the InN or GaN, high resolution TEM micrographs were recorded along the <0001> zone axis of the area containing it, as shown in Fig 4.2(a).
Figure 4.2. a) HRTEM micrograph of a QD in plan view orientation; in the white square, area where the TD is present b) Associated diffractogram. Circled in yellow, the spots corresponding to InN, and in red, those corresponding to GaN.
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In the associated digital diffraction pattern (Fig. 4.2(b)) Bragg masks were applied separately to the spots corresponding to the two materials and Fourier filtered images were reconstructed revealing solely the InN crystal (Figure 4.3(a)) and the GaN one (Figure 4.3(b)). For the latter, the projection of two extra { 11 00 }half-planes can be seen indicated by white arrows. The Burgers circuit drawn around the area containing these half planes shows a closure failure, whereas applying the same procedure for the InN no closure failure is observed. Therefore, at this point we can conclude that i) the TD is located into the GaN and does not propagate into the InN QD and ii) the TD has at least an edge component of the Burgers vector b = 1 3 1120 .
a)
b)
Figure 4.3. Fourier filtered image where Burgers circuits have been drawn around the area containing the TD for a)InN and b)GaN.
Figure 4.4. Cross-sectional micrographs taken under two beam conditions using the reflexions a) g=0002 and b) g= 2 110.
To determine whether the TD has a mixed or pure edge character, diffraction contrast images were recorded in conventional cross-section TEM under two beam conditions. We used the two characteristic reflexions for the wurtzite system in cross-section geometry, i.e., g=0002 and g= 2 110, near the zone axis <01 1 0>. For the first case (Fig 4.4(a)), no threading dislocation is visible, however, for the second condition (Fig 4.4(b)) a TD is clearly observed
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in the GaN substrate, not in the InN, confirming the previous results. Applying the g.b invisibility criterion summarized in Table 4.I, we finally conclude that the TD is pure edge type ( b = 1 3 1120 ). Table 4.I. Pure edge (a) g=0002 g= 2 110
b=1/3<11 2 0> 0 ≠0
Pure screw (c) b=<0001>
≠0 0
Mixed (a+c) b=1/3<11 2 3> ≠0 ≠0
This was not an isolated observation, but a general rule for every QD observed in large areas of plan view specimens (an example of some of them is shown in Figure 4.5). Therefore, it seems clear that these pure edge TDs in GaN have to be closely related to the nucleation mechanisms of the InN QDs.
Figure 4.5. PVTEM micrograph of several InN QDs in plan view. In all of them, a TD is visible in the center of the area corresponding to the QD.
There exist three main epitaxial growth models typically found in the literature: FrankVan der Merwe (FM), Volmer-Weber (VW) and Stranski-Krastanov (SK), schematically shown in Figure 4.6 (a), (b) and (c) respectively. The first one[22] is a bidimensional growth model where the structure is fabricated monolayer by monolayer, whereas with the VW model[23], three dimensional islands are directly obtained. The occurrence of one or another model is determined by a balance among the surface energy, the elastic energy and the
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interface energy. If the surface energy of the substrate is equal or less than the sum of both the elastic energy due to the formation of the overgrown layer and the energy associated to the interface, a VW growth will occur; otherwise, a FM growth will take place. a)
b)
c)
Figure 4.6. Growth model scheme a)VW, b) FM y c) S-K.
An intermediate case is the one proposed by Stranski-Krastanov[24], where after the deposition of a few epitaxial monolayers (“wetting layer”), a three dimensional growth of islands starts. The general model that explains this growth is based on a balance between the surface energies of the substrate and of the layer, the formation energy of the interface, the strain energy of the layer and the deformation energy of the substrate[25]. According to this, the SK growth may occur in systems where the formation of two-dimensional layers is favourable during the deposition of the first few monolayers. The strain energy in the layer increases with increasing its thickness, and above a critical value, tc3D, the formation of 3D islands spontaneously occurs in order to relax the accumulated elastic strain. In the final configuration, a 2D wetting layers is obtained with 3D islands on top. However, for InN QDs on GaN, as will be shown in the next section of the chapter, the relaxation of the strain due to the lattice mismatch occurs via the formation of a misfit dislocations network in the interface InN/GaN from the very beginning of the growth instead of by surface islanding, and thus, the SK model is not applicable. Besides this, none of the models consider the existence of superficial defects in the substrate, and thus do not explain the previously described experimental results.
Figure 4.7. Scheme of threading dislocation with a screw component.
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A kinematical model that takes into account the influence of surface defects in the material serving as substrate in order to explain the subsequent growth of islands is the proposed by Burton, Cabrera and Frank[26] (BCF). Initially, Frank predicted the formation of a superficial step when a threading dislocation with a screw component (either pure screw or mixed) reaches the free surface of a crystal[27] (Figure 4.7). This type of dislocations produces a vertical displacement of the atoms in the surface with a magnitude equal to the component of the Burgers vector normal to the surface. The originated step results energetically favorable for the subsequent growth around it, as it saturates a larger number of bonds than it generates and, furthermore, it repeats the preexisting configuration as the material nucleates. So, a spiral growth is obtained around the dislocation, known as BCF growth model. However, as explained, for a dislocation to be considered as a nucleation center following this model it must have at least a screw component of the Burgers vector, whereas the threading dislocations in the GaN substrate on top of which the InN QDs nucleate were demonstrated to be pure edge type, and therefore the BCF model does not either apply. A more similar nucleation mechanism was described by Rouviere et al.[28] for GaN QDs grown on AlN. In this case the QDs nucleated adjacent to edge TDs that propagated in the AlN barrier (Figure 4.8). The edge component of the TD introduces on one of its sides an additional {2 1 1 0} plane, and therefore the neighborhood of the dislocation that contain this extra half-plane is in compression. This region has locally a smaller lattice parameter than that for AlN, and the GaN, whose lattice parameter is about 2.5% larger, tends to nucleate at the opposite side of the dislocation where the AlN lattice is stretched. Nevertheless, this model does not fully correspond to the observed for InN QDs on GaN, since for this the QDs nucleate on top of the TDs and not adjacent to them.
Dislocation line
Extra half-plane
{2 1 1 0} Figure 4.8. Scheme of the nucleation mechanism proposed by Rouvier et al.
Consequently, the nucleation mechanism for InN QDs on GaN described in this section does not correspond to our knowledge to any of the previously reported in the literature. It seems clear that the strain fields associated to the pure edge TDs when they reach the free surface of the substrate is the process that governs the latter nucleation location of the InN QDs. Regarding this, Bauser and Strunk[29] demonstrated that all types of threading dislocations, not only those with screw component, can create a protuberance in the surface of the substrate. Later, E.A. Beam[30], investigating the replication mechanism of dislocations coming from the substrate in InP epilayers, proposed basing on computational simulations,
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that a pure edge dislocation creates a superficial wave parallel to the dislocation line due to the contraction and expansion of the free surface atoms located inside the strain field associated to the dislocation. Thus, the constricted area would correspond to the peak, and the expanded area to the valley, producing a sinusoidal wave in the surface, as the one shown in Figure 4.9. Thus, in our case, the InN QD would tend to nucleate in the strained area to relax it.
Dislocation line, ξ
Figure 4.9. Surface wave produced by a pured edge TD on the free surface, following the model proposed by Beam et al.
This mechanism offers a possible means to increase and control InN QD density through control of the surface morphology and defect density of the substrate material (including the pure edge TDs). However, in the present case this is clearly related to the low growth rate of InN, linked to the poor dissociation of NH3 at low temperature. As a result, the available chemical potential difference, which is the driving force for growth, is low, and nucleation occurs on sites where local conditions lower the energy barrier for nucleation. Clearly, the TDs we have observed fulfill these conditions. This indicates that an improvement of InN QDs density may be realized through the increase of InN growth rate, i.e., by operating in the gas diffusion-limited growth mode in MOVPE. In this respect, the use of alternative precursors, decomposing at lower temperatures, is promising.
V. Determination of the Degree of Plastic Relaxation In mismatched heteroepitaxial growth, it is known that the strain can be accommodated elastically or plastically depending on the lattice mismatch and the surface energy of the materials involved[31]. In many systems, such as InAs on GaAs quantum dots, the growth of the nanostructures occurs pseudomorphically, i.e., with the same lattice parameter than the substrate, and therefore certain strain is exerted on the overgrown material[32,33]. However, in systems with a large lattice mismatch, like InN on GaN, (aGaN=0.3189 nm and aInN=0.3533 nm) the pseudomorphic growth is really difficult and generally the structures are dislocated from the very early stages of the growth. On the other hand, it has been shown that the optical emission peaks, mainly in nitrides-based nanostructures, are significantly shifted as a consequence of the piezoelectric polarization induced by the stress[34,35]. Thus, an accurate determination of the strain state is important not only to know the structural properties of the nanostructures but also the relationship with their optoelectronic properties. Here we will
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present a detailed study of the plastic relaxation of uncapped InN quantum dots grown on GaN by using an effect directly observed by TEM: the moiré fringes patterns. The results are corroborated studying the structure at atomic scale of the QDs by high resolution TEM.
III.1. Determination by Moiré Fringes As mentioned in the previous section of the chapter, all the QDs observed in plan view orientation present a well defined set of three directional moiré fringes patterns in the three main directions <11 2 0>, as a consequence of the overlapping of the {1 1 00} planes in InN and GaN. These fringes were applied to estimate the degree of plastic relaxation of the InN QDs individually. Measuring directly on the micrographies corresponding to a large numer of QDs, as the one shown in Figure 5.1, we obtained an average distance for the moiré fringes of Dm=2.9±0.2 nm. So, from the well known expression for the moiré fringes:
Dm =
QD d InN d GaN QD d InN − d GaN
(1)
where dGaN is the distance between {1 1 00} planes in GaN, that we suppose to be fully relaxed and thus dGaN = aGaN·cos(π/6) = 0.276 nm, we can obtain the value for {1 1 00} planes QD
in the InN QD, d InN .
Figure 5.1. Plan view micrograph of an InN QD showing the three families of intersenting moiré fringes.
The percentage of plastic relaxation, δ is obtained from the expression:
⎛
δ = 100 ⋅ ⎜⎜1 − ⎝
εr f
QD − d InN d ⎞ ⎟⎟ = 100 ⋅ GaN d GaN − d InN ⎠
(2)
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where εr is the residual strain, dInN is the interplanar distance for the {1 1 00}planes in fully relaxed InN and f is the lattice mismatch, defined as:
f =
d GaN − d InN d GaN
(3)
By simple substitution, we obtain, in average, δ = 97±6 %, and thus, InN in the QDs is almost fully relaxed. The process that gives rise to this high degree of plastic relaxation will be discussed in the following section.
III.2. Determination of the Density of Misfit Dislocations by High Resolution TEM In heteroepitaxial systems, due to the difference between the reticular parameters of the substrates and overgrown materials, the introduction of misfit dislocations (MDs) is the main mechanism that plastically relaxes the strain. When a low lattice mismatch exists, the classical model of Matthews-Blakelee[36], based on force balance equilibrium, is the most often used: the epilayer grows pseudomorphically with respect to the substrate until a critical thickness is reached, hc. At this thickness, the generation of MDs occurs due to the bending of threading dislocations propagating from the substrate, and the plastic relaxation of the heterostructure takes place. However, in systems with a high reticular mismatch, such as InN/GaN, this model is not applicable since now the critical thickness becomes less than one monolayer[37]. Other type of dislocations is then defined[38], called “geometrical misfit dislocations”, generated from a different physical mechanism in comparison with the classical Van der Merwe-Matthews[39] and that accommodate the majority of the initial strain due to lattice mismatch. These MDs were studied by high resolution TEM recording images in cross section orientation along the <11 2 0> zone axis containing InN QDs on GaN, as shown in Figure 5.2., where interruptions in the reticular planes of GaN can be observed due to the introduction of MDs. They appear sequentially every a certain number of {1 1 00} planes, more clearly observed in the inset, that shows a Fourier filtered image of that region and where the extra half-planes are indicated by white arrows. In average, a MD is introduced every 10.5 planes of GaN or equivalently 9.5 planes of InN. The lattice mismatch for InN on GaN, as expressed in (3) results f = -0.1074, what means that a compressive strain is exerted on the InN QD, whereas the GaN substrate is supposed to be fully relaxed due to its larger thickness. It can also be expressed in a different way as:
f = δ ef + ε r where δef is the effective reticular mismatch in a relaxed configuration, and can be calculated from:
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δ ef =
be Dd
being Dd the spacing between misfit dislocations, obtained from the relationship between planes formerly mentioned, and |be| is the edge component of the Burgers vector module along the <11 2 0> direction in GaN,. For 60º MDs this results:
be =
3 aGaN = 0.2762 nm 2
Figure 5.2. Cross sectional high resolution TEM micrograph of an InN QD on GaN. In the inset, Fourier filtered area of the interface, where the MDs are indicated by white arrows.
Substituting, we obtain δef = (-) 0.10526. This implies that a 97.6% of the initial strain due to lattice mismatch is relieved by the introduction of MDs in the InN/GaN interface, in very good agreement with the moiré results. A very important fact worth to mention is that, in spite of the high density of MDs, no threading dislocations are observed inside the InN QDs. As was mentioned in the introduction of this chapter, TDs behave as free electron traps and thus, they have deleterious effects on the optical properties of the QDs. In this sense, it is a very interesting result for the posterior fabrication of optoelectronic devices based on these heterostructures.
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VI. Characterization of the Misfit Dislocations Network As mentioned in the previous section of the chapter, due to the high lattice mismatch between the two crystals, the majority of the initial strain is relieved by the introduction of a geometrical misfit dislocations network in the interface InN/GaN. These dislocations are not easily observable by diffraction contrast in conventional TEM, unlike other well known structures such as SiGe/Si and InGaAs/GaAs, since they are too closely spaced to be resolved. Cross-sectional high resolution TEM is the main technique commonly used to characterize these MDs network, where the edge component of each MD is seen as an extra half-plane in the material will smaller lattice parameter, providing a linear dislocation density. Additionally, a Burgers circuit drawn around the area containing the dislocation provides information about the sign and magnitude of the Burgers vector. However, this description results incomplete, since many relevant features such as changes in the line direction of the dislocations or interaction between them remain unknown. As a consequence, there is a variety of different models for dislocation networks that could accommodate the lattice mismatch in the case of interfaces formed by close-packed planes such as (111) and (0001) for face centered cubic and compact hexagonal systems, respectively[40]; which vary from different kinds of hexagonal honeycomb networks to a net of independent lines that could form various mosaic structures. These questions would be answered when the dislocation array at the interface could be observed in plan-view orientation, , whenever the dislocation width in diffraction contrast is close to or even shorte than the spacing between them. Recently, the analysis of high resolution TEM images using peak finding[41,42,43] and/or geometric phase (GP) methodologies[44,45,46], that allows quantitative strain mapping at very high spatial resolution in crystalline materials[47], are widespreadly used. Here we have applied this technique to high resolution TEM images of InN QDs on GaN in plan view orientation, and it will be shown that a complete characterization of the MDs network can be achieved. Additionally, information about the behaviour of the MDs when they are close to the edge of the QDs is obtained. Figure 6.1 shows a high resolution TEM micrograph along the <0001> zone axis of an InN QD and the corresponding digital diffractogram, obtained after applying a fast Fourier transform to the image. First, a Wiener filter was performed, which locally estimates the noise in the Fourier transform and, for each spatial frequency, its amplitude, reducing the noise in the image. Then, a Bragg filter was applied to the digital diffractogram, considering the peaks corresponding to InN and GaN and neglecting double diffraction peaks, which lead to the formation of the moiré fringes. Avoiding these doubly diffracted beams, the filtered image using only InN and GaN peaks does not contain information about the moiré fringes. Six symmetric Gaussian masks were applied around the {1 1 00} InN and GaN peaks with radius of 0.256 nm-1, small enough to exclude the double diffraction contribution but wide enough to keep all the fine details and do not remove any information.
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Figure 6.1. High resolution TEM micrograph of an InN QD in plan view orientation, and its corresponding diffractogram.
For remembering, the GP algorithm is based on the calculation of the displacement field and subsequently the strain map by numerical derivatives, from the phase images for different and non-collinear vector. A full description of the methodology can be found in elsewhere[44]. GaN away from the InN quantum dot was chosen as reference material. By combining the phase images, the displacement can be calculated and subsequently the strain field by numerical differentiation using:
u (r ) = −
[
1 Pg1 (r )a1 + Pg 2 (r )a 2 2π
1 ⎛ ∂u
]
∂u j ⎞
⎟ ε ij = ⎜⎜ i + 2 ⎝ ∂x j ∂xi ⎟⎠ where Pg is the phase image and ai and gi the real space and reciprocal lattice vectors respectively. The image processing and calculation on the strain field were performed using routines written in Matlab. The scale corresponds to the relative strain, ε’ with zero corresponding to the GaN buffer without the InN QD and defined as:
ε′ =
0 a − aGaN 0 aGaN
where a is the experimentally measured lattice parameter, and the superscript 0 correspond to unstressed crystal. The distortion field determined in this work is therefore calculated using the GaN as reference, what is slightly different from the absolute strain:
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ε InN =
285
0 0 QD a InN a − aGaN − a InN ∝ = ε′ 0 0 a InN aGaN
a)
b)
c)
d)
e)
Figure 6.2. a–c) visualization of the three families of 60° misfit dislocation in the heterostructure. In the reference system the b directions lie parallel the <1 1 00> directions and the a directions would correspond to the <11 2 0> directions. d) combined image (a)–(c), and e) HTREM images superimposed with the misfit dislocation network.
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Figures 6.2 (a-c) show the obtained strain maps, where the red lines correspond to the more relatively strained areas and therefore are related to three different sets of misfit dislocations in the InN/GaN system. These three sets of MDs are rotated with respect to each other by an angle of 60º. The combination of the three maps is displayed in Figure 6.2(d), which provides a clear visualization of the whole MDs network that accommodates the lattice misfit in the InN/GaN heterosystem, in good agreement with previous reports[48]. The average spacing between dislocations is 2.70±0.02 nm, what gives a linear density in the interface, defined as the inverse of the average spacing, of (3.7±0.1)·108 cm-1 along the three <1 1 00> directions. This symmetry in the dislocations distribution indicates that the geometrical MDs nucleate along the close packed <11 2 0> directions, a fact that reflects the 6-fold point symmetry of the two crystals in the same way as the pronounced difference in the linear interface dislocation densities along the two <110> directions in InGaAs/GaAs is consistent with the 2-fold point symmetry of that system. Figure 6.2 (f), that displays the superposition of the whole 60º MDs network with the original high resolution TEM micrograph, shows alternating regions that correspond to areas of better and worse fit between the two crystal lattices. As can be seen, the sharp contrast associated to the atomic columns correspond to the better fit regions, whereas the blurred contrast would correspond to areas of worse fit between the lattices and therefore are related to the MDs. Thus, the strain map shows a regular distribution of hexagonal regions with pseudomorphic growth separated by a MDs network with the six-fold symmetry of the (0001) plane. We can clearly observe that these misfit dislocations do not interact between them, so no dislocation nodes are created in the network junctions, but form a sort of ‘David’s star’ network instead of a hexagonal honeycomb observed in similar systems[49,50]. Sharp atomic column contrast, i.e., good fit between the InN and the GaN substrate is observed in both the hexagonal and triangular areas. The area of the hexagons is ~ 4.7 nm2, what gives a 70% of the whole interface free of strain. We also have to mention that, due to its large dimensions, there exist some areas where the noise affects the image quality for quantitative analysis. For instance, in Figure 6.2, blue lines inside the quantum dot can be seen, a fact that would indicate areas under a very tensile strain. They undoubtly correspond to artefacts from the geometric phase methodology that do not correspond to a real situation in the QD areas. In this kind of images, the most important noise is due to thin foil irregularities, something that is almost unavoidable. These irregularities can also give rise to contrast errors, and the quantitative analysis fails in these areas. In addition, this analysis has also revealed details about the behaviour of these misfit dislocations when they are close to the edge of the quantum dot. From the obtained deformation maps and the corresponding elastic constants[51], we calculated the stress components following the isotropic elastic theory approach[52], and the σxx, σyy and σxy components of the stress fields obtained from the experimental measurements are displayed in Figure 6.3(a-c). These stress components turned out to be in very good agreement with the stress distribution around an edge threading dislocation[52] when observed in planar view orientation. Therefore, we interpret these images as showing a network of threading dislocations surrounding the InN QDs. These threading dislocations have a Burgers vector b=1/3< 1 2 1 0> with edge component, indicated by a white arrow in Figure 6.3, and therefore they have an edge orientation.
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(a) y z
b σxx
(b)
σyy
(c)
σxy
Figure 6.3. a) σxx, b) σyy, and c) σxy stress field component of the threading dislocation network. These distributions correspond to typical edge dislocations. The x and y directions correspond to <1 1 00> and <11 2 0> respectively being the dislocation vector b parallel to the latter.
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This threading dislocation network around the InN QD was also confirmed by conventional TEM under two beam conditions. Figure 6.4 (a) and (b) correspond respectively to bright field and weak beam PVTEM images recorded for g=11 2 0 near the <0001> zone axis. For the second case, a threading dislocations arrangement surrounding the InN QD is clearly visible. Taking into account the diffraction contrast invisibility criterion, these dislocations may possess Burgers vector b=1/3< 1 2 1 0> or b= b=1/3< 1 2 1 3>, consistent with the results obtained by the analysis of the high resolution TEM images. (b)
(a)
Figure 6.4. TEM planar view images of the sample: a) Bright field and b) weak beam showing the threading dislocations with an a-component Burgers vector around the InN QDs. a3= [ 1 1 20 ]
a2= [ 1 2 1 0 ]
a1=[ 2 1 1 0 ]
p=( 1 010 )
p=( 1 011 )
p=( 1 012 )
b=1/3[ 1 2 1 0 ]
b=1/3[ 1 2 1 0 ]
b=1/3[ 1 2 1 0 ]
ξMD=[ 2 1 1 0 ] ξTD=[ 0001 ]
ξMD=[ 2 1 1 0 ]
ξMD=[ 2 1 1 0 ]
ξTD=[ 2113 ]
ξTD=[ 1 011]
Figure 6.5. Schematic of the slip systems in wurtzite structure. a) 1/3< 1 2 1 0>( 1 010), b) 1/3< 1 2 1 0>( 1 011), and c) 1/3< 1 2 1 0>( 1 012). p indicates the plane, b the Burgers vector and the misfit and threading dislocation lines direction are ξMD and ξTD, respectively.
Thus, these results demonstrate a tendency of the misfit dislocations to bend and form edge threading dislocations due to the influence of the free surface. The slip system 1/3< 1 2 1 0>{0001} in the InN/GaN wurtzite heterostructure, being the basal plane parallel to the interface, is the main source of geometrical misfit dislocations. When these dislocations
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are close to the edge of the island, they are attracted towards the free surface reducing the dislocation energy[53], and giving rise to a threading dislocation network surrounding the QD, as was mentioned above. Given that the Burgers vector for these dislocations were demonstrated to lie along the < 1 2 1 0> direction, the possible slip systems are shown schematically in Figure. The ( 1 0 1 0), ( 1 011) and ( 1 012) planes are operative since the driving force acting on such slip planes is due to the nearness of the surface. This force decreases slowly with increasing distance from the surface, so in areas close to the interface the force induced by the surface is considerably stronger. Three possibilities may occur regarding the behaviour of a misfit dislocation running along the direction ξMD = [2 11 0] with a Burgers vector b= b=1/3< 1 2 1 0>, which are displayed in Figure 6.5. ξTD denotes the line direction for the threading dislocation segment, and p the slip plane. Among them, the small Peierls force acting on the dislocation is expected in the 1/3< 1 2 1 0> { 1 010} slip system, since it presents the highest d/b ratio, which would give a line direction of [0001]. Taking into account both observed phenomena, on one side the 60º misfit dislocation network at the InN/GaN interface and, on the other side, the threading dislocations network surrounding the InN QD, we suggest the mechanism shown schematically in Figure 6.6, where a misfit dislocation running along the < 1 2 1 0> direction at the InN/GaN interface has been displayed. When this misfit dislocation is close to the edge of the quantum dot, it is attracted towards the free surface and the dislocation bends becoming a threading dislocation that propagates along the <0001> direction in the (10 1 0) prismatic plane.
InN QD
MD
TD b 90º
57º
GaN
Figure 6.6. Proposed mechanism for the bending of the interfacial misfit dislocation network into threading segments.
Further studies are required in order to understand this TDs network. It would be interesting to analyse the coalescence between two islands to clarify the TDs movement along the boundary as well as the origin of the high TDs density observed in InN thick layers[38]. Also, the influence of the sample thickness in the strain maps must me analysed. All the
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experimental strain and stress calculations have been obtained from a very thin TEM specimen, so to quantify the real state of the stress in the system a more complex process than the elastic theory approach should be used, using the nonlinear modelling starting from the experimental deformation measurements[51,54]. Thus, the measured distortions are probably different from a bulk sample due to the reduction of the substrate thickness, and these effects have to be taken into account to obtain an accurate measurement of the distortion field. However, we consider that this fact does not affect the dislocation network distribution that highlights the InN/GaN system. The misfit dislocations with b=1/3<11 2 0> hardly slip in the basal plane (0001) in hcp structures, providing its characteristic brittle behaviour. Further studies taking into account all these factors are being developed in order to obtain more accurate distortion maps of these highly mismatched heterosystems.
VII. Effect of the Growth of the GaN Capping Layer The posterior stage in the fabrication of QDs to be integrated in active layers of functional optoelectronic devices is their overgrowth with a capping layer. This is necessary to confine the charge carriers by providing a potential barrier, and thus create a quantum dot where the energetic levels for the nanostructure become discrete. Typically, a material with the same crystalline structure and wider bandgap than the nanostructure is chosen. In our case, a low temperature grown GaN was used to cover the InN QDs. As mentioned in the experimental section, the temperature at which this LT-GaN capping layer is grown is very close to the used for the deposition of the QDs. This may result a problem for InN, since its low dissociation temperature may result in In evaporation or atomic rearrangement during the capping layer growth process, and thus changes in the morphology or strain state of the nanostructures may be expected. In this section, the effect of the growth of the LT-GaN capping on these aspects is discussed.
VII.1. Effect on the Morphology Important morphological changes were observed after the LT-GaN capping process. As expected the average density of quantum dots remains in the same order of magnitude, since both samples were grown initially under the same conditions, and the only difference between them is the existence of the capping layer. However, the average diameter, d, has considerably increased, being now d = 120±30 nm. This is evinced in Figure 7.1, recorded under the same magnification than Figure 3.1. Moreover, an important spreading on the area distribution of the QDs takes places (the standard deviation is now σ = 3700±200), as shown in the corresponding histogram in Figure 7.2. For this, we have an average perimeter, p=450±60 nm and an average area, A=15000±400 nm2. Thus, we obtain an average value for the roundness, R=1.12±0.08, very close again to the roundness for a perfect hexagon. Furthermore, from the corresponding histogram (Figure 7.3) we conclude that not only an increase in the average size of the QDs takes place, but also the capping layers promotes an homogeneization in their shapes, with more than a 60% of the QDs under the value R = 1.10.
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Figure 7.1. Plan view micrograph of InN QDs capped with low-temperature GaN.
Counts
6 4 2 0 2000
4000
6000
8000 10000 12000 14000 16000 18000 20000 22000 24000
Capped quantum dots area, nm
2
Figure 7.2. Histogram showing the area distribution for the capped InN QDs. 14
12
10
Counts
8
6
4
2
0 1.0
1.1
1.2
1.3
1.4
1.5
Roundness
Figure 7.3. Histogram corresponding to the parameter R for the capped QDs.
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It would be reasonable to think that, since both samples were grown under the same conditions, with the exception of the incorporation of the GaN capping layer, the QDs in both cases would have the same volume. Therefore, an increase in the average area would suppose a decrease in the average height of the QDs. However, cross-sectional TEM micrographs determined that the height was the same than for the uncapped InN QDs, i.e., h = 12±2 nm, and the aspect ratio is now 1/10. This means that the volume of the capped QDs is around three times the volume of the QDs before the LT-GaN capping process. Possible mechanisms that would explain this fact would be the existence of In/Ga intermixing phenomena or ii) InN QDs coallescence. The viability of one or another mechanism will be discussed in the next section.
VII.2. Effect on the Strain State After the LT-GaN capping process, not only morphological changes, but also variations in the strain state of the QDs can be observed. First, from the analysis of the moiré fringes pattern that appear in all the heterostructures studied in planar view orientation, we can conclude that a clear increase of the distance between the fringes takes place (Figure 7.4). Now, the average cap
distance between fringes is Dm
= 3.2±0.2 nm, and applying again Eqs (1) and (2), we obtain
that the degree of plastic relaxation of the heterostructure has decreased down to a value δcap=87±5%.
Figure 7.4. Plan view micrograph of a GaN-capped InN QD showing a family of translational moiré fringes.
As for the case of the increase in the volume of the QDs, a possible mechanism that would explain this difference would be the existence of In/Ga intermixing between the capping/substrate and the InN QD. In this sense, we would suppose that the system is almost fully relaxed, as was demonstrated to occur for the uncapped QDs, but it would be now constituted by a ternary alloy InxGa1-xN. We used Vegard’s law to determine the concentration, x, that would give the lattice parameter experimentally obtained above. This
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semiempiric law relates the lattice parameter, a, and the composition, x, through a expression that is assumed to be linear for compound semiconductors: a(AxB1-x) = x.a(A)+(1-x).a(B) For this case, x would be the In content and (1-x) the Ga content, and thus:
x=
a ( In x Ga1− x N ) − a(GaN ) a( InN ) − a(GaN )
Substituting, we obtain x = 0.86. This would correspond to an In-rich ternary allow. However, this hypothesis was ruled out after applying compositional analysis techniques, namely HAADF/EDX. As can be observed in Figure 7.5 (a) and (b), the hexagonal shape of
(a)
(c)
(b) (d)
(b)
In peak
(e)
Figure 7.5. HAADF micrographs of GaN capped InN QDs a) in plan view and b) in cross section with the corresponding EDX spectra measured in c) the GaN capping layer,d) the QD and e) the GaN substrate.
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the quantum dots remains, with clearly delimited edges. Moreover, EDX spectra do not contain information about presence of In in the capping layer nor in the substrate in areas close to the quantum dot (Figures 7.5(c-e)). On the other hand, no wetting layer is present in the samples, and therefore the increase in the volume of the QDs cannot be due to the incorporation of material from this wetting layer. So, we can conclude that the most possible mechanism that promotes the increase in the volume of the QDs is the existence of coalescence phenomena. Fourier-filtered high resolution TEM micrographs of the interface between the GaN substrate and the capped InN QDs (Figure 7.6(a)), were used to corroborate the previous moiré results. As for the moiré fringes, in this case an increase of the distance between misfit dislocations has been observed. Now, a MD is introduced in average every 10.2 {10 1 0} planes of InN, or equivalently 11.2 planes of GaN; what results in an interplanar spacing for these planes in the InN QD dcap = 0.3031 nm. Substituting again in Eq (2), we obtain an average degree of plastic relaxation of the InN in the heterostructure δcap = 89%, in good agreement with the moiré analysis results. In light of these experimental results, it seems reasonable to expect for this case the existence of a second MD network, which would be now located in the interface between the InN QD and the LT-GaN capping layer. To clarify this, we constructed strain maps obtained by applying the Geometric Phase Algorithm to the previous high resolution TEM micrograph. In Figure 7.6(b), the MD networks at both interfaces, InN/GaN and GaN/InN, are visible, consisting of regularly spaced blue and red lobular shapes that correspond to the strain distribution around the edge component of a misfit dislocation. Again, we used Fourier filtered images of the resolved upper interface to estimate the degree of plastic relaxation of the capping layer. For this case, a MD is introduced every 10.4{10 1 0} planes of InN, and following the same procedure described above, considering now the obtained value for dcap, we deduce that the LT grown GaN in the capping layer is ~99% relaxed. Therefore, in the final configuration, the capping layer is fully relaxed by the introduction of a misfit dislocation network, while the quantum dot increases its residual strain. In this sense, we do not expect any variation in the strain state of the QD with increasing thickness of the GaN capping layer. Unfortunately, the existence of the capping layer that provides an extra thickness and the two MD networks have not allowed us up until now to obtain HRTEM micrographs with enough information to carry out a complete description of the MD network at the InN/GaN interface in capped QDs. In the ideal case, a perfect correspondence in the location and density of both MD networks might be expected. In an unstressed InN layer, the lower MD network would be replicated in the upper interface, now with opposite Burgers vector, that would generate a GaN capping layer with a similar stress state than the GaN substrate. However, the introduction of the MDs is gradual during the epitaxial growth, and the first GaN monolayers in the capping layer would exert a compressive strain in the underneath InN quantum dot. As a result of this, a small increase in the residual strain of the quantum dot is energetically favourable, since it decreases the strain in the GaN capping layer; in other words, a stress balance can occur between the layers. This is observed experimentally as a reconfiguration of the MD network at the InN/GaN substrate interface, decreasing the density of MDs.
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(a)
(b)
Figure 7.6. (a)High resolution XTEM micrograph of a capped InN quantum dot . The inset is a Fourier filtered image of the interface where an increase of the distance between MDs after the capping process is observed. (b)Strain mapping of the image, showing the MD networks at both InN/GaN interfaces.
Actually, the lateral movement of the MDs at this interface is an open question because the MDs would not be expected to glide in the basal plane (0001) in wurtzite systems due to biaxial stress from the lattice misfit[55]. The high atomic mobility and temperature instability of InN may be the main reasons for the rearrangement of the MD network during the growth of the capping layer. Further investigations are needed in order to clarify the mechanisms of the rearrangement of the MD network.
VIII. Conclusion A complete study of the morphology, strain state, and nucleation mechanism has been developed for samples consisting of uncapped and LT-GaN capped InN quantum dots. The
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uncapped QDs present a truncated pyramid hexagonal shape, with an aspect ratio of 1/6, a gaussian area distribution and a low density. The moiré fringes pattern appearing in all the QDs when observed in plan view orientation allowed us to estimate the degree of plastic relaxation of InN in the heterostructure, showing that in average the QDs are almost fully relaxed (97% of the initial strain due to lattice mismatch). This high degree of relaxation is directly related to the generation of a misfit dislocation network at the InN/GaN interface, a fact confirmed by high resolution TEM which, in spite of its high density, does not generate threading dislocations inside the QDs. This MD network was fully characterized by applying the Geometric Phase Algorithm to high resolution TEM micrographs of InN QDs in plan view geometry, demonstrating that it consists of three sets of misfit dislocations propagating along the three main directions <11 2 0> without interaction between them nor nodes generation. These MDs also tend to bend when they are close to the edge of the QD forming each a short segment of threading dislocation and thus generating a TDs network surrounding the system. The inclusion of a LT-GaN capping layer promotes significant morphological changes in the QDs, with a decrease in the aspect ratio down to 1/10. Even though the height does not change, the average diameter increases and a spreading of the area distribution takes place. On the other hand, the capping layer, also fully relaxed, produces a rearrangement of the MDs at the interface QD/substrate with the subsequent diminution in the degree of plastic relaxation of the heterostructure. It has also been shown that all the QDs that do not propagate into the InN QD nucleate preferentially on top of threading dislocations located in the GaN, and have pure edge character. Therefore, the nucleation mechanism of InN QDs on GaN has to differ notably from the classical Stranski-Krastanov of Burton-Cabrera-Frank models.
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In: Quantum Dots: Research, Technology and Applications ISBN: 978-1-60456-930-8 Editor: Randolf W. Knoss, pp. 299-329 © 2008 Nova Science Publishers, Inc.
Chapter 8
ELECTRONIC STRUCTURE AND PHYSICAL PROPERTIES OF SEMICONDUCTOR QUANTUM DOTS Xiu-Wen Zhang, Yuan-Hui Zhu and Jian-Bai Xia State Key Laboratory for Superlattices and Microstructures, Institute of Semiconductors, Chinese Academy of Sciences, Beijing 100083, China
Abstract Along with growing of synthesizing methods of semiconductor quantum dots, they are widely investigated experimentally and theoretically. The electronic structure and optical, magnetic property of colloidal quantum dots are investigated in the framework of effective mass envelope function theory by expanding the envelope function in spherical Bessel functions and spherical harmonic functions. On the basis of calculating the energy levels and envelope functions the various physical properties of semiconductor quantum dots are investigated. The chapter will be organized as following: 1. Effective-mass envelope function theory for quantum dots. 2. Polarization properties of emission, including: strong linear polarization along the c-axis of wurtzite quantum ellipsoids, circular polarized property of wurtzite quantum dots ensemble in the magnetic field. 3. Electon g factors, including: electron g factors as functions of size and shape of dots, direction of magnetic field, and electric field tunable electron g factor of quantum dots. 4. Highly anisotropic Stark effect of quantum ellipsoids. 5. Giant Zeeman splitting, including: Zeeman splitting energies as functions of radius of dots, Mn ion concentration, magnetic field, highly anisotropic Zeeman splitting in wurtzite quantum dots, and radius sensitive Zeeman splitting of zero-gap quantum dots. 6. Curie temperature of DMS quantum dots, including: definition of Curie temperature in quantum dots, effect of hole number on the Curie temperature, room temperature ferromagnetism of (Zn,Mn)O quantum dot, electric field tunable ferromagnetism of quantum dots, and highly anisotropic ferromagnetism in oblate quantum dots.
I. Introduction Low-dimensional systems such as semiconductor quantum dots and quantum wires have fascinating and technologically useful optical and electric properties. Studies on these systems advance our knowledge on low-dimensional physics and chemistry. Semiconductor
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quantum dots exhibit novel electric and optical properties owing to their quantum confinement effects in three dimensions. Nowadays, high-quality semiconductor quantum dots were synthesized by the colloidal method [1-4]. Their physical properties were widely investigated. For example, emission spectra and luminescence decay [3], fine structure of the exciton [5], few particle effects [6], and optical transitions [7] in a single CdTe quantum dot were investigated. Size-dependent energy levels of CdTe quantum dots were measured [8]. InAs short quantum rods were synthesized which have ellipsoidal shape as shown in the TEM images [9]. Using the same method, other quantum ellipsoids were also synthesized [10-12], whose shape can be controlled. Linearly polarized emission from InAs ellipsoids in the absence of magnetic field were observed [9]. Circularly polarized emissions under circularly polarized excitations were measured [13,14]. Electron g factors of these lowdimensional systems were investigated experimentally [15,16] and theoretically [17]. The electronic structures of quantum dots have been calculated using the effective-mass model [18,19], the pseudopotential method [20], or the empirical tight-binding description [21]. Recently, much of the research in semiconductor physics has been shifting towards diluted magnetic semiconductors (DMSs), which have extensive applications in spintronics. Manganese-doped II-VI and III-V compound semiconductors have been widely studied. The method to dope Mn ions into CdSe quantum dots was achieved [22]. DMS quantum dots were used as spin-polarized light-emitting diodes [23], and the electron spins in DMS quantum dots can be used as qubits for quantum information processing [24]. In this paper, we introduce our recent theoretical results on the electronic structures and optical properties of semiconductor quantum dots. The remainder of this paper is organized as follows. In Sec. II we give the effective-mass envelope function model. The physical properties: polarization properties of emission, g factors of quantum dots, giant Zeeman splitting in DMS quantum dots, Curie temperatures of DMS quantum dots, are given in Sec. III, IV, V, and VI, respectively. Finally, we draw a brief conclusion in Sec. VII.
II. Effective-Mass Envelope Function Model 2.1. Hole Effective-Mass Hamiltonian The hole effective-mass Hamiltonian for wurtzite semiconductors in the case of zero spinorbital coupling is given by Xia et al. [25,26],
Lpx2 + Mp y2 + Npz2 1 Hh = Rpx p y 2m0 Ap1 px + Qpx pz
Rpx p y Lp + Mpx2 + Npz2 2 y
Ap1 p y + Qp y pz
Ap1 px + Qpx pz Ap1 p y + Qp y pz
,
(1)
S ( px2 + p y2 ) + Tpz2 + 2m0 Δ c
where the valence band basic functions are X-like (Γ6), Y-like (Γ6) and Z-like (Γ1) functions, respectively, L, M, ⋯ , S, T are effective-mass parameters to be determined, Δc is the crystal field splitting energy. By comparing the valence bands near the top calculated by the effective-mass Hamiltonian (1) and by the empirical pseudopotential method, we determined
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uniquely the effective-mass parameters in Hamiltonian (1), as given in related reference papers, together with the electron effective masses perpendicular to and along the c axis, mx* and mz*, respectively. To make the coefficient A of the linear term dimensionless, we introduced p1 =
2m0δ , δ=10 meV.
Figure 1. Energy bands of ZnO near the top of valence band.
Figure 2. Energy bands of GaN near the top of valence band.
Figs. 1 and 2 show the valence bands near the top calculated by the effective-mass Hamiltonian (1) and by the empirical pseudopotential method along several symmetry directions for ZnO and GaN, respectively. From the figures we see that the agreement is so good, that we nearly cannot see the difference between two bands, except for the [101]
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direction. It is noticed that the linear terms Ap1px and Ap1py in Hamiltonian (1) are important in determining the valence band structure near the top, for example the Γ1 bands of ZnO and GaN along the [100], [110], and [101] directions show non-parabolicity, due to the linear dependence of k. The hole effective-mass Hamiltonian for semiconductors with zinc-blende structure is similar to that of wurtzite structure Eq. (1). Because the zinc-blende structure has cubic symmetry, the effective-mass parameters M=N=S, L=T, R=Q, A= Δ c =0.
2.2. Effective-Mass Theory in the Spherical Coordinate [27,28] In order to calculate the electronic structure of quantum spheres we write the electron and hole Hamiltonians in the spherical coordinate. The hole Hamiltonian (1) in the spherical coordinate is written as
P1 1 Hh = − S∗ 2m0 ∗ T
S
T
P3 S∗
S , P1
(2)
where
P1 = γ 1 p 2 −
2 γ 2 P0(2) , 3
P3 = γ 1′ p 2 + 2
2 γ 2′ P0(2) + 2m0 EΔ , 3
(3)
(2) S = AP−(1) 1 + 2γ 3 P−1 ,
T = η P2(2) + δ P2(2) , γ1, γ2, …are effective-mass parameters related to L, M, …, P(2) and P(1) are the second- and first-order momentum tensor operators, respectively. The Bloch basic functions of the valence band
top
1−1 = 1
have
been
transformed
to
11 = 1
2 ( X + iY ), 10 = Z ,
and
2 ( X − iY ) . The envelope wave functions are expanded with the spherical
Bessel functions and spherical harmonic functions,
( ) ( ) ( )
⎛ a L ,n C L ,n j L k Ln r YL , M −1 (θ , φ ) ⎞ ⎜ ⎟ ΨM = ∑ ⎜ bL ,n C L ,n j L k Ln r YL , M (θ , φ ) ⎟. L ,n ⎜ ⎟ n ⎝ d L ,n C L ,n j L k L r YL , M +1 (θ , φ )⎠
(4)
Because of the hexagonal symmetry of crystal, only the z component of the angular momentum M is a good quantum number. The quadratic terms in the Hamiltonian couple the
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L state with L±2 states due to the second-order tensor operator P(2). The spin-orbital coupling (SOC) can be considered by adding a spin-orbital coupling Hamiltonian for the basic functions 11 ↑, 10 ↑, 1 − 1 ↑, 11 ↓, 10 ↓, 1 − 1 ↓ ,
H so =
−λ
0
0
0
0
0
0
0
0
2λ
0
0
0
0
λ
0
− 2λ
0
0
2λ
0
λ
0
0
0 0
0 0
− 2λ 0
0 0
0 0
0 −λ
,
(5)
where λ=Δso/3, Δso is the spin-orbital splitting energy of the valence band. In this case the envelope wave function (4) becomes 6 rows, keeping the z component of the total angular momentum (orbit+spin) as a constant. By using of the property of the k-order tensor operator of the momentum [29],
L′M ′ Pq( k ) LM = ( −1)
L′− M '
⎛ L′ ⎜ ' ⎝ −M
k L⎞ (k ) ⎟ ( L′ || P || L ) , q M⎠
(6)
where (L’||P(k)||L) is the reduced matrix element of the k-order momentum tensor, which is not zero only for L’=L or L’=L±k, we can calculate the matrix elements of hole Hamiltonian (3). The coefficient before (L’||P(k)||L) is the 6j coefficient of the vector coupling. The reduced matrix elements of the first-order and second-order momentum tensors and the derivative formulas of radial wave functions are given in the Appendix.
2.3. Effective-Mass Theory of Quantum Ellipsoids (Rods) [30] Some quantum dots grown by the chemical colloidal method have the shape of ellipsoid or rod, so we extended the effective-mass theory of quantum spheres to the case of quantum ellipsoids. For the case of quantum ellipsoids, we introduce a coordinate transformation, which transforms the ellipsoidal boundary to the spherical one: x’=x, y’=y, z’=z/e, where e is the aspect ratio of the ellipsoid (long axis/short axis). In the new coordinate system x’, y’, z’, the boundary is spherical, and the electronic Hamiltonian changes into,
p2 1 − He = 2ma 2mb where
2 ( 2) P0 , 3
(7)
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1 1 ⎛ 1⎞ 1 1 ⎛ 1⎞ = ∗ ⎜ 2 + 2 ⎟, = ∗ ⎜1 − 2 ⎟ . ma 3m ⎝ e ⎠ mb 3m ⎝ e ⎠
(8)
Similarly we can transform the hole Hamiltonian to the new coordinate, for example the P1 term in the hole Hamiltonian (3),
(
)
(
)
⎡ (γ 1 + γ 2 ) 1 + e2 γ 2 ⎤ 2 2 ⎡γ 2 1 − e2 (γ 1 + γ 2 )⎤ (2) − 2 ⎥p − ⎢ 2 − P1 = ⎢ ⎥P0 , 2 2 3 3 e e e 3 e ⎣ ⎦ ⎣ ⎦
(9)
etc.
2.5. Effective-Mass Theory of Narrow Gap Semiconductor Quantum Dots [31] Some semiconductors are narrow energy gap semiconductors, for example InAsInSb of zincblende structure, HgTe of wurtzite structure, etc. In these semiconductors there are strong interactions between the conduction band and the valence bands, so that we should use the effective-mass theory of 8 bands model, including the conduction band and the valence bands. Taking the Bloch functions of the conduction band bottom and the valence band top
S ,
11 = ( X + iY )
2,
10 = Z ,
1 − 1 = ( X − iY )
2
as basis functions, the Hamiltonian is written as
⎛ ε g + Pe ⎜ (1) 1 ⎜ − ip 0 P−1 H= 2m0 ⎜⎜ − ip 0 P0(1) ⎜ ip P (1) ⎝ 0 1
− ip 0 P1(1)
ip 0 P0(1)
− P1 − S∗
−S − P3
−T∗
− S∗
ip 0 P−(11) ⎞ ⎟ −T ⎟ , − S ⎟⎟ − P1 ⎟⎠
(10)
Where Pe is the electron kinetic energy term, P1, P3, T, S are given in Eq. (3), εg=2m0Eg， Eg is the energy gap， p 0 =
2m0 E p ，Ep is the matrix element in Kane’s theory.
As we have taken into account the coupling of valence and conduction bands, the Luttinger parameters γiL (i=1,2,3) should minus the contribution from the conduction band to the hole effective masses, and the electron effective mass minus the contribution from the valence band. Taking the isotropic approximation of the valence bands, γ2L=γ3L=γL, then
γ 1 = γ 1L − E p 3E g , γ = γ L − E p 6 E g . And the electron effective mass,
(11)
Electronic Structure and Physical Properties of Semiconductor Quantum Dots
α=
1 1 ⎡Ep − ⎢ mc m0 ⎢⎣ 3
⎛ 2 1 ⎜ + ⎜E ⎝ g E g + Δ so
⎞⎤ ⎟⎥. ⎟⎥ ⎠⎦
305
(12)
They are modified further due to the nonlocal contribution which are absent in narrow gap nanocrystallites, but existed in bulk material [32]. The nonlocal contributions are
Δγ 1 = −5δ nl , Δγ = −4δ nl , Δα = −10δ nl , δ nl =
2
EB E g
15πε r Eg
3
,
(13)
where EB=27.2 eV is the Bohr energy, εr is the dielectric constant.
2.5. Effective-Mass Theory of Quantum Rods in the Electric Field [33,34] The electric and magnetic fields offers many possibilities to modulate the electronic structure and optical properties of quantum dots and quantum wires. There have been new physical behaviors discovered, for example, quantum Stark effect, size-dependent and anisotropic g factor in quantum dots, etc. Due to the symmetry along the long axis of the dot (z axis), we assume that the electric field is in the x-z plane. F is the electric field strength inside the rod, due to the dielectric effect it is not equal to the external electric field Fext,
Fz =
ε0
ε r nz + (1 − nz ) ε 0
Fz ,ext ,
⎞ 1 − e′2 ⎛ 1 + p nz = ln − 2 p ⎟, 2 ⎜ 2e′ ⎝ 1 − p ⎠
Fx =
ε0
nxε r + (1 − nx ) ε 0
(14)
1 p = 1− 2 , e
Fx ,ext ,
1 − nz nx = , 2
(15)
where εr and ε0 are the dielectric constants of the rod and the surrounding material, respectively, e is the aspect ratio, and p is the partiality of the rod. The electric field potential
G G G V ( r ) = −qF ⋅ r = −qr ′ ( eFz cos θ ′ + Fx sin θ ′ cos ϕ ′ ) ,
(16)
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where q is the charge of the electron or hole, r’, θ’ and ϕ’ are the transformed spherical coordinates of the electron or hole. To simplify the calculation, we use the spherical harmonic functions to describe the electric field potential term,
G 12 V ( r ) = − qFz eR ( r ′ R )( 4π 3) Y1,0 − qFx R ( r ′ R )( 2π 3)
12
(Y
1, −1 − Y1,1 ) ,
(17)
where R is the transverse radius.
2.6. Effective-Mass Theory of Quantum Dots in Magnetic Field [35] We extended the Luttinger’s effective-mass theory of semiconductors of zinc-blende structure in magnetic field to the case of spherical quantum dots of wurtzite structure. The hole effective-mass Hamiltonian in the zero SOC and zero magnetic field case is given in Eqs. (2) and (3). Assume that the magnetic field is applied in the x-z plane of the crystal structure. The vector potential of the magnetic field is,
G ⎛ 1 1 1 1 ⎞ A = ⎜ − Bz y, Bz x − Bx z , Bx y ⎟ . 2 2 2 ⎝ 2 ⎠
G
G
(18)
G
The momentum in the Hamiltonian p ⇒ p + eA c , whose components do not commute. Following Luttinger, we divided the Hamiltonian into two parts. The symmetric part is the original Hamiltonian, in which the operator pαpβ is replaced by the symmetric product,
{ p p } = 12 ( p α
β
α
pβ + pβ pα ) .
(19)
The matrix elements in the symmetric part become more complicated, which are given in the appendix of the paper [35]. The antisymmetric part is written as
G G H asym = K μ B I ⋅ B,
(20)
G
where I is an angle momentum matrix. If the basic functions are |1,1〉, |1,0〉, and |1,-1〉, it’s components are
⎛ 0 −1 0 ⎞ ⎛0 1 0⎞ ⎛1 0 0 ⎞ 2⎜ 2i ⎜ ⎟ ⎟ ⎜ ⎟ Ix = −1 0 1 ⎟ , I y = −1 0 − 1 ⎟ , I z = ⎜ 0 0 0 ⎟ . ⎜ ⎜ 2 ⎜ 2 ⎜ ⎟ ⎟ ⎜ 0 0 −1 ⎟ ⎝ 0 1 0⎠ ⎝0 1 0⎠ ⎝ ⎠
(21)
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307
The total electron and hole Hamiltonian can be written as
H e = H e 0 + H mm _ e + H Zeeman _ e , H h = H h 0 + H so + H mm _ h − H asym − H Zeeman _ h .
(22)
III. Polarization Properties of Emission Linearly polarized emission of slightly elongated quantum dots (quantum rods) has been reported with an experiment and a theoretical explanation of empirical pseudopotential calculations [10]. This discovery gives colloidal quantum dots a much more promising future because linearly polarized emissions have a much wider range of applications, such as biological labeling and optoelectronic devices. We calculated the electronic and hole states of CdSe rods for different respect ratio e [30]. Taking into account the Boltzmann distribution of the electronic and hole states, and summing up all contribution to the transitions of z and x, y polarizations, we obtained the strengths of optical transition for two polarizations: Iz and Ix (polarization along the x and z direction, respectively), and the polarization factor P,
P=
Iz − Ix . Iz + Ix
(23)
Figure 3. Polarization factor P, optical transition strengths of the CdSe rods for two polarizations Iz and Ix as functions of e for T=300 K, Δc=25 meV, and R=2.1 nm [30].
Fig. 3 shows the P, Iz, and Ix as functions of the e, assuming that the temperature T= 300K, the radius of CdSe rods R=2.1nm, and the crystal field splitting energy Δc=25meV. P increases rapidly as the e increases from 1 to 3, then approaches to a saturation value 0.5, in agreement with the experimental result [30].
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We considered some factors affecting the polarization, the crystal field splitting energy Δc, the temperature T, and the radius R. Fig. 4 shows the P as functions of the e for different R at T= 300K, Δc=25meV. From Fig. 4 we see that the linear polarization factor is larger for dots of smaller radius. The difference of P for different cases can be explained by the positions of hole energy levels and the Boltzmann distribution.
Figure 4. P of the CdSe rods as a function of e in the case of R=1.5, 1.7, 1.9, and 2.1 nm for T= 300 K, Δc=25 meV [30].
Figure 5. P of the CdSe rods as functions of e, (b) R=2.0 nm, for different T; (c) T=100 K, for different R [36].
Fig. 5 shows P of the CdSe rods as functions of e [36]. Form Fig. 5 we see that when e increases from 0.8 (oblate ellipsoid) to be larger than 1, P increases gradually from negative value (~-0.9) to positive. When e=1, P does not equal zero, it is still negative (-0.6~-0.8). Only at a critical aspect ratio, for example, e=1.39, P=0. The critical aspect ratio changes with the temperature and the radius of the ellipsoid. When T=100K, the critical aspect ratios for R=2.5nm and 3nm are 1.57 and 1.92, respectively. That the critical aspect ratio does not equal 1 is caused by the non-equivalence of the c-axis and the a-axis of the wurtzite structure.
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Figure 6. Polarization factors P, light strength Ix, and Iz of the CdSe rods as functions of electric field strength K for R=2.1 nm and (a) e=2, (b) e=3 [33].
Fig. 6 shows the polarization factors P, light strength Ix, and Iz of the CdSe rods as functions of electric field strength along the z direction K for R=2.1 nm and (a) e=2, (b) e=3 [33]. The polarization factor P increases as the electric field K increases, and the P of the e=3 case is larger than that of the e=2 case. It means that the longitudinal electric field can increase P, especially for elongated rods. If the quantum dots are in the magnetic field, then in the magnetic field direction the strengths of the left and right circular polarization emissions (σ- and σ+) are different. It was found experimentally that [13], the polarization factor of emission from the random oriented CdSe quantum spheres of R=2.85 nm in the magnetic field is 0.8. We sum the normalized intensities of σ- and σ+ transitions of randomly oriented dots Iσ±, and define the circular polarization factors,
(
Pc = Iσ − − Iσ +
) (I
σ−
)
+ Iσ + .
(24)
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Xiu-Wen Zhang, Yuan-Hui Zhu and Jian-Bai Xia
Figure 7. (a) Normalized intensities of σ - and σ + transitions of CdSe dots with R=2.85 nm as functions of magnetic field b, (b) circular polarization factors Pc as functions of b at different temperatures [35].
Fig. 7 shows (a) the normalized intensities of σ - and σ + transitions of CdSe dots with R=2.85 nm as functions of b, (b) circular polarization factors Pc as functions of b at different temperatures [35]. Pc increase as the magnetic field increases, and the saturation value is not 1, but about 0.8. At lower temperature the Pc approaches to the saturation value faster. The saturation value 0.8 is caused by the asymmetry of the crystal of the wurtzite semiconductor, the c-axis is not equivalent to the a-axis. Hence our results is in agreement with the experimental result [13].
Figure 8. Pc of single CdSe and GaAs dots (R=28.5 nm) as functions of cos θ.
Fig. 8 shows the circular polarization factor Pc of a single CdSe dot and GaAs dot with R=2.85 nm as a function of cosθ, θ is the angle between the magnetic field and the z axis. The Pc of CdSe dot increases from 0 at cosθ=0 to about 1 at cosθ=1 (B||z). While for the GaAs dot,
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311
the P is nearly a constant 1 independent of θ, due to the cubic symmetry. Hence, it is, in principle, possible to use the polarization spectroscopy to determine the orientation of the caxis of the individual wurtzite dots.
IV. g Factors of Quantum Dots 4.1. g Factors of CdSe [37] and InSb [38] Quantum Dots The g factor is an important physical quantity, which determines the energy level splitting in the magnetic field. The g factors of semiconductor bulk materials are well known, but those of nanostructures are mostly not measured in experiments. There have been only few theories, for example, the g factor of CdTe quantum dots were investigated theoretically by Prado et al. [7]. The ground state of the conduction band splits in the magnetic field B, if the splitting energy is ΔE, then the electron g factor is defined as
g = ΔE μ B B,
(25)
where μB is the Bohr magneton.
Figure 9. Electron g factors of CdTe spheres as functions of R [37].
Fig. 9 shows the electron g factors of CdTe spheres as functions of the radius R. Due to the equivalence of the three axes of the crystal structure, gz=gx (for magnetic field along the z and x directions). As R increases, the electron g factors decrease. When R is very small, the g is nearly 2 (1.71). When R is very large, the g approaches -0.74, which is the bulk-material value. It was also found that the g factor depends on the shape of the ellipsoid. Fig. 10 shows (a) the g factors of CdTe ellipsoids as functions of e for R=2.0 nm, (b) the g factors of CdTe ellipsoids as functions of R for length L=4.0 nm. For the sphere (e=1), gz=gx=1.03. For R=2.0 nm, as e increases, gz and gx both decrease, but gx decreases more quickly. When e is very large, similar to the nanowire case, gz=0.78, gx=0.69. For L=4.0 nm, as R increases (oblate ellipsoids), gz and gx both decrease, but gz decreases more quickly. When R is very large,
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Xiu-Wen Zhang, Yuan-Hui Zhu and Jian-Bai Xia
similar to the quantum well case, gz=0.015, gx=0.26. So the dimensions perpendicular to the direction of the magnetic field affect the g factors more than the other dimension due to the confinement of the cyclotron movement.
Figure 10. Electron g factors of CdTe ellipsoids (a) as functions of e for R=2.0 nm; (b) as functions of R for L=4.0 nm [37].
Figure 11. Electron g factors of InSb spheres as functions of R [38].
Fig. 11 shows the electron g factors of InSb spheres as functions of R, the inset shows the enlarged electron g factors in the smallest sized spheres. The g factors decrease as the increasing R. When R is very small, the g factors are about 2. As R increases, the saturation g
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313
factors are about -47.2, corresponding to the bulk-material value. The g factor inverts the sign at R=2.25 nm, it means that the g factor can be tuned to be positive, zero, and negative values near certain radius of quantum dots.
Figure 12. Electron g factors gz and gx of InSb ellipsoids (a) as functions of e for R=2.0 nm, (b) as functions of R for L=4.0 nm [38].
Fig. 12 shows the electron g factors gz and gx of InSb ellipsoids (a) as functions of e for R=2.0 nm, (b) as functions of R for L=4.0 nm. As e increases, the gz and gx approach to -0.399 and -0.704, respectively. As R increases, the gz and gx approach to -8.455 and -3.836, respectively, similar to the case of CdTe ellipsoids.
4.2. Electric Field Tunable Electron g Factor and Highly Anisotropic Stark Effect It was found that the applied electric field F can modulate the g factors [34]. Fig. 13 shows the electron gz factor of InAs ellipsoids with R=3 nm (a) as functions of e at F=0, (b) as functions of F at e=3 for F||z and F||x. At F=0 the g factor decreases as e increases and equals zero at e=1.96. When F||x, the g factor is almost not affected, while when F||z it increases a lot with the increasing electric field. For the InAs ellipsoid of R=3 nm and e=3 the g factor can
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Xiu-Wen Zhang, Yuan-Hui Zhu and Jian-Bai Xia
be tuned from negative (-0.17) to positive, and crosses zero at F=12.3 mV/nm. This electric field tunable g factor is independent of the temperature and magnetic field, which is different from the magnetic field tunable g factor, and may be used to tune the electron spin to be polarized, unpolarized, or antipolarized.
Figure 13. Electron gz factor of InAs ellipsoids with R=3 nm (a) as functions of e at F=0, (b) as functions of F at e=3 for F||z and F||x [34].
Figure 14. (a) Electron (b) hole Stark shifts of InAs quantum ellipsoids with R=3 nm, e=3 as functions of F [34].
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315
The Stark effect of semiconductor quantum dots has been studied experimentally [39] and theoretically [33]. We found the high asymmetry of Stark effect in quantum ellipsoids [34]. Fig. 14 shows the electron and hole Stark shift of InAs quantum ellipsoids with R=3 nm, e=3 as functions of the electric field strength F. For the InAs ellipsoid of R=3 nm, e=3, the Stark shift of electron and hole ground states are nearly zero when F||x, while when F||z the Stark shifts can be tens of meV at large electric field. This high asymmetry is caused by the simultaneous dielectric effect and quantum confinement effect. Due to the dielectric effect, the electric field in the quantum dots is smaller than the external electric field. When e=3, nz=0.102, and nx=0.449, the inner field when F||x is about three times smaller than that when F||z [see Eqs. (14) and (15)]. As the quantum confinement of the quantum ellipsoid along the x axis is larger than that along the z axis, the Stark shift in the x axis is smaller than that in the z axis. Define the asymmetry factor as the ratio of the Stark shifts for F||z and F||x. When F=20 mV/nm, the asymmetry factor of electron Stark shift is 319, and that of hole Stark shift is 40. Therefore, though the asymmetry factor of the shape of quantum dot is e=3, the asymmetry factor of the electron Stark shift is as high as 319. With this high asymmetrical Stark effect, the quantum ellipsoids in electric fields with different orientations can emit light with quite different wave lengths.
V. Giant Zeeman Splitting in DMS Quantum Dots The nature of the spin of carriers in confined nanocrystals has received considerable attention. With the rapid progress in semiconductor fabrication techniques, it is now possible to dope magnetic impurities inside the nanocrystals to produce the diluted magnetic semiconductor (DMS) quantum dots or wires. An interesting feature of Mn ions doped nanocrystals is the exchange field, which results in the giant Zeeman splitting of the spin sublevels of the electron and hole. On the other hand, it was found that the Curie temperature TC in DMS nanodots or nanowires can be higher than the TC of the corresponding bulk material, which can be used as an efficient spin injector from the ferromagnetic DMS into semiconductor.
5.1. Giant Zeeman Splitting in ZnMnSe Quantum Spheres [40] For the DMS nanostructures the electron and hole effective-mass Hamiltonian (2) should be added a term Vexch, which is the sp-d exchange interaction between the carriers and magnetic ions Mn2+, in the mean-field approximation, e Vexch = − xN 0ασ z S z , h Vexch = − xN 0 βσ z S z ,
(26)
for electron and hole, respectively. x is the effective Mn2+ ions concentration, N0α (N0β) is the exchange integral for the conduction band (valence band). 〈Sz〉 is the thermal average of the Mn2+ spin, given by the Brillouin function B5/2,
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Xiu-Wen Zhang, Yuan-Hui Zhu and Jian-Bai Xia
Sz =
⎛ 5μ B B ⎞ 5 B5 2 ⎜⎜ ⎟⎟ . 2 ⎝ k B (T + T0 ) ⎠
(27)
Figure 15. Energy levels of electron states (a) and hole states (b) of ZnSe quantum spheres as functions of R in B=2T. The inset shows the enlarged energies of the ground electron states [40].
Figure 16. Energy levels of electron states (a) and hole states (b) of ZnMnSe (x=0.018) quantum spheres as functions of R in B=2T [40].
Electronic Structure and Physical Properties of Semiconductor Quantum Dots
317
Fig. 15 and Fig. 16 show the energy levels of electron states (a) and hole states (b) of ZnSe and ZnMnSe (x=0.018) quantum spheres as functions of R in B=2T, respectively. The energy unit is ε0=(1/2m0)( /R)2. The splitting of energy levels in the doped case is much larger than that in the undoped case under the same magnetic filed strength, which is called giant Zeeman splitting effect.
Figure 17. Total Zeeman splitting energies in ZnMnSe quantum spheres with different R and x as functions of the magnetic field B [40].
Figure 18. Total Zeeman splitting energy of ZnMnSe quantum spheres as functions of R in B=2 T [40].
The total Zeeman splitting energy ΔE is the difference between the transition energies from two splitting hole ground states to two splitting electron ground state, which is the sum of the electron and hole Zeeman splitting energies. Fig. 17 shows the total Zeeman splitting energies in ZnMnSe quantum spheres with different R and x as functions of the magnetic field B. The square points are the magnetic circular dichroism results [41]. The calculated results are qualitatively in agreement with experimental data. The splitting energies increase linearly
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Xiu-Wen Zhang, Yuan-Hui Zhu and Jian-Bai Xia
with fields at low fields, depending on doping concentration, and saturate as modest fields decided by the formula of S z
[Eq. (27)].
Fig. 18 shows the total Zeeman splitting energy of ZnMnSe quantum spheres as functions of R in B=2 T, the curve of the undoped case is scaled by 10. The Zeeman splitting energy of the Mn doped case is much larger than that the undoped case. The total Zeeman splitting energies are basically independent on the R, but mainly depend on the effective Mn concentration x.
5.2. Anisotropic Giant Zeeman Splitting in InMnAs Quantum Dots [42] We studied the Zeeman splitting of In1-yMnyAs oblate quantum dots [42], and found that the Zeeman splittings of the hole states are highly anisotropic. Fig. 19 shows the energies of hole states as functions of B for the In1-yMnyAs quantum rod with R=3.5 nm, the aspect ratio e=0.245, yeff=0.08, at T=1 K and F=0 in the case of B||z and B||x, respectively. Most of states do not split in the case of B||x, except the h2 state. While for the h1, h3, h4, and h5 states, the Zeeman splittings in the two cases are totally different, i.e. the Zeeman splitting is highly anisotropic. Physically, the highly anisotropic Zeeman splitting is induced by the spin-orbit coupling effect, which couples the spin states with the space-wave functions. Sometimes, the space-wave functions are anisotropic due to the anisotropic shape of the quantum dot, resulting in anisotropy of the Zeeman splitting. For example, the wave functions of the h1 degenerate state are mainly 0SX+↑ (m=0, l=0, Bloch state |11〉, MJ=3/2) and 0SX-↓ (m=0, l=0, Bloch state |1-1〉, MJ=-3/2), respectively. When B||z, the main Zeeman term is Aσz, where A is a constant and σz is the Dirac operator, and the states 0SX+↑ and 0SX-↓ split normally. When B||x, the main Zeeman term is Aσx, and it has zero matrix element between the states 0SX+↑ and SX-↓, as their space-wave functions are orthogonal. The reasons for the highly anisotropic Zeeman splittings of the h3, h4,and h5 states are similar. It is found that the electron states have also the highly anisotropic Zeeman splitting.
Figure 19. Energies of hole states as functions of B for the In1-yMnyAs quantum rod with R=3.5 nm, the aspect ratio e=0.245, yeff=0.08, at T=1 K and F=0. (a) B||z (b) B||x [42].
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We define an anisotropy factor of the Zeeman splitting as ΔEz/ΔEx. Fig. 20 shows the anisotropy factor of the Zeeman splitting of the hole ground state as a function of B for the In1-yMnyAs quantum dot same as Fig. 19. The anisotropy factor decreases as the magnetic field increases, which is 918 at B=0.1 T and can be larger than 3000 at very small magnetic field.
Figure 20. Anisotropy factor of the Zeeman splitting as a function of B [20].
5.3. Giant and Size-Sensitive g Factor in HgMnTe Quantum Spheres [43] HgTe has an inverted (zero-gap) band structure, the conduction band Γ6 is under the valence band Γ8. In quantum dots the conduction band states becomes higher than the valence band states due to the quantum confinement effect.
Figure 21. Energies of the lowest electron and hole states as functions of the sphere radius for HgMnTe quantum spheres in the external magnetic field B=2T [43].
Fig. 21 shows the energies of the lowest electron and hole states as functions of the sphere radius for HgMnTe quantum spheres of x=0.06 in the external magnetic field B=2T at
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T=2 K. The energy unit is ε0=(1/2m0)( /R)2. The lowest degenerated electron states with the S character split in the presence of magnetic field. When the spherical radius R is smaller than 13.6nm, the electron state (1/2)Se1↑+PX1↑+PZ1↑ is separated from valence bands, indicating the transition from the semimetal to the semiconductor structure. When R decreases to Rc=8.87 nm, the electron state (-1/2)Se1↓+PX⁻ 1↓+PZ1↓ becomes lower than the state (1/2)Se1 ↑+PX1 ↑+PZ1 ↑, and turns out to be the electron ground state. This crossing behavior indicates that electron g factor could change from a negative value to a positive one, due to the quantum confinement effect.
Figure 22. Electron g factors as functions of R at B=2.0 T for different x [43].
Figure 23. Electron g factors as functions of R at B=2.0 T and x=0.06 for different T [43].
Electronic Structure and Physical Properties of Semiconductor Quantum Dots
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Fig. 22 shows the size dependences of the g factors of mobile electron states with Γ6 symmetry in the applied magnetic field B=2.0 T at T=4.2 K for different x. When the radius increases, the g factor decreases, and changes sign at a critical spherical radius Rc. Due to the exchange interaction, the g factor is greatly enlarged and it decreases from 580 to -374 as the radius increases from 1 to 13.6nm for x=0.06. The absolute value of the g factor increases linearly with the doped effective concentration x as shown in the inset. Fig. 23 shows the size dependence of the g factors of electron states at temperature T=4.2, 17 and 30K for x=0.06 and B=2.0T. The g factors depend sensitively on the temperature, and they are larger at lower temperature and decrease more quickly with increasing temperature. The temperature dependence of the g factor is shown in the insets. The absolute value of the g factor decreases linearly with increasing temperature at low temperature, and then decreases with a smaller slope at higher temperature.
VI. Curie Temperature of DMS Quantum Dots The mean-field model was first proposed by Zener, and then extended by Dietl et al. [44] to study the Curie temperatures of Mn-doped DMSs. We have extended the mean-field theory to the Fe and Co doped DMSs [45]. Here we extended this theory to the DMS quantum dots.
6.1. Curie Temperature of ZnO Quantum Dots [46] We studied the hole mediated TC of quantum dots, and found that the Helmholtz free energy in DMS quantum dot is a linear function of the magnetization due to the discrete levels, which is quite different from the quadratic function in higher dimension structures of DMS. Because of the discrete levels in quantum dots, the Helmholtz free energy is defined as
Fc ( M ) = −
1 V
∑ E (M ) n
n
1 , ⎡ En ( M ) − EF ⎤ 1 + exp ⎢ − ⎥ k BT ⎣ ⎦
(28)
where En(M) is the n-th hole energy level. The magnetization of the localized spins of magnetic ions in the absence of external magnetic field is
⎡ Sg μ B ( − ∂Fc ( M ) ∂M ) ⎤ M = Sg μ B N 0 xBS ⎢ ⎥. k B (T + TAF ) ⎢⎣ ⎥⎦
(29)
If the temperature T and the number of holes p in the dot are given, we can calculate out a self-consistent magnetization as a function of T and p, M(T,p), using above two equations. It was found that the magnetization-temperature curve of a DMS dot does not cross with the abscissa axis (M=0), while has a long tail, and approaches to zero gradually, so that the Curie temperature in DMS dots should be defined anew. We define the temperature at which the M has a very small value as the Curie temperature of the quantum dot.
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Fig. 24(a) shows the magnetization M(T,p) (M||z) of (Zn,Mn)O quantum sphere with R=2 nm, x=0.05 as functions of T, where the curves correspond to different number of the hole p in the dot. The magnetization decreases with increasing temperature T and becomes nearly zero at the Curie temperature TC. It is noticed that when there are more than 3 holes in the dot, the Curie temperature is expected to be much higher than the room temperature. It suggests that a few holes can mediate high temperature ferromagnetism in DMS quantum dot. The case of M||x (Fig. 24(b)) is similar to the M||z case. Fig. 24(f) shows the M(T,p) of R=3 nm sphere, the TC is lower than that of R=2 nm. Hence the ferromagnetism is enhanced by the quantum confinement effect.
Figure 24. Magnetization M(T,p) of (Zn,Mn)O quantum sphere with R=2 nm, x=0.05 as functions of T, (a) M||z, (b) M⊥z. (c) Hole levels (b) electron levels as functions of M for M||z. M(T,p) as functions of T for (e) electron, (f) hole of R=3 nm sphere.
Fig. 24(c) shows the hole levels of the dot as functions of M (M||z). Due to the level crossing in the small M range, there are many step decreases of the magnetization when T is high [see Fig. 24(a) and 24(b)]. Fig. 24(d) shows the electron levels as functions of M. The lowest two electron levels split as M increases and do not cross with upper levels. Thus when there are two electrons in the dot, the magnetization can not reduce the Helmholtz free energy [Eq. (28)], and the calculated self-consistent magnetization is nearly zero, as shown in Fig. 24(e). When there is one electron, the magnetization is large at low temperature, similar to the hole case [see Fig. 24(a)], the Curie temperature is about 20 K [see Fig. 24(e)]. So the dependence of electron-mediate ferromagnetism on the parity of the number of electrons [47] is confirmed for the ground electron states in our calculation. Therefore, the Curie
Electronic Structure and Physical Properties of Semiconductor Quantum Dots
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temperature of the hole-mediate ferromagnetism is much higher than that of the electronmediate one. The magnetization can be measured in experiment as the residual magnetization in the magnetization hysteresis loop [48], and the direction of the magnetization is determined by the direction of the magnetic field in the experiment. Fig. 25(a) shows the calculated Curie temperature TC as functions of the hole number p for the dot with R=2 nm, e=1, and x=0.05. The TC increases as the hole numbers in the dot increases, and are higher than that in the bulk. When p=18, TC=772 K. Fig. 25(b) shows the TC as functions of the dot radius for the dot of e=1 and the fixed hole number (p=1). The TC decreases as the radius increases. That is because as R increases the volume of the dot V increases and the Helmholtz free energy decreases, so the ferromagnetism is reduced. Fig. 25(c) shows the TC as functions of the aspect ratio e when the volume and the hole number are fixed. The TC decreases as e increases. Fig. 25(d) shows the corresponding hole levels as functions of e for M=0. As e increases, the energy differences between the hole levels increase, so the ferromagnetism decreases. Figs. 25(e) and 25(f) show the TC as functions of the electric field for the dot with R=4 nm, e=0.2, in the case of F||x and F||z, respectively. When the external electric field increases, the TC decreases, and the decrease in the F||x case is very larger than that in the F||z case. When F||x, the TC can be tuned from much higher than the room temperature to lower than it.
Figure 27. (a) Calculated Curie temperature TC as functions of the hole number p for the dot with R=2 nm, e=1, and x=0.05. (b) TC as functions of the dot radius for the dot of e=1 and p=1. (c) TC as functions of the aspect ratio e. (d) Corresponding hole levels as functions of e for M=0. (e) F||x, (f) F||z TC as functions of the electric field F [46].
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6.2. Anisotropic Curie Temperature of InMnAs Quantum Dots [49] We found that the (In,Mn)As quantum dot can be ferromagnetic at room temperature when there is only one hole in the dot. The (In,Mn)As oblate quantum dot have highly anisotropic Zeeman splitting and ferromagnetism due to the spin-orbit coupling effect. The Curie temperature depends on the shape of the quantum dot and the measurement direction of magnetization. The (In,Mn)As oblate quantum dot can be used as an uniaxis spin amplifier which amplifies the spin polarization along the z direction. Fig. 26 shows the magnetization M as functions of T for the (In,Mn)As oblate quantum dot with V=V0 (D=4 nm, e=1), e=0.3, x=0.1, and one hole in the dot. The unit of M is M0=10-3N0gMnμB. The cases of M||z (a) and M||x (b) are quite different. The magnetization in the case of M||x approaches to zero more quickly than the magnetization in the case of M||z. Though at low temperature, the former is close to the latter. Therefore, the room temperature ferromagnetism is easy to happen in the case of M||z, while is hard to happen in the case of M||x. So the (In,Mn)As oblate quantum dot have highly anisotropic ferromagnetism (TC). The Curie temperature decreases with increasing diameter, and also decreases as the effective composition of magnetic ions decreases. For a zinc-blende InMnAs quantum sphere (e=1), the (M||z) case is identical with the (M||x) case, because the z axis is equivalent with the x axis. We see from Fig. 26(a) that the Curie temperature TC≈350 K. Thus InMnAs quantum dot can be ferromagnetic at room temperature, which is qualitatively in agreement with the experimental result [47].
Figure 26. Magnetization M as functions of T for the (In,Mn)As oblate quantum dot with V=V0, e=0.3, x=0.1, and one hole in the dot, (a) M||z, (b) M⊥z [49].
In order to explain the highly anisotropic ferromagnetism, we calculated the hole levels of (In,Mn)As oblate quantum dot with V=V0 (volume of D=4 nm sphere) and e=0.3 as functions of M for M||z and M||x, similar to Fig. 19. As M increases (from zero), the double degenerate states split as Zeeman splitting. The Zeeman splittings of many states in the cases of M||z and M||x are very different, i.e. the Zeeman splitting is highly anisotropic. The reason has been discussed in the Sec. 5.2. The Helmholtz free energy is related to the hole levels, and the highly anisotropic Zeeman splitting causes highly anisotropic ferromagnetism.
Electronic Structure and Physical Properties of Semiconductor Quantum Dots
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When there is no hole in the DMS dot, it is paramagnetic at T
before tunneling. If a hole with
sz = ±δ (0<δ<<1) tunnels into the dot, its spin will be changed to be sz = ±1 . Thus the spin polarization along the z direction is amplified, and we can use this InMnAs oblate quantum dot as an uniaxial spin amplifier.
VII. Summary There are many freedoms to modulate the physical properties of semiconductor quantum dots by, for example, shape: spheres, ellipsoids (long, oblate); size: radius, length; crystal structure: zine-blende, wurtzite; energy gap: wide, middle, narrow; doping: N, dilute magnetic ions; external field: strain, electric field, magnetic field, etc. The physical properties of quantum dots have been investigated theoretically, including: electronic structure, optical property, polarization of emission, Stark effect, g factor, ferromagnetism, Curie temperature, etc. The theoretical model is the effective-mass theory, but is developed to be formulation in the spherical coordinate, and under the electric and magnetic field. The main results are: 1. The emission of elongated quantum dots has strong linear polarization, and the electric field along the long axis direction can increase the polarization. In the magnetic field the random oriented quantum dots emit light of left and right polarizations with different strength, the circular polarization factor approaches to 0.8 at large magnetic field. 2. The electron g factors of quantum dots depend on the radius of dots, approaches to 2 at small radius, and approaches to the bulk material values at large radius. The gz and gx factors of quantum ellipsoids decrease as the longitudinal length L or the transverse radius R increase, the gx decreases more quickly when the L increases, and the gz decreases more quickly when the R increases. Hence, the dimensions perpendicular to the direction of the magnetic field affect the g factors more than the other dimension. The InAs quantum ellipsoid has electric field tunable g factor and high anisotropic Stark shift for electric field along the z and x directions. 3. The DMS quantum dots have giant Zeeman splitting of electron and hole energy levels, which is much larger than the general Zeeman splitting in undoped dots. The g factor of HgMnTe quantum dots decrease from a large positive value to a large negative value when the radius decreases from 1nm to 13nm. The Zeeman splitting of hole states in InMnAs oblate quantum dots is highly anisotropic for B||z and B||x. The anisotropy factor of the Zeeman splitting of the hole ground state is 918 at B=0.1T. 4. The mean-field theory of quantum dots to calculate the Curie temperature is different from that of bulk material. The spontaneous magnetization M is a function of temperature T and hole number p. When T increases, M decreases, and does not cross with the abscissa axis (M=0), while has a long tail, and approaches to zero gradually, so that the Curie temperature in DMS dots is defined anew. The Curie temperatures of quantum dots can be larger than that of corresponding bulk material due to the density of states at the Fermi level.
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The Curie temperatures of the oblate quantum dots can be highly anisotropic caused by the quantum confinement effect, and the anisotropic shape of quantum dots. The transverse electric field can modulate the g factor and TC. All these demonstrate application prospect in the microelectronics, optoelectronics, and spintronics.
Acknowledgments The authors would like to acknowledge our cooperators Prof. W. J. Fan, Prof. S. S. Li, Prof. K. Chang, Prof. J. B. Li, Dr. Y. H. Zheng, Dr. X. Z. Li, and the special funds for Major State Basic Research Project No. 069C031001, the support from the National Natural Science Fundation No. 60521001, 90301007，863 Project 068C041001, and A*STAR of Singapore (Grant No. 0421010077).
Appendix Matrix elements of first order tensor momentum operators,
( l + 1 P l ) = =i (1)
⎛d l⎞ l +1⎜ − ⎟, ⎝ dr r ⎠ = ⎛ d l +1⎞ l =− l⎜ + ⎟. i r ⎠ ⎝ dr
(l −1 P ) (1)
(A1)
Matrix elements of second order tensor momentum operators,
)
( l + 2 )( l + 1) ⎛ d 2
)
( l − 1) l ⎛ d 2
(
l + 2 P (2) l = −3
(
l − 2 P (2) l = −3
2l + 3 ⎜ 2l − 1 ⎝ dr 2
3 ( 2l + 2 )( 2l + 1) l l = ( 2l + 3)( 2l − 1)
(l P ) (2)
⎜ 2− ⎝ dr +
2l + 1 d l ( l + 2 ) ⎞ + ⎟, r dr r2 ⎠
2l + 1 d l 2 − 1 ⎞ + 2 ⎟, r dr r ⎠
(A2)
⎛ d 2 2 d l ( l + 1) ⎞ − ⎜ 2+ ⎟. r dr r2 ⎠ ⎝ dr
Derivative formulas of radial functions,
⎛d l⎞ ⎜ − ⎟ jl = − jl +1 , ⎝ dr r ⎠ ⎛ d l +1 ⎞ ⎜ + ⎟ jl = jl −1 , r ⎠ ⎝ dr
(A3)
Electronic Structure and Physical Properties of Semiconductor Quantum Dots
⎛ d 2 2l + 1 d l ( l + 2 ) ⎞ ⎛ d l + 1 ⎞⎛ d l ⎞ + ⎜ 2− ⎟ jl = ⎜ − ⎟⎜ − ⎟ jl = jl + 2 , 2 dr r dr r dr r ⎝ ⎠⎝ dr r ⎠ ⎝ ⎠ ⎛ d 2 2l + 1 d l 2 − 1 ⎞ ⎛ d l ⎞ ⎛ d l +1⎞ + 2 ⎟ jl = ⎜ + ⎟ ⎜ + ⎜ 2+ ⎟ jl = jl − 2 , r dr r ⎠ r ⎠ ⎝ dr r ⎠ ⎝ dr ⎝ dr
327
(A4)
⎛ d 2 2 d l ( l + 1) ⎞ ⎛ d l + 2 ⎞⎛ d l ⎞ − ⎜ 2+ ⎟ jl = ⎜ + ⎟ ⎜ − ⎟ jl = − jl . 2 r dr r r ⎠ ⎝ dr r ⎠ ⎝ dr ⎝ dr ⎠ where jl is the l-th spherical Bessel function.
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In: Quantum Dots: Research, Technology and Applications ISBN: 978-1-60456-930-8 Editor: Randolf W. Knoss, pp. 331-370 © 2008 Nova Science Publishers, Inc.
Chapter 9
GE NANOCLUSTERS IN GEO2 FILMS: SYNTHESIS, STRUCTURAL RESEARCH AND OPTICAL PROPERTIES V.A. Volodin1,2,* and E.B. Gorokhov1 1
Institute of Semiconductor Physics, Siberian Branch of Russian Academy of Sciences, Pr. Lavrent’eva 13, 630090, Novosibirsk, Russia 2 Novosibirsk State University, Pirogova Street, 2, 630090, Novosibirsk, Russia
Abstract Semiconductor nanostructures, namely, quantum dots and quantum well wires attract a lot of interest due to its new electronic and optical properties that can be modified artificially. The quantum size effects in semiconductor quantum dots lead to possibility of application of semiconductors with indirect band structure (Si and Ge) in optoelectronics. The germanium has several advantages comparing with silicon (relatively low temperature of processing, bigger Bohr radius, bigger electrical permittivity). In this chapter, germanium nanoclusters in GeO2 films have been obtained with the use of two methods. The first method of Ge nanocluster formation is a film deposition from supersaturated GeO vapor with subsequent dissociation of meta-stable (in solid phase) GeO on hetero-phase system Ge:GeO2. The second method is growth of anomalous thick native germanium oxide layers with chemical composition GeOx(H2O) during catalytically enhanced Ge oxidation, x~1. The obtained films were studied with the use of photoluminescence, Raman scattering spectroscopy, IR-spectroscopy, ellipsometry, high-resolution electron microscopy. Strong photoluminescence signals were detected in GeO2 films with Ge nanocrystals at room temperature. “Blue-shift” of the photoluminescence maximum was observed with reducing of Ge nanocrystal size in anomalous thick native germanium oxide films. So, the correlation between reducing of the Ge nanocrystal sizes (estimated from position of Raman peaks) and photoluminescence “blue-shift” was observed. The Ge nanocrystals presence was confirmed by high-resolution electron microscopy data. The optical gap in Ge nanocrystals was calculated with taking into account quantum size effects and compared with the position of the experimental photoluminescence peaks. It can be concluded that a Ge nanocrystal in GeO2 matrix is a quantum dot of type I. It was shown that “band gap *
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V.A. Volodin and E.B. Gorokhov engineering” approaches can lead to creation of Ge:GeO2 heterostructures with required properties. The possibility of relatively low-temperature crystallization of dielectric GeO2 based film was demonstrated, the crystallized films have device quality. This heterostructures can be perspective for using in opto-electronics, for creation of elements of quasi-nonvolatile MOS memory, etc.
Key words: Germanium nanoclusters, GeO2, Raman scattering, photoluminescence, Gebased MOS structures
Introduction Semiconductor nanocrystals (NCs) are intensively studied recently years. The great interest to these studies is caused by that the nanometer-sized semiconductor NCs, embedded in widegap insulating matrices, have shown significant promises for a wide range of nanoelectronics and optoelectronics applications. Quantum effects in such heterosystems become brightly apparent at room temperature, and because in some experiments a single NC reveals deltafunction-like energy spectra [1] it can be called quantum dots (QDs). So, these heterostructures are interesting for both fundamental and applied sides. The Ge NCs in GeO2 films have practically not been studied, and very little work has been performed on relative Ge:GeO2 heterostructures, unlike quite long and well studied Si and Ge NCs in other matrixes. For example, free Si nanoparticles obtained using plasma enhanced chemical vapor deposition (PECVD) method [2] or using power electron beam evaporation [3]; Si nanowires in porous Si [4]; Si NCs formed in SiO2 matrix using Si+ ions implantation [5-7]; Si NCs formed by co-sputtering of Si and SiO2 by electron beam gun [8]; Ge NCs formed in SiO2 matrix using Ge and SiO2 rf-magnetron co-sputtering [9,10]; Ge NCs formed in SiO2 matrix [11-13] and in Si3N4 matrix [14] using Ge+ ions implantation; and A2B6 NCs in bulk glasses [15] were studied earlier. A comparison of group IV semiconductors (Si and Ge) reveals that, from the technological point of view, germanium has several advantages over silicon. First of all, the melting point of Ge is sufficiently lower as compared to Si (940°C versus 1420°C), causing lower crystallization temperature of Ge. So, the temperature of Ge-based technology processes is relatively low, what is important in planar microelectronic technology. For example, thermal annealing with high thermal budget (1100oC during one or two hours) needed for formation Si NCs in SiO2 after ion implantation is not compatible with forming of shallow p-n junction in silicon, because leads to impurity diffusion. The germanium monoxide, GeO, is volatile and readily sublimates at sufficiently low temperature (~600°C), forming the bases for the development of different technologies utilizing the mass-transfer reactions. Besides, according to theoretical calculations [16], the energy gap values of Ge and Si NCs become equal when their sizes reach 2.3 nm. Further decrease of sizes down to 1.5 nm causes an abrupt increase in Ge gap up to 3 eV, which exceeds by 0.7 eV the gap of a Si NC with the same size. This is the reason for an intensive and useful blue shift of Ge-based light-emitting nanostructures. Ge has a larger exciton Bohr radius (4.9 nm for Si and 24.3 nm for Ge). Thus, the quantum-confinement-related positive effects in Ge manifest themselves at larger sizes, which can be reached with less difficulty. The barriers for electrons and holes in Ge:GeO2 heterostructures is more preferable for its injection. This important feature can increase the efficiency of the injection-type devices produced on the basis of Ge-GeO2 system. Germanium is a high-refractive-index
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semiconductor (n=4.12 at 2.0 micrometer), valuable for filling the interstitial voids in opals and producing photonic crystals with a complete photonic band gap. Germanium, germanium oxides and nitrides also have higher dielectric constant then silicon based dielectrics, for example, measured dielectric constant for GeO2 is 5.1-5.3 [17], so the exciton binding energy in Ge NCs is smaller then in Si NCs and limit for “blue-shift” for excitons in Ge NCs is higher and energy of optical transitions can reach UV diapason. The technology of these Ge:GeO2 film deposition is simple, not expensive and compatible with common silicon technology. So, the relatively old obtained Ge:GeO2 heterostructures [18] attract more attention to researchers recent years [19-28].
Experimental The GeO2 films with Ge NCs were obtained on sapphire, silicon and germanium substrates with the use of two methods. The first one is a film deposition from supersaturated germanium monoxide - GeO vapor with subsequent dissociation of meta-stable (in solid phase) Ge monoxide on heterophase system Ge:GeO2. The scheme of quarts tube reactor is shown in figure 1. The reactor contains evaporation zone (A) and deposition zone (B) placed consequently in one quartz tube. The mono-crystalline Ge plates are placed in evaporation zone; it can be heated to temperature up to 700-800oC. The deposition zone contains substrate for deposition and its temperature is lower than temperature in evaporation zone, but higher than room temperature (usually the temperature was in range 400-500oC). The bearing gas is Ar (helium is also can be used) with impurities of O2 and H2O. Oxygen reacts with Ge plates in evaporation zone with formation of GeO gas, according to next chemical reactions: 2Ge (solid) +O2 (gas) → 2GeO(gas) Ge (solid) +H2O (gas) → GeO(gas) + H2 (gas)
(1)
Figure 1. The scheme of growth reactor. 1- quartz tube; 2- Ge plates; 3- substrates for Ge:GeO2 films.
The GeO is volatile, in contrast to Ge or GeO2, the last can be appreciably evaporate only at high temperatures, 1000oC and higher. Because the temperature in deposition zone is quite low, the pressure of GeO gas in deposition zone is higher than saturation pressure, and GeO precipitate on substrates. The solid GeO films is meta-stable and decompose in Ge and GeO2 at relatively low temperatures (about 350oC and higher) during several minutes. The growth
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rate depends on GeO vapor pressure and temperature of substrates. The process of formation of Ge clusters is schematically shown in figure 2. The growth of Ge clusters is limited by diffusion. Because the surface diffusion rate is higher than the volume one, the more rapid growth rate – the lesser sizes of Ge clusters. Depending on growth condition one can obtain GeO2 films with amorphous Ge nanoclusters or Ge NCs. To obtain amorphous Ge nanoclusters the lower substrate temperatures should be used. Because there is gradient of temperature and gradient of GeO gas pressure in deposition zone, in some conditions one can grow the Ge:GeO2 hetero-film with gradient of thickness and gradient of structural parameters. But, it should be noted, that our hetero-structure have remarkable property – due to reverse chemical reaction (2GeO(solid) →Ge+GeO2) and consequence Ge precipitation procedure it always has molar ratio between Ge and GeO2 exactly 1:1, independently on growth condition. More detailed the growth procedure is described elsewhere [20, 21].
Figure 2. Illustrations of diffusion limited growth of Ge clusters: a) surface diffusion, b) bulk diffusion.
The second method is growth of anomalous thick native germanium oxide layers with chemical composition GeOx(H2O) during catalytically enhanced Ge oxidation. Wet chemical treatments in HF acid were carried out before catalytically enhanced oxidation of Ge (100) plates. In the first method, Ge NCs formation takes place during growth via surface diffusion of Ge atoms (fig. 2). In the second method, Ge NCs were formed by post-annealing treatments of amorphous GeOx films. Because of suppressed Ge diffusion in bulk (compared with surface diffusion), it was supposed that in this case one can forms Ge NCs of very small sizes, limited by diffusion length (Ldiff, fig. 2). It should be noted, that for the second method, for all samples mole composition ratio of Ge and GeO2 was about 1:1 (as it was previously said, for the first method, this ratio was exactly equal to 1:1). So, in such hetero-system, the average distance between Ge clusters is about the value of its radius. The obtained films were studied with the use of photoluminescence (PL) spectroscopy, Raman scattering spectroscopy, IR-spectroscopy, ellipsometry, and high-resolution transmittance electron microscopy (HRTEM) techniques. Experimental Raman spectra were registered in quasi back-scattering geometry, the 514.5 nm Ar+ laser lines was used. The PL spectra were registered at room and low temperatures using cw He-Cd (325 nm), Ar+ (488 nm and 514.5 nm lines) He-Ne (633 nm), pulse N2 (337 nm) lasers and mercury lamp for pumping. The normal incidence IR absorption measurements were carried out with a resolution of 4 cm-1. Some samples were annealed in high-vacuum quartz tube with a tubular
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oven. The heating rate was 10 oC min-1. When the annealing temperature was reached, the oven was removed and the films cooled naturally. To avoid the evaporation of Ge:GeO2 films its was covered by cap layer from 50 nm thick SiO2 deposited at temperature 100 oC and pressure equal to 10-8 Torr using electron beam gun evaporation of silica. Some films were specially prepared for HRTEM studies. The HRTEM studies were carried out with the use of a JEOL 4000EX (Japan) electron microscope operated at an accelerating voltage of 250 keV, for which the lateral point resolution is 0.19 nm. The optical constant of the films were studied using scanning ellipsometry with step δl=0,5 mm, He-Ne laser (633 nm) and the results were interpreted using described method and specially developed algorithm. Spectral dependence of refraction and absorption coefficients from diameter of Ge nanoclusters n(D) and k(D) was studied using many-angle, many-thickness spectral (250-800 nm) scanning ellipsometry. All elipsometers were manufactured in Institute of Semiconductor Physics, Siberian Branch of Russian Academy of Sciences.
Results and Discussion 1. Quantum Size Effects in Ge:GeO2 Films Visible by the Naked Eye It should be noted for some historical review, that the first Ge:GeO2 films were obtained by one of the authors (EBG) at the end of 1970s. The first method (described earlier) of film deposition from supersaturated germanium monoxide - GeO vapor on transparent substrates (sapphire and glasses) had been used. This work was part of big project – developing of technology of growth of dielectric films on germanium. The films should had mechanical and electrical properties high enough to be used in Ge based MOSFET devices. As a result of this project, it becomes convinced that it is double-layered film compositions like SiO2/Si3N4 or GeO2/Si3N4 that were most preferable for FET applications on Ge substrates. The typical energy density of interface states in these compositions was (1-2)⋅1012 cm-2⋅eV-1 and ~(24)⋅1011 cm-2⋅eV-1, respectively. Both systems were electrically and mechanically stable, exhibited low hysteresis in electrical characteristics and small built-in charges, and displayed some other useful properties. Using this technology, not only working prototypes of Ge-MOS transistors, but also a small integrated circuit made by a planar Ge photodiode and an amplifier composed by 3 MIS transistors had been successfully fabricated. Additionally, a Ge photodiode array (32 photodiodes, each 50x50 µm2 in size) was fabricated. In the photodiode array, double-layer metallization was used, and each of the photodiodes was provided with an additional metal electrode used to exert control over the surface potential. To produce GeO2 film on Ge several approaches were used. One of these was GeO film deposition for its subsequent oxidation. During this work, one very interesting fallout result was observed. The relatively thick (up to micron volume) GeO films contained effective thick film of germanium (according to IR-spectroscopy data GeO material was totally decomposited into Ge and GeO2) and therefore should be not transparent in visible spectral range. But it was surprisingly transparent, namely because quantum-size effect – blue shift of absorption edge in small Ge clusters. Unfortunately, there was no possibility to study the structure of the films that time, and also there was no possibility to study quantitatively the optical properties of the films. It becomes possible some later.
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2. Transmittance Spectroscopy, Raman and Photoluminescence Studies The following studies allow to gathering of evidence of quantum-size effect influence on optical properties of GeO2 films with Ge NCs. Strong blue-shift for absorption edge was observed for GeO2 film with Ge nanocrystals grown on sapphire substrate with the use of the first method (GeO decomposition). In that series of growth big amount of mono-crystalline Ge plates were placed in evaporation zone of growth reactor (up to 12 plates). The pressure of GeO gas was relatively high and the growth rate was relatively high (about 100 nm per minute). So, as it was mentioned above, the average size of Ge NCs was relatively small (lesser than 10 nm). The experimental transmission spectrum of an Ge:GeO2 film on sapphire substrate is shown in figure 3. Calculations of absorbance edge (taking into account influence of interference) were carried out, and one can see from figure 3, that the absorbance edge is about 650-700 nm. It is much less than the optical gap for bulk germanium (about 1700 nm). The PL measurement was also carried out for this sample. As one can see from figure 3 (down), the maximum of PL signal is close to the absorbance edge. One can assume that the absorbance and the emission of light originate from electron transfers in small Ge NCs, and the observed blue shift is due to quantum size effect [19]. If so, the size-dependence of PL signal should take place for GeO2 films containing Ge NCs of various sizes.
Figure 3. Spectra of transmittance (up) and photoluminescence (down) for GeO2 film with Ge nanocrystals grown on sapphire substrate.
Normalized PL intensity
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1-Al2O3 2-Ann1 2-Ann2
*1/3
1,4 1,6 1,8 2,0 2,2 2,4 2,6 2,8 3,0
Photon Energy (eV) Figure 4. PL spectra of Ge:GeO2 films with Ge NCs with different sizes. The film deposited from GeO (first method) and anomalous thick native germanium oxide films were studied.
To confirm that assumption, the PL spectroscopy studies of GeO2:Ge films with different sizes of Ge nanocrystals were carried out [20]. Pulse N2 (337 nm) lasers was used for pumping. Strong PL signals at room temperature were detected in GeO2 films with Ge NCs with sizes less than 3 nm. “Blue-shift” of PL maximum was observed with reduction of Ge NCs sizes (figure 4). The curves for films obtained using deposition from supersaturated GeO vapor (the first method) is marked by number 1 (sapphire was used as substrate, temperature of deposition was 500 oC). The curves for anomalous thick native germanium oxide layers (the second method) is marked by number 2. The curve marked as Ann1 corresponds to an anomalous thick native germanium oxide film annealed in rare gas atmosphere at temperature 550 oC, 15 minutes. The curve marked as Ann2 corresponds to a sample without thermal annealing. This sample also have PL signal in near IR region. Origination of this signal is not quite clear, probably it can be caused by radiative transmission related to defects or interface states. To obtain the value of average size of Ge NCs the analysis of Raman spectra was used. The Raman spectra of the samples are shown on figure 5. One can see the Raman peak due to scattering on optical phonons confined in Ge NCs in Raman of all samples. Some samples contain also a shoulder at 270-280 cm-1 due to signal from amorphous Ge. The arrow shows the position of Raman peak for not stressed volume Ge. The confinement effect shifts the position of Raman peak to low frequencies. The dot-and- dash curve corresponds to films
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grown in Si substrate with relatively low growth rate and contains Ge NCs with sizes about 69 nm. This (as-grown) sample has no PL signal, the reasons of that will be discussed below.
Normalized Raman intensity
1-Si 1-Al2O3 2-Ann1 2-Ann2
260
270
280
Bulk Ge
290
300
310
320
-1
Raman shift, cm
Figure 5. Raman spectra of deposited Ge:GeO2 films and anomalous thick native germanium oxide films. The PL spectra of some of them are shown in figure 4.
The NC sizes were estimated from position of Raman peaks [20, 21]. We use the model of efficient density of folded vibration states in NCs of spherical shapes. The model is described elsewhere [29, 30]. The Raman intensity can be written as: 2
q2r 4πq ⋅ exp(− 0 ) 1 4 dq ∫0 [n(ωi (q)) + 1] ⋅ Γ (ω − ω i (q)) 2 + ( ) 2 2 2
6
I (ω ) = A ⋅ ∑ i =1
n – Bose-Einstein factor, r0- nanocrystal radius.
(2)
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Intensity, arb. un.
Bulk Ge nc-Ge 10.0 nm nc-Ge 5.0 nm nc-Ge 3.0 nm nc-Ge 2.0 nm
280
290
300
310
-1
320
Raman shift, cm . Figure 6. Calculated Raman spectra of Ge NCs in frame of model of effective folding of vibration states.
The phonon dispersion for volume Ge was taken from low energy neutron scattering data [31]. Some calculated Raman spectra for Ge NCs with different diameters are shown in figure 6. The dependence of Raman main peak position from NCs diameter is shown in figure 7. For comparison, the calculated main peak positions are shown (circles). In this case the microscopical valence force field model was used [32]. It should be noted, that the Raman peak width is mainly defined by dispersion of NCs sizes. Especially the widening of Raman signal is seen for sample 1 deposited on sapphire substrate (figure 5). A correlation between reducing of the NC sizes and PL “blue-shift” was observed. The temperature of deposition for sample 1-Al2O3 (sapphire substrate) was 500 oC. According to our estimate from Raman spectrum, the average size of NCs in this case is about 2.6 nm. The temperature of deposition for sample 1-Si (silicon substrate, Raman peak is shown in figure 5) was 560 oC. According to our estimate, the average size of NCs in this case is about 6-9 nm. The NCs sizes formed in anomalous thick native germanium oxide layers by thermal annealing are smaller. The higher annealing temperature (sample 2-Ann1), the bigger NCs size. For the sample 2-Ann2 (without post-annealing) the average NCs size is about 1.2 nm. For that sample there is also PL signal in near-IR region (fig. 4). As it was already supposed, it can be connected with radiation transfers with participation of interface or defect electron levels, or exited levels in the NCs.
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Figure 7. Calculated Raman shift versus Ge-NC average diameter. Line - model of effective folding of vibration states, circles – literature data calculated from microscopic valence force field model [32].
Figure 8. A sample of HREM image (left) and electron diffraction pattern (right) of Ge nanocrystals.
The NCs presence was confirmed by HRTEM data (figure 8). The sample for HRTEM studies was deposited at a special transparence for electrons membrane in condition close to growth condition for sample 1-Si (Raman spectrum is shown in figure 5). As one can see, the
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average NCs size obtained from Raman data is in good correlation with direct HRTEM data (6-7 nm average size).
Figure 9. Raman spectra of deposited Ge:GeO2 films in optic (I) and acoustic (II) phonon region. The average size of Ge nanoparticles is A - 8 nm, B – 11 nm, C – 30 nm.
It should be noted the effects of acoustical phonon localization were observed for the first time for Ge:GeO2 hetero-films in work [18] (figure 9). From shift of Raman peaks in acoustical phonon region one also can obtain the average sizes of Ge NCs. The optical gap in Ge-NCs calculated with taking into account quantum size effects and the position of the experimental PL peaks are also correlated. The calculations of optical gap of Ge NCs taking into account quantum confinement effect in model of effective mass were carried out. The Shrödinger equation in the case of quantum well with spherical form is:
(
)
2m(r ) 1 ∂2 2 L( L + 1) Ψ (r ) + ( E − U ( r ))Ψ (r ) = 0 r Ψ (r ) − 2 2 2 =2 r ∂r r
(3)
and for the basic state (quantum number for momentum of pulse L=0) can be reduced to onedimensional equation with known solution [33]. Then, if one will measure energy in eV and radius of Ge NCs in nanometers, for infinite barriers, the energy gap between lowest electron level and highest hole level is:
Etr = E g +
4.1 r0
2
(4)
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Ψ=A
sin (α ⋅ r ) 2mE π , where α = = r = r0
(5)
E – the additional energy due to quantization for electrons and for holes. For our calculations we used electron effective masses for L-valley. Because electron are localized and electron wave functions with wave vector k both parallel and perpendicular to (111) direction are intermixed, one can used effective isotropic mass for electrons as: me=3me⊥me║/(2me║+me⊥). Using effective electron masses for L-valley (parallel and perpendicular to (111) direction - me⊥=0.082, me║=1.58 [34]) on can calculate me=0.123 (in free electron masses). The level for holes was calculated for heavy holes with mass mhh=0.33 [34].
Figure 10. Calculated optical gap in Ge NCs in GeO2 matrix with comparison to experimental PL and Raman data (crosses).
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The decision for finite energy of barriers can be derived from wave function matching in the interface. One can do not take into account the difference in effective masses in well and in barrier, and the boundary condition is:
Ψwell = Ψbar
(6)
If to take into account the difference in effective masses in well and in barrier, the additional boundary condition is: ' ' Ψwell Ψbar = m well m bar
(7)
where mwell and mbar – effective masses in well and in barrier consequently. We take the effective mass in barrier equal to mass of free electron. The equation 3 can be solved numerically. The results of calculations with comparing to experiment are shown in figure 10. As one can see, for NCs diameter 2.6 nm (from Raman data), the calculated optical gap for finite barriers is in very good agreement with experimental PL data. From obtained data it can be concluded that a Ge-NC in GeO2 matrix is a quantum dot of type I. Some deviation of calculated from experimental data for NCs of small sizes can be explained as result of dependence of effective masses for electrons and holes from NCs size. For such small NCs (sizes lesser than 2 nm), the simple effective mass approximation is not accurate. For more precise calculation “ab initio” methods should be used.
3. Thin (2D) Massifs of Ge NCs – HREM Studies For some applications (for example for creation of elements of quantum dots based quasinonvolatile MOS memory) one should obtain very thin (tunnelling thin) films with dense 2D massifs of semiconductor NCs. The NCs in dielectric films can be traps for electrons and holes, and many-bit memory cell based on such system can be realized. The density of NCs in such films should be 1012 per squared centimetre or higher. The first method (deposition from supersaturated GeO gas) allows to obtain very thin GeO2 films with dense array (massive) of Ge NCs. For example, in figure 11 a electron microscopy image of 7 nm thick Ge:GeO2 film is presented. One can see a 2D dense array of Ge quantum dots (NCs). For studying such films by HREM to obtain the density and the histogram of NCs sizes the special technology of membrane preparing was developed [35]. For this case, the Si/SiO2 substrate was used. The SiO2 film contain nano-pores, and selective etching was used to obtain membranes. Then, the Ge:GeO2 films were grown on the films with membranes, and can be easily transfer on special cop net for HREM studies. All technology is schematically shown in figure 12. This technology allows to carry out express analysis of the films using HREM. Some results are presented in figure 13. The electron diffraction data A and C were obtained from the different part of a film grown in condition of temperature gradient in deposition zone of reactor. HREM data (B and D) confirm that the higher temperature of deposition – the bigger average sizes of Ge NCs. The histograms of the Ge NCs sizes were obtained using the HREM data
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(figure 13, E and F). One can see that the density of Ge NCs can exceed the 1012 cm-2 (figure 13, E). For films grown in optimal conditions the Ge NC density can reach 1013 cm-2. Some advantages of our approach for using in flash memory elements were discussed upper and will be discussed bellow.
Figure 11. HREM image of a sample of dense packed Ge NCs (2D-layer).
Figure 12. а) Scheme of process of a cop set glue on SiO2/GeO2:(Ge-QDs) sample with a special membrane. On a cross-section one can see etching hole and a free film - membrane. B) The image of SiO2/GeO2:(Ge-QDs) structure: free two-layer film glued on a cop set with sell 70 × 210 micrometers. The thickness of layers SiO2/GeO2:(Ge-QDs) are 40/30 nm, correspondingly. C) Regular array of membranes in Si substrate.
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Figure 13. HREM images from Ge:GeO2 films grown in condition of temperature and GeO concentration gradient. Microscope JEM-4000EX, 250kV, plan-view configuration.
4. Elliposmetry Studies of Ge:GeO2 Films To study quantum-size effects for semiconductor QDs in dielectric matrix one need precise methods for control of structural, electro-physical and optical properties of the heterostructures. One need to know QD concentration, it’s average size, dispersion of sizes, and, for the case of multiphase structure – ratio of amorphous and crystalline parts. It is also needed to know how these structural properties impact on electrical and optical properties of the heterostructures. It is also desirable that such methods were express and non-destructive, what is possible mainly for non-direct methods. In present work we have tried to developed such methods, mainly based on ellipsometry and, for control, also Raman scattering spectroscopy and high resolution transmission electron microscopy (HRTEM) were used.
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c Figure 14. a) Ellipsometry data for GeOx:(H2O) films (falling ang.-70о, λo= 632,8 nm): a) 1(о) - without annealing; 2(х), 3(□), 4(∆) – after annealing 400, 525, 650 oC correspondingly; solid lines (___) – calculations: 1) - nλo= 1,58, kλo= 0,00 2) - nλo= 1,66 ± 0,01 kλo= 0,01 ± 0,005 3) nλo= 1,77 ± 0,02 kλo= 0,04 ± 0,005 4) nλo= 1,84 ±0,01 kλo= 0,11 ± 0,005; b) Ψ and Δ angles, corresponding to GeOx:(H2O) films (1), (2) и (3) in Figure 4, measured using spectral ellipsometer (for the λ=632,8 nm); c) Dependence of absorption coefficient versus photon energy for anomalous thick native germanium oxide film (Тann=650 oC, 15 min, sample 3), calculated from spectral ellipsomertry data.
The optical properties of films obtained using both first and second growth procedure were studied. Anomalous thick native germanium oxide (ATNGO) layers seem to represent a material of type GeOx:(H2O) in which Ge and O atoms are bound together by atomic bonds similar to those in solid GeO; this composition, however, is heavily hydrated and has a very
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loose structure. Anneals performed at Тann=200–650 С in He ambient during 15-20 min was found to affect the properties of the layers in close agreement with the expectations drawn from assumptions concerning the initial composition and structure of the material. The IR peak at 770 cm-1 gives way to a peak at 830 cm-1, close to the 860 cm-1 peak due to valence vibrations in thermal amorphous GeO2 [39], in line with the reaction 2GeO(solid)→Ge+GeO2. The process proceeds until being almost fully completed at all Тann. With increasing annealing temperature, the oxide shrinks dramatically, and all forms of bound moisture and OH groups rapidly disappear, partly, as supposed, due to the oxidation of the Ge segregating in the film [39]. The latter is evident from the IR absorption intensity reduction in the broad absorption band at 2800-3600 cm-1. The ellipsometry data obtained by means of ordinary with λo=632.8 nm (figure 14a) and spectral ellipsometry (see figure 14b) seem to agree with the above concept. Multi-thickness changes obtained when interpreting these data in the model of a single-layered film covering a Ge substrate showed the ATNGO layers to be layers uniform over their depth (d) [39]. The emergence of the additional light absorption in the heat-treated layers points to the nucleation of Ge NCs in them. As the diffusion processes in the oxide proceed with increasing Тann more vigorously, the mean particle size of Ge nanoparticles increases, causing, due to the size-quantization effect, an increase in kλo and nλo, the absorption and refractive indices of the layers. The optical constants of anomalous thick native germanium oxide layers in the spectral range λ=260-650 nm were extracted from their spectra of ellipsometric parameters within the framework of the Bruggemann model [36, 37], assuming the material to be a mixture of Ge and GeO2 (both amorphous; their optical constants were borrowed from reference sources). The values of gGeO2 , gGe (the volume contents of GeO2 and Ge) and d, representing adjustable parameters, were chosen so that to obtain a best fit of the predicted spectral dependences of ellipsometric parameters to measured spectra. Three films of anomalous thick native germanium oxide were studied. We will refer these films as 1, 2 and 3 (figure 14). According to Raman data the average size of Ge NCs was lesser for samples 2 and 3 (analog of sample Ann2 in figure 5), than for sample 1 (analog of sample Ann1 in figure 5). The thicknesses of three films 1-3 were found to fall into the range 75 to 90 nm, in close agreement with the data obtained from multi-thickness measurements, while the values of gGe were found to be 15 % (1), 14 % (2) and 9 % (3). Making allowance for the reduction in k(E) (E is the photon energy) due to the size-quantization effect in Ge nanoclusters is expected to yield greater values of gGe. With d layers installed, we calculated the spectral dependences of k(λ) and n(λ) for these layers, numerically solving, in succession for all values of λ, the nonlinear equation tg Ψλ ⋅ exp( iΔ λ ) = f (λ , n, k ) . It is clearly seen from the dependence k(E) calculated for one of the films (figure 14c) that the total absorption in the layer increases with increasing E, exhibiting three maxima. The broad maximum at 3.7 eV coincides with the absorption maximum of amorphous Ge [38], while the narrower maximum at E=4.37 eV is presumably - due to the fundamental-edge absorption in GeO2. Another absorption peak is centered at E=2.28 eV; this peak has a half-width of 0.12 eV and can be attributed to the Ge quantum dots. Similar maxima in the same spectral regions were also displayed by the k(E) dependences of other two anomalous thick native germanium oxide layers. It should be noted, that according to IRspectroscopy data, the GeOx(H2O) anomalous thick native germanium oxide layers have composition parameter x close to 1 [39]. The observed underestimating of x parameter in spectral ellipsometry data supposed to be the result of quantum size effects. In the frame of
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Penn model [40], the quantum confinement led to more tight binding of electrons and, consequently, to reduction of polarizability and dielectric constant. So, without taking into account this effect, the Bruggemann model leads to underestimating of x parameter. The above mentioned approach was applied for study of Ge:GeO2 films deposited from GeO vapor (first method) in conditions of gradient of substrate temperature and GeO concentration. To find the optimal regimes of growth of a heterostructure with QDs it is often needed to carry out the growth in gradient of temperature and concentration of reagent gases. In this case, thickness and optical parameters of the grown film have smooth gradient. Let’s choose some trajectory (l) on our structure. Scanning ellipsometry gives a set of ellipsometry angles Ψ(li), Δ(li). These functions are smooth and it’s derivative are also smooth. The using of theory of ellipsometry allow us to calculate from these angles the dependences of refractive index, absorption index and thickness: n(l), k(l) и h(l). For our growth condition, the dependences of n(l) and k(l) are results of changes of QDs density (ρ (l)), sizes (D (l)), and phase composition (γ (l)) due to temperature and GeO concentration gradient along l. The thickness of film h(l) also depends on growth condition gradient. Knowing the dependence of optical parameters from ρ (l), D (l), and γ (l), one can try to solve inverse problem and to find these functions. The main interest is how the optical constant of heterofilm depends on the QDs sizes – the quantum size effect. Therefore, the n(l) and k(l) should be defined with maximal possible accuracy.
Figure 15. Raman spectra from different parts of a Ge:GeO2 film grown in condition of temperature and GeO concentration gradient. The line along Raman data were measured is coincide with line along ellipsometry data were obtained.
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Usually, there is a problem to define volume ratio between QDs and matrix in heterostructures. For example, it is difficult to find volume ratio of Si NCs formed by Si+ ions implantation and consequent annealing for further definition of optical properties of composite Si/SiO2 films from Bruggemann formulae [41]. As it was already mentioned, our heterostructure have remarkable property – due to (2GeO(solid) →Ge+GeO2) growth procedure it has molar ratio between Ge and GeO2 as 1:1, independently on growth condition [21, 35, 42]. So, from Ge and GeO2 densities one can define, that the volume part of c-Ge or a-Ge clusters is ~ 30,7%. Therefore, the concentration of Ge QDs and average distance between QDs depends only on the QDs average size. Some deviation in volume ratio can be due to difference in mass density of c-Ge and a-Ge. The ratio c-Ge/a-Ge can be measured from Raman spectra (figure 15). The Raman spectra were registered from different part of the Ge:GeO2 film grown in condition of temperature and GeO concentration gradient. The Ge:GeO2 heterofilm which Raman spectra were shown in figure 15 was studied using scanning ellipsometry with step δl=0,5mm and the results were interpreted using above described method and specially developed algorithm. As one can see from figure 16, the ellipsometry angles Ψ(li) and Δ(li) are depended on n(l), k(l) and h(l) in the film. It should be mentioned, that ellipsometric data were registered at wavelength 632.8 nm (He-Ne laser). In our model the experimentally measured ellipsometry angles Ψ(li) and Δ(li) are in very good correspondence in theoretically calculated angles - Psi and Delta in figure 16, upper case. The thickness of film smoothly reduced in direction of flow of gas because of depletion of GeO vapor. Consequently, the Ge nanocluster size and ratio of c-Ge/a-Ge also reduce in that direction. As one can see from figure 16, the functions n(l), k(l) and h(l) are smooth, and there is some plateau in the middle of the film. The data obtained from ellipsometry are in good correspondence with Raman and HREM data (figures 13 and 15). The line along which the ellipsometry data were obtained corresponds to line along which the Raman data were measured. 360 320 280
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Figure 16. Ellipsometric nomogramm and dependence of optical constant in the Ge:GeO2 film grown in condition of temperature and GeO concentration gradient. Upper case – experimental and calculated nomogram; middle case – dependence of the film thickness along line of GeO gas flow; bottom case – dependence of optical constants of the film along the same line.
The analysis of experimental and calculated ellipsometry data reveal the following: – Reducing of h(l) with reducing of substrate temperature and direction of flow of the GeO gas. It is because depletion of GeO gas in growth zone. – The absorption of light (632.8 nm) by our heterostructures is result of presence of Ge clusters. Reducing of absorption index k(l) simultaneously with reducing of h(l) in direction of gas flow is the influence of quantum size effect. The smaller Ge particles is more transparent. This is in good correlation with HREM data (figure 13). With reducing of average Ge cluster size, the part of amorphous Ge particle grow (comparing with c-Ge clusters). It is also confirmed by Raman data (figure 15) data. Comparing deposit in
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absorption index of quantum size effect and phase composition effect one can conclude that for this case the first effect is more important. - Because of for wavelength 632.8 nm amorphous Ge have bigger refractive index n than crystalline Ge, some growth of n in direction of gas flow was observed. - The smooth plateau in functions n(l), k(l) and h(l) in middle of the film (despite of temperature gradient) show the importance of technological factors in controllable variation of structural properties of the films. As it was already mentioned, spectral dependence of refraction and absorption coefficients from diameter of Ge nanoclusters n(D) and k(D) can give useful information about the quantum size effect influence. But usually, the accuracy of measurement of n and k is not enough to reveal this influence for all spectral diapasons. Original special approach for many-angle, many-thickness spectral (250-800 nm) scanning ellipsometry was developed. The developed mathematical approach allows measuring the ellipsometry data with an order of magnitude better accuracy, comparing in the case of measuring in one point (not scanned by wedge). The accuracy worsens if the ellipsometry nomogramms pass close to each other. But changing the wavelength one can shift in the area of good accuracy. On the film with different thickness h(l) but almost identical n(l) and k(l) one can chose the area of good accuracy for spectral measurements for any given spectral region. In this area the spectral ellipsometry angles ψ(h) and Δ(h) can be measured with high accuracy, and, consequently, one can get n and k with high accuracy for all spectral diapasons.
Figure 17. Spectral dependence of refraction and absorption indexes for c-Ge, a-Ge, GeO2 and Ge:GeO2 films.
Knowing spectral dependencies of n(l) and k(l) for pure GeO2 (media a) and for crystal Ge (c-Ge) and amorphous Ge (a-Ge) (media b) (which are shown in figure 17) and also volume ratio of Ge (fb Ge~30,7 %), one can calculate from Bruggeman model [36, 37] n(l) and k(l) for Ge:GeO2 hetero-films:
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0 = fa
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In our case fa=0.693, fb=0.307. One can calculate real and imaginary parts of dielectric permittivity ε from n and k. The difference in calculated (for a-Ge) and experimentally measured data is easily seen (figure 18) and is supposed to be due to quantum size effect, because in Bruggemann model this effect is not taken into account. Due to enhanced accuracy it was possible to find the influence of quantum size effects in Ge QDs on optical properties of hetero-films. The difference is qualitative clear, the experimental absorption in long-wave region is lower than the calculated absorption. Even for amorphous Ge nanoclusters, the quantum size effects lead to broadening of optical gap in absorbance. For quantitative analysis the well known theoretical models, for example Penn model [40] are planned to use in future.
Figure 18. Comparative analysis of experimental optical constants of a Ge:GeO2 film and calculated optical constants using Bruggemann model without taking into account the quantum size effects.
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5. Studies of Annealed Ge:GeO2 Films It is known, that thermal annealing leads to transformation of structural properties of the semiconductor NCs / dielectric matrix heterostructures. The annealing can vary NCs sizes, its phase composition, transform defects in the heterosystem. The Ge:GeO2 film grown with the use of the first method (deposition from supersaturated geO gas) was annealed in highvacuum quartz tube with a tubular oven. The details of annealing is described in section “experimental” of present work.
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Figure 19 presents the IR absorption spectrum of initial Ge:GeO2 film in range from 700 to 1200 cm-1. The observed peak is assigned to Ge-O-Ge stretching vibration mode. The peak position is about 870-880 cm-1, so, one can assume, that even initial film (deposited from supersaturated GeO vapor at substrate temperature about 500-550 oC) is totally dissociated on Ge and GeO2. It is known, that in GeOx films this peaks approximately linearly shift from about 820-830 cm-1 for x=1 to about 870-880 cm-1 for x=2 [22, 28, 43]. The Raman spectra of the initial and annealed films are shown in figure 20. One can see the Raman peak due to scattering on optical phonons confined in Ge NCs in Raman spectra. It is known, that Raman signal from amorphous Ge is a broad peak with maximum at 270-280 cm-1. The Raman signal from crystalline Ge is narrower peak with position depending on NCs sizes. The confinement effect shifts the position of Raman peak to low frequencies from bulkGe Raman peak position (about 300.3-300.5 cm-1). So, from figure 20 one can see that in initial Ge:GeO2 film practically all Ge clusters were crystalline. The NCs sizes can be estimated from position of Raman peaks [20, 32]. We use the above mentioned model of efficient density of folded vibration states in NCs of spherical shapes [29, 30]. According to dependence of Raman peak position from NCs diameter, the diameter of Ge NCs is about 6-8 nm. It should be noted, that the Raman peak width is mainly defined by dispersion of NCs sizes, and the size dispersion in our case is relatively small, but we could not quantitatively define this dispersion. One can see very small shift of Raman peaks with annealing (thin line is drawn for convenient). It can be caused by some ordering of Ge-GeO2 interface and also by annealing of defects in Ge NCs. According to theoretical calculations of quantum confinement effects in Ge NCs with the sizes 6-8 nm, the maximum of PL signal should be in range 1.0-0.8 eV [44]. Our simple effective-mass approach model [20] also gives similar values. So, to study the PL signal we used IR-region PL setup with monochromator equipped with a 600 grooves/mm grating and InAs cooled photodiode as a detector. He-Ne (633 nm) and He-Cd (325 nm) cw lasers with power about 30 mW were used for excitation. For study PL in visible diapason we used other PL setups – one equipped by mercury lamp as an excitation source and silicon based CCD line as a detector and second with Ar+ and N2 lasers for excitation and photomultiplier tube as a detector. The PL spectra in visible and IR diapason of the initial and annealed films are shown in figure 21. In region near 1.1 eV at temperature 4 K we saw only peaks due to recombination in free and bound excitons in Si substrate, because our Ge:GeO2 film with thickness about 50 nm was semi-transparent even in UV diapason. In diapason 0.8-1.0 eV we have observed two peaks (see insert in figure 21). But intensity of this peaks increase with decreasing of thickness of our initial films and also was more intensive in the case of He-Ne laser excitation comparing with He-Cd laser excitation. So, one can assume, that these peaks have origination from silicon substrate. And really, we ascertained that our silicon substrates contain dislocations, which can be irradiative recombination centers, and position of our peaks good coincide with well-known D1 (1523 nm) and D2 (1420 nm) peaks [45] (see inset in figure 21, PL spectrum of initial Ge:GeO2 film was excited by He-Cd laser). To grow our film on clear ”fresh” surface we used silicon substrates with thick thermal silicon dioxide (used as a cap layer), which we had removed by HF acid immediately before the film deposition. Obviously, the thermal oxidation of silicon at very high temperatures had created the dislocations in our substrates. The PL signal from dislocation at low temperatures is relatively high, so we could not see PL signal from Ge NCs even if we had any. The PL peaks D1 and D2 have some
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different temperature dependence but both were diminished to zero at temperatures higher than 40 K. And at temperatures 40 K and higher we had not observed PL signal (which can be associated with Ge NCs) in IR diapason.
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According to theoretical calculations, radiative lifetime in Ge NCs of spherical shape with diameter 6-8 nm should be quite large – about 10-2 second [44]. So, if we have notradiative recombination centers with much shorter lifetime, the radiative recombination would be negligible. In Ge NCs such centers can be defects inside NCs, interface defects and molecules of water. It is known, that GeO2 films and low temperature porous SiO2 films can absorb H2O molecules and evaporate it during thermal annealings [46]. Using the thermal annealing we have tried to anneal defects, to evaporate water and to order the Ge NCs – GeO2 matrix interface. To avoid the evaporation of Ge:GeO2 films its was covered by cap layer from 50 nm thick SiO2. Using PL setup equipped with mercury lamp we have observed some week PL signal in green region, but the lines of mercury lamp do not allow us to study it accurately. So, we used Ar+ laser excitation and results are presented in figure 21. After annealings the green-red PL signal appears (the blue-green 488 nm Ar+ laser line was used for excitation). The power of this cw Ar laser line was 200 mW. The spot about 200 micrometers, so the density of excitation was about 5·1012 W/m2. One can see in spectra small peak from 514.5 nm Ar+ laser line (marked by triangle). When we used the last line for excitation, similar PL signal was observed. It should be noted that the intensity of signal was 3-5 times lower comparing with PL signal from reference sample – 75 nm thick SiO2 films with Si NCs with sizes about 5 nm (similar samples were studied in work [6] – PL signal in 750-780 nm was observed). When we used for excitation UV laser (N2, wavelength 337 nm) the signals
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from Ge:GeO2 films were very low, several orders less than signal from the reference sample. It is also important to mention, that we have reference Si/SiO2 sample (the SiO2 film was deposited on Si substrate without Ge:GeO2 film in the same process – “side-by-side”) and we have no PL signal for this sample. If the observed PL signal (2-2.4 eV) was from quasi-direct excitons in Ge NCs, the sizes of NCs should be about 2 nm [44]. As it was mentioned above, according to our Raman data and also according to direct electron microscopy data for samples grown in similar conditions [26, 35], average diameter of ge NCs in our case is 6-8 nm. So, one can assume, that we have PL assisted with some defects in GeO2 matrix or Ge-GeO2 interface, or some other excitations in Ge NCs. However, GeO2 film obtained by Ge oxidation (without Ge NCs) has no PL in this region. Paying attention to band structure of bulk Ge (for example figure 1 in work [44]), one can see that energy of direct transition between L valleys for electron and holes is about 2.1 eV. From this energy of photons the abrupt growth of light absorbance take place. In the case excitation of bulk Ge, the holes relax very fast to Γ-valley emitting acoustical and optical phonons through continuous energy band states. In NCs electron (and phonon) states is discrete, not continuous, and such relaxation can be not very fast, and part of holes can radiative recombine with electrons through direct transitions in L valleys. Also, because the average quasi-pulse of electron and holes in NCs is zero (localized, standing waves), the 4 L valleys is “folded” and electrons and holes from all that valleys can recombine, so probability of such processes can be compared with probabilities of notradiative transitions. It can be presumable explanation of observed PL signal in our case. Some shift (compared to bulk Ge L1-L3’ transition energy) can be due to quantum-size effects. The broad PL peak can be due to Ge NCs size dispersion, and also due to complex energy structure of discrete electron states in relatively big Ge NCs. It should be noted, that at temperature 600oC and higher the intermixing of SiO2 and GeO2 films can begin with forming of GexSi(1-x)O2 thin layer and origination of PL signal from this layer can be also possible. In further studies, the proposed mechanism of PL should be tested using the GeO2 films with bigger and smaller Ge NCs. The similar PL signal in visible range was observed also for Ge NCs obtained using laser pulse deposition [47], but author of that work have not presented data concerning sizes of the Ge NCs.
6. Some Aspects of “Band Gap Engineering” of Ge:GeO2 Based Films and Its Possible Applications The developed technology of deposition Ge:GeO2 based films has several possibilities for structural changes. Because electrical and optical properties of dielectric films with semiconductor NCs depend on its structure, the controllable modification of its structural parameters is in frame of well-known approach – “band gap engineering”. For, example, for modifying the structural parameters of the Ge:GeO2 based films one can use: (i) In situ modification of the Ge NC size by varying the temperature and rate of condensation of germanium monoxide when synthesizing the film; (ii) Ex situ controllable increase in the Ge nanocrystal size and in the width of the barriers between the nanocrystals by annealing in an inert atmosphere (without changing the Ge:GeO2
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ratio in the film) to provide the diffusion-limited transformation of small Ge clusters into larger ones; (iii) Controllable decrease of the Ge NC sizes by oxidation (with all of the treatments performed below 600°C); (iv) Variations in the height of the potential barriers of quantum wells in the heterosystem using chemical and phase transformations of the constituent components; in particular, the fraction x in germanium silicate glass GeO2(x):SiO2(1–x) can be changed in the range from unity to zero, simultaneously resulting in near-linear changes in the optical and electronic parameters of the dielectric [48]; in this case, the band gap Eg increases from ~5.5 to ~9 eV; (v) The use of glasses of complex composition and their crystallization at low temperatures (below 750°C [49]) to modify the properties of the dielectric barriers; it is worth noting that, on crystallization, the permittivity of the films increases by an average of ~1520%, and this is favorable for quasi-nonvolatile MOS memory devices based on the trapping of charge carriers by QDs. (vi) The possibility to vary, in the heterosystem of QDs in GeO2 (amorph), barrier parameters and the dielectric constant of the insulating matrix by organizing a proper GeO2Si3N4 intermixing. Such composite glasses are close in their physical and chemical properties to thermally grown SiO2; at Т>650-750 oC these solutions may crystallize, giving rise to laterally uniform and even coatings. The system offers considerable potentialities in optoelectronics and can be employed for fabricating quasi-non-volatile MOS memory elements where Ge NCs are used as electron or hole traps. The elements for flash memory based on Ge NCs formed in GeO2(0.4):SiO2(0.6) plasma enhanced CVD deposited glass were fabricated recently [50]. Our method of deposition is fully compatible with silicon based MOSFET technology. Moreover, with sufficiently high electroluminescence intensity achieved, these layers will enable fabrication of light-emitting elements on Ge whose functioning can be controlled by a Ge-MOS transistor IC. Unlike the case of silicon substrates, such elements can be obtained without depositing on the substrate additional layers, like GaAs-based epitaxial layers or simpler layers such as SiNx(H) layer enriched with silicon. The non-stability and hysteresis effect in conductivity of the Ge:GeO2 films grown on sapphire substrate reveal that the charge and discharge of Ge nanoclusters take place in process of electrical transport. The International Technology Roadmap for Semiconductors (ITRS) states that thin equivalent oxide thicknesses (EOTs) <0.5 nm will be required by 2016 to sustain gate terminal control of the charge carrier in the semiconductor channel. The density of tunneling current through SiO2 with such thicknesses is too big and new dielectrics with high dielectric constants (high-K dielectrics) should be used. The new high-K dielectric films are not naturally connected with “God made” SiO2-Si hetero-pair, so the interest to Ge-based MOSFET technology rises from the oblivion last years [51]. The main advantages of Ge compared with Si are the higher charge carrier mobility (caused by little effective masses of electrons and holes for Ge) and well match of Ge and GaAs lattice constants (what gives possibility to integrate in one chip the very fast electrical and opto-electronics devices). So, the problem of formation of good quality Ge/high-K dielectric interface is actual. As in the case of Si/high-K dielectric, the very thin native oxide transmission layer should be needed, and consequently, the problem of enhancement of quality Ge/Ge oxide (GeO2 or GeO) is also actual. The typical interface state density for Ge/high-K dielectric hetero-system is 1011-1012
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cm-2 now [51]. Because GeO2 is not chemically stable, absorb H2O and its defect system is not stable that cause the non stability of electrical characteristics, it becames convinced that it is double-layered film compositions like GeO2/Si3N4 or GeO2/SiO2 are more preferable for FET applications on Ge substrates. Amorphous SiO2 and GeO2 are widely known to be the main glassy materials readily dissolvable in each other in arbitrary proportions at temperatures above ~700o. But it remains practically unknown that, through thermal mutual dissolution of GeO2 and SiO2 films, one can obtain a homogeneous GeO2:SiO2 glassy system more dense and much more resistant to chemical agents than either of the two initial films. Indeed, we found that at temperatures T>700oC GeO2 and Si3N4 films may enter the reaction 3GeO2+Si3N4 -> Ge3N4+3SiO2 [48]. The resulting GeO2:SiO2:Ge3N4 glassy film possesses remarkably high chemical stability. The specific electronic feature of these films is the variability of the parameters Eg , Ec and Ev in them, depending on the particular content of the constituent substances. This feature is a result of different energy band-gap parameters of GeO2, SiO2, and Ge3N4. (Eg in GeO2 and Ge3N4 is ~5-5.5 eV, whereas Eg in SiO2 is ~9 eV). It was found that the compound glassy films on Si and Ge substrates undergo crystallization at T>750oC, this crystallization offers an additional means for controllable variation of Eg, Ec and Ev. The described chemical and structural modifications in the glassy films present a very attractive way in band-gap engineering of quantum-dot systems primarily due to the temperature range of these processes, which is mostly below 900-1000oC. Finally, based on the short review of our origin results and also known from literature results, one can conclude that heterosystems with Ge quantum dots in amorphous GeO2 is quite a new and almost unexplored research area in the field of dielectric films on solid substrates, and developed a new method for synthesis of such films is perspective for future applications.
Figure 22. IR spectra of initial GeO2-SiO2 film, this film after annealing and consequent etching.
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Some examples of formation of GexSi(1-x)O2 and GeO2-Si3N4 films are reported bellow. The GeO2 films were grown on Ge substrate by oxidizing in oxygen flow at temperature 600 oC. Then 100 nm thick GeO2 film was covered by 50 nm thick pyrolytic SiO2 layer. The IR spectrum (curve 1) of initial hetero-film is shown in figure 22. The spectrum contain 2 peaks related to absorbance by valence vibrations of Si-O (1060-1070 cm-1) and Ge-O (860880 cm-1) bonds. Then the film was annealed 3.5 hours at temperature 750 oC in He atmosphere. As on can see, in the spectrum of annealed film the Ge-O peak have transformed to Si-O-Ge peak (1000-1020 cm-1) and subsequent removing of parts of the films (curves 3-8) do not lead to radical changes in IR spectra. So, one can assume, that annealing leads to intermixing of the initial GeO2-SiO2 hetero-film in homogeneous GexSi(1-x)O2 film. The results of measurements of rate of chemical etching of the film (figure 23) confirm this supposition. The buffer etchant was used, and the thickness of residual film was measured using ellipsometry technique. As on can see from figure 23, on first stage of etching the etching rate is equal to etching rate of pyrolytic SiO2 (curve 2 is very close to dots on curve 3). One can assume, that the top of the film remain SiO2. But surprisingly, the etching rate of the GexSi(1-x)O2 film is lower than etching rate of pyrolytic SiO2. As for GeO2 film, it should be momentary removed even by water (curve 1), never mind the buffer etchant. It should be noted, that the GexSi(1-x)O2 film was amorphous (but in our supposition it is in pre-crystalline phase), the etching rate of crystalline GexSi(1-x)O2 film should be much lower. So, the possibility of forming of chemically durable based GeO2-SiO2 based films were demonstrated.
Figure 23. Thickness of annealed GeO2-SiO2 film versus time of etching. Curve 1 – the dependence for GeO2 film, curve 2 – the dependence for SiO2 film, curve 3- the experimental dependence for GexSi(1x)O2 film.
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Figure 24. IR spectra of initial GeO2-Si3N4 film, this film after annealing and consequent etching.
The similar results on intermixing of GeO2-Si3N4 films were also obtained (figure 24). The 100 nm thick GeO2 film on Ge substrate was covered by 40 nm thick Si3N4 layer. This layer was deposited from ammonolysis of NH3-SiH4 gas mixture diluted by Ar at temperature 700 oC. The IR spectrum (curve 1) of initial hetero-film is shown in figure 24. One can see broad band caused by absorption of valence vibrations of Si-N (about 800-900 cm-1) and GeO (860-880 cm-1) bonds. These peaks are not resolved in spectrum. Annealing leads to structural transformation revealed in IR spectrum (curve 2). One can see that additional peak at 1065 cm-1 have appeared, and main peak at 865 cm-1 have shifted to 845 cm-1. The first peak can be only due to absorbance by valence vibrations of Si-O (1060-1070 cm-1). So, one can assume the next reaction take place [52]: 3GeO2 +Si3N4 → 3SiO2 + Ge3N4
(9)
For comparison the IR spectra of amorphous (curve 1) and poly-crystalline (curve 2) Ge3N4 films on Ge substrate [53] are shown in figure 25. Amorphous film is characterized by broad asymmetric band with maxima at 750 cm-1. The crystalline Ge3N4 film has narrower peaks with peculiarities at 910, 830, 770, and 740 cm-1.
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Figure 25. IR spectra of amorphous (curve 1) and poly-crystalline (curve 2) Ge3N4 films on Ge substrate. Data from work 53.
Because the initial GeO2 film was relatively thick, obviously, the obtained film is not pure Ge3N4 but germanium oxinitride. The dependence of IR spectra transformations on etching (figure 24) show that the film is quite homogeneous. The low-frequency shift of absorbance peak is caused by transformation of GeO2 into GeO2-Ge3N4 mixture film. After etching, the absorbance peak due to valence vibrations of Si-O diapers, so, one can assume, that Si-O bonds present only in upper part of the film. The lower part of film is germanium oxinitride with absorbance band in range 750-870 cm-1 (curves 7-11, figure 24).
Figure 26. Changing of IR spectra of Ge-GeO2 structure during annealing in NH3 atmosphere at temperature 750 oC. Time of annealing: curve 1 – initial film; curve 2 – 10 minutes; curve 3 – 30 minutes; curve 4 – 40 minutes; curve 5 – 50 minutes.
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The germanium nitride and germanium oxinitride films can be obtained also using another method – annealing in NH3 atmosphere. The chemical reaction of GeO2 and nitrogen with formation of GeOxNy films is confirmed by IR data (figure 26). One can see that initial peak related to absorbance by valence vibrations of Ge-O (870 cm-1) bonds during annealing shifts to lower frequencies. This fact is consequence of growth of nitrogen in GeOxNy films. The similar behavior was observed for SiOxNy films when the part of silicon nitride grow in silicon oxinitride films [54]. So, one can assume that the structure of GeOxNy and SiOxNy films is very similar. As one can see, even after 50 minutes annealing (figure 26, curve 5) the IR absorbance band maximum is about 780 cm-1, what is not correspond to pure amorphous Ge3N4 (figure 25, curve 1), so, the film is germanium oxinitride.
Figure 27. Dependence of thickness of the film in structure Ge-GeO2-Si3N4 from etching time (HF etchant): curve l – initial structure Ge-GeO2(25nm)-Si3N4(100nm); curve 2 – the structure GeGeO2(25nm)-Si3N4(100nm) after 1 hour annealing at 700 oC; 3 – hypothesized dependence of thickness of the structure Ge-GeO2(25nm)-Si3N4(100nm) from etching time without taking into account the interaction between germanium oxide and silicon nitride.
Such kind of films is also chemically durable and has relatively big dielectric constant. This fact is confirmed by data shown in figure 27. It should be noted, that the structure with relatively thin (25 nm) GeO2 film and relatively thick (100 nm)-Si3N4 film was annealed and studied. The thickness of films were controlled using ellipsometry data. As one can see from figure 27, the rate of etching of annealed film is lower than that of initial film. Moreover, some interaction take place even on stage of film growth. It is seen from difference of etching rate of initial film (curve 1) and hypothesized etching rate for Ge-GeO2(25nm)-Si3N4(100nm) structure without taking into account the interaction between germanium oxide and silicon nitride (curve 3). One can also see some not homogeneous behavior of etching rate of annealed structure (curve 2). Obviously, the upper part of the structure contains more chemically durable Ge3N4, the lower part contains more GeO2. Summarize these data, one can conclude, that the possibility of formation of chemically durable germanium oxinitride films was demonstrated.
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In final part of this section it should be mentioned about possibility of crystallization of films of thermal GeO2 films. Usually, the films of thermal GeO2 are evaporated during annealings in noble gas atmosphere, according to next reaction: Ge (solid) +GeO2 (solid) → 2GeO(gas)
(10)
Figure 28. Crystallization of thermal oxides on silicon and germanium in various conditions: a – growth of spherulites of α-cristobalite in Si/SiO2 structure (annealing 1200 оС, 16 hours, Ar atmosphere); bgrowth of spherulites of α-quartz in Ge/GeO2 structure (annealing 730 оС, 6 minutes, Ar+O2 atmosphere); c- homogenious film from grown texture GeO2(hexagonal) in Ge/GeO2 structure (annealing 670 оС, 15 minutes, Ar+O2 atmosphere).
The approaches to suppress the evaporation were successfully developed, that makes it possible to study the processes of crystallization. The processes of GeO2 film growth, crystallization and evaporation were detailed studied in work [55]. The micro-images (optical microscope with Nomarskiy contrast) are shown in figure 28. One can see growth of spherulites of α-cristobalite in Si/SiO2 structure (figure 28a – data from book [56]), growth of spherulites of α-quartz in Ge/GeO2 structure (figure 28b), and homogenious textured film of GeO2(hexagonal) in Ge/GeO2 structure (figure 28c). The crystallization of thermal GeO2 films begins with nucleation of flat spherulites of low-quartz. Then, the overgrown spherulites form texture with <001> axes directed to normal of the film. Distances between crystallographic planes in polycrystalline GeO2(hex) were taken from handbook [57] and were compared with distances between crystallographic planes for our studied samples (Table 1). From these data obtained using electron diffraction one can conclude that our crystallized GeO2 films have hexagonal structure. Activation energy for phase transformation GeO2(amorphous) – GeO2(hex) is about 2.2 kilocalories per mole. It should be noted, that for crystallization of SiO2 films the high temperature (1200 oC) annealing with big thermal budget (16 hours) is needed. The GeO2 films can be crystallize at relatively low temperatures during several minutes. Moreover, the thermally grown GeO2 films can spontaneously crystallize at room temperature – as one can see from figure 29. The spontaneous crystallization is enhanced by influence of water in atmosphere. The presence of heterogeneous structures (at substrate and into film) also takes place and can influence on the crystallization process. The amorphous thermally GeO2 films grown on not polished
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substrates crystallize faster than the films grown on polished surfaces. The structure of spontaneously crystallized GeO2 films is more complex than the structure of the films crystallized by annealings. Usually, these are agglomerates of micro-dispersed poly-crystals or poly-crystals .of dendrite types (figure 29). Such films is solvable by water (approximately 200 nm thick film can be solved during 25 minutes) but it is much slower comparing with amorphous GeO2 films. It is worth to pay attention that GeO2 films crystallized by thermal annealings have device quality. They durable to solution both water and buffer etchant. The temperature dependence of crystallization rate in thermal GeO2 films is shown in figure 30. One can see that activation energy for crystallization is about 0.5 eV for this case.
Figure 29. Crystallization of films of thermal germanium oxide during keeping in air atmosphere. a) – agglomerates of micro-disperse crystallites, the film was kept in air atmosphere 4 months after synthesis on planar side of Ge substrate; b) – flat crystallites with dendrite form, the primarily amorphous GeO2 film was grown on polished Ge substrate and was kept in air atmosphere 2 years after synthesis.
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Table 1. Distances between crystallographic planes in polycrystalline GeO2(hex) and the studied samples (Å) The studied polycrystalline GeO2 №
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Polycrystalline GeO2(hex) data from [57]
4,31 3,41 2,48 3,35 2,28 2,15 2,00 1,87 1,71 1,62 1,56 1,495 1,445 1,410 1,386 1,339 1,301 1,248 1,228
Room temperature crystallization 4,3 3,37 2,46 2,37 2,25 2,13 2,00 1,86 1,73 1,70 1,55 1,49 -1,40 1,33 1,23
Crystallization by annealings Planar side
Not planar side
4,36 3,41 2,55 2,18 2,04 1,914 1,70 1,56 1,46 1,43
4,32 3,41 2,52 2,35 2,28 2,14 2,01 1,86 1,68 1,66 1,55 1,50 1,41 1,39 1,35 1,29 1,24 1,23
1,39 1,36 1,23
Figure 30. The temperature dependence of crystallization rate in thermal GeO2 films – regime of growth of flat spherulites of low-quartz-like phase in Ge/GeO2 structure.
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Figure 31. Crystallization of GeO2 based films with complex chemical composition. a) crystallized GeO2-SiO2 film of (α-quartz) – entire film consisted from grown spherulites on Si(111) (parameters of GeO2:SiO2 structure ~ 105:40 nm, time of annealing 3 hours, annealing temperature 750 oC); b) crystallization of GeO2:Ge3N4 structures on Ge(111) substrate (parameters of GeO2/Si3N4-structure 30:100 nm, time of annealing 5 hours, annealing temperature 750 oC). The cracks below amorphous Si3N4 are seen.
The GeO2 based films with complex chemical composition can be also crystallized at relatively low temperatures. The micro-images (optical microscope with Nomarskiy contrast) of such complex structures are shown in figure 31. One can see in figure 31a grown spherulites of crystallized GeO2-SiO2 film. The homogeneous GeO2-Ge3N4 film with cracks below amorphous Si3N4 are seen in figure 31b. In both systems crystallization process is similar to growth of flat spherulites of GeO2(hex) phase in thermal GeO2 films The
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GeO2:SiO2:Ge3N4 layer on Ge substrate (figure 31b) was formed under annealing due to interaction (intermixing) of thin layer of thermal GeO2 (~20 nm) with deposited (~100 nm) Si3N4 layer. Not intermixed part of Si3N4 layer cover the mixed GeO2:SiO2:Ge3N4 layer. Consolidations in Si3N4 layer during the annealing leads to its cracking, the edges of the cracks were centers of nucleation for crystallization of 3-component glass GeO2:SiO2:Ge3N4. Note, that material of not crystallized GeO2:SiO2 and GeO2:SiO2:Ge3N4 layers (the part of films behind the front of crystallization spreading from the cracks and heterogeneous formations) is kept in pre-crystallized phase very long time. This phase has glass structure (without far ordering) but characterized by low defectness. The films in pre-crystallized phase are more dense and more chemically durable than initial amorphous films. Noteworthy, that crystal space lattices of GeO2(hex), GeO2:SiO2(hex) GeO2:SiO2:Ge3N4(hex) films are isomorphic to low-quarts, so the appearance of piezoelectric effects is quite possible in these films. It can be interesting for device applications. It is also notable, that the lattice parameters (lattice constants) of these crystallized films can be controllable varied by changing of its chemical composition. It can be interesting for epitaxy of various films on these films with matching of needed lattice constant. Finally, the possibility of relatively low-temperature crystallization of GeO2 based film was demonstrated.
Summary So, the optical properties of GeO2 films with Ge nanocrystals were studied, the influence of quantum size effect on optical gap of such heterostructure was observed. The structural parameters of Ge NCs in GeO2 films can be controllable changed both during deposition (in situ) and using post-deposition treatments. The use of scanning ellipsometry methods (with analytical interpretation and calculations based on Bruggeman model) consequently with using of HREM and Raman data allows us to obtain experimental evidence of the quantumsize effect on optical properties of heterostructure Ge:GeO2. Broadly speaking, the developed methods can be applied for study of others heterostructures like semiconductor QDs in dielectric matrix. The optical properties of initial and annealed GeO2 films with Ge NCs with average sizes 6-8 nm (according to Raman data) were studied, the green-red PL was observed for annealed in high vacuum samples. The annealings can improve structure of the Ge NCs and the interface. The mechanism of direct radiative transitions for confined in the NCs electrons and holes (it’s wave functions are originated from L valleys of bulk Ge) is proposed for explanation of experimental effect. It is supposed that “band gap engineering” approaches can lead to creation of Ge:GeO2 heterostructure with required properties. The possibility of relatively low-temperature crystallization of dielectric GeO2 based film was demonstrated. This heterostructore can be perspective for using in opto-electronics, for creation of elements of quasi-nonvolatile MOS memory using Ge nanocrystals as traps for electrons or holes, e.t.c.
Acknowledgements The Russian Fund for Basic Researches supports this study (project № 07-08-00438). Authors are thankful to D.V. Marin for help in PL and ellipsometry studies, to A.N. Borisov for help in spectral ellipsometry measurements, to Dr. A.K. Gutakovskiy and A.G. Cherkov for
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HREM studies, to Dr. S.V. Golod for help in membrane preparation. Some PL and IR measurements were made in Laboratoire de Physique des Materiaux, Universite de Nancy, and V.A. Volodin and E.B. Gorokhov are grateful to Professor Michel Vergnat for visit grant.
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In: Quantum Dots: Research, Technology and Applications ISBN 978-1-60456-930-8 c 2008 Nova Science Publishers, Inc. Editor: Randolf W. Knoss, pp. 371-392
Chapter 10
M ODEL FOR THE C OHERENT O PTICAL M ANIPULATION OF A S INGLE S PIN S TATE IN A C HARGED Q UANTUM D OT Gabriela M. Slavcheva University of Surrey, Guilford, Surrey,UK
Abstract The optically-driven coherent dynamics associated with the single-shot initialization and readout of a localized spin in a charged semiconductor quantum dot embedded in a realistic structure is studied theoretically using a new Maxwell-pseudospin model. Generalized pseudospin master equation is derived for description of the time evolution of spin coherences and spin populations in terms of the real state pseudospin (coherence) vector including dissipation in the system through spin relaxation processes. The equation is solved in the time domain self-consistently with the vector Maxwell equations for the optical wave propagation coupled to it via macroscopic medium polarization. Using the model the long-lived electron spin coherence left behind a single resonant ultrashort optical excitation of the electron-trion transition in a charged QD is simulated in the low- and high-intensity Rabi oscillations regime. Signatures of the polarized photoluminescence (PPL) resulting from the numerical simulations, such as the appearance of a second echo pulse following the excitation and a characteristic non-monotonic PPL trace shape, specific for initial spin-up orientation, are discussed for realization of high-fidelity schemes for coherent readout of a single spin polarization state.
1.
Introduction
One of the challenges for the development of future technologies is the realization of next generation devices that control not only the electron charge, as in contemporary state-ofthe-art electronics, but also its spin degree of freedom. Among the huge variety of proposed physical realizations of a solid-state qubit and respective schemes to realize quantum computing based on it, a single electron (hole) spin confined in a charged quantum dot (QD) [1], [2] has recently attracted significant attention. This proposal is particularly
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Figure 1. Schematic representation of the initial electron states in a single singly-charged quantum dot (a,b) and the ground trion state created from it by σ − (c) and σ + (d) excitation of the heavy-hole excitonic transitions. (e) Energy-level diagram of a negatively charged exciton (trion) state in a single quantum dot. The levels are labeled by the total angular momentum projection along the propagation and quantization z-axis. Dipole-allowed optical hω0 ; Coherent optitransitions correspond to ∆Jz = ±1 and the fundamental energy gap is ¯ cal transitions excited by left (σ − ) - and right(σ + )-circularly polarized light are designated by solid arrows, spontaneous optical transitions with rate Γ (dashed arrows), transitions due to electron and hole spin-relaxation γ1 and γ3 , respectively; spin decoherence rates for electrons γ2 and holes Γτ . promising due to a combination of advantages: It enables the implementation of ultrafast all-optically based quantum computation schemes through the optical spin orientation mechanism using circularly polarized ultrashort optical pulses, a technique of key importance for generation and manipulation of spin-polarized states in semiconductors [3]. This, in turn is a prerequisite for realization of quantum coherent control of the spin since a necessary condition for quantum coherence is the use of sufficiently short optical pulses so they can interact with the quantum system before it can be affected by its environment. On the other hand, QDs have the advantage of being intermediate between ordinary atoms and the higher-dimensional semiconductor structures characterized by complex many-body physics. Owing to the 3D-carrier localization and the discrete-level electronic structure the electron spin confined in a QD is far less affected by the semiconductor environment, compared to the QWs and bulk, resulting in relatively long spin-coherence times. Spin states in QDs have been initially studied by measuring the average signal from a large ensemble of QDs. Long ensemble spin coherence time T2∗ ∼ 100 ns in bulk semiconductors have been reported [4]. Recent experiments on single-shot readout of an individual spin in GaAs and In(Ga)As single QDs have demonstrated long single electron spin energy relaxation times (spin-flip time) T1 in the ms range (1-20 ms) [5]. Since T2 may last as long as 2T1 [6], long single spin coherence times are anticipated. This has been recently confirmed by the extended single spin coherence times T2 > 1 µs found in a double QD in the high-intensity coherent regime [7] employing the techniques of quantum coherent
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control, such as spin-echo pulse sequences, to suppress hyperfine-induced dephasing, due to interactions with the nuclear spins of the lattice ions. In this regime of resonant nonlinear coherent light-matter interactions Rabi oscillations of the population between the discrete levels occur. In this respect QDs are advantageous to the ordinary atoms since population flopping over many periods is possible for them, being systems with longer coherence times and larger dipole moments. Spin decoherence times T2 on the order of magnitude of 1 − 100 µs [8] are theoretically predicted. The long spin relaxation and spin decoherence times recently observed ensure a long-lived quantum state required for performing a large number of spin manipulations (quantum operations) during which coherence needs to be retained. On the other hand, owing to the angular momentum conservation, the polarization state of photons (the so called flying qubit) can be converted into a localized spin (stationary qubit) and the process is reversible. This has direct implications for the transport of qubits from one location to another and allows for building up scalable architectures, in contrast with proposals based on conditional exciton dynamics (see e.g. [9]) which have problems with scalability. One promising approach is to optically address individual carrier spins in semiconductor QDs and to manipulate them through optically excited states (charged excitons) by employing the techniques of coherent quantum control and optical orientation.There is strong evidence from both recent experimental data [7], [21], and theoretical computations [15], [19] that the intense resonant excitation of the trion transition suppresses the electron spin relaxation due to hyperfine interaction with nuclear spins of the lattice ions which limits the ability to accurately measure the electron spin orientation at low temperatures. On the other hand, Rabi oscillations in QDs correspond to one-qubit rotation and therefore represent a step towards the implementation of quantum information processing (QIP) in QDs. Despite the recent progress achieved in coherent optical spin state preparation with near-unity fidelity, using laser cooling (optical pumping) techniques [10], single-spin qubit detection in solid-state systems has only been achieved using transport measurements [11], magnetic resonance force microscopy [12], and most recently off-resonant optical Faraday rotation [13], and the all-optical preparation, manipulation and detection still remains a challenging task. A diversity of theoretical schemes of how the charged exciton (trion) permits the readout of a single electron spin during the spin relaxation time [14, 15], or using spin-flip Raman transitions [17], [19] have been discussed. The most widely applied model for coherent optical control of a localized electron spin is based on master equation for the density matrix. The system Hamiltonian is in dipole approximation in the four-level basis describing the trion state [15], or Luttinger-type Hamiltonian with negligible heavy-hole -light-hole mixing in the rotating wave approximation (RWA) [19, 23]. An alternative method based on the spin-density matrix (pseudospin formalism) [3, 16] has been applied to describe the spin dynamics after initialization by a short pulse [15, 21]. Theoretical models using solely the density matrix, without taking into account the optical wave propagation and radiationmatter interactions, are limited by approximations and generally not valid for ultrashort pulses. The proposed approach is based on a self-consistent solution of originally derived semiclassical coherent Maxwell-master pseudospin equations, thereby taking into account the optical wave polarization, propagation within device boundaries, spontaneous emission,
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spin relaxation and decoherence beyond the RWA and the slowly-varying envelope (SVEA) approximations. A major drawback of the approaches applied up to now is their inability to describe the system dynamics taking into account the real macroscopic boundary conditions and the shape of the guided/cavity modes. It is clear that the non-uniformity of the spatial profile of the modes and the local spatio-temporal dynamics will be crucial for the physical realization of coherent optical control. A serious advantage of the proposed method is that it provides nonperturbative (with respect to laser-dot interaction) description of the coherent nonlinear dynamics and allows for the treatment of arbitrarily-shaped pulses. The purpose of this work is to investigate the fundamental problem of the optical initialization and detection of a single electron spin confined in a negatively charged QD embedded in realistic device geometries (e.g. nonlinear optical waveguides, semiconductor microcavities) by employing our new model, towards the realization of basic quantum logic operations. The results from our theory predict two novel distinct signatures in the simulated dynamics specific to the initial spin-up polarized state: the appearance of a photon echo pulse, and non-monotonicity of the PL as a function of time. These permit a unique determination of the resident electron initial spin orientation and can be exploited in high-fidelity single-shot initialization/readout optical experimental setups.
2. 2.1.
Dynamical Model Theoretical Background
We shall study quantum dots charged with one single electron. Injection of a single electron into a quantum dot can be achieved by using e.g. modulation doping in the barrier region adjusting the impurity doping level within the delta-doped layer to transfer on average one electron per dot and to populate only their lowest states, or by electrical injection. We focus on a model of lens-shaped quantum dots with lateral dimensions largely exceeding the height that is sufficiently general to represent a wide class of zero-dimensional systems (e.g. self-assembled InAs QDs grown by molecular beam epitaxy, GaAs natural island-like QDs, formed by GaAs quantum well thickness fluctuations, or nanocrystal CdSe QDs) . Due to the quasicylindrical symmetry of the QD about the quantization axis z, the singleparticle states of the electrons in the conduction band and the holes in the valence band can be approximated by those of a pair of harmonic oscillators [20] (Fig.1(a,b)). The ground electron, heavy hole and light hole states are degenerate with respect to their total angular momentum projections ±1/2, ±3/2, and ±1/2, respectively. We assume that the lowest heavy hole state (with total angular momentum projection Jz = ±3/2) and light hole state ( Jz = ±1/2 ) are split by an energy ∆hh−lh and the band mixing of heavy hole and light hole states is negligible. In what follows we shall consider a resonant circularly polarized optical excitation that is restricted to the heavy hole states only. The electromagnetic wave incident to the singly-charged quantum dot is propagating along the quantization direction z and is circularly polarized in a plane perpendicular to it. A single electron in the lowest orbital state is either in the spin-up or spin-down ground state (Fig.1 (a,b)). The resonant circularly polarized optical excitation of the charged dot leads to the formation of a negatively charged exciton (trion) consisting of two electrons sitting at the same lowest (conduction band) electron quantum level, forming a spin singlet state, and a hole occupying the lowest
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valence band hole level. According to the exciton optical selection rules, which reflect the angular momentum conservation along z-axis, σ − -polarized light couples | − 3/2i heavy hole states and | + 1/2i (spin-up) electron state with total angular momentum projection -1 (Fig. 1 (c)), whereas σ + light couples | + 3/2i heavy hole state with | − 1/2i electron state with total angular momentum projection +1 (Fig.1 (d)). It should be noted that the ground trion state optically created by adding σ + (σ − )- exciton to the single electron in the dot can decay spontaneously only into the same initial electron state emitting circularly polarized light with the same polarization as the one of the stimulated resonant coherent excitation. Therefore the polarization of the photoluminescence uniquely determines the initial electron spin projection ( σ + polarized photoluminescence implies electron spin-up |↑i projection, whereas σ − polarized photoluminescence determines electron spin-down |↓i projection). As it has been pointed out [15] the selection rules permit many-cycle repetitions without loosing information about the electron spin state and the electron and the trion spin relaxation limits the measurement time of the photoluminescence with a given polarization. It is obvious that the longer the spin lifetime is the the better are the chances to optically manipulate a particular spin state and to read out the state of a single electron spin during the spin-relaxation time. Therefore we shall be interested in the spin dynamics at low temperatures since in this regime the localized electron has a long spin coherence time. It has been pointed out [22] that the most-likely dominant mechanism of electron spin relaxation in quantum dots at low temperatures is the hyperfine interaction with the frozen configuration of randomly oriented spins of the lattice nuclei. There are strong indications, however, that in the strong excitation regime the electron spin relaxation in the nuclear field is suppressed by the intense resonant optical excitation through a mechanism similar to ”motional (dynamical) narrowing [3], [15]. In this paper we investigate both the low- and the high-intensity nonlinear excitation regimes. We have adopted the usual statistical interpretation of quantum mechanics that uses the ensemble point of view. In this view the state vector or density matrix describes not a single system but an ensemble of identically prepared systems. Therefore, we shall model the resonant nonlinearity associated with the coherent optical trion transitions in a single charged quantum dot by an ensemble of multi-level systems (degenerate 4-level systems (Fig. 1(e)) with density Na obtained by replication of the single dot system. It should be noted that the individual members of the ensemble are independent: there is no interaction between the replicas within the ensemble and there is no transfer of population outside each replica, so that the conservation of the total occupation probability within each replica is preserved (ρ11 + ρ22 + ρ33 + ρ44 = 1) at all times. The initial assumption is that at t = 0 we create (by e.g. optical pumping with circularly polarized light) simultaneously electrons in the lowest orbital state, all having the same spin orientation. The systems have exactly the same occupation probability distributions under the driving laser field, so that all possible configurations (realizations) are going to be equivalent and the ensemble averaging procedure is straightforward. The justification of such an approach is based on the assumptions of validity of the ergodic hypothesis, namely the equivalence of the time averages of an observable (in this case the single-dot polarized photoluminescence, determined by the dynamics) and the ensemble average, that is the average at one time over a large number of systems all of which have identical properties. In what follows, we apply proper normalization of the dot density in order to ensure a single dot within the simulation domain and show
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Table 1. Nonvanishing components of f-tensor (174 nonvanishing elements grouped in 29 i,j,k permutations) ijk fijk ijk fijk ijk fijk ijk fijk ijk fijk
1, 2, 9 1, 3, 8 1, 4, 11 1, 5, 10 1, 7, 13 2, 3, 7 2, 5, 12 1/2 1/2 1/2 1/2 1 −1/2 1/2 2, 6, 11 2, 8, 13 2, 8,.14 3, 4, 12 3, 6, 10 3, 9, 13 3, 9,.14 √ √ 1/2 −1/2 3 2 1/2 1/2 1/2 3 2 4, 5, 7 4, 6, 9 4, 10, 13 4,.10, 14 4, 10, 15 5, 6, 8 5, 11, 13 √ √ .√ 1/2 1/2 1/2 1 2 3 2 3 1/2 −1/2 5,.11, 14 5, 11, 15 6, 12, 6, 12, .√14 √ .√15 7, 8, 9 7, 10, 11 8, 11, 12 √ √ .√ 1 2 3 2 3 −1 3 2 3 −1/2 1/2 1/2 9, 10, 12 1/2
that there are strong indications that the ergodicity holds in the problem considered. Under these conditions, it has been demonstrated [18] that quantum mechanics allows predictions of single systems based on macroscopic time averages of observables. Therefore we shall assume that the time-dependence of the optically-induced coherent spin generation and subsequent relaxation in a single quantum dot, averaged over a large number of successive measurements is equivalent to the corresponding spin dynamics of an ensemble of identical quantum dots. The ensemble of homogeneously broadened degenerate four-level systems is resonantly coupled to an optical wave propagating along the z - direction, circularly polarized in a perpendicular plane to z . During its propagation, the optical wave interacts with the four-level medium schematically shown in Fig.1, thereby inducing polarizations Px and Py along the x and y directions, respectively. The one-dimensional Maxwell’s curl equations in an isotropic medium read: ∂Hx (z,t) ∂t ∂Hy (z,t) ∂t ∂Ex (z,t) ∂t ∂Ey (z,t) ∂t
= = = =
1 ∂Ey (z,t) µ ∂z − µ1 ∂Ex∂z(z,t) ∂Hy (z,t) − 1ε ∂z − 1ε ∂Px∂t(z,t) ∂P (z,t) 1 ∂Hx (z,t) − 1ε y∂t ε ∂z
(1)
The time evolution of a discrete 4-level system in an external time-dependent dipolecoupling perturbation is governed by the Liouville equation of motion for the complex density matrix ρˆ: i h ˆi ∂ ρˆ = ρˆ, H (2) ∂t h ¯ ˆ is the system Hamiltonian in the 4-level basis (Fig.1 (e)), constructed phewhere H nomenologically assigning complex Rabi frequencies to the coherent optical dipole-allowed transitions in the quantum system [34], [35]. Assuming dipole-coupling perturbation, the ˆ 0 (in the absence of lightsystem Hamiltonian is a sum of the unperturbed Hamiltonian H ˆ int : matter interaction) and the dipole interaction perturbation Hamiltonian H ˆ int (t) ˆ ˆ0 + H H(t) =H
(3)
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with the dipole interaction Hamiltonian given by: ~ Q ˆ ˆ int (t) = −eE. H
(4)
ˆ being the local displacement operator. and Q Since the initial electron spin can be either in spin-down state |1i, or spin-up state |3i, the corresponding Hamiltonian for each case is given by:
ˆ1 = ¯ H h
0 − 12 (Ωx − iΩy ) 1 ω0 − 2 (Ωx + iΩy ) 0 0 0 0 0
0 0 0 0
0 0 0 ω0
0 0 0 0 0 ω0 0 0 − 12 (Ωx + iΩy ) ω0 0 − 12 (Ωx − iΩy )
0 ˆ2 = ¯ h H 0
0
(5)
where ω0 is the resonant electron-trion transition frequency. The Rabi frequencies are defined according to: ℘ ℘ (6) Ωx = Ex ; Ωy = Ey h ¯ h ¯ where ℘ is the trion optical dipole transition matrix element. The dynamics associated with the spin-flip (longitudinal) and spin-decoherence (transverse) relaxation processes and the trion radiative decay via spontaneous emission need to be taken into account in Eq.(2). Spin-flip relaxation of the electron spin due to hyperfine interaction with the lattice ions nuclear spin induces population transfer between the lower-lying levels, while the hole spin relaxation in the trion state through phonon-assisted processes induces population transfer between the upper-lying levels (Fig.1 (e)) [15]. For the sake of generality we shall denote the electron spin-down population transfer rate via spin-flip processes from level |1i to level |3i by γ13 and the opposite transition spin-up population transfer rate by γ31, similarly the trion state hole spin population transfer rates between level |2i and |4i will be denoted by γ24 and γ42, respectively. Electron spin decoherence, trion spin decoherence and spontaneous emission rate are denoted by γ2, Γτ , and Γ, respectively. Writing down the rate equations for the population change of each level, the following explicit equations for the diagonal components of the density matrix are obtained: ∂ρ11 ∂t
=
i ¯ h
∂ρ22 ∂t
=
i ¯ h
∂ρ33 ∂t
=
i h ¯
∂ρ44 ∂t
=
i ¯ h
h
ˆ ρˆ, H
h
ˆ ρˆ, H
h
ˆ ρˆ, H
h
ˆ ρˆ, H
i 11
i 22
i 33
i 44
− γ13ρ11 + γ31ρ33 + Γρ22 − γ24ρ22 + γ42ρ44 − Γρ22 (7) + γ13ρ11 − γ31ρ33 + Γρ44 + γ24ρ22 − γ42ρ44 − Γρ44
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The above equations can be unified with the explicit equations for the off-diagonal density matrix elements, describing the spin decoherence, resulting in the following generalized master equation for the density matrix ρ: ˆ i h ˆi ∂ ρˆ ˆ t ρˆ = ρˆ, H + σ ˆ−Γ ∂t ¯ h
(8)
where we have introduced σ ˆ matrix accounting for the spin population transfer, according to:
σ ˆ=
ˆ 1ρˆ) T r(Γ 0 0 0 ˆ 0 0 0 T r(Γ2ρˆ) ˆ 3ρˆ) 0 0 0 T r(Γ ˆ 4ρ) ˆ 0 0 0 T r(Γ
(9)
defining the matrices:
Γ1 =
Γ3 =
−γ13 0 0 0 Γ 0 0 0 γ31 0 0 0 γ13 0 0 0
0 0 0 0
0 0 0 0 0 0 0 −γ31 0 0 0 Γ
; Γ2 =
; Γ4 =
0 0 0 −Γ − γ24 0 0 0 0 0 0 0 γ24 0 0 0 0
0 0 0 0 0 0 0 γ42
0 0 0 0 0 0 0 −Γ − γ42
(10)
ˆ t accounts for the dissipation in the system due to and the spin decoherence matrix Γ loss of spin coherence:
ˆt = Γ
0 Γτ γ2 γ2
Γτ 0 Γτ Γτ
γ2 Γτ 0 Γτ
γ2 Γτ Γτ 0
(11)
Note that due to symmetry relationships ( ρij = ρ∗ji ) there are only 10 independent components of the density matrix. Employing the generalized pseudospin formalism based on the commutator Lie algebra of SU(N) group [24], [25] we expand the density matrix and the system Hamiltonian of a discrete 4-level quantum system in terms of the λ-generators of SU(4) algebra, calculated using the definitions given in [24]. A possible choice of the SU(4) λ-generators which satisfy the commutation relation ˆ k ] = 2ifjkl λ ˆl ˆj , λ [λ
(12)
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is given by:
0 1 ˆ λ1 = 0 0
0 0 ˆ λ4 = 0 1
0 0 0 0
0 0 0 0
1 0 ˆ λ5 = 0 0
ˆ 13 λ
0 0 ˆ λ = 0 2 0
0 −i ˆ7 = λ 0 0 0 0 = 0 −i
0 0 0 0
ˆ 10 λ
1 0 0 0
−1 0 = 0 0
i 0 0 0
0 0 0 0
0 1 0 0
0 0 ˆ λ = 0 3 0
0 0 0 0
0 0 0 1
0 0 0 0
0 0 0 0
0 0 0 0
i 0 ˆ λ = 0 11 0
0 1 0 0
0 0 0 0
0 0 ˆ λ14 = 0 0
0 0 1 0
0 0 0 ˆ 0 λ8 = 0 0 0 0
0 0 0 0
0 0 −i 0 0 0 0 0
0 0 0 −i
−1 0 1 √ 3 0 0
0 0 1 0
0 0 0 0
1 0 0 0
0 0 0 0
0 1 ˆ λ6 = 0 0
0 0 0 0
0 0 0 0
0 0 0 1
0 0 1 0
0 0 0 ˆ 0 λ9 = 0 −i 0 0
0 0 0 0
i 0 0 0
0 i 0 0 0 0 0 0
0 −1 0 0
0 i ˆ λ = 0 12 0 0 0 2 0
0 0 0 0
0 0 ˆ λ15 = 0 0
0 0 0 0
0 0 0 0 0 0 0 −i
−1 0 1 √ 6 0 1
(13)
0 0 i 0 0 −1 0 0
0 0 −1 0
0 0 0 3
Substituting back in (2), a pseudospin master equation for the real 15-dimensional state vector is derived, governing the stimulated dynamics, spin-flip relaxation and spin decoherence: 1 1 ˆ f Γ S + T r σ ˆ λ j − Tj (Sj − Sje ) , jkl k l 2 ∂Sj (14) = j = 1, 2, ..., 12 ∂t f Γ S + 1Tr σ ˆ ˆ λj , j = 13, 14, 15 jkl k l 2 where Γ is the torque vector, f is the fully antisymmetric tensor of the structure constants of SU(4) group, whose non-vanishing components are listed in Table I, and Tj are the phenomenologically introduced non-uniform spin decoherence times describing the relaxation of the real state vector components Sj toward their equilibrium values Sje . The matrix σ in (14) is the diagonal matrix from Eq. (9), expressed in terms of the real state vector components, as follows: σ11 σ22 σ33 σ44
√ √ 1 = 12 − −3 + √ 6S15 (Γ − γ13 + γ31 ) + 2√ 3S14(−Γ + γ13 + 2γ31) 6S13 (Γ + γ13) √ 1 3 + 6S13 − 2 3S14 = − 12 ) + 14 1 + 6S √ − 6S15 (Γ + γ42√ √15 γ42 1 γ γ = 12 3Γ + 3 − 6S13 − 2 3S − 3 + 4 3S + 6S15 (3Γ − γ13 + γ31 ) 14 13 14 31 √ √ √ 1 = 12 3 + 6S13 − 2 3S14 − 6S15 γ24 − 3 1 + 6S15 (Γ + γ42) (15)
Expressions for the longitudinal spin population relaxation times T13, T14, T15 due to both spin relaxation processes and spontaneous emission have been derived through the second term in (14), giving:
T13 =
4 12 6 ; T14 = ; T15 = 2Γ + γ13 + γ24 3γ13 + γ24 + 6γ31 γ24 + 3 (Γ + γ42)
(16)
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From the equations of motion for the off-diagonal density matrix components, using the relationship between the real state vector components and the density matrix components, the following spin decoherence times are obtained, namely: T1 = T2 = T5 = T6 = T7 = T8 = T11 = T12 = 1/Γτ , T3 = T4 = T9 = T10 = 1/γ2. The torque vector Γ components are expressed in terms of the λ-generators of the SU(4) Lie algebra according to [24]. Using the Hamiltonian (5) the torque vector is calculated for the two initial spin orientations considered, giving correspondingly:
ωo , Γ1 = −Ωx , 0, 0, 0, 0, 0, −Ωy, 0, 0, 0, 0, 0, ωo, − √ 3
Γ2 =
q
2 ωo q3 ωo 0, 0, 0, 0, 0, −Ωx, 0, 0, 0, 0, 0, −Ωy, ωo, − √ , 23 ωo 3
(17)
Substituting back in (14) taking into account all non-vanishing components of the ftensor, a system describing the time-evolution of the 15-dimensional state vector is obtained, which in matrix form reads:
ˆ − diag (1/T1, 1/T2, ..., 1/T12) (S − S∗ ) ˆ S + 1Tr σ ˆλ ∂S M E 2 = 1 ˆ ˆ ∂t ˆλ M S + 2Tr σ
(18)
∗ = (S , S , S , ..., S where SE 1e 2e 3e 12e) denotes a subset of the equilibrium vector SE = (0, 0, 0, ..., S13e, S14e, S15e) responsible for the spin decoherence. Note that the first 12 components of SE vanish because of the incoherent nature of the energy input that maintains the system at a definite level of excitation [26] and only the population terms components are nonzero. In the above equation M is 15 × 15 antisymmetric block matrix with 13 independent components, given by:
P 6×6 Q6×6 R6×3 T ˆ M = −Q6×6 P 6×6 S 6×3 −RT3×6 −S T3×6 03×3
(19)
and the block matrices have the following explicit form given for initial spin-down populated level (the derivation for initial spin-up populated level is similar):
ˆ P =
ˆ= R
0 0 0 0 0 0 Ωy 0 0 0 0 0
0 0 Ωy 2
0 0 0 0 0 0 0 0 0
0 Ω − 2y 0 0 0 0
0 0 0 0 − Ω2y 0
0 0 0 0 0 0
ˆ S =
0 0 0 Ωy 2
0 0 −Ωx 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0
ω0 0 0 Q ˆ= 0 0
0
0 0 0 0 0 0
0 −ω0 Ωx 2
0 0 0
0 Ωx 2
0 0 0 0
0 0 0 ω0 Ωx 2
0
0 0 0 Ωx 2
0 0
0 0 0 0 0 ω0
(20)
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For the special case of a two-level system the Hamiltonian is 2 × 2 and the λ-generators of the SU(2) Lie algebra are simply the Pauli matrices. In this case the pseudospin system is reduced to the one derived by Feynman et al. [27]. During its propagation in the resonantly absorbing or amplifying medium the polarized optical pulse induces polarization along x- and y-axes, perpendicular to the propagation direction. The macroscopic medium polarization is given by: ˆ (21) P = −eNa T r ρˆ.Q where Na is the density of the ensemble of resonantly absorbing/amplifying four-level systems. Taking into account the form of the interaction Hamiltonian (5), expanding the local ˆ components and the density matrix in terms of λ-generators of displacement operator Q SU(4) Lie algebra [25], the Cartesian polarization components are expressed in terms of the real pseudospin (state) vector components: Px = −℘Na S1 Py = −℘Na S7
(22)
The above polarizations act as source terms in the vector Maxwell’s equation for the optical wave propagation (1). Maxwell curl equations (1) are solved self-consistently with the master pseudospin equations (18), using (22), for the fields and the real-vector components in the time domain employing the Finite-Difference Time-Domain (FDTD) technique [36] without invoking any approximations, such as slowly-varying wave approximation (SVEA) and rotatingwave approximation (RWA). The initial boundary value problem requires the knowledge of the whole time history of the initial field along some characteristic, e.g. at z=0 (the left boundary of our simulation domain). The circularly-polarized optical pulse is modeled by two orthogonal linearly polarized optical waves, phase-shifted by π/2 [33]- [35]:
σ
−
σ+
( (
Ex (z = 0, t) = Eosech (10Γ) cos(ωo t) Ey (z = 0, t) = −E0 sech (10Γ) sin(ωo t) (23) Ex (z = 0, t) = Eosech (10Γ) cos(ωo t) Ey (z = 0, t) = E0 sech (10Γ) sin(ωo t)
where E0 is the initial field amplitude, Γ = [t − (Tp)]/(Tp/2) and Tp is the pulse duration.
2.2.
FDTD Numerical Implementation
The system under investigation is a GaAs/AlGaAs self-assembled modulation-doped MBEgrown QD with 5 nm height (see e.g. [29]) sandwiched between two 50 nm Al0.3Ga0.7As barriers with refractive indices nGaAs = 3.63 and nAlGaAs = 3.46 at the trion transition resonance wavelength (λ = 787 nm) (Fig. 2). The circularly polarized pulse center frequency ω0 = 2.39 × 1015 rad.s−1 is tuned in resonance with the energy splitting between the ground electron and the singlet trion states of 1.58 eV [29], corresponding to a wavelength λ = 787 nm and the pulse shape is given by a hyperbolic secant envelope
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Gabriela Slavcheva
(23) with duration Tp = 1.3 ps. Throughout the simulations the initial field amplitude is varied from a low-field E0 = 550 Vm−1, through E0 = 3 × 106 Vm−1 , representing a π-pulse, completely inverting the spin population, to E0 = 4 × 107 Vm−1 , corresponding to a ∼ 12π-pulse,inducing 6 full Rabi flops. An estimate for the trion dipole moment is obtained using the exciton Bohr radius, giving a value of ℘ = 4.8 × 10−28 Cm. The active simulation domain is 5 nm long, corresponding to a typical volume of a cylindrical quantum dot (with diameter d=10 nm and height h=5 nm) of 3.93 × 10−25m−3 . The quantum dot (resonant 4-level system) density Na = 2.5 × 1024 m−3 is selected to give on average one dot within this microscopic volume. Therefore, although the volume density is a very large number, the simulations are restricted to this particular spatial domain which contains only a single dot. The full-wave vector Maxwell’s equations coupled to the time evolution equations of the degenerate four-level quantum system are discretized on a Yee-grid with fine discretization ˚ in order to resolve the QD, which implies a timestep ∆t = 3.336 × in space ∆z = 1 A, −4 10 fs through the Courant stability condition and solved numerically in the time domain using the FDTD method with predictor-corrector iterative scheme [33]. We start to propagate a source circularly-polarized pulse through the left boundary of the structure and monitor the time evolution of the E-field components and populations of all four levels at a point within the QD (Figs. 3-7), or monitor their spatial distribution across the structure at different time moments (Fig. 2). We shall be interested in the low-temperature regime. Transitions between the lowerlying initial electron levels occur due to the hyperfine interaction of the electron spin with the frozen random configuration of the nuclear spins of the lattice ions [15], [22]. In what follows, we shall assume that the electron spin population transfer rates between the lowerlying energy levels are equal, namely γ13 = γ31 = γ1 and the hole spin population transfer rates between the upper-lying levels are also equal, namely γ24 = γ42 = γ3. Throughout the simulations the following parameters are kept constant: the electron spin relaxation rate due to electron spin precession in the frozen random configuration of lattice ions nuclear spins is set to γ1 = 500 ps−1 as a lower limit of the theoretical rates calculated in [8], [22]. Transitions between the upper-lying levels also occur due to the holespin relaxation in the trion state which is a phonon assisted process and yields relaxation rates that can be comparable to, or even longer than those for electrons [30], [31], [32] with relaxation rate γ3 = 170 ps−1 in agreement with [21]. The trion radiative decay (recombination) rate is set to Γ = 400 ps−1 as inferred from time-resolved photoluminescence [21]. The electron spin decoherence (transverse) rate is taken to be γ2 = 450 ps−1 (see e.g. [23]), and the trion state spin decoherence rate is assumed to be Γτ ∼ 2γ3 = 340 ps−1 . In Fig.2 a snapshot of the spatial distribution of the E-field components and the spin populations of all four levels across the structure is given for a 2π σ − -polarized pulse driving the system through a full Rabi flop of the spin population. In Fig. 2 (a) the spin population is still almost entirely residing in level |1i. Fig. 2 (b) shows depletion of level |1i and accumulation in level |2i with negligible population transfer to the second two-level system at a later time. In Fig. 2 (c) population inversion occurs in the first two-level system and the population transferred to the second two-level system increases. We shall be interested in the long-lived spin coherence left behind an ultrashort pulse when the excitation intensity is varied. The time evolution of the left-circularly polarized
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pulse Cartesian components Ex and Ey and the spin population of all four levels at a point within the QD is given in Fig. 3 at the initial field amplitude E0 = 550 Vm−1 . The initial spin orientation is assumed to be spin-down (state |1i), taking the initial state of the system to be its equilibrium state (Fig.3(a)). The left-circularly ( σ − ) polarized ultrashort optical pulse initially slowly excites the spin population in agreement with the optical dipole selection rules. As the time elapses, the propagating resonant pulse experiences amplification in the QD-medium and the amplified pulse excites nearly completely the spin population into the upper level |2i. Note that the pulse split-up is due to the transfer of spin population between the lower- and upper-lying levels which creates polarization density causing re-emission back to the field. The evolution of the population ρ22 of state |2i (blue curve) describes the spin population of | − 3/2i trion state excited by σ − - polarized pulse, and therefore represents a measure of the intensity of the σ − -polarized photoluminescence, since the rate of the σ − - photon emission is Γρ22 [15]. The long-time dynamics develops at a time scale much longer than the pulse duration due to the long spin-relaxation times involving spin population transfer between the spin-down and spin-up states. If the initial spin population resides in |3i with spin-up orientation, the optical pulse with the same σ − -polarization does not affect the second system (|3i −→ |4i) (Fig. 3(b) initial section of the green curve representing ρ33) and the spin population of the | − 3/2i trion state remains low (Fig.3 (b) blue curve), slowly varying in time due solely to transitions between levels |1i → |3i, |2i → |4i and spontaneous emission. Note that at the simulation time t = 7 ps (Fig. 3(b)) the spin population of the state |3i is exactly equal to the spin population of level |4i and therefore a dipole is formed due to the spin-relaxation processes. The dipole acts as a source in Maxwell’s equations leading to an additional electric field pulse which appears at later times following the excitation. The time traces of the σ − -polarized photoluminescence for the two cases considered above is plotted on the same plot in Fig. 3(c) showing that a sufficiently long time interval exists ( ∼ 400 ps) within which a differentiation between the two initial spin-states can be made with great fidelity, thereby allowing for uniquely detecting the spin polarization through time-dependent polarized photoluminescence experiments. Note that the σ − -polarized PL decay for spin-down and spin-up initial states is considerably different. While the polarized PL for initial spindown state exhibits exponential decay (Fig. 3(c) blue curve), the PL for initial spin-up state is a non-monotonic function of time with characteristic rising time, reaching a maximum and a subsequent decay (Fig. 3(c) red curve). This opens up the possibility of exploiting the time-resolved polarized PL characteristic shape in the time domain for high-fidelity determination of the initial spin state. On the other hand, the appearance of a second pulse after the initial excitation in the case of initial spin-up state could be used as a probe for differentiation between the two initial spin states thus determining the initial spin-up orientation with high accuracy. In our simulations, we pay special attention to the optical excitations with pulse area of π (or odd multiples of π) since it completely inverts the population in a two-level system. In Fig. 4 (a) the coherent time evolution of the E-field components and the corresponding spin populations of the four levels after a passage of an ultrashort π pulse is plotted. At this relatively high-intensity the field amplitude exhibits saturation although has not reached the stationary value. The pulse is split into two and its trailing edge is due to the population transferred to the second two-level system (|3i-|4i). The spin population ρ22 of level |2i
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Gabriela Slavcheva
Figure 2. A snapshot of the spatial distribution of E-field components and the corresponding spin population of all four states in Fig.1(e) across the QD and its vicinity after the passage of a 2π σ − -polarized optical pulse (Tp = 1.3 ps, initial field amplitude E0 = 6.68673 × 106 Vm−1 ) superimposed on the refractive index profile of the QD structure (black line right scale). (a) at the simulation time t = 0.3369 ps showing the entire initial population ρ11 (red curve) in state |1i with | ↓i orientation (Fig.1 (e)); (b) at the simulation time t = 0.54037 ps the initial spin population has decreased and the population ρ22 of level |2i (blue curve) has increased while the populations of level |3i and |4i are almost unchanged; (c) at the simulation time t = 30.021 ps population inversion occurs (compare with (b)) and the population of level level |3i and |4i slowly increase.
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Figure 3. Time evolution of the σ − -polarized optical pulse (Tp = 1.3 ps, initial field amplitude E0 = 550 Vm−1 ) E-field components and the corresponding spin population of all four states in Fig.1(e); ρ22 (blue curve) represents the trion | − 3/2i state spin population (σ − -polarized photoluminescence) for (a) initially | ↓i populated state |1i (Fig.1 (e)); (b) initial | ↑i state excited by a pulse with the same σ − -helicity; (c) Time-resolved σ − polarized photoluminescence of the trion | − 3/2i state for | ↓i (blue curve) and | ↑i (red curve) initial electron spin states
(blue curve in Fig. 4(a)) represents the σ − -photoluminescence decay. If the initial spin population is in spin-up state residing in state |3i, a σ − π-pulse initially does not affect the (|3i-|4i) system until sufficiently large population is transferred to the first two-level system ((|1i-|2i) and a dipole is created which emits radiation back to the field (Fig. 4(b)). This in turn leads to the appearance of a second pulse at later times. The second pulse experimental detection would provide means to distinguish between the two initial spin states with great accuracy. A comparison of the photoluminescence traces for both initial spin orientation cases is plotted in Fig. 4(c) showing relatively long time interval of ( ∼ 400 ps) within which the initial spin state can be detected with great accuracy after which the two states become indistinguishable. In the high excitation regime, the time evolution of the left-circularly polarized pulse
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Cartesian components Ex and Ey and the spin population of all four levels at a point within the QD is given in Fig. 5 at the initial field amplitude E0 = 4 × 107 Vm−1 , corresponding to an even multiples of π pulse (12π-pulse). At this high excitation intensity the optical field amplitude saturates and the short-time dynamics exhibits Rabi oscillations of the spin population. The number of the full Rabi flops is determined by the pulse area 12π yielding 6 full Rabi flops (Fig. 5(a)). It becomes apparent that the Rabi oscillations over the width of the pulse suppress the spin relaxation, leading to a slightly longer coherence decay time (in the case of a single ultrashort pulse) in agreement with [15]. It is clear that the longer the pulse duration is the more extended in time the suppression of the time decay of the coherence is. Fig. 5(b) shows the corresponding time traces when the initial electron spin state is | ↑i. In accordance with the optical selection rules, the spin population ρ33 residing in state |3i remains unaffected by the propagating pulse at short times. However at longer times (∼ 4 ps) the spin populations ( ρ33 and ρ44) equalize, leading to the formation of an electric dipole between the states which in turn emits back radiation to the field, thus creating an after-pulse, similar to the previous case (with much lower amplitude). The two time traces of the photoluminescence of the trion | − 3/2i state are plotted in Fig. 5(c) showing a long enough time interval ( ∼ 400 ps) within which differentiation between the two electron spin states can be made. In this particular case Rabi oscillations occur at short times for initial | ↓i orientation, as opposed to the smooth curve for initial | ↑i orientation, thereby allowing for differentiation between the two initial spin orientations with great accuracy. In Fig. 6 the time trace of the right-circularly polarized electric field σ + -components and the spin population of the levels is shown for the case of initial spin-up population residing in level |3i. The simulation is performed keeping all parameters the same as in the previous simulations. Comparison with Fig. 5 shows that the population of the trion |3/2i spin state ρ44 exhibits Rabi flops similar to the ones in Fig. 5 and its time decay represents the σ + -polarized photoluminescence. In this case, the second system is excited by the rightcircularly polarized pulse (in agreement with the dipole optical selection rules) and the first system is only affected through the spin-flip and trion spontaneous decay processes. Comparison of the short-time dynamics of the polarized PL following an ultrashort σ − pulse excitation of initial spin-down state for the three cases considered above is given in Fig. 7. It shows decrease of the initial rising times with increasing the pulse area (pulse intensity). For the special case of a 2π pulse excitation the spin population of level |2i performs a full Rabi flop (blue curve). Obviously the greater the difference between the PPL intensity for the two initial spin orientations remains in time, the better are the chances that the two initial states could be experimentally detected and differentiated. This naturally invokes the idea of undoing (time-reversing) the time evolution of the spin coherence using techniques similar to spinecho pulse sequences. However, we note that this kind of techniques are not applicable for undoing the spin population evolution since a population transfer is involved between the two two-level systems rather than solely spin decoherence processes. These techniques would be useful, however, for re-phasing of the pseudospin vector transverse dephasing. Finally, turning to the model justification, besides the proper normalization of the dot density aiming to ensure a single dot in the microscopic simulation domain, the validity of
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Figure 4. Time evolution of the σ − -polarized optical pulse (Tp = 1.3 ps, initial field amplitude E0 = 3 × 106 Vm−1 , giving a pulse area of π) E-field components and the corresponding spin population of all four states in Fig.1(e); ρ22 (blue curve) represents the trion | − 3/2i state spin population (σ − -polarized photoluminescence) for (a) initially | ↓i populated state |1i (Fig.1 (e)); (b) initial | ↑i state excited by a pulse with the same σ − helicity; (c) Time-resolved σ − -polarized photoluminescence of the trion | − 3/2i state for both cases (a) and (b)
the ergodic hypothesis and hence of our approach is reinforced by the following conditions: i) the system of simultaneous equations is linear and the time-integration is performed up to very long times, when the system has approached dynamical equilibrium; ii) the simulated dynamics of a laser-driven dot plus radiation field system is Markovian since the dynamics at any future time moment is calculated solely from the previous time moment (by virtue of the FDTD method of calculation); iii) the density matrix relaxes to a stationary value which does not depend on the initial conditions [15]. In fact, these are all well-known conditions
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Figure 5. Time evolution of the σ − -polarized optical pulse (Tp = 1.3 ps, initial field amplitude E0 = 4 × 107 Vm−1 , giving a pulse area of 12π) E-field components and the corresponding spin population of all four states in Fig.1(e); ρ22 (blue curve) represents the trion | − 3/2i state spin population (σ − -polarized photoluminescence) for (a) initially | ↓i populated state |1i (Fig.1 (e)); (b) initial | ↑i state excited by a pulse with the same σ − helicity; (c) Time-resolved σ − -polarized photoluminescence of the trion | − 3/2i state for both cases (a) and (b) under which the ergodicity is valid although a rigorous mathematical proof still does not exist.
3.
Conclusion
In conclusion, we have presented a novel dynamical model for the coherent spin dynamics induced by an ultrashort optical excitation through optical orientation mechanism. The
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Figure 6. Time evolution of the σ + -polarized optical pulse (Tp = 1.3 ps,initial field amplitude E0 = 4 × 107 Vm−1 ) E-field components and the corresponding spin population of all four states in Fig.1(e); ρ44 (magenta curve) represents the trion | + 3/2i state spin population (σ + -polarized photoluminescence) for initially | ↑i populated state |3i (Fig.1 (e))
Figure 7. Short-time dynamics of the σ − -polarized PL (ρ22) induced by an optical pulse with pulse duration Tp = 1.3 ps,for initial field amplitude E0 = 550 Vm−1 (green);E0 = 3 × 106 Vm−1 , corresponding to a pulse area θ = π (red); E0 = 6.68673 × 106 Vm−1 , θ = 2π (blue); E0 = 4 × 107 Vm−1 , θ = 12π yielding 6 full Rabi flops of the spin population (magenta).
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adopted approach models the coherent light-matter interaction exploiting the SU(N) Lie group symmetries in a discrete multi-level quantum system (particularly adapted to the QD description) and the full vector treatment of the electromagnetic wave propagation, thereby accounting for the electromagnetic field polarization. The model has been applied to the ultrafast spatio-temporal dynamics involved in the trion state of a singly charged QD. Selective generation of specific spin states by circularly polarized light with predefined helicity and subsequent detection through the polarized time-resolved photoluminescence have been demonstrated. The simulations imply two distinct ways of reliable coherent initial spin state detection, namely through the pulse echo appearing at later times following the initial excitation, specific only to initial state with spin-up orientation, and through the shape of the polarized PL trace in time showing a maximum again for this initial state in contrast with the single-exponential decay characterizing the spin-down initial state. The simulations show the onset of the high-intensity optical Rabi oscillations regime suppressing the spin relaxation processes. The role of the effects such as inhomogeneous broadening [37] and pure dephasing [38] in a more realistic case of ensemble of QDs is tractable within the framework of the present formalism and will be discussed in a separate paper.
References [1] Loss D. and DiVincenzo D. P., Phys. Rev. A, 57, 120 (1998) (Preprint condmat/9701055) [2] Semiconductor Spintronics and Quantum Computation , Editors, Awschalom D.D., Loss D., and Samarth N. ; Eds.; Springer-Verlag, Heidelberg, 2002 [3] Optical Orientation, Editor, Meier B. and Zakharchenya B. P.; North-Holland, Amsterdam, 1984 [4] Kikkawa J. M., et al. Science 227, 1284 (1997); Kikkawa J. et al. Phys. Rev. Lett. 80, 4313 (1998); Gupta J. A., et al. Phys. Rev. B 59, R10421 (1999) [5] Elzerman J. M., et al. Nature (London) 430, 431 (2004); Kroutvar M., et al. Nature (London) 432, 81, (2004); Johnson A.C., et al. Nature (London) 435, 925 (2005) [6] Coish W.A. and Loss ., Phys. Rev. B 70, 195340 (2004) [7] Petta J. R., Johnson A. C., Taylor J. M., Laird E.A., Yacoby A., Lukin M. D., Marcus C. M., Hanson M. P., Gossard A. C., Science, 309, 2180 (2005) [8] Khaetskii A. V. et al. Phys. Rev. Lett. 88, 186802 (2002); Merkulov I.A. et al. Pphys. Rev.B 65, 205309 (2002); de Sousa R., and Das Sarma S., Phys. Rev.B 67, 033301 (2003) [9] Li X. , Wu Y., Steel D., Gammon D., Stievater T. H., Katzer D. S., Park D., Piermarocchi C., Sham L. J. , Science, 301, 809 (2003) [10] Atat¨ure M., Dreiser J., Badolato A. ,H¨ogele A., Karrai K., Imamoglu A., Science, 312,551 (2006)
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[11] Elzerman J.M. et al., Nature 430, 431 (2004); Koppens et al., Nature 442, 766 (2006); Xiao et al. Nature, 430, 435 (2004) [12] Rugar et al. Nature 430, 329 (2004) [13] Atat¨ure M., Dreiser J., Badolato A., and Imamoglu A., Nature Physics 3, 101 (2007) [14] Cortez S., et al. Phys. Rev. Lett. 89, 207401 (2002); Pazy E., et al. Europys. Lett. 62, 175 (2003) [15] Shabaev A., Efros Al. L., Gammon D., and Merkulov I. A., Phys. Rev. B 68, 201305(R) (2003) [16] R. I. Dzhioev et al., Phys. Rev. B, 56, 13405 (1997) [17] M.V. Gurudev Dutt et al., Phys. Rev. Lett. 94, 227403 (2005) [18] Hegerfeld G., Fortschr. Phys. 46, 595 (1998) [19] Chen P., Piermarocchi C., Sham L.J., Gammon D., and Steel D. G., Phys. Rev. B 69, 075320 (2004) [20] Wojs A., Hawrylak P., Fafard S., and Jacak L., Phys. Rev. B 54, 5604 (1996) [21] Greilich A., Oulton R., Zhukov E.A., Yugova I. A., Yakovlev D. R., Bayer M., Shabaev A., Efros Al. L., Merkulov I.A., Stavarache V., Reuter D., and Wieck A., Phys. Rev. Lett. 96, 227401 (2006) [22] Merkulov I.A., Efros Al. L., and Rosen M., Phys. Rev. B,65, 205309 (2002) [23] Economou S.E., Liu R-B., Sham L.J., and Steel D.G., Phys.Rev. B 71, 195327 (2005) [24] Hioe F.T. and Eberly J. H., Phys. Rev. Lett. 47, 838 (1981) [25] Hioe F. T., Phys. Rev. A 28, 879 (1983). [26] Aravind P.K., J. Opt. Soc. Am. B, 3, 1025 (1986) [27] Feynman R. P., Vernon F. L., and Hellwarth R. W., J. Appl. Phys. 28, 49 (1957). [28] Kroutvar M., Ducommun Y., Heiss D., Bichler M., Schuh D., Abstreiter G., and Finley J., Letters to Nature 432, 81 (2004). [29] Hartmann A., Ducommun Y., Kapon E., Hohenester U., and Molinari E., Phys. Rev. Lett. 84, 5648 (2000). [30] Bulaev D.V. and Loss D., Phys. Rev. Lett. 95, 076805 (2005) [31] Takagahara T., Phys. Rev. B 62, 16840 (2000) [32] Flissikowski T., Akimov I.A., Hundt A., and Henneberger F., Phys. Rev. B, 68, 161309(R) (2003)
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[33] Slavcheva G., and Hess O., Phys. Rev. A 72 ,053804 (2005) [34] Slavcheva G., and Hess O., Physica Status Solidi (C), 3, 2414 (2006) [35] Slavcheva G. and Hess O., Optical and Quantum Electronics (Springer), 38, 973 (2006) [36] Taflove A., Computational Electrodynamics: The Finite-Difference Time-Domain Method Norwood, MA:Artech, 1995 [37] Schneider S., Borri P., Langbein W., Woggon U., F¨orstner J., Knorr A., Sellin R. L., Ouyang D., and Bimberg D., Appl. Phys. Lett., 83,3668 (2003) [38] Krummheuer B., Axt V.M., Kuhn T., D’Amico I., and Rossi F.,Phys. Rev. B, 71, 235329 (2005)
In: Quantum Dots: Research, Technology and Applications ISBN: 978-1-60456-930-8 Editor: Randolf W. Knoss, pp. 393-412 © 2008 Nova Science Publishers, Inc.
Chapter 11
SUB-DIFFRACTION QUANTUM DOT WAVEGUIDES Chia-Jean Wang and Lih Y. Lin Department of Electrical Engineering University of Washington, Seattle, Washington 98195
Abstract Quantum dots (QD) have been popularized in biological tagging applications and low threshold lasers. However, the unique 3D confinement, size and surface chemistry properties may also be employed for high component density photonic circuit applications. With conventional dielectric waveguides subject to the diffraction limit, we proposed the QD cascade array, which operates on the principle of stimulated emission of a signal light given a pump excitation source. The device is designed to guide light within several hundreds of nanometers or smaller. In the chapter, we focus on the modeling, fabrication and experimental results, which together form a comprehensive discussion. In particular, simulation of the gain, inter-dot coupling and overall transmission behavior provide theoretical insight. Furthermore, two different fabrication processes are outlined, implemented and compared. Finally, a presentation of the measured loss and crosstalk characteristics under a near field optical test setup reveals that the QD nanophotonic waveguide is a technique with high potential for subdiffraction guiding and opens up an opportunity to create wavelength specific, nanoscale optical logic structures.
Introduction While optical transmission technology holds a number of advantages above electronics, the application of Moore’s law from integrated circuits to the photonic domain has been curbed by the diffraction limit. Recent work to scale down to sub-wavelength feature dimensions and capitalize on the extended modulation and capacity property rests largely on plasmon propagation using negative dielectric materials. Example structures proposed in the last decade include the 1D fiber [1], strip waveguide [2,3], nanoparticle array [4,5] and various metal and insulator sandwiched junctions [6-8]. As may be expected, the width, thickness, and nanoparticle periodicity as applicable, control the propagation and confinement characteristics such that larger structures trend towards improved loss figures [2,6].
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Canvassing published work for theoretical and the smaller subset of experimental results gives 3 dB attenuation lengths of 1.76 μm for a 200 nm wide × 50 nm thick gold nanowire [2], 1.03 μm for a 300 nm base with 40° angle gold wedge [3], 97 nm for a 50 nm diameter silver nanoparticle array with 50 nm inter-dot distance [5] and 3.57 μm for a 150 nm wide × 250 nm thick gold insulator-metal-insulator (MIM) junction embedded in silicon [7]. On the other hand, waveguides with nanoscale widths without sub-diffraction performance may be designed from oxide based nanoribbons [9], photonic crystals [10-12], and high refractive index contrast materials [13]. However, reducing the active size of the latter devices would lead to high crosstalk between neighboring lines. Accordingly, the self-assembled quantum dot (QD) nanophotonic waveguide was proposed as a solution, which established the possibility of gain combined with a chemically selective fabrication process [14]. The key to operation is that the QDs are first excited by a pump source to generate electron-hole (e-h) pairs and subsequent introduction of a signal light prompts stimulated emission of a photon, as depicted in Figure 1. Choosing the pump energy to be equivalent to the difference between the second electron and hole states in the conduction and valence bands while the signal energy is set at the first levels prevents wavelength mixing. The emitted photons then interact with neighboring dots in the near-field domain to cause additional e-h pair recombination and sub-wavelength energy transfer. In depositing the nanoparticles (NP) to form an elongate array, the overall output will be determined by the cascaded effects of pump-induced gain and the coupling efficiency as a function of inter-dot separation upon the input light [15].
Reprinted with permission from Nano Letters. Copyright 2006 American Chemical Society.
Figure 1. Sub-diffraction quantum dot waveguide in operation.
To address the QD device, the Chapter is divided into the three major sections of modeling, fabrication and testing. The theoretical work discusses the gain behavior of an optically pumped quantum dot as well as the propagation result derived from Monte Carlo simulation of near-field coupled randomly spaced particles. Both findings are applicable for wider purposes involving quantum dots and near-field energy exchanges. For assembly, we discuss a DNA based method in addition to a two-layer approach which trades programmable deposition for rapid prototyping and improved homogeneity. Demonstration of the nanophotonic waveguide provides the final portion and allows for performance comparisons to be drawn with plasmonic structures. Crosstalk and 3 dB loss values indicate sub-diffraction capability.
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Quantum Dot Waveguide Model We begin the analysis of the quantum dot waveguide, described by Figure 2, at the single nanoparticle to determine the gain coefficient generated by optical pumping [15,16]. The required pump power must account for the non-radiative Auger process [17-18], which depletes the photons available to the system. Next, the dot-to-dot interaction is approached separately by using finite difference time domain (FDTD) simulations to calculate the field distribution as a function of position and distance separating neighboring NPs. With the coupling efficiency and gain characteristics, waveguide transmission is addressed by formulating a Monte Carlo model to capture the effect of the aperiodic placement of QDs. Both one and two dimensional array formations are considered and the implications of the outcome, where wider structures require less gain for the equivalent output of narrower ones, are born out in the experimental loss behavior.
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Figure 2. Progression of model from (a) single quantum dot level to establish gain, (b) dot-to-dot interaction to find coupling efficiency, to (c) assembled array (2D distribution shown although 1D will also be discussed).
Quantum Dot Gain Model Since optical pumping may take either pulsed or continuous wave (CW) form, the gain model may be derived for the two conditions [15]. For the purpose of the waveguide, however, the CW pump represents the practical approach in monitoring and testing purposes. From another perspective, activating gain inside the quantum dot using a laser with periodic emission would require a pulse duration within the natural decay time of the exciton, which is nominally less than tens of nanoseconds, resulting in a high system cost and additional experimental challenges to detect the point of photon transmission. Hence, further discussion considers only continuous wave pumping. Under CW operation, linear gain, calculated by subtracting the absorption spectrum from that of emission, is found using a set of equilibrium equations. Specifically, there are two criteria in that excitation must be countered by relaxation events and the number of electrons raised to the conduction band equals the number of holes left in the valence band.
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Reprinted with permission from IEEE J. of Select. Topics Quantum Electron. Copyright 2005 IEEE.
Figure 3. Three level transition system [15].
Using a three state system, depicted in Figure 3, the first steady state equation becomes: rabs ,02 = rst .ems ,20 + rsp.ems ,20 + rsp.ems ,10
(1)
where the rate of photon absorption prompted by the pump light, rabs,02, is dissipated through multiple methods, two of which involve spontaneous decay from the second, rsp.ems,02, or first, rsp.ems,01, levels with the third being stimulated emission back to ground, rst.ems,02. Next, the balance of electron, N, and holes, P, may be described as:
N =P⇒∑ lmn
⇒∑ lmn
2 f c ( Eclmn ) 2(1 − f v ( Ehlmn )) =∑ V V lmn
2 2 =∑ ⎡ − ⎤ ⎡ E E E ⎛ clmn ⎛ fv − Evlmn ⎞ ⎤ lmn fc ⎞ ⎢1 + exp ⎜ ⎥V ⎢1 + exp ⎜ ⎟ ⎟⎥V kT kT ⎝ ⎠⎦ ⎝ ⎠⎦ ⎣ ⎣
(2)
given V as the QD volume, fc, fv being quasi Fermi levels describing conduction and valence occupation and the factor of two in the numerator accounting for spin. The summation over l, m, n addresses the all excited states specified in x, y, z coordinates as determined by the solution to a 3D potential well [19] for a cubic nanoparticle structure. Given that the material constants and nanoparticle parameters are known, Eqn. (1) may be formulated in terms of the quasi Fermi energies, Efc, Efv, as the two unknowns, which are found by simultaneously solving both formulas. Then, gain as a function of angular frequency, ω, may be found through: G (ω ) = e (ω ) − α (ω ) =
ω nr
μ0 ε0
∞
∑∫
lmn E g
2 R ch
g ch [ f c ( E 2 ) − f v ( E1 )]= / τ in dE ch ( E ch − = ω ) 2 + ( = / τ in ) 2
(3)
where τin is the intraband relaxation time giving rise to a Lorentzian linewidth broadening of the energy state from particle collisions [20], nr is the refractive index, Eg is the bandgap energy, and E2 and E1 are the relative energy positions of electrons and holes in the quantum
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dot. The dipole moment is
, the density of states is gch [19], ħ is the reduced Planck’s constant, and μ0 and ε0 are the magnetic and electric constants. An important note is that switching to the spherical potential well and corresponding coordinates enables calculation of gain for a sphere-shaped nanoparticle. Subsequently, applying the material and dimension parameters for the QDs used in fabrication, the gain spectra for a core/shell CdSe/ZnS nanoparticle are shown in Figure 4. The box (Figure 4a) and sphere (Figure 4b) models are unified by the location of the peak wavelength at 655 nm and distinguishable by the difference in required pump for the onset as well as the magnitude of gain coefficient. Focusing on the 1 nW point, the peak value at 1.51×107 m-1 for the core/shell box is smaller than 4.24×107 m-1 for the core/shell sphere due to the impact of NP volume and allowed eigenenergies. In particular, to provide identical emission wavelengths of 655 nm using the same 1.1 nm shell thickness on each QD shape, the box is sized with 7.4 nm length edges while the sphere diameter is set at 7 nm. Specifying the pump energy as the difference between the second excited electron and hole states, the pump wavelength is 594 nm for the cubic and 572 nm for the spherical dot. Figuring the discrepancy into the summations of Eqn. (1) and (2) at the quasi Fermi level results leads to increased saturated gain for the sphere, but a reduced pump threshold for the box model.
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Figure 4. Gain spectra for a optically CW pumped CdSe/ZnS quantum dot using a core/shell (a) box, and (b) sphere model. The box dimensions are 7.4 × 7.4 × 7.4 nm3, and sphere radius is 3.5 nm (both with 1.1 nm shell).
Now, to correct the gain output for Auger recombination requires taking a step back and revising the rate equation for absorption and emission events. In particular, we add another emissive term to Eqn. (1): rabs ,02 = rst .ems ,20 + rsp .ems ,20 + rsp .ems ,10 + rA
(4)
where rA describes the Auger rate inside a quantum dot in which a biexciton is formed. Instead of generating a photon, taken as implicit in the first model, the energy released from the relaxation of an electron back to ground is donated to increase the excited state of the electron or hole from the second exciton. As a result, the threshold pump power required to generate gain and thus transmit the signal light increases. Accordingly, the Auger rate may be expressed as:
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f c ( E1e )[1 − f v ( E1h )] ∞ =/τ ∫ [ E − ( E − E )]in2 + (= / τ )2 dE τA 1e 1h in Eg
(5)
where E1e and E1h are first electron and hole state energies and the Auger time constant follows the form, τA = β R 3, with R as the radius and β = 5 ps/nm3 for a quantum dot composed of cadmium selenide [21]. Although the NPs used in fabrication and the model include a zinc sulfide shell, the impact of a capping layer is observed to have negligible impact. Maintaining the spherical gain model, τA becomes 214 ps for R = 3.5 nm. Using equilibrium equations (2) and (5) to determine the quasi Fermi energies, the revised gain curves in Figure 5a are contrasted to the original ones in Figure 5b. As predicted, the threshold pump power to bring the quantum dot out of the absorption region rises from 0.067 nW/QD to 7.8 nW/QD although the maximum saturation gain is consistent.
(a)
(b)
Figure 5. Gain for a spherical QD with 3.5 nm radius and 1.1 nm shell under continuous wave optical pumping (a) with and (b) without Auger recombination.
To arrive at the pump power absorbed by the quantum dot, we multiply the absorption of the 405 nm pump wavelength at threshold, calculated to be 3.98×107 m-1, with the particle diameter of 7 nm to get a fractional absorption of 0.28. Factoring the contribution of absorption into the pump power gives 0.28 × 7.8 nW/QD = 2.16 nW per QD, which means each particle requires 2.16 nW of impinging light at 405 nm wavelength in order to achieve gain. In addition, the theoretical value is supported by experiment findings through the threshold optical intensity relation, Ith ≈ ħωp/(σabsτA) [17]. The pump light frequency is ωp and σabs = 2303ελ / NA is the absorption cross-section, where ελ is the extinction coefficient measured in [(M-cm)-1] units and NA is Avogadro’s constant [22,23]. From the manufacturer specification sheet for the CdSe/ZnS nanoparticles, the extracted ελ produces an absorption cross-section of 2.14 nm2. Working backwards and using a τA of 214 ps, Ith turns out to be 1.07 mW/μm2, which implies a threshold pump power of (Ith ×σabs =) 2.3 nW/QD at 405 nm excitation and closely compares to 2.16 nW/QD.
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Inter-Dot Coupling Calculation of the coupling efficiency is the next link before we can address waveguide transmission behavior. While Förster [24] and optical near field [25,26] energy transfer mechanisms represent two ways to find inter-dot behavior, neither accounts for the directional component of photons in stimulated emission. Instead, Förster transfer depends only upon the absorption and emission spectrum overlap of the two particles of interest. Optical near field, however, examines the interaction between the emitting QD, its neighbors and the surroundings through exciton-polariton coupling. As a third and more effective method, finite difference time domain (FDTD) simulation provides a handle on inter-dot coupling [27]. Primarily, we look for the field distribution between two particles by modeling a photon emitted from a quantum dot as a directional point source placed at the center of a dielectric sphere. To keep the dimensions and material parameters true to experimental realization, an 8 nm diameter NP with the refractive index of CdSe is specified along with a 655 nm wavelength line source set at 0.4 nm full width at half maximum (FWHM) directed to propagate forward in the + zˆ direction. As the narrow linewidth produces higher divergence away from the dot, the coupling efficiency tends toward conservative values. The 3D simulation volume is optimized at 8000 nm3 with a mesh size of 0.15 nm such that Maxwell’s equations are iteratively solved and matched at the boundaries of cells with (0.15) 3 nm3 volume. Then, the x, y, z component solutions of the electric and magnetic fields lead to the Poynting vector distributions, S, which provides a method to arrive at the inter-dot coupling: JG JJG Pabsorb ∫ S ⋅ da . η= = JG JJG Ptotal ∫ S ⋅ dA
(6)
Here, Ptotal is the sum of Poynting vectors distributed at the six faces of a box symmetric about the nanoparticle defined to be the top, bottom, front, back, left and right sides while Pabsorb is the sum for only where the adjacent quantum dot interface lies in the propagation direction. Broadly, η is the ratio describing the power incident on the neighboring QD compared to that dissipated over all space. With the specified model criteria, Figure 6a shows the coupling efficiency curve as a function of the inter-dot separation along the z axis given in nanometers. A peak in the profile, which is directly related to a resonance in the field distribution [28], monotonically decreases as the light travels away from the dot. On the other hand, Figure 6b shows related curves, namely, the lateral coupling efficiencies which signify the crosstalk or power lost to the upward ( + xˆ ) and downward ( − xˆ ) directions. Due to an offset in the number of meshes across the volume, there is a slight asymmetry between ηup and ηdown, where the respective positive and negative values denote the direction of energy flow. However, the data is still effective to derive a functional relationship for η, ηup and ηdown, and determine the interaction between neighboring QDs may be generalized to determine waveguide throughput. Multiple curve fits are necessary to adhere to the forward coupling, giving the lowest R-squared value of 0.9993. On the other
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hand, the lateral efficiencies may be easily converted into distance-dependent exponential formulas with: η down = 0.065d -0.4564 and η up = 0.0674d -0.5537
(7)
given d as the separation between the center and neighboring dot located above or below. The associated R2 values are 0.9668 for ηup and 0.9807 for ηdown and indicate close approximation to the FDTD results.
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Figure 6. FDTD simulation of inter-dot coupling efficiency as a function of inter-dot separation: (a) Coupling along forward propagating direction. (b) Cross-coupling in upward and downward (lateral) directions.
Waveguide Transmission From the foundation of having characterized the quantum dot gain and inter-dot coupling efficiency, a model for waveguide transmission may now be formulated. The key variables in the process are the waveguide length, width, QD diameter, D, and maximum inter-dot distance, dmax. In addition, placement of the nanoparticles with respect to one another cannot be assumed to be uniform as the deposition process is guided by self-assembly and successive chemical interactions. Instead, random positioning must be built into the description, and Figure 7 depicts an example of the particle distribution for a structure with a width of 4D. As highlighted by the arrows denoting the energy transfer between NPs, the coupling efficiencies do not necessarily fall distinctly into the forward or cross-coupling directions. Rather, the z and x components need to be factored together in determining the power propagated to dots placed at an angle from the original emitter. Accordingly, a Monte Carlo (MC) model provides a way to combine all the elements. The computation flow begins with creating a population of non-overlapping, randomly distributed quantum dots in a 1D or 2D array. The transmission is then calculated successively along the dots in the + zˆ direction accounting for the contribution of gain, G, within each QD and the transfer efficiency between particles, η. Consequently, the overall throughput is given by the signal emitted from the dot(s) at the exit edge. The simulation is
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run for a large number of cycles, each using a newly initialized quantum dot distribution, in order to arrive at a statistically significant outcome.
Figure 7. Example of cross coupling near field energy transfer in 2D array.
To provide more detail on the transfer function, we first define the x, y and z axes to follow the width, length and height directions of a waveguide. Moreover, the height is limited by the particle diameter as only one layer of QDs are deposited, which is affirmed in the fabrication results by atomic force microscopy. Therefore, choosing indices i and j to denote the z and x position of the dot within the randomized array, the signal transmission immediately after the dot at the (i,j) position may be set as: down Ti,j = eG D (Ti-1,jηi,jWi + ∑ Tk , j −1η up j −1Lk + ∑ Tk , j +1η j +1 Lk ) , k ≤i
(8)
k ≤i
where L and W act as the overlap coefficients in lateral and forward propagation directions, which we take to linearly decrease with the x and z position offset between the QD of interest and the adjacent particle. Individually, the first part of the sum describes the weighted forward contribution given by the immediately preceding dot while the second and third parts account for the energy transfer from NPs situated above and below as revised by the crosscoupling efficiencies and lateral spatial overlap. Then, gain from the quantum dot enters into (8) through the multiplicative factor eGD, with a diameter D of 8 nm. For a one-dimensional array, the crosstalk components disappear and what is left is a simple recursive formula: Ti = eG D ⋅ Ti-1 ⋅ηi ,
(9)
requiring only the forward coupling and gain coefficients to be known. To note, the initial condition is T1 = 1 or in the prior discussion, T1,j = 1, and the final output is found by dividing out the total input signal. Comparing both 1D and 2D cases, we simulate the transmission behavior across QD arrays of 155 × 1 particles and 155 × 40 particles with a random inter-dot spacing maximized at 10 nm. Figure 8 provides the median output for 500 MC simulation cycles, which is a better indicator than the averaged value since the latter may still be skewed by one or two outlier points. From Figure 8a, we see that the relative output varies from 10-6 to 108 when the gain ranges between 1×107 to 5×107 m-1 in a 500 nm width by 2 μm length waveguide. More
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importantly, unity transmission occurs at G = 3.1×107 m-1, which is below the saturation point provided by the gain result in Figure 4b and within the bounds set by the model. On the other hand, a 1D, 2 μm long waveguide requires a gain coefficient ~11.6×107 m-1 to enable lossless propagation. The factor of four difference reveals several important details. Namely, there is a tradeoff between necessary gain, waveguide width and the desired throughput. Although cross-talk may be undesirable between waveguides, the concentration of the lateral coupling to within a few nanometers spacing enables a reduction in the gain threshold for low loss output within the array itself. In addition, there will be a point of diminishing returns on the waveguide width when sub-diffraction operation no longer holds. On the whole, Monte Carlo provides a framework for modeling the system sensitivity to small parameter adjustments.
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Figure 8. Relative transmission result from Monte Carlo simulations using randomized particle spacing as averaged over 500 cycles. (a) 2D array output with 155 × 40 particles corresponding to a 500 nm wide, 2 μm long waveguide. (b) 1D array output with 155 particles corresponding to a 2 μm long waveguide.
Fabrication Methods Quantum dot nanophotonic waveguides have been fabricated with two distinct processes. The first uses deoxyribonucleic acid or DNA as a programmable template for depositing nanoparticles. Through hybridization, the base layer DNA, anchored through self-assembly to the substrate, reacts only with the complementary strand sequence, which is then tethered to a specific set of QDs. The second method trades the complexity of DNA binding for a two monolayer approach to patterning the device. The fabrication details of both procedures follow.
DNA-Mediated Assembly Technique The procedure for capitalizing on the programmable nature of DNA to create a quantum dot waveguide is illustrated in Figure 9 [16]. Oxidized silicon is chosen as the substrate due to its ubiquity in integrated circuit fabrication. Using a small sample, a xylene, acetone, isoproypl
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alcohol (IPA) and de-ionized (DI) water wash cleans and prepares the surface for selfassembly. A positive resist, polymethylmethacrylate (PMMA) diluted to 3%, is spin-coated onto the coupon to produce a 90 nm thick layer. Following a pre-bake at 180° C for 90 seconds, the piece is placed inside a scanning electron microscope (SEM) vacuum chamber and waveguide patterns are written by e-beam lithography (EBL). Next, the exposed PMMA areas are developed with 1:3 methyl isobutyl ketone (MIBK):IPA for 70 seconds, and the sample is rinsed with IPA and blown dry with nitrogen. As an indicator of a successful EBL process, diffraction of light across the waveguide trenches will be observable under the optical microscope.
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(e)
(f) Figures 9(a)-(e) reprinted with permission from Nano Letters. Copyright 2006 American Chemical Society.
Figure 9. DNA-mediated self-assembly of QD waveguides: (a) EBL pattern PMMA coated substrate, treat with O2 plasma; (b) deposit MPTMS monolayer; (c) covalently bind with 5’acrydite-DNA; (d) hybridize with biotin-modified cDNA; (e) bind streptavidin-QDs to biotin-cDNA sites; remove PMMA with dichloromethane; (f) detail of DNA sequences used in hybridization.
The ensuing step is to prime the surface with hydroxyl (-OH) groups for further chemical bonding via an oxygen plasma treatment. Since the process simultaneously strips the PMMA thus removing the EBL pattern, a brief 1 minute duration using a low RF power level of ~20W balances hydroxylation. Afterwards, the first self-assembled monolayer (SAM) of 3’mercaptotrimethoxysilane (MPTMS) is deposited by gas phase in a vacuum chamber. Over
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a period of two hours, the silane terminals in the SAM covalently bind with the -OH groups to present the mercapto (-SH) end at the dangling ends. At the end of the reaction time, the vacuum is turned on and left for 1 hour to eliminate excess MPTMS, which is followed by removing the sample from the container, rinsing with IPA, N2 drying, and curing on a hot plate at 80°C for 10 minutes [29,30]. To note, only N2, which helps preserve the surface chemistry, is used to blow dry the chip. Subsequently, the second monolayer, consisting of a 10 μM DNA dilution where a 500 μL volume contains 440 μL de-ionized water (DI), 50 μL 1x phosphate buffer solution (PBS), 5 μL 3 mM MgCl2 and 5 μL of 1 mM of base layer DNA (sequence: 5’acryditeATCCTGAATGCG-3'), is added to the surface and left to interact overnight. Washing and drying the coupon with 1x PBS and gaseous nitrogen prepares the coupon for immersion in buffered acrylic acid to passivate the -SH termini that did not bind with the acrydite molecules. Then, the surface is treated with a 2 μM solution form of 5’biotin conjugated complementary DNA (cDNA) in 2x SSPE, consisting of 0.3 M sodium chloride, 0.02 M sodium phosphate, 0.002 M ethylene-diaminetetraacetic acid, 0.2% sodium dodecyl sulfate. The third SAM forms over a 30 minute period [31] and is then rinsed off with 2x SSPE and dried. Now, the biotinylated cDNA provides the connection for streptavidin-bound quantum dot deposition and construction of the waveguide. The avidin-biotin linking mechanism is wellunderstood from biological studies, such that the streptavidin protein contains four binding sites which preferentially attract biotin molecules. Thus, the sample is exposed for 30 minutes to a 0.1 μM solution of streptavidin conjugated QDs in 1x PBS and later rinsed and immersed 3 times for a five minute interval each period in 2x SSPE, which removes the salts from the PBS and clears the surface. As a final step, the PMMA layer is removed using a 3 minute toluene submersion, followed by an IPA rinse and N2 dry, which leaves only the quantum dot waveguide formations on the surface.
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Figure 10. (a) SEM image demonstrating QD deposition confined in the patterned region; (b) Fluorescence image of a 1 μm width 655 nm emission CdSe/ZnS QD waveguide.
Figure 10 demonstrates the resulting assembly by scanning electron and fluorescence microscopy. The individual nanoparticles are visible in Figure 10a where the boundary of the waveguide is defined while the QD luminescence within the pattern is clear from Figure 10b.
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Furthermore, by depositing different base layer DNA sequences across the surface and then reacting the particles with the corresponding complementary DNA strands ahead of time, we can program the final NP placement through the hybridization process. Hence, multiple quantum dot type devices are enabled through the flexibility and base-pair selectivity of DNA.
Two-Layer Assembly Technique Trading the intricacy of the former fabrication method, a two monolayer technique was developed to simplify the number of steps and improve the density distribution of the nanoparticles [16]. Similar to the DNA-based process, sample cleaning, e-beam lithography and oxygen plasma treatment take place. However, MPTMS gas deposition is replaced by solution immersion of 0.1~0.2% v/v 3’aminopropyltriethoxysilane (APTES) in 95% IPA and 5% DI H2O to provide the first SAM. A rinse of IPA removes excess APTES and after N2 drying, the sample is cured at 110 ºC for 7.5 minutes to solidify the monolayer formation. As the upper surface is now composed of amine (-NH3) groups, the next step to deposit carboxylated quantum dots takes advantage of the carboxyl-amine bond to anchor and create the QD waveguide. Subsequently, the sample surface is coated with droplets of 125 μM quantum dot solution with DI water and 1 mM 1-ethyl-3-(3’dimethylaminopropyl)-carbodiimide (EDC), a coupling reagent. After at least an one hour reaction time, the coupon is immersion rinsed in 1x PBS and 0.3 M ammonium acetate and nitrogen dried. Fabrication of the waveguide ends with PMMA removal with a 3 minute immersion in dichloromethane, CH2Cl2 [32], DI H2O rinse and N2 dry. Although the DNA process described the use of toluene to release the polymer resist, in the evolution of the two-layer method, dichloromethane was found to yield less residue and a better result.
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(c)
Reprinted with permission from Nano Letters. Copyright 2006 American Chemical Society.
Figure 11. Fabrication of two-layer self-assembled QD waveguides: (a) use e-beam lithography to write and then develop a pattern on substrate coated with PMMA, treat with oxygen plasma to create hydroxyl groups on surface; (b) deposit a monolayer of APTES and then (c) covalently bind carboxylated QDs to the amine terminal group, and strip PMMA with dichloromethane.
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With 655 nm emission CdSe/ZnS core/shell QDs, SEM, atomic force microscopy (AFM) and fluorescence images of the fabrication outcome are depicted in Figure 12. Specifically, 500 nm wide waveguides spaced 200 nm apart are shown with high nanoparticle packing and continuity within the pattern, although the fluorescence picture in Figure 12c appears to have a reduced separation due to diffraction limited imaging. In terms of testing, the closely spaced structures are used to determine the crosstalk behavior between waveguides.
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Figure 12. 500 nm wide QD waveguide pair with 200 nm separation: (a) scanning electron, (b) atomic force and (c) fluorescence micrographs. Scale bar in (b), (c) is 1 μm.
As an extension, the entire two-layer process may be repeated on the same substrate to create additional structures. Essentially, after PMMA release, a new resist layer is be deposited and followed by EBL patterning and surface treatment. Multiple type quantum dot waveguides, shown in Figure 13, [33] enhance the overall capability of the photonic circuit and improve crosstalk suppression, since the interaction between mismatched nanoparticles and their respective emission and absorption profiles is reduced.
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Figure 13. (a) Fluorescence micrograph and (b) AFM of zoomed-in region confirming multiple quantum dot type waveguide deposition. Emission wavelengths are 655 nm and 565 nm corresponding to larger and smaller QDs.
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Experimental Results In order to test the nanophotonic waveguides, an optical microscope was adapted to deliver collimated pump light and xyz stages holding input and output tapered fibers provided the signal source and detection means. A dichroic mirror replaced the beam splitter used in the microscope to direct the 405 nm beam to the sample surface while transmitting longer wavelengths to the eyepiece and overhead CCD camera. Focusing lenses were also added to the optical path to reduce the beam diameter and supply a higher pump power per QD. On the signal laser side, a fiber-pigtailed 639 nm emission diode source is optically modulated at 470 Hz by a chopper and passed into a multimode fiber connectorized to the input probe. On the output end, measured throughput passes into a femtoWatt photoreceiver attached to a lock-in detector, which distinguishes the CW pump from the modulated signal. Additionally, source based fluctuations may be removed from the measurements through power monitoring with 99/1 split ratio couplers inserted at the pump and post-modulated signal paths. For further clarification, Figure 14 depicts the relative equipment placement and experimental setup. As a key point, the construction and quality of the fiber probes have a critical role in the ease of testing. While heat pulling was found to greatly reduce the throughput and pickup, thereby limiting the measured signal to noise ratio, switching to a chemical etching with hydrofluoric acid and a protection layer of isooctane enhanced transmission by a factor of 100 to 1000 [34,35]. Moreover, optimization of the system led to use of an etched multimode fiber with a 200 μm core to maximize signal input and a silver-metallized, tapered singlemode fiber as the output probe for adequate near-field detection. In terms of alignment between the fibers and waveguide, a rough placement within the patterned area took place under brightfield followed by fine tuning via fluorescence imaging under the pump source.
Figure 14. Block diagram of experimental setup.
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Tying the whole setup together is a custom LabVIEW program, which monitors the tapped powers, controls the lasers and records the lock-in measurements. Data logging takes place over a test period, nominally between 2 to 10 minutes for each combination of signal and pump sources. Analysis is then performed offline to find averages and standard deviations of measurements and extract trends.
Loss and Crosstalk Measurements For waveguides, the fundamental figure of merit is the loss as measured in dB per length. The common value is to calculate the propagation distance where the input power is halved, or the 3 dB value. To keep as many variables constant as possible between tests, only one probe is moved to extend the measured waveguide length prior to taking new data. Consequently, a comparison between the same pump powers and overlapping array areas over a range of lengths may be drawn. Figures 15a and 15b illustrate the pump dependent transmission results for 500 nm and 100 nm wide quantum dot structures from 4 μm to 10 μm length. Applying an exponential fit to the data as the output power follows the relation, P ( z ) = P0 exp(−α z ) , with respect to the input power Po and the loss coefficient, α, we
determine an average loss value of 3 dB per 2.26 μm and 4.06 μm for the 100 nm and 500 nm width waveguides across the 1.18 to 2.03 nW pump power/QD curves. The fact that higher pump leads to increased propagation is consistent with previous work [16] and the gain model. 60 2.03 nW/QD 1.53 nW/QD 1.18 nW/QD
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Figure 15. Quantum dot waveguide loss behavior measured over multiple lengths and pump powers using (a) 500 nm wide and (b) 100 nm wide QD waveguides.
In addition to determining the loss figure, crosstalk between waveguides is another key value. Measurement of energy transfer to adjacent arrays gives further proof for subdiffraction guiding. Although FDTD modeling showed superior crosstalk performance when compared to conventional dielectric waveguides due to the non-linear relationship with distance and cross-coupling [28], experimental demonstration is provided here.
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Figure 16. Measured throughput and crosstalk for 500 nm wide quantum dot waveguides with variable length and separation distances: (a) 5 μm length, 500 nm separation. (b) 5 μm length, 200 nm separation. (c) 2 μm length, 200 nm separation. (d) 8 μm length, 200 nm separation. Red crossbars on the crosstalk data are the error bars.
To characterize leakage of the propagating mode, the throughput across a straight waveguide is found first and then one of the fiber probes is moved to align with the center of the neighboring structure. Using 500 nm wide waveguides, Figure 16 presents the signal and corresponding crosstalk responses over a number of different device lengths and separations. In particular, Figures 16a and 16b contrast the results over the same 5 μm length for 500 nm or 200 nm separation while Figures 16c and 16d provide the outcomes for 500 nm separation in 2 μm and 8 μm length nanoparticle arrays. From all data sets, where the red crossbars on the crosstalk denote the error range, we find that crosstalk is negligible for spacings at 200 nm and 500 nm, which promotes the idea of sub-diffraction behavior. As an aside, the downward turn for higher pump powers in Figures 16a and 16b are indicative of possible photobleaching of the quantum dot, but does not obscure the overall trend. To compare the quantum dot waveguide test data with related art, Table 1 lists the reported theoretical and experimental loss values for investigated sub-diffraction propagation methods. With the exception of the gold insulator-metal-insulator (IMI) junction, the QD device provides the lowest loss.
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Table 1. Theoretical and experimental loss values for sub-diffraction waveguiding methods. Experimental loss N/A
Device dimensions
λoperation
Theoretical loss
Ag pin 1D fiber [1] Au nanowire [2] Ag wedge [3] Ag nanoparticle array [4]
20 nm diameter core 200 nm width 50 nm thickness 300 nm base 40° angle 50 nm diameter 25 nm inter-dot separation
633 nm
7.31 dB/μm =3 dB/410 nm N/A
Ag nanoparticle array [5]
50 nm diameter 50 nm inter-dot separation
570 nm
30 dB/μm =3 dB/100 nm
31 dB/μm =3 dB/97 nm
Au IMI [6] Au IMI in Si [7] Au clad MIM [8] Au clad indexguided MIM [8]
2D coverage 45 nm thickness 150 nm width 250 nm thickness 150 nm width 100 nm thickness 150 nm width 100 nm thickness
1.55 μm
0.00076 dB/μm =3 dB/3.9 mm 0.55 dB/μm =3 dB/5.45μm 12.2 dB/μm =3 dB/246 nm 3.1 dB/μm =3 dB/968 nm
N/A
QD waveguide
500 nm width 100 nm width
Method
800 nm 632 nm 488 nm
1.55 μm 633 nm 633 nm 639 nm
1.9 dB/μm =3 dB/1.58 μm 4.8 dB/μm =3 dB/614 nm
N/A
1.7 dB/μm =3 dB/1.76 μm 2.9 dB/μm =3 dB/1.03 μm N/A
0.8 dB/μm =3 dB/3.75 μm N/A N/A 3 dB/4.06 μm 3 dB/2.26 μm
Conclusion With the goal of finding a gain-enhanced sub-diffraction method for energy transfer, the quantum dot based nanophotonic waveguide was developed and qualified through modeling, fabrication and testing. Beginning with calculation of the gain behavior under optical pumping and adding Auger recombination into the description, we found the required threshold pump power to be about 2 nW/QD. Then, the inter-dot coupling efficiency was derived through the Poynting vector ratios from FDTD to determine both forward and crosstalk components. Combining the gain and coupling profiles enabled an estimate for waveguide transmission, where a Monte Carlo approach using numerous simulation cycles yielded a statistically significant result. For unity transmission, the outcomes for a 1D to 2D quantum dot array required gain coefficients of 11.6×107 m-1 and 3.1×107 m-1, respectively. In addition to the reduced threshold, which may be realized as dictated by the CdSe/ZnS spherical gain model, the two dimensional waveguide is more practical in terms of physical realization and fabrication.
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Accordingly, 2D waveguides were demonstrated using both DNA-based and two-layer self-assembly techniques. Devices made from the latter method were selected for testing as the nanoparticle packing better resembled the spacing needed for high inter-dot interaction. The throughput under CW pumping revealed a correlation between increased pump power and higher output. Moreover, characterization of transmission over constant width but variable length structures produced loss values of 3 dB/4.06 μm for 500 nm wide and 3 dB/2.26 μm for 100 nm wide waveguides. The trend in which the narrow waveguide experiences greater loss per length reflects the Monte Carlo findings where unity transmission shifts towards larger gain values going from the 2D to 1D case. In other words, at the same gain coefficient, wider structures will encounter additional lateral cross-coupling components to allow for enhanced throughput. However, if the ultimate objective is to arrive at lossless and even amplified waveguiding in sub-wavelength devices, a quantum dot system capable of producing high gain with low pump power is critical. Recent work with tuning the bandgap structure shows promise in reducing the threshold pump by circumventing the Auger effect [36]. Consequently, the gain model would revert back to the original two steady state formulas to lead to a single exciton, sub-nW/QD threshold pump requirement. Furthermore, sub-diffraction performance would be enabled at a lower power cost, giving rise to an improved generation of quantum dot devices that may be effectively applied to photonic integration. Indeed, we can be sure that the path for nanoscale optics will continue to be illuminated with advances in materials, fabrication and experiment processes.
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[12] Notomi, M.; Shinya, A.; Mitsugi, S.; Kuramochi, E.; Ryu, H.-Y. Opt. Exp. 2004, 12, 1551-1561. [13] Xu, Q.; Almeida, V. R.; Panepucci, R.; Lipson, M. Opt. Lett. 2004, 29, 1626-1628. [14] Wang, C.-J.; Lin, L. Y.; Parviz, B. A. IEEE/LEOS International Optical MEMS Conf. Proc. 2004, Kagawa, Japan, 24-25. [15] Wang, C.-J.; Lin, L. Y.; Parviz, B. A. IEEE J. of Select. Topics in Quantum Electron. 2005, 11, 500-509. [16] Wang, C.-J.; Huang, L.; Parviz, B. A.; Lin, L. Y. Nano Lett. 2006, 6, 2549-53 [17] Klimov, V. I.; Mikhailovsky, A. A.; McBranch, D. W.; Leatherdale, C. A.; Bawendi, M. G. Science 2000, 290, 314-317. [18] Wang, L.-W.; Califano, M.; Zunger, A.; Franceschetti, A. Phys. Rev. Lett. 2003, 91, 056404. [19] Asada, M.; Miyamoto, Y.; Suematsu, Y. IEEE J. Quantum Electron. 1986, 22, 19151921. [20] Benson O.; Yamamoto, Y. Phys. Rev. A 1999, 59, 4756-4763. [21] Klimov, V. I.; Mikhailovsky, A. A.; McBranch, D. W.; Leatherdale, C. A.; Bawendi, M. G. Science 2000, 287, 1011-1013. [22] Klimov, V. I. J. Phys. Chem. B 2000, 104, 6112-6123. [23] Leatherdale, C. A.; Woo, W.-K.; Mikulec, F. V.; Bawendi, M. G. J. Phys. Chem. B 2002, 106, 7619-7622. [24] Förster, T. Disc. Faraday Soc. 1959, 27, 300-320. [25] Sangu, S.; Kobayashi, K.; Ohtsu, M. J. Microscopy 2001, 202, 279-285. [26] Nomura, W.; Yatsui, T.; Kawazoe, T.; Ohtsu, M. J. Nanophotonics 2007, 1, 011591. [27] Yee, K. IEEE Trans. on Antennas Prop. 1966, 14, 302–307. [28] Huang, L.; Wang, C-J.; Lin, L. Y. Opt. Lett. 2007, 32, 235-237. [29] Ramanath, G.; Cui, G.; Ganesan, P. G.; Guo, X.; Ellis, A. V.; Stukowski, M.; Vijayamohanan, K.; Doppelt, P.; Lane, M. Appl. Phys. Lett. 2003, 83, 383-385. [30] Kurth, D. G.; Bein, T. Langmuir 1993, 9, 2965-2973. [31] Demers, L. M.; Ginger, D. S.; Park, S.-J.; Li, Z.; Chung, S.-W.; Mirkin, C. A. Science 2002, 296, 1836-1838. [32] Hu, W.; Sarveswaran, K.; Lieberman, M.; Bernstein G. H. IEEE Trans. Nanotech. 2005, 4, 312-316. [33] Wang, C.-J.; Lin, L. Y. Nano. Res. Lett. 2007, 2, 219-229. [34] Hoffman, P.; Dutoit, B.; Salathé, R-P. Ultramicroscopy 1996, 61, 165-170. [35] Lambelet, P.; Sayah, A.; Pfeffer, M.; Philipona, C.; Marquis-Weible, F. Appl. Opt. 1998, 37, 7289-7292. [36] Klimov, V. I.; Ivanov, S. A.; Na, J.; Achermann, M.; Bezel, I.; McGuire, J. A.; Piryatinski, A. Nature 2007, 447, 441-446.
In: Quantum Dots: Research, Technology and Applications ISBN: 978-1-60456-930-8 Editor: Randolf W. Knoss, pp. 413-425 © 2008 Nova Science Publishers, Inc.
Chapter 12
THREE-DIMENSIONAL IMAGINGS OF THE INTRACELLULAR LOCALIZATION OF MRNA AND ITS TRANSCRIPT USING NANOCRYSTAL (QUANTUM DOT) AND CONFOCAL LASER SCANNING MICROSCOPY TECHNIQUES Akira Matsuno1,*, Akiko Mizutani2, Susumu Takekoshi2, R. Yoshiyuki Osamura2, Johbu Itoh3, Fuyuaki Ide1, Satoru Miyawaki1, Takeshi Uno1, Shuichiro Asano1, Junichi Tanaka1, Hiroshi Nakaguchi1, Mitsuyoshi Sasaki1, Mineko Murakami1 and Hiroko Okinaga4 1
Department of Neurosurgery, Teikyo University Chiba Medical Center, 3426-3 Anesaki, Ichihara City, Chiba 299-0111, Japan 2 Department of Pathology, Tokai University School of Medicine, Boseidai, Isehara City, Kanagawa 259-1100, Japan 3 Teaching and Research Support Center, Tokai University School of Medicine, Boseidai, Isehara City, Kanagawa 259-1100, Japan 4 Vice-President, Teikyo University, 2-11-1 Kaga, Itabashi-ku, Tokyo 173-8605, Japan
Abstract Confocal laser scanning microscopy (CLSM) combined with computed imaging analysis enables us to observe subcellular organelles, mRNA and protein, three-dimensionally, in routinely processed light microscopic specimens. Meanwhile, recently developed semiconductor nanocrystals (Quantum dots, Qdots), which do not fade upon exposure to light, enable us to obtain multicolor images of molecules due to a narrow emission peak that can be *
E-mail address: [email protected]. Tel: 81-436-62-1211, Fax:81-436-62-1357. Correspondence: Akira Matsuno, M.D., Ph.D., Department of Neurosurgery, Teikyo University Chiba Medical Center, 3426-3 Anesaki, Ichihara City, Chiba 299-0111, Japan
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Akira Matsuno, Akiko Mizutani, Susumu Takekoshi et al. excited via a single wavelength of light. Qdots have recently been used in biological research, and they are utilized to detect signals of immunohistochemistry and fluorescence in situ hybridization (FISH). Recently, we successfully applied the above-mentioned advantages of Qdots and CLSM to three-dimensional imagings of the intracellular localization of mRNA and protein. In this paper, we describe our new technique of three-dimensional imaging using Qdots and CLSM and discuss the advantages of this method. In situ hybridization and immunohistochemistry using Qdots combined with CLSM can optimally illustrate the relationship between protein and mRNA simultaneously in three dimensions. Such an approach enables us to visualize functional images of proteins in relation with mRNA synthesis and localization.
Key Words: in situ hybridization, immunohistochemistry, mRNA, Quantum dot, confocal laser scanning microscopy
Introduction Electron microscopic (EM) in situ hybridization (ISH) (EM-ISH) is used to examine the intracellular distribution and role of mRNA in protein synthesis [1-12]. In recent years, we developed a non-radioisotopic EM-ISH method using biotinylated synthesized oligonucleotide probes [10-12]. We used this method in pathophysiological studies of the pituitary xells, and we successfully visualized the ultrastructural localization of growth hormone (GH) and prolactin (PRL) mRNA in rat pituitary cells [10-12]. In addition, we developed a combined method of EM-ISH and immunohistochemistry (IHC) for the simultaneous identification of pituitary hormone and its mRNA in the same cell [13-20]. The EM-ISH method combining IHC and a non-radioisotopic preembedding ISH method is very useful for the study of the spatial relationship of mRNA and the encoded protein. However, it provides only a two-dimensional image of the mRNA and protein. Confocal laser scanning microscopy (CLSM) can facilitate the intracellular identification of subcellular organelles, mRNA, and protein [21], by using non-fluorescent signals, such as horseradish peroxidase (HRP) and diaminobenzidine (DAB) [22, 23]. CLSM combined with computed imaging analysis enables us to observe these structures three-dimensionally in routinely processed light microscopic specimens [24-33]. Meanwhile, semiconductor nanocrystals (Quantum dots, Qdots) have recently been developed. Qdots do not fade upon exposure to light and enable us to obtain multicolor images due to a narrow emission peak that can be excited via a single wavelength of light [34, 35]. Qdots have recently been used in biological research. They have been shown to detect immunohistochemical signals and signals of fluorescence in situ hybridization (FISH) [3641]. We successfully applied the above-mentioned advantages of Qdots and CLSM in order to obtain three-dimensional images of the intracellular localization of mRNA and protein simultaneously using Qdots [42, 43]. In this paper, we describe our new technique of threedimensional imaging using Qdots and discuss the advantages of this method as compared with the EM-ISH method and the combined EM-ISH and IHC method.
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Materials and Methods Tissue Preparation Male and female Wistar-Imamichi rats (8 weeks from birth, body weight ranging from 240 to 330 g, purchased from Charles River Japan Inc., Yokohama, Japan) were studied. Male rats were used to study the intracellular localization of GH mRNA and GH protein. Female rats were treated intramuscularly with 5 mg estradiol dipropionate (E2 depot: Ovahormon Depot; ASKA Pharmaceutical Co. Ltd., Tokyo, Japan), and after 4 weeks, they were trated with additional injection of 5 mg estradiol dipropionate. Three weeks after the second injection, the female rats were sacrificed and used to study the intracellular localization of PRL mRNA and PRL protein. The pituitary glands were removed, and the anterior lobes were immediately fixed overnight at 4℃ in 4% paraformaldehyde dissolved in 0.01 M phosphate buffered saline, pH 7.4 (PBS). After immersion in graded concentrations of sucrose dissolved in PBS at 4℃(10% for 1 h, 15% for 2 h, 20% for 4 h), the tissues were embedded in Optimal Cutting Temperature (OCT) compound (Tissue-Tek; Miles Laboratories Inc., Elkhart, Ind., USA). Ribonuclease-free solutions treated with 0.02% diethylpyrocarbonate (DEPC) were used routinely, and gloves were used when handling all the tissue specimens and glass slides.
Biotinylation of Synthesized Oligonucleotide Probes for ISH The sequence of the oligonucleotide probe for rat GH mRNA is 5'-dATC GCT GCG CAT GTT GGC GTC, and the sequence of the oligonucleotide probe for rat PRL mRNA is 5'dGGC TTG CTC CTT GTC TTC AGG [44]. The antisense, sense, and scramble oligonucleotide probes were synthesized with a DNA synthesizer (Applied Biosystems model 392; Applied Biosystems, Foster City, CA, USA) and biotinylated by 3'-end labeling method using ENZO's terminal labeling kit (ENZO Diagnostics Inc., Farmingdale, NY, USA), according to the manufacturer's protocol. The specificities of the biotinylated probes for both hormone mRNAs were confirmed by Northern blot hybridization, using total RNA extracted from normal male Wistar-Imamichi rat pituitary glands [10, 11].
Combined ISH and IHC Using HRP-DAB for the Detection of mRNA and Qdot for the Detection of Protein Six μm thick tissue specimens were mounted on 3-aminopropylmethoxysilane-coated slides. After air drying for 1 h, tissue sections were washed with PBS for 15 min. Subsequently they were treated with 0.1 μg/ml proteinase K at 37℃ for 30 min, followed by treatment for 10 min with 0.25% acetic anhydride in 0.1 M triethanolamine. Following this treatment, the slides were washed in 2x sodium chloride sodium citrate (SSC) at room temperature for 3 min and then prehybridized at 37℃ for 30 min. The prehybridization solution consists of 10% dextran sulfate, 3xSSC, 1x Denhardt's solution (0.02% Ficoll / 0.02% bovine serum albumin (BSA) / 0.02% polyvinylpyrrolidone), 100 μg/ml salmon sperm DNA, 125 μg/ml yeast
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tRNA, 10 μg/ml polyadenylic-cytidylic acid, 1 mg/ml sodium pyrophosphate pH 7.4, and 50% formamide. The biotinylated probe for rat GH or PRL mRNA with the concentration of 0.1 ng/μl was diluted with this solution, and hybridization was carried out overnight at 37℃. After hybridization, the slides were washed at room temperature with 2xSSC, 1xSSC, and then 0.5xSSC for 15 min each. The hybridization signals were detected with streptavidinbiotin-horseradish peroxidase (ABC-HRP) for 30 min, using Vectastain's ABC kit (Vector Laboratories Inc., Burlingame, CA, USA), and thereafter developed with DAB and 0.017% H2O2 for 5 min. The slides were washed in PBS, and immunohistochemical staining for rat GH or PRL was carried out for 1 h at room temperature. The antibodies used were anti-rat GH antibody (rabbit, polyclonal, 1:400 diluted in BSA - PBS, from Biogenesis, Ltd., Poole, Dorset, United Kingdom), or anti-rat PRL antibody (rabbit, polyclonal, diluted 1:100 with PBS, supplied by the National Institute of Diabetes and Digestive and Kidney Diseases (NIDDK), Bethesda, MD, U.S.A.). The slides were washed in PBS, and Qdot 655 conjugated with anti-rabbit IgG (Invitrogen Corporation, Carlsbad, CA, U.S.A.) was applied as the second antibody for 30 min at room temperature. The slides were washed in PBS, and nuclear staining was carried out with methyl green. The negative control experiments for ISH included hybridization studies with probes of sense or scramble sequence, as well as studies without probes. The negative control experiments for IHC involved the substitution of normal rabbit immunoglobulin fraction (DAKO, Carpinteria, CA, U.S.A.) for primary antibodies.
Combined ISH and IHC Using Qdots for the Detection of mRNA and Protein Six μm thick tissue specimens were mounted on 3-aminopropylmethoxysilane-coated slides. After air drying for 1 h, tissue sections were washed with PBS for 15 min. Subsequently they were treated with 0.1 μg/ml proteinase K at 37℃ for 30 min, followed by treatment with 0.25% acetic anhydride in 0.1 M triethanolamine for 10 min. The slides were washed in 2x SSC at room temperature for 3 min and then prehybridized at 37℃ for 30 min. The biotinylated probe for rat GH or PRL mRNA with the concentration of 0.1 ng/μl was diluted with the prehybridization solution and hybridization was carried out overnight at 37℃. After hybridization, the slides were washed with 2xSSC, 1xSSC, and then 0.5xSSC for 15 min each at room temperature. Then the slides were washed in PBS, and immunohistochemical staining for rat GH or PRL was carried out for 1 h at room temperature. The antibodies used were anti-rat GH antibody or anti-rat PRL antibody, as described in the previous section. The slides were washed in PBS, and both Qdot 655 conjugated with anti-rabbit IgG and Qdot 605 conjugated with streptavidin (Invitrogen Corporation, Carlsbad, CA, U.S.A.) were applied for 30 min at room temperature. Qdot 655 conjugated with anti-rabbit IgG was used for the detection of GH or PRL protein, and Qdot 605 conjugated with streptavidin was used for the detection of GH or PRL mRNA. The slides were washed in PBS, and nuclear staining was carried out with methyl green. The negative control experiments for ISH included hybridization studies with probes of sense or scramble sequence, as well as studies without probes. The negative control
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experiments for IHC involved the substitution of normal rabbit immunoglobulin fraction for primary antibodies.
Detection of Emission Signals of Qdot 605 and 655 An excitation light of 488 nm evoked a specific emission curve for Qdot 605, Qdot 655, and methyl green. Signals that had emission curves equal to these standards were detected under CLSM and were displayed in the same image.
Results Combined ISH and IHC using HRP-DAB for the detection of mRNA and Qdot for the detection of protein:
Figure 1. With the CLSM reflection mode, GH mRNA was observed as a DAB signal (yellow), and with the confocal mode, GH protein was observed as a 655 nm emission signal using Qdot 655 (red). (stereo-images, bar =5μm).
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Figure 2. With the CLSM reflection mode, PRL mRNA was observed as a DAB signal (yellow), and with the confocal mode, PRL protein was observed as a 655 nm emission signal using Qdot 655 (red). (stereo-images, bar =2μm).
Hybridization signals for rat GH mRNA were demonstrated with light microscopy using ABC-HRP. With the CLSM reflection mode, GH mRNA was observed as a DAB signal, and with the confocal mode, GH protein was observed as a 655 nm emission signal (Fig. 2). When GH mRNA and protein were located in the same or adjacent places, their signals were detected in the mixed color image (Fig. 1). Similarly, hybridization signals for rat PRL mRNA were demonstrated with light microscopy using ABC-HRP. With the CLSM reflection mode, PRL mRNA was observed as a DAB signal, and with the confocal mode, PRL protein was observed as a 655 nm emission signal (Fig. 2). When PRL mRNA and protein were located in the same or adjacent places, their signals were detected in the mixed color image (Fig. 3).
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Figure 3. With the confocal mode, GH protein was observed as a 655 nm emission signal using Qdot 655 (a, red), and GH mRNA was observed as a 605 nm emission signal using Qdot 605 (b, green). Merged images of both signals revealed that GH mRNA and protein were located in the same or adjacent places. Their signals were detected in the mixed color images (c, yellow). (a, b, c: stereoimages, bar=2μm).
Compared to IHC using conventional fluorophores, such as fluorescein isothiocyanate (FITC) and Texas Red, for the detection of proteins, Qdots had no fading upon exposure to light.
Combined ISH and IHC Using Qdots for the Detection of mRNA and Protein With the confocal mode, GH protein was observed as a 655 nm emission signal using Qdot 655 (Fig. 3-a), and GH mRNA was observed as a 605 nm emission signal using Qdot 605 (Fig. 3-b). Merged images of both signals revealed that GH mRNA and protein were located
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either in the same or adjacent places. Their signals were detected in the mixed color image (Fig. 3-c).
Figure 4. With the confocal mode, PRL protein was observed as a 655 nm emission signal using Qdot 655 (a, red), and PRL mRNA was observed as a 605 nm emission signal using Qdot 605 (b, green). Merged images of both signals revealed that when PRL mRNA and protein were located in the same or adjacent places, their signals were detected in the mixed color images (c, yellow). Compared with GH, PRL had less of an association with PRL mRNA. (a, b, c: stereo-images, bar=2μm).
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Figure 5. Negative control experiments, namely ISH with sense probe and IHC with substitution of normal rabbit immunoglobulin fraction for primary antibody, showed only signals for methyl green. (left: confocal image, right: combined image of transmittance and confocal images, bar=20μm).
With the confocal mode, PRL protein was observed as a 655 nm emission signal using Qdot 655 (Fig. 4-a), and PRL mRNA was observed as a 605 nm emission signal using Qdot 605 (Fig. 4-b). Merged images of both signals revealed that when PRL mRNA and protein were located either in the same or adjacent places, their signals were detected in the mixed color image (Fig. 4-c). Negative control experiments, namely ISH with sense probe and IHC with substitution of normal rabbit immunoglobulin fraction for primary antibody, showed only signals for methyl green (Fig. 5). The GH signals were more abundant than those of PRL. GH was localized in the vicinity of GH mRNA, and thus abundant combined signals were noted. On the other hand, PRL, whose intracellular signals were more sparse than those of GH, had less of as association with PRL mRNA. These findings were more prominently shown in the combined ISH and IHC method using Qdots for the detection of mRNA and protein than in the combined ISH and IHC method using HRP-DAB for the detection of mRNA and Qdot for the detection of protein.
Discussion Qdots are nanometer scale particles that absorb light, and then quickly re-emit the light in a different color. Although other organic and inorganic materials exhibit this phenomenon as fluorescence, Qdots are bright and non-photobleaching. As well, Qdots have narrow, symmetric emission spectra with multiple resolvable colors that can be excited simultaneously using a single excitation wavelength. The most striking property is that the color of Qdots can be tuned to any chosen wavelength by simply changing their size. This property enables multiple labeling of subcellular molecules. Even though they may be larger molecules than conventional fluorophores, such as FITC and Texas Red, Qdots have more stable signals and do not fade upon exposure to light, Xiao et al. stated that significantly less signal loss was observed for Qdot probes than for FITC or Texas Red probes[41]. Furthermore, they found that Qdot signals were more than 11-fold stronger than those of fluorescein [41]. Photostability and prominent signal intensity give Qdots a very useful advantage over conventional fluorophores in histochemical studies of intracellular molecules.
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Using these Qdot properties, we successfully visualized the intracellular localization of pituitary hormones (GH and PRL) and their mRNA by using different sized Qdots with CLSM. This analysis has several merits, in that it can be used with light microscopic specimens; it can be observed in any chosen cells and any chosen depth of the section; it can reconstruct three-dimensional images [42, 43]. Immunohistochemical studies using multicolored images of Qdots have been demonstrated by the manufacturer, however, there have been no reports that have applied Qdot imaging to the detection of mRNA ISH signals. Thus, our reports is the first description of three-dimensional imaging of the intracellular localization of GH and PRL and their mRNA using Qdots with CLSM [42, 43]. As we reported previously, EM-ISH is essential for the visualization of the intracellular distribution, which then leads to an understanding of its role in protein synthesis [10-12]. The EM-ISH method, which includes the combined use of IHC and the non-radioisotopic preembedding ISH method, is very useful in the study of the spatial relationship between mRNA and the encoded protein [13-20]. The EM-ISH method can provide higher resolutional images of subcellular organelles than the ISH and IHC using Qdots. However, EM-ISH and IHC has some limitations, as: it can be used only with EM specimens; it can observe only a small number of cells; and it provides only a two-dimensional image of the mRNA and protein. Intracellular organelles, such as the rough endoplasmic reticulum and secretory granule, have a three-dimensional structure and localization. Three-dimensional images of the intracellular localization of the mRNA and the encoded protein can be obtained by using combined ISH and IHC with Qdots for the detection of the mRNA and protein. These images may therefore enhance our three-dimensional understanding of the localization of the mRNA and the secreted protein. In our study, GH was more abundant than PRL, and GH was localized in the vicinity of GH mRNA, whereas PRL had less of an association with PRL mRNA. These findings could suggest several possibilities; 1) PRL is being transported to the plasma membrane and secreted more rapidly than GH. 2) The intracellular site of protein synthesis may differ between GH and PRL. 3) The turn-over of mRNA my differ between these two proteins, etc. These observations were more prominently shown using the combined ISH and IHC with Qdots to detect the mRNA and protein than using the combined ISH and IHC with HRP-DAB to detect the mRNA and Qdot to detect the protein. Two different sized Qdots can discriminate between two molecules that are located in the three-dimensional distance more than 25±13nm [45, 46]. Therefore, the mixed color images of GH mRNA and protein conjugated with different sized Qdots mean that these molecules are located within the three-dimensional distance of 12-38nm. By using the ISH and IHC with Qdots and CLSM, one can optimize the visualization of the relationship between the protein and the mRNA simultaneously and three-dimensionally. It may enable us to visualize functional images of proteins as they relate to mRNA synthesis and localization.
Conclusion Qdots do not fade upon exposure to light and enable us to obtain multicolor images due to a narrow emission peak that can be excited via a single wavelength of light. By using the ISH and IHC with Qdots and CLSM, we can optimize the visualization of the relationship
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between the protein and the mRNA simultaneously and three-dimensionally. It may enable us to visualize functional images of proteins as they relate to mRNA synthesis and localization.
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In: Quantum Dots: Research, Technology and Applications ISBN 978-1-60456-930-8 c 2008 Nova Science Publishers, Inc. Editor: Randolf W. Knoss, pp. 427-491
Chapter 13
U NIFIED D ESCRIPTION OF R ESONANCE AND D ECAY P HENOMENA IN Q UANTUM D OTS Ingrid Rotter1 and Almas F. Sadreev2 1 Max Planck Institute for the Physics of Complex Systems, D-01187 Dresden, Germany 2 Kirensky Institute of Physics, 660036, Krasnoyarsk, Russia Abstract We exploit the analogy between light nuclei and quantum dots (QDs) for applying the Feshbach projection operator (FPO) formalism onto the description of the transmission through QDs with a small number of states. In the first part of the review, the exact solutions of the formalism as well as the S matrix are derived. The spectroscopic information on the system is contained in the complex eigenvalues and eigenfunctions of a non-Hermitian Hamilton operator that describes the localized part of the system. It depends explicitly on energy. The eigenfunctions are biorthogonal. The eigenvalues give the positions as well as the decay widths of the resonance states. The unitarity of the S matrix is guaranteed at all parameter values (including energy). Very often, it is achieved by the parameter dependence of the eigenvalues, above all of their imaginary parts. The properties of branch points (exceptional points) in the complex plane are considered and their role for physical processes is discussed. Avoided level crossings lead to level repulsion at small coupling strength between system and environment and to widths bifurcation at larger coupling strength. They cause an internal impurity of an open quantum system which quantitatively can be expressed by the phase rigidity of the wave function that varies between 1 and 0. It does not vanish at zero temperature. Due to the widths bifurcation, bound states in the continuum (BICs) may appear. They do not decay although they lie above particle decay thresholds and their decay is not forbidden by any selection rule. In the second part of the review, the FPO formalism is applied to the description of QDs. By means of analytical and numerical studies, it is shown that the generic properties of open quantum systems can be seen also in QDs. The topology of the branch points is compared to that of diabolic points. The geometrical phase of a branch point is half of the Berry phase. The role of the branch points for the spectroscopic properties of different QDs is discussed. They cause avoided level crossings of resonance as well as of discrete states. In double QDs, resonance
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Ingrid Rotter and Almas F. Sadreev states with vanishing widths (BICs) appear when the system is symmetrical, and with almost vanishing widths when the symmetry is somewhat disturbed. The branch points govern, generally, the crossover from standing to traveling modes in the transmission. Here the phase rigidity is reduced and the transmission probability is enhanced. Some results obtained in experimental studies of high accuracy, which cannot be explained in the framework of the standard theory, are qualitatively discussed.
1.
Introduction
Transmission through a quantum dot (QD) occurs by electrons moving from one of the attached leads to another one by staying for a certain time in the dot. This process is analogue to the scattering of nucleons on nuclei via compound nucleus states: the incident nucleon stands, for a certain time, together with the A nucleons of the target nucleus in one of the resonance states of the compound nucleus consisting of A + 1 nucleons. The compound nucleus decays by emitting one nucleon and leaving the target nucleus in its ground (or an excited) state. These processes are described usually by the resonance part of the S matrix. The matrix elements describing the excitation of the system into resonance states and their subsequent decay are involved in the numerator of the S matrix while the denominator contains, respectively, the excitation and decay characteristics (positions and widths) of the resonance states. The S matrix embodies therefore a unified description of resonance and decay phenomena. Nuclei and QDs differ, of course, by many specific features. For example: the shape of the nucleus is not fixed and the emission of nucleons takes place everywhere at the (badly defined) surface of the nucleus. The transmission through QDs occurs, however, via the leads attached to the dot while no electrons are emitted from other regions of the surface. Another difference consists in the extension of the spectrum. It is bounded from below in a nucleus while it is bounded from below and above ( energy band) in a QD. Furthermore, the interaction between the nucleons bound in the nucleus and those of the environment (continuum) is determined by the two-body residual interaction. In QDs however, the coupling between system and environment is described mostly by using the tight-binding approach. At strong coupling strength between system and environment, the individual resonance states overlap. In this regime, individual resonances in the cross section and in the transmission probability, respectively, can no longer be identified. Due to the overlapping of the resonance states, the so-called external interaction of the resonance states via the common continuum of scattering states becomes important. It appears additionally to the so-called internal interaction between the substituents of the (closed) system. More than 40 years ago, the Feshbach projection operator (FPO) formalism has been worked out in nuclear physics in order to include both the internal and external interaction into the theoretical description. This formalism allows a unified description of fast and slow processes. The fast processes take place directly without participation of the compound nucleus states while the slow ones occur via excitation of the long-lived resonance states of the compound nucleus [1]. In this formalism, a non-Hermitian Hamilton operator Heff appears that contains both the internal and external interaction of the nuclear states. For heavy nuclei, it is impossible to calculate all the coupling matrix elements between discrete and scattering states which are involved in the theory. The wavefunction ΨE C found analytically
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for the solution of the problem (H − E)ΨE C = 0 was considered therefore to be a formal solution. Many theoretical papers are devoted to this problem. They lead to an expansion of the S matrix in terms of energy-independent poles and residues from which the spectroscopic information on the compound nucleus states can be received. Most studies are performed by using statistical assumptions for the compound nucleus states as well as for the coupling coefficients between system and environment. Some years ago it became possible for light nuclei, to calculate all the matrix elements involved in the theory, and to provide a solution for the wavefunction that is no longer formal but obtained in a numerically exact manner [2]. It relies on a unified description of structure and reaction aspects. Knowing the ΨE C , an expression for the S matrix could be derived by using the Lippmann-Schwinger-like relation between the wavefunctions of the resonance states and the eigenfunctions of Heff . It contains the eigenvalues and eigenfunctions of the effective Hamilton operator Heff [3, 4]. These values provide the spectroscopic information on the compound nucleus states. They are energy dependent since the Hamilton operator Heff depends explicitly on energy. The unitarity of the S matrix [5] is guaranteed also in the case of resonance states with complicated structure in realistic systems. Resonance and decay processes are described in a unified manner. The present-day version of the FPO formalism is based on a unified description of structure and reaction aspects as sketched above [4]. It allows to describe the dynamics of open quantum systems. The dynamics is determined by branch points in the complex plane at which the eigenvalues of two states of the effective Hamiltonian coalesce. The Jordan structure of the branch points (called mostly exceptional points in the mathematical literature) is studied recently in [6]. During last ten years, results for concrete open quantum systems controlled by an external parameter are obtained for atoms in a laser field [7, 8, 9], for loosely bound nucleons in nuclei [10, 11, 12, 13, 14, 15, 16, 17], and for the transmission through QDs [18, 19, 20, 21, 22, 23, 24]. In the effective Hamiltonian Heff , the external interaction via the continuum is a secondorder effect. It may become dominant in the regime of overlapping resonances. Here, the influence of branch points in the complex energy plane onto the dynamics of the system can be seen immediately: they cause the phenomenon of avoided crossing of resonance states when controlled by an external parameter. Avoided crossing of discrete and narrow resonance states means the well-known level repulsion in energy while it induces a bifurcation of the decay widths at stronger coupling between system and environment [4]. The avoided crossings lead to some specific features of open quantum systems which are no longer characteristic of the (closed) system itself (determined by the so-called internal interaction). Instead, they are generic for open quantum systems (i.e. for quantum systems embedded into an environment of scattering wave functions). They can be seen most clearly at temperatures being small relatively to the (internal and external) interaction forces between the substituents of the system. Otherwise, the generic features may be disguised by temperature depending effects. The transmission through QDs can be simulated well by studying the results obtained by using microwave cavities [26, 25]. The advantage of the microwave studies consists, above all, in the fact that they are not (or only a little) masked by temperature dependent effects. The results obtained from these studies at room temperature can be compared therefore more directly with those of nuclear reaction studies as well as with those obtained from the
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study of laser induced structures in atoms. In this manner, the results on microwave cavities give a valuable contribution to our understanding of the physics of open quantum systems, including QDs at low temperature. Very much theoretical as well as experimental results exist for quantum systems (including QDs) with many levels that are coupled to many open decay channels. In order to describe these systems, statistical approaches are used mostly in the calculations. Much less attention is devoted to the description of the properties of QDs with a small number of states and a small number of decay channels. In such cases, it is possible to perform the calculations by using the FPO formalism with an accuracy that is numerically exact. Sometimes it is meaningful to perform the calculations by using toy models. Although the results can not always be compared directly with experimental data, they give very much insight into the properties of open quantum systems. Most important is the possibility to trace the system properties as a function of control parameters. It is the aim of this paper to discuss the generic features of open QDs in the regime of overlapping resonances without using statistical assumptions. The discussion is based on the present-day FPO formalism which allows to control the physics of quantum systems (localized in space) when they are opened by coupling to the (extended) environment of scattering wavefunctions. Due to this coupling, the discrete states of the quantum system turn over into resonance states the lifetime of which is, as a rule, finite. In nuclei, the coupling of most low-lying states to the environment can not be switched off (or varied parametrically) such that only a ”snapshot” of the process can be received. A QD, however, can be opened by attaching leads (or antennas) to it, and the coupling strength as well as its shape can be varied. This allows to control the system when external parameters are varied. In Sect. 2., the basic equations of the FPO formalism are derived, including the S matrix used in present-day calculations. Although formally similar to the standard expression of the S matrix, there are some differences to the standard theory as, e.g., the energy dependence of all its ingredients. The S matrix is always unitary. Spectroscopic information on the states of the system is obtained from the eigenvalues and eigenfunctions of the effective non-Hermitian Hamilton operator Heff describing the localized part of the system. It is the central part of the formalism. The mathematical and physical peculiarities of a nonHermitian operator are discussed in Sect. 3.. Of special interest are the branch points in the complex plane (called mostly exceptional points in the mathematical literature). Their geometric phase is half of the Berry phase. They cause level repulsion of discrete and narrow resonance states as well as widths bifurcation at stronger coupling between system and environment. Also bound states in the continuum (BICs) may appear due to widths bifurcation. They can be seen in the cross section only as a phase jump by π appearing at their position. Phase lapses appear, in general, due to interferences between the resonance states (if the system is more complicated than a 1d chain). In Sect. 4., the solution of the Schr¨odinger equation in the whole function space and its localized part are given. In the regime of overlapping resonances, neighboring resonance states disturb each other. The disturbance can be simulated by an internal impurity that does not vanish at zero temperature. It can be expressed quantitatively by the phase rigidity of the wave function. In the second part of the review (Sects. 5. and 6.), the FPO formalism is applied to the description of open QDs. Branch points are shown to appear. They have all the generic properties discussed in Sect. 3.. Level repulsion as well as widths bifurcation can be ob-
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served under appropriate conditions. Also BICs appear. Further, the internal impurity of an open quantum system which is caused by the mutual distortion of neighboring resonance states can be expressed by the phase rigidity of the wave function. Some remarks are given on the electron-electron interaction in the FPO formalism worked out for the description of the nuclear many-body problem with strong nucleon-nucleon interaction. Furthermore, the electron phase coherence time is considered qualitatively on the basis of the generic results received for resonance states at low level density as well as at high level density. In Sect. 6., the transmission through different QDs is studied in the crossover from the weak-coupling to the strong-coupling regime. The crossover from standing to traveling waves can be controlled. Most interesting result is that conductance and phase rigidity are strongly correlated: the conductance is enhanced when the phase rigidity is reduced. The last value characterizes the internal impurity of the open quantum system as well as the spectroscopic redistribution processes taking place in the crossover regime. The origin of phase lapses in the transmission through QDs is discussed. Some concluding remarks can be found in the last section.
2. 2.1.
Feshbach Projection Operator (FPO) Formalism and S Matrix Solution of the Schr o¨ dinger Equation with Discrete and Continuous States
The description of open quantum systems meets the problem to consider simultaneously the wavefunctions of discrete and scattering states. Mathematically, both types of wavefunctions are completely different from one another. The discrete states λ characterize the spectrum of the system and are normalized according to the Kronecker delta δλλ0 while the scattering states are continuous in energy E and can be normalized according to the Dirac delta function δ(E − E 0). A powerful method to solve this mathematical problem is the Feshbach projection operator (FPO) technique [1]. In this formalism, the total function space is split into two parts: the function space of discrete states and the function space of scattering states into which the discrete states are embedded. Due to the embedding into the continuum, the discrete states pass into resonance states having, generally, a finite lifetime. The basic equation of the FPO method is the Schr¨odinger equation with the Hermitian Hamilton operator H (H − E)ΨE C =0
(1)
which is solved by means of the two projection operators Q and P that project onto the subspace of discrete and scattering states, respectively. In each of the two subspaces, the problem is solved by standard methods such that the solution ΨE C of (1) can be obtained, as will be shown in the following. To begin with, we define two sets of wave functions by solving the Schr¨odinger equation in the two subspaces. Firstly (HB − EiB ) ΦB i =0
(2)
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for the discrete states of the closed system, and secondly X E(+) (HCC 0 − E) ξC 0 = 0
(3)
C0
for the scattering states of the environment. Here, HB = H0 +VB (with the central potential H0 and the (residual) interaction VB ) is the standard Hamiltonian describing the closed system, and HCC = H0 + VC (with the channel channel coupling VC ) is the standard Hamiltonian used in coupled-channel calculations. The channels C are determined by the motion of one unbound particle relative to the closed system with A − 1 bound particles in a } of the residual system are discrete states, the one-particle certain state k. The states {ΦA−1 k C are scattering states and unbound states ζE C(+)
χE
C(+)
= ζE
⊗ ΦA−1 i
(4)
are the basic (uncoupled) channel wave functions. The channel numbers C are defined by the quantum numbers of the states k of the residual system and those of the unbound particle which are coupled to the total quantum number J π of the channel. By means of the two function sets obtained, the Q and P operators can be defined by Q=
N X
B |ΦB i ihΦi |
P =
Λ Z X
∞ E E dE |ξC ihξC |
(5)
C=1 C
i=1
E = 0 ; P · ΦB = 0. We identify H with QHQ ≡ H with Q · ξC B QQ and HCC with i P HP ≡ HP P where H is given by (1). Assuming Q + P = 1, we can determine a third wave function by solving the coupledchannel equations with source term (+)
ω ˆ i = GP HP Q · ΦB i ,
(6)
(+)
(7)
where GP
= P (E − HP P )−1P
is the Green function in the P subspace and HP Q ≡ P HQ. Using the representation of the P operator, we get X C0
1 E (HCC 0 − E) hξC ωi i = − √ · γˆiC (E) . 0 |ˆ 2π
(8)
Here γˆiC (E) =
√ √ C † C 2π hξE |V |ΦB 2π hΦB i i= i |V |ξE i ,
(9)
are the coupling matrix elements between the wave functions of the two subspaces. C ωi }, the solution Ψ = QΨ + P Ψ in the Using the three function sets {ΦB i }, {ξE } and {ˆ total function space can be obtained in the following manner. From (1) follows E (HP P − E) · P ΨE C = −HP Q · QΨC ;
E (HQQ − E) · QΨE C = −HQP · P ΨC
(10)
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and (+)
E E P ΨE C = ξC + GP HP Q · QΨC ;
−1 E QΨE · HQP · ξC , C = (E − Heff )
(11)
where (+)
Heff = HQQ + HQP GP HP Q
(12)
is an effective Hamiltonian appearing in the function space of discrete states. With Q+P = 1 follows from Eq. (11) (+)
E E E ΨE C = (P + Q) ΨC = ξC + (1 + GP HP Q ) · QΨC .
(13)
Using the ansatz [2] QΨE C =
X
Bi ΦB i
(14)
i
and Eq. (11), one gets Bi =
X hΦB i | j
1 E |ΦB ihΦB j |HQP |ξC i , E − HQQ j
(15)
and E ΨE C = ξC +
X (ΦB ˆi )hΦB i +ω i | ij
1 E |ΦB ihΦB j |HQP |ξC i . E − Heff j
(16)
Using the complex eigenfunctions φλ and eigenvalues zλ of Heff , (Heff − zλ ) φλ = 0 ,
(17)
odinger equation (1) in the total function space of discrete and the solution ΨE C of the Schr¨ scattering states reads N 1 X γλC E √ = ξ + Ω · . ΨE λ C C E − zλ 2π i=1
(18)
Here (+)
Ωλ = φλ + ωλ = (1 + GP HP Q ) φλ
(19)
with ωλ defined by (+)
ωλ = GP HP Q · φλ
(20)
in analogy to Eq. (6), and γλC (E) =
√ √ E E 2π hφ∗λ|HQP |ξC i = 2π hξC |HP Q|φλ i
(21)
in analogy to Eq. (9). The function Ωλ is the wavefunction of the resonance state λ. According to (19), Ωλ ≈ φλ in the interior region while the asymptotic behavior is given (+) by GP HP Q φλ . The Hamilton operator Heff , Eq. (12), depends explicitly on energy. Therefore, also the eigenvalues zλ and eigenfunctions φλ as well as the coupling coefficients γλC depend on energy, generally.
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The S Matrix
The S matrix is defined by the relation between the incoming and outgoing waves in the asymptotic region. Its general form is i h E (22) SCC 0 = ei(δC −δC 0 ) δCC 0 − 2iπhχE C 0 |V |ΨC i , where the χE C [see Eq. (4)] are uncoupled scattering wave functions obtained from X ([H0]CC 0 − E) χE C0 = 0
(23)
C0
and ΨE C is given by (18). Eq. (22) can be written as h i (1) (2) SCC 0 = ei(δC −δC 0 ) δCC 0 − SCC 0 − SCC 0
(24)
where (1)
E SCC 0 = 2iπhχE C 0 |VP P |ξC i
(25)
is the smooth direct reaction part related to the short-time scale, and (2)
SCC 0 = i
N X √ 2π hχE C 0 |VP Q |Ωλ i · λ=1
γλC E − zλ
(26)
is the resonance reaction part related to the long-time scale. It contains the excitation of the resonance state Ωλ from the channel C 0 with wave function χE C 0 (incoming wave) as well as the decay of the eigenstate φλ of Heff into the channel C which is described by γλC (outgoing wave). The relation (19) between the wave functions Ωλ of the resonance states and the eigenfunctions φλ of Heff is completely analogous to the Lippman-Schwinger equation (+)
E = (1 + GP · VP P ) χE ξC C
(27)
between the two scattering wave functions. One arrives therefore at [3] : E hχE C 0 |VP Q |Ωλ i = hξC 0 |VP Q |φλ i .
(28)
Using this relation, the resonance part (26) of the S matrix reads (2) SCC 0
0 N X γλC γλC =i . E − zλ
(29)
λ=1
0
Here, γλC is related to the incoming wave in channel C 0 while γλC is related to the outgoing wave in channel C. It should be underlined that the zλ in the denominator of (29) are energy dependent functions (eigenvalues of Heff ) and, moreover, the γλC in the enumerator [being the coupling matrix elements between the states λ and the scattering states, see Eq. (21)] show a resonance-like behavior at the energy of the branch point [5]. Although (29) contains only 0 the product γλC γλC , Eq. (26) shows that one of these factors stands for the excitation of the (extended) resonance state Ωλ and the other one for the decay of the (localized) eigenstate φλ of Heff .
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2.3.
435
Tight-Binding Approach for Transmission through Quantum Billiards
The S matrix theory can be applied to the description of the transmission through QDs by using discretized schemes. According to (29), the transmission is t = −2πi
X hξ E |V |φλihφ∗ |V |ξ E i L
λ
E − zλ
λ
R
.
(30)
In difference to (29), there are at least two channels. They differ by the geometry of the E are the wavefunctions in, respectively, the left and right two attached leads: ξLE and ξR semi-infinite lead. The numerical solution of the Schr¨odinger equation ~2 2 (31) ∇ + U (x, y) ψ(x, y) = Eψ(x, y) − 2m for electron transport through a QD is often found by the method of finite differences x = a0 i, y = a0 j where i, j are integers. Then the Schr¨odinger equation becomes 2ma20 (Uij − E) + 4 ψi,j . ψi+1,j + ψi−1,j + ψi,j+1 + ψi,j−1 = ~2 This equation can be rewritten as Hψi,j = ψi,j where H is the tight-binding Hamiltonian [27, 18] X t(i, j, µ)|i, jih(i, j) + µ| . (32) H =− i,j,µ
Here t are the hopping matrix elements, i, j run over the two-dimensional sites of the lattice model, and µ runs over the nearest neighbors of the site (i, j). The hopping matrix elements might be different, in general, at the different sites. Very often, they are assumed to be equal, t = 1. Then in the tight-binding formulation of the electron transport through the QD, the electron can jump only from one site to the nearest neighboring one by hopping as shown in Fig. 1. The advantage of the tight-binding scheme for computation is that the matrix of the Hamiltonian (32) is sparse. The effective Hamiltonian (12) is [28, 4] Heff = HB +
X C=L,R
VBC
E+
1 VCB . − HC
(33)
where C = L, R stands for the left and right leads. The calculation of (33) is simplified enormously in the site representation. Indeed, the matrix elements of the coupling operator VBC are calculated between the C-th lead and the QD with equal hopping matrix elements (shown by solid red lines in Fig. 1). With the exception of these boundary sites jC (adjacent to the C-th lead via the solid red lines) the Hamiltonian is Heff = HB in all sites inside the R P QD. Using the identities ij |i, jihi, j| = 1, dE|C, EihC, E| = 1 in (33) we obtain for the boundary sites jC XZ 1 hjC |Heff |jC0 i = t2 . (34) dE 0φ2C (jC ) + E − E0 C
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Ingrid Rotter and Almas F. Sadreev
Here we used that hjC |VBC |jC0 i = −t only for those jC and jC0 which are connected to the interior by bold red lines in Fig. 1. The wave functions of the straight semi-infinite leads are specified by the quantum numbers p = 1, 2, 3, . . . [18] r πp i 2 sin sin kp(E)(j − jC − 1). (35) φp(i, j; E) = NL + 1 NL + 1 Each lead has many propagation bands given by = −2 cos kp − 2 cos[πp/(NL + 1)].
(36)
It is presented by the number p of different continua which depends on the (dimensionless) energy . Note, both leads are assumed to have the same width d.
B
L
R
Figure 1. The QD is opened by attaching two semi-infinite leads. The system is mapped onto discrete sites i, j and described by the tight-binding Hamiltonian (32). The coupling between leads and QD is realized via the hopping matrix elements shown by legs. After specifying the energy spectrum and the wave functions in the leads, the integral in (34) can be obtained by the standard approach E +1−E 0 = P E +1−E 0 + iπδ(E 0 − E). Substituting (35) and (36) into (34) we finally obtain the surprisingly simple formula [18] X φp (jC )φp(jC0 ) exp(ikpa0), (37) hjC |Heff |jC0 i = hjC |HB |jC0 i − t2 p
q j describe the transverse eigenfunctions of the lead where the φp (j) = NL2+1 sin Nπp L +1 with the numerical width NL = d/a0. This result was obtained first by Datta [27] by using the Green function approach. The overall Green function in the tight-binding lattice method is G = (E + − H)−1 where H
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is the full Hamiltonian in the whole function space with discrete and scattering states. H is hermitian. Further, in the site representation G can be subdivided in the following manner + −1 VCB E − HC GC GCB (38) = GBC GB VBC E − HB where E + − HC represents one of the isolated leads and E − HB the closed billiard. From (38), an expression for the submatrix GB can be derived. It follows immediately (E + − HC ) GCB + VCB GB = 0
(39)
(E − HB ) GB + VBC GCB = 1 .
(40)
GCB = −GC VCB GB
(41)
GC = (E + − HC )−1
(42)
Further from (39)
where
is the Green function for the isolated semi-infinite lead C. Substituting (41) into (40), the expression for GB reads GB = (E − HB − VBC GC VCB )−1 .
(43)
The Green function (43) takes an especially simply form in the site representation: VBC = −t only for those boundary sites coupled by the solid red lines in Fig. 1. The Green function of the semi-infinite lead can be calculated analytically [27]. As a result, the Green function for the sites in the interior of the QD takes the form GB = (E − Heff )−1
(44)
where the effective Hamiltonian Heff is derived in (37). That means, the tight-binding lattice Green function method developed by Datta [27] is equivalent to the FPO technique with the non-Hermitian effective Hamiltonian (33). In order to calculate the transport through QDs, GB is the only component of the overall Green function G(jL, jR) [27]. The effects of the leads are taken into account by means of the terms VBC GC VCB , C = L, R. Although for computations the site representation of the effective Hamiltonian (37) is the most simple one, it is worthy to present also the presentation of Heff in the basis of the eigenstates of HB (defined by HB |ψbi = Eb |ψbi). Using the projection operator Q one obtains X X C C ikpC Vb,p Vb0 ,p e , (45) hψb |Heff |ψb0 i = Eb δbb0 − p C=L,R
where [18] in accordance to (37) C =t Vb,p
X
ψb (jC )φp(jC )
(46)
jC
and jC runs over the boundary sites of the C-th wire. This equation shows that the overlapping of the eigenfunctions of the QD with the wavefunctions of the leads defines the coupling strength.
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Ingrid Rotter and Almas F. Sadreev
The non-Hermitian Effective Hamilton Operator Heff Non-Hermiticity
The open quantum system is characterized by two Hamilton operators: H defined by (1) and Heff given in (12). The operator Heff characterizes the part of the problem that is localized in the Q subspace and embedded into the P subspace, while the operator H describes the problem in the whole function space P + Q = 1. Therefore, Heff is non-Hermitian and H is Hermitian. The eigenvalues zλ of Heff are complex. The eigenfunctions are biorthogonal, i.e. the right and left eigenfunctions φλ and ψλ, respectively, are different from each other (see, e.g., [29] for the special case of Heff ). Suppose the right eigenfunctions |φλi follow from Heff |φλi = zλ |φλi .
(47)
Then, by multiplying (47) to the left with the left eigenfunctions hψλ0 |, one gets hψλ0 |Heff |φλi = zλ hψλ0 |φλ i = zλ δλλ0
(48)
where the left and right eigenfunctions are assumed to be orthonormalized. From (48) follows that hψλ0 |Heff = zλ hψλ0 |
(49)
which is the Schr¨odinger equation for the left state. From (49), one gets † |ψλi = zλ∗ |ψλi . Heff
(50)
In the case of a Hermitian operator H † = H, it immediately follows from Eqs. (47) and † 6= Heff . (50) that zλ is real and ψλ = φλ . However, for the non-Hermitian Heff holds Heff † B ∗ In our calculations, the Φi are real and Heff is symmetric: Heff = Heff leading to ψλ = φ∗λ .
(51)
With vanishing coupling γλC between system and environment, the non-Hermitian Hamilton operator Heff characterizing the open quantum system, passes into the Hermitian operator HB that characterizes the corresponding closed system. In the case the operator HB describes a many-particle system (such as a nucleus), it contains the interaction u of the discrete states which is given by the nondiagonal matrix elements of HB . This interaction characterizes the corresponding closed system and may be called internal interaction. The operator Heff , Eq. (12), contains additionally the interaction v of the resonance states via the common continuum (v is used here instead of the concrete matrix elements of the second term of Heff ). This part of interaction is, formally, of second order and may be called external interaction. As the effective Hamilton operator Heff depends explicitly on energy E, so do its eigenvalues zλ and eigenfunctions φλ. Far from thresholds, the energy dependence is weak, generally, in an energy interval of the order of magnitude of the width of the resonance state. It can, however, not be neglected when the resonance states overlap (Sects. 3.2. and 3.3.). At the branch point where zλ = zλ0 for two different states λ and λ0, the coupling coefficients γλC show a resonance-like behavior (Sect. 3.4.).
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439
The Eigenvalues
The eigenvalues zλ = Eλ − i/2 Γλ of Heff , Eq. (12), provide not only the energies Eλ of the resonance states but also their decay widths Γλ (inverse lifetimes). This holds true, however, only if the two subspaces are defined in an appropriate manner. That means, the Q subspace contains all wavefunctions that are localized inside the system and vanish exponentially outside while the wavefunctions of the P subspace are extended up to infinity and vanish inside the system. This fact calls for a special consideration of broad short-lived resonance states the wavefunctions of which do not vanish asymptotically. An example are the single-particle resonances in nuclei [2]. The solutions of the fixed-point equations Eλ = Re(zλ)|E=Eλ and of Γλ = −2 Im(zλ)|E=Eλ are numbers that coincide (approximately) with the poles of the S matrix. In the FPO formalism, however, it is not necessary to consider the poles of the S matrix since the spectroscopic information on the system follows directly from the complex eigenvalues zλ and eigenfunctions φλ of Heff . Moreover, these energy dependent eigenvalues zλ are involved in the physical observables related to the S matrix, see (29). Due to this fact, information on the vicinity (in energy) of the considered resonance states such as the position of decay thresholds and of neighboring resonance states is involved in the S matrix and can be received. Such an information can not be obtained from the poles of the S matrix. The situation near a threshold is illustrated in [30] where the line shape of a resonance is shown to pass into a cusp at the position of the threshold of an inelastic channel to which the resonance state is coupled strongly. The results are shown in Fig. 2. Cusps in nuclear reactions are observed experimentally and discussed in many papers. Another example for the influence of a decay threshold onto the decay width of states and the cross section is shown in Fig. 3. This figure demonstrates clearly that states below thresholds can influence directly the cross section and that they are able even to trap resonance states. The latter result can be seen in the fact that the resonance part of the cross section and the width of the resonance are reduced [note the different scales of σ tot in Figs. 3(a) and (c)]. The regime of overlapping resonances is governed by the resonance trapping phenomenon: a few resonance states receive large decay widths due to their alignment with the channel wavefunctions while the other ones decouple, to a great deal, from the continuum and become long-lived (Fig. 4). This phenomenon was observed first in theoretical nuclear reaction studies [33] and then in a schematic model [34]. It is discussed also in the vibrational predissociation in a linear molecule [35] and in laser induced continuum structures in atoms [8, 9]. Calculations for an open microwave billiard showed the collective character of the short-lived states [36] and the relation of the decay widths of all resonance states (including the trapped ones) to the time delay function [37]. Meanwhile the resonance trapping phenomenon is proven experimentally in a microwave cavity [38]. The situation in the regime of overlapping resonances is best illustrated by the following fact. As shown in [39] it is impossible clearly to interpret the reaction cross section by means of the existence or non-existence of a (short-lived) doorway state in the theory with energy independent positions (in energy) and widths of the resonance states. The authors of this paper draw the conclusion that the experimental investigation of an intermediate structure phenomenon must always be accompanied with a specific dynamical model for
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Ingrid Rotter and Almas F. Sadreev
Figure 2. Influence of the energy dependence of Γλ onto the line shape of a resonance. At the position of the threshold Ethr for opening a decay channel, the width Γλ of the resonance state λ is strongly energy dependent as shown in the upper part of the figure. This energy dependence influences the line shape of a resonance lying in the very neighborhood of the threshold. The shape may pass from a Breit-Wigner-like shape below the threshold to a cusp at the threshold as shown in the lower part of the figure. The calculation is performed in the framework of the continuum shell model for nuclear reactions [2]. The position Ei of the discrete state is varied by hand such that Eλ varies from Eλ1 < Ethr < Eλn where Eλk (k = 1, .., n) are the different positions of the resonance state obtained by varying Ei . All the other parameters of the system are fixed. Figure taken from [30].
the doorway state. An illustration of this fact is given in [40]. In the FPO formalism being a dynamical model, the positions and widths are no longer numbers but energy dependent functions. Controlling these functions by external parameters allows to receive unique information even on the role of doorway states. An example are the whispering gallery modes appearing in microwave cavities with convex boundary and leads attached such that these modes are supported [41]. They are regular and short-lived as a shot-noise analysis has been shown [42]. In contrast to the eigenvalue trajectories EiB of a Hermitian Hamilton operator, the eigenvalue trajectories zλ (X) of a non-Hermitian one may cross in the complex plane (where X is a certain parameter). The crossing points are branch points and have a strong influence on the dynamics of open quantum systems. In the physical literature, they are called mostly double poles of the S matrix, while they are named exceptional points in the mathematical literature [43]. Their properties will be discussed in Sects. 3.5. and 3.6. The crossing points are responsible for the avoided level crossing phenomenon appear-
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Figure 3. Influence of a bound state (lying slightly below the threshold) onto the cross section. The nuclear reaction cross section σ tot is calculated in the framework of the continuum shell model [2] for the case with one open neutron channel and one resonance state [solid line in (a)], no resonance state but one bound state [solid line in (b) and dash-dotted line in (c)], one bound state and one resonance state [solid line in (c)]. The dashed lines show the direct reaction part. Because of the neighborhood to the (elastic) threshold, the resonance shape is nonsymmetric with a comparably long tail to larger energies. The interference between the resonance state and the ”tail” of the bound state at E > 0 (where Γλ > 0) can clearly be seen. Figure taken from [31].
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Ingrid Rotter and Almas F. Sadreev
Figure 4. Resonance trapping related to a branch point in the continuum. The eigenvalues Eλ − i/2 Γλ of four resonance states coupled to one channel are traced as a function of the coupling strength α in the schematical model Heff = H0 − iαV V + . At α = 0, the four states are equidistant (top) and randomly distributed (bottom), respectively. In the first case, the widths of the two middle states bifurcate after crossing at the branch point. In the second case, the two middle states avoid crossing. In any case, three of the states become trapped by one state at large α (width bifurcation). Figure taken from [32].
ing in their vicinity (Fig. 4). This phenomenon will be discussed in some detail in Sects. 3.5. and 3.7.. Among other interesting effects, it causes the appearance of bound states in the continuum (BICs) [44], i.e. of resonance states the widths of which vanish although they are lying above decay thresholds and their decay is not forbidden by any selection rule. For a detailed discussion of BICs see Sect. 3.7..
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3.3.
443
The Eigenfunctions
The eigenfunctions φλ of Heff can be represented in the set {ΦB λ } of eigenfunctions of the Hamiltonian HB of the corresponding closed system, X dλk ΦB (52) φλ = k k
P 2 with complex coefficients dλk (normalized according to |dλk |2 / M λ”=1 |dλ0 λ” | due to the B biorthogonality of the φλ ). In the case the wavefunctions Φk describe a many-body problem, the eigenfunctions φλ of Heff characterize the many-body open quantum system. The eigenfunctions of the non-Hermitian symmetrical Hamilton operator Heff are complex, energy dependent and biorthogonal. The left and right eigenvectors, ψλ and φλ respectively, differ from one another: ψλ = φ∗λ (Eq. (51), Sect. 3.1.). In contrast to hφλ |φλi, the value hφ∗λ|φλi is complex. Nevertheless, it can be used to normalize the biorthogonal wavefunctions [4, 6]. Choosing the orthonormality conditions, as usually, as hφ∗λ|φλ0 i = δλ,λ0
(53)
the transition with v → 0 is smooth from the wavefunctions of an open quantum system to those of the corresponding closed one (with Γλ → 0 and real wavefunctions that are normalized in the standard manner, and v is the coupling strength between system and environment). That means hφ∗λ |φλi → hφλ |φλi = 1 if the coupling vectors in the nonHermitian part of (12) vanish. As a consequence of (53) [45] hφλ|φλi ≡ Aλ ≥ 1 0 Bλλ
≡ hφλ|φλ06=λ i =
−Bλλ0
(54)
≡ − hφλ0 6=λ |φλi 0
|Bλλ | ≥ 0 .
(55)
The normalization condition (53) entails that the phases of the eigenfunctions in the overlapping regime are not rigid: the normalization condition hφ∗λ|φλ i = 1 is fulfilled only if Imhφ∗λ|φλi = 0 .
(56)
Since φλ and, as a consequence, also the value Imhφ∗λ |φλi depend on parameters, the condition (56) corresponds generally to a rotation of the eigenvector by a certain angle βλ (phase change of the wavefunction by βλ) if the parameters are varied. Let us fix the phases of the wavefunctions of the original states when the external coupling strength v of the states via the continuum vanishes [i.e. when the non-diagonal matrix elements of the second term of (12) vanish according to v = 0]. Choosing βλ0 = 0 or ±π, so that Im φ0λ = 0. The influence of a neighboring state is described by v 6= 0 [i.e. by the non-diagonal matrix elements of the second term of (12)]. At v 6= 0, the angle βλ is different from βλ0 , generally. The difference |βλ − βλ0 | may be ±π/4 at most, corresponding to Re φλ = ± Im φλ (as compared to Im φ0λ = 0). This maximum value occurs at an exceptional (branch) point where two eigenvalues zλ , zλ0 of Heff coalesce. Here [4, 19, 20, 6, 45] EP ; φEP λ → ± i φλ0
EP φEP λ0 → ∓ i φλ .
(57)
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Ingrid Rotter and Almas F. Sadreev
This relation between the two wave functions at the branch point has been found also in numerical studies for a realistic system (laser-induced continuum structures in atoms [9]). The phase rigidity defined by rλ =
hφ∗λ|φλi 1 1 = = 2 2 hφλ|φλi (Re φλ) + (Im φλ) Aλ
(58)
is a useful measure [23, 24, 46] for the rotation angle βλ. If the resonance states are distant from one another, it is rλ ≈ 1 due to hφλ |φλi ≈ hφ∗λ|φλi. In approaching a branch point in the complex energy plane [32, 20], we have hφλ|φλi ≡ Aλ → ∞ and rλ → 0. Therefore 1 ≥ rλ ≥ 0. The phase rigidity rλ is a measure for the degree of alignment of one of the overlapping E of the environment. This alignment resonance states with one of the scattering states ξC takes place at the cost of the other states that decouple, to a certain extent, from the environment (widths bifurcation or resonance trapping [4]). The reduction of the phase rigidity rλ in approaching the branch point agrees with experimental data [47], according to which the phase rigidity drops smoothly from its maximum value rλ = 1 far from the branch point to its minimum value rλ = 0 at the branch point, see Sect. 3.6.. It should be underlined that, after defining the normalization condition (53), the values rλ are fixed by the coupling matrix elements v of Heff which determine the degree of overlapping of the resonance states. They can be varied by controlling the system by means of external parameters, e.g. a laser in the case of an atom with many levels [8, 9] or the shape of the QD [19, 20, 21, 22, 23, 24]. As one of the consequences of the bi-orthogonality of the φλ, the expectation value of ˆ with Xφ ˆ λ = xλ φλ is [2] an operator X ˆ λi = xλ ; hφ∗λ|X|φ
ˆ λi = xλ · Aλ with Aλ ≥ 1 . hφλ|X|φ
(59)
That means observable values are enhanced in the regime of overlapping resonances where the phase rigidity rλ is reduced.
3.4.
The Coupling between Localized and Extended States
According to (12) B B B ˆ hΦB i |Heff |Φj i = hΦi |HB |Φj i + Wij ∞ X Z γˆiC γˆjC 1 i X C C ˆ ij ≡ P dE 0 − γˆi γˆj W 0 2π E−E 2 C
(60)
C
C
with the coupling matrix elements γ ˆic, Eq. (9), between discrete states and scattering states. One may call Wij the external interaction appearing additionally to the internal interaction B ˆic γˆjC is real, we have V involved in the first term hΦB i |HB |Φj i with HB = H0 + V . Since γ B B B ˆ Re {hΦB i |Heff |Φj i} = hΦi |HB |Φj i + Re (Wij )
Z γˆiC γ ˆjC 1 X + P dE 0 2π E − E0 ∞
=
B hΦB i |HB |Φj i
C
C
(61)
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and 1 B ˆ Im {hΦB i |Heff |Φj i} = Im (Wij ) = − 2
X
γˆiC γˆjC .
(62)
C
The principal value integral in (61) does not vanish, in general. It creates energy shifts of ˆ ij ) in (62) is related to the widths of the states. In standard the states. The term Im (W calculations, the principal value integral is approximated by introducing effective forces in ˆ ij ) is neglected. HB simulating, in this manner, the expression (61) while Im (W ˆ and the non-Hermiticity of the effective Hamiltonian The correlations induced by W play an important role for closely-lying resonance states which, generically, avoid overlap0 ping. Here, Aλ > 1 and |Bλλ 6=λ | > 0, see Eqs. (54) and (55). For overlapping resonances, B B ΦB i is ill defined and Im {hΦi |Heff |Φi i} does not determine the width of any of the states of the system. Instead, the eigenfunctions φλ of Heff are meaningful and Re {hφ∗λ|Heff |φλi} = Eλ ;
Im {hφ∗λ|Heff |φλi} = −
1 Γλ , 2
(63)
if the two subspaces are defined in an adequate manner (see Sect. 3.2.). Here, the zλ = Eλ − i/2 Γλ give the energies and widths of the resonance states, respectively, and the Ωλ, Eq. (19), are their wave functions. Furthermore, the γλC , Eq. (21), are the coupling coefficients between the localized resonance states and the extended scattering states of the environment. The relation between Γλ and the coupling matrix elements γλC is [4] P X |γ C |2 ≤ |γλC |2 . (64) Γλ = −2 Im(zλ) = C λ Aλ C
This expression holds true at all energies. The inequality in (64) is caused by the biorthogonality (expressed by Aλ , Sect. 3.3.) of the functions φλ . It is a special case of the enhancement of observable values according to (59). In calculations on the basis of P the R matrix theory, the width Γλ of a resonance state is ˜ λ = |γ C |2 at the energy of the state. According to (64) this calculated from the relation Γ λ is a good approximation only for Aλ ≈ 1, i.e. for isolated resonance states. In analogy to (60), we define the coupling matrix elements between two resonance states via the scattering states C of the common continuum by Z γ C γ C0 1 X i X C C = P dE 0 λ λ 0 − γ λ γ λ0 . 2π E−E 2 ∞
Wλλ0
C
C
(65)
C
They consist of a principal value integral and the residuum. The values UλC = (γλC )2 can be displayed in different equivalent representations. The advantage of the energy dependent representation is that it can be used also at the branch (exceptional) point where two eigenvalues zλ coalesce [5, 48]. For two neighboring resonance states λ and λ0 and one channel, it is ! Γλ0 C=1 C=1 2 . (66) = (γλ ) = Γλ 1 − i Uλ 2E − (Eλ + Eλ0 ) + 2i (Γλ + Γλ0 )
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Ingrid Rotter and Almas F. Sadreev
This expression shows a resonance behavior at E = (Eλ + Eλ0 )/2. However, the phase jump by 2π will not influence the phase of the S matrix. The only exception is the case with two coalesced eigenvalues. Here (at the energy of the branch point), the phase jump is reduced to π [5].
Figure 5. The quantity |1 − S|2 determining the total cross section for two resonance states coupled to one channel for three values of the coupling strength α in the schematical model Heff = H0 − iαV V + (full lines): α = 0.08 (a), α = 1 where two eigenvalues zλ coalesce (branch point) (b), and α = 4 (c). The dashed curves are the Breit-Wigner line shapes calculated from the complex eigenvalues of the two resonance states for the same α. The difference between the full and dashed curves arises from the interferences between the two resonance states. The formation of a long-lived state trapped by a short-lived one at large α can clearly be seen. The cross section at the branch point ( α = 1) is described by (68). Figure taken from [32]. The S matrix (24) with (29) and with two closely lying resonance states λ = 1, 2 and one channel reads S = 1−i
X
γλC γλC . i E − E + Γ λ λ 2 λ=1,2
(67)
At a branch point where Γλ ≡ Γ1 = Γ2 and Eλ ≡ E1 = E2, Eq. (67) can be rewritten
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447
[5, 48], S = 1 − 2i
Γλ − E − Eλ + 2i Γλ
Γ2λ E − Eλ +
i 2 Γλ
2 .
(68)
The two resonance states cause two interference peaks in the cross section at E < Eλ and E > Eλ while the two resonance contributions in (68) cancel each other at E = Eλ. The scattering phase varies by π in the neighborhood of E = Eλ according to the value Γλ , and phase lapses appear at the energies of the interference peaks [49]. They are a hint to the change of the regime from level repulsion at low level density to widths bifurcation at high level density, which takes place at the branch point. For illustration, the cross section with two resonance states is shown in Fig. 5 for small, critical and strong coupling strength to one continuum of scattering wave functions (one channel). At small coupling strength ( α = 0.08), we see two narrow resonances while at large coupling strength (α = 4), there is only one narrow (trapped) resonance. It appears as a dip for which the broad resonance state is the background. At the critical value ( α = 1) of the coupling strength to the continuum, the eigenvalues of the two resonance states coalesce and the eigenvalue trajectories cross in the branch point (compare Fig. 4). In this case, the cross section is described by (68). The interference minimum at E = 0 can clearly be seen.
3.5.
Branch Points and Avoided Level Crossings
An exceptional (branch) point is defined by the coalescence of (at least) two eigenvalues: zλEP ≡ zλ = zλ0 where zλ = Eλ − i/2 Γλ . According to Sect. 3.3., the eigenfunctions EP φEP λ and φλ0 of the two states at the branch point have the following properties: EP 1. the eigenfunctions φEP λ and φλ0 are linearly dependent according (53) to (57) with 0 Aλ → ∞ and |Bλλ | → ∞,
2. the phase rigidity is zero, rλ = 0 according to (57) and (58), i.e. Re(φEP λ ) = EP Im(φλ ), 3. the phases of the eigenfunctions are ill defined due to the jump by π/4 at the exceptional point [6, 20]. ∗ are supplemented by the correspondand φEP Furthermore, the eigenvectors φEP λ λ ing associated vectors defined by Jordan chain relations, see equation (13) in [6]. The wavefunction at the branch (exceptional) point is chiral-like since it can be represented by EP φEP λ ± i a φλ0 with real a according to (57). Only for a = 1 (neglecting the phase jump), the wavefunction can be considered to be chiral. The S matrix at the branch point is given by (68). At and in the neighborhood of a branch point of an N -level system, the influence of the rest of N − 2 levels onto the two crossing ones can be assumed to be small. Then the setup for studying exceptional (branch) points can be modeled by an effective complex symmetric non-Hermitian 2 × 2 matrix Hamiltonian e1 ω (69) , H = HT . H= ω e2
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Ingrid Rotter and Almas F. Sadreev
The complex energies e1,2 and the complex channel coupling matrix elements ω are, in general, parameter dependent. The eigenvalues ε± = E± − 2i Γ± and eigenfunctions Φ± of H are [6] p (70) ε± = E0 ± ω Z 2 + 1 and Φ± =
1 √ −Z ± Z 2 + 1
c±
(71)
where 1 e1 − e2 (e1 + e2 ), (72) Z= 2 2ω and c± 6= 0 is complex. The eigenfunctions are biorthogonal according to (53) with (54) and (55). According to (70), it is Z = ± i at the exceptional point and, consequently [6], 1 1 c± or Φ± = c± (73) Φ± = −i i E0 =
with |c± | → ∞. Further [6] c+ /c− → ± i and Φ+ /Φ− = ± i [in agreement with (57)]. When Z 2 + 1 decreases as a function of a parameter, the eigenvalue trajectories of the two levels approach each other. They may or may not cross when only one parameter is varied. In the first case, an exceptional point is met while in the second case, the two trajectories avoid crossing. With further increasing value of the parameter√such that |Z 2 +1| 2 increases, the eigenvalue equation √ (70) shows level repulsion when Re (ω Z + 1) 6= 0 and widths bifurcation when Im(ω Z 2 + 1) 6= 0. For illustration, we show in Fig. 6 the motion of the poles of the S matrix (eigenvalues E± − i/2 Γ± of Heff ) in dependence on increasing coupling strength Re(ω) (left) and Im(ω) (right). In the first case, the two eigenvalues show level repulsion in energy at large coupling strength while in the second case, the widths bifurcate. In any case, the eigenvalue trajectories avoid crossing. At an avoided level crossing holds 0
1. Aλ > 1 (but finite) and |Bλλ | 6= 0, 2. the phase rigidity is reduced, 0 < rλ < 1, 3. the height of the phase jump is π/4 and takes the form of a Heaviside step function (see equation (49) in [6]). The avoided level crossing phenomenon is directly related to the existence of a branch point in the complex plane since, by considering a second parameter, it is always possible to find the corresponding exceptional point where the two trajectories really cross. Due to this relation, the branch points strongly influence the physical properties of open quantum systems in the regime of overlapping resonances in a large parameter range. They separate the regime of level repulsion (and small influence on the widths) from that of widths bifurcation (accompanied by level clustering). The regime of level repulsion is very well known, especially for discrete states. It is related to quantum chaos [50, 25]. Much less known and much debated is the regime of widths bifurcation. Meanwhile it is proven experimentally in studies on a microwave cavity [38].
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Figure 6. The avoided level crossing phenomenon: eigenvalue trajectories traced as a function of increasing coupling strength for real ω = ω R (left) and for imaginary ω = ω I (right). In the first case, the two levels are close to each other in energy for small ω R and repel each other at large ω R . In the second case, the widths of the levels are similar at small ω I but bifurcate at large ω I . The general picture with complex ω is similar. In any case, the levels avoid crossing. Figure taken from [51].
3.6.
Geometric Phases of Diabolic and Exceptional Points
In the pioneering papers [52, 53], the Berry phase is introduced. It is an essential part of quantum mechanics. For nonvanishing (internal) interaction u between the eigenstates, two eigenvalue trajectories of the (Hermitian) Hamilton operator do not cross. This follows from the eigenvalue equation for a Hermitian Hamilton operator (which is equivalent to (70) but with real energies ε± and real non-vanishing interaction u ≡ ω such that Z 2 + 1 6= 0 in this case). The avoided level crossings appearing at certain critical parameter values in the function space of discrete states, are called usually diabolic points. The topological structure of a diabolic point is characterized by the Berry phase. The geometric phase of eigenvectors of non-Hermitian complex symmetric operators has been considered recently in different theoretical papers for paths in parameter space that encircle an exceptional point. In [54], Gamow states are considered. In these studies, an additional part to the Berry phase arises which vanishes with vanishing coupling to the continuum, i.e. when the Gamow states pass into discrete states. This result is related to the fact that, in these studies, the interaction of the Gamow states via a common continuum is not considered. In other papers [20, 55, 56, 6, 57], the interaction of the resonance states via a common continuum is taken into account. Due to this interaction, the resulting geometric phase differs from that obtained in [54], and the limiting case of vanishing coupling to the continuum is not trivial. A cycle around the exceptional point has to be passed four times in order to produce one full 2π circle in the geometric phase. That means, the exceptional (branch) point has to be encircled two times more than a diabolic point in order to restore the wavefunction including its phase.
450
Ingrid Rotter and Almas F. Sadreev For illustration, the fourfold winding around the exceptional point, W (α) Φ(0) = e−iα/4 Φ(0)
(74)
with the transformation matrix W (α) = Φ(α)/Φ(0) [6], can be represented in the following manner [according to (57)] 1. cycle :
ε± → ε∓
2. cycle :
ε∓ → ε±
3. cycle :
ε± → ε∓
4. cycle :
ε∓ → ε±
Φ± →
± i Φ∓
± i Φ∓ → −Φ± − Φ± → ∓ i Φ∓ ∓ i Φ∓ →
Φ±
(75)
Thus, the geometric phase of the exceptional (branch) point is half of the geometric phase (Berry phase) of a diabolic point. In the last case, the system is described by the Hamiltonian HB of a closed system with discrete quantum mechanical states. It contains only the internal interaction of the states which is of first order ( HB = H0 + VB ). In the case of the branch point, however, the Hamiltonian is Heff which contains, additionally to HB , a second-order term arising from the coupling of the states via the continuum. This (+) (+) (+) term is the external interaction HQP GP HP Q = VQP GP VP Q = VBC GC VCB (where P Q = 0, the identification of P with the channels C, and Q with the closed system B [see (12)] are used). At the branch point, this second-order term becomes the leading term. The difference between the geometric phases in the two cases with HB and Heff , respectively, can therefore be related to the different type (first and second order, respectively) of the leading interaction term. It illustrates the importance of the interaction of the quantum states via the continuum when the quantum system is open. More than 10 years ago, the geometric phases of real wave functions in nonintegrable quantum billiards are measured by using microwave resonators [58]. The results showed the Berry phase. The cyclic excursion around the diabolic point is achieved by means of parameters that control the shape of the resonator. The geometric phases appear as a sign change of the wave function after one cycle. According to expectations it builds up whenever a double degeneracy is encompassed. However, also triple degeneracies lead to a sign change. This last observation caused theoretical studies aiming to explain the data, e.g. [59]. In triple degeneracies, an additional mirror symmetry comes into play. Also the topological structure of an exceptional point has been studied in a microwave cavity experiment [60, 61, 47]. To get access to and encircle an exceptional point in the experiment, an absorptive system is used. It consists of two semicircular cavities of slightly different size which can be coupled by adjusting the opening of a slit between them. The second parameter is given by the distance between the centers of the cavity and a teflon semicircle placed on one side of the cavity. The experiment is performed in such a manner that the eigenvalues and eigenvectors of the non-Hermitian Hamilton operator Heff could be traced on a closed path around the exceptional point. Along this path the eigenvalue trajectories avoid crossing in the complex energy plane. The experimental results [60] confirm the expectations: a cycle around the exceptional point in parameter space has to be passed four times in order to produce one full 2π cycle in the geometric phase. In a next experiment, the authors studied the phase difference between the two eigenvectors in approaching the exceptional point [47]. As a result, the phase difference between
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the two modes changes from π at large distance between them to π/2 in approaching the exceptional point. This result has been explained by the authors [47] by the assumption that the state at the exceptional point is a chiral state. The experimental results [47] can be explained also by means of the phase rigidity rλ of the complex eigenfunctions φλ of the non-Hermitian Hamilton operator Heff [46]. The phase rigidity drops smoothly from its maximum value r± = 1 far from the exceptional point [with the phase difference π (or 2π) between the wave functions of isolated resonance states] to its minimum value r± = 0 at the exceptional point [with the phase difference ±π/2 according to (57)]. This interpretation explains, in a natural manner, the experimentally observed smooth reduction of the phase difference in a comparably large parameter range. Further, the phase jump by π/4 occurring in passing the exceptional point [20, 6], is directly related to the fact that this point has to be encircled four times in order to restore the wavefunction including its phase. It corresponds therefore to the geometric phase of this point that is measured in [60]. In this manner, the experimental results can be considered to demonstrate the (parametric) dynamics of open quantum systems which is generated by the interaction of resonance states via the continuum.
3.7.
Widths Bifurcation and Bound States in the Continuum (BICs)
In the regime of overlapping resonances, widths bifurcation may occur. This phenomenon has first shown to appear in nuclear reactions [33] more than 20 years ago. It is studied in many subsequent papers (see e.g. the reviews [4]) and is proven experimentally on a microwave cavity [38]. It causes the appearance of narrow (trapped) resonance states together with a few short-lived ones. The resonance trapping phenomenon occurs hierarchically, see Fig. 7 for an example obtained in a schematical model. The width of a trapped resonance state may even vanish although no selection rule forbids its decay. Such a state is called usually bound state in the continuum (BIC). In 1985 Friedrich and Wintgen [44] considered the problem by using the FPO formalism and related the existence of BICs to avoided level crossings. Numerical examples showing the strong parameter dependence of the decay widths are obtained for atoms in a laser field [8, 9] and for the transmission through quantum dots [21, 22]. In these cases, the exact disappearance of the decay width of a resonance state is related to some symmetries involved in the system. In the transmission through a quantum billiard (Fig. 12), BICs appear at those energies at which the resonant transmission crosses a transmission zero [21, 22, 64, 65, 66]. In the case of laser induced continuum structures in atoms, the BICs may appear at realistic values of the strength of the laser field if both the Hermitian and the non-Hermitian parts of Heff are considered straightforwardly [8, 9]. This result differs from those obtained in a schematical model, such as Fig. 7, where BICs appear only at α → ∞. Since BICs are states that do not decay, the population probability of these states is constant in time (population trapping ) [63]. This time dependent representation is equivalent to the time independent approach by using the FPO technique (where population trapping in a certain level is described by the existence of a BIC [8, 9]). Let us now consider the study of BICs in the FPO formalism in detail [67]. According to the definition, a BIC is a resonance state with vanishing width, Γλ0 |(E=Eλ0 ) = 0 .
(76)
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Ingrid Rotter and Almas F. Sadreev
Figure 7. The resonance trapping phenomenon occurs hierarchically. The calculation is performed with the schematical Hamiltonian Heff = H0 − iαV V + for two groups of altogether 16 resonance states coupled to two open decay channels. The energies of the states are symmetrical (left) and asymmetrical (right). In any case, two broad resonance states occur at strong coupling strength by trapping the remaining 14 resonance states. Figure taken from [62]. Its energy is obtained from the solution of the fixed-point equation Eλ0 = Eλ|(E=Eλ0 ) . It follows from (64) that a state being decoupled from all channels C of the continuum according to E |V |φλ0 i → 0 hξC
(77)
Ei → 0 is a BIC with Γλ0 ≡ −2 Im(zλ0 ) → 0 [the condition (77) is equivalent to hφ∗λ0 |V |ξC due to the symmetry of Heff and the biorthogonality of the φk ]. The opposite case follows by considering the S matrix
SCC 0
E = e2iδC δCC 0 − 2iπhχE C 0 |V |ξC i
−2iπ
N X hξ E0 |V |φλihφ∗ |V |ξ E i C
λ=1
λ
E − zλ
C
(78)
where the χE C are the uncoupled scattering wave functions (4). At the position of a BIC, we have E − zλ0 → 0 and consequently (77) for all C. That means: the decoupling from all channels of the continuum described by (77) is a necessary and sufficient condition for a resonance state to be a BIC, i.e. a state with vanishing decay width Γλ0 = 0. It should be mentioned here that the scattering phase characteristic of resonances, passes into a phase jump by π at the energy of the BIC. An example is shown in Fig. 13. The wave function of a BIC is, according to (19), eigenfunction of Heff and, consequently, localized. At sufficiently small coupling strength v between system and environment, the condition Γλ0 = 0 cannot be fulfilled if the spectrum of the closed system (described by HB ) is not degenerated [68]. However, BICs exist at arbitrary (including small) coupling strength if the spectrum of HB is degenerated. We refer here to two concrete examples studied analytically with the postulation Γλ0 = 0. In [22, 65], an open quantum billiard with variable
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shape is studied. The two-level approximation is used what is justified in the avoided level crossing scenario. In another study [66], an open Aharonov-Bohm ring with degenerated spectrum of HB is investigated. For more details see Chapter 15 of the present book. By means of the complex eigenvalues zλ of Heff , the appearance of a BIC can be traced as a function of a control parameter X, i.e. by controlling the trajectories Eλ(X) and Γλ (X). The BIC appears at the point X = X0 where Γλ (X0) = 0. It is also possible to consider the neighborhood of the BIC including the cases if Γλ (X 0) is always different from zero and Γλ (X00 ) corresponds to the minimum of Γλ (X 0) with a small but nonvanishing value Γλ (X00 ) ≈ 0. This feature of the FPO technique is invaluable for applications since the stabilization of the system (caused by the vanishing width Γλ ) must be known not only at the single point X0 but also in its neighborhood (where Γλ & 0) in order to estimate the possibility of an experimental observation. Examples of Γλ (X) trajectories with Γλ (X0) = 0 as well as with Γλ (X00 ) ≈ 0 are studied on the basis of the FPO method for concrete systems, see [8, 9] for atoms and [21] for quantum dots. In both cases, the reason for the strong parameter dependence of Γλ (X) and Γλ (X 0) in the neighborhood of X0 and X00 , respectively, is discussed and the condition for the exact or approximate appearance of a BIC is understood (it is related to the avoided level crossing phenomenon). As a result of the studies on laser induced continuum structures in atoms [8], it should be underlined once more that the interplay between the real and imaginary parts of the complex non-diagonal matrix elements of the second term of Heff makes possible the appearance of BICs at physical (finite) values of the coupling strength between system and environment, and that the system is stabilized in a broad range of the parameter values. Every BIC appears together with at least one other state whose width is enhanced around X = X0 and whose energy is, generally, different from Eλ0 . Recently, a similar theoretical study is performed for optical microcavities [69]. Here, very long-lived scar-like modes appear near avoided crossings of resonance states. We mention that BICs can be studied in the framework of the FPO formalism also in the many-body problem with many states. In this formalism, the BICs are nothing else than special many-body eigenstates of Heff .
4. 4.1.
Solution of the Schr o¨ dinger Equation in the whole Function Space Internal Impurity of an Open Quantum System
The solution of the problem in the whole function space P + Q = 1 with the Hermitian Hamilton operator H is (18). According to this expression, the eigenfunctions φλ of the non-Hermitian Hamilton operator Heff give the main contribution to the scattering wave ˆ E in the interior of the system, function Ψ C ˆE ΨE C → ΨC =
X λ
cλE φλ ;
cλE =
Ei hφ∗λ|V |ξC . E − zλ
(79)
The weight factors cλE contain the excitation probability of the states λ. The representation of ΨE C (being solution of (1) with the Hermitian operator H) in the set of wave functions
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{φλ} [being solutions of (12) and (17) with the non-Hermitian operator Heff ] is characteristic of the consideration of localized states in the FPO formalism. In the FPO method supplemented by the normalization condition (53), the definition of the two subspaces (system and environment) appears in a natural manner: HB describes the closed system which becomes open when embedded in the continuum of scattering wave E described by H . All spectroscopic values characteristic of resonance states functions ξC C can be traced to the corresponding values of discrete states by controlling the coupling to the continuum. That means with v → 0, the transition from resonance states (described by the non-Hermitian Heff ) to discrete states (described by the Hermitian HB ) can be controlled. ˆE Let us consider the one-channel case, C = 1, and ΨE C → Ψ in the interior of the system. From (79) follows for the right and left wave functions X ˆ Ei = cλE |φλi |Ψ λ
X
ˆ E| = hΨ
dλE hφ∗λ|
(80)
λ E |V |φ i/(E − z ) = c according to (51) and dλE = hξC λ λ λE according to (21) when excitation and decay of the state λ occur via the same mechanism (and the same channel wave function). It follows X X ˆ Ei = ˆ E |Ψ (cλ0E )2 hφ∗λ|φλ0 i = (cλE )2 (81) hΨ λλ0
λ
due to (53). Since (cλE )2 is a complex number, the normalization ˆ Ei = 1 ˆ E |Ψ hΨ
(82)
corresponds to a rotation in analogy to (53) and (56), Re(cλE ) Im(cλE ) = 0. The normalization has to be done separately at every energy E due to the explicit energy dependence of the cλE . Moreover, X ˆ Ei = ˆ E ∗|Ψ c∗λE cλ0E hφλ|φλ0 i hΨ λλ0
=
X
=
X
|cλE |2 Aλ +
X
c∗λE cλ0 E Bλλ
0
λ6=λ0
λ
|cλE |2 Aλ +
X
0
(c∗λE cλ0 E − c∗λ0 E cλE ) Bλλ .
(83)
λ<λ0
λ 0
According to (54) and (55), Aλ and iBλλ = −iBλλ0 are real numbers. In correspondence to 0 the definition (58), it is Aλ > 1 and Bλλ 6= 0 if rλ < 1. ˆ E may be defined by The phase rigidity ρ of the wavefunctions Ψ ρ = e2iθ
˜ E∗|Ψ ˜ Ei hΨ ˜ E |Ψ ˜ Ei hΨ
(84)
ˆ E by θ being determined by in analogy to (58). The value ρ corresponds to a rotation of Ψ 0 the ratio between its real and imaginary parts. The values Aλ and Bλλ = −Bλλ0 are inherent
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in the expression for ρ. In spite of the complicated structure of ρ, it holds 1 ≥ ρ ≥ 0 [since 1 ≤ (a2 − b2)/(a2 + b2 ) ≤ 0 for every summand (a + ib)2 in (84)]. The equations (81) and (83) show that the definition of ρ is meaningful only if the sum of all the overlapping states λ at the energy E is considered and the average over energy E of the system is performed. The value ρ is uniquely determined by the spectroscopic properties of the system that are expressed by the coupling coefficients to the environment and the level density, or by the positions and widths of the resonance states and the degree of resonance overlapping. The last value is characterized by the phase rigidities rλ. According to (84), we have the following border cases. 1. The resonances are well separated from one another, Γλ ∆E ≡ Eλ −Eλ0 : rλ ≈ 1 and (cλE )2 ≈ |cλE |2 = 1 for E → Eλ . In such a case hρi → 1. 2. The resonances overlap and rλ < 1 (but different from 0) for a certain number of neighboring resonances: it may happen that ρ = 0 in a finite energy interval, see [19, 23, 24] for numerical examples. 3. The eigenvalues zλ of two resonance states coalesce at E → Eλ: rλ → 0 and (cλE )2 → 0 at this energy (Fig. 5). Therefore ρ is finite at E → Eλ. The results of a numerical example (double quantum dot) are shown in [23, 24]. 4. K out of N wave functions ΨE C are aligned with the K scattering wave functions E of the environment while the remaining N − K wave functions are more or less ξC decoupled from the continuum and well separated from one another. In such a case, hρi → 1. In difference to the first case, the N − K trapped (narrow) resonance states are superposed by a background term that arises from the K aligned (short-lived) resonance states. This behavior of the phase rigidity hρi is traced in a numerical study for different quantum billiards [23, 24]. ˆ E are the exact solutions of the It should be underlined that the wave functions Ψ Schr¨odinger equation (1) in the interior of the system and that the phase rigidity hρi obtained for these wave functions is related to the individual rλ (i.e. to the corresponding 0 values Aλ and Bλλ ). These relations become important only in the regime of overlapping resonances where rλ < 1 and the individual wave functions φλ align with the scattering E of the environment. That means, the phase rigidity (84) may be reduced, wave functions ξC in an open quantum system, due to the biorthogonality of the eigenfunctions φλ of the nonHermitian Hamilton operator Heff . The value rλ characterizes an internal impurity of an open quantum system (Sect. 3.3.). The value ρ can be calculated, see e.g. [23, 24] for open quantum billiards. The internal impurity does not vanish at zero temperature. This result is in contrast to the definition of ρbr given by Brouwer [70] by means of an ˜ although arbitrary wave function Ψ R 2 2 ˜ ˜ 2iΘ dr(|ReΨ(r)| − |ImΨ(r)| ) (85) ρbr = e R 2 + |ImΨ(r)| 2) ˜ ˜ dr(|ReΨ(r)| is formally analog to the definition (84). In the case of ρbr, the source for the reduction of the phase rigidity is an external one, e.g. a magnetic impurity. It is expressed quite generally by the value ρbr in analyzing experimental data.
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In any case, the phase rigidity of the scattering wave function is a quantitative measure for the existence of an impurity. This holds true for magnetic as well as for internal impurities. In the last case, ρ = 1 for well separated states while ρ → 0 if many states are almost aligned each to one of the scattering states of the environment. In the last case, the system is (almost) transparent. An example is the transmission through a Bunimovich stadium with support of whispering gallery modes (Sect. 6., Fig. 17).
4.2.
Phase Transition in an Open Quantum System
Let us consider first a schematical model in which the effective Hamilton operator is approximated by ˜ = H ˜ 0 − iαV V + H ˆ); ˜ 0 = HB + Re (W H (+)
ˆ) αV V + = Im (W
(86)
ˆ = VBC G VCB is non-Hermitian [compare (12)] and V V + is a Hermitian where W C ˜ 0 describes the internal structure of the system in the Q subspace operator. The first term H while the second term is the residuum of the coupling term between the two subspaces with the parameter α characterizing the mean coupling strength between discrete and continuum ˜ is written in the eigenbasis of H ˜0 , ˜ 0 is supposed to be diagonal, i.e. H states. Further: (i) H (ii) the number M of resonance states is large, (iii) the number K of open decay channels is restricted to K = 1, and (iv) the energy dependence of the eigenvalues and eigenfunctions of the effective Hamiltonian is weak in spite of the large number M of states. It is therefore ˜ 0 is equal to the number M of states considered. The coupling neglected. The rank of H matrix V is a K × M matrix and the matrix element Vkc describes the coupling of the discrete state k to the channel c ( k = 1, ..., M ; c = 1, ..., K) due to which the resonance state is excited or decays, respectively, into channel c. Thus, the rank of V V + is K = 1. Let us consider the case with real α. If |α| is small, the second term in (1) can be ˜ has M almost real eigenvalues. If, however, regarded as a small perturbation. In this case, H 0 ˜ appears as a small perturbation and the matrix V V + provides K |α| 1, the first term H ˜ has K eigenvalues with large imaginary part. In between these eigenvalues. Therefore, H two limiting cases, a transition occurs between both regimes. Crucial for this transition is the distribution of the exceptional points in the complex energy plane which is exclusively ˜ 0 and V V + . fixed by the distribution of the matrix elements of H In [56, 71], the Hamilton operator (86) has been used in order to investigate if and under which conditions the crossover from low to high level density can be understood as a phase transition. The study is performed with K = 1, meaning that finally one short-lived mode is formed after M − 1 avoided or true crossings with M − 1 resonance states. Thus, M − 1 exceptional points are expected to appear. It is shown analytically [71] that, in the limit M → ∞, a simultaneous coalescence of all eigenvalues occurs at a finite real value of ˜ 0 and the coupling matrix elements ˜k of H α, if the distribution of the real eigenvalues E vk (i.e. the elements of the vector V ) are appropriately chosen. In that case, all M − 1 exceptional points accumulate at one single point in the complex parameter plane. The most illustrative case is a picket-fence model with equal distance between the states and equal coupling strength of all the states to the continuum, v ≡ vk for all k. More generally, an appropriate condition can be achieved when regions with a smaller level density of the
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unperturbed states are stronger coupled to the decay channel than those with a higher level ˜ 2 ≈ xt and the coupling strength v 2 ≈ xr , density. For example for the level distribution E k k such a situation appears [71] when 2(r + 1) = t. Here, αcr = (r + 1)/π = t/ (2π). For the picket-fence model, it is t = 2, r = 0 and αcr = 1/ π. If 2(r+1) > t, αcr → 0 in the limit M → ∞. That means, there exists a state with large decay width at any finite value α > 0. If however 2(r + 1) < t, it follows αcr → ∞, i.e. the reorganization process occurs always locally and does not finish for any finite arbitrary high value α. In this case also an eigenvalue with large imaginary part appears, but now via a successive but infinite chain of level repulsions. Only the case 2(r + 1) = t corresponds to a simultaneous coalescence of all eigenvalues such that a phase transition occurs at the critical value αcr and a short-lived state appears coherently from all M states. Although mathematically the limit M → ∞ is required for the simultaneous coalescence of all eigenvalues, the evolution of the system traced by varying α along the real axis resembles nicely all features of a second order phase transition even for M = 102 states [when 2(r +1) = t] [71]. Here the coupling strength α acts as a control parameter while the imaginary part of the large eigenvalue plays the role of an order parameter. Furthermore, it could be shown that the relation between the distribution of unperturbed states and the coupling strength (i.e. between r and t) has to be fulfilled only approximately. If either ˜ 0 or the coupling matrix V V + (or both) are additionally altered by the level density of H noisy perturbations, an abrupt transition occurs at αcr numerically even if only a comparably small configuration space is considered [71]. This means that even under the condition 2(r + 1) ≈ t, all exceptional points of the system accumulate at some finite real value of the parameter α = αcr (see Fig. 2 of [56]). In the limit M → ∞ a perfect coalescence of an infinite number of exceptional points is succeeded. It is interesting to remark that, in the case of a phase transition, the short-lived eigenstate λ0 is collective in the sense that the number of principle components of its eigenfunction jumps abruptly to its maximal value at the critical value αcr . That means, its wave function ˜ 0 . The wavefunctions of all consists of a (constructive) superposition of all eigenstates of H ˜ however, stay almost pure in this basis. In this sense, the the other M − 1 eigenstates of H, short-lived eigenmode with a large imaginary part Γ0 /2 of its eigenvalue is an extremely collective state. This is true, although Γ0 is much smaller than the extension of the spectrum at α = αcr . It is Γ0 ∝ ln (M ). A more realistic situation [according to the general expression (12) and Sect. 3.4.] is obtained when the coupling parameter is chosen to be complex: α → αeiϕ in (1). In this case, the system can no longer evolve through the accumulation point of the exceptional points [71]. The reason is that the accumulation point of the exceptional points is at the real α-axis. The system can therefore not hit the accumulation point, but has to pass it at a certain distance in the complex parameter space. Hence, the reorganization process is washed out. That means, a critical region of reorganization of the system can be observed as a function of α, but a strict phase transition cannot occur as long as α is complex.
4.3.
Peculiarities of the FPO Method
The characteristic features of the FPO formalism consist, above all, in the fact that the solution ΨE C in the whole function space can be represented in the set of (biorthogonal)
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wave functions {φλ } that describe the localized part of the problem, Eq. (79). The localized wave functions represent a subspace of the whole function space with the consequence that the corresponding Hamilton operator Heff is non-Hermitian. The cross section is independent of the manner the two subspaces are defined as long as P + Q = 1 is fulfilled. The eigenvalues of Heff have, however, a physical meaning only if the total function space is divided into the two subspaces ( system and environment) according to the following criteria [2]: the system ( Q subspace) is localized and contains all resonance-like phenomena while the environment (P subspace) is extended and describes the smooth (direct) reaction part in the energy region considered. Such a division of the whole function space into two subspaces was considered by Feshbach [1] about 40 years ago for heavy nuclei with excitation of neutron resonances. Here, the level density is very high (104 to 106 states in an energy interval typical for the corresponding particle-particle interaction) and the neutron resonances are isolated from one another due to their extremely long lifetimes. They are treated successfully by means of statistical methods. Eq. (18) is the basic relation of the unified theory of nuclear reactions [1]. The division into the two subspaces is more difficult for light nuclei due to the existence of single-particle resonances [2]. Due to the low level density in light nuclei, the resonance states keep, to a large extent, their individual features, and can not be treated by statistical methods. All the coupling matrix elements have to be calculated. The situation for quantum dots corresponds to that for nuclei. The so-called mesoscopic features known to characterize small systems, are washed out in larger systems with many levels, see e.g. [83]. However, the numerical calculations for small quantum dots are easier to perform than those for light nuclei. The two subspaces can be defined in a natural manner: the localized part (Q subspace) is the dot and the extended part ( P subspace) consists of the attached leads (single-particle resonances exist only in nuclei due to the strong nuclear forces). The main advantages of the FPO formalism being involved in the present-day [72] calculations, consist in the following. (i) The spectroscopic information on the resonance states is obtained directly from the complex eigenvalues zλ and eigenfunctions φλ of the non-Hermitian Hamilton operator Heff . The zλ and φλ are energy dependent functions, generally, and contain the influence of neighboring resonance states as well as of decay thresholds onto the considered state λ. This energy dependence allows to describe decay and resonance phenomena also in the very neighborhood of decay thresholds and in the regime of overlapping resonances. Since also the coupling coefficients between system and continuum depend on energy, generally, the unitarity of the S matrix is guaranteed for all parameter values [5]. (ii) The resonance states are directly related to the discrete states of a (many-body) closed system described by standard quantum mechanics (with the Hermitian Hamilton operator HB ). They are generated by opening the system what is achieved by coupling the discrete states to the environment of scattering states by means of the second term of the Hamilton operator Heff . Therefore, they are realistic localized (long-lived many-particle) states of an open quantum system. The transition from resonance states (described by the non-Hermitian Heff ) to discrete states (described by the Hermitian HB ) can be controlled. (iii) The properties of branch points and their vicinity can be studied relatively easy. At these points, two (or more) eigenvalues zλ of Heff coalesce. Since it is not necessary to
Unified Description of Resonance and Decay Phenomena...
459
consider the poles of the S matrix in the FPO formalism, additional mathematical problems at and in the vicinity of branch points in the complex plane are avoided. (iv) The phases of the eigenfunctions φλ of Heff are not rigid in the vicinity of a branch point. This fact allows to describe the spectroscopic reordering processes in the system that E of the environment into take place under the influence of the scattering wave functions ξC which the system is embedded.
5. 5.1.
Spectroscopy of Open Quantum Dots Electron-Electron interaction in Open Quantum Dots
The FPO method is formulated originally [1] for the description of nuclear reactions induced by slow nucleons on a target nucleus consisting of A nucleons. In the reaction, a compound nucleus is formed that consists of A + 1 nucleons. In the compound nucleus, the interaction between all A + 1 nucleons is strong and single narrow resonances can be seen in the cross section which correspond to the excitation of individual resonance states of the compound nucleus. The FPO method was successfully applied to the description of compound nucleus reactions on heavy nuclei in which the level density is very high. In this case, it was possible (and meaningful) to use statistical methods in order to describe the states of the compound nucleus as well as the coupling matrix elements between the states of the compound nucleus and those of the nucleon scattering continuum [1]. In the present-day calculations [72], the FPO method is applied also to the description of reactions on light nuclei the level density of which is small such that a statistical treatment is not justified [2, 4]. Instead, all states of both the target nucleus with A nucleons and the compound nucleus with A + 1 nucleons as well as the coupling matrix elements of the discrete states to the scattering states have to be calculated [2, 10, 11, 12, 13]. The nucleon-nucleon interaction is included in these calculations by starting from the eigenvalues EiB and eigenfunctions ΦB i of the Hamiltonian HB , Eq. (2), and calculating the coupling matrix elements by means of them. Since the nuclear forces are not known, the calculations for nuclei can be done also by identifying the interaction involved in HB with the well-established effective forces involved in Re (Heff ) such that only the non-Hermitian part Im(Heff ) has to be taken into account additionally [14, 15, 16, 17]. In any case, the FPO method allows to take into account the many-body effects in a straightforward manner. The excitation of the levels of a QD occurs in a manner being similar to the excitation of the compound nucleus levels in low-energy nucleon-induced nuclear reactions. In the FPO method, the electron-electron interaction should be included into HB by standard methods used for closed QDs with discrete states. In the regime of (almost) isolated resonances (at low level density), the electron-electron interaction causes (almost) the same effects as they are known from the study of closed QDs. At high level density, however, the spectroscopic properties of open quantum systems are determined mainly by the second-order term of Heff (i.e. by the interaction of the levels via the common continuum). In this regime, the degree of overlapping of neighbored resonance states is decisive for the behavior of the open system. This behavior is generic. Here, we are interested mostly in the properties of QDs at high level density. We will not consider therefore the electron-electron interaction in detail.
460
5.2.
Ingrid Rotter and Almas F. Sadreev
Branch Points in a Double Quantum Dot
Branch points play an important role in open quantum systems (Sect. 3.5.), including in QDs that are opened by attaching leads to them. In this section, we follow the study performed in [20] and consider branch points in the case of a double QD that consists of two single QDs coupled to each other by a wire, Fig. 8. The S matrix theory for transmission through such a QD can be formulated by using the non-hermitian effective Hamilton operator Heff that appears in the framework of the FPO technique, Sect. 2.3..
w
v
u
QD
u
v
reservoir
reservoir
QD
Figure 8. The double dot system is connected to the reservoirs by the coupling constants v. The single dots are coupled to the wire by the coupling constants u. Figure taken from [20]. The effective Hamilton operator for QDs in the tight-binding approach contains the spectroscopic properties of the closed QD as well as the coupling matrix elements between the dot and the two attached leads. In the subspace of localized states, the effective Hamilton operator has the general form (12) where HB is the Hamiltonian of the closed double dot system (in a basis in which HB is diagonal), HC is the Hamiltonian of the left (C = L) and right (C = R) reservoir and E + = E + i0. The matrix elements of Heff are calculated in the basis of the eigenstates of HB . The second term of Heff takes into account the coupling of the eigenstates of HB via the reservoirs (continuum of incoming and outgoing waves) when the system is opened. The corresponding coupling matrix elements are denoted by VBC and VCB , respectively. For illustration, we restrict the consideration to the case with only one state with energy ε1 in each single dot, one mode propagating in the wire, and one channel (scattering wave function) in each of the two attached leads. We consider as a linear function of the length L of the wire. The dependence of L may be replaced by a dependence on, e.g., the diameter of the wire without any influence on the discussion of the physical results. For simplicity, we consider first a symmetrical system: the coupling of the two single dots to the internal wire, denoted by u, is assumed to be the same for the two single dots. Also the coupling strength v between the whole double dot and the attached leads is taken to be the same for both leads. The effective Hamiltonian of such a system is (12) where
ε1 u 0 HB = u (L) u 0 u ε1
(87)
Unified Description of Resonance and Decay Phenomena...
461
is the Hamiltonian of the closed double dot system. The coupling matrix between the closed double dot system and the reservoirs can be found after specifying both systems. We take the reservoirs (leads) as semi infinite one-dimensional wires in tight-binding approach (Sect. 2.3.). The connection points of the coupling between the system and the reservoirs are at the edges of the one-dimensional leads. Then the coupling matrix elements take the following form [73] r sin k ψm (1), Vm(E, L) = vψE,L(xL)ψm (j = 1) = v 2π r sin k ψm (3), (88) Vm(E, R) = vψE,L(xR )ψm (j = 3) = v 2π where k is the wave vector related by E = −2 cos k to the energy, ψm(j), j = 1, 2, 3 are the eigenfunctions of (87), and v is the hopping matrix element between the edge of the lead and the QD. The v will be varied in our calculations. The eigenvalues of the Hamiltonian (87) are real, ε1 + (L) ∓ η, 2
B = E1,3
E2B = ε1 ,
(89)
where η 2 = ∆ε2 + 2u2, ε1 − (L) , ∆ε = 2
(90) (91)
and the eigenstates read −u 1 η + ∆ε , |ΦB 1i = p 2η(η + ∆ε) −u
1 1 |ΦB 0 , 2i = √ 2 −1 u 1 η − ∆ε . |ΦB 3i= p 2η(η − ∆ε) u
(92)
Using (88) to (92), we get the following expression for the effective Hamiltonian [73], B v 2 u2 eik v 2√ueik E1 − η(η+∆ε) 0 2η 0 ε1 − v 2eik 0 (93) Heff = , v 2√ueik v 2 u2 eik B 0 E − 3 η(η−∆ε) 2η which is symmetric. Its complex eigenvalues zλ and eigenvectors φλ are [73] z2 = ε1 − v 2 eik , z1,3 = and
ε1 + (L) − v 2eik ∓ 2
a |φ1 i = 0 , b
s
(L) − ε1 + v 2eik 2
0 |φ2 i = 1 , 0
2
+ 2u2
b |φ3i = 0 , −a
(94)
(95)
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Ingrid Rotter and Almas F. Sadreev
where f , a = −p 2ξ(ξ + ω) f
=
v 2ueik √ , 2η
b=
s
ξ+ω 2ξ
∆εv 2eik , 2η
ω = −η +
ξ2 = ω2 + f 2.
(96)
The eigenfunctions are biorthogonal according to (53). The energy is related to the wave number by E = − 2 cos k. As a consequence, the eigenvalues of the effective Hamiltonian are k dependent (inside the band). It is this k-dependence by which our model differs from the standard S matrix theory formulated in, e.g., [28]. It is interesting to remark that the eigenfunction |φ2i of Heff does not depend on the parameters. This is a consequence of the symmetry involved in HB as well as in Heff . The eigenvalue z2 depends on the coupling strength √ v between system and environment. The two eigenvalues z1,3 of Heff differ by 2 F where F =
2
v 2eik − ∆ε 2
+ 2u2 = ξ 2 .
(97)
The point at which F = 0, is a branch point. The two equations for the branch point take the following form 1 2 v2 cos kcr = − cr Ecr, (98) ∆ε(Lcr) = vcr 2 4 4 2 Ecr vcr 2 1− , (99) 2ucr = 4 4 which define a surface of branch points for the four parameters of the system (as shown in Fig. 9). For the energy at which the eigenvalues zλ coalesce, the fixed-point equation Eλ = Re(zλ)|E=Eλ and the equation Γλ = −2 Im(zλ)|E=Eλ can be easily solved analytically. We obtain 4∆ε(Lcr) (100) Ecr = Eλ = (Lcr ) = − 2 vcr and u2cr
∆ε(Lcr )2 = 2
4 −1 . (Lcr)2
(101)
These conditions reduce the number of physical parameters from four to three, vcr, ucr, Lcr, related to one other by two equations. We underline that the coalescence of two eigenvalues of Heff at a certain energy E of the system does not mean that also two poles of the S matrix coalesce at this energy. energy dependent functions. Only the The point is that the eigenvalues zλ of Heff are solutions of the fixed-point equations, Eλ = Re zλ E=E , and the widths defined by Γλ = λ −2 Im zλ E=E are numbers that correspond to the poles of the S matrix (Sect. 3.2.). λ In the general case, the two levels whose (energy dependent) eigenvalues coalesce at the energy E = Ecr, avoid crossing.
Unified Description of Resonance and Decay Phenomena...
2
1
E
v
2
0 2 0
∆ε
0 −2
−2
u
0
−2 −2
2
(a)
463
(b)
2 0
0
∆ε
2
−2
u
Figure 9. The surfaces of the branch points in the four-manifold parameter space ∆ε(L) ≡ (ε1 − (L))/2, u, v, E, defined by Eqs. (98) and (99). Figure taken from [20].
5.3.
Approaching Branch Points of a Double Quantum Dot
Here, we are interested in the question how the eigenvalues zλ , Eq. (94), and eigenvectors φλ , Eq. (95), of the effective Hamiltonian Heff , Eq. (93), behave if we trace them along a certain line that touches the surface of the exceptional points. In order to find an answer, we follow (as in Sect. 5.2.) the study performed in [20]. For a fixed energy E the surface of exceptional points reduces to a line. If we start at v = 0 (and u = ucr, ∆εcr ≡ ∆ε(Lcr)), the path will cross the line at v = vcr. At the line, the absolute values of the eigenvector components |a|, |b| are singular and the phases of the components α = arg(a), β = arg(b) are not determined. We can therefore not trace the path when crossing the line. We can choose, however, paths that cross the very neighborhood of the line. For illustration, the behavior of the eigenvalues zλ and eigenvectors φλ as a function of the coupling strength v along the two paths with L = Lcr + ∆L is shown in Fig. 10 where ∆L = 0.01 is small as compared to Lcr. The parameters are chosen as ε1 = 1, (L) = 2 − L/5, √ u = ucr, E = Ecr but L = Lcr + 0.01. Here Lcr = 1.4645, ucr = 1/4 and Ecr = 2 are the critical physical parameters which define the line of branch points shown in Fig. 10 provided that v = vcr = 1. The real parts of z1 and z3 repel each other at v < 1 and cross at v = vcr0 ≈ vcr = 1. The imaginary parts of z1 and z3 are similar if v < 1 but |Im(z1) − Im(z3)| 6= 0 for all v, including the critical value. If v > 1, the widths bifurcate. The figure shows further that also the complex amplitudes a and b of the eigenvector components, defined by (95), (96) have characteristic features at v = vcr0 (vcr00 ) : |a| 1, |b| 1, and the phases α = arg(a) and β = arg(b) jump by +π/4 if L = Lc + 0.01 (Fig. 10). They jump by −π/4 if L = Lc − 0.01 [20]. Note that the features observed in the amplitudes a and b at v ≈ vcr are the more pronounced the smaller ∆L is. According to (95), (96), |a| → ∞, |b| → ∞, vcr0 (vcr00 ) → vc if ∆L → 0. The phase jumps at L = Lcr ± ∆L are of different sign when traced as a function of increasing v [20]. When traced, however, in one case as a function of increasing v and in the other case as a function of decreasing v, the two phase jumps add to ±π/2. This last case corresponds to a connecting of the two paths with L = Lcr ± ∆L at v = ±∞, i.e. an
464
Ingrid Rotter and Almas F. Sadreev 4 0
Im(z1,3)
Re(z1,3)
3 2 1 0 0
(a)
−0.5 −1 −1.5
0.5
1
1.5
2
(b) −2
0
0.5
1
0.2
4
0
α/π, β/π
|a|, |b|
5
3 2 1
(c)
0 0
1.5
2
1.5
2
v
v
−0.2
v
−0.4 −0.6 −0.8
0.5
1
1.5
v
−1 0
2
0.5
(d)
1
v
Figure 10. The evolution of the eigenvalues z1 (solid lines) and z3 (dashed lines) (a, b) and of the components a = |a| eiα (dashed lines) and b = |b| eiβ (solid lines), defined by (95), of the eigenfunctions φ1 and φ3 (c, d) of the effective Hamiltonian Heff as a function of v √ for E = Ec = 2. The parameters are u = ucr = 1/4, L = Lcr − 0.01, Lcr = 1.4645. At the critical value of v, |a| 1, |b| 1 and the phases jump by −π/4. Figure taken from [20]. encircling of the branch point along a path that is very different from a circle. Analogues results are obtained when the evolution of the eigenvalues zλ and of the components a and b of the eigenfunctions of the effective Hamiltonian Heff are considered as a function of another parameter. The parameter may even be the energy E of the system [20]. In all cases, the real and imaginary parts of the components of the wave functions evolve differently in the regime of avoided level crossing. This result corresponds to the loss of phase rigidity in approaching a branch point (Sect. 3.3.). It is a generic phenomenon.
5.4.
Encircling Diabolic and Branch Points
Let us now consider the geometric phases of the diabolic and branch points of the double QD looked at in Sect. 5.2.. The two cases are (i) the real eigenvalues E1B and E3B of HB coalesce and (ii) the complex eigenvalues z1 and z3 of Heff coalesce. The first case is a diabolic point while the second one is a branch point. Both points with coalesced eigenvalues are related to avoided level crossings. The condition for a branch
Unified Description of Resonance and Decay Phenomena...
465
point to appear is F = 0, Eq. (97), while that for a diabolic point is η = 0, Eq. (90). Discrete states of the double QD can cross therefore only if the interaction between the single QDs and the internal wire vanishes, u = 0. In contrast to this, resonance states may cross, according to (97), also when the interaction between them is different from zero. This holds for the direct internal interaction u as well as for the external interaction v of the resonance states via the continuum of scattering wave functions due to their overlapping. 1/2
X
u
2
mapping
θ DP
BP
∆ε
Y
Y
mapping 0.1
BP u
φ
0.05
BP
X DP 0
−0.05
0
∆ε
0.05
Figure 11. The encircling of the diabolic point u = 0, ∆ε = 0, given by Eq. (102) [top left] and its mapping onto the plane X, Y [top right] according to Eq. (105). The encircling of the branch point X = 0, Y = 0, given by Eq. (104), [bottom left] and its mapping onto the plane ∆ε, u [bottom right] according to Eq. (107) for E = 0, v = 0.5. Dashed line: R = v 4/4, solid line: R = v 4/6. The point marked by an open circle corresponds to the branch point X = 0, Y = 0. The point marked by a full circle corresponds to the DP ∆ε = 0, u = 0. Figure taken from [20]. The topological structure of the diabolic points is completely different from that of the branch points as an encircling of the two singular points shows. Let us first analyze the diabolic point, at which two real eigenvalues of the hermitian Hamilton operator HB coalesce. The properties of these points are well known [53]. They are related to avoided crossings of discrete levels, and an encircling of them causes the well-known Berry phase [52]. In our case, the diabolic point is defined by η = 0, i.e. by u = 0, ∆ε = 0 according to Eq. (90). Both values u and ∆ε = 0 are real. By encircling the diabolic point according
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Ingrid Rotter and Almas F. Sadreev
to ∆ε = η cos θ,
√ 2u = η sin θ,
(102)
we obtain that E1B and E3B vary as cos θ and
− sin θ/2 cos θ/2 √ √ 1 1 |Φ1i = √ 2 cos θ/2 , |Φ3i = √ 2 sin θ/2 . 2 2 − sin θ/2 cos θ/2
(103)
We see immediately that after each encircling of a diabolic point, the eigenvalues of HB are restored and the eigenfunctions change their sign. That means, the eigenstates of the closed system are restored after two cycles. Eqs. (103) express the well-known Berry phase in our model system. Let us now consider the branch points that appear in the open system when F = ξ 2 = 0, Eq. (97). We encircle the branch point by defining F = X + iY = R exp(iφ), R = |F |. Substituting these expressions into (97) gives X = |F | cos φ = η 2 + Y
= |F | sin φ =
v4 cos 2k − v 2 cos k ∆ε, 4
v4 sin 2k − v 2 sin k ∆ε. 4
(104)
We see that by encircling a branch point, the eigenvalues zλ of the effective Hamiltonian Heff behave as z1 − z3 ∼ exp(iφ/2) while the components of the eigenstates behave as a, b ∼ exp(−iφ/4). This means that the eigenvalues of the effective Hamiltonian are restored after two cycles while the eigenstates are restored only after four cycles. This result corresponds to the discussion in Sect. 3.6.. It shows that the difference between the geometric phases obtained in our model system for the diabolic and branch points, respectively, is generic. The next question is: does the encircling of the diabolic point give rise to a nontrivial phase behavior of the eigenstates of Heff ? From (102) and (104) we obtain X = r2 + Y
=
v4 cos 2k − v 2 cos k r cos θ, 4
v4 sin 2k − v 2 sin k r cos θ 4
(105)
√ where r = η. The mapping of the encircling of a diabolic point (in the plane ∆ε, 2u) onto the complex plane X, Y is shown in Fig. 11. Irrespective of the choice of the parameters E and v of the open system, the encircling of the diabolic point maps onto a straight line in the X, Y plane. It does not cross the branch point X = 0, Y = 0. Hence the encircling of the diabolic point has no consequence in the open system. We will now consider the opposite case by starting from the encircling of the branch point in the complex plane X, Y . Some simple algebra gives us, according to (105), the
Unified Description of Resonance and Decay Phenomena...
467
following mapping
2u2
v2 E− 4
Y q 2 v 2 1 − E4 E2 Y2 v4 1− − = X+ 4 4 v2 1 −
∆ε = −
E2 4
.
(106)
This mapping simplifies by choosing E = 0, ∆ε = −
Y v2
2u2 = X +
v4 Y 2 − 2 . 4 v
(107)
From these expressions, we obtain the condition R ≤ v 4/4 for the encircling radius. The mapping (107) is shown also in Fig. 11 by the dashed line for a fixed value of the coupling constant v. The case E 6= 0 shifts the mapping to the plane (∆ε, u) but never encircles the diabolic point. Thus, irrespective of the choice of v and E, the encircling of the branch point does not encircle the diabolic point. The conclusions from the two mappings shown in Fig. 11 are the following. (i) The encircling of a diabolic point gives rise to a geometric phase in the closed system, and does not cause any phase in the open system. (ii) The encircling of the branch point in the complex plane gives rise to a geometric phase in the open system but has no effect in the closed system. The difference between the two cases is, as discussed in Sect. 3.6., the following. The diabolic point is defined in the closed quantum system that is described by the Hermitian Hamilton operator HB . It contains the direct (first-order) interaction between the states. The branch point however, characterizes an open quantum system described by the nonHermitian Hamilton operator Heff in the extreme situation that the (second-order) term of interaction via the environment is decisive. This difference is generic. It expresses the relation of the geometric phases to the interaction type involved in the Hamilton operator. The difference between branch points in the complex plane and diabolic points in the real plane is related to nonlinear effects caused by the overlapping of resonance states. In order to show this, let us rewrite the effective Hamiltonian (93) as 0 0 f 0 0 + W = Heff + 0 0 0 (108) Heff = Heff f 0 0 0 is the diagonal part of H where Heff eff and the nondiagonal matrix elements f describe the coupling between the two resonance states 1 and 3 due to their overlapping. Then, the 0 reads Schr¨odinger equation with the Hamiltonian Heff 0 0 f 0 − zλ )|φλi = − 0 0 0 |φλ i ≡ −W |φλ i . (109) (Heff f 0 0
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Ingrid Rotter and Almas F. Sadreev
From the biorthogonality relations (54) and (55) follows hφλ |φλi = Rehφλ |φλi , hφλ|φλ0 i = i Imhφλ |φλ0 i = −hφλ0 |φλi ,
Aλ ≡ hφλ|φλi ≥ 1 0
|Bλλ | ≡ |hφλ|φλ0 i| ≥ 0 , λ 6= λ0 .(110)
Using these equations, the r.h.s. of Eq. (109) reads [45, 4] X X hφλ |W |φλ0 i hφλ|φλ00 i|φλ00 i W |φλ0 i = λ=1,3
λ00 =1,3
0 = W 1λ A|φ1 i + iB|φ3 i + W 3λ A|φ3 i − iB|φ1 i 0
(111)
0
with W λλ ≡ hφλ |W |φλ0 i, λ, λ0 = 1, 3. This relation gives |φλ0 i → ± i |φλ0 i, λ0 6= λ, in approaching the branch point due to Aλ0 =→ ∞, Bλλ0 =→ ∞ what agrees with (57). Furthermore, we see that nonlinear terms, caused by the interaction f , appear in the Schr¨odinger equation (109) as soon as A 6= 1 and B 6= 0 (i.e. as soon as the resonance states overlap). This means, nonlinear terms in the Schr¨odinger equation appear due to the overlapping of resonance states. They are large in the neighborhood of a branch point. An analogous effect does, of course, not occur in the neighborhood of a diabolic point. These results show that diabolic and branch points exist also in open QDs. They determine the spectroscopic properties of open quantum systems, see Sect. 3.6., and therefore also those of QDs.
5.5.
Bound States in the Continuum (BICs) in a Double Quantum Dot
In order to illustrate the appearance of bound states in the continuum (BICs) in QDs, we consider a double QD consisting of two single QDs and a wire connecting the two single dots, Fig. 8. In difference to Sect. 5.2., each of the two single dots has two states (since BICs do not appear in a 1d chain of dots). We follow here the study performed in [21]. The Hamiltonian of the closed double dot is ε1 0 u 0 0 0 ε2 u 0 0 (112) HB = u u (L) u u . 0 0 u ε2 0 0 0 u 0 ε1 For simplicity we have assumed in (112) that all the coupling constants between the wire and the single QD are the same and are given by the constant value u. The Hamiltonian (112) is written in the energy representation. In order to specify the connection between the reservoirs and the single QDs, we have however to know the eigenstates of (112) also in the site representation. The Hamiltonian of the single QD in the site representation is ε0 ub . (113) Hb = ub ε0 The eigenfunctions and eigenvalues of Hb are the following 1 1 1 1 , hj|ε2i = √ hj|ε1i = √ 2 1 2 −1 ε1,2 = ε0 ∓ ub .
(114) (115)
Unified Description of Resonance and Decay Phenomena... We introduce the projection operators X |εbL ihεbL | , Pw = |1w ih1w | , PL = bL
PR =
X
|εbR ihεbR |
469
(116)
bR
where bL = 1, 2, bR = 1, 2, and |1w i is the one-dimensional eigenstate of the wire. Let Em and |mi with m = 1, ..., 5 denote the five eigenenergies and eigenstates of (112), HB |mi = Em |mi. The elements of the left coupling matrix are hL, E|V |mi = hL, E|V PL |mi =
X hL, E|V |εbL ihεbL |mi bL
=
X X hL, E|V |jL ihjL|εbL ihεbL |mi.
(117)
jL =1,2 bL
Similar expressions can be derived for the right coupling matrix. Here we used the assumption that the left reservoir is connected only to the left single QD and the right reservoir only to the right single QD. As previously, the reservoirs are assumed to be semi-infinite one-dimensional wires. Next we have to specify which sites of the left (right) single QD are connected to the left (right) reservoir. There are two possibilities. (i) Assume the left reservoir is connected only to the first site jL = 1 of the left single QD. Then, with account of (114), (117) becomes r sin k X hεbL |mi. (118) hL, E|V |mi = v 2π bL
A corresponding expression can be written down for the right coupling matrix if the right reservoir is connected to the first site of the right single QD. (ii) We can assume that the reservoirs are connected to both sites of the single QDs with the same coupling constant v. Then the elements of the coupling matrices (118) are the following r sin k hε1|mi (119) hL, E|V |mi = v 2π provided that the energy level ε1 is the lowest in energy, see (115). Knowing the Hamiltonian (112) of the closed system, diagonalizing it and calculating the coupling matrix elements (118) or (119) to the reservoir, the effective Hamiltonian Heff , Eq. (12), can be obtained. After diagonalizing Heff , the transmission through the double QD is given by (30). The transmission probability is T = |t|2. The expression (30) is unitary at all energies E also in the regime of overlapping resonances (see Sect. 2.2.). In Fig. 12, the transmission probability versus energy E and length L of a double QD is shown for the case that the single dots are equal, each single dot has two states and both sites of the single QD are connected to the reservoir with the same coupling matrix elements (119). The figure shows a zero in the transmission probability, Fig. 12(b). According to Fig. 12(c) and (d), the positions and decay widths of the eigenstates 2 and 4 of the effective Hamiltonian are independent of the length L of the wire while those of the other states depend strongly on L. The state 3, lying in the middle of the spectrum, crosses the
470
Ingrid Rotter and Almas F. Sadreev
Figure 12. (a) The transmission through a double QD with two identical single QDs that are connected by a wire according to Fig. 8 [0 (black) ≤ |t| ≤ 1 (white)]. The eigenvalues of HB are shown by full lines. ε1 = −1.7, ε2 = −1.4 and (L) = −1 − L/5 (dashed line), v = 0.3, u = 0.1. (b) The modules of the transmission amplitude |t(E, L)| for the same double QD as in (a) for fixed lengths L = 2.75 (solid line) and L = 4 (dashed line). The energies of the two single QDs are shown by circles. The real part (c) and imaginary part (d) of the 5 eigenvalues zk of the effective Hamiltonian as a function of L for E = −1.5. Thin solid line: z1 , dashed line: z2 , thick solid line: z3, dotted line: z4 , and dash-dotted line: z5 . At L = 2.75 the imaginary part of the third eigenvalue is equal to zero at a all energies E. Figure taken from [21].
transmission zero at L = 2.75. Here, the decay width of this state approaches zero for all energies E. We consider now the evolution of the modules of the transmission amplitude and the corresponding phase shifts when the decay width of one of the states approaches zero. The results shown in Fig. 13 are performed for the double QD the transmission of which is shown in Fig. 12 together with the eigenvalues zλ of the effective Hamiltonian Heff as a function of L. At L = 2.75, the third eigenstate crosses the energy of the transmission zero, and its decay width Im(z3 )/2 approaches zero. As long as L 6= 2.75 and Im(z3 ) 6= 0, the phase of the transmission amplitude varies by π more or less smoothly, according to the phase shift caused by a resonance state with a finite decay width. If L → 2.75 and Im(z3 ) → 0, the phase jumps by π due to the vanishing decay width of the resonance state (Fig. 13). Therefore, we have also in this case a phase jump of the transmission amplitude by π. It is the only signature of the resonance state when its width vanishes. We mention here that resonance states with vanishing decay width are considered also
Unified Description of Resonance and Decay Phenomena...
471
2.5
|t(E)|, arg(t(E))/ π
2
1.5
1
L=2.25 L=2.5
0.5
L=2.75
0 −1.62
L=3.0 L=3.25 −1.6
−1.58
−1.56
−1.54
−1.52
−1.5
−1.48
−1.46
−1.44
E
Figure 13. The energy dependence of the modules |t(E)| (bottom) and of the phase arg(t(E))/π (top) of the transmission amplitude for L = 2.25, 2.5, 2.75, 3.0, 3.25. The other parameters are the same as those in Fig. 12. The transmission zeros are denoted by stars. They are of second order. The ordinate is shifted every time by 0.1 when L is changed by 0.25. All phases are shifted by π. Figure taken from [21].
by other authors. In [74], they are called ”ghost” Fano resonances that appear in a double QD attached to leads. In [75], the appearance of discrete levels in the continuum is shown to correspond to the occurrence of special localized electron states that appear due to a ”collapse” of Fano resonances. In laser-induced continuum structures in atoms, the phenomena related to resonance states with vanishing decay width are known as population trapping [8, 63]. They result from the interplay of the direct coupling of the states and their coupling via the continuum under the influence of a strong laser field. In the case considered in the present paper, they appear due to some constraint onto the system as a consequence of the unitarity of the S matrix. On the one hand, the position of the transmission zeros is determined by the spectroscopic properties of the single dots. On the other hand, however, the transmission is resonant and related to the spectroscopic properties of the double QD. These two facts cause some nontrivial constraint onto the system in order to fulfill the condition of unitarity of the S matrix for the double QD as a whole. In order to achieve the transmission zero of the double dot, the width of the state in the middle of the spectrum is strongly parameter dependent since it has to vanish when it crosses the energy of the transmission zero as a function of the length L of the wire, Let us now consider the case that the symmetry involved in the double QD is distorted by assuming that the two single dots have each two states, but different from one another, εiL 6= εiR , i = 1, 2 (Fig. 14). In contrast to the foregoing case, the width of the state in the middle of the spectrum does not approach zero. Although it depends strongly on energy, it remains different from zero for all L. The constraint onto the middle state is reduced due
472
Ingrid Rotter and Almas F. Sadreev
Figure 14. The same as Fig. 12, but with two different single QDs: ε1L = −1.7, ε2L = −1.4, ε1R = −1.6, ε2R = −1.3. Figure taken from [21]. to the different spectra of the two single dots. The width of this state is reduced only in such a manner that the state with this value of the width is able to interfere destructively with the other four states in order to achieve the two transmission zeros. At each of these transmission zeros, the phase jumps by −π. If each single QD has N states, the number of zeros in the transmission through the double QD is 2(N − 1) when the spectra of the two single dots are different from one another. This conclusion is demonstrated by the results of numerical calculations [21]. These results show that transmission zeros in a double QD show some nontrivial behavior since two conditions for their appearance have to be fulfilled which are independent from one another. On the one hand, the transmission zeros are related to the spectroscopic features of the single dots due to the fact that full reflection is determined by the area of attachment, and the leads are attached to the single dots. On the other hand, however, the resonance states of the system are characteristic of the double QD as a whole. As a consequence, even the number of transmission zeros differs, as a rule, from the number of resonance states. This result does not agree with the simple Fano interference picture where each resonance state creates a zero in the reaction cross section due to its interference with the smooth background. The formation of the BIC and of the state with small (but nonvanishing) width, respectively, occurs together with the formation of states with larger widths. Thus, also in the complicated case of a double QD, the BIC appears due to widths bifurcation. It originates, eventually, from the avoided level crossing phenomenon. We underline that the unitarity of the S matrix at all energies is ensured by the strong parameter (and energy) dependence of the decay widths of the resonance states.
Unified Description of Resonance and Decay Phenomena...
5.6.
473
Internal Impurity of an Open Quantum Dot (QD)
The spectroscopic properties of the open QD are determined by the non-Hermitian Hamilton operator Heff , Eq. (12). It contains the internal as well as the external interaction of the electrons. The internal interaction is of standard type. It is real and contained in the eigenvalues EiB and eigenfunctions ΦB i of the Hermitian Hamilton operator HB of the closed QD, Eq. (2). The external interaction is caused by the coupling of the eigenstates of HB to the continuum of scattering wave functions into which the QD is embedded. It consists of the principal value integral and the corresponding residuum of the second part of Heff and is therefore complex. The real part determines the shift in energy of the eigenstates of Heff relative to the eigenstates of HB , Eq. (61). Mostly, this energy shift is taken into account eff with the matrix effectively by considering the effective Hermitian Hamilton operator HB elements Z∞ γ ˆiC γˆjC 1 X B eff B B B eff eff P dE 0 . (120) hΦi |HB |Φj i = hΦi |HB |Φj i + Wij ; Wij ≈ 2π E − E0 C
C
The imaginary (non-Hermitian) part of Heff gives the widths (inverse lifetimes) of the eigenstates of Heff . It is mostly neglected. That means, the mutual distortion of the resonance states that is directly related to the branch points, is usually not considered. eff are discrete, their widths are vanishing and their lifeThe eigenstates of HB and HB times are infinitely large. This statement holds true as long as the QD is clean, i.e. as long as it does not contain any magnetic (or other) impurities. The lifetime of the states is expected therefore to diverge as temperature is lowered, being infinite at zero temperature [76]. The situation is however another one for open quantum systems when not only the eff is considered but also the non-Hermitian part effective Hermitian Hamilton operator HB eff ) and have, generally, a Im(Heff ). The eigenstates of Heff are localized (as those of HB finite lifetime also at zero temperature. There is no need for the lifetimes of the states to diverge in approaching zero temperature. In open quantum systems, there is still another effect that is unknown in standard quantum mechanics for closed systems with discrete states. Although the eigenstates of Heff are characterized also by discrete quantum numbers, they are not orthogonal to one another in the standard manner. The eigenfunctions are rather biorthogonal according to (54) and (55). Under the influence of neighboring states, the phases of the eigenfunctions φλ of Heff are not rigid. The phase rigidity rλ = 1/Aλ of the eigenfunction φλ is a quantitative measure for the perturbation of the state λ by a neighboring state (Sect. 3.3.). It is rλ = 1 for a state well separated from others, while rλ → 0 when a branch point, characterized by zλ0 = zλ , is approached. Thus, a reduced value of the phase rigidity rλ describes an internal impurity of an open quantum system that arises from the impact that one resonance state has on a neighboring one. Due to this mutual influence of neighboring states, the phase relations between the different states of the system are changed. It has to be underlined here that the internal impurity involved in an open quantum system, differs fundamentally from a magnetic impurity that is considered, in many papers, to be the source of dephasing observed experimentally at low temperatures. The internal impurity is an internal property of the open system (Sect. 4.1.). It depends on the degree of overlapping of the states and does not vanish at zero temperature (in contrast to the magnetic
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Ingrid Rotter and Almas F. Sadreev
impurity that is assumed usually to vanish at T = 0). Moreover, the lifetimes of the states of an open QD do not diverge in approaching zero temperature even for vanishing internal impurity (i.e. for rλ = 1). The sole observation of saturation of the phase coherence time in experimental data at low temperature can therefore not be considered to be a signature of the existence of an impurity. It expresses only the fact that most states of an open quantum system have a finite lifetime. Only if the eigenstates of Heff overlap (i.e. if rλ < 1) an internal impurity appears. For numerical examples see Sect. 6.4..
5.7.
Electron Phase Coherence Time
Comparing the basic ingredients of the theory of open quantum systems with the experimental results on dephasing at very low temperature, it should firstly be stated that the concept dephasing is used differently in the different papers. Within the framework of Landau’s theory of Fermi liquids, dephasing is related to the time an electron can travel in the system before losing its phase coherence and thus its wave-like behavior. In open quantum systems, however, the spectroscopic properties of localized states are considered which are described by the non-Hermitian Hamilton operator Heff . The phases of the eigenfunctions of Heff are well defined but not rigid, generally (Sect. 3.3.). The phase rigidity rλ, Eq. (58), represents a measure for the dependence of the phases of these states on (external) parameters. It represents therefore an internal property of an open quantum system (see Sect. 5.6.) that may be controlled by varying external parameters such as, e.g., the coupling between QD and leads attached to it. In the following, a short discussion of the results obtained experimentally on dephasing will be given from the point of view of an open quantum system. The discussion is qualitatively by using the results obtained in different recent studies. It avoids to comment the many controversial discussions that exist in the literature to this question. In the proceedings [76] of a recent conference, Saminadayar et al. review the experimental progress on the saturation problem in metallic quantum wires. As a conclusion of this analysis, based on all presently available measurements of the phase coherence time τφ in very clean metallic wires, it is hard to conceive that the apparent saturation of τφ is solely due to the presence of an extremely small amount of magnetic impurities. Hackens et al. [77] are interested in the determination of the absolute value of τφ , and not just its temperature dependence. They find that the electron dwell time is the central parameter governing the saturation of phase coherence at low temperature. The condition for the occurrence of saturation is found to be τφsat ≈ τd where τφsat is the saturated coherence time and τd is the dwell time. This simple behavior holds over the three orders of magnitude covered by the available data in the literature. According to the authors, τφ is found to be intrinsic to the physics of the QDs, but not due to the coherence time of the electrons themselves. Hackens et al. find furthermore [77] that τφ is strongly influenced by the population of the second electronic subband in the quantum well. According to Lin et al. [76], one consensus has been reached by several groups, saying that the responsible electron dephasing processes in highly disordered and weakly disordered metals might be dissimilar. That is, while one mechanism is responsible in weakly disordered metals, another mechanism may be relevant for the saturation or very weak temperature dependence of τφ found in highly disordered alloys. According to the authors, the
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intriguing electron dephasing is very unlikely due to magnetic scattering. It may originate from specific dynamical structure defects in the samples. Golubev and Zaikin [76] collected experimental data from many different publications for τφ0 obtained in metallic samples with different diffusion coefficients. The conclusion is that low temperature saturation of τφ is universally caused by electron-electron interactions. They found seemingly contradicting dependencies of τφ0 on the diffusion coefficient D in weakly and strongly disordered conductors. While the trend ”less disorder – less decoherence” for sufficiently clean conductors is quite obvious, the opposite trend ”more disorder – less decoherence” in strongly disordered structures is unexpected. All these statements obtained from the results of many experimental studies fit qualitatively into the expectations received by considering the QD as an open quantum system. First of all, the saturation of τφ appears in a natural manner since most states of an open QD have a finite lifetime at zero temperature. The value of the lifetime can be obtained from the imaginary part of the complex eigenvalue zλ of the non-Hermitian Hamilton operator Heff [i.e. from Im(zλ )]. It expresses the time the electron stays in the QD. This time is called usually dwell time. Thus, the result obtained by Hackens et al. [77] supports the description of the QD as an open quantum system. Also the more complicated result of different processes in weakly and strongly disordered systems is by no means in contradiction to the properties known for the eigenstates of open quantum systems. In some cases, τφ0 depends only weakly on the electron diffusion constant D: it is somewhat smaller when D is larger. That means, states with a large lifetime give only a small contribution to the diffusion – a result which is very well known. In other cases, the relation between τφ0 and the diffusion constant D shows the opposite trend. Also in this case the states with a large lifetime give, of course, a small contribution to the diffusion. In contrast to the foregoing case, however, the main contribution to the diffusion arises obviously from short-lived states. This follows from the resonance trapping phenomenon (widths bifurcation) characteristic of the regime of overlapping resonances. In this regime the widths of these states increase and the widths of the trapped resonance states decrease with increasing degree of overlapping (see Sects. 3.2. and 6.2.). Finally, the short-lived resonance states form some background for the long-lived resonance states (see Sect. 6.3.). The diffusion constant is determined mainly by the contribution of the background states. Therefore, the diffusion constant D increases with increasing τφ0 of the (long-lived) resonance states – a result being counterintuitive in the same manner as the resonance trapping effect. The last one is directly proven experimentally [38]. In this respect another experimental result obtained by Hackens et al. [77] is interesting. It shows that, in the systems considered, the value τφ is strongly influenced by the population of the second electronic subband in the quantum well. Obviously this means that the degree of overlapping of the states plays an important role for the lifetimes of the states – according to one of the basic properties of the eigenstates of Heff (see Sect. 3.2.). Further experimental studies related to this question would be very useful.
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Ingrid Rotter and Almas F. Sadreev
Transmission through Quantum Dots in the Few-Channel Case Isolated Resonances
In an exact description of resonance phenomena by using the FPO formalism, the nonhermitian effective Hamiltonian (12) appears. In the case of a QD, C consists of the waves in the right (R) and left (L) leads attached to the dot (Sect. 2.3.). In this representation, the resonance picture is determined by the eigenvalues zλ and eigenfunctions φλ of Heff which differ, generally, from those of HB . The amplitude for the transmission through a QD is (30). The eigenvalues zλ and eigenfunctions φλ of Heff are involved in (30) with their full energy dependence (Sect. 2.2.). According to (30), the transmission is resonant in relation to the real part of the eigenvalues of Heff . As long as the resonance states are more or less well isolated from one another, Heff is determined mainly by the Hamiltonian HB of the closed system [i.e. by the first term in (12)]. The transmission peaks appear at the positions Eλ ≡ Re(zλ)|E=Eλ ≈ EkB of the resonance states. Using the relation (64) for non-overlapping resonance states ( Aλ = 1), it follows E E |V |φλ ihφ∗λ|V |ξR i} Γλ = 2π{hξLE |V |φλ ihφ∗λ|V |ξLE i + hξR E ∗ E = 4πhξC |V |φλihφλ|V |ξC i
(121)
in the case of a QD with well separated resonance states and one channel in each of the two identical (semi-infinite) leads. The peak height is |t(E→Eλ )| =
4π E E |hξ |V |φλ ihφ∗λ|V |ξR i| = 1 . Γλ L
(122)
Except for threshold effects, the profile of the transmission peak is of Breit-Wigner type, determined by the width Γλ ≡ − 2 Im(zλ )|E=Eλ of the resonance state λ. We underline once more that the simple relations (121) and (122) hold true only for isolated resonances of Breit-Wigner shape (i.e. for the case that interferences between the resonance states, and also with some background, are not important). Under these conditions, transmission peaks appear at the energies Eλ ≈ EkB where EkB denote the energies of the discrete states of the corresponding closed system. One may say therefore that the transmission in the non-overlapping regime is caused by waves ”standing” in the cavity at the energies Eλ of the resonance states during the time determined by τλ ∝ 1/Γλ .
6.2.
Regime of Overlapping Resonances
The situation is another one when the resonance states overlap, i.e. when interferences between the individual resonance states can not be neglected. In the overlapping regime, the resonance states avoid crossings with neighbored resonance states. In contrast to (121), it holds E i (123) Γλ < 4π hξLE |V |φλihφ∗λ|V |ξR
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in the case with one channel in each of the two identical leads due to the biorthogonality of the eigenfunctions [A > 1 in (64)]. At E → Eλ, the transmission amplitude is t(E→Eλ ) = −2πi
Ei X hξLE |V |φλ0 ihφ∗ 0 |V |ξR λ λ06=λ
E − zλ0
− 4π
E hξLE |V |φλihφ∗λ|V |ξR i .(124) Γλ
It follows from (123) that the contribution of the state λ to t(E→Eλ ) is larger than 1. The unitarity condition is fulfilled due a phase change of the wave functions φλ [described by the biorthogonality of the {φλ }]. Moreover, the minima in the transmission between two resonance peaks may be filled up due to phase changes of the wave functions φλ and φλ0 of the two neighboring resonance states λ and λ0. Since interferences appear at and between the energies of every two overlapping resonance states, the profile of the transmission in the overlapping regime is different from that in the regime of isolated resonance states. The transmission does not necessarily show peaks at the positions Eλ of the single resonance states. Let us rewrite therefore the transmission amplitude (30) by means of (79), E ˆ Ei |V |Ψ t = −2πi hξC C
(125)
ˆ E being complex, in general. The advantage of this representation consists in the with Ψ R fact that it does not suggest the existence of Breit-Wigner peaks in the transmission probability. Quite the contrary, the transmission is determined by the degree of alignment of ˆ E with the scattering wave function ξ E in the leads (i.e. by the value the wave function Ψ C C E E ˆ i). hξC |V |Ψ C ˆ E is possible due to the interaction of the individual resonance states The alignment of Ψ R via the continuum in the overlapping regime (Sect. 4.1.). It is described by the nondiagonal matrix elements of the second term of (12). It is maximum in a certain energy region E ˆ E i is maximum in ∆E. The two limiting cases are ReΨ ˆ E = ±ImΨ ˆE |V |Ψ ∆E if hξC C C C ˆ E = 0. The first case describes a traveling mode through the cavity while the and ImΨ C second case means complete reflection (zero transmission). The alignment is a coherent collective phenomenon as shown in Sects. 4.1. and 4.2.. It causes resonance trapping that is accompanied by the formation of a few short-lived (aligned) resonance states. This effect is proven experimentally [38]. For illustration, we consider the case of extremely strong overlapping of two resonance states (corresponding to rλ1 = rλ2 = 0) which occurs at the branch point in the complex energy plane. Here two eigenvalues of Heff coalesce, Eλ1 = Eλ2 ≡ Eλ , Γλ1 = Γλ2 ≡ Γλ . In the case of one channel in each of the two identical leads, it follows from (30) 4π E E i + hξLE |V |φλ2 ihφ∗λ2 |V |ξR i → 0(126) hξLE |V |φλ1 ihφ∗λ1 |V |ξR t(E→Eλ ) = Γλ at E → Eλ due to |φλi → ± i |φλ06=λ i at the branch point, Eq. (57). That means, the transmission vanishes at the energy E = Eλ of the two resonance states. The transmission profile can be derived from (30). In analogy to (68) it is Γλ − t = −2 i E − Eλ + 2i Γλ
Γλ E − Eλ + 2i Γλ
2
.
(127)
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The interference between both terms in (127) causes two transmission peaks in an energy region ∆E that is characteristic of the first term of (127). The resulting ”antiresonance” at E = Eλ is narrower than a Breit-Wigner resonance, and the two transmission peaks are nonsymmetrical. Let us compare the transmission in the energy region ∆E when (i) there are two coalesced eigenvalues of Heff as discussed above and (ii) there are two (more or less) isolated resonance states resulting in two symmetrical transmission peaks of BreitWigner shape. In both cases we have two transmission peaks, however with a different profile. As a consequence, the transmission is different in the two cases. It is larger in the first case than in the second one. This example illustrates that the transmission t is controlled by the individual phase rigidities rλ. Nevertheless most interesting is, generally, the relation of t to the phase rigidity ρ, Eq. (84), of the scattering wave function. Both values are characterized by a sum over (overlapping) resonance states, Eqs. (79) and (30), with the same weight factors E i/(E − zλ ). In the case considered above, ρ is reduced but different from zero hφ∗λ|V |ξC in a certain region around the branch point (since the contributions from both states with rλ = 0 annihilate each other at the branch point [23, 24]), while t is enhanced but different from the maximum value. It is interesting to consider the two limiting cases ρ = 0 and ρ = 1 (with one channel ˜ ˜ in each of the two identical leads). In the first case, Ψ(r) is complex with Re Ψ(r) = ± ˜ ˜ ImΨ(r) according to (84). In the second case however, Ψ(r) is real. The relation of these two limiting cases to the transmission is as follows. (i) When ρ = 0, many resonance states are almost aligned with the scattering states E i are large and add (mainly) conin the leads. Many single overlap integrals hφ∗λ |V |ξC structively (since there are only two identical channels according to which the alignment takes place). An example with three resonance states is considered in [23, 24]. Since ˜ ˜ ReΨ(r) = ± ImΨ(r), the transmission is maximum. For the case of one channel in each of the two identical leads follows at most |t(∆E)| → 1
(128)
due to unitarity. The difference between the case of an isolated resonance state, Eq. (122), and that with overlapping resonance states, Eq. (128), consists in the profile of the transmission. While in the first case, |t| reaches the maximum value only at the energy E = Eλ , this value is obtained in the second case in the whole energy region ∆E where ρ ≈ 0. Narrow (not aligned) resonance states lying in ∆E, lead to dips in the transmission due to the unitarity condition. An example are the dips caused by the trapped resonance states, if the transmission is determined by whispering gallery modes formed together with long-lived (trapped) resonance states in small cavities with convex boundary [41]. (ii) The case ρ = 1 is related to zero transmission as can be seen in the following manner. When crossing the energy E = E0 of a transmission zero by varying a certain parameter, the resonance state becomes a bound state in the continuum whose decay width Γλ vanishes (Sect. 5.5.). The system is decoupled from all channels and the input flux is ˆ E is real and ρ = 1. That completely reflected. As a consequence, the wave function Ψ C means, |t| and 1 − ρ are correlated at the energy E = E0. We underline that the phase rigidity ρ is related to the spectroscopic redistribution processes taking place in the interior of the system in the regime of overlapping resonances. In
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contrast to the individual rλ, the reduction of the phase rigidity ρ is however related only to those redistribution processes which can be seen in the enhancement of the transmission. It is completely insensitive to the redistribution processes as long as the transmission is zero, i.e. when the system is decoupled from the continuum of scattering wave functions. In such a case, ρ = 1 since ΨE C is real, and the individual rλ are the only measure for the redistribution processes taking place in the regime of overlapping resonances.
6.3.
Regime with Well Separated Time Scales
In the overlapping regime, the coupling of the different states of the system via the continuum is, generally, large: in (12) the second term becomes dominant and the widths bifurcate. Eventually, a few short-lived states φλ are created which are aligned, to some E . The remaining resonance states extent, each with one of the channel wavefunctions ξC decouple, to a great extent, from the continuum of scattering wave functions and become long-lived (resonance trapping, Sects. 3.2. and 3.5.). These long-lived resonance states are E . They appear as (more or less) not at all aligned with the scattering wave functions ξC isolated narrow Fano [78] resonances on a smooth background created by the short-lived (aligned) resonance states. For the phase rigidity holds therefore ρ → 1 in a similar manner as in the non-overlapping regime. As a consequence of these redistribution processes in the regime of overlapping resonances, the transmission takes place eventually by two different mechanisms: one of them occurs in the short-time scale and the other one in the long-time scale. This statement is proven for a QD with convex boundary by a shot-noise analysis [42]. The transmission via the short-lived states may be interpreted as caused by waves ”traveling” directly through the cavity from one of the leads to the other one (since the wave functions of the corresponding short-lived resonance states are aligned with the scattering wave functions in the leads). Superposed on these ”traveling” waves are, of course, ”standing” waves at the energies Eλ of the long-lived trapped resonance states. It should be mentioned here that redistribution processes do not always take place. When the leads are attached to the cavity in such a manner that, at a certain energy E, transmission is possible on a direct (short) path between the two leads, then two of the E in the two wave functions φλ are (almost) aligned with the scattering wave functions ξC leads already at very small coupling strength between cavity and leads. It may even happen that the two channels shrink to one channel that is coupled to the cavity at the place of attaching the leads. Examples for large cavities are considered in [79]. Formally, such a situation is similar to the general case with ρ being constant (ρ ≈ 1), in which the individual resonances do not overlap (at small coupling strength) or all the redistribution processes have taken place and short-lived and long-lived resonance states have been formed (at large coupling strength). The relation of these redistribution processes to a phase transition is discussed in Sect. 4.2..
6.4.
Crossover from Standing to Traveling Waves
The relation of the results obtained from (30) and (125), respectively, to the idea of ”standing” waves in almost closed systems and ”traveling” waves in open systems [80] is the
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Ingrid Rotter and Almas F. Sadreev 1.5
c v
1
0.5
0 −0.4
−0.2
0 Re(zk)
0.2
0.4
1.5
d
v
1
0.5
0 −2
−1.5
−1 −0.5 Im(z )
0
k
Figure 15. Left: the transmission probability [ 0 (black) ≤ |t| ≤ 1 (white)] through a double QD versus coupling strength v and energy E (top) for the case with altogether 3 states, one in each dot and another one in the wire connecting the two dots, see Fig. 8. The same (bottom) but for fixed v = 0.2 (dashed line), v = 0.53 (solid line), and v = 0.83 (dot-dashed line). At v = 0.53, the double QD is a perfect filter. Right: the evolution of Re(zk ) (top) and Im(zk ) (bottom), k = 1, 2, 3, of the three eigenvalues of the effective Hamiltonian Heff as a function of v at E = Ec = 0. The parameters u = 1/4, L = 10, ε1 = 0, (L) = 2 − L/5 of the double QD system are chosen in such a manner that (L) = ε1 = 0 at E = 0. Here, the two eigenvalues coalesce. vcr = 81/4u1/2 = 0.8409. Figure taken from [19]. following. In an almost closed system it is Heff ≈ HB , while in a strongly opened sysP + −1 tem, Heff is determined mainly by the coupling term C=L,R VBC (E − HC ) VCB via the continuum, Eq. (12). The first case is realized in the regime of non-overlapping resonances while in the second case a smooth background (arising from the short-lived resonances) is superimposed by long-lived narrow (non-overlapping) resonances. Furthermore, the monochromatic source considered in Ref. [80] corresponds to the two identical one-channel continua described by the one-channel scattering wave functions in (12). It is therefore not astonishing that the results obtained from (30) for the transmission amplitudes in the two limiting cases fit well to the picture described in Ref. [80] for waves propagating in a random medium. However, the crossover between the two limiting cases is described differently in the two methods. In [80] an interpolation between the two limiting regimes is proposed, while in our formulation the equivalent exact expressions (30) and (125) are used for the description of, respectively, isolated resonances and the crossover regime with overlapping resonances. The crossover regime is dominated by coherent collective phenomena
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1
v
0.8 0.6 0.4 0.2 0
−0.2
−0.1
0
0.1
0.2
E, Re(z ) λ
Figure 16. The transmission |t| (left) and the landscape of the phase rigidity |ρ| (right, thin lines) for a double quantum dot over energy E and coupling strength v [0 (black) ≤ |t| ≤ 1 (white)]. The distance between the contour lines is ∆|ρ| = 1/30. The minimal value ρ = 0 is surrounded by a high density of contour lines. The highest shown contour line corresponds to |ρ| = 1 − 1/30. The Re(zλ ) of the three eigenstates (thick lines in both panels √ of the figure) are calculated at E = 0. The branch point is at vcr = 1/2, Ecr = 0. u = 2/16. Around v = 0.345, the phase rigidity is minimal and the transmission maximal with a plateau |t| = 1 (compare Fig. 15). Figure taken from [23]. as discussed above. Results of calculations characteristic of the crossover regime are shown in Figs. 15 to 17. The transmission through a QD with three levels shows three resonance peaks at small coupling strength and one peak at large coupling strength (Fig. 15, left). The reason is resonance trapping: at strong coupling strength v, two resonance states become short-lived by trapping the third resonance state (Fig. 15, right). Fig. 16 shows that the enhanced transmission in the critical region is correlated with a reduced phase rigidity ρ. The transmission |t| as well as the phase rigidity ρ may be plateau-like. An enhancement of the transmission in the critical region and its correlation to the reduction of the phase rigidity ρ occurs also in the realistic case of microwave cavities [24]. As an example, the transmission through a microwave cavity of Bunimovich type and the corresponding phase rigidity ρ are shown in Fig. 17. In this case, the effect is especially large since the whispering gallery modes along the convex boundary are very stable. In the critical region, the transmission is plateau-like with some dips arising from the long-lived trapped resonance states. Corresponding to this picture, the phase rigidity ρ is almost zero in the critical region. These results for the transmission can be explained by means of the simple picture with only three resonances (Figs. 15, 16).
6.5.
Phase Lapses
In experiments [81, 82, 83] on Aharonov-Bohm rings containing a quantum dot in one arm, both the phase and magnitude of the transmission amplitude T = |T |eiθ of the dot can be extracted. The results obtained caused much discussion since they did not fit into
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Ingrid Rotter and Almas F. Sadreev
Figure 17. The transmission |t| (top) and the phase rigidity |ρ| (bottom) over energy E and coupling strength v for a billiard of Bunimovich type to which the leads are attached in such a manner that transmission via whispering gallery modes is supported [ 0 (black) ≤ |t| ≤ 1 (white)]. The shape of the billiard is given by radius R = 3 and distance D = 2 between the centers (in units of the width of the leads). A similar relation between |t| and ρ is obtained also for billiards with another shape [24]. Figure taken from [24].
the general understanding of the transmission process. As a function of the plunger gate voltage Vg , a series of well-separated transmission peaks of rather similar width and height has been observed and, according to expectations, the transmission phases θ(Vg ) increase continuously by π across every resonance. In contrast to expectations, however, θ always jumps sharply downwards by π in each valley between any two successive peaks. This jump called phase lapse, was observed in a large succession of valleys for every many-electron dot studied. The problem is considered theoretically in many papers [84, 85, 86, 87, 88, 89,
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|T|
|T|
1
0.5
0 −1
0.5
0
1
0 −1
0
E 5
3
4
2
3
θ/π
θ/π
E 4
1
2
0 −1 −1
1
1
0
1
E
0 −1
0
1
E
Figure 18. The transmission |T |eiθ [modulus |T | (top) and phase θ/π (bottom)] over energy E for a one-dimensional chain of 11 sites (left) and for a rectangle of 6 sites along the x direction and 3 sites along the y direction (right). The arrows in the insets (bottom) show the sites at which the two (left and right) wires are attached. The hopping matrix elements inside the chain (rectangle) are 0.5 (0.3) and those inside the attached wires are t = 1. The coupling matrix elements are vL = vR = 0.7 for the chain and vL = vR = 0.3 for the rectangle.
90, 91, 92]. In the most recent experiment [83], the transmission is studied not only through manyelectron dots but also through few-electron ones. In the last case, the expected so-called mesoscopic behavior is observed: the phases are sensitive to details of the dot’s configuration such as, e.g., the potential. In this regime, universal phase lapses between every two resonances are not observed. The main difference between few-electron and many-electron dots is that the level spacings are smaller in the latter case than in the first one such that the degree of resonance overlapping (ratio of average level spacing δ to average level width Γ) is different in the two cases [91]. Using the numerical and functional renormalization group approaches, systems with up to 4 levels are studied for different values δ/Γ. If δ ≤ Γ, one of the renormalized effective single-particle levels becomes wider than all the other ones in the regime of Vg for which the phase lapses occur. For δ ≥ Γ, the phase α(Vg ) behaves mesoscopically. That means, universal phase lapses appear only in the regime of overlapping resonances. This result is in qualitative agreement with the experimental ones. The formation of broad (short-lived) resonance states together with narrow (long-lived) ones in the overlapping regime is discussed in Sects. 3.5. and 3.7.. It is a generic effect
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studied analytically as well as numerically in different small open quantum systems such as nuclei, atoms, QDs. It is found even experimentally in a microwave cavity [38]. This trapping of resonance states at high level density is directly related to the phenomenon of avoided or true crossing of resonance states in the complex energy plane that separates the regime of widths bifurcation from that of level repulsion. Most studies are performed by using the FPO method and solving directly the basic equations of the formalism. The cross-over from low to high level density is traced (Sect. 6.4.). The resonance part of the S matrix, Eq. (29), shows the standard phase behavior as long as the individual resonance states are well isolated: the phase increases by π across every resonance state. Usually, it increases S-shape-like according to the finite width Γλ 6= 0 in the energy region in which the cross section shows a resonance. At the position of BICs (having vanishing width), the phase jumps by π (Fig. 13). This phase jump is the only reminiscent of the resonance state (since it can not be seen in the cross section due to its vanishing width Γλ = 0). At high level density, narrow (trapped) resonance states are superposed by (at least) one broad resonance state (Sect. 3.5.). In this case, interferences determine the cross section and the transmission picture through the QD. The trapped (narrow) resonance states appear as dips in the cross section (Sect. 3.4.). Here, the transmission vanishes and phase jumps appear. When there is additionally a background term in the transmission, then the dips may convert to resonances [48]. In such a case, the transmission picture feigns the existence of well isolated (non-overlapping) resonance states. However, phase jumps appear between neighboring resonances. In the crossover regime from low to high level density the resonance states overlap and interferences between the resonance states can, generally, not be neglected (Sect. 6.4.). Here, phase lapses may appear in the valleys between resonance peaks. For illustration, the results of two different calculations with one channel in each of the two (identical) attached leads are shown in Fig. 18. In the first case (left), the transmission through a chain is considered with two leads attached to two sites in the interior of the chain (see the inset of the bottom figure). The hopping matrix elements between chain and lead are larger than those inside the chain (0.7 and 0.5, respectively). As can be seen from the transmission picture, a broad state is formed (see also Figs. 7 to 10 in [24] for similar cases). We see phase jumps by −π at the energies of all transmission zeros. In the second case (right), the hopping matrix elements inside the system (rectangle) and those between system and leads are equal. The broad state is not fully developed as can be seen from the transmission picture (see also Fig. 11 in [24]). Also in this case, transmission zeros appear due to interferences between the resonance states. At all transmission zeros, phase jumps by −π can be seen. Around two energies, two of them are very near to one another according to the very close positions of the corresponding transmission zeros (forming together a broad valley of almost vanishing transmission). In the cases shown in Fig. 18, the phase jumps are less regular in the second case than those in the first one where two broad (short-lived) states exist due to the stronger coupling between system and leads.
7.
Conclusion
The results presented in the present review show clearly that a thorough study of QDs is of great value. On the one hand, the results of these studies are important for engineering
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questions such as the construction of quantum computers. On the other hand, new valuable information on basic questions of quantum mechanics can be reached due to the fact that QDs can be manipulated in a unique manner. It is possible to control the system by means of many different external parameters such that its properties can be studied not only under different conditions, but also the crossover from one regime to another one can be traced. The present experimental studies on QDs provide results of high accuracy the explanation of which is a challenge for the theory. For example, the phase lapses observed in the transmission through QDs [83, 82, 81] and the results on the electron phase coherence time [76, 77] have to be explained in the framework of a general theory. In this review, we used the FPO formalism in its present-day [72] representation for the description of open quantum systems. That means, we solved the basic equations without any statistical assumptions such that the results are numerically exact. The basic assumption of the formalism is that the system is localized in space and embedded into the always existing subspace of extended scattering states (in the case of nuclei) or coupled to a current that flows through the system by means of leads attached to it (in the case of a QD). The equations in the two subspaces Q and P can be solved by using standard methods (diagonalization of the Hamilton operator in the Q subspace and coupled channel calculations in the P subspace). Using the solutions (eigenfunctions and eigenvalues) in the two subspaces, the coupling matrix elements between the two subspaces can be calculated. In rewriting the Schr¨odinger equation (H − E)ΨE C = 0 in the whole function space in order to , the non-Hermitian Hamilton operator Heff appears (Sect. 2.1.) which find the solution ΨE C describes the spectroscopic properties of the open quantum system (Sect. 3.). The influence of decay thresholds and of neighboring resonance states is taken into account in a natural manner since the spectroscopic values obtained from the eigenvalues and eigenfunctions of the energy dependent Hamilton operator Heff are not numbers but energy dependent functions. The S matrix is derived in Sect. 2.2. by using the Lippmann-Schwinger-like relation between the wavefunctions of the resonance states and the eigenfunctions of the non-Hermitian Hamilton operator Heff . It is always unitary. The poles of the S matrix need not to be considered. An important feature of the FPO formalism is the possibility to consider also the borderline cases which are described successfully by using standard methods. There are two such cases: (i) the borderline case of an almost closed system with almost vanishing coupling strength between system and environment and (ii) the many-channel case with a lot of narrow resonance states which is described well by using statistical assumptions for the individual states and the coupling coefficients. ˆ ij ) → 0 In the first case, the effective Hamilton operator Heff becomes real when Im(W ˆ ij ) 6= 0 according to (61) in this limit. This fact is according to (62). However, Re(W taken into account in the standard theory by considering effective forces in the (Hermitian) Hamilton operator that describes the system. The method works well for the description of 0 almost closed systems. Here, Eλ ≈ EkB and hφ∗λ|φλi ≈ δλ,λ0 with Aλ ≈ 1, Bλλ ≈ 0, see Eqs. (53) to (55). The method fails, however, in describing systems with stronger coupling 0 between system and environment and with overlapping resonances (where Aλ > 1, Bλλ 6= 0). In this regime, the system properties are determined, above all, by the branch points in the complex energy plane. Also the second case is well justified when narrow resonance states are coupled to
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many different channels (corresponding to the different scattering wavefunctions of the environment) such that the alignment of the eigenfunctions of Heff each with one decay channel, leads to interferences. Numerical calculations for a microwave cavity have shown that these interferences become important already at about five different channels. In such a case, the statistical approach with energy independent ingredients of the S matrix is not only justified, but provides results that are difficult to obtain when all the coupling matrix elements appearing in the exact solution of the problem, are calculated. Most interesting application of the FPO formalism is therefore the study of small QDs (with a small number of resonance states) coupled to a small number of decay channels in the regime of resonance overlapping. Here, the different approximations of the standard theory can not be justified. The unitarity of the S matrix is guaranteed only if the energies Eλ and decay widths Γλ of the resonance states are energy dependent functions. The resonance-like behavior of the coupling coefficients γλC between the states λ and the continuum can not be neglected at the branch point. Furthermore, a bifurcation of the widths may appear instead of level repulsion observed for narrow resonance states and discrete states. Related to the widths bifurcation are several interesting physical effects, e.g. the strong parameter dependence of the decay widths in order to fulfill the requirement of unitarity of the S matrix and, as a special case, the appearance of BICs. In this respect, it is interesting to consider theoretically in more detail the experimental results obtained for the phase lapses as well as for the electron phase coherence time. The phase lapses observed experimentally at high level density may be considered to be a further hint to the resonance trapping phenomenon appearing in the regime of overlapping resonances [49]. Also the many experimental results of different type on the electron phase coherence time are in qualitative agreement with the expectations according to the FPO formalism with the non-Hermitian effective Hamilton operator Heff (Sect. 5.7.). The point is that the non-Hermiticity of Heff induces some internal impurity in an open quantum system (Sect. 4.1.). Concluding, we state the following. The study of the phenomena observed in the transmission through small QDs coupled to a few decay channels, is important not only for QDs and their application under different conditions. It will surely give an answer also to some fundamental questions of quantum mechanics due to the unique possibility to control the dynamics of the system under different conditions. Some problems need further investigation. Examples are the redistribution processes occurring in the crossover from the regime at low level density to that at high level density. These redistribution processes might be related to the opening of a new decay channel and to a phase transition. Another interesting problem is the time asymmetry of physical processes that is studied in very many papers in a controversial manner. In any case, new insights are expected when the non-Hermiticity of the effective Hamilton operator Heff is taken into account in the theory from the very beginning.
Acknowledgments Valuable discussions with Evgeny Bulgakov, Markus M¨uller and Konstantin Pichugin are gratefully acknowledged. We thank the Max Planck Institute for the Physics of Complex Systems for its hospitality.
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In: Quantum Dots: Research, Technology and Applications ISBN: 978-1-60456-930-8 Editor: Randolf W. Knoss, pp. 493-544 © 2008 Nova Science Publishers, Inc.
Chapter 14
THEORETICAL STUDY ON QUANTUM DOTS USING EFFECTIVE-MASS ENVELOPE FUNCTION THEORY Shu-Shen Li* and Jian-Bai Xia National Laboratory for Superlattices and Microstructures, Institute of Semiconductors, Chinese Academy of Sciences, P. O. Box 912, Beijing 100083, People’s Republic of China
Abstract In this chapter, we will review the results of our theoretical research on quantum dots. Based on the effective-mass envelope function theory, our investigation primarily covers single quantum dots, coupled quantum dots, and N quantum dot molecule. For single quantum dots, we will mainly present our study on the InAs/GaAs single quantum dots and the hierarchical self-assembly of GaAs/AlxGa1-xAs quantum dots. We will discuss the electronic states, valance band structures, quantum-confined Stark effects, properties in magnetic field, and application as single-electron dot qubit of InAs/GaAs quantum dots. Then we will turn our attention to the electronic and optical properties, and the asymmetric quantum-confined Stark effects of the hierarchical self-assembly of GaAs/AlxGa1-xAs quantum dots. As to the coupled quantum dots, we will mainly focus on our research into InAs/GaAs strained coupled quantum dots, and the properties of coupled quantum dots arranged as superlattice. Finally, we will discuss the electronic structures of N quantum dot molecule.
1. Introduction Quantum dots (QDs) have attracted much attention due to their unique electronic and optical properties as well as potential applications in electronic and optoelectronic devices. It is difficult to fabricate QDs by using conventional techniques. Thus, a variety of methods have been invented to make nano-sized particles form self-assembly. QDs formed in this way are generally called self-assembly, self-assembled, or self-organized QDs. Thanks to the
*
E-mail address: [email protected]
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appearance of self-assembled QDs (SAQDs), it is convenient for researchers to study the lowdimensional semiconductor materials. Generally, SAQDs are formed in the materials which have large mismatch of lattice parameters, and SAQDs have island structures which are formed naturally in the process of growing materials due to strain. The length scale of SAQDs can be controlled within 100 nm, which is approaching the de Broglie wavelength of carriers in the materials. Meanwhile, a SAQD includes several hundred or several thousand atoms. The most widely used method to produce SAQDs is the Stranski-Krastanow (SK) growth model [1], in which defect-free three-dimensional (3D) islands spontaneously form on top of a thin wetting layer (WL) during lattice-mismatched heteroepitaxial growth. SK-grown QDs are strained and significant intermixing usually occurs both during island formation [2] and overgrowth [3], rendering QD geometry and composition uncertain. InAs/GaAs SAQD is the most often investigated system formed by SK growth mode due to high, 7%, mismatch of the InAs and GaAs lattice constants [4]. In section 2.1, we will discuss the electronic properties of InAs/GaAs SAQDs systems. Furthermore, due to SAQDs incorporated into semiconductor devices are often working under electric field or magnetic field. Thus, it is very essential to study the influence of electric filed and magnetic field on SAQDs systems. Therefore, in section 2.1 we also discuss the quantum-confined Stark effect and electronic properties in magnetic field. As one example of application, in section 2.1 we also investigate InAs/GaAs single-electron quantum dot qubit, which is one of hot topics in the field of quantum information and computation. Compared with InAs/GaAs SAQDs, GaAs/AlxGa1-xAs QDs show different characteristics. For example, GaAs/AlxGa1-xAs QDs cannot be produced by SK growth model due to the perfect match of lattice constants. In section 2.2, we will turn our attention to the electronic and optical properties, and the asymmetric quantum-confined Stark effects of the hierarchical self-assembly of GaAs/AlxGa1-xAs quantum dots. QDs are often called artificial atoms, because QDs show many characteristics similar to those of atoms. For instance, QDs have discrete energy levels; thus have much more remarkable quantum effects compared with quantum wells and quantum wires. Accordingly, when two or more than two QDs are coupled with each other, they show some characteristics similar to molecules, thus are called artificial molecules. Coupled QDs (CQDs) have new electronic properties and are also very important in application. Therefore, we will investigate InAs/GaAs strained coupled quantum dots, and the properties of CQDs arranged as superlattice in section 3. In the final section, section 4, we will discuss the electronic structures of N quantum dot molecule.
2. Single Quantum Dot 2.1. InAs/GaAs Self-assembled Quantum Dots This section is arranged in the following way. In section 2.1.1, we will discuss energy states of InAs/GaAs SAQDs. Then we will present the study on quantum-confined Stark effects in 2.1.2, electronic properties in magnetic filed in 2.1.3. Finally, in 2.1.4, as an example of
Theoretical Study on Quantum Dots Using Effective-Mass Envelope Function Theory 495 application, we will investigate the application of InAs/GaAs SAQDs as single-electron quantum dot qubit.
2.1.1. Energy Levels The electronic properties of InAs/GaAs SAQD are strongly affected by its size and composition. Generally speaking, InAs/GaAs SAQD has the following characteristic features [4]. (1) Lateral confining potentials are within a good approximation parabolic [5,6] (2) Lateral quantization energy is much lower than the vertical quantization energy. (3) Quantization energies of both electrons and holes are larger than energies of Coulomb interaction between electrons or an electron and a hole, which means that Coulomb effects can be treated as a perturbation to a single-particle structure [7]. Marzin and Bastard first studied the electronic properties of InAs quantum dots embedded in GaAs theoretically [8]. In their calculation, the quantum dots are assumed to be cone shape with height h and base radius rc as shown in Fig. 1. The base angle of the cone is assumed to be 12o. Due to the very small angle, they further took the strain to be a constant in InAs material and zero in the surrounding GaAs barrier. By using single-band effective-mass theory, they calculated the energy levels of electrons and holes, which show that (1) For small Rt values, the electronic levels saturate towards the energy of the first bound level in an InAs quantum well of thickness d (as it should). At high Rt values, where h + d becomes significantly larger than d, quite similar results are obtained for the different values of d. (2) The electron is indeed quite localized on the cone and does not feel the surrounding InAs quantum well. (3) The large energy distance exists between the two first electron level for Rt > 5nm. (4) The energy distance between the fundamental and first excited electron and hole states are significantly larger than the longitudinal optical phonon energy for actual Rt values.
Figure 1. Schematic cross-section of an InAs quantum dot. (From [8])
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Figure 2. Schematic drawing of the dot geometry. (From [9])
Grundmann et. al. studied a more realistic pyramidal geometry using single-band theory [9]. The schematic geometry is shown in Fig. 2. They determined the strain distribution in and around the InAs island using elastic continuum theory. They found that for typical dot sized investigated (pyramid base length 6-20 nm) only one confined electron exists. Groundstate wave functions of electron and heavy holes have a large overlap (about 90%); however, a quite large local charge non-neutrality remains. Excited hole level wave functions are classified by their nodes in x, y, and z directions. The exciton binding energy can be calculated in first-order perturbation theory, and typically amounts 20 meV. Piezoelectric effects lift degeneracies and distort the hole wave functions, but affect the energies of allowed optical transitions rather weakly. The researchers above all neglected valence-band mixing and the strain dependence of the effective masses which were considered by Cusack et. al. in their theoretical study on the electronic structure of InAs/GaAs self-assembled quantum dots as shown in Fig. 3 [10].
Figure 3. Schematic diagram of InAs/GaAs self-assembled quantum dots. (From [10])
Theoretical Study on Quantum Dots Using Effective-Mass Envelope Function Theory 497 In their calculation [10], the strain modification to the confinement potential, valenceband mixing, and the conduction-band mass in the InAs dot and the surrounding GaAs barrier are all taken into account. The structure and the variation of strain were determined by a valence force field method, and the confined levels are calculated by using multiband effective-mass method. Furthermore, the conduction and valence bands are assumed to be decoupled for simplicity. Then the general solutions for the electron states are
ψ nc ( x, y, z ) = u cφ nc ( x, y, z ) ,
(1)
φnc is an envelop function satisfying the simple single-band SchrÖdinger equation. By invoking periodic boundary conditions, φ can c
where u is a bulk band-edge Bloch function and
be expanded in terms of normalized plane-wave states, and then diagonalized to get the energy levels of electrons. While the valence-band states are defined by solutions of the fourband SchrÖdinger equation, and the hole states ψ n ( x, y, z ) can by expanded in plane waves v
in the following form: 4
ψ nv ( x, y, z ) = ∑ u vφ nv ( x, y, z ) ,
(2)
v =1
v
where u are the J=3/2 angular momentum states. Their work shows that [10]: (1) In the dot material, the compressive stress alters the curvature of the bulk bands causing the effective masses to differ from those of unstrained InAs; their calculations yield a value for the effective mass of 0.04 me compared to the value for unstrained InAs of 0.023 me . (2) Since the strain varies form cell to cell, the confining potentials will also vary from cell to cell. Furthermore, degeneracies in the valence-band edge will be lifted due to deviations of the unit cells from cubic symmetry. (3) The light-hole band edge is higher in energy than the heavy-hole band edge in the barrier, and towards the apex of the pyramid; the heavy-hole band is the uppermost band at the base of the pyramid. (4) The direction and magnitude of the splitting of the light- and heavy-hole bands-in the absence of appreciable shear strain components-is dependent solely on the magnitude and sign of the biaxial strain. In those regions of the structure where the biaxial strain is negative the lighthole band will be shifted upwards in energy and the heavy-hole band downwards; in those regions where the biaxial strain is positive, the heavy-hole band will be uppermost. When the biaxial strain function is zero, the light- and heavy-hole bands will be degenerate. (5) For dot base diameters smaller than approximately 6 nm, no bound-electron states exist. This number increases to three for structures larger than 12 nm. In the valence band there are many confined hole states. This is due to the larger effective mass associated with these carriers, and to the nature of the light-hole confinement potential the smoothly varying form of which leads to a quasicontinuum of tenuously bound states. (6) The first and second electron excited states are degenerate, while the first and second excited hole levels are split due to mixing between different bulk states. (7) The relatively isotropic character of the confining potential for electrons coupled with the small effective mass results in a state that permeates throughout
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Shu-Shen Li and Jian-Bai Xia
the dot and penetrates the sides of the pyramid. Charge does not significantly sample the apex or the base corners of the pyramid. (8) Unlike the ground conduction state, the ground hole state is confined to the base of the dot due to the larger effective mass, and the anisotropic nature of the heavy-hole confining potential. (9) The energy splitting between the ground and the first excited hole state is about 30 meV. A larger number of conduction states are predicted than those predicted by previous calculation [8,9]. Furthermore, quantum dots in the shape of lenses [5, 11-13], and disks [14-16] are also studied. In the following, we will present our investigation on a more universal case, quantum ring (QR) [17, 18], which has a cylindrical shape. We suppose the inner radius, outer radius, and high of QR are R1, R2, and l, respectively. If R1 = 0, R2 → ∞ and l is finite, the quantum rings become quantum wells; if l → ∞ , R1 and R2 are finite, the quantum rings become quantum wires; if R1=0, R2 and l are finite, the quantum rings become quantum dots; Therefore, our model can be used to calculate the electronic properties of quantum wells, quantum wires, and quantum dots. We choose the z direction of our coordinate system as perpendicular to the plane of quantum rings. According to Burt and Foreman’s effective-mass theory and taking into account the differences of the effective masses between InAs and GaAs materials [19, 20], the electron Hamiltonian can be written as the equation below (neglecting the second order and higher order terms in the approximation)
He = P
1 P +V e (x,y,z ) , 2m e (x,y,z )
(3)
∗
in the above equation,
⎧⎪0 R12 ≤ x 2 + y 2 ≤ R22 and z ≤ l , Ve ( x , y , z ) = ⎨ ⎪⎩ E c others,
(4)
⎧⎪m1∗ R12 ≤ x 2 + y 2 ≤ R22 and z ≤ l , m ( x, y , z ) = ⎨ ∗ ⎪⎩m2 others, ∗ e
∗
(5)
∗
where Ec is the conduction band offset between GaAs and InAs, m1 and m2 are the electron effective masses in InAs and GaAs materials, respectively. The effective-mass Hamiltonian of the hole can be written as
R − Q− 0 ⎤ ⎡ P+ ⎢ + ⎥ P− C + − Q++ ⎥ 1 ⎢ R H0 = + Vh , 2m0 ⎢− Q−+ C P− − R ⎥ ⎢ ⎥ ⎢⎣ 0 − Q+ − R + P+ ⎥⎦
(6)
Theoretical Study on Quantum Dots Using Effective-Mass Envelope Function Theory 499
⎧⎪0 R12 ≤ x 2 + y 2 ≤ R22 and z ≤ l , V h ( x, y , z ) = ⎨ ⎪⎩Vh 0 others.
(7)
Using the periodic boundary condition, we assume that the electron and hole wave functions can be expanded in term of plan wave states as follows:
Ψe (r ) =
Ψh (r ) =
1 Lx L y L z
∑a
nml
e
i[( k x + nK x ) x + ( k y + mK y ) y + ( k z + lK z ) z ]
,
(8)
nml
⎡anml ⎤ ⎢b ⎥ 1 ⎢ nml ⎥ × e i[( k x + nK x ) x + ( k y + mK y ) y + ( k z +lK z ) z ] , ∑ Lx Ly Lz nml ⎢cnml ⎥ ⎥ ⎢ ⎣d nml ⎦
(9)
with K x = 2π / L x , K y = 2π / L y , and K z = 2π / L z ; n, m, l = 0, ± 1, ± 2,... . L x , L y and
Lz are the periods of the large units along x, y, and z directions, respectively. We can then calculate the matrix elements of Hamiltonian (3) and (6). Including the effect of conduction band nonparabolicity, the electron effective mass can be written as
me∗ ( E ) = mb∗ (1 +
2E ), Eg
(10)
where mb * and Eg are the effective mass of conduction band bottom and band gap, respectively. We have calculated the electron and hole energy levels as a function of inner radius of quantum rings.[17,18]. The results for electron energy levels are shown in Fig. 4, 5, and 6. From Fig. 4(a) and 4(b), one may find that the confined electronic energy levels are sensitively dependent on the inner radiuses for the small outer radius. The numbers of confined energy levels increase when the outer radiuses increase. The results of R1=0 are the electronic energy levels of disk quantum dots with the radius R2 and height l. The dotted lines give the theoretical results of Ref. [21] ( ω = 10 meV), which cannot show the energy change with the radius of the quantum ring. From Fig. 5, we find that the electronic energy levels are not sensitively dependent on the outer radiuses, if the outer radiuses are larger than 30 nm. This is the reason that experimental spectroscopy is almost the same for many quantum rings with different outer radiuses in samples [22]. From Fig. 4 and 5, we find that the energy levels decrease and the energy spacing between energy levels increase as the inner radius decreases. When increasing the outer
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Shu-Shen Li and Jian-Bai Xia
radius, the energy levels and the energy spacing decrease simultaneously. Therefore, if one wants to increase the energy spacing between energy levels but keep the ground state energy level from changing, he can decrease the inner and outer radii simultaneously. If one changes only one of the two radii (inner or outer radius), the ground state energy level and the energy spacing will change simultaneously.
Figure 4. The electron energy levels as a function of inner radius of quantum ring. The height of quantum rings is l=2 nm.(a) and (b) for outer radiuses are R2=10 nm and R2=30 nm, respectively. The dotted lines are the results of Ref. [21] with ω = 10 meV. (From [17])
Figure 5. The electronic energy levels as a function of outer radius of quantum rings with height l=2 nm. The inner radius are R1=0 nm (quantum dots) and R1=10 nm in (a) and (b), respectively. (From [17])
Theoretical Study on Quantum Dots Using Effective-Mass Envelope Function Theory 501
Figure 6. The electronic energy levels as a function of quantum ring heights. The inner and outer radiuses are 10 and 30 nm, respectively. (From [17])
The electronic energy levels as a function of height are shown in Fig. 6. When l is very large, the quantum rings will become quantum wires, which have two forms: one form is ‘‘real core’’ quantum wires with R1=0, and another form is ‘‘empty core’’ quantum wires with R1 ≠ 0 .
Figure 7. The hole energy levels as a function of inner radius of quantum ring. The height of quantum rings is l=2 nm.(a) and (b) for outer radiuses are R2=10 nm and R2=30 nm, respectively. (From [18])
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Shu-Shen Li and Jian-Bai Xia
Table 1. The electronic excited energies (meV) with l=2 nm, R1=10 nm, and R2=45 nm (From [17])
a
See Ref. [22]
If taking shape parameters to be l=2 nm, R1=10nm, and R2=45 nm, we can get the electronic excited energies. The calculated results are listed in Table 1. These calculated results are very close to the available experimental data in Ref. [22].
Figure 8. The hole energy levels as a function of outer radius of quantum rings with height l=2 nm. The inner radius are R1=0 nm (quantum dots) and R1=10 nm in (a) and (b), respectively. (From [18])
Fig. 7, 8, and 9 show the hole energy levels. One can find that: (1) the confined hole energy levels are sensitively dependent on the inner radiuses. For the small outer radius, the holes have the large confined energies. (2) The hole energy levels are not sensitively dependent on the outer radiuses, if the outer radiuses are larger than 30 nm. (3) The energy levels decrease and the energy spacing between energy levels increase as the inner radius decreases. When increasing the outer radius, the energy levels and the energy spacing decrease simultaneously. Therefore, if one wants to increase the energy spacing between energy levels but keep the ground state energy level from changing, he can decrease the inner and outer radii simultaneously. If one changes only one of the two radii (inner or outer radius), the ground state energy level and the energy spacing will change simultaneously. (4) If we fix the inner and outer radius, When l is very large, the quantum rings will become quantum wires, which have two forms: one form is ‘‘real core’’ quantum wires with R1=0, and another form is ‘‘empty core’’ quantum wires with R1 ≠ 0 .
Theoretical Study on Quantum Dots Using Effective-Mass Envelope Function Theory 503
Figure 9. The hole energy levels as a function of quantum ring heights. The inner and outer radiuses are 10 and 30 nm, respectively. (From [18])
Figure 10. The hole energy levels as a function of the width of the quantum rings r0 [ = ( R2 − R1 ) / 2 ]. The average radius R0 [ = ( R2 + R1 ) / 2 ] and the height of the quantum rings are taken as 10 and 2 nm, respectively. ( From [18])
Fig. 10, 11 and 12 show that: (1) the hole energy levels monotonously decrease as the increasing quantum ring width and the difference energies increase when the quantum rings widths increase; (2) the first heavy hole energy level (the first solid line in Fig. 11) is close to the first light hole energy level (the second solid line in Fig. 11), which is very different from the results of the simple model (the first dotted and the first dashed lines in Fig. 11); (3) the transition energies from the first electron energy level to the first heavy and light hole energy levels are very close (solid lines in Fig. 12).
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Shu-Shen Li and Jian-Bai Xia
Figure 11. The hole energy levels as a function of the average radius R0 .The dotted and dashed lines are the heavy and light hole energy levels ignoring the effects of the valence band mixing between light and heavy holes and supposing the effective masses of the heavy and light holes are 0.335 and 0.099 m0 , respectively. The width and height of the quantum rings are 10 and 2 nm, respectively. (From [18])
Figure 12. The transition energies from the first electron energy level to the first heavy (solid line) and light hole (dashed line) energy levels as a function of the average radius R0 . The electron energy level is taken from Ref. 8. The dotted and dot-dashed lines are the results of ignoring the effects of the valence band mixing between light and heavy holes and supposing the effective masses of the holes are 0.335 and 0.099 m0. The width and height are 10 and 2 nm, respectively. (From [18])
Theoretical Study on Quantum Dots Using Effective-Mass Envelope Function Theory 505
2.1.2. Quantum-Confined Stark Effects Quantum devices which are based on quantum dots usually work under electric field and that is why it is very important to study the effects of the electric field on the quantum dot. An additional potential is produced when the quantum dot is under applied bias, namely electric field; thus an energy shift of the interband optical spectra is produced. Quantum-confined Stark effect has attracted much attention. Fry et. al.[23] found that the hole is localized towards the top of the dot, and the excited state transitions arise from lateral quantization and that tuning through the inhomogeneous distribution of dot energies can be achieved by variation of electric field. Based on eight-band strain-dependent k ⋅ p Hamiltonian, Sheng and Leburton [24] studied vertically stacked and coupled InAs/GaAs SAQDs which are predicted to exhibit strong hole localization even with vanishing separation between the dots, and a non-parabolic dependence of the interband transition energy on the electric field, which is not encountered in single SAQD structures. Their work also indicates that this anomalous quantum confined Stark effect is caused by the three-dimensional strain field distribution which influences drastically the hole states in the stacked SAQD structures. They also reported significant deviations from the usual quadratic dependence of the groundstate interband transition energy on applied electric fields in single InAs/GaAs self-assembled quantum dots [25]. Sheng and Leburton show that earlier works that used conventional second-order perturbation theory to claim a negative dipole moment in the presence of external electric field fails to correctly describe the Stark shift for electric field below F=10 kV/cm in high dots. Based on eight-band k ⋅ p calculations, their results demonstrate that this effect is predominantly due to the three-dimensional strain field distribution which for various dot shapes and stoichiometric compositions drastically affect the hole ground state. To study an InAs SAQD in the electric field, we choose the growth direction (100) as the z direction of our coordinate system, and assume the InAs SAQD as a cylinder [26]. In the z direction, the height of the InAs dot is L. In the parallel direction, the radius of InAs dots is R. For an electric field F in the z direction In our study, the effect of finite band offset, valence band mixing, and strain are all taken into account. The effective Hamiltonians of electrons and holes are [26]
He = P
1 2m ∗ (r )
P +Ve (r ) − eFz ,
(11)
and
− Q− R 0 ⎤ ⎡ P+ − D ⎢ ⎥ + + − Q++ ⎥ C 1 ⎢ R P− + D Hh = + D′ + Vh - eFz . 2m0 ⎢ − Q−+ C P− + D − R ⎥ ⎢ ⎥ ⎢⎣ 0 − Q+ − R + P+ − D ⎥⎦
(12)
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Shu-Shen Li and Jian-Bai Xia
Figure 13. The ground state energy level (slid lines) and the first excited state energy level (dotted lines) of electron as a function of electric field along the growth direction (a) and along the parallel direction (b). The diameter and height of the QD are 5 and 3 nm, respectively. (From [26])
Figure 14. The first four energy levels of the hole as functions of electric fields along the growth direction (a) and along the parallel direction (b). The diameter and height of the QD are 5 and 3 nm, respectively. The solid lines, dotted lines, dashed lines, and short-dashed lines correspond to the first heavy-hole, the first light-hole, the second heavy-hole, and the third heavy-hole energy levels, respectively.(From [26])
The results are shown in Fig. 13, 14, and 15. From Fig. 13, we find that the electronic ground state energy level is weakly affected by the vertical electric field when the electric field is lower than 300kV/cm because the QDs in our calculation have a small height (only 3 nm). Along the parallel direction, the QDs have a large diameter (10 nm), and the electronic energy levels are strongly affected by the parallel electric field; this trend can be found in Fig. 13(b). From Fig. 13, we also find that the electronic excited state energy levels are affected by
Theoretical Study on Quantum Dots Using Effective-Mass Envelope Function Theory 507 an electric field stronger than the ground state energy levels. The energy difference between the ground state and the first excited state decreases as the electric field increases. Fig. 14 shows that the hole energy levels have complicated structures due to the valence band mixing.
Figure 15. The transition energies of the first electron energy level to the first heavy-hole energy level along the growth direction. The diameter and height of the QD are 5 nm. The black circles are the experimental results of Refs. 27 and 28. (From [26])
Fry et al. studied the electronic states as a function of applied vertical electric field in InAs/GaAs self-assembled QDs using photocurrent spectroscopy [27,28]. For comparison, we have calculated the transition energies of the first electron energy level to the first heavy-hole energy levels along the growth direction. The results are shown in Fig. 15. The black circles are the experimental results. From this figure, we find that the optical transition energies have clear redshifts in the perpendicular electric field. The theoretical results are very close to the experimental data.
2.1.3. Energy States in Magnetic Field Study on energy states of QDs in magnetic filed is also very essential because QDs incorporated into devices are sometimes subject to magnetic filed. Using the effective mass theory, Pedersen and Chang [29] calculated the one- and two-hole energies in parabolic GaAs QD in the presence of perpendicular magnetic filed. Their theoretical model did not include the effect of strain on the energy levels. Their results indicated that the single-hole levels showed strong anticrossing due to the valence-band mixing, and that as a result these levels have in general a weaker filed dependence compared with the corresponding uncoupled levels. Reuter et. al. studied the hole charging spectra of InAs SAQDs in perpendicular magnetic fields by capacitance-voltage spectroscopy [30]. From the magnetic-field dependence of the individual peaks they concluded that the s-like ground state is completely filled with two holes but that the fourfold degenerate p shell is only half filled with two holes before the filling of the d shell starts, and the resulting six-hole ground state is highly
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Shu-Shen Li and Jian-Bai Xia
polarized. This incomplete shell filling can be explained by the large influence of the Coulomb interaction in this system. In the framework of the Burt-Foreman theory, Mlinar et. al. derived a nonsymmetrized eight-band effective-mass Hamiltonian for nanostructures in the presence of a magnetic field [31]. The Hamiltonian was also tested for the case of a cylindrical quantum dot with parabolic in-plane confinement potential in a perpendicular magnetic field. They found that in structures with a large difference of Luttinger parameters between the constituent materials, such as InAs/GaAs QDs, the conventional multiband models lead to unphysical high magnetic-filed solutions that were substantially different from those obtained from the nonsymmetrized Hamiltonian and single-band model for the ground state. They attributed this discrepancy to an overestimation of band mixing in conventional models because of the inappropriate treatment of the boundary. Nguyen, et. al. studied the effect of a strong magnetic filed applied parallel to the growth direction of InAs/GaAs semiconductor QDs embedded in a GaAs/AlAs superlattice [32]. They predicted that the flatness of the InAs/GaAs dots lead to a midinfrared absorption which was almost insensitive to the magnetic filed. Larsson et. al. presented a photoluminescence study of self-assembled InAs/GaAs quantum dots under the influence of magnetic fields perpendicular and parallel to the dot layer [33]. Their results show that the magnetic field perpendicular to the dot layer alters the in-plane transport properties due to localization of carriers in wetting layer (WL) potential fluctuations at low temperatures. Also, the effect of the magnetic field exhibits a considerable dot density dependence, which confirms the correlation to the in-plane transport properties. Furthermore, they observed an interesting effect at temperatures above approximately 100 K that magnetic fields, both perpendicular and parallel to the dot layer, induced an increment of the quantum dot photoluminescence. They attributed this effect to the magnetic confinement of the exciton wave function, which increases the probability for carrier capture and localization in the dot, but affects also the radiative recombination with a reduced radiative lifetime in the dots under magnetic compression. In this section, we will study the electronic structures of InAs SAQDs in an axial magnetic field [34]. In our study, the effect of finite offset, valence-band mixing, and strain are taken into account. We choose the growth direction (100) as the z direction of our coordinate system, and assume the InAs SAQD to be a cylinder. In the z direction, the width of an InAs dot is L. In the parallel direction, the radius of InAs dots is R. For a magnetic field B in the z direction, let A be the vector potential. In the symmetric gauge,
A = B × r / 2 = (− y, x,0) B / 2 .
(13)
In cylindrical coordinates, we take the lateral confinement potential be parabolic, viz.
V (ρ ) = The larger
1 mω 02 ρ 2 . . 2
(14)
ω0 is, the smaller the quantum dot size will be. It should be pointed out that,
in the case of a real experiment, the order of
ω0 can be chosen in such a way that
Theoretical Study on Quantum Dots Using Effective-Mass Envelope Function Theory 509
0 ρ 2 0 = /( mω 0 ) is roughly equal to dot sizes (diameter of cylinder) in the parallel direction. Here, 0 indicates the ground state of the carrier. The quantum well potential in the Z direction is accounted for by a finite square potential well of width L,
⎧⎪0 V⊥ ( z ) = ⎨ ⎪⎩ΔV0
for z ≥ L / 2, for z < L / 2.
(15)
The electron Zeeman energy introduced by magnetic field is
H ze =
e Bjz , me
(16)
where j z = ±1 / 2 is the Z component of electron spin angular momentum, and me is the electron effective mass. In the parallel direction, the electron Hamiltonian is
He =
1 ( P − eA) 2 + V . 2me
(17)
In the growth direction, the electron Hamiltonian is
H ⊥e =
1 2 Pz + V⊥ . 2me
(18)
Therefore, the electron total Hamiltonian is
H e = H e + H ⊥e + H ze .
(19)
Including the effect of conduction-band, nonparabolicity, the electron effective mass can be written as [35]
me ( E ) = mb (1 +
2E ), Eg
(20)
where mb and Eg are the effective masses of the conduction-band bottom and band gap, respectively. From the electron envelope function equation, theelectron energy levels can be calculated. Including the effect of strain, the hole Hamiltonian can be written as [36]
H h = H Lh + H ε + V h + H zh + V⊥ .
(21)
510
Shu-Shen Li and Jian-Bai Xia h
In the above equation, HL is the same as the Eq. (1) of Ref. [29]. H ε is the strain h
Hamiltonian. H z is the hole Zeeman energy introduced by magnetic field [29]. We choose the basis as the same as that of Ref. [29] (see Eq. (16) of Ref. [29]). The hole energy is found variationally by minimizing with respect to the expansion coefficients.
Figure 16. The lowest energy levels of electrons with S1+/ 2 and S −+1 / 2 symmetry as a function of magnetic field. (From [34])
Figure 17. Hole energy levels with S3+/ 2 symmetry as a functions of magnetic field. (From [34])
Theoretical Study on Quantum Dots Using Effective-Mass Envelope Function Theory 511 +
+
Fig.16 shows the lowest energy levels of electrons with S1 / 2 and S −1 / 2 symmetry as a function of magnetic field B. Fig. 17 shows the hole energy levels of the InAs quantum dot +
with S3 / 2 symmetry. It is the same as the case of no strain in that, as the symmetry forbids the coupled levels to cross, we also see strong anticrossings of the levels.
Figure 18. The lowest hole energy levels of the two states with S3+/ 2 and S −+3 / 2 symmetry. The dot lines are uncoupled levels.(From [34]) +
+
Fig.18 shows the lowest energy levels of the two states with S3 / 2 and S − 3 / 2 symmetry, which are degenerate at B=0 and split at nonzero magnetic field. In comparison, we have also included in Fig. 18 related uncoupled levels (dot lines), which are obtained with the off-diagonal terms in HL set to zero. We see that the inclusion of coupling due the offdiagonal terms of HL lowers the hole energies of the two states. In the above material and structure parameters, the effect of coupling due to the off-diagonal terms of HL is not very large. There are two reasons. The first one is that the difference of subband energy levels is somewhat large, and the second is that the strain enlarges the energy difference between j = ±3 / 2 and j = ±1 / 2 . The second reason is easily found from the strain Hamiltonian Eq. (13) of Ref. [37]. Furthermore, as levels of the same symmetry are forbidden to cross, they show in general a weaker field dependence compared to the uncoupled ones. Note that +
+
for B>0, the state S3 / 2 is always lower in energy than S − 3 / 2 both for the coupled and uncoupled levels.
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Shu-Shen Li and Jian-Bai Xia
Figure 19. The effect of strain on the first two energy levels of S3+/ 2 symmetry. The solid and dashed lines are the results of including and excluding the strain.(From [34])
Figure 20. The lowest energy levels for six different symmetries S ±+3 / 2 , S ±+1 / 2 , and P±+5 / 2 . (From [34]) +
Fig. 4 shows the effect of strain on the first two energy levels of S ± 3 / 2 symmetry. From +
this figure, we find that the strain raises the energy levels of S ± 3 / 2 symmetry. This comes mainly from the second term D′ of the strain Hamiltonian Eq. (13) of Ref. [37]. +
+
Fig. 20 displays the lowest energy levels for six different symmetries S ± 3 / 2 , S ±1 / 2 , and
P±+5 / 2 , all of which have even parity. The corresponding energy levels of odd-parity symmetry are shown in Fig. 21.
Theoretical Study on Quantum Dots Using Effective-Mass Envelope Function Theory 513
Figure 21. Same as Fig. 5, but now for the lowest energy levels of the odd-parity states S ±−3 / 2 , S ±−1 / 2 , and P±−5 / 2 .(From [34])
Figure 22. The transition energies of the lowest electron S ±+1 / 2 states to lowest hole S ±+1 / 2, ±3 / 2 states as a functions of magnetic field. Solid, dotted, dashed, and dash-dotted lines are the transition energies of electron S±+−1 / 2 to hole S ±+−1 / 2 , electron S ±+1 / 2 to hole S ±+1 / 2 , electron S ±+−1 / 2 to hole S ±+−3 / 2 , and electron
S±+1 / 2 to hole S ±+3 / 2 , respectively. (From [34]) +
+
The transition energies of electron lowest S ±1 / 2 states to lowest hole S ±1 / 2, ±3 / 2 states as a +
functions of magnetic field are shown in Fig. 22. As the electron S ±1 / 2 states and hole
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Shu-Shen Li and Jian-Bai Xia
S ±+1 / 2, ±3 / 2 degenerate at B=0, the electron S1+/ 2 state to hole S1+/ 2 state and electron S −+1 / 2 +
+
state to hole S −1 / 2 state give the same transition energy, and the electron S1 / 2 state to hole
S3+/ 2 state and electron S −+1 / 2 state to hole S −+3 / 2 give the same transition energy. 2.1.4. Application as Single-Electron Quantum Dot Qubit The elementary unit of quantum information in a quantum computer (QC) is the quantum bit (qubit). A single qubit can be envisaged as a two-state system such as a spin-half particle or a two-level atom. The potential power of a QC is based on the ability of quantum systems to be in a superposition of its basic states. In order to perform quantum computations, one should have the following basic conditions [38]: (i) a two-level system ( 0 and 1 ) as a qubit; (ii) the ability to prepare the qubit in a given state, say 0 ; (iii) the capability of measuring each qubit; (iv) the ability to perform basic gate operations such as a conditional logic gate (the control-not gate); and (v) a sufficient long decoherence time. It is very important for a QC to be well isolated from any environmental interaction which would destroy the superposition of states. Furthermore, one has to use quantum error correction. Several schemes, like trapped ions [39], quantum optical systems [40], nuclear and electron spins [41-43], and superconductor Josephson junctions [44-47] have been proposed for realizing quantum computation. However, in order to show its superiority over the most advanced classical computers, quantum computers need to be composed of at least thousands of qubits to be feasible. To this end, it is clear that quantum computation with a significant number of qubits would be more realizable in solids [48], especially by invoking semiconductor nanostructures or quantum dots (QDs) [49]. The ground state ( 0 ) and the first excited state ( 1 ) of an electron in a QD may be employed as a two-level quantum system. An electromagnetic pulse can be applied to drive an electron from 0 to 1 or to the superposition state of 0 and 1 . To perform a quantum-controlled “not” manipulation, one may simply apply a static electric field by placing a gate near the QD. However, before quantum computation can be realized using QDs, two main obstacles must be overcome. First, high-quality, regularly spaced, uniform semiconductor QDs must be fabricated. Today, using the Stranski–Krastanov method, the fabrication of InAs/GaAs SAQDs of high quality may not be very difficult by various types of modern epitaxy technologies like molecular-beam epitaxy, but the growth of regularly spaced, uniform, SAQDs remains a severe challenge for such a technology. The second key issue is how to prolong the decoherence time in semiconductor QDs when there exist innumerable degrees of freedom which dephase the systems very fast. Bertoni et al. studied the oscillation of the electron density between two coupled quantum wires, which can be used to realize the universal set of quantum logic gates [50]. In this section, we shall study the dephasing rate, time evolution of the quantum state of the electron in an InAs/GaAs QD, and the interaction of the two electrons located in different QDs [38].
Theoretical Study on Quantum Dots Using Effective-Mass Envelope Function Theory 515 To study an InAs/GaAs self-assembled QD in an electronic field, we choose the growth direction (100) as the z direction of our coordinate system, and assume the InAs SAQD to be a cylinder. The height of the QD in the z direction is L. In the parallel direction, the radius of the QD is R. The effective Hamiltonian of electrons is
He = P
1 2m ∗ (r )
P +V e (r ) − eFz ,
(22)
and the electron envelope function equation is
H eψ e = E eψ e ,
(23)
where ψ e is wave function of electrons, and can be expanded in terms of normalized planewave states. In literature [50-56], the issue of dephasing in QDs is discussed. Impurities and thermal -9
vibration (phonons) can reduce the lifetime to ~10 s or even worse, but, in principle, their effects can be minimized by a more precise fabrication technology, by cooling the crystal, and by choosing the state and the physical parameters properly [49]. It must be pointed out that the decoherence time decreases if the temperature decreases, but, when we reach the E2–E1 KT limit, the decoherence time does not decrease any more. In the present model, we assume a large energy difference between 0 and 1 , so we can neglect the acoustic and optical phonon scattering and only take into account the decoherence coming from the vacuum fluctuation. Under the dipole approximation, based on the Fermi Golden Rule [57], the spontaneous emission rate can be written in the following form:
τ −1 =
e 2 ΔE 3πε 0 2 m02 C 3
2 ε 0r 1 , ε0
(24)
The time evolution of the quantum state of the electron can be written as
ψ e (t , r ) =
1
ψ e1 (r )e −itE / + 1
2
1 2
ψ e2 (r )e −itE
2
/
(25)
The Coulomb interaction energy between two electrons located in different QDs can be calculated using 2
Eij = ∫
ψ i (r1 ) ψ j (r2 ) 4πε r1 − r2
2
dr1 dr2 .
(26)
516
Shu-Shen Li and Jian-Bai Xia We obtained the following results. (1) The energy of 0 does not depend on the parallel electric field sensitively. However,
as long as the electric field is larger than 5 kV/cm, the energies of 1 and ΔE decrease substantially with increasing electric field. (2) The static electric field induces a change of the electron charge distribution in the QD which is opposite for 0 and 1 . The induced dipole moment points in the same direction as the electric field for 0 , but is in the opposite direction for 1 . (3) (i) The oscillation period decreases as the radius of the QD decreases because the energy difference (E2-E1) increases; (ii) there is a minimum in the oscillating period curve; and (iii) if the radius is smaller than the critical value, the period of oscillation increases as the radius decreases. The reason is that there is only one confined quantum state (the ground state 0 ) in the QD when the radius is smaller than the critical value. The excited quantum states are continuous states located above the barrier. The energies of the continuous states are close to the top of the barrier and do not depend sensitively on the radius of the QD. (4) The interaction energy between electrons located in different QDs is a very important parameter in designing quantum gates. The interaction energies decrease as the distance increases. (5) A QD that can be used in quantum computation must consist of at least two binding states. Fig. 23 shows the parameter-phase diagram of one InAs/GaAs QD. The gray region in Fig. 23 indicates that only one binding state resides in the QD. In contrast, there are at least two binding states in the white region so that the QD may be used as a qubit.
Figure 23. Shaper of the parameter-phase of one InAs/GaAs QD used as a qubit; the gray region indicates that the QD cannot be used as a qubit. (From [38])
Theoretical Study on Quantum Dots Using Effective-Mass Envelope Function Theory 517 (6) The decoherence time does not depend on the electric field sensitively until the strength of the electric field is lower than 5 kV/cm. The decoherence time then increases very fast as the electric field goes beyond 5 kV/cm. The decoherence time may reach an order of magnitude of milliseconds under a 20 kV/cm static electric field for the QD with 5 nm radius and 4 nm height.
2.2. Hierarchical self-assembly of GaAs/AlxGa1-xAs Single Quantum Dot Due to the almost perfect match of lattice parameters, GaAs/AlxGa1-xAs QDs cannot be created by Stranski-Krastanow (SK) growth mode which is widely used to produce QDs. According to Rastelli et. al. [58], however, GaAs/AlxGa1-xAs system has several advantages compared to others. First, the grown material is ideally unstrained, and sharp interfaces with reduced intermixing can be obtained. Another advantage is that AlGaAs heterostructures can be designed to emit light in the optimum spectral range of the most advanced optical components available nowadays. Nevertheless, such QDs have small confinement energy (about 10 meV). Using modified-droplet epitaxy method, Wanatabe et. al. have grown selfassembled 3D QDs with larger confinement energy [59]. The disadvantage is that this method requires low temperatures and the size homogeneity of the obtained QDs is very poor. Due to these reasons, several elaborate methods are also used to fabricate GaAs QDs [60]. By combining SK growth and in situ etching, Rastelli et. al. obtained self-assembled, unstrained, inverted GaAs QDs with tunable size, large confinement energy, and good size homogeneity. Photoluminescence (PL) spectroscopy revealed the light emission of this structure with very narrow inhomogeneous broadening and clearly resolved excited states at high excitation intensity. The dot morphology was determined by scanning probe microscopy and, as combined with single-band and eight-band k ⋅ p theoretical calculations, was used to interpret PL and single-dot spectra with no adjustable structural parameter [58]. In this section, we will present our theoretical study on hierarchical self-assembly of GaAs/AlxGa1-xAs QDs [61,62]. In our calculation, the effect of finite offset, valence-band mixing, the effects due to the different effective masses of electrons and holes in different regions, and the real quantum dot structures are all taken into account.
2.2.1. Energy States We consider the GaAs/AlxGa1-xAs self-assembled QD sketched as in Fig.24 according to the experimental results in Ref. [58]. The real section of the QD is a parabolic section. While we take a rectangular section model as shown in Fig.24, the width can be taken to be the average value of the real QD, so the calculated energy levels are very close to the real ones. In our coordinate system, we choose the growth direction [100] as the z direction. In the z direction, the height of the GaAs dot is h, and the width of GaAs quantum well (QW) is d. In the perpendicular direction, the width of GaAs dot are Wx and Wy, respectively, along the x and y direction. Lx , Ly, and Lz denote the widths of the large unite cell along x, y, and z directions, respectively.
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Shu-Shen Li and Jian-Bai Xia
Figure 24. The structures of the hierarchical self-assembly of GaAs/AlxGa1-xAs QDs (a) along the growth direction and (b) along the direction perpendicular to the growth direction. (From [61])
The effective Hamiltonian of electrons and holes can be written as [61]
He = P
and
1 2m ∗ (r )
P +V e (r ) ,
R − Q− 0 ⎤ ⎡ P+ ⎥ ⎢ + P− C + − Q++ ⎥ 1 ⎢ R Hh = + Vh(r ) . 2m0 ⎢− Q−+ C P− − R ⎥ ⎥ ⎢ ⎢⎣ 0 − Q+ − R + P+ ⎥⎦
Figure 25. Continued on next page.
(27)
(28)
Theoretical Study on Quantum Dots Using Effective-Mass Envelope Function Theory 519
Figure 25. The first five energy levels of electrons as a function of (a) QD height, (b) QD width along the x direction, and (c) QW width, respectively. The other structure parameters are indicated in the corresponding figures. (From [61])
Figure 26. Same as Fig.2, except for the holes. (From [61])
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Shu-Shen Li and Jian-Bai Xia
Using the normalized plan-wave expansion method, the results can be calculated, which are shown in Fig. 25, 26, 27. The results show that for energy states of electrons, (1) the electron energy levels decrease monotonically and the energy difference between the different neighboring energy levels increase as the GaAs QD height increase; (2) the energy levels of electrons decrease as GaAs QD widths along the x direction increase when other structure parameters are fixed; (3) the strong energy mixing exists between the different energy levels; (4) the electron energy levels decrease monotonically as the GaAs QD width increases, but the energy differences between the energy levels are almost not affected by the variation of the GaAs QD width. The hole energy levels have similar variation trends as those of electrons as the structure parameters change. However, in the hole energy states, there are other two characteristics. One is that the hole energy levels decrease more quickly as the GaAs size increases. The other is that the hole every levels of excited states are very close to each other.
2.2.2. Optical Transition Energy
Figure 27. The energies of transition from the first electron energy level to the first heavy- and lighthole energy levels as function of (a) QD height, (b) QD width along the x direction, and (c) QW width (From[61])
Theoretical Study on Quantum Dots Using Effective-Mass Envelope Function Theory 521 The energies of transition from the first electron-energy levels to the first heavy and light hole energy levels as function of QD height, QD width along the x direction, and QW width are shown in Fig. 27(a)-4(c), respectively. The results show that (1) the difference between the first heavy- and light-hole levels increases slightly as the QD size increases; (2) the first heavy- and light hole transition energies are very close to each other for the self-assembly of GaAs/AlxGa1-xAs QDs. The transition energies for the first electron energy level to the first heavy-hole energy levels have been compared with the experimental results in Ref. [58]. The structure parameters are taken to be the same as those in Ref. [58] The results are shown in Fig. 28, in which solid circles are the experimental results of Ref. [58]. Our theoretical results are very close to the experimental data.
Figure 28. The energy of transition from the first electron energy level to the first heavy-hole energy level. The structure parameters are the same as those in Ref. [58]. The solid circles are the experimental results in Ref.6.(From [61]
2.2.3. Asymmetric Quantum-Confined Stark Effects The structure and our coordinate system are the same as Fig.24, except that we apply an electric field to the QDs. The angle between the z axis and the direction of electric field is θ . The electric field will point to the direction of z, x, -z, when θ equals 0, π / 2 , and π , respectively. In the presence of electric field, the effective Hamiltonians of electrons and holes are [62]
He = P
1 2m ∗ (r )
P +V e (r ) − eF ( z cos θ + x sin θ ) ,
(29)
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Shu-Shen Li and Jian-Bai Xia
and
R − Q− 0 ⎤ ⎡ P+ ⎥ ⎢ + P− C + − Q++ ⎥ 1 ⎢ R Hh = + Vh(r ) − eF ( z cos θ + x sin θ ) , 2m0 ⎢− Q−+ C P− − R ⎥ ⎥ ⎢ ⎢⎣ 0 − Q+ − R + P+ ⎥⎦
(30)
where F is the electric field. The transition energy is ET = E e + E h + E G ,
(31)
where E e , E h and EG are electron energy, hole energy, and the band gap of GaAs material, respectively.
Figure 29. The energy of transition from the first electron energy level to the first heavy (solid line) and light (dashed line) hole energy levels for the electric field parallel to the growth direction.(from [62])
Figure 30. The energies of transition from the first electron energy level to the first five hole energy levels as functions of the angle θ between the electric field and z direction. The electric field is 100kV/cm. The structure parameters of QD are W x = W y = 7 nm, and h=4nm, and d=2nm.(From [62])
Theoretical Study on Quantum Dots Using Effective-Mass Envelope Function Theory 523 Using similar method as described before, the results are calculated and shown in Fig.29 and 30. In Fig. 29, the positive and negative values of the electric field indicate the field applied along the growth direction and along the opposite of the growth direction, respectively. According to our results, we find that: (1) The transition energy has redshifts for the electric field along the growth direction. If the field along the opposite of the growth direction, the transition energy first blueshifts, then reaches the extremum of the curve, and then redshifts as the electric field increases. The reason is that the aluminum content x1 is larger in the QD structures than x2 in our calculation. The larger the electric field along the opposite of the growth direction, the smaller the distance between the carriers (electron and hole) and the Alx1 Ga 1−x1 As region. Then, the carriers will be confined more strongly. But, if the electric field along the opposite of the growth direction is larger than a critical value (about170 kV/cm at our chosen parameters [62]), the barrier in the Alx1 Ga 1−x1 As region will reduce more quickly, so the quantum confinement effects will reduce accordingly. (2) The higher the transition energy level, the larger the asymmetric quantum confinement effects (3) The valence-band mixing effects, coming form the hole 4 × 4 Hamiltonian, exist under an electric field, The off diagonal in the 4 × 4 Hamiltonian arose the mixing between light and heavy holes.
3. Coupled Quantum Dots Coupled quantum dots are also called artificial molecules which are defined by more than one QD coupled with one another electrostatically or by tunneling barriers. In coupled QDs, carriers can move between spatially separated dots; thus, the distribution of charges is strongly influenced by electrostatic interaction. In this section, we will present our understanding of electrical properties of InAs/GaAs strained coupled quantum dots [63,64] and coupled QDs arranged as superlattice [65].
3.1. InAs/GaAs Strained Coupled Quantum Dots Strain due to the lattice mismatch at the interfaces between two semiconductors is the driving force for the growth of self-assembled quantum dots and is known to play an important role in determining the electronic and optical properties of single and multiple SAQDs [66-69]. Many theoretical works have focused on the study of such systems. Atomistic approaches taking strain effects into consideration have been applied mainly to single quantum dots [7074], while coupled dots have been treated usually by simplified, continuous-medium models [69]. Recently, coupled and strained dots have been investigated in the framework of the pseudopotential approach [75-77]. Pryo studied the electronic structure of an infinite 1D array of vertically coupled InAs/GaAs strained quantum dots using an eight-band strain-dependent k ⋅ p Hamiltonian
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Shu-Shen Li and Jian-Bai Xia
[78]. In his study, the coupled dots form a unique quantum wire structure in which the miniband widths and effective masses are controlled by the distance between the islands, d. The miniband structure is calculated as a function of d, and it is shown that for d>4 nm the miniband is narrower than the optical phonon energy, while the gap between the first and second minibands is greater than the optical phonon energy. This leads to decreased optical phonon scattering. These miniband properties are also ideal for Bloch oscillations. Taddei et. al. investigated the effect of vertical coupling on the electronic levels and transition energies in multilayer InAs/GaAs quantum-dot structures, grown by ALMBE, as a function of the GaAs interlayer spacer thickness [79]. Their results show that either a blueshift or a redshift of the fundamental transition energy can be observed in different coupling conditions, which can be straightforwardly explained by including strain, indium segregation, and electron-hole Coulomb interaction. By using the adiabatic approximation, Korkusiński and Hawrylak calculated the electronic energy levels in the vertically coupled double quantum dot system [16]. Their procedure, besides geometric parameters of the system, requires knowledge of the band edge discontinuity between the quantum well and the barrier and the electron effective mass. They calculated the former using the continuum elasticity theory, and the latter by comparing the energy spectrum to that obtained from the numerical k•p calculation, but treated it as a fitting parameter. Their investigation shows that the change of the QD layer distance, D, strongly modifies the electronic energies, leading to a splitting (of order of 30 meV for small D) between the symmetric and antisymmetric levels, and causes crossings between levels belonging to different shells. These crossings are removed by a magnetic field perpendicular to the growth direction. Using an empirical tight-binding formalism (ETB), Jaskólski et. al. [66] investigated the electronic structure and optical properties of lensshaped, InAs/GaAs self-assembled, vertically stacked, double quantum dots situated on 2 monolayer thick wetting layers. Their study shows that for intermediate separation distances between the dots, the tight-binding theory confirms the effect of strain-induced localization of the ground hole state in the lower dot, as predicted in other approaches. However, the tightbinding calculations predict weaker localization at large separation distances and no localization for closely spaced and overlapping dots. Moreover, an anomalous reversal of the bonding character of the ground hole state for large separation distances, found previously for unstrained systems, is present for strained dots. Their results also show that in double quantum dots there may exist bound and localized electron and hole states with energies above the edge of the wetting layer continuum. In the following section, using the effective-mass envelope-function theory and planewave expansion method, we will present our study on InAs/GaAs strained coupled quantum dots [63-65]. In our study, the effects due to the different effective masses of electrons and holes in different materials are included.
3.1.1. Energy States To study the InAs/GaAs quantum dots grow on (100) GaAs substrate [63], we assume that the strain only arises in the InAs dots. In our calculation, we choose the growth direction (100) as the z direction of our coordinate system. The InAs dots are periodically arranged boxes [80]. In the z direction, the width of the InAs dot is l, and the distance between two adjacent dots is d, the period is then l+d. In the parallel direction, the radius of InAs dots is R,
Theoretical Study on Quantum Dots Using Effective-Mass Envelope Function Theory 525 the distance between two nearby dots is L-2R, and L is the period. We choose the boxed region as the unit cell in Fig. 31.
Figure 31. The structures of coupled InAs monolayer quantum dots: (a) along the parallel direction and (b) along the growth direction. (From [63])
The electron Hamiltonian can be written as the equation below
He = P
1 P +V e (x,y,z ) , 2m e (x,y,z ) ∗
(32)
For the valence subbands, including the effects of strain, the hole Hamiltonian can be written as
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Shu-Shen Li and Jian-Bai Xia
H h = H 0 + H ε + V h ( x, y , z ) ,
(33)
where Vh ( x, y, z ) is the hole potential of coupled quantum dots. H 0 is the Foreman effective-mass Hamiltonian for the hole state (excluding spin-orbit splitting ) [81], and H ε is the strained energies of the hole [82, 83].
R ⎡ P+ ⎢ + P− 1 ⎢ R H0 = 2m0 ⎢− Q−+ C ⎢ ⎢⎣ 0 − Q+ H ε = − Dd (ε xx + ε yy + ε zz ) − −
− Q− C+ P−
− R+
0 ⎤ ⎥ − Q++ ⎥ , −R ⎥ ⎥ P+ ⎥⎦
(34)
1 2 1 1 Du [( J x2 − J 2 )ε xx + ( J y2 − J 2 )ε yy + ( J z2 − J 2 )ε zz ] 3 3 3 3
2 ′ Du [2{J x , J y }ε xy + 2{J y , J z }ε yz + 2{J z , J x }ε zx ], 3 (35)
where Dd , Du , Du
′
are the deformation potentials,
ε xx , ε yy ,… are the strain tensor
components. The substrate of the InAs dots is GaAs, we assume
ε xx = ε yy =
a0 − a and ε ij = 0 (i ≠ j ) , a
(36)
in InAs where a and a 0 are the lattice parameters of bulk InAs and GaAs, respectively. The energy density of stain is U =
1 C11 (2ε xx2 + ε zz2 ) + C12 (2ε xx ε zz + ε xx2 ) . 2
U should have minimum in the condition (36), so
ε zz = −2
C12 ε xx C11
(37)
C12 and C11 are the elastic moduli of InAs. Using the representation of J x , J y and J z in Eq. (39) of Ref. [36], we obtain the hole effective-mass Hamiltonian,
Theoretical Study on Quantum Dots Using Effective-Mass Envelope Function Theory 527
R − Q− 0 ⎤ ⎡ P+ − D ⎢ ⎥ + − Q++ ⎥ P− + D C+ 1 ⎢ R Hh = + D ′ + Vh( z h ) . 2m0 ⎢ − Q−+ C P− + D − R ⎥ ⎢ ⎥ ⎢⎣ 0 P+ − D ⎥⎦ − Q+ − R +
(38)
Using the normalized plan-wave expansion method, the electron and hole energy states can be calculated. The treatment of strain effects in Eqs. (36) and (37) is a good approximation for larger R, but in the limit of small R, the strain will approach the case of ε xx = ε yy = ε zz = (a 0 − a) / a . So, our simplified treatment of strain will break down for small R. Due to the matrix elements of strain energy include the factor S j , and S j will approach zero when R approaches zero; the transition energies calculated using Eq. (36) is close to those calculated using ε xx = ε yy = ε zz for the small R. The R=0 transition energies is equal to the transition energies of bulk GaAs. The exciton Hamiltonian can be written as
e2 H = He + Hh − εr
.
(39)
Assuming that the exciton wave function is of the following form:
Ψex = Ψe Ψh G ( ρ , z ,θ ) = i, j ,
(40)
Ψe and Ψh are electron and hole wave function, respectively. G ( ρ , z ,θ ) = ∑ Aij ( ij
2α i
π
1/ 2
)
(
2β j
π
)1 / 4 exp(−α i ρ 2 − β j z 2 ) ,
ρ 2 = ( xe − x h ) 2 + ( y e − y h ) 2 , and z 2 = ( ze − z h ) 2 .
( 41)
The exciton energies can then be determined by
det( H i′j′,ij − ES i′j′,ij ) = 0 , with
(42)
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Shu-Shen Li and Jian-Bai Xia
H i′j ′,ij = i ′j ′ H e + H h −
e2 ij , εr
(43)
and S i′j ′,ij = i ′j ′ ij . The exciton binding energies are therefore given by
Eb = Ee + E h − E .
(44)
In our calculation, the number of plane waves is n x , n y , n z = 0,± 1, ± 2, ± 3 . Taking More plane waves, the calculated results will be slightly improved.
Figure 32. The heavy- and light-hole energy transitions as functions of the radius R (a) and width l (b). (From [63])
Figure 33. The distribution of electron and hole wave functions along the coordinate axes. The dotted line is the interface of InAs and GaAs. (From [63])
Theoretical Study on Quantum Dots Using Effective-Mass Envelope Function Theory 529 The results are shown in Fig.32 and Fig.33. The results indicate narrow bandwidths and sharp peak density of states in coupled quantum dots. The width of the first band is only about 1 meV for both electron and hole. The width of bands will become lager when the dots are closer to each other or the dots are wider in size. The mixing between the LH and HH is determined by R, Q, and C in Eq. (34). The mixing still exists even when k=0 (at the Γ point) in coupled quantum dots, while it does not exist in superlattices. The energy levels at the Γ point (k=0) are shown in Table 2. Form this table, we can see that the energy levels are strongly affected by the material parameters and the mixing effect between LH and HH in coupled quantum dots. Table 2. The energy levels (meV) at the Γ point (k=0). (From [63])
Using the envelope-function approach, one can indeed show that the excitonic wave functions are mostly confined in InAs dots. For simplicity, we assume here an effective mass, close to that of the InAs, for the whole InAs/GaAs coupled quantum dots system, and perform the same effective-mass calculation. When the volume of the InAs dots is not very small, this is a good approximation. Including the excitonic effects, Fig.32 gives the HH and LH energy transitions as functions of the radius R (a) and width l (b). The energy transition will decrease as the size of InAs dots increase. Table 3. The squared optical transition matrix elements from the first electronic energy level to the first heavy-hole and light-hole energy levels. (From [63])
For the monolayer coupled quantum dots, l=0.283 nm. Setting k=0, and taking structure parameters in Ref. [63], as well as the optical transition matrix constant
2 P 2 / m0 = 18.71 eV [84], we have calculated the squared optical transition-matrix elements
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Shu-Shen Li and Jian-Bai Xia
from the first electronic energy level (CB1) (where CB denotes conduction band) to the first LH energy level (LH1) and the first HH energy level (HH1). The results are given in Table 3, in comparison with the results of the superlattice model. For our calculation, we found the optical transitions are clearly anisotropic, and the squared optical transition-matrix elements of the dot model are close to those of the superlattice mode. Fig.33. shows the distribution of electron and hole wave functions of the ground states along coordinate axes for coupled InAs monolayer quantum dots. Here the upper lines correspond to electrons and the lower lines correspond to holes. From this figure, we see that the wave functions of electrons and holes penetrate very deeply into the GaAs barrier along the growth direction. Also the penetration of the wave functions of LH into GaAs is deeper than those of HH. We can calculate the exciton states. We find that the LH exciton binding energies are very close to the value of bulk GaAs and weakly dependent on the structure parameters of the InAs dots. This is also due to the very deep penetration of the LH wave functions into the GaAs barrier. For HH, when d = 50 Å and L=200Å, the binding energies are close to the value of bulk GaAs and weakly dependent on the radius R; with a fixed value of d=50Å and R=50Å, and L increasing, the binding energies form the minimum value (close to the value of the bulk GaAs) increase to a maximum (only about 4.5 meV), and then decreases to a Value again, close to the value of bulk GaAs. The L value corresponding to the maximum binding energy is about 65 nm. When R and L are very large, the results will be similar to the results of InAs/GaAs superlattices. In a word, the excitonic binding energies are close to the value of bulk GaAs.
3.1.2. Intraband Optical Absorption In this section , we study the intraband optical absorption of InAs/GaAs strained couple quantum dots [64]. We choose the structure and coordinate system same as those in section 3.1.1. The electron envelope function equation is
[P
1 P +V e (x,y,z )]Fn (r ) = E n Fn (r ) , 2m e (x,y,z ) ∗
(45)
∗
where Ve and me are defined in Ref. [63]; n=0,1,2… denote the ground state, the first excited state, the second excited state, etc. Fn (r ) is the electron envelope function.
Ψ0 and Ψn denote the electron wave functions of the ground state and the nth subband state, respectively. . In the effective mass approximation, the matrix element between the two subband states ( Ψn and Ψ0 ) in coupled quantum dot can be expressed as follows:
Ψn H ′ Ψ0 =
m Fn H ′ F0 , m∗
(46)
Theoretical Study on Quantum Dots Using Effective-Mass Envelope Function Theory 531 where H ′ is the interaction Hamiltonian between a free electron with mass m and a radiation field. Let us focus our attention on the transitions 0 → n and exclude other possible transitions such as 1 → n , etc. The absorption coefficient is given by
8π 2 e 2 α ( ω) = ηcωm 2 where
2
dk ∑n ∫ (2π ) 3 × Ψn H ′ Ψ0 δ ( En − E0 − ω ) × [ f 0 (k ) − f n (k )] , (47)
η is the refractive index, and f 0 (k ) and f n (k ) are the occupancy of the electron at
the ground state and n state, respectively. In order to obtain a smooth absorption spectrum, we replace the with a Lorentzian function with a half-width Γ , viz.,
δ (E − ω) ≈
Γ
π [( E − ω ) 2 + Γ 2 ]
δ function in Eq. (47)
.
(48)
The magnitude of Γ is roughly equal to the energy spacing of the eigenstates.
Figure 34. The conduction subbands and minigaps as a function of QD radius R. The two lines marked with same symbols are the bottom and top of corresponding subband, respectively. (From [64])
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Shu-Shen Li and Jian-Bai Xia
Figure 35. Absorption coefficient spetra for InAs/GaAs QD at room temperature with -3 N InAs = 1.5 × 1018 cm and L=200Å, R=50Å, l=30Å, and d=50Å. The solid and dotted lines are the results for electric vector of incident light perpendicular to the growth direction ( θ = π / 2 , and β = 0 ) and parallel to the growth direction ( θ = 0 ), respectively. The latter has been multiple by 10. (From [64])
Figure 36. (a) Maximum absorption coefficient and (b) resonance energy as a function of R for the -3 0 → 1 transition in InAs/GaAs QD at room temperature with N InAs = 1.5 × 1018 cm and L=200Å,
l=30Å, and d=50Å. The solid and dotted lines are the results of electric vector of incident light perpendicular to the growth direction ( θ = π / 2 , and β = 0 ) and parallel to the growth direction ( θ = 0 ), respectively. (From [64])
Theoretical Study on Quantum Dots Using Effective-Mass Envelope Function Theory 533
Figure 37 Maximum absorption coefficient as a function of electron concentration NInAs for the 0 → 1 transition in InAs/GaAs QD at room temperature with L=200Å, R=50Å, l=30Å, and d=50Å. The meaning of solid and dotted lines are the same as in Fig. 36 (From [64])
The results show that the absorption coefficient is weakly dependent on β ( β is the angle between the electric vector of light and the X direction of the QD). From Fig 35, we can find that the absorption coefficient for the electric vector perpendicular to the growth direction is larger than that for the electric vector parallel to the growth direction. This result is very different from the case of superlattice (SL). In SL, the absorption coefficient is always zero when the electric vector of incident light is perpendicular to the growth direction. We think that it may be more convenient to fabricate an infrared detector using a QD device rather than using a SL device. The energy band of the ground and the first excited states has a small width due to the small coupling between dots. So, the resonance energy of the 0 → 1 transition is almost located at the same site when changing the directions of incident light. Fig. 36(a) and (b) show maximum absorption coefficient α p and resonance energy E r as a function of QD radius R for the 0 → 1 transition at room temperature with
N InAs = 1.5 × 1018 cm-3 and L=200Å, l=30Å, and d=50Å. The α p value decreases and the E r value increases with decreasing R (form 8 nm) and deviate from the trend at a critical radius Rc . When R is smaller than Rc , the first excited state will exceed the potential barrier height, and the energy band of the first excited state becomes broad. So the
α p value will
increase and the E r value will decrease with decreasing R when R is smaller than Rc . From Fig. 36(a), we can find that the maximum absorption coefficient is basically constant when
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Shu-Shen Li and Jian-Bai Xia
the electric vector of the incident light is parallel to the growth direction. This is due to the constant width of the QD along the growth direction. Form Fig.37, we can find that the α p values increase with increasing N and deviate from the line at higher N values. This deviation is though to imply the onset of occupation by electrons of the first excited state to which the transition from the ground state should occur. The critical N value for this deviation approximately coincides with the value at which the Fermi function goes into the first excited state to some extent. This characteristic is the same as the results in SL.
3.2. Coupled Quantum Dots Arranged as Superlattice In our consideration [65], we suppose that the QDs are cubic, the side length is b, and the faces are parallel to the corresponding faces of the superlattice cell of the coupled QDs. The constant of the coupled QD superlattices is a. The electron and hole Hamiltonian can be written approximately as
He =
1 2 Pe +V e (x,y,z ) , 2m e∗
H h = H 0 + V h ( x, y , z ) ,
(49)
(50)
where H 0 is the Luttinger effective mass Hamiltonian, which can be written as ⎡ ⎢ 1 ⎢ H0 = 2m 0 ⎢ ⎢ ⎢⎣
P1 Q R
∗
0
∗
Q
R
P2
0
0
P2
R∗ − Q∗
0 ⎤ ⎥ R ⎥ . − Q⎥ ⎥ P1 ⎥⎦
(51)
Below, we give some numerical results for GaAs/Ga1-xAlxAs QD superlattices. The confinement is caused by the band offset. Fig. 38 gives the electron subband structures of GaAs/Ga0.7Al0.3As Coupled QDs arranged as simple cubic (sc) superlattice. Form Fig.38, we can find the following properties: (1) Due to the coupling between QDs, the energy levels of quantum dots are broaden to form the energy band structures. (2) The first energy band is very narrow; its width is only about 4.7 meV for the chosen structural parameters. (3) There is only one confined energy band, which is lower than the potential barrier height. (4) The second and the other higher electron states are continuous states, with energies higher than the potential barrier height. (5) Along different directions in k space, the electron subband structures are different from each other due to the different coupling between QDs.
Theoretical Study on Quantum Dots Using Effective-Mass Envelope Function Theory 535
Figure 38. Electron subband structures of GaAs/Ga0.7Al0.3As coupled QDs arranged as simple cubit (sc) superlattice with structural parameters a=200Å and b=50Å.(From [65])
Figure 39. Electron subband structures of GaAs/Ga0.7Al0.3As coupled QDs arranged as body-centered cubic (bcc) superlattices with structural parameters a = 200 × 21 / 3 Å and b=50Å.(From [65])
Fig. 39 and 40 show the electron subband structures of GaAs/Ga0.7Al0.3As coupled QDs arranged as a body-centered cubic (bcc) and face-centered cubic (fcc) superlattice, respectively. For the bcc QDs, there are two confined energy bands, lower than the electron barrier, which are too close to separate form each other in Fig. 39. The reason is that there are
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two QDs in one cubic superlattice cell. For the fcc QD superlattice, there are four confined energy bands, lower than the electron barrier, which are very close in energies.
Figure 40. Electron subband structures of GaAs/Ga0.7Al0.3As coupled QDs arranged as face-centered cubic (fcc) superlattices with structural parameters a = 200 × 41 / 3 Å and b=50Å.(From [65])
Comparing Fig.38, 39 and 40, we find that the electron energies of bcc, sc, and fcc superlattices are the lowest, the highest, and the middle, respectively, for the same electron subband under the same QD density. In other words, in the same QD density, the confinement effects in sc, fcc, and bcc superlattices are the largest, the middle, and the smallest, respectively. Table 4. The Γ point energy levels (in units of meV) of the first electron subband for the same QD density (same ρ ) and the same superlattice constant (same a=200Å ) and b=50Å (From [65])
Table 4. gives the Γ point energy levels of the first electron subband. It shows that the electron energies will decrease when QD density increase since the coupling between QDs will increase when increasing QD density. But for a fixed superlattice constant, the electron, the electron energies in bcc, sc, and fcc structures are still the lowest, the highest, and the middle, respectively. The hole subband structures are similar to those of the electron subband structures.
Theoretical Study on Quantum Dots Using Effective-Mass Envelope Function Theory 537 Table 5 gives the squared optical transition matrix elements from the first electron level (E1) to the first heavy hole (H1) and to the second hole (H2) energy levels at Γ point for a superlattice QD with a=200Å and b=50Å. We can find that the squared optical transition matrix elements from the first electron energy level (E1) to the second hole (H2) energy levels at Γ point are very small for bcc and fcc QD superlattices. Table 5. The Γ point squared optical transition matrix elements (in units of eV) from the first electron level (E1) to the first heavy hole (H1) and the second hole (H2) energy levels with a=200Å and b=50Å. (From [65])
Figure 41. The Γ point transition energies from the first electron subband to the first hole subband as functions of the QD size b. The solid, dotted, and the dashed line are the results of sc, fcc, and bcc superlattice, respectively. (From [65])
The transition energies at Γ point from the E1 to the H1 as function of the QD size b are shown in Fig.41 with a fixed as 200Å. When the size of QDs is very small (smaller than about 20 Å), the transition energies in all three structures are almost equal. The differences between the transition energies become significant as the size of the QDs increases. For example, when a=200Å and b=60Å, the differences of transition energy are about 90 (between fcc and sc), 142 (between bcc and fcc), and 232 (between bcc and sc), respectively.
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4. N Quantum Dot Molecule(N QDM) A single QD is usually considered as an artificial atom due to its features characteristic of an atom. Likewise, a chain of coupled QDs has many similarities to molecules; thus it is generally called an artificial molecule or a quantum dot molecule (QDM). QDM is a subject of considerable interest due to possible application for photoelectric devices and quantum computing [85, 86]. The electron and hole energies are strongly affected by the number of quantum dots in the molecule and the shape as well.
Figure 42. The schematic picture of 2 QDM. (From [16])
Many attentions have been paid to more than one quantum dot molecule. In the framework of effective mass approximation, Korkusiński and Hawrylak [16] studied 2 QDM theoretically. They used the adiabatic approximation to calculate the electronic energy levels in the vertically coupled double quantum dot system, as shown in Fig. 42. The procedure, beside geometric parameters of the system, requires knowledge of the band edge discontinuity between the quantum well and the barrier and the electron effective mass. They calculated the former using the continuum elasticity theory, and the latter by comparing the energy spectrum to that obtained from the numerical k•p calculation, but treated it as a fitting parameter. They found that the change of the QD layer distance strongly modifies the electronic energies, leading to a splitting (of order of 30 meV for small D) between the symmetric and antisymmetric levels, and causes crossings between levels belonging to different shells. These crossings are removed by a magnetic field perpendicular to the growth direction. Using correlated pseudopotential calculation, Bester et al.[77] studied an exciton in a pair of vertically stacked InGaAs/GaAs dots. They found that competing effects of strain, geometry, and band mixing led to many unexpected features missing in contemporary models. Their calculations show that the first four excitonic states are all optically active at small interdot separation, due to the broken symmetry of the single particle states. They also quantified the degree of entanglement of the exciton wave functions and showed its sensitivity to interdot separation. Using single-particle pseudopotential and many-particle configuration interaction methods, He and Zunger [87] compared various physical quantities of (In,Ga)As/GaAs QDMs made of dissimilar dots (heteropolar QDMs) with QDMs made of identical dots (homopolar QDMs). The geometry the used is shown in Fig. 43. Their calculations show that
Theoretical Study on Quantum Dots Using Effective-Mass Envelope Function Theory 539 the electronic structures of hetero-QDMs and homo-QDMs differs significantly at large interdot distance. In particular, (1) unlike those of homo-QDMs, the single-particle molecular orbitals of hetero-QDMs convert to dot-localize dorbitals at large interdot distance; (2) consequently, in a hetero-QMD the bonding-antibonding splitting of molecular orbitals at large interdot distance issignificantly larger than the electron hopping energy whereas for a homo-QDM, the bonding-antibonding splitting is very similar to the hopping energy; (3) the asymmetry of the QDM increases significantly the double occupation for the two-electron ground states and therefore lowers the degree of entanglement of the two electrons.
Figure 43. The geometry of (In,Ga)As/GaAs QDMs (From [87])
Xie [88] studied three electrons confined in one-, two-, and three-layer quantum dots, by using the exact diagonalization method. His calculation shows that, for three-electron QDs, the series of the magic numbers in one-, two-, and three-layer QDs, are different. These magic numbers can be understood from symmetries. He also concluded that the composite fermion model provides an alternative for the explanation of magic numbers in one-layer quantum dots [89], which was found to be quite successful in the regime of weak interactions or for short-range interactions, but less so for the long-range Coulomb interaction and/or in the regime of strong interaction. Furthermore, Emary et al. [90] described a mechanism for the production of polarizationentangled microwaves using intraband transitions in a pair of quantum dots (4QDM). To study a general case, namely N QDM, taking the effects of finite offset and valenceband mixing into consideration, we choose the symmetry center of the N QDM as the origin of our coordinate system [91]. The electron and hole envelope function in the framework of the effective-mass approximation is
[ H 0 + V (r )]ψ (r ) = E ψ (r ) , e,h e,h e,h e,h e,h
52
where the letters e and h indicate electron and hole states, respectively. For electron states,
H e0 = −
1 d2 d2 d2 [ + + ], me∗ dx 2 dy 2 dz 2
(53)
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and for hole states
⎡ ⎢ 0 H h = ⎢⎢ ⎢ ⎢⎣
0 ⎤ ⎥ R P− 0 − Q ⎥ , Q ∗ 0 P− R ⎥ ⎥ 0 − Q ∗ R ∗ P+ ⎥⎦ P+
R
Q
∗
(54)
with
P± = (γ 1 ± γ 2 )( p x2 + p y2 ) + (γ 1 ∓ 2γ 2 ) p z2 ,
Q = −i 2 3γ 3 ( p x − ip y ) p z , R = 3[γ 2 ( p x2 − p y2 ) − 2iγ 3 p x p y ] , ∗
where m0 and
(55)
γ 1 , γ 2 , and γ 3 are the electron effective-mass and Luttinger parameters,
respectively. Using the normalized pane-wave expansion method, the electron and hole states in the N QDM can be easily calculated form the matrix elements. In the calculation, the matrix elements of Ve , h (r ) depend on the QD shapes and position in N QDM.
Figure 44. Ground state energy levels of electrons as a function of quantum dot radius. (From [91])
In the following, we will give some numerical results for the electronic structure of an electron and hole in GaAs/Ga0.65Al0.35As N ball QDM, i.e., the shape of the QDs in N QDM is
Theoretical Study on Quantum Dots Using Effective-Mass Envelope Function Theory 541 a ball with radius R0. Our calculation method can be easily extended to other QD shapes. We adopt a square potential energy model in our calculation, i.e., Ve , h (r ) = 0 inside and
Ve ,h (r ) = Ve0,h outside of the quantum dots. We calculate N QDM for N=1, 2, 3, 4, 5, 6, 7. We assume that the quantum dots in 1 QDM are in a plan, the shape of quantum dots is a ball with radius R0 , and the ball are tangent. Our method of calculation is easily generalized to other more complex N QDM.
Figure 45. Same as Fig. 44 except for the holes (From [91])
Fig. 44 gives the ground state levels of electrons as functions of QD radius for 1-7 QDM. As shown in Fig. 44, the electron energy levels decrease monotonically and the energy difference between the different QDMs decreases as the QD radius increases. This is because the quantum confinement effect is smaller for the larger quantum dot. Another characteristic is that the energy level is lower and quantum confinement is smaller for the larger N QDM. Fig. 45 shows the ground energy levels of the hole as function of QD radius for 1-7 QDM. Compared with electrons, the greatest difference is that the ground state energy level of the hole is lower for the one dot QDM than N (greater than 1) QDMs if QD radius is larger than about 5 nm. The reason is that the valence-band mixing effect is stronger for the larger QDs and larger N QDMs. Since the light-hole energy level is higher than heavy-hole energy level, the valence-band mixing effect of heavy and light holes induces the ground state include more light-hole component, so it causes the ground state energy level going up. If we 0
cancel the valence-band mixing (let the non-cross terms of the H h equal 0), the change trend of the energy levels will be consistent with Fig. 44 and the energy level cross will disappear.
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In: Quantum Dots: Research, Technology and Applications ISBN 978-1-60456-930-8 c 2008 Nova Science Publishers, Inc. Editor: Randolf W. Knoss, pp. 545-575
Chapter 15
T RANSMISSION THROUGH Q UANTUM D OTS WITH VARIABLE S HAPE : B OUND S TATES IN THE C ONTINUUM Almas F. Sadreev1 , Evgeny N. Bulgakov1, Konstantin N. Pichugin 1, Ingrid Rotter2 and Tatyana V. Babushkina 3 1 Institute of Physics, Academy of Sciences, 660036 Krasnoyarsk, Russia 2 Max-Planck-Institut f¨ur Physik Komplexer Systeme, D-01187 Dresden, Germany 3 Siberian Federal University, Krasnoyarsk 660090, Russia Abstract We consider open quantum dots (QD) whose spectra can be varied continuously by variation of gate voltage. We show that bound states in the continuum (BICs) may occur for discrete values of the voltage and energy of incident electrons. They are localized inside the QD and superposed by the transport solution. However superposition coefficient depends on the way the BIC point is approached. For integrable QD this phenomenon occurs, if the QD spectrum is degenerated incidentally. However a BIC might occur for irregular shape of QD. Both types of QDs are considered analytically in the simplest case of a two level QD and are complemented by numerical calculations for the realistic QB. Although each eigen state of QD is coupled to waveguide, the coupling of BIC with propagating mode of the waveguide turns to zero because of interference with other resonances. As a result, resonance width tends to zero for approaching to the BIC point. In order to find explicitly BICs we are looking for the complex eigenvalues of the effective non hermitian Hamiltonian which respond for positions and widths of the resonance states. In particular we show that BIC is an eigenstate of the effective Hamiltonian with real eigenvalue. We present a few numerical examples of BICs in realistic QDs and in systems of double QDs coupled by a wire with variable spectrum. In the framework of the impurity Anderson model we took into account Coulomb effects. Such an approach allows one to find the Green function of the closed QD exactly. Further, the solution of the Dyson equation for full Green function describes the open QD. We show that the Coulomb repulsion does not eliminate the BIC, but on the contrary, replicates BICs as two-electron BICs.
PACS numbers: 03.65.-w, 03.65.Ge, 03.65.Nk, 73.23.Ad
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Almas F. Sadreev, Evgeny N. Bulgakov, Konstantin N. Pichugin et al.
Introduction
In 1929, von Neumann and Wigner [1] firstly pointed to the existence of discrete solutions of the single-particle Schr¨odinger equation embedded in the continuum of positive energy states. The analysis has been examined later by Stillinger and Herrick [2] in the context of possible bound states (BICs) in atoms and molecules (see also, e.g., [3, 4, 5]). In 1973 Herrik [6] predicted BICs in semiconductor superlattices which later were observed by Capasso et al [7]. In the framework of the Feshbach’s theory of resonances Friedrich and Wintgen [8] have shown that BIC occurs due to the interference of resonances. If two resonances pass each other as a function of a continuous parameter, then for a given value of the parameter one resonance has exactly vanishing width. Later this result in the twolevel approximation was obtained in application to many physical systems [9, ?, 11, 12, 13, 14, 15]). Numerical calculations for realistic quantum open system variable structures beyond of the two-level approximation confirmed that the resonance width, of at least one of the poles may, in fact, turn to zero, for example, in laser induced continuum structures in atoms [16], in bent waveguide [17, 18], in the molecular system [19], in quantum dot (QD) (or resonator) with variable shape for transmission of free electrons [20, 21]. Moreover BICs were shown to exist in a system of two realistic dots (resonators) coupled by wire with variable spectrum [22, 23] which is a counterpart of system of coupled model two-level QDs [14], in the two-dimensional Aharonov-Bohm ring [24], counterpart of model onedimensional ring [25, 24], in stubbed waveguides [26] etc. Therefore, examples of BICs can more easily be found if one goes beyond the onedimensional Schr¨odinger equation studied in [1, 2, 4, 5] . In hard wall approximation QDs with attached wires are described by the Helmholtz equation −∇2 ψ(x, y) = ψ(x, y)
(1)
with Dirichlet boundary conditions if to disregard the Coulomb interactions between electrons. Typical space structure of the system is shown in Fig. 1. Here = E/E0, E0 = b
a y
x d
d
Figure 1. Typical structure of open QD with two attached wires with the width d. By application of gate potentials [44] a shape of QD can vary. Here we show evolution of the system from asymmetrical position of wires (a) into symmetrical one (b). ¯ 2 /2md2 is the dimensionless energy related to the particle energy E and the width d of h
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the wires. Below, the well-defined threshold π 2 for the propagating states in the wires the energy eigenvalues are discrete and the corresponding eigenfunctions are square-integrable bound states. Above the threshold the eigenvalues are distributed continuously and the corresponding eigenfunctions are normalized via the delta-function in energy. We consider that square-integrable solutions of the Schr¨odinger equation with isolated discrete energy above the continuum threshold might appear for different shapes of the QBs. These solutions are BICs having an infinitely long lifetime. In present paper we advocate the approach of effective Hamiltonian for the open QDs which is the result of projection of the hermitian total Hamiltonian onto eigenstates of the closed QD. A condition that imaginary part of complex eigenvalues of the effective Hamiltonian turns to zero defines BIC. This approach is equivalent to that a solution of the Helmholtz equation (1) is the square-integrable complex function [1, 2, 4, 5, 27, 28].
2.
BICs in the One-Dimensional Ring Pierced by Magnetic Field
The Aharonov-Bohm (AB) oscillations of conductance with changing magnetic flux [29], realized in normal metallic and semiconductor rings [30, 31], is an important achievement of mesoscopic physics. Although a dimensionality of ring is important for AB oscillations [33], the 1d ring is remarkable by that the consideration of BICs does not need numerical calculations. Moreover, as was shown in [24] a solutions for BICs in the 2d ring are quite similar to the solutions in the 1d ring. Following Xia [32] we write the wave functions in the segments of the structure shown in inset of Fig. 2 as ψ1 (x) ψ2 (x) ψ3 (x) ψ4 (x)
= = = =
exp(ikx) + r exp(−ikx), a1 exp(ik− x) + a2 exp(−ik+ x), b1 exp(ik+ x) + b2 exp(−ik− x), t exp(ikx),
(2)
where k− = k − γ, k+ = k + γ, γ = 2πΦ/Φ0, Φ = BπR2 is the magnetic flux, hc/e. The ring length 2πR is chosen as unit. The boundary conditions (the Φ0 = 2π¯ continuity of the wave functions and the conservation of the current density) allow to find all coefficients in (2). We write the corresponding equation in matrix form → → − Fˆ ψ = − g,
(3)
where Fˆ (k, γ) is the following matrix
−1 0 −1 0 0 −1 0 −1 1 0 0
−1
1 0 ik − /2
e
0 k− k − k − i k2 e k
1 0 −ik + /2
e
0 + − kk +
− kk e−i
0 1 0 eik
k+ 2
+ /2
k+ k + k + i k2 e k
0 1 0 −
e−ik /2 − − kk −
− kk e−i
k− 2
,
(4)
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2
−
2.5
γ/2π
−
m=
2
3
m=
4
1 3
−
2 1.5
1
m=
0
m=
2
m=
1 0.5 0 0
−1
1
m=
0.5
m=
1
k/2π
1.5
2
Figure 2. Transmission zeros |t| = 0 and ones |t| = 1 of the one-dimensional ring as function of the wave number k and flux γ. The zeros (ones) are shown by dashed (solid) lines. The thin solid lines represent the eigenenergies of the closed ring. − →T − → g T = (1 1 0 0 1 0). The vector ψ = (r t a1 a2 b1 b2) is the solution for the scattering wave function: r = 2(3 cos k − 4 cos γ + 1)/Z, t = 16i(sin k2 cos γ2 )/Z (5) a1 = 2(2eiγ − 3e−ik + 1)/Z, ik iγ a2 = 2(e + 1 − 2e )/Z, Z = 8 cos γ − 9e−ik − eik + 2, b1,2(k, γ) = a1,2(k, −γ). In Fig. 2 we show lines of the transmission zeros ( |t(k, γ)| = 0, dashed lines) which cross the lines of the transmission ones ( |t(k, γ)| = 1, solid lines) at points km = 2πm, m = ±1, ±2, . . . , (6) γn = 2πn, n = 0, ±1, ±2, . . .. As can be seen from the expression for the denominator Z in Eqs. (5), the imaginary part of the poles vanishes at these points. Simultaneously, there is a degeneracy of eigenenergies (km − γ)2 of closed ring. Here m is the magnetic quantum number that defines the eigen functions of the closed ring ψm (x) = exp(ikmx). The point k = 0 is excluded from the consideration since it gives zero conductance. The peculiar points (6) were shown in [25] for the case of single wire attached to the 1d ring. To show that the BICs appear at the points (6), let us consider one of the points, say, s0 = (k1, γ1) = 2π(1, 1). All the other points are equivalent because of the periodical dependence of the system on k and γ. In the vicinity of the point s0 we write Eqs (5) in the following approximated form i(3∆k2 − 4∆γ 2) ∆k , r≈ , 2 ∆k + i(∆γ) /2 4(2∆k + i∆γ 2) 3∆k + 2∆γ ∆k − 2∆γ , a2 ≈ , a1 ≈ 2 4∆k + 2i∆γ 4∆k + 2i∆γ 2
t≈
(7)
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where ∆k = k − k1 , ∆γ = γ − γ1. The transmission amplitude in the vicinity of the point s0 in (7) is similar to the expressions obtained for a shifted von Neumann-Wigner potential [4, 5]. One can see that all amplitudes a1,2, b1,2 of the inner wave functions are singular at the point s0. Such a result for the BIC points was firstly found by Pursey and Weber [5]. Eqs (3) and (4) allow to show that the point s0 corresponds to the BIC in an open one-dimensional ring. At this point the matrix (4) takes the following form
ˆ F (s0) =
−1 0 1 1 0 −1 0 0 0 1 0 −1 1 1 0 0 −1 0 0 1 1 0 0 −2 2 0 −1 0 −2 2
0 1 0 1 0 0
.
(8)
The determinant of the matrix Fˆ (s0) equals zero. Therefore, there is a vector that is a → − →T − solution of equation Fˆ (s0) f0 = 0. By direct substitution of the vector f0 = 12 (0 0 1 −1 − → − 1 1) one can verify that f 0 is such a solution which is the null vector. The corresponding → − ˜ left null eigenvector is f 0 = 12 (−1 1 1 − 1 0 0). It is well known from linear algebra, that if the determinant of matrix Fˆ is equal to zero, then the necessary and sufficient condition → − ˜ for the existence of the solution of the equation (3) is that the vector f 0 is orthogonal to − → ˜ → → g = 0. Thus, the solution of Eq. (3) at the point s0 vector − g [34]. It holds, indeed, f 0 · − can therefore be presented as → − − → → − (9) ψ (s0) = α f 0 + ψp, − → where α is an arbitrary coefficient and ψp is particular transport solution of Eq. (3). By − →T direct substitution one can verify that ψp = 0 1 34 14 34 14 is the particular solution of Eq. (3). It is worthwhile to note that this result completely agrees with the scattering theory on graphs [35, 36]. Texier [35] has shown that for certain graphs the stationary scattering state gives the solution of the Schr¨odinger equation for the continuum spectrum apart for discrete set of energies where some additional states are localized in the graph and thus are not probing by scattering, leading to the failure of the state counting method from the scattering. In the vicinity of the BIC point s0 the scattering state becomes by use (7) → − → − ∆γ f 0 + ∆k ψ p − → , ψ (s) ≈ ∆k + i∆γ 2/2
(10)
to leading order of ∆k, ∆γ where s = (k, γ). Thus, the scattering state in the nearest → − vicinity of the BIC point also is a superposition of the BIC vector f 0 and of the particular → − → − solution ψ p . Eq. (10) shows that the limiting scattering wave state ψ depends on the way → − − → s → s0 . If we at first take ∆γ = 0, then obtain ψ = ψ p which is the transport solution. → → − 2 − f 0, i.e. the scattering state is If we, however, choose at first ∆k = 0, then have ψ = i∆γ → − diverging inside the ring. This formula shows that the BIC state f 0 can be extracted from the scattering state by a special limit in (10).
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It is worthwhile to present wave function solution (2) at the BIC points (6). Substituting →T − the solution f0 = 12 (0 0 1 − 1 − 1 1) into (2) we obtain ψ1(x) ψ2(x) ψ3(x) ψ4(x)
= = = =
0, i exp(−iγx) sin(kx), −i exp(iγx) sin(kx), 0.
(11)
One can see that this solution is fully localized inside the ring and, therefore corresponds to BIC. At the BIC energies with the wave vector k = 2πm, m = ±1, ±2, . . . the interior BIC function (11) has nodes at the points of attachment of wires x = 0, x = 1/2. That shows that a coupling of BIC state with wires equals zero that makes BIC invisible for electron incident in wires. Moreover the BIC solution (11) equals zero also at any commensurate points x = m/2m0, m0 = 1, 2, . . .. Therefore, BIC might appear for wires attached at these commensurate posisitions in the ring. Similar for the particular transport solution we obtain the wave function (2) as follows ψ1(x) ψ2(x) ψ3(x) ψ4(x)
= exp(ikx), = 14 [3 exp(ikx) + exp(−ikx)] exp(−iγx), = 14 [3 exp(ikx) + exp(−ikx)] exp(iγx), = exp(ikx).
(12)
This wave function corresponds to completely transparent ring ( |t| = 1) and has antinodes at the points of attachment of wires, i.e. has maximal coupling of the ring with the wires. As a result at the BIC point this transport solution corresponds to maximal openness of the ring.
3.
The Concept of Effective Hamiltonian in Numerics
Numerically the scattering wave function satisfied to the Helmholtz equation (1) and the conductance can be computed in many ways. For our purpose of study of BICs we choose the method of finite differences which transforms the Hamiltonian into the tight-binding one [37, 38]. Next, we split the total Hilbert space into two subspaces with Q+P=1 where Q projects onto the subspace of discrete states of closed QD and P onto the supplementary subspace of scattering states in the wires. Assume that solutions of both subsystems are known: X |bihb| (13) HB |bi = Eb |bi, hb|b0i = δbb0 , Q = b
for QD, and 0
0
HC |C, Ei = E|C, Ei, hE|E i = δ(E − E ), P =
Z
dE|EihE|
(14)
C
for wires correspondingly, where the index C involves here as the type of wires (the left or right) as well the type of propagating mode in the wires enumerated by quantum number p with energy (15) E = E0(d2kp2 + π 2 p2), p = 1, 2, 3, . . ..
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By use of the projection operators P and Q one can split the total Hamiltonian as follows [37, 39, 38, 40] X X HC + HB + (VBC + VCB ) (16) H= C
C
where VBC is the operator of coupling between the closed QD and propagating modes C (the C-th continuum). Finally, we can project the total Hamiltonian (16) onto space Q to formulate the effective Hamiltonian [39, 40] Heff = HB +
X
VBC
C
E+
1 VCB . − HC
(17)
The concept of the effective Hamiltonian appeared first in Feshbach’s papers [41]. Heff takes the most simple form in the site representation of the tight-binding Hamiltonian [37, 38] X |jihj + µ| (18) H = −t jµ
where t are the hopping matrix elements, j runs over two-dimensional sites of the lattice model, and µ runs over the nearest neighbors of the site j. Then the matrix elements of the coupling operator VBC exist only between wires and QD and equal −t. After integration over the spectrum of propagating modes in the wires (15) we obtain that Heff = HB in all sites inside the QD except those boundary sites jC that are adjacent to the C-th wire. For the last sites X φp (jC )φp (jC0 ) exp(ikp), (19) hjC |Heff |jC0 i = HB − t2 p
q
describe the transverse eigenfunctions of the wire with where φp (j) = NL2+1 sin Nπpj L +1 numerical width NL . We assume that both wires are identical. In the presentation of the eigenstates of closed QD (13) matrix elements of the Heff are hb|Hef f |b0i = Eb δbb0 −
X X
C
C C ikp Vb,p Vb0 ,pe ,
(20)
p C=L,R
where in accordance to (19) we obtain [38] C =t Vb,p
X
ψb (jC )φp(jC ),
(21)
jC
jC runs over boundary sites of the C-th wire. Heff is non-hermitian. The complex eigenvalues of the effective Hamiltonian determine the positions and widths of the resonance states. Because of energy dependence of the effective Hamiltonian the positions and widths of the resonance states are defined by the following nonlinear fixed point equations [40] Eλ = Re(zλ (Eλ)), 2Γλ = −Im(zλ (Eλ)).
(22)
Here zλ are the complex eigenvalues of the effective Hamiltonian (17) Hef f |λ) = zλ |λ)
(23)
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with right eigenstates |λ). The solutions (22) coincide approximately with the poles of the S matrix. However coincidence becomes exact when the pole becomes real. The case of Im(zλ) = 0 at which the inverse of operator E − Hef f becomes singular defines the condition for BIC as it will be considered in the next section. The transmission amplitude and the scattering wave function become especially simple and transparent in the biorthogonal basis of the eigen states of the effective Hamiltonian. In particular, the transmission amplitude is given [38] t = −2πi
X λ
VλL V˜λR , E − zλ(E)
(24)
where VλL = (λ|V |L, pi =
X
V˜λ = hR, p|V |λ) =
X
ψλ(jL) sin
jL
jR
ψeλ(jR) sin
πpjL , NL + 1
πpjR . NR + 1
(25)
The equation for the scattering wave function projected onto the QD |ψB i can be derived from the Lippmann-Schwinger equation [39, 38, 40] and takes the following form (Hef f − E)|ψB i = VBL|L, pi.
(26)
This formula states that scattering takes place for electron incident from the left wire. If Det(Hef f − E) 6= 0, then in the biorthogonal basis |λ) the scattering wave function inside the QD takes the simple form [38, 40] |ψB i =
X λ
VλL |λ). E − zλ(E)
(27)
Armed by the formalism of the effective Hamiltonian given in (24) and (27) we can analyze BICs in QD.
4.
General Consideration of BICs
Let the QD be specified by eigenenergies Eb and eigenstates |bi as defined in (13). If quantum wires are attached to the QD, it becomes opened. So the quantities Eb and |bi seize to be appropriate characteristics of QD. However still Eb may respond for positions of resonances in conductance of QD provided that transmission of electrons through QD can be considered in tunneling regime. Quantitatively the tunneling regime means that the resonant widths are to be much less than energy distances between the nearest energy levels Γb |Eb − Eb±1 | and easily can be achieved for small coupling constants of the QD eigenstates with the wires Vb . Because of coupling of the QD eigenstates with extended states of the wires they become extended. However that is truth not for all QD states. First, if the propagation band of the wires has a finite width D, those QD states whose eigenenergies are beyond the propagation band, remain localized. They slightly exceed the
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p
QD by a distance of order 1/ |δb | where δb is an energy distance between the propagation band and the b-th eigenenergy of QD. Second, assume that the total system possesses by ˆ = 0 where L ˆ is the symmetry transformation operator. For a symmetry so that [H, L] ˆ as the coordinate transformation y → −y (see Fig. 1 (b)). Then illustration we consider L the eigen functions of, both, QD and wires both can be only odd or even relative to y → −y. If the electron can propagate in the wires only in the first even channel, all odd eigen states of QD have zero couplings with wires, and therefore are not visible for the probing wires. Formally we can consider these odd states as BICs because they have discrete eigenenergies inside the continuum of propagation band of wires. When the wires are attached to the QD non symmetrically as shown in Fig. 1 (b) all eigen states of the QD, odd and even both, are coupled to the wires. For wires approaching to the middle of the QD (Fig. 1 (a)) the coupling constants (21) of the QD odd eigenstates with the even first channel of wire tend to zero Vb,even → 0 resulting in narrowing the resonance widths. Or one can take the symmetrical case in Fig. 1 (a) but imply small external magnetic field pierced the QD which lifts the symmetry relative to y → −y and thereby transforms all BICs into the narrow resonances similar to cross structure [46, 47]. Moreover recently in [48, 49] considered BICs or quasi BICs in a single-level FanoAnderson model for that the effective coupling constant Vb (E) = Vb f (E) might turn to zero because of specific properties of continua. As a result at some energy f (E) = 0. In the following we exclude here these cases resulting by the zero coupling constant Vb = 0 or effective constant f (E) = 0 and consider the BICs appearing at finite coupling constants Vb (E) 6= 0. The question raises whether or not BIC can appear in open QD for that case? This general question was formulated recently by Miyamoto [27] and who answered to this question for the case of non degenerated spectrum of QD and small |Vb| |Eb − Eb0 | in the framework of the N-level Friedrichs model. The answer is that BICs are impossible for the case with small coupling constants. We try to answer this general question using the effective Hamiltonian approach outlined in section III. Let us denote a set of physical parameters of the system as s = (E, γ). For example, for the case of the Aharonov-Bohm ring the energy and magnetic flux present the set of parameters. For the quantum dot s might be energy and the gate voltage changing the confined potential. Let us consider the point s0 = (E0, γ0) at which Eq. (22) is fulfilled such that (28) E0 = zλ0 (E0, γ0), Γλ0 = 0, i.e. one (λ0) of the complex eigenvalues of Hef f becomes real at this point. That condition is as shown in [42] a necessary and sufficient one for the BIC to exist. To the best of our knowledge, Pursey and Weber have shown that imaginary part of poles of the Jost function turns to zero at the BIC point [5]. For E = E0 one has (29) (Hef f − E)|λ0) = 0. Comparing this equation to (26) we see that the eigen state |λ0) corresponds to the solution of the Lippmann-Schwinger equation if there is no ingoing current in the left wire. Respectively, the state |λ0) can not give rise to outgoing currents because of the continuity equation for the current density. In order to fulfill that condition we have to consider that the right eigen function ψλ0 of the effective Hamiltonian does not overlap with the first channel of
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the left wire, i. e. VλL0 (s0) = 0.
(30)
In other words, the coupling of such a state with wire turns to zero. The eigen states of closed QD |bi and the eigen states of Hef f belong to the subspace Q. Therefore we can P expand the eigen state |λ0) = b cb |bi. Substituting this expansion into (30) we obtain that if the resonance width turns to zero, then X
cb VbL (s0) = 0.
(31)
b
Because of identity of the wires we have the same equation for the coupling matrix elements with the right wire. Comparing to the equation derived by Miyamoto [27] one can see that two approaches, the N-level Fridrichs approach and the effective Hamiltonian one, give the same equation for BIC [42]. The result (30) may be also established by consideration of the transmission amplitude (24). In fact, because of symmetry of the system relative to the left and right wires |VλL0 | = |VλR0 |. In approaching the point s → s0 the denominator E − zλ0 (s) → 0. In order that the ratio |VλL0 (s)|2/(E − zλ0 (s)) remained finite in (24), it is therefore necessary that |VλL0 (s)| → 0 for s → s0 . Thus, at the BIC point we have orthogonality of the righthand state ( V |E, Li) in Eq. (26) to the left eigen state (λ0|. Then, in full correspondence to the consideration of the 1d ring (Eq. (9)), we have the following solution for the scattering state inside the scattering region, QD or ring, (32) |ψB (s0)i = α|λ0(s0)) + |ψp(s0)i, where coefficient α is arbitrary. Right eigen function ψλ0(s0 ) of the effective Hamiltonian is squared integrable and therefore is the BIC function shown in Fig. 3 (a). This figure clearly shows that the BIC is an odd function relative to y → −y, therefore has zero coupling with the even wave function of the wire. Thus, the BIC is invisible for that first even continuum, since it is decoupled. However if, at once, the energy of the incident electron will approach the threshold 4π 2, the BIC will be exponentially extended with the characteristic length (4π 2 − E/E0)−1/2 and will be extended for the two-channel transmission ( E/E0 ≥ 4π 2). The reason is clear for that. It is impossible to have a bound state in QD which could be orthogonal to odd and even propagating states both simultaneously. Moreover the BIC as was shown for the case of 1d ring is degenerated with the continuum and exists parallel to the transport solution as formula (32) shows. Numerically a degeneracy of the BIC to the continuum was shown by Olendski and Mikhailovska [17]. The mere form of the transport solution for the 2d AB ring is shown in Fig. 3 (b). In the vicinity of s0 a value E − zλ0 (E, γ) is small. Then we can split the summation over λ in (27) by two parts, λ = λ0 and λ 6= λ0 and similar to (9) write the scattering state inside the QD as (33) |ψB (s)i = α(s)|λ0(s)) + |ψp(s)i, where α(s) =
Vλ0 (s) , E − zλ0 (s)
(34)
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Figure 3. The BIC function |ψλ0 | which is the eigen function of the effective Hamiltonian (17) (a) and the transport solution |ψp| (b) at BIC point which substitute the general solution (32). ( Figures are taken from [24]). and |ψpi is contribution of all other resonances. Eq.(33) shows that the scattering wave function (51) diverges inside the QD when approaching the BIC’s point meaning localization of the incident particle inside the QD. The divergence of the scattering wave function in the interior of the scattering region is typical for approaching the BIC point and was observed in many structures [50, 51, 43, 22]. However as will shown below there might be a special line over which the coefficient (34) is close to zero. As a result the BIC state can not be observed if the BIC point os approaching by special line [24].
5.
Analytical Results
In this section we present cases which allow an analytical treatment of BICs. For example, the scattering theory in one-dimensional graphs [35, 36, 24, 43] is such a case. In particular the scattering theory in the one-dimensional ring as was shown in section II shows existence of BICs at discrete series of magnetic flux with discrete energies defined by Eq. (6). First, we reproduce the result by Miyamoto [?] of absence of BICs for small coupling constants if the spectrum of QD is non degenerated. For the half infinite propagating band of the continual wires E = [0∞) we obtain hb|Hef f |b0i = Ebδbb0 − Ω
X
VbC VbC0
(35)
C
for the matrix elements of the effective Hamiltonian, where [39, 38] VbC =
Z d/2
dyφC (y)
d/2
Ω=P
Z ∞ 0
∂ψb(x, y) , ∂x |x=xC
dω + iπ. E−ω
(36)
(37)
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Here the C-th wire has a width d with transverse functions φC attached at the QD at coordinates xC . If the second contribution in the right-hand term in (35) can be considered as a perturbation for the non degenerated spectrum Eb, i.e. the coupling constants P |Ω C VbC VbC0 | |Eb − Eb0 |, then the perturbation theory gives zb ≈ Eb − Ω
X
(VbC )2.
(38)
C
From (38) we obtain the condition for BIC: Im(zb ) = 0 can not be satisfied for VbC 6= 0 as was firstly established in [27]. It is easy to see that for tight-binding models of continua this result holds too. Now let some QD energies are crossing under variation of some physical parameter of the system, for example, for change of shape of QD under effect of gate voltage. Numerical examples presented below easily show such a possibility. We restrict ourselves to only one continuum, or, to many continua with the same matrix elements VbC . Then equation for the complex eigenvalues z takes the following form [38]
E1 + ΩV12 − z ωV1V2 ΩV1V3 .. .
ΩV1V2 E2 + ΩV22 − z ΩV2V3 .. .
ΩV1V3 ΩV1V3 E3 + ΩV32 − z .. .
= 0. ...
... ... ...
(39)
This equation can be transformed to Y
X
b
b
(z − Eb ) 1 + Ω
Vb2 z − Eb
!
= 0.
(40)
Let some pair of the energy levels coincide, say, E1 = E2. Then it immediately follows from Eq. (40) that z = E1, i.e. Im(z) = 0. Therefore at the point of degeneracy of eigenenergies of QD the BIC of energy E1 appears. Similar to quantum closed systems in the vicinity of the degeneracy point = E2 − E1 = 0 we can truncate the effective Hamiltonian provided that ||, |ΩVbV1 |, |ΩVbV2| |Eb − E1|, b = 3, 4, . . . and write (35) as ! √ − iΓ u − i Γ Γ 1 1 2 √ . (41) Heff = HB − iπW + W = u − i Γ1 Γ2 − − iΓ2 Here, the radiation shifts resulting from the connection of the QB to the two continua (left and right wires) are considered to be involved in HB = E2 − E1, dω , i = 1, 2, E i = Ei − Vi2P E −ω 0 Z ∞ dω . Γi = πVi2, u = V1V2P E −ω 0 Z ∞
(42)
Such a model is explored in the description of different scattering phenomena [9, 39, 12, 40, 52]. Although the parameter u in (41) is result of the radiation shifts as seen from
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Eqs (37) and (42), it may be introduced in order to transform from the integrable QD with degeneracy to the non integrable one in which the eigenenergies show an avoiding crossing behavior. The transmission amplitude is [12] √ EΓ + ∆Γ + u Γ1 Γ2 , (43) T =2 (E − z1 )(E − z2 ) where z1,2 are the eigenvalues of the effective Hamiltonian (17) p
z1,2 = −iΓ ± R, R2 = ( − i∆Γ)2 + (u − i Γ1 Γ2 )2, Γ1 − Γ2 Γ1 + Γ2 , ∆Γ = . Γ= 2 2
(44)
With Im(zλ) = 0 characteristic of a BIC, from (44) follows directly the condition Γ = Im(R) for its existence. The solution of this equation in general form was found by Volya and Zelevinsky [12]: p (45) 2u = Γ1 Γ2 /∆Γ. If the parameter u is purely the result of the radiation shift (42), one can verify that Eq. (45) is fulfilled for crossing of pure eigenenergies of the closed QD E1 = E2 (Fig. 4(a, b)). In general the transmission probability displays the avoided level crossing behavior as shown in Fig. 4 (c -f). In the vicinity of the BIC’s point = 0, E = 0 for the particular case u = 0, ∆Γ = 0 the eigenvalues of Heff can be approximated as z1 ≈ −i2 /2Γ, z2 ≈ −2iΓ. Correspondingly the transmission amplitude (43) takes the simple form T (E, ) ≈ −
2iEΓ . 2EΓ + i2
(46)
It follows |T | = 0 for E = 0, 6= 0, and |T | = 1 for = 0, E 6= 0. Therefore, the BIC is a singular point in the sense that the value of the transmission amplitude depends on the way to approach this point. If ∆Γ 6= 0 the transmission zero follows E = ∆Γ/Γ. The behavior of the transmission |T (E, )| and of the resonance widths is shown in Fig. 4 (the left and right panels correspondingly). Figs. 4 (a, c, e) completely reproduce fragments of the transmission probability numerically computed for realistic QDs shown in Figs. 6 and 10. As Fig. 4 (a) shows for u = 0 the eigenenergies of the closed integrable QB as well as the resonant positions defined by Re(zλ ) of the eigenvalues zλ cross. Next, we consider the scattering wave function ψ which is solution of the Schr¨odinger equation in the total function space (involving wires and QB) provided that electron incidents from the left wire [38, 40, 39]:
|ψi = |E, Li + 1 +
X 1 |λ)(λ|VL|L, Ei E − HL λ E − zλ
(47)
where |L, Ei is the state of the left wire. Here we used that the biorthogonal basis of right and left eigenstates of the nonhermitian matrix, Heff |λ) = zλ |λ), (λ|λ0) = δλλ , (λ| = |λ)T , is complete. Inside the QD the scattering wave state (47) is simplified as shown in (27).
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Figure 4. (color online). Left panel: The transmission probability |T | through the two-state QB versus incident energy E and , and Re(zλ ) (solid lines). The dark areas correspond to low transmission probability. T and zλ are defined by (43) and (44), respectively. The eigenenergies ± of the QB are shown by thin solid lines. Right panel: the resonant widths Im(zλ) as function of (solid lines). (a) and (b): u = 0, Γ1 = 0.1, Γ2 = 0.05. (c) and (d): u = 0.05, Γ1 = Γ2 = 0.1. (e) and (f): u = 0.05, Γ1 = 0.1, Γ2 = 0.05. The left eigenstates of the effective Hamiltonian (17) are equal to (1| = (β γ), (2| = (−γ β) 1 1 β = (u − i Γ1 Γ2 ), γ = (− + i∆Γ + R), A A p A2 = (− + i∆Γ + R)2 + (u − i Γ1 Γ2 )2.
(48)
p
Substituting (48) and (41) into Vλ =
P
m
Vmhm|λ) and taking for simplicity Γ1 = Γ2 we
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q
(49)
obtain Vλ = (λ|V |C, Ei =
Γ/π(β + γ, β − γ) .
In the vicinity of the BIC’s point = 0, E = 0, the eigenstates (48) can, for u = 0, ∆Γ = 0, be approximated as
iµ 1 iµ 1 , γ ≈ √ 1− β ≈ −√ 1 + 2 2 2 2
(50)
where µ = /Γ. The formulas (48), (49) and (50) allow to write the scattering wave function (16) in the vicinity of the BIC. In the wires it is of the order 1. In the interior of the QB the scattering wave function (27) takes the following form s
ψ≈ where a=
Γ (a + b, − a + b) 4π
(51)
2 iµ , b= 2 E + iΓµ /2 E + 2iΓ
. At = 0, E = 0 the right eigen vectors of (41) become 1 |1) = √ 2Γ
! √ 1 Γ 2 √ , |2) = √ − Γ1 2Γ
! √ Γ 1 √ Γ2
(52)
according to (50). The eigenvectors |1) corresponds to the eigenvalue z1 = 0, i.e. to the BIC. Taking Γ1 = Γ2 and comparing (51) with (33) we immediately obtain p
i 1/2Γ . α(, E) = E + i2 /2Γ
(53)
However even for that oversimplified two-level approach with equal coupling constants Γi , i = 1, 2 the behavior of the coefficient (53) in the vicinity of the BIC point is rather complicated and crucially depends on way to approach the BIC point shown in Fig. 5 (a) by star. To show that let us encircle the BIC point as E = r cos φ, = r sin φ as shown in Fig. 4 (a) where the radius of encircling r is small. Angular behaviors of quantities defining the parameter (34) are shown in Fig. 5 (b). In particular one can see that the numerator in (34) turns to zero at line = 0 while the denominator equals zero just at line of zero transmission. Therefore in order to extract the |ψpi from the scattering wave function (27) we should put at first = 0 and then limit E → 0. If take limit to the BIC point along the zero transmission, the scattering state transforms to the BIC state |λ0). It is worthy to present mappings of the encircling of the BIC points given by angle φ in Fig. 4 (a) onto phase of the transmission amplitude T = |T | exp(iθ) and phase of the BIC. The first mapping is easily obtained from (46) to be tan θ = −2 tan(φ) and shows usual π phase laps for θ at those φ where the transmission equals zero. The second mapping is cos φ . The obtained from formula (53) for α = |α| exp(iϑ) and has the form tan ϑ = 2Γ r sin2 φ angle ϑ displays similar phase laps at the same φ and for r → 0.
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ln(|α|
|V1|, |E−z1|
0.03
0.01
0 −2 −4
0 0
0.5
1
1.5
2
θ/pi
0
1
θ/pi
2
Figure 5. Angular behavior of quantities defining the parameter (34) around the BIC point marked by star. The parameters of the two-level system in (41) are chosen as Γ1 = 0.1, Γ2 = 0.05, u = 0. The radius of encircling r = 0.025.
6.
Numerical Examples of BICs
A special shape of the QB has no importance for the existence of BICs. For the numerical study we have chosen the QB of rectangular shape with variable width W , see the inset in Fig. 1. In order to vary the width of the QB we place a split-gate wire above the semiconductor heterostructure. Application of negative voltage to that wire is responcible for an auxiliary potential in dimensionless form [44] V (y) = V0{1 + 0.5[tanh(C(y − W/2)) − tanh(C(y + W/2))]}. In numerics we have chosen V0 = 100, C = 17. The system is coupled to two continua represented by left and right single-channel wires that are attached to the QB symmetrically over the axis y as shown in inset Fig. 1. The eigenvalues and eigenstates of the closed rectangular QB are specified by the two quantum numbers (m, n). Further, the eigenstates have a definite parity relative to the x and y-axes. Since only the even eigenstates relative to the y-axis participate in the first-channel transmission, we consider BICs among those states which are even relative to y → −y (n = 1, 3, 5, . . .) as shown in Fig. 1. The odd states are not coupled to the first-channel continuum, and are therefore, beyond of our interest as was discussed in section IV. We solve numerically the problem of particle transmission through the billiard with incident particles from the left wire using the tight-binding lattice model [37] for sufficiently large grids. In order to find the positions and widths of the resonance states, we solve the fixed-point equations for the complex eigenvalues zλ (28). In Fig. 6 we present the transmission probability in log scale in order to show clearly the transmission zeros. They appear between the energies of the eigenstates with the same parity relative to the transmission axis, as shown by Lee [45]. The most interesting features appear at the points at which the degenerated states of the closed QB touch the transmission zeros (three such points are marked by bold circles in Fig. 6). At such a point, the width of the narrower resonance state vanishes for a certain value of W as shown in Fig. 8 (left panel). In numerics the QD is considered in tight-binding approximation for rather large numerical grids, around hundreds by hundreds of sites. Therefore, the total number of points is of order tens of thousand. Then the rank of the effective Hamiltonian and
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its number of eigenvalues amounts to the same number. It is obvious, we can present here only a few. Thereby, Fig. 8 (left panel) we show only imaginary part of zλ which turns to zero and the nearest imaginary part, which shows the Dicke superradiant resonance [12]. Approaching this point, the scattering wave function (solution of the Schr¨odinger
Figure 6. The log scale probability for the transmission through the rectangular billiard shown in the inset, versus energy E and width W of the billiard (in terms of the width of the wire). The dark areas correspond to low transmission probability. The length of the QB along the x-axis is 4. The eigenenergies of the closed billiard are marked by crosses. The positions of the BICs are shown by bold circles. The patterns of the two BICs A and B are shown in Fig. 9. equation Hψ = Eψ in the whole function space) diverges in the interior of the billiard. Similar to Fig. 4 we present in Fig. 7 the angular behavior of all relevant parameters defined contribution of the BIC into the scattering wave function (33). One can see that the encircling the BIC point B in Fig. 6 is very similar to the case of two-level approximation. The divergence of the scattering wave function in the interior of the QB shows that the probability to find a quantum particle in the interior of the QB becomes prevailing. In fact, the incident quantum particle is localized inside the QB at the BIC’s point with corresponding square integrable wave function as two examples show in Fig. 9. Although the BICs are mostly localized inside the QB, exponentially small tails remain in the wires. They originate from coupling of BICs with evanescent modes. The numerical study shows that the magnitude of the tails is of the order of magnitude 10−3. Without evanescent modes in the wires, the BIC would be localized completely in the interior of the QB. As can be seen further from Fig. 1, the BIC A consists mainly of the two QB eigenstates (1,5) and (5,1). The moduli of the superposition coefficients are 0.991 and 0.131. If to insert
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r,φ
0
BIC
0.015
∆E 0.005 0
0
0.5
1
0
0.5
1
−3
x 10
9
log(α)
0
|E−zλ |
4
2
0
0
−7 0
0.5
1
φ/2π
Figure 7. Angular behavior of quantities defining the parameter (34) around the BIC point B in Fig. 6. this superposition into (25) one could find that the coupling matrix element equals zero just for these superposition coefficients although each eigen function is the superposition has the same parity as the parity of the wire’s eigen function. The BIC B consists of the states (4,3) and (2,5) with the superposition coefficients 0.374 and 0.927, respectively. The contributions of the other eigenstates of the QB with eigenenergies above π 2 are of the order of magnitude of 10−5.
Moreover the coupling of the 2d QD with the evanescent modes gives rise to the BIC points are close to but different from points at which two eigen functions of closed 2d ring classified by two quantum numbers m, n have the same energy. The evanescent modes have imaginary wave numbers kp which change effectively the Hamiltonian of closed QB by matrix X X C C −|kp | Vb,p Vb0 ,p e ∼ (d/R)2 p6=1 C=L,R
via Eq. (17). Fig. 10 shows the transmission probability through the same rectangular billiard but weakly connected to the wires via diaphragms due to which the closed billiard becomes non-integrable. Correspondingly, the eigenstates of the billiard repel each other in energy provided that they have the same parity relative to the y-axis being perpendicular to the
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0 −0.01
Im(z) −0.03
−0.05
−0.07
4.2
4.3
4.4
W
4.5
4.6
−4
Im(z)
0
x 10
−4
−8 13.6
14
14.4
14.8
Re(z)
Figure 8. Left panel. The dependence of the imaginary parts of the eigenvalues zλ of the effective Hamiltonian on the width of the QD shown in the inset of Fig. 6. Right panel. The evolution of z of one of resonance states in the vicinity of the BIC B in Fig. 6 with the width of QD in the region [4.3, 4.6]. Im(z) vanishes at W = 4.45 (marked by a cross).
Figure 9. The patterns of the two BICs A and B marked in Fig. 6 by bold circles.
transport axis. In this case, BICs appear at those places at which the lines of the resonant transmission (which almost coincide with eigenenergies of the closed QB in the vicinity of
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Figure 10. The same as in Fig. 6 but the QB is weakly connected to the wires by diaphragms. Due to the diaphragms the billiard becomes non-integrable. As a result the eigenenergies of the closed QB slightly repel each other. They are marked by a line of small open circles, and the parity relative to the y-axis is indicated by plus or minus.
BICs) touch the transmission zeros (shown by bold circles in Fig. 10). One can notice that still the two-level approximation (41) with u 6= 0 well describes numerical picture of the avoided level crossing scenario in Fig. 10. The system of two identical QDs with variable shape and coupled in between by a wire with variable spectrum is interesting by many aspects. If we restrict ourselves to the single level QDs the system effectively is reduced to the two-level system considered in Section IV. Even this very simple case shows BICs for variation of coupling constant as was considered in [13, 52, 21]. For fixed energy of the intermediate wire (that is equivalent to the fixed coupling constant between QDs) the system shows BICs in each QD as in the former case of the single dot with degenerated spectrum. However coupling between QDs will give rise to molecular type BICs [21]. It was shown numerically [22, 42] as well as analytically by model Hamiltonians [14, 15, 22] the BIC occurs in such a system for the energy of the wire equaled to the half of distance between energy levels of the identical QDs. As a result BICs might be mostly localized between the QDs in the wire provided that the coupling between the wire and the QDs is weak, as shown in Fig. 11. At last, we can vary as the shape of QDs as the shape of the intermediate wire to have two-parametric case.
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Figure 11. The patterns of the two BICs in the system of two identical QDs for different widths of the intermediate wire: W = 0.638 (left) and W = 0.74 (right). The details of computation and parameters of the system are given in [22].
7.
Two-Electron Bound States in Continuum in Quantum Dots
We consider quantum dot (QD) coupled to leads (left and right) which support one propagating mode (the case of two continuums) with the following total Hamiltonian X
H=
HC + HD + V.
(54)
C=L,R
The leads, left (L) and right (R) in (54) are presented as the non interacting electron gas HC =
X
(k)c+ kσC ckσC , C = L, R.
(55)
kσ
A continual spectrum (k) defines the propagating band of leads. The Hamiltonian of many level QD is that of the impurity Anderson model [53], HD =
X
m a+ mσ amσ +
mσ
X
Um nmσ nmσ .
(56)
mn
Here a+ mσ is the creation operator of an electron on the m-th level of the QD, Um takes into account the Hubbard repulsion at the level m, σ = −σ and nmσ = a+ mσ amσ . The interaction X Vm (k)(c+ (57) V = kσC amσ + h.c) kσmC
describes couplings between the leads and the QD. Here a+ mσ is the creation operator of an is the creation operator of an electron in the leads electron on the m-th level of QD, c+ kσC C. In order to calculate transport properties of the QD with account of the Coulomb interactions interior the QD we use a technique of the equations of motion for retarded and
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advanced Green functions which successfully used to consider the Fano and Kondo resonances in the Anderson model [54, 55, 56, 57, 58, 59]. Following Laxroix [60] we use + + a Hartree-Fock approximation in the wires hhckσC a+ nσ anσ |amσ 0 ii ≈ hnnσ ihhckσC |amσ 0 ii. The approximation is justified for weak couplings compared to the Coulomb interactions: Vm Um . As a result we obtain the following equation G−1 (E) = G−1 QD (E) + iΓ
(58)
−1 in the form of the Dyson equation for the Green functions Gmσ,nσ0 (E) = hhamσ |a+ nσ 0 ii [56]. Here GQD (E) is the Green function of the isolated QD
GQD,mm0 ,σ,σ0 (E) = GQD,mσ (E)δmm0 δσ,σ0 GQD,mσ (E) =
1−hnmσ i E−m +hσ
+
hnmσ i E−m +hσ−Um .
(59)
These Green functions are exact for isolated QD. As usually we take wide band wires and approximate the self-energy as [61, 56] X Vm (k)Vn (k) k
E − (k)σ + i0
p
= −iπVm Vn ρC (E) = −i Γm Γn
(60)
where ρC (E) is the density of states of the left and right wires. The average values of the occupation numbers hnmσ i = ha+ mσ amσ i which enter the expressions for the Green functions are calculated self-consistently via the formulas [60] 1 hnmσ i = π
Z
dEImGmσ,mσ (E).
(61)
The form of the self-energy (60) and the QD Green function (59) allows to proceed to the case of free electrons with Um = 0. In this case BIC appears if QD acquires accidental degeneracy 1 = 2 [20]. In the vicinity of the degeneracy point ε = 2 − 1 = 0 we restrict ourselves to the two-level approximation for QD [12]. Then the occupation numbers (61) are given by four poles of the Green function (58). At zero temperature, the transmission amplitude can be expressed in terms of the Green function T = ΓG(E)Γ+ , Γ = (Γ1, Γ2 ).
(62)
The results of numerical self-consistent calculation of the conductance (62) are presented in Fig. 12. For the case of zero Hubbard repulsion Umn = 0 (no electron correlations) the QD is given only by one electron energy levels. As shown in section V (Fig. 4 (a) the BIC occurs at the point of degeneracy of electron states in QD for = 0. As the Hubbard repulsion is included, the QD is given not only by one electron states but also by two electron states as shown in Fig. 12 (left) by solid lines. As a result we obtain that the number of degenerated points becomes four as seen from Fig. 12 (left). One can see that lines of zero conductance cross the lines of maximal unit conductance at these points. Therefore, one can expect the BICs at four points of degeneracy. In order to show this result we present in Fig. 12 (right) the resonance widths of the energy levels defined as Γλ = −2Im(zλ), λ = 1, 2, 3, 4, where zλ (E, ) are the poles of the Green function or zeros of the right hand expression
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in the Dyson equation (58). The points at which Γλ = 0 define BICs [24, 42]. One can see that these points coincide with the points of degeneracy of the QD given by equations c1 = 0, c2 = U1 /2, c3 = −U2 /2, c4 = (U2 − U1 )/2. The widths Γλ depend on two parameters, the energy E and the level splitting . However the BIC point is defined by the only parameter [8, 24]. Although specific values of Γm are of no importance for BIC’s 0.15
Γ
0.1
0.05
0 −0.2
−0.15
−0.1
−0.05
0
ε
0.05
0.1
0.15
0.2
Figure 12. The left panel: the conductance ln G of QD versus energy of incident electron and energy splitting for the case of strong Hubbard repulsion U1 = 0.2, U2 = 0.3 in comparison to the resonance widths Γ1 = Γ2 = 0.05. The one-electron and two-electron energy levels in closed QD are shown by thin lines. Black regions correspond to those where the conductance close to zero. The right panel: The resonance widths defined as −2Im[zλ(E, )], λ = 1, 2, 3, 4 versus for E = 0 where zλ are the poles of the Green function (58). points defined by crossings of the energy levels of QD, they are important for appearance of the Dicke superradiant state which accumulates the total width [12] as seen from Fig. 12 (right). Since the resonance width turns to zero with approaching the BIC point, we expect singular behavior of occupation numbers (61) at the energy of BIC. In fact, Fig. 13 (a, b, c) demonstrate this effect. Let us consider the first BIC at = 0 with discrete energy E = 0 at which the one electron energies in the QD are crossing as shown in Fig. 12. One can see from Fig. 13 (a) that at the energy E=0 both energy levels are sharply and simultaneously populated till one half. The next resonances with finite widths correspond to the two-electron energies of the QD that are populated smoothly at the Hubbard repulsive energies U1 = 0.2 and U2 = 0.3 by usual scenario as seen from Fig. 13 (a). The next BIC happens for the one electron state crosses the two electron state at points = −0.15 and = 0.1. These cases are shown in Fig. 13 (b) and (c). The BIC’s discrete energies for that case equal to E = 0.15 and E = 0.1 correspondingly (Fig. 12). Again we see that for approaching this energy the BIC populates sharply. However the populations of the one-electron level and two-electron one are well separated because of the Hubbard repulsion of the two electron state. The last figure Fig. 13 (c) refers to the crossing of two
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1
1
a 0.8
b
0.6
〈n 〉 1σ
〈n 〉
0.4
0.2
0
〈 n2σ〉
0.6
〈 n2σ〉
0.4
|T|
0.8
|T|
1σ
0.2
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0.2
0.3
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0.4
−0.2
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E
0.1
0.2
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E
1
1
c
d
0.8
〈 n1σ〉
|T|
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|T| 0.6
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〈 n2σ〉
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0.4
〈 n2σ〉
0.2
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0.2
−0.1
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E
0.2
0.3
0.4
0
〈n 〉 1σ −0.2
−0.1
0
0.1
0.2
0.3
0.4
E
Figure 13. Color online. The electron populations as dependent on the energy of incident electron defined by (61) for the parameters of the system given in Fig. 1. (a) = 0, (b) = −0.15, (c) = 0.1,and (d) = −0.05. The conductance is shown by thin green line.
electron states at the point = −0.05. As seen from Fig. 12 the two-electron BIC has energy E = 0.25. As a result for approaching this energy we observe sharp population of this state similar to the case in Fig. 13 (a). Are BICs critical to energy level crossing? Similar to [12, 20] we lift a degeneracy in QD by transitions between levels, i.e. we add a hopping term between one electron states + into a Hamiltonian of the two-level QD, HD → HD − va+ 1σ a2σ − va2σ a1σ which evolves the picture of energy crossing into the picture with an avoided crossing. Fig. 14 (a) shows the conductance of QD in which the energy levels repel each other because of the hopping between QD levels. In order to show clearly the zero and unit conductance we present in this figure double log scale for the transmission ln(− ln(1 − |G|)). One can see the avoided level crossings shown by white lines correspond to T = 1. BICs shown by open circles are located at those points where the unit transmission T = 1 (white lines) crosses the zero one G = 0 (black lines) similar to the case of non interacting electrons [20]. Fig. 14 (b) shows that the resonance widths turn to zero at four critical values of . It is clear that the Coulomb interaction results in not only the Hubbard repulsion in each
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0.16 0.14 0.12
Γ
0.1 0.08 0.06 0.04 0.02 0
−0.2
−0.1
0
ε
0.1
0.2
Figure 14. (a) The conductance ln(− ln(1 − |G|)) of QD versus energy of incident electron and energy splitting in the avoiding crossing scenario v = 0.05, U1 = 0.2, U2 = 0.3, Γ1 = Γ2 = 0.05. Black regions correspond to those where the conductance close to zero while the white ones do to those where the conductance is near unit. Thereby the white regions follow the one-electron and two-electron energy levels in closed QD. Positions of BICs are shown by open circles. (b) The resonance widths dependent on . + energy level but also interaction U12a+ 1σ a1σ a2σ 0 a2σ 0 between the levels. Even for this case the motion equations for the Green functions can be solved to give 10 × 10 matrix of inner Green functions and correspondingly 10 resonances. Numerical results for the transmission are shown in 15.
8.
Concluding Remarks and Open Questions
The BIC arises at those points at which the line of resonant transmission crosses or touches the transmission zero. For integrable QBs (or for QBs close to integrable ones), the BICs are close to the points of degeneracy (or quasi degeneracy) of eigenenergies of the closed quantum system provided that the eigenenergies are in the continuous part of the spectrum and the interaction u is small. The model consideration shows the existence of BICs also at strong repulsion u ∼ 1. However, then the two-level approximation can not be applied to consider the eigenvalues of the effective Hamiltonian. There are many studies for model systems, while the numerical calculations of BICs grasp only a few realistic systems [17, 51, 20, 21, 22, 26]. In particular, we do not know if BIC exists in open QD of irregular shape or the QD with randomly distributed impurities. The general question is: Is the symmetry of total system or the symmetry of attached wires relative to the QD relevant for the BIC to exist? The BIC appears at a single point of some physical parameter, for example, of gate potential responsible for the shape of QD. It is clear that such an isolated point is not achievable because of inevitable physical processes of decoherence (finite temperature, impurities
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0.1 0.09 0.08 0.07
Γ
0.06 0.05 0.04 0.03 0.02 0.01 0
−0.2
−0.1
0
ε
0.1
0.2
Figure 15. Color online. (a) The transmission probability |T |2 of QD versus energy of incident electron and energy splitting for the case of Coulomb interaction between levels U12 = 0.1. All other parameters coincide with those given in Fig.12. (b) The resonance widths dependent on .
etc). These processes result in a finite width on the background of that which we can not observe-the phenomenon that the resonance width turns to zero. Nevertheless, signatures of the BIC can be seen. First, as shown in Figs 6, 10 and 12, the BIC point is related to the typical Fano conductance behavior [18, 26]. The evolution of the Fano resonance when the maximum and zero of the conductance are approaching each other by variation of the physical parameter, indicates that we are approaching the BIC point. Second, in accordance to formula (33) we could achieve rather large dominance of the BIC in the scattering wave function provided that we path to the BIC points in special direction, as Fig. 7 demonstrates. Using the microwave systems equivalent to the quantum mechanical scattering [62] we can directly observe this phenomenon. As was first discussed in [63] for solving the continuum shell-model equations and in
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[64] for solving potential resonance scattering task, the choice of the effective Hamiltonian is not unambiguous. For example, for the system shown in Fig. 1 we can expand the closed system as shown in Fig. 16 while the wires remain the same. Then the complex eigenvalues of Hef f are changed compared to the case shown in Fig. 1 (b). However, since the total system is remained the same, the scattering properties of the system, the transmission amplitude, in particular, remain invariant under such an expansion of space Q which describes the closed dot. At the BIC point scattering matrix has a singularity. Therefore it follows that the BIC point can not change under this variation of the boundary of the closed system. In fact, we tested the condition for BICs Im(zλ) = 0 under an expansion of the closed system as shown in Figs 1 (b) and 16 numerically and observed that points of BICs have not changed.
Figure 16. The boundaries between the closed QD and wires (dashed line) is shifted in comparison to Fig. 1) (b). Although such a shift does not change transport properties of the system, the effective Hamiltonian is changed.
Acknowledgments ENB, KNP and AFS, thank the Max Planck Institute for the Physics of Complex Systems in Dresden for hospitality.
References [1] von Neumann J. and Wigner E. (1929). Uber merkwurdige diskrete Eigenwerte. Phys. Z., 30, 465-467. [2] Stillinger F.H. and Herrick D.R. (1975). Bound states in the continuum. Phys. Rev. A, 11, 446-454. [3] Newton R.G. (1982). Scattering Theory of Waves and Particles. Berlin: Springer.
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In: Quantum Dots: Research, Technology and Applications ISBN: 978-1-60456-930-8 Editor: Randolf W. Knoss, pp. 577-622 © 2008 Nova Science Publishers, Inc.
Chapter 16
OPTICAL PROPERTIES OF QUANTUM DOTS: POSSIBLE CONTROL OF THE IMPURITY ABSORPTION SPECTRA AND FACTOR OF GEOMETRIC FORM V.D. Krevchik1,2, M.B. Semenov1,2 and R.V. Zaitsev3 1
Penza State University of Russia, Physics Department, Penza, Russia 2 Institute of Basic Research, Palm Harbor, USA 3 Penza State Pedagogical University of Russia, Physics Department, Penza, Russia
Abstract Research of the magnetic freezing effect for D(-) – states in quasi-zero-dimensional structure with parabolic confinement potential has been fulfilled in frames of common theoretical approach, which is based on the zero-range potential method. It has been shown that the D(-) – state binding energy for quantum dot (QD) in magnetic field can exceed by many times its “bulk” value, because of hybrid quantization. The magneto-optical impurity absorption spectra in quasi-zero-dimensional structure with D(-) – centers has been also calculated. It has been shown, that for such structures there is absorption dichroism, which is connected with change in selection rules under optical transitions of electron from the D(-) – center ground state to hybrid-quantizing states of quasi-zero-dimensional structure. It has been demonstrated, that possible control of the magneto-optical impurity absorption spectra has been provided by the spectrum parameters dependence from characteristic frequencies: the confinement potential frequency, cyclotron and hybrid frequencies. The light impurity absorption features, which are connected with the geometric form change for quasi-zero-dimensional structures of two types: QD with the ellipsoid of revolution shape, and the disk-shaped QD, have been theoretically investigated. The dispersion equation for electron, which is localized on D(0) – center in QD with the ellipsoid of rotation shape with parabolic confinement potential, has been obtained in the zero-range potential model. It has been shown that character of the binding energy spatial anisotropy for D(-) – state is comparable with case of D(-) – state in the sphere-shaped QD under influence of external magnetic field. The optical impurity absorption coefficient for quasi-zero-dimensional structure with the ellipsoid of rotation – shaped QD has been calculated in dipole approximation. It has been demonstrated that for quasi-zero-dimensional structure with nonspherical QD there is the impurity absorption dichroism, which is connected with the selection rules change for magnetic quantum number in radial direction and for oscillator quantum
578
V.D. Krevchik, M.B. Semenov and R.V. Zaitsev number in z-direction of QD. Under this situation, spectral dependence of the impurity absorption coefficient has oscillating character with the oscillation period, which is determined by corresponding characteristic frequencies of the confinement potential. The light impurity absorption in quasi-zero-dimensional structures with the disk-shaped QD has been also theoretically investigated. Theoretical approach is based on the D(-) – state energy spectrum investigation in model of the zero-range potential with account of the logarithmic divergence in the one-electron Green function. For simulation of the quantum disk confinement potential in radial direction the potential of “rigid wall” has been used; and in zdirection – potential of the one-dimensional harmonic oscillator. It has been shown, that there is spatial anisotropy for the D(-) – state binding energy in quantum disk, that is due to feature of the quantum disk geometric shape. Calculation of the optical impurity absorption coefficient has been maid in dipole approximation for quasi-zero-dimensional structure with the disk-shaped QD with account of their characteristic sizes dispersion. It has been shown, that in the case of transversal light polarization (in relation to the quantum disk axis) optical transitions are possible only to the dimensionally-quantizing states of quantum disk with even values of the oscillator quantum numbers and with values of magnetic quantum number ± 1. It has been also demonstrated that spatial dependence for the absorption coefficient has oscillating character with pronounced peaks, position of which is determined by characteristic sizes of quantum disk and by amplitude of confinement potential in z-direction. It has been revealed that factor of the QD geometric form essentially influence as on coordinate dependence of the D(-) – state binding energy, as also on the optical properties of structures with QD. It is very important, because the non-uniform broadening for energy levels in the QD set can be connected with factor of the QD-nonidentity; and the QD-set can be used as active environment for laser structures.
I. Magneto-Optical Properties of Quantum Dots with Impurity Centers Introduction (I) Electro-optical [1-4] and magneto-optical [5-7] properties for quasi-0D-structures, formed by the semiconductive sphere-shaped nanocrystals with radii ≈1 – 102 nm, synthesized in a transparent dielectric matrix, now, are intensively investigated. Such research is due to situation, that similar hetero-phase systems are the new perspective materials for the nonlinear opto-electronic active elements design, and, in particularly, for the controlled (by optical signals) elements in quantum computers and lasers. The quantum dots (QD) magnetooptics causes great attention because of the new effects observation possibility. This effects are connected with hybridization of dimensional and magnetic quantization [8]. On other side, impurity centers existence in quasdi-0D-structures stimulates an interest to the impurity centers binding energy controlled modulation problem [9], and, correspondingly, to the problem for hetero-phase systems with QD impurity magneto-optical absorption control. This investigation is devoted to theoretical research for the light impurity magneto-optical absorption for longitudinal and transversal polarization (with respect to the applied magnetic field direction), in QD, synthesized in a transparent dielectric matrix, with consideration of the QD-size dispersion [10-13]. Theoretical approach is based on the investigation for the quasi – 0D energy spectrum in quantizing magnetic field.
Optical Properties of Quantum Dots
579
I1. Features for the Quasi – 0D Energy Spectrum in Quantizing Magnetic Field As known [14], D(–)-states are the solid state analogous for H- - ion. Such states in QD gives new interesting possibilities for the low-dimensional systems correlation effects research [14]. The D(–)-centers - 0D-structure appears in the binding energy considerable increase [15] in compare with 3D-case. As it will be shown in this chapter, that the D(–)-states (in QD) “population” enhance can be waiting for because of the hybrid quantization. The
R0
semiconductive sphere-shaped QD with radius
in quantizing magnetic field is
considered. The subsequent calculations will be made in cylindrical system of reference with
B
origin O in QD-center, and the magnetic induction vector (B
↑↑ k , k
– unit vector along
is directed along
O z -axis
O z -axis). For the one-electron states in QD description,
the oscillator spherical well confinement potential has been used:
V0 (ρ, z ) = where
m∗
frequency;
– is the electron effective mass;
ρ, ϕ, z
(
m ∗ ω 02 ρ 2 + z 2 2
ω0
– are cylindrical coordinates;
)
(1)
– is the QD binding potential characteristic
ρ ≤ R0; − R0 ≤ z ≤ R0.
In the effective mass approximation, in the symmetrical gauge fixing of vector-potential
A , the Hamiltonian operator H QD in cylindrical system of reference can be written as HQD =−
where
⎛ 1 ∂ ⎛ ∂ ⎞ 1 ∂ 2 ⎞ i ωB ∂ m ∗ ⎛ 2 ωB2 ⎞ 2 + ⎜ ⎟− ⎜ω0 + ⎟ ρ + Hz QD , (2) ⎜ρ ⎟+ 2m ∗ ⎝ ρ ∂ρ ⎝ ∂ρ ⎠ ρ 2 ∂ϕ 2 ⎠ 2 ∂ϕ 2 ⎝ 4⎠ 2
ωB = e B / m ∗
– is cyclotron frequency;
e
– is the electron charge absolute value;
B – is the magnetic induction absolute H z QD = − 2 / ( 2m ∗ ) ( ∂ 2 / ∂z 2 ) + m ∗ω02 z 2 / 2 . Eigen-values for
E n 1, m, n 2
value
and corresponding eigen-functions
for
vector
B;
Ψn 1 , m , n 2 (ρ, ϕ, z )
for Hamiltonian (2) are given by expressions of the next view [16]:
E n 1 , m, n 2 =
ω2 ωB m 1⎞ ⎛ + ω0 ⎜ n2 + ⎟ + ω0 1 + B2 (2 n1 + m + 1) , 2 2⎠ 4 ω0 ⎝
(3)
580
V.D. Krevchik, M.B. Semenov and R.V. Zaitsev 1
⎞2 ⎛ ⎟ ⎛ ρ2 n 1! 1 ⎜ Ψn 1 , m , n 2 (ρ, ϕ, z ) = ⎟ ⎜ ⎜ 2 a 1 ⎜ n 2 +1 32 ⎟ ⎜⎝ 2 a 1 ( ) 2 ! ! n n m a π + 2 1 ⎠ ⎝ 2 ⎛ z ⎞ m ⎛⎜ ρ × H n 2 ⎜ ⎟ L n1 ⎜ 2 a 12 ⎝a⎠ ⎝
n 1 , n 2 = 0,1, 2, ...
where
m
⎞ ⎟ ⎟ ⎠
2
⎡ ⎛ ρ2 z2 exp ⎢− ⎜⎜ 2 + 2 ⎣⎢ ⎝ 4 a1 2 a
⎞ ⎟ exp(i m ϕ) ⎟ ⎠
the magnetic quantum number;
(
/ m∗ ωB
(4)
– are quantum numbers, corresponding to Landau levels and to
energy levels for spherically-symmetric oscillator potential well;
aB =
⎞⎤ ⎟⎟⎥ × ⎠⎦⎥
(
(
a 12 = a 2 / 2 1 + a 4 / 4 a B4
m = 0, ± 1, ± 2, ...
)) ;
(
– is
/ m∗ ω0
a=
);
) – magnetic length; H (x) – are the Hermite polynomials [17]. Let n2
us consider D(–)-center is localized in point
R a = (ρ a , ϕ a , z a ) .
γ = 2π
imitated by the zero-range potential with intensity
2
(
Impurity potential is
/ αm∗
). In cylindrical
system of reference (or coordinate system) this potential has the next view
Vδ (ρ, ϕ, z; ρa , ϕa , z a ) = γ
where
α
is determined by the binding state energy
semiconductor; (QD )
Ψλ B
δ(ρ − ρa ) ⎡ ∂⎤ ∂ δ(ϕ − ϕa ) δ( z − z a ) ⎢1 + (ρ − ρa ) + ( z − za ) ⎥ , (5) ∂z ⎦ ∂ρ ρ ⎣
δ( x )
–
the
Dirac
Ei
for the same D(−)-center in massive
delta-function.
The
(ρ, ϕ, z; ρ a , ϕ a , z a ) for electron, which is localized on D
(−)
wave-function
-center, in the effective
mass approximation, satisfies the Shrödinger equation
(E where
0 λB
)
− HQD Ψ(λB ) ( ρ, ϕ, z; ρa, ϕa, za ) =Vδ ( ρ, ϕ, z; ρa, ϕa, za ) Ψ(λB ) ( ρ, ϕ, z; ρa, ϕa, za ) , (6) QD
E 0 λB = −
QD
2
(
λ 2B / 2 m ∗
)
–
are
δ H (QD ) B = H QD +Vδ ( ρ , ϕ , z; ρ a , ϕ a , za ) .
(
G ρ, ϕ, z, ρ1 , ϕ1 , z1 ; E0 λ B
E0 λ B
and to source in point
eigen-values
for
One-electron
Hamiltonian Green-function
) for Shrödinger equation (6), which is corresponding to energy r1 = (ρ1 , ϕ1 , z1 ) , can be written as
Optical Properties of Quantum Dots
(
) ∑
G ρ, ϕ, z , ρ1 , ϕ1 , z1 ; E 0 λ B =
581
Ψn∗1, m , n 2 (ρ1 , ϕ1 , z1 ) Ψn 1, m , n 2 (ρ, ϕ, z )
(E
n 1, m , n 2
0λB
− E n 1, m , n 2
)
. (7)
The Lippman-Schwinger equation for D(−)-state in QD, which is positioned in external magnetic field, has the next view (QD )
Ψλ B
∞ 2 π +∞ (ρ, ϕ, z; ρ a , ϕ a , z a ) = ∫ ∫ ∫ ρ1 dρ1 dϕ1 dz1 G ρ, ϕ, z, ρ1 , ϕ1 , z1 ; E 0 λ B × −∞ 0 0 ) (ρ1 , ϕ1 , z1 ; ρ a , ϕ a , z a ) . × Vδ (ρ1 , ϕ1 , z1 ; ρ a , ϕ a , z a ) Ψλ(QD (8) B
(
+
)
After substitution of the zero-range potential expression (5) into (8), we obtain
(
)
) (ρ, ϕ, z; ρ a , ϕ a , z a ) = γ G ρ, ϕ, z, ρ a , ϕ a , z a ; E0 λ B × Ψλ(QD B
(
) × T Ψ (λQD B
)
( ρ a , ϕ a , za ; ρ a , ϕ a , za ) ,
(9)
where
(T Ψ ( ) ) QD
λB
( ρ a , ϕ a , za ; ρ a , ϕ a , za ) ≡
⎡ ∂ ∂ ⎤ ) (ρ, ϕ, z; ρ a , ϕ a , z a ) . ≡ lim ⎢1 + (ρ − ρ a ) + ( z − z a ) ⎥ Ψλ(QD B ρ →ρ a ∂ρ ∂ z⎦ ⎣ ϕ→ϕ
(10)
a z → za
E0 λ B
(for D(−)-center) dependence
R a = (ρ a , ϕ a , z a )
and magnetic field value В ,
Equation, which determines the binding state energy from QD-parameters, impurity position can be obtain under operator
T action to both parts of expression (9):
2 π α= m∗
2
Let us consider impurity level ( E 0λ
B
< 0 ),
(T G ) ( ρ , ϕ , z , ρ , ϕ , z ; E ) , a
E0 λ B
then the Green - function
effective Bohr energy
Ed
a
a
a
a
a
(11)
0 λB
is situated lower than the QD potential well bottom
(
G ρ, ϕ, z, ρ a , ϕ a , z a ; E 0 λ B
and the effective Bohr radius
), (in units of the
ad = 4 π ε 0 ε
where ε – is the QD relative static dielectric permeability), can be written as
2
(
/ m∗ e
2
),
582
V.D. Krevchik, M.B. Semenov and R.V. Zaitsev
(
G ρ, ϕ, z , ρ a , ϕ a , z a ; E 0 λ B
+
×
∞
∫ 0
m
×
where
1 ⎞ ⎤ ∞ ⎛ e −t + w 1 + ⎟ t ⎥ ∑ ⎜⎜ 2 ⎠ ⎦ n 2= 0 ⎝ 2
exp[− m w t ]exp [(i(ϕ − ϕ ) − β ∑ = ∞ +
×
⎡ ⎛ dt exp ⎢− ⎜ β1 η 02 B ⎣ ⎝
)
⎡ ⎛ ρ2 + ρ2 z 2 + z 2 exp ⎢− ⎜ a 2 + a 2 =− 3 2a ⎢⎣ ⎜⎝ 4 a 1 2 π 2 a a 12 E d β1
∞
1
−
∞
⎛ ρ 2a m ⎜ L ∑ n1 ⎜ 2 ( ) + n m ! n1 = 0 1 ⎝ 2a1
n1 !
)
(
β1 = R ∗0 / 4 U 0∗
w1 = 1+ β12 a∗
−4
;
1
a
;
⎞ m ⎛ ρ2 ⎟Ln ⎜ ⎟ 1⎜ 2a 2 ⎠ ⎝ 1
⎞ ⎟⎟ ⎠
n2
⎛z ⎞ ⎛z⎞ Hn ⎜ a ⎟ Hn ⎜ ⎟ 2 2 ⎝a⎠ ⎝ a ⎠ × n 2!
)]
⎛ρ ρ⎞ t m ⎜ a 2⎟ ⎜ 2a ⎟ ⎝ 1 ⎠
m
⎞ ⎟ exp − 2 n 1 w 1 t ⎟ ⎠
],
a
∗ −2
⎞⎤ ⎟⎥ × ⎟⎥ ⎠⎦
[
×
(12)
R ∗0 = 2 R 0 / a d ; U 0∗ = U 0 / E d ; η 02B = E 0 λ B / E d ;
a∗ = aB / a d . n2
Summation in (12) over quantum number
can be fulfilled with help of the Mehler
formula [18]
∞ ⎛ e −t ∑ ⎜⎜ n2 =0⎝ 2
⎞ ⎟⎟ ⎠
n2
⎛z ⎞ ⎛z⎞ H n 2 ⎜ a ⎟H n 2 ⎜ ⎟ ⎧ 2 z z e − t − z a2 + z 2 e − 2 t ⎫ 1 ⎝a⎠ = ⎝a⎠ exp⎨ a ⎬. n2! a 2 1 − e − 2t 1 − e − 2t ⎩ ⎭
(
(
)
)
(13)
With usage of Hille – Hardi formula for bilinear generating function [18], series
n1
summation over quantum number
∞
can be represented as
⎛ρ ρ⎞ ⎛ ρ a2 ⎞ m ⎛ ρ 2 ⎞ ⎜⎜ ⎟L ⎜ ⎟ exp − 2 n 1 w 1 t = ⎜ a 2 ⎟ L ∑ 2 ⎟ n1 ⎜ 2 ⎟ ⎜ 2 a1 ⎟ n 1 = 0 (n 1 + m )! ⎝ 2a 1 ⎠ ⎝ 2a 1 ⎠ ⎝ ⎠ n1 !
[
m n1
[
× (1 − exp − 2 w 1 t
])
−1
[ [
]
(
−m
[
)
⎡ ρ a2 + ρ 2 exp ⎢− exp − 2 w 1 t 2 a 12 (1 − exp − 2 w 1 t ⎢⎣
[
]
⎛ ρ a ρ exp − w 1 t ×Im ⎜ 2 ⎜ a (1 − exp − 2 w t 1 ⎝ 1
⎞ ⎟ ) ⎟⎠ .
]
]
[
]
exp m w 1 t ×
⎤ )⎥⎥⎦ ×
]
(14)
Optical Properties of Quantum Dots
583
Series summation over magnetic quantum number m can be written as
[(
∞
[ [
)]
]
⎛ ρ a ρ exp − w 1 t ∗ −2 ⎜ ( ) ϕ − ϕ − β exp i a t m I ∑ a 1 m ⎜ 2 m= −∞ ⎝ a 1 (1 − exp − 2 w 1 t +
( [
]
[
⎞ ⎟= ) ⎟⎠
]
])
[ [
]
⎡1 ρ a ρ exp − w 1 t ⎤ −2 −2 = exp⎢ exp i(ϕ − ϕ a ) − β 1 a ∗ t + exp − i (ϕ − ϕ a ) + β 1 a ∗ t ⎥. 2 a 1 (1 − exp − 2 w 1 t )⎥⎦ ⎢⎣ 2
]
(15) Accounting of (13) – (15), the Green - function (12) can be written as
(
)
G ρ, ϕ, z, ρ a , ϕ a , z a ; E 0 λ B = −
⎡+ ∞ ⎡ ⎛ 1⎞ ⎤ 2 ⎢ ∫ dt exp⎢− ⎜ β 1η0 B + w1 + ⎟ t ⎥ × 2⎠ ⎦ ⎣ ⎝ ⎢ β 1 Ed a 3d ⎣ 0
1 3
23 π 2
⎛ ⎡ z a2 + z 2 ⎤ − 2t ⎜ × 2 2 w 1 exp ⎢− ⎥ 1− e 2 ⎜ ⎢⎣ 4 β 1 a d ⎥⎦ ⎝
(
) (1 − exp[− 2 w t ] ) −
1 2
1
(
)
[
−1
×
⎧⎪ 2 z z e − t − z a2 + z 2 e − 2 t ⎫⎪ ⎡ ρ 2a + ρ 2 w 1 (1 + exp − 2 w 1 t exp × exp⎨ a ⎢− ⎬ 2 β 1 a 2d 1 − e − 2 t 4 β 1 a d2 (1 − exp − 2 w 1 t ) ⎪⎩ ⎪⎭ ⎢⎣
( (
( [
) )
] [
[
])
]
[ [
] )⎤ × ⎥ ⎥⎦
]
⎡1 ρ a ρ w1 exp − w1 t −2 −2 × exp⎢ exp i (ϕ − ϕa ) − β 1a ∗ t + exp − i (ϕ − ϕa ) + β 1a ∗ t β 1 a 2d (1 − exp − 2 w1 t ⎢⎣ 2
−t
−
3 2
⎡ (ρ − ρ a ) 2 w 1 + ( z − z a ) 2 ⎤ ⎞ exp ⎢− ⎥ ⎟+ 2 4β 1 a d t ⎥⎦ ⎟⎠ ⎢⎣
⎡ exp ⎢− ⎢ ⎣ + 2 πβ 1 a d
(2 β η 1
2 0B
)(
]
⎤ )⎥⎥⎦ −
)
2 2 + 2 w 1 + 1 (ρ − ρ a ) w 1 + ( z − z a ) ⎤ ⎥ ⎥ 2 β 1 a 2d ⎦
(ρ − ρ a ) 2 w 1 + (z − z a ) 2
⎤ ⎥ ⎥ ⎥ ⎥. ⎥ ⎥ ⎦
(16) With account of (16), equation (11) will have the next view:
584
V.D. Krevchik, M.B. Semenov and R.V. Zaitsev
η
2 0B
+ (2 β 1 ) + w1 β −1
+
2 =ηi − πβ 1
−1 1
∞
∫ 0
⎡ ⎛ 1⎞ ⎤ dt exp⎢− ⎜ β 1 η02 B + w1 + ⎟ t ⎥ × 2⎠ ⎦ ⎣ ⎝
( (
1 ⎡ 1 ⎡ z∗ 2 1 − e −t −1 −2 t − 2 (1 − exp − 2 w1 t ) exp ⎢− a ×⎢ − w1 1 − e −t ⎢⎣ 2 t 2 t ⎢⎣ 2 β 1 1 + e 2 ⎡ ρ ∗a w 1 (1 + exp − 2 w 1 t − 2 exp − w 1 t × × exp ⎢− ⎢⎣ 2 β 1 (1 − exp − 2 w 1 t )
(
( [
× exp − β 1 a ∗ where
[
]
−2
[
)
[
]
[
t + exp β 1 a ∗
]
−2
t
]
[
) ⎤⎥ × )⎥⎦
]
] ))]]
,
(17)
η i2 = E i / E d ; z a∗ = z a / a d ; ρ ∗a = ρ a / a d .
The QD electron states cardinal
modification, which is conditioned by dimensional quantization over three-dimensional directions, gives the D(−)-center binding energy anisotropy: in plane, which is perpendicular to magnetic field direction, there is dimensional quantization. For D(−)-centers, which are (QD ) can be represented as situated in radial plane, the binding energy E λ
(
(E ( ) ) QD λB
ρ
B
)
ρ
⎧ ω2 ⎪ ω 0 1 + B2 + E0 λ B , E0 λ B < 0, 4 ω0 ⎪ =⎨ 2 ⎪ ω 1 + ωB − E 0 λ B , E0 λ B > 0 0 2 ⎪ 4 ω 0 ⎩
In magnetic field direction the D(−)-center binding energy
(E ) (QD )
λB
z
⎧ ⎪⎪ =⎨ ⎪ ⎪⎩
ω0 2 ω0 2
+ E0 λ
B
(E ( ) ) QD λB
z
has the next view:
, E0 λ < 0, B
− E0 λ , E0 λ > 0 B
(18)
(19)
B
As numerical analysis for expression (18) shows, the binding energy
(E ( ) ) QD λB
ρ
(−)
dependence, (for the quantum dots with D -centers, based on InSb), from polar radius
ρ ∗a = ρ a / a d
(in the Bohr units) for
E0 λ B < 0 ,
practically reproduces corresponding
Optical Properties of Quantum Dots
E QD OB
585
0.095
, eV U
2 0.09
0.085
0.08
0.075
1 0.07
0.065
4 0.06
0.055
0.05
3 0.045
0.04
0
0.05
0.1
0.15
U Figure 1. The binding energy
(E ( ) ) QD λB
ρ
( E0λ
B
a
0.2 0.225
U a / ad
< 0 ) dependence, (for the quantum dots with
D(−)-centers, based on InSb), from the impurity polar radius
ρ ∗a = ρ a / a d
for different values of
magnetic field B (lines 3 and 4 shows the two-dimensional oscillator ground state energy levels positions, for B=0 T and for B=12 T, correspondingly;
E i = 3.5 × 10 − 2 eV ,
R 0 = 35.8 nm , U 0 = 0.2 eV ): 1 – B=0 T; 2 – B=12 T . dependence for case of quantum wire with D(−)-centers (see Fig. 1) [22]. Fig. 2 shows the (QD ) dependence, (for the quantum dots with D(−)-centers, based on binding energy E λ
(
B
InSb), from coordinate
)
z
z a∗ = z a / a d
(in the Bohr units) for
E0 λ B < 0 . As one can see
from Fig. 2, in the case, when impurity levels are positioned lower than QD bottom, the D(−)-
586
V.D. Krevchik, M.B. Semenov and R.V. Zaitsev
center binding energy slightly decreases with the magnetic field increase (compare curves 1 and 2). It is due to absence of the ground oscillator state (in Oz – axis direction) dependence from magnetic field, i.e. due to absence of magnetic quantization. Hence, in this direction magnetic field gives instable action to the QD D(−)-states. 0.022
E , eV QD OB
z
0.021
1
0.02
0.019
0.018
0.017
0.016
0.015
0.014
0.013
0.012
2
3
0.011
0.01 0
0.05
0.1
0.15
z
Figure 2. The binding energy InSb), from coordinate energy level);
(E ( ) ) QD λB
z
z a∗ = z a / a d ,
( E0 λ
B
a
0.2
0.25
z a / ad
< 0 ) dependence, (for D(−)-center in QD, based on
(line 3 shows the one-dimensional oscillator ground state
E i = 1.38 × 10 − 2 eV , R 0 = 71.6 nm , U 0 = 0.2 eV ):
B = 0 T ; 2 – B = 15 T
.
1 –
Optical Properties of Quantum Dots
587
I2. The Hybridization Effect for Dimensional and Magnetic Quantization in the Light Impurity Absorption Spectrums An interest to the hetero-phase systems magneto-optics is, first of all, due to experimental observation possibility for the hybridization effect for dimensional and magnetic quantization in the light impurity absorption spectrums. As it will be shown in this chapter, this effect carries important information about zone – structure, as also about the QD impurity states. This information can be obtained, for example, from analysis of the Zeeman energy shift and oscillations period in the magneto-optical impurity absorption spectrums. Let us consider the light impurity absorption by the quantum dots with impurity centers in case of longitudinal polarization with respect to the applied magnetic field direction,
↑↑ B , eλ – is the unit light polarization vector). Let us also consider the D(-) –center,
( eλ
which is localized in point
Ra = (0, 0, 0) . The impurity binding state energy level E0 λ B
is
positioned lower, than the sphere - shaped oscillator well bottom, (this quite well describes the QD confinement potential, ( E0 λ ) (ρ, ϕ, z;0) Ψλ(QD B
B
< 0 )).
In this case the wave function
for electron, which is localized at the short-range potential for D(-) –
center, can be written as (see (16)): ) (ρ, ϕ, z;0) = C1QDB Ψλ(QD B
+
( (
∞
∫ 0
⎡ ⎛ 1⎞ ⎤ dt exp⎢− ⎜ β 1η02 B + w1 + ⎟ t ⎥ 1 − e − 2 t 2⎠ ⎦ ⎣ ⎝
(
) (1 − exp[− 2 w t ]) −
1 2
−1
1
) )
⎡ ρ 2 w 1 (1 + exp[− 2 w1t ]) ⎤ ⎡ z 2 1 + e −2 t ⎤ × exp ⎢− exp ⎢− ⎥, 2 2 −2 t ⎥ ⎢⎣ 4β1 a d (1 − exp[− 2 w1t ]) ⎥⎦ ⎣ 4β1 a d 1 − e ⎦
here
(QD )
Ψλ B
(ρ, ϕ, z;0) ≡ Ψλ B (ρ, ϕ, z;0, 0, 0) . (QD )
Coefficient
QD 1B
C
×
(20)
−
3 2
= π w 1C BQD
is
determined by the next expression
C BQD
⎡ ⎢ ⎢ −1 −3 3 ⎛1 ⎞ = ⎢− 2 2 π 2 β12 a d3 w 1Γ ⎜ − w 1 ⎟ ⎝2 ⎠ ⎢ ⎢ ⎢⎣
⎛ β 1η02 B + w 1 5 ⎞ + ⎟ Γ⎜ ⎜ 2 4 ⎟⎠ ⎝ × 2 ⎛ β 1η02 B + w 1 1 ⎞ ⎛ β 1η02 B − w 1 3 ⎞ ⎜ + ⎟ Γ⎜ + ⎟ ⎜ ⎟ ⎜ 2 4 2 4 ⎟⎠ ⎝ ⎠ ⎝
⎡ ⎛ β 1η02 B + w 1 1 ⎞ ⎡ ⎛ β 1η02 B + w 1 5 ⎞ ⎛ β 1η02 B − w 1 3 ⎞⎤ ⎤ ⎤ + ⎟⎥ − 1⎥ ⎥ + ⎟ − Ψ⎜ ×⎢⎜ + ⎟ ⎢Ψ ⎜ ⎜ 2 4 ⎟⎠ ⎢⎣ ⎜⎝ 2 4 ⎟⎠ 2 4 ⎟⎠⎥⎦ ⎥ ⎥ ⎢⎣ ⎜⎝ ⎝ ⎦⎦
−
1 2
. (21)
588
V.D. Krevchik, M.B. Semenov and R.V. Zaitsev The effective Hamiltonian
eλ s
longitudinal polarization
( ) H int B s
of interaction with the light wave field in case of
, (with respect to the applied magnetic field direction), can be
written as ( ) H int B = −i
where
λ0
2π 2α ∗ I 0 exp m∗2ω
λ0
s
frequency
ω , wave vector q s
operator. The matrix elements
qs r )
the longitudinal light polarization
∇r ) ,
(22)
ε ; I 0 – is intensity for electromagnetic wave with
and the polarization unit vector
) M (fsQD , λB
( eλs
α ∗ – is the fine structure constant with account of
– is the local field coefficient;
the static relative dielectric permeability
(i
eλ s ; ∇ r
– is Hamiltonian
calculation for the dipole optical transitions in case of
eλ s , leads to integrals of the next view
⎧ 0, если m ≠ 0,
2π
∫ exp(− imϕ) dϕ = ⎨⎩2π, если m = 0, 0
∞
⎛ ⎡ ⎤ z2 z ⎜ − dz z exp H ⎢ ⎥ n −2 t 2 ∫−∞ 2 ⎜ ⎣ 2β1a d 1 − e ⎦ ⎝ 2 β1 a d
⎞ ⎟= ⎟ ⎠ 0, если n 2 ≠ 2n + 1, n = 0,1, 2,..., ⎧ ⎪ 3 =⎨ 3⎞ n 2 n+ 2 2 ⎛ −2 t 2 ( ) [ ] − β Γ + − − 1 2 a n exp 2 n t 1 e , если n 2 = 2n + 1. ⎜ ⎟ 1 d ⎪ 2⎠ ⎝ ⎩ +
(
)
(
)
(23)
Due to (23), the optical transitions from the D(−)-center ground state occurs only to the QD – states with m=0 and the odd quantum number out selection rules, the matrix elements
n 2 values. With account of the pointed
) M (fsQD , λB
expression for considered optical
transitions, can be written as (s )
3 2
M f QD, λ B = π i λ 0 ×
α∗ I 0 3⎞ ⎛ n 2 n+ 2 E d a d4 β1 w 1−1 C1QD Γ⎜n + ⎟× B C n 1 , 0, 2 n +1 (− 1) 2 2⎠ ω ⎝ 5
(2 n + 3 / 2 + (2 n
1
+ 1) w 1 + β 1η 02 B
)
2 ⎛ β 1η 02 B 1⎞ 1 ⎞ ⎛ β 1η 0 B 1⎞ ⎛ ⎛ ⎜ + n + ⎜ n1 + ⎟ w1 + ⎟ ⎜ + n + ⎜ n 1 + ⎟ w1 + ⎜ 2 ⎟ ⎜ 2⎠ 4 ⎠⎝ 2 2⎠ ⎝ ⎝ ⎝
where the normalizing multiples product has the next view:
5 ⎞⎟ 4 ⎟⎠
,
(24)
Optical Properties of Quantum Dots
QD 1B
C
C n 1 , 0, 2 n +1 = 2
− n −1
π
−
3 2
−
β1
3 2
589
⎛ β 1η02 B + w 1 1 ⎞ + ⎟× a w1 ⎜ ⎜ 2 4 ⎟⎠ ⎝ −3 d
⎡ ⎛ β 1η 02 B − w 1 3 ⎞ Γ⎜ + ⎟ ⎢ ⎜ 2 4 ⎟⎠ ⎢ ⎝ × ⎢− × 2 ⎛ ⎞ β η + w 1 5 1 ⎢ (2 n + 1)!Γ ⎛ − w ⎞ Γ ⎜ 1 0 B + ⎟ ⎜ 1⎟ ⎢ ⎜ 2 2 4 ⎟⎠ ⎝ ⎠ ⎝ ⎣ 1
⎤2 ⎥ ⎥ 1 ⎥ . (25) × ⎡ ⎛ β 1η02 B + w 1 1 ⎞ ⎡ ⎛ β 1η02 B + w 1 5 ⎞ ⎛ β 1η02 B − w 1 3 ⎞⎤ ⎤ ⎥ ⎢⎜ + ⎟⎥ − 1⎥ ⎥ + ⎟ − Ψ⎜ + ⎟ ⎢Ψ ⎜ ⎜ 2 4 ⎟⎠ ⎣⎢ ⎜⎝ 2 4 ⎟⎠ 2 4 ⎟⎠⎦⎥ ⎥ ⎥ ⎢⎣ ⎜⎝ ⎝ ⎦⎦ Let us supposed, that the QD – sizes dispersion u arises under the phase decay process in resaturated solid solution [19, 20] and has been satisfactorily described by the Lifshits – Slezov formula [20, 21]
⎧ 3 4 e u 2 exp[− 1 / (1 − 2 u / 3)] 3 ,u< , ⎪ 5 7 11 2 ⎪ P(u ) = ⎨ 2 3 (u + 3) 3 (3 / 2 − u ) 3 ⎪ 3 u> , ⎪⎩ 0, 2 where
u = R0 / R0
,
R 0 and R 0
(26)
– are QD – radius and the mean value of QD –
radius, respectively; е – is the natural logarithm base. The light impurity absorption coefficient polarization
eλ s
K B( s ) (ω)
in the case of longitudinal
, (with respect to the applied magnetic field direction), with an account of
QD – sizes dispersion, can be represented as
K B( s ) (ω) =
2 π N0 I0
3 2
∑ ∑ δ ∫ du P(u ) M ( )
s f QD , λ B
m, 0
m n 1, n
0
2
1
⎛ 2 n + 3 / 2 + (2 n + 1) 1 + β ∗ 2 a ∗ − 4 u 2 ⎞ ⎜ ⎟ 1 × δ⎜ −u⎟ , 2 ∗ β X − η0 B ⎜ ⎟ ⎝ ⎠
(
)
(
ω0 (u )β ∗ X − η 02 B
)× (27)
590
V.D. Krevchik, M.B. Semenov and R.V. Zaitsev
where
X = ω / Ed
– is photon energy in the effective Bohr energy units;
δ ( x ) – is the Dirac delta - function; δ m , 0
concentration in dielectric matrix; symbol;
(
β ∗ = R 0∗ / 4 U 0∗
)
;
– is QD
– is Kronecker
R 0∗ = 2 R 0 / a d . ωth( s )B
The light impurity magneto-optical absorption band edge polarization
N0
in case of longitudinal
eλ s , (with respect to the applied magnetic field direction), is determined by the
impurity level position depth, by the cyclotron frequency and by the QD – sizes dispersion value:
ωth( s )B ≈ E0, 0,1 (u 0 ) + E0 λ B
ωth( s )B
Fig. 3 shows the photon energy cutoff value
(28)
dependence, in case of the
longitudinal polarization light magneto-optical absorption, for quantum dots with impurity centers (based on InSb), which is synthesized in borosilicate glass matrix, from magnetic induction value B. As one can see from Fig. 3, this dependence has monotonously increasing character, and the light impurity absorption band edge displacement is more than 0.03 eV , (in external magnetic field with induction value B=12 T). For the integration fulfillment in (27) it is necessary to find out the Dirac delta – function argument roots. As result we obtain the equation
2 n + 3 / 2 + (2 n 1 + 1) 1 + β ∗ a ∗ 2
(
β ∗ X − η02 B
−4
)
u2
−u = 0.
(29)
The equation (29) roots finding leads to decision of the next system
⎧ ⎪ ⎪ ∗2 X − η 02 B ⎪β ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩
[(
where
) − (2 n 2
X th( sB) = ωth( s )B / E d
next view
+ 1) a ∗ 2
1
−4
]u
2
(
)
− 2 β ∗ (2 n + 3 / 2 ) X − η 02 B u +
+ (2 n + 3 / 2 ) − (2 n 1 + 1) = 0, (2 n + 3 / 2) , u> ∗ β X − η 02 B X ≥ X th( sB) . 2
2
(
(30)
)
. It is easy to show, that the system (30) decision
un1n
has the
Optical Properties of Quantum Dots
591
0.132 0.13
s
= Zth B , eV 0.125
0.12
0.115
0.11
0.105
0.1
0.095
0
2
4
6
8
10
12
B, T
ωth( s )B
Figure 3. The photon energy cutoff value
dependence, in case of the longitudinal polarization
light magneto-optical absorption, for quantum dots with impurity centers (based on InSb,
Ra = (0, 0, 0) , E i = 5.5 × 10 −2 eV , R 0 = 71.6 nm , U 0 = 0.3 eV
), which is
synthesized in borosilicate glass matrix, from magnetic induction value B .
(
u n 1, n =
)
(
3⎞ ⎛ 2 2 ⎜ 2 n + ⎟ X − η0 B + (2 n1 +1) X − η0 B 2 ⎝ ⎠
[(
β ∗ X − η02 B
)
2
2 ⎡⎛ ⎤ −4 3⎞ 2 + ⎢⎜ 2 n + ⎟ − (2 n1 +1) ⎥ a ∗ 2⎠ ⎢⎣⎝ ⎥⎦
) − (2 n +1) 2
1
2
a∗
−4
]
With account of (24) – (26) and (31), the light impurity absorption coefficient in case of longitudinal polarization direction), can be written as
eλ s
(31)
K B( s ) (ω)
, (with respect to the applied magnetic field
592
V.D. Krevchik, M.B. Semenov and R.V. Zaitsev
K B( s ) (ω) = K 0 β ∗ X
N (1)
( )(2 n
N (2 )
∑= ∑ =
n1
0
0 n
[
u n21 , n P u n 1 , n
× β u n 1 , n (2 n 1 + 1) a ∗
2
∗ −4
(
− X −η
+ 1)(2 n + 2 ) Γ (2 n + 2)
(
2
1
Γ 2 (n + 2)
)
2 2 0B
]
)
× 2 w n 1, n ×
(
3⎞ ⎛ + ⎜ 2 n + ⎟ X − η02 B 2⎠ ⎝
)
−1
×
3⎞ ⎛ Γ ⎜ δ n1, n − w n1, n + ⎟ 4⎠ ⎝ × × ⎡ ⎛1 5⎞ ⎞⎤ ⎛ ⎢− Γ ⎜ 2 − 2 w n 1 , n ⎟⎥ Γ ⎜ δ n 1 , n + w n 1 , n + 4 ⎟ ⎠⎦ ⎝ ⎠ ⎣ ⎝ ×
1 ⎡⎛ 1⎞⎛ ⎛ 5⎞ 3 ⎞⎞ ⎤ ⎛ ⎢⎜ δ n 1 , n + w n 1 , n + ⎟ ⎜⎜ Ψ ⎜ δ n 1 , n + w n 1 , n + ⎟ − Ψ ⎜ δ n 1 , n − w n 1 , n + ⎟ ⎟⎟ − 1⎥ 4⎠⎝ ⎝ 4⎠ 4 ⎠⎠ ⎦ ⎝ ⎣⎝
1⎞ ⎛ ⎜ δ n 1, n + w n 1, n + ⎟ 4⎠ ×⎝
(β
where
∗2
)
u n21 , n X 2 − 1
2
2
,
(32)
[ ] – is the integer part of value for C ( ) = 3 (β (X − η ) − 1)/ ⎛⎜ 4 1 + 9 β a / 4 ⎞⎟ − 1 / 2 ; ⎝ ⎠
K 0 = 2 4 π 2 λ 20 α ∗ a d2 N 0 ; N (1) = C (1) 1
expression
∗
[ ] – is the integer part of C ( ) = 3 / 4 × (β (X − η ) − 1) − (n + 1 / 2) 1 + 9 β ∗
2 0B
2
w n 1, n = 1 + β ∗ a ∗
−4
1
∗ −4
∗2
2 0B
N (2 ) = C (2 ) 2
×
value ∗2
a∗
−4
for
expression
/4 ;
u n21 , n / 2 ; δ n 1 , n = β ∗ η 02 B u n 1 , n / 2 .
Fig. 4 shows the spectral dependence for the light magneto-optical impurity absorption (s ) coefficient K B ω , (for longitudinal polarization), in the case of borosilicate glass, which
( )
is pigmented by the InSb crystallites. With the magnetic field increase, (compare curves 1 and 2), the light impurity absorption band edge shifts to the short-wave spectrum region, that is connected with corresponding dynamics for impurity level and Landau levels. The oscillations period (see qurve 2), (under the quantum number determined by the hybrid frequency
Ω = 4 ω02 + ω 2B
n1
changing on 1), is
and equals to
Ω
. Distance
between two nearest bands for the light impurity magneto-optical absorption spectrum, (in case of longitudinal polarization), equals to
2 ω0 .
Optical Properties of Quantum Dots
593
500
K B s Z , sm – 1
1 450
400
350
2 300
250
200
150
100
50
0
0.12
0.14
0.16
0.18
0.2
0.22
=Z , eV Figure 4. The spectral dependence for the light magneto-optical impurity absorption coefficient
K B( s ) (ω)
, (for longitudinal polarization), in case of borosilicate glass, which is pigmented by InSb
crystallites (
E i = 3.5 × 10 −2 eV , R 0 = 35.8 nm , U 0 = 0.2 eV , N 0 = 10 15 sm−3 ),
for different magnetic induction values
B:1– B = 0 T ;2– B =5 T
.
Let us consider the light impurity absorption by quantum dots with impurity centers, (in case of transversal polarization ( eλ
⊥ B ),
with respect to the applied magnetic field
direction). Let us also consider the binding energy level in point
Ra = (0, 0, 0) ),
bottom ( E0 λ
B
< 0 ).
E0 λ B
, (for D(−)-center, is localized
which is positioned lower than the QD parabolic potential well
594
V.D. Krevchik, M.B. Semenov and R.V. Zaitsev The effective Hamiltonian
vector
qt
() H int B t
for interaction with the light wave field, (with wave
and the unit transversal polarization vector
eλ t
, with respect to the applied
magnetic field direction), can be written as
2π 2α ∗ I 0 exp m∗2ω
(t )
H int B = −i λ0
( iqt r )
In dipole approximation the matrix elements
Ψn 1 , m, n 2 (ρ, ϕ, z ) ,
(
)
) M (ft QD , λB
optical transitions from D(−)-center ground state discrete spectrum
⎛ ⎞ ieB ⎡⎣eλ t , r ⎤⎦ ⎟ . (33) ⎜ eλ t ∇ r − z 2 ⎝ ⎠ , which determines the electron
) (ρ, ϕ, z;0) Ψλ(QD B
to states of the QD
for transversal polarization
eλ t
case, can be
represented as the next sum of two parts: ) M (ft QD , λB = M1 + M 2 ,
(34)
where
M1 = i λ 0
(
)
2πα∗ I 0 ) (ρ, ϕ, z; 0) , (35) En 1 , m, n 2 − E0 λ B Ψn∗1 , m, n 2 (ρ, ϕ, z ) (eλ t , r ) Ψλ(QD B ω
M 2 = − λ0
2πα∗ I 0 ω
ωB ) (ρ, ϕ, z; 0) . (36) Ψn∗1 , m , n 2 (ρ, ϕ, z ) [eλ t , r ] z Ψλ(QD B 2
Expression (35) with account of the energy spectrum (3), as also of the QD electron wave functions (4) and of the D(−) -center binding state wave function, can be written as 1 2
− α∗ I 0 n (2 n )! QD C1 B Cn 1 , ±1, 2 n × exp (∓ i ϑ)E d a d4 β1 w 1 2 (− 1) n! ω
1
M1 = 2 π i λ0 ×
where
2
(mβ a 1
∗− 2
+ (2 n + 1 / 2 ) + (2 n 1 + 2 )w 1 + β 1η02 B
)
⎛ β 1η02 B ⎞ ⎛ β 1η02 B 1 1 3⎞ 1⎞ ⎛ ⎞ ⎛ ⎜ + n + ⎜ n 1 + ⎟ w1 + ⎟ ⎜ + n + ⎜ n 1 + ⎟ w1 + ⎟ ⎜ 2 2⎠ 4 ⎟⎠ ⎜⎝ 2 2⎠ 4 ⎟⎠ ⎝ ⎝ ⎝ ϑ
– is polar angle for the transversal polarization unit vector
eλ t
,
(37)
in cylindrical
system of reference. Under calculation of (37), integrals of the next view are appeared
Optical Properties of Quantum Dots
595
⎧π exp(∓ i ϑ), если m = ±1, ( ) ( ) d cos exp im ϕ ϕ − ϑ − ϕ = ⎨ ∫0 0, если m ≠ ±1 ⎩
(38)
2π
∞
⎛ ⎡ ⎤ z2 z ⎜ − dz exp H ⎢ ⎥ n −2 t 2 ∫ 2 ⎜ ⎣ 2β1 a d 1 − e ⎦ −∞ ⎝ 2 β1 a d +
⎞ ⎟= ⎟ ⎠ 0, если n 2 ≠ 2n, n = 0,1, 2,..., (2 n )! exp[− 2 n t ] 1 − e −2 t , если n = 2n, 2 π β1 a d 2 n!
(
⎧ ⎪ =⎨ (− 1)n ⎪⎩
)
The selection rules for quantum numbers The normalizing factors product
QD 1B
C
C n 1 , ± 1, 2 n = π
−
3 2
−
β1
3 2
m
and
C1QD B C n 1 , ± 1, 2 n
n2
(39)
are followed from (38) and (39).
in (37) is written as
⎛ β 1η02 B + w 1 1 ⎞ + ⎟× a w1 ⎜ ⎜ 2 4 ⎟⎠ ⎝ −3 d
⎡ ⎛ β 1η02 B − w 1 3 ⎞ Γ⎜ + ⎟ ⎢ ⎜ 2 4 ⎟⎠ ( ) 1 n + ⎢ 1 ⎝ × × ⎢− 2 n+1 2 ( ) 2 2 ! n ⎞ ⎛ w β η + 1 5 ⎛ ⎞ 1 0 B 1 ⎢ Γ ⎜ − w1 ⎟ Γ ⎜ + ⎟ ⎜ ⎢ 2 2 4 ⎟⎠ ⎝ ⎠ ⎝ ⎣ 1
⎤2 ⎥ ⎥ 1 ⎥ . × 2 ⎡ ⎛ β 1η02 B + w 1 1 ⎞ ⎡ ⎛ β 1η02 B + w 1 5 ⎞ ⎤ ⎥ ⎤ ⎛ β 1η 0 B − w 1 3 ⎞ ⎢⎜ + ⎟⎥ − 1⎥ ⎥ + ⎟ − Ψ⎜ + ⎟ ⎢Ψ ⎜ ⎜ 2 4 ⎟⎠ ⎢⎣ ⎜⎝ 2 4 ⎟⎠ 2 4 ⎟⎠⎥⎦ ⎥ ⎥ ⎢⎣ ⎜⎝ ⎝ ⎦⎦ (40) With account of (4) and (20), expression (36) for factor
M 2 of matrix elements (34) can
be represented as 1
M 2 = 2 2 π 2i λ 0
− α∗ I 0 −2 n (2 n )! QD C1 B C n 1 , ±1, 2 n × exp (∓ i ϑ)E d a d4 m a ∗ β12 w 1 2 (− 1) ω n!
⎛ β 1η 02 B 1⎞ 1⎞ ⎛ ×⎜ + n + ⎜ n 1 + ⎟ w1 + ⎟ ⎜ 2 2⎠ 4 ⎟⎠ ⎝ ⎝
1
−1
−1
⎛ β 1η 02 B 3⎞ 1⎞ ⎛ ⎜ + n + ⎜ n 1 + ⎟ w1 + ⎟ , ⎜ 2 2⎠ 4 ⎟⎠ ⎝ ⎝
(41)
596
V.D. Krevchik, M.B. Semenov and R.V. Zaitsev Under calculations in (41) integrals of the next view are appeared
⎧∓ π i exp(∓ i ϑ), если m = ±1, ( ) ( ) d sin exp im ϕ ϕ − ϑ − ϕ = ⎨ ∫0 0, если m ≠ ±1. ⎩
2π
(42)
From (34), (37) and (41), for the considered optical transitions matrix elements (t ) M f QD, λ we will obtain B
− α∗ I 0 n (2 n )! QD C1 B C n 1 , ±1, 2 n × exp (∓ i ϑ)E d a d4 β1 w 1 2 (− 1) ω n! 1
1 2
(t )
M f QD , λ B = 2 π i λ 0
×
here
2
(2 mβ a
∗− 2
1
+ (2 n + 1 / 2 ) + (2 n 1 + 2 )w 1 + β 1η 02 B
)
⎛ β 1η 02 B ⎞ ⎛ β 1η 02 B 1 1 3⎞ 1⎞ ⎛ ⎞ ⎛ ⎜ + n + ⎜ n 1 + ⎟ w1 + ⎟ ⎜ + n + ⎜ n 1 + ⎟ w1 + ⎟ ⎜ 2 2⎠ 4 ⎟⎠ ⎜⎝ 2 2⎠ 4 ⎟⎠ ⎝ ⎝ ⎝
C1QD B C n 1 , ± 1, 2 n
is determined by the expression (40), and
m = ±1 .
,
(43)
As one can see
from (38), (39) and (42), that selection rules for the magnetic quantum number m ( m = and quantum number
n 2 ( n 2 = 2 n, n = 0, 1, 2,...
±1 )
) are such ones, that the optical
m = ±1 and with (t ) expression K B (ω) for
transitions from impurity level are possible only to the QD states with
n2.
even values of
The light impurity absorption coefficient
transversal polarization
eλ t
case, (with respect to the applied magnetic field direction), can
∑
1
be written as
K B(t ) (ω) =
2 π N0 I0
3 2
(
)
) δ E n 1 , m, 2 n + E 0 λ B − ω . (44) ∑ δ m ,1 ∫ du P(u ) M (ft QD ,λB
n 1 , n m = −1
As it leads from condition
2
0
ω0 (u ) =
Ed β∗ u
and (3), that eigen-values
E n 1 , m, 2 n
for
Hamiltonian (2) are the decreasing functions of the QD – sizes dispersion u , (0 < u
< 3 / 2 ):
Optical Properties of Quantum Dots
En 1 , m, 2 n
here
597
−4 2 1 ⎛ ⎞ 2 n + + (2 n 1 + m + 1) 1 + β ∗ a ∗ u 2 ⎟ ⎜ −2 2 ⎟ , (45) = En 1 , m, 2 n (u ) = Ed ⎜ m a ∗ + β ∗u ⎜ ⎟ ⎜ ⎟ ⎝ ⎠
m = ±1 .
It allows to represent the light impurity absorption coefficient
K B(t ) (ω)
in the next
manner
K B(t ) (ω) =
2 π N0 I0
3 2
1
∑ ∑ δ ∫ du P(u ) M ( )
n 1 , n m = −1
t f QD , λ B
m ,1
2
ω0 (u )β
0
1 ∗
(X − η ) × 2 0B
2 −4 ⎛ ⎞ 2 n + 1 / 2 + (2 n 1 + 2 ) 1 + β ∗ a ∗ u 2 mu ⎜ ⎟ × δ⎜ 2 + −u⎟. ∗ 2 β X − η0 B ⎜ a ∗ X − η02 B ⎟ ⎝ ⎠
(
(
)
The photon energy cutoff value
ωth(t )B
)
for the transversal polarization light
eλ t
(46)
case
can be written as 2
(t )
X th B ≈ η
where
2 0B
+
1/ 2 + 2 1 + β ∗ a ∗
−4
u 02
∗
β u0
−a∗
−2
,
(47)
X th(t )B = ωth(t )B / E d ; u 0 = 3 / 2 .
Fig. 5 shows the photon energy cutoff value
ωth(t )B
dependence, in case of the
transversal polarization light magneto-optical absorption by the “QD - D(–)-center” – complexes (based on InSb), which is synthesized in borosilicate glass matrix, from magnetic induction value B. This dependence has nonmonotonous character with the clear pronounced minimum. Equation for the Dirac delta-function argument roots search (in (44)) has the next view
2
(
mu
a ∗ X − η 02 B
2 n + 1 / 2 + (2 n 1 + 2 ) 1 + β ∗ a ∗ 2
)
+
(
β ∗ X − η 02 B
)
−4
u2
−u = 0.
(48)
598
V.D. Krevchik, M.B. Semenov and R.V. Zaitsev
= Zth t B , eV
0.118 0.116
0.114
0.112
0.11
0.108
0.106
0.104
0.102
0.1
0.098
0.096
0.094
0.092
0.09
0
2
4
6
8
10
12
B, T Figure 5. The photon energy cutoff value
ωth(t )B
dependence, in case of the transversal polarization
light magneto-optical absorption by the quantum dots with impurity centers (based on InSb), which is synthesized
in
borosilicate
glass
R 0 = 71.6 nm , U 0 = 0.3 eV
matrix,
( Ra
= (0, 0, 0) ,
E i = 5.5 × 10 −2 eV
,
), from magnetic induction value B .
For case of the transversal polarization light
eλ t
absorption, ( X
(48) roots calculation is equivalent to decision of the next system:
≥ X th(t )B ), the equation
Optical Properties of Quantum Dots
599
⎧ ⎪ ⎪β ∗ 2 ⎡ X − η 2 − m a ∗ − 2 2 − (2 n + 2 ) 2 a ∗ − 4 ⎤ u 2 − 2 β ∗ (2 n + 1 / 2 ) × 1 0B ⎥⎦ ⎢⎣ ⎪ ⎪ 2 2 2 ∗ −2 u + (2 n + 1 / 2 ) − (2 n 1 + 2 ) = 0, × X − η0 B − m a ⎨ ⎪ (2 n + 1 / 2 ) u> , ⎪ −2 β ∗ X − η 02 B − m a ∗ ⎪ ⎪ X ≥ X th(t )B , ⎩
(49)
(
where
)
(
) (
)
m = ±1 . The square equation roots in (49) is written as
un1, n, m =
⎤ 1 ⎡⎛ 1⎞ 2 2 +⎢⎜2n+ ⎟ −(2n1 +2) ⎥ 4 ⎥⎦ a∗ ⎢⎣⎝ 2⎠ , (50) 2 ⎡⎛ ⎤ ⎞ m 1 2 β∗ ⎢⎜⎜X −η02B − 2 ⎟⎟ −(2n1 +2) 4 ⎥ ⎢⎣⎝ a∗ ⎠ a∗ ⎥⎦
m⎞ ⎛ 1⎞⎛⎜ 2 ⎜2n+ ⎟⎜X −η0B − ∗ 2 ⎟⎟+(2n1 +2) ⎝ 2⎠⎝ a ⎠
un(21), n, m =
⎛ m⎞ ⎜⎜X −η02B − 2 ⎟⎟ a∗ ⎠ ⎝
2
2 ⎤ 1 ⎛ m⎞ ⎡ 1⎞ ⎜⎜ X −η02B − 2 ⎟⎟ +⎢⎛⎜2n+ ⎟ −(2n1 +2) 2 ⎥ 4 2⎠ ⎥⎦ a ∗ a ∗ ⎠ ⎢⎣⎝ ⎝ , (51) 2 ⎡⎛ ⎤ m⎞ 2 1 ⎥ β∗ ⎢⎜⎜ X −η02B − 2 ⎟⎟ −(2n1 +2) 4 ∗ ⎢⎣⎝ a ⎠ a ∗ ⎥⎦
1⎞⎛ m⎞ ⎛ 2 ⎜2n+ ⎟⎜⎜ X −η0B − 2 ⎟⎟−(2n1 +2) 2⎠⎝ ⎝ a∗ ⎠
2
here m = ±1 . It is easy to show, that only expression (50) is the system (49) decision, and, hence, it is decision for equation (48). As result, accounting of (26), (43) and (50), the light impurity absorption coefficient (t ) K B ω in case of the transversal light polarization eλ t , (with respect to the applied
( )
magnetic field direction), can be represented as
⎡ −2 (t ) ∗ −1 ⎢ K (ω) = K01 β X X −a∗ ⎢ ⎣
(
)
2
0 P1
0 P2
∑= ∑ =
n1
2
0 n
0
Γ (2 n + 1) (n1 + 1)(2 n1 + 2) 22 n Γ 2 (n + 1)
2
(
)
u n21, n, − 1 P u n1, n, − 1 ×
3⎞ ⎛ 1⎞ 2⎛ 3 ⎞ ⎛ ⎜ 2 γ n 1, n, − 1 + ⎟ Γ ⎜ γ n 1, n, − 1 + ⎟ Γ ⎜ γ n 1, n, − 1 + + n ⎟ 2⎠ ⎝ 4⎠ ⎝ 4 ⎠ ⎝ × × ⎡ 7⎞ ⎛ 3 ⎞⎛ ⎛ 7⎞ 1 ⎞⎞ ⎤ ⎛ ⎛ Γ ⎜ γ n 1 , n , − 1 + ⎟ ⎢⎜ γ n 1 , n , − 1 + ⎟ ⎜⎜ Ψ ⎜ γ n 1 , n , − 1 + ⎟ − Ψ ⎜ γ n 1 , n , − 1 + ⎟ ⎟⎟ − 1⎥ 4 ⎠ ⎣⎝ 4 ⎠⎝ ⎝ 4⎠ 4 ⎠⎠ ⎦ ⎝ ⎝
600
V.D. Krevchik, M.B. Semenov and R.V. Zaitsev
(
)
(
)
−1
k+
3
−4 −2 2 ⎤ −2 1⎞ ⎛ 2 × β u n 1 , n , − 1 ⎡(2 n 1 + 2) a ∗ − X − η02 + a ∗ + 2 n + ⎟ X − η02 + a ∗ ⎢⎣ ⎥⎦ ⎜⎝ 2⎠ ∗
⎡ ⎢ k k k +1 ⎢ n 1 (− 1) C n 1 2 Γ (k + 2) × ⎢∑ 11 ⎞ ⎢ k =0 Γ ⎛⎜ γ +n+k⎟ n 1 , n, −1 + ⎢ 4 ⎝ ⎠ ⎢⎣
(
)
×
⎡ ∗ 1 ⎞⎤ 2 ⎛ ∗ −2 2 β − η + − + 2 X a u n ⎜ ⎟ n 1 , n , −1 0 ⎢ 2 ⎠⎥⎦ ⎝ ⎣ × k +2 − 2 3 ⎡ ∗ ⎤ ∗ 2 u n 1 , n, −1 + 2 (n 1 − n ) + ⎥ ⎢⎣β X − η0 + a 2⎦
(
)
2
⎛ ⎞⎤ ⎜ ⎟⎥ 2 (2n1 + 2) 3 11 ⎟⎥ + × F ⎜ γ n1, n, −1 + + n, k + 2; γ n1, n, −1 + + n + k;1− 3⎟⎥ 4 4 ⎜ ∗ ∗ −2 2 un1, n, −1 + 2 (n1 − n) + ⎟ β X − η0 + a ⎜ 2 ⎠ ⎥⎦ ⎝
(
(
+ X +a
)
2 ∗ −2
0 P1
0 P2
∑= ∑ =
n1
0 n
)
Γ (2 n + 1) (n 1 + 1)(2 n 1 + 2 ) 2
0
2n
Γ
2
(
2
u n21 , n , + 1 P u
(n + 1)
n 1, n, + 1
)×
2
3⎞ ⎛ 1⎞ 2⎛ 3 ⎞ ⎛ ⎜ 2 γ n 1, n, + 1 + ⎟ Γ ⎜ γ n 1, n, + 1 + ⎟ Γ ⎜ γ n 1, n, + 1 + + n ⎟ 2⎠ ⎝ 4⎠ ⎝ 4 ⎠ ⎝ × × 7 ⎞ ⎡⎛ 3 ⎞⎛ ⎛ 7⎞ 1 ⎞⎞ ⎤ ⎛ ⎛ Γ ⎜ γ n 1 , n , + 1 + ⎟ ⎢⎜ γ n 1 , n , + 1 + ⎟ ⎜⎜ Ψ ⎜ γ n 1 , n , + 1 + ⎟ − Ψ ⎜ γ n 1 , n , + 1 + ⎟ ⎟⎟ − 1⎥ 4 ⎠ ⎣⎝ 4 ⎠⎝ ⎝ 4⎠ 4 ⎠⎠ ⎦ ⎝ ⎝
(
)
(
−4 −2 2 ⎤ −2 1⎞ ⎛ 2 × β u n 1 , n , + 1 ⎡(2 n 1 + 2 ) a ∗ − X − η02 − a ∗ + 2 n + ⎟ X − η02 − a ∗ ⎢⎣ ⎥⎦ ⎜⎝ 2⎠ ∗
)
−1
×
3 k+ ⎡ 2 2 − ⎡ ⎤ 1 ⎛ ⎞ 2 ∗ ∗ ⎢ β − η − − + X a u n 2 ⎜ ⎟ n 1 (− 1)k C k 2 k +1 Γ (k + 2 ) 0 n 1 , n, +1 ⎢ ⎢ n1 2 ⎠⎥⎦ ⎝ ⎣ × ⎢∑ × k +2 11 ⎛ ⎞ 2 − 3 0 k = ⎡ ⎤ ⎢ Γ⎜ γ + n + k ⎟ β∗ X − η02 − a ∗ u n , n, +1 + 2 (n 1 − n ) + n , n, +1 + ⎢ 1 4 ⎝ 1 ⎠ ⎢⎣ 2 ⎥⎦ ⎢⎣
(
)
(
)
⎞⎤ ⎛ ⎟⎥ ⎜ 2 (2n1 + 2) 3 11 ⎟⎥ × F ⎜ γ n1, n, +1 + + n, k + 2; γ n1, n, +1 + + n + k;1 − 3⎟⎥ 4 4 ⎜ ∗ 2 ∗ −2 un1, n, +1 + 2(n1 − n) + ⎟ β X − η0 − a ⎜ 2 ⎠ ⎦⎥ ⎝
(
)
2
⎤ ⎥ ⎥ ⎥ ⎥ ⎦
(52) where
[ ]
K 0 = 2 4 π 2 λ 20 α ∗ a d2 N 0 , P1 0 = C0(3)
( (
C 0(3 ) = 3 β ∗ X − η02 + a ∗
−2
)− 1)/⎛⎜⎝ 4
2
1 + 9β ∗ a ∗
−4
– is the integer part for number
[ ]
0 (4 ) – is the integer / 4 ⎞⎟ − 1 ; P2 = C 0 ⎠
Optical Properties of Quantum Dots
( (
part for number C 0(4 ) = 1 / 4 × 3 β ∗ X − η02 + a ∗
−2
)− 1)− (n
601
+ 1) 1 + 9 β ∗ a ∗ 2
1
are determined by the formula (50), in which coefficient
η02 B
−4
/4 ;
u n 1, n, ± 1
should be replaced by
η 02 ;
γ n 1 , n , ± 1 = β ∗ η 02 u n 1 , n , ± 1 / 2 . 1400
K B t Z , sm – 1
1 1200
2 1000
800
600
400
200
0
0.1
0.15
0.2
0.25
0.3
0.35
=Z , eV Figure 6. The spectral dependence for the light magneto-optical impurity absorption coefficient
K B(t ) (ω)
, (for transversal polarization), in case of borosilicate glass, which is pigmented by InSb
crystallites ( E i = 5.5 × 10 −2 eV , R 0 = 71.6 nm , different values of magnetic induction
U 0 = 0.3 eV , N 0 = 10
B:1– B = 0 T ;2– B = 4 T
.
15
sm − 3 ), for
602
V.D. Krevchik, M.B. Semenov and R.V. Zaitsev 300
K t Z , sm – 1
280
1
2
260
240
220
200
180
160
140
120
100
80
60
40
20
0
0.12
0.14
0.16
0.18
0.2
0.22
0.24
=Z , eV Figure 7. The spectral dependence for the light magneto-optical impurity absorption coefficient
K B(t ) (ω)
, (for transversal polarization), in case of borosilicate glass, which is pigmented by InSb
crystallites
(
E i = 5.5 ×10 −2 eV ,
N 0 = 10 15 sm − 3 ), B = 3.7 T .
R 0 = 35.9 nm ,
for different values of magnetic induction
B:
U 0 = 0.2 eV , 1 –
B=0 T;
2 –
Optical Properties of Quantum Dots
603
Fig. 6 represents the spectral dependence for the light magneto-optical impurity (t ) absorption coefficient K B ω , (for transversal polarization), in case of borosilicate glass,
( )
which is pigmented by InSb crystallites. Fig. 7 shows the spectral dependence for the light (t ) ω for transversal polarization, in magneto-optical impurity absorption coefficient K
( )
case when the magnetic field influence to the QD impurity ground state is negligible. The impurity absorption band (curve 1) in external magnetic field is splitted into Zeeman doublet (curve 2). Height of absorption peak, related to the electron optical transition to state with
m = −1 , is several times smaller than the peak, related to the electron optical transition to state with m = +1 . Such an doublet asymmetry is due to displacement from the spherically symmetrical potential well for the electron wave function, which corresponds to state with
n 1 = 0 , m = −1
and
n2 = 0,
[13]. Distance between peaks in Zeeman doublet is
ω B ; and distance between two nearest doublets depends from the confinement potential character frequency ω 0 and equals
determined by the cyclotron frequency, i.e. equals to
to
2 ω 0 . Under quantum number n 1
becoming equal to
changing on 1, distance between nearest doublets is
Ω , i.e. is determined by the hybrid frequency.
I3. Conclusions Thus, in this article, in frames of the zero-range potential model in the effective mass approximation, the problem of binding states in quantum dots with impurity centers under magnetic field influence is decided analytically exactly. It is found that the QD electron states cardinal modification, which is caused by double quantization, leads to the D(–)-center binding energy spatial anisotropy: i.e., the binding energy dependence from polar radius in QD for impurity levels, which are positioned lower than QD – bottom, is analogous to correspondent dependence in quantum wire, (magnetic field existence leads to the binding states stabilization), and the impurity centers binding energy slightly decreases in the applied magnetic field direction. It is shown that spectral dependence for the light impurity absorption coefficient (for longitudinal polarization) has an oscillating character. The oscillations period is determined by the hybrid frequency, if the Landau level number changes on 1; but under constant Landau level number (without changes), oscillations period is determined by the oscillator character frequency. It is also shown that the light impurity absorption coefficient spatial dependence (in case of transversal polarization) is characterized by the quantum-dimensional Zeeman effect with asymmetric doublet. It is found that the distance between peaks in doublet is determined by the cyclotron frequency, and distance between two nearest doublets under the constant Landau level number is determined by the oscillator character frequency, and if the Landau level number changes on 1, this distance is determined by the hybrid frequency.
604
V.D. Krevchik, M.B. Semenov and R.V. Zaitsev
II. Optical Properties of the Disk – Shaped Quantum Dots with D − - Impurity Centers Introduction (II) The modern tendencies for semiconductive nano-electronics show the necessity to take into account influence of the nano-structures geometric form features to the electronic energy spectrum, including impure states. Experimental observation for the spread of InAs quantum dots on GaAs underlying surface show [23], that quantum dots (InAs) are strongly flattened disk – shaped clusters. Cardinal modification in electron spectrum during transition “sphere – shaped QD → quantum disk” leads to essential changes in magnetic and optical properties of QD [24]. High sensitivity of the impure carrier binding energy to energy spectrum of QD allows, in principle, observe evolution of the binding energy during change of the QD geometric form. It is actual, because, as experiments show [14], impurities affect transport and optical properties of nano-structures. On other hand, in real systems sizes and shape of separate QD are deviated from equilibrium ones, that gives changes in optical properties for systems with QD [23], as also open possibility to realize opto-electronic devices [23, 25]. Hence, necessity to investigate influence of the QD geometric form on the light impurity absorption spectra in quasi-zero-dimensional structures is arisen. −
The aim of this work is to calculate the binding energy of D -center of quantum disk in frames of the zero-range potential method [26, 27] and to investigate the light impurity absorption by systems of disk-shaped QD, which have been synthetized in transparent dielectric matrix. For simulation of the quantum disk confinement in radial direction the “hard wall” potential is used:
⎧0, ρ ≤ R0 , U (ρ ) = ⎨ ⎩∞, ρ > R0 ,
(53)
where R0 - is the quantum disk radius. And in case of QD with the ellipsoid of revolution shape we can use confinement potential as 2D oscillator sphere well:
m∗ω12 2 ρ , U1 ( ρ ) = 2 ∗
where m - effective mass of electron, in radial direction;
(54)
ω1 -character frequency for the confinement potential
ρ ≤ R0 , R0 - radius for non-spherical QD in radial direction.
The one-dimensional harmonic oscillator potential U ( z ) is used in
z
- direction:
Optical Properties of Quantum Dots
U ( z) = ∗
where m - effective mass of electron;
ω0
605
m∗ω02 2 z , 2
(55)
– characteristic frequency of oscillator.
1D harmonic oscillator can be used as model for the non-spherical QD in z - direction
m∗ω22 2 U2 ( z ) = z , 2 where
(56)
ω2 - characteristic frequency for the non-spherical QD confinement potential in z -
direction. It is easy to show, that Schrodinger equation for considering model of quantum disk
allow the separation of variables, and one – electron wave functions Ψ n , m , k ( ρ , ϕ , z ) and energy spectrum En ,m ,k can be written as −
1
Ψ n,m,k ( ρ , ϕ , z ) =
2n n !π 2 aR02 J m +1 (ξ km ) 3
e
where
n
ρ ⎞ imϕ ⎛z⎞ ⎛ Η n ⎜ ⎟ J m ⎜ ξ km ⎟ e , (57) R0 ⎠ ⎝a⎠ ⎝
(ξkm ) , 1⎞ ⎛ = ω0 ⎜ n + ⎟ + 2 ⎠ 2m∗ R02 ⎝ 2
En ,m ,k
z2
2 a2
2
(58)
= 0, 1, 2,… - quantum numbers, which are corresponded to the energy levels of
one-dimensional potential well;
m
= 0, ±1, ±2,… - magnetic quantum number;
Bessel function of the first kind for order
m ( J m (ξ km ) = 0 ) ; k
number of the Bessel function roots; a =
ξ km - roots of
= 1, 2, 3,… - ordinal
m∗ω0 - characteristic length of oscillator;
ρ , ϕ , z - cylindrical coordinates; Η n ( x ) - Hermite polynomials. Impurity
γ = 2π
2
potential
is
simulated
by
the
zero-range
potential
with
intensity
(α m ) . This potential with account of logarithmic divergence for one – electron ∗
Green function can be written as (in cylindrical coordinate system)
Vδ (ρ , ϕ , z; ρ a , ϕ a , z a ) = γ
δ (ρ − ρ a ) δ (ϕ − ϕ a )δ (z − z a ) × ρ
⎡ ∂ ∂⎤ × ⎢1 − (ρ − ρ a ) ln (ρ ∗ − ρ a∗ ) + ( z − z a ) ⎥ , ∂ρ ∂z ⎦ ⎣
(59)
606
V.D. Krevchik, M.B. Semenov and R.V. Zaitsev
where
α
is determined by the binding energy
Ei
for
D−
- state in bulk semiconductor;
ρ ∗ = ρ ad ; ρ a∗ = ρ a ad ; ad - effective Bohr radius; ρ a , ϕ a , za - coordinates of D − center in quantum disk.
II1. Binding Energy of
D−
- State in Quantum Disk
D−
The Lippmann – Schwinger equation for
Ψλ (ρ , ϕ , z; ρ a , ϕ a , z a ) =
- state in quantum disk can be written as
R0 2π ∞
∫ ∫ ∫ ρ dρ dϕ dz G(ρ ,ϕ , z; ρ , ϕ , z ; Eλ ) × , 1
1
1
1
1
1
1
0 0 −∞
× Vδ (ρ1 , ϕ1 , z1; ρ a , ϕ a , za )Ψλ (ρ1 , ϕ1 , z1; ρ a , ϕ a , za )
(60)
where Ψ λ ( ρ , ϕ , z; ρ a , ϕ a , za ) - wave function of electron, which is localized on
D0
-
center in quantum disk, G (ρ , ϕ , z; ρ1 , ϕ1 , z1 ; E λ ) - one – electron Green function, which is corresponded to source in point
(ρ a , ϕ a , z а ) and to energy
G ( ρ , ϕ , z; ρ1 , ϕ1 , z1 ; Eλ ) =
∑
Eλ :
Ψ ∗n , m, k ( ρ1 , ϕ1 , z1 ) Ψ n ,m ,k ( ρ , ϕ , z ) Eλ − En , m, k
n,m,k
.
(61)
Substituting (59) into (60), we obtain
(
Ψ λ ( ρ , ϕ , z; ρ a , ϕa , za ) = γ G ( ρ , ϕ , z; ρ a , ϕ a , za ; Eλ ) Tˆ Ψ λ
) ( ρ ,ϕ , z ; ρ ,ϕ , z ) , a
a
a
a
a
a
(62)
where operator Tˆ is defined as
(TˆΨ ) ( ρ ,ϕ , z ; ρ ,ϕ , z ) ≡ lim ⎡⎢⎣1 − ( ρ − ρ ) ln ( ρ λ
a
a
a
a
a
a
ρ → ρa ϕ →ϕa
a
∗
− ρ a∗ )
∂ ∂⎤ + ( z − za ) ⎥ . (63) ∂ρ ∂z ⎦
z → za
Acting by operator Tˆ to both parts of (62), we obtain equation, which determines dependence of the binding energy for
D
−
D−
- state from characteristic sizes of quantum disks,
- center coordinates and parameters of confinement potential:
α=
2π 2 ˆ TG ( ρ a , ϕ a , za ; ρ a , ϕ a , za ; Eλ ) . m∗
( )
(64)
Optical Properties of Quantum Dots
607
Usage of overt view of one – electron wave functions (57), as also (58), for Green function in (64), gives
G ( ρ , ϕ , z; ρ a , ϕa , za ; Eλ ) = −
∞
(
)
1 1 dt 2 −4 β t − 2 − η + β − × exp t 1 e ( ) ( ) 2ad2 aEd π 3 2 ∫0 t
⎡ ⎡ 4 za∗ z ∗e −2 β t − ( za∗2 + z ∗2 )(1 + e −4 β t ) ⎤ β ⎤ ⎦ ⎥× × exp ⎢ ⎣ −4 β t ⎢ ⎥ 2 (1 − e ) ⎣ ⎦ ∗ ∗ ⎡ ⎛ ρ a∗ ⎞ ⎛ ρ ⎞ ⎛ R0∗ ⎞ ⎛ ρ a∗ ⎞ ⎛ ρ ⎞ ⎛ R0∗ ⎞ ⎤ ⎜ ⎟ ⎜ ⎟K m ⎜ ⎟ ⎥ I m ⎜⎜ ⎟⎟ I m I 0 ⎜⎜ ⎟⎟ I 0 K 0 ⎜⎜ ⎟⎟ ⎢ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ + ∞ ⎢ ⎝ t⎠ ⎝ t⎠ ⎝ t⎠ ⎝ t ⎠ ⎝ t ⎠ ⎝ t ⎠⎥ × ⎢ K 0 (w) − − 2∑ ⎥, ∗ ∗ ⎛ ⎞ ⎛ ⎞ R R = m 1 0 0 ⎥ ⎢ I m ⎜⎜ ⎟⎟ I 0 ⎜⎜ ⎟⎟ ⎥ ⎢ t t ⎝ ⎠ ⎝ ⎠ ⎦ ⎣
where
η2 =
Eλ Ed
; Ed
- effective Bohr energy;
amplitude of confinement potential in
R0∗ =
R0
∗
ad
, za =
za
ad
z
β=
U 0∗
- direction; U 0 =
∗
L
;U 0∗ =
m∗ω02 L2
2
U0
Ed
(65)
;U 0 -
; L∗ = L
ad
;-
; I m ( x ) and K m ( x ) - modified Bessel functions of integer order
of the first and second kind, correspondingly; w =
ρ a∗2 + ρ ∗2 − 2 ρ a∗ ρ ∗ cos (ϕ − ϕa ) .
Selecting of diverging part in (65) gives:
⎛ z − za ⎞ exp ⎜ − η 2 + β ⎟ a ⎠ 1 ⎝ G ( ρ , ϕ , z; ρ1 , ϕ1 , z1 ; Eλ ) = − − 32 × 2 4π a Ed z − za 4π ad aEd ⎧ ∞ dt × ⎨ ∫ exp − (η 2 + β ) t ⎩0 t
(
)
1 ⎡ ⎛ ( z 2 + za2 ) ⎞ ⎡ 2 za ze −2 β t − ( za2 + z 2 ) e −4 β t ⎤ −4 β t − 2 ⎢ exp ⎜ − ⎟ ⎢ ⎥× − 1 exp e ( ) 2 −4 β t ⎜ 2a 2 ⎟ a 1 e − ⎢ ⎢ ( ) ⎝ ⎠ ⎣ ⎦⎥ ⎣
∗ ∗ ⎡ ⎛ ρ ∗ ⎞ ⎛ ρ ⎞ ⎛ R∗ ⎞ ⎛ ρ ∗ ⎞ ⎛ ρ ⎞ ⎛ R∗ ⎞ ⎤ I 0 ⎜⎜ a ⎟⎟ I 0 ⎜ ⎟ K 0 ⎜⎜ 0 ⎟⎟ I m ⎜⎜ a ⎟⎟ I m ⎜ ⎟ K m ⎜⎜ 0 ⎟⎟ ⎥ ⎢ ⎜ ⎟ ⎜ ⎟ +∞ ⎢ ⎝ t⎠ ⎝ t⎠ ⎝ t⎠ ⎝ t ⎠ ⎝ t ⎠ ⎝ t ⎠⎥ × ⎢ K 0 (w ) − − 2 ∑ ⎥− ∗ ∗ ⎛ ⎞ ⎛ ⎞ R R = m 1 ⎢ ⎥ I 0 ⎜⎜ 0 ⎟⎟ I m ⎜⎜ 0 ⎟⎟ ⎢ ⎥ ⎝ t⎠ ⎝ t⎠ ⎣ ⎦
⎛ ( z − z a )2 − exp⎜⎜ − 4a 2 t t ⎝ 1
⎞⎤ ⎫⎪ ⎟⎥ ⎬ . ⎟ ⎠⎦⎥ ⎪⎭
(66)
608
V.D. Krevchik, M.B. Semenov and R.V. Zaitsev Substituting (66) into (64), we obtain dispersion equation for electron, which is localized
on the
D0
- center in quantum disk:
1 ⎡ β ⎧∞ dt 2 − 4 βt − ∗2 η + β = ηi − ⎨∫ exp[− (η + β )t ]⎢(1 − e ) 2 exp(− z a β thβ t )× π ⎩0 t ⎣ 2
∗ ⎤⎫ ⎡ 2 ⎛ ρ a∗ ⎞ ⎛ R0∗ ⎞ ⎤ ⎛ R0∗ ⎞ 2 ⎛ ρa ⎞ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ I m ⎜ ⎟ K m ⎜⎜ ⎟⎟ ⎥⎪ ⎢ I 0 ⎜ ⎟K 0 ⎜ ⎟ ⎥ +∞ t⎠ ⎝ t⎠ t⎠ ⎝ t⎠ 2 t ⎥ 1 ⎥⎪ ⎝ ⎝ ⎢ × + 2∑ − ln − ⎥⎬ , ⎢ γ ⎥ ⎛ R0∗ ⎞ ⎛ R0∗ ⎞ t ⎪ m =1 ⎥ ⎢ ⎥ I 0 ⎜⎜ ⎟⎟ I m ⎜⎜ ⎟⎟ ⎥⎦ ⎪ ⎝ t⎠ ⎝ t⎠ ⎣⎢ ⎦⎥ ⎭
γ
where
EλCD
-
the
= Eλ +
ρ
2
Euler
constant.
( ξ ) ( 2m R ) 2
∗
1,1
- center in radial plane and in
z
2 0
Fig.
D−
for
8
shows
the
binding
(67)
energy
- state dependence from coordinates of
- direction EλCD
z
= Eλ +
2
U0
( 2m L ) ∗ 2
D−
in quantum
disk, based on InSb, (this dependence is derived from (67)). And in case of QD with the ellipsoid of revolution shape dispersion equation for electron, which is localized on the
η + ( 2 β ) + wβ −1
2
D0 −1
- center in non-spherical QD:
= ηi −
2
∞
⎡ ⎛
dt exp ⎢ − ⎜ βη πβ ∫ ⎣ ⎝ 0
− w (1 − e
where
1 −2 t − 2
) (1 − e )
−2 wt −1
2
1⎞ ⎤⎡ 1 + w + ⎟t⎥ ⎢ − 2 ⎠ ⎦ ⎣ 2t 2t
⎧ za∗2 t ⎫ ⎧ ρ a∗2 w wt ⎫ ⎤ th ⎬ exp ⎨− th ⎬⎥ , exp ⎨− 2 ⎭⎦ ⎩ 2β 2 ⎭ ⎩ 2β
z a∗ = z a a d ; ρ a∗ = ρ a a d
(68)
.
From (68) we can obtain passage to the limit of sphere – shaped QD, and under w → 1 we have ∞ ⎡ ⎛ 3 −1 2 3⎞ ⎤⎡ 1 − η + β = ηi − dt exp ⎢ − ⎜ βη 2 + ⎟ t ⎥ ⎢ ∫ 2 2 ⎠ ⎦ ⎣ 2t 2t βπ 0 ⎣ ⎝ 2
− (1 − e ∗
where Ra
= ρ a∗2 + z a∗2 .
3 −2 t − 2
)
⎧ Ra∗2 β −1 t ⎫ ⎤ th ⎬⎥ , exp ⎨− 2 2 ⎭⎦ ⎩
(69)
Optical Properties of Quantum Dots
609
As we see from comparison of curves 1a and 3a on fig. 8, there is spatial anisotropy for −
the D - state binding energy in quantum disk. This anisotropy is conditioned by feature of geometric and potential confinement of quantum disk. Moreover, not only character of the coordinate dependence for binding energy is changed, but also its value is changed too. We −
can also see, that the binding energy for D - state is essentially increased with decrease of characteristic sizes of quantum disk (compare curves 2a and 1a, 4a and 3a). This is because of the quantum dimensional effect.
II2. The Light Impurity Absorption Coefficient in Structures with the Disk – Shaped Quantum Dots Let us consider the light impurity absorption in structure, which is the transparent dielectric matrix, with synthesized disk – shaped QD inside of this matrix. It is supposed, that
D−
-
center is positioned in point Ra = (0,0,0 ). , and impurity level is situated lower, than bottom of quantum disk ( Eλ < 0 ) . Then, because of (62) and (65), wave function Ψ λ ( ρ , ϕ , z ) for electron, which is localized on short-range potential, can be written as
Ψ λ ( ρ , ϕ , z ) = ÑN
∞
R0∗2 2π
3
2
1 ⎡ z ∗2 β ⎤ dt 2 −4 β t − 2 − + − t e η β exp 1 exp ( ) ( ) ⎢ − 2 th 2β t ⎥ × ∫0 t ⎣ ⎦
(
)
⎡ ⎛ ρ a∗ ⎞ ⎛ R0∗ ⎞ ⎤ I ⎢ 0⎜ ⎜ t ⎟⎟ K 0 ⎜⎜ t ⎟⎟ ⎥ ∗ ⎛ ⎞ R 0 × ⎢ K 0 ⎜⎜ ⎟⎟ − ⎝ ⎠ ∗ ⎝ ⎠ ⎥ , ⎢ ⎝ t⎠ ⎥ ⎛R ⎞ I 0 ⎜⎜ 0 ⎟⎟ ⎢ ⎥ ⎢⎣ ⎥⎦ ⎝ t⎠
(70)
where Ñ N - is normalized factor: −
1
⎡ aR 2 ∞ Γ ( f k ′ ) ⎤ 2 0 , ⎡ ϕ ⎤ ÑN = ⎢ f Ψ − Ψ ⎥ ( ) ( ) k′ k′ ⎦ 3 ∑ ⎣ ⎢⎣ π ( 4β ) k ′=1 Γ (ϕk ′ ) ⎥⎦ Here
2 ⎛ ⎛ ξ k ′0 ⎞ ⎞ 2 f k ′ = ⎜η + β + ⎜ ∗ ⎟ ⎟ ⎜ ⎝ R0 ⎠ ⎠⎟ ⎝
2 ⎛ ⎛ ξ k ′0 ⎞ ⎞ 2 ( 4β ) ;ϕk ′ = ⎜⎜η + β + ⎜ ∗ ⎟ ⎟⎟ ⎝ R0 ⎠ ⎠ ⎝
(71)
( 4β ) + 1 2; Ψ ( x )
-
logarithmic derivative of gamma function Γ( x ) .
The effective Hamiltonian of interaction with the light wave field Ĥint in case of transversal light polarization (in respect to axis of quantum disk), in cylindrical coordinate system can be written as
610
V.D. Krevchik, M.B. Semenov and R.V. Zaitsev
⎛ 2π 2α ∗ ∂ 1 ∂ ⎞ ˆ H int = −i λ0 I 0 eiqz z ⎜ cos (θ − ϕ ) + sin (θ − ϕ ) ⎟, ∗2 ∂ρ ρ ∂ϕ ⎠ m ω ⎝ where q = (0,0, q z ) - wave vector of photon;
α∗ =
e
(72)
λ0 - coefficient of local field;
2
(4πε
0
ε c
) - the fine structure constant with account of static relative dielectric
permeability ε ; с – speed of light in vacuum; I 0 - light intensity; ω - frequency of light;
θ
- polar angle for unit vector of polarization eλ in the cylindrical coordinate system. Matrix element М iλ , which determines the oscillator force for the dipole optical transition from
D−
- state Ψ λ ( ρ , ϕ , z ) to states of dimensional-quantizing Ψ n , m , k ( ρ , ϕ , z ) in quantum disk, can be written as 1
М iλ = i λ0 a ×π å
± iθ
2 d
− 3 R0∗2 ⎛ 2 n 2π 2α ∗ 2 2 ⎞ 2 ( ) ( ) I С 2 2 n ! π aR J ξ × ⎜ 0 н 1 0 m +1 km ⎟ ∗2 ⎠ 2π ⎝ m ω
( 2n )! ∞ dtF (η , t ) ⎡⎛ 1 + a∗2 F β , t ⎞−1 − 1⎤ δ m,±1 1 ∫ 1 )⎟ ⎥ ⎢⎜ 2( ( n1 )! 0 t ⎠ ⎢⎣⎝ 2 ⎥⎦
n1
(
−
1
⎛ 1 ⎞ 2 ⎜ ∗2 + F2 ( β , t ) ⎟ × ⎝ 2a ⎠
)
(
)
∗ ∗ ∗ 2 −1 ⎡ ⎧⎛ ⎛ R0∗ ⎞ ⎤ R0 K 0 R0 t ξ k 1 J 0 (ξ k 1 ) I1 R0 t ⎫⎪ ⎪ 1 ⎛ ξ k1 ⎞ ⎞ ⎢ ξ k1 t ξ J ξ K × ⎨⎜ + ⎜ ∗ ⎟ ⎟ + + ( ) ⎬ ⎟⎥ 1⎜ k1 2 k1 2 ∗ I 0 R0∗ t (ξk1 ) − R0∗ t ⎪⎭ ⎝ t ⎠⎦ ⎪⎝⎜ t ⎝ R0 ⎠ ⎠⎟ ⎣⎢ R0 ⎩
(
)
(73) In case of the ellipsoid of revolution – QD, anisotropy for the binding energy of D(-) –
( Å( ) ) NQD
state in non-spherical QD takes place. It should be noted, that binding energy
λ
ρ
(-)
for D – center, which is positioned in radial plane can be determined as
(E
( NQD )
λ
)
ρ
2 2U10 = Eλ + , m∗ R02
(74)
or in the Bohr units
( Å( ) ) NQD
λ
Ed
ρ
= β −1w + η 2 .
(75)
Optical Properties of Quantum Dots 0 .4
0.2
611
0 .6
Rȡaa**
0 .8
E Ȝ , m eV 1a
2a
1 20
1b 1 00 2b 3a 3b
80
4a
60
4b
0 .1
Figure 8. Dependence of binding energy InSb ) from radial
ρ a* = ρ a a d
0 .4
0 .3
0 .2
( 0)
< 0 ) in quantum disk (based on
for D–- state ( Eλ
Eλ
z a*
(curves 1а and 2а for disk with radius 51 nm and 68 nm
correspondingly, za=0, L=13,6 nm) and axial
z a* = z a a d
(curves 3а and 4а for disk with
thickness 17 nm and 34 nm correspondingly, ρa=0, R0=68 nm) coordinates of impurity under
U 0 = 0.25 eV
(dashed curves 1b - 4b show corresponding energies of the ground state in
quantum disk).
In
(
( z - direction the D(-) – center binding energy Åλ
(E
( NQD )
λ
or in the Bohr units
)
z
= Eλ +
NQD )
)
has the next view
z
2
U 20 2m∗ L2
,
(76)
612
V.D. Krevchik, M.B. Semenov and R.V. Zaitsev
( Å( ) ) NQD
λ
Ed
z
= ( 2β ) + η 2 . −1
(77)
Figures show results of numerical analysis of eq. (75) and (77). From fig. 9 a, b we can see anisotropy for D(-) – state binding energy: in z - direction binding energy more than in radial plane of non – spherical QD approximately on 0,01 eV. It is connected with the form feature of non – spherical QD: QD – size in z - direction smaller, than size of non – spherical QD in radial plane. As result, dimensional quantizing effect is more strong in z - direction; that gives increase in D(-) – state binding energy. From comparison of curves 1, 1’ and 2, 2’ at both figures we can see, that D(-) – state binding energy increases with the confinement potential amplitude increase (for non-spherical QD); because of the shift for the ground state of non-spherical QD. EO
NQT
EO
, eV
NQT
, eV
0.05
1c 2c
0.04
3c
0.03
4c
0.02
0.01
0
U D
UD / ad
a
0.2
0.4
0.6
zD
zD / a d
0.8
1
b
Figure 9. Dependence of D(-) – state binding energy on coordinates of the non-spherical QD D(-) – center under η i = 7, R0* = 1 (а), L* = 0.5 (b), for different amplitudes of confinement potential: 1,3 – U0* = 300; 2,4 – U0* = 200. Lines 3, 4 and 3`, 4` show position of the ground state level for nonspherical QD.
Optical Properties of Quantum Dots
E ,eV
E , eV
NQD
O
613
NQD
O
U
z
0.05
0.05
1c
0.04
0.04 1c 1c
2c
0.03
0.03
0.02
0.02
0.01
0.01
3c
4c
0
0.5 U D
1U / a D d
U UDD
1.5
0
2
0.5
zD
zD
1 zD / a d1.5 zD / a d
2
z z /a zD DzD /ad D d
UUD /D ad/
ɚ
ɛ
Figure 10. Dependence of D(-) – state binding energy on coordinates of D(-) – center: а) in radial direction and b) in z – direction of non-spherical QD (based on InSb) under
Ei == 0, 007 eV , U 01 = U 02 = 0, 2 eV
for different values of parameters R0* and L*: a) 1 –
R0* = 1; 2 – R0* = 2; b) 1` – L* = 0.5; 2` – L* = 1. Lines 3, 4 and 3`, 4` show position of corresponding levels for the ground state of non-spherical QD.
Figures 10 а, b gives dependencies of the D(-) – state binding energy in non – spherical QD for different values of parameters
R0*
and
L* ,
spherical QD (in the Bohr units) in radial plane and in
which can determine sizes of non-
z - direction, correspondingly. We can
see, that in conditions of strong dimensional quantizing ( R
*
, L* ≤ 1 )
the D(-) – state
binding energy essentially increases; and hence the existence region for D(-) – state considerably decreases. As result, non - sphericity for the QD – form leads to considerable modification of impure states, that is due to corresponding sensitivity of electronic energy spectrum to the geometric form of nanostructures. Situation in this case is analogous to case of D(-) – state in sphere – shaped QD under influence of external magnetic field.
614
V.D. Krevchik, M.B. Semenov and R.V. Zaitsev Under calculation of М iλ integrals of the next view are appeared +∞
∗ ⎛ ∗2 ⎛ 1 ⎞⎞ ⎛ z ⎞ ∫ dz exp⎜⎜⎝ − z ⎜⎝ 2a ∗2 + F2 (β1t )⎟⎠ ⎟⎟⎠H n ⎜⎜⎝ a∗ ⎟⎟⎠ = −∞ ∗
⎧0, åñëèn ≠ 2n1 , ⎪⎪ 1 n1 −1 − ,(78) ⎤ ⎛ 1 = ⎨ π ( 2n1 ) ! ⎡⎛ ∗2 ⎛ 1 ⎞⎞ ⎞ 2 + − + = β β a F , t 1 F , t , åñëèn 2 n , ⎢ ⎥ ( ) ( ) 2 2 1 ⎪ ⎟⎟ ⎜ ∗2 ⎟ ⎜ ⎜ ∗2 ⎠⎠ ⎠ ⎪⎩ n1 ! ⎣⎢⎝ ⎝ 2a ⎦⎥ ⎝ 2a 2π
∫ dϕe
−imϕ
cos(θ − ϕ ) = δ m, ±1π exp(∓ iθ ) ,
(79)
0
where sign “-“ in the exponent index exp(∓ iθ ) corresponds to value m=+1, and sign “+” to m=-1.
Functions F1 (η , t ) and F2 ( β , t ) in (17) are determined as 1 − 1 −(η 2 + β )t F1 (η , t ) = e 1 − e−4 β t ) 2 , ( t
F2 ( β , t ) =
β 2
th2 β t .
(80)
(81)
We can see from (78) and (79), that optical transitions from impure level can occur only to states of quantum disk with even values of oscillator quantum number n=2n1 (n1=0, 1, 2,…) and with values of magnetic quantum number m = ±1 . Coefficient K (ω ) of the light impurity absorption by structure with quantum disk, with
account of their characteristic sizes dispersion, is determined by expression of the next view
2π N 0 K (ω ) = I0
3
2
∑∑ δ m,±1 ∫ duP ( u ) Ì m
n
δ ( En ,m ,k + Eλ − ω ) .
2
iλ
(82)
0
It is supposed in (82), that dispersion u = R0 R0 = L L of characteristic sizes of quantum disks arises during process of phase decay in resaturated solid solution and can be satisfactorily described by the Lifshitz – Slezov formula
Optical Properties of Quantum Dots
615
⎧ 4 2 ⎡ −1 ⎤ ⎪ 3 eu exp ⎢ ⎥ ⎣1 − 2u 3 ⎦ , u < 3 , ⎪ 11 3 ⎪ 2 P(u ) = ⎨ 2 5 3 (u + 3)7 3 ⎛⎜ 3 − u ⎞⎟ ⎪ ⎠ ⎝2 ⎪ 3 ⎪0, u > , 2 ⎩
(83)
where R0 - mean value of the quantum disk radius; e - natural logarithm base; 2 L - mean value of the quantum disk height. In (82) N 0 - concentration of quantum disks in dielectric matrix. After integration over u in (82), for coefficient of impurity absorption K ( ω ) , we obtain
K (ω ) = K 0 ( β )
7 2
X
−2
( 2n1 )! ⎧⎪ F (η , L ,U k ,1 ) P U ⎡U − β ( 2n + 1 2 ) ⎤ ( k ,1 ) ⎢ k ,1 ⎥ ∑ ∑ 2 ⎨ 32 2n X n = 0 k =1 2 ( n1 !) ⎪⎩ (U k ,1 ) ⎣ ⎦ N
K
−1
×
1
1
⎡⎛ 1 2 2 dt F1 (η , L ,U k ,1 ; t ) ⎢⎜ + ( L∗ ) (U k ,1 ) F2 (η , L , U k ,1 ; t ) ×∫ t ⎢⎣⎝ 2 0 ∞
(
2 2 ⎡ × ⎢ ( L∗ ) (U k ,1 ) ⎣
U 0∗
)
−1
⎤ + F2 (η , L ,U k ,1 ; t ) ⎥ ⎦
−1 2
n1
−1 ⎤ ⎞ U ⎟ − 1⎥ × ⎠ ⎥⎦ ∗ 0
−1
2 ⎛1 ⎛ ξ ⎞ ⎞ k ,1 ⎜ +⎜ ⎟ ⎟ × ⎜ t ⎜⎝ R0∗U k ,1 ⎟⎠ ⎟ ⎝ ⎠
2 ⎡ ∗2 ∗ ∗ ⎫ ∗ R K U R t ξ J ξ I U R t ⎤ ⎛U R ⎞ 0 0 k ,1 0 k ,1 0 ( k ,1 ) 1 k ,1 0 ⎢ξ t ⎪ × ⎢ k∗1 + ξ k 1 J 2 (ξ k 1 ) K1 ⎜⎜ k ,1 0 ⎟⎟ ⎥ + ⎬ × 2 2 ∗ t ⎠ ⎥⎦ I 0 U k ,1 R0 t ⎝ ⎪ ξ k ,1 ) − U k ,1 R0∗ t ⎢ R0 U k ,1 ( ⎭ ⎣
(
(
)
)
(
(
)
⎡ 1 1 ⎤⎫ ×⎢ 2 + 2 ⎥⎬ , ⎣ J 0 (ξ k 1 ) J 2 (ξ k 1 ) ⎦ ⎭ where
C1 = 3 X
)
(84)
K 0 = 28 π N 0 ad2 λ02α ∗ ; N = [C1 ]
( 2 β ) − 1 2 − (ξ ) ( 6 β R ) 2
∗2 0
1,1
;
integer
part
of
number
β = U 0∗ L∗ ; K - is the integer part of
solution for transcendental equation of the next view
(ξ ) k ,1
2
=
9 ∗2 R0 X − 3β ( 2n + 1 2 ) R0∗2 , 4
(85)
616
V.D. Krevchik, M.B. Semenov and R.V. Zaitsev Here X =
ω Ed - the photon energy in units of the effective Bohr energy; functions
F (η , L , uk ,1 ) , F1 (η , L , uk ,1 ; t ) and F2 (η , L , uk ,1 ; t ) are determined as
(
))
(
(
)
2 Γ ⎡ η 2 + U 0∗ ( L∗uk ,1 ) + ξ k ,0 ( R0∗uk ,1 ) L∗uk ,1 4 U 0∗ ⎤ ⎢⎣ ⎥⎦ , (86) F (η , L , uk ,1 ) = ∑ 2 ⎡ ⎤ 2 ∗ ∗ ∗ ∗ ∗ k =1 Γ ⎢ η + U 0 ( L uk ,1 ) + ξ k ,0 ( R0 uk ,1 ) L uk ,1 4 U 0 + 1 2 ⎥ ⎣ ⎦ ∞
(
))
(
1 −(η 2 + β F1 (η , L , uk ,1 ; t ) = e t F2 (η , L , uk ,1 ; t ) =
)
uk ,1 t
(1 − e
− 4 β t uk ,1
(
)
)
−1 2
,
⎛ 2β t ⎞ th ⎜ ⎟, 2uk ,1 ⎜⎝ uk ,1 ⎟⎠
β
(87)
(88)
where
uk ,1 =
β ( 2n + 1 2 ) X
+
β 2 ( 2n + 1 2 ) X2
2
(ξ ) +
2
k ,1
R0∗2 X
.
(89)
Fig. 11 shows spectral dependence of coefficient for the light impurity absorption by quasi-zero-dimensional structure with the disk-shaped QD (based on InSb). As we can see from fig. 11, this absorption coefficient has non-monotone spectral dependence, which is conditioned by dimensional quantization. States, which are corresponded to energy with magnetic quantum number m = ±1 are degenerated ones. Because of it, oscillator force for dipole optical transition of electron from impure level to dimensional-quantizing states with m = ±1 is quite large (compare peaks on fig. 11). Besides, the characteristic sizes dispersion of quantum disks gives up limitation for possible values of oscillation quantum number n, because of condition u < 3 2 . So, for example, for parameters values, under which curve on fig. 11 has been obtained, N=0, and oscillations of the absorption coefficient are connected, in general, with optical transitions of electron between levels of dimensional quantizing for 2dimensional potential well, (this well simulates confinement potential of quantum disk in radial direction). Hence, in this article essential role of the QD geometric form factor in coordinate dependence for the binding energy of D–- state, as also in the light impurity absorption spectrum under transition “sphere-shaped QD → disk-shaped QD”, has been demonstrated. Unlike the case of sphere-shaped QD [15], binding energy of D–- state in quantum disk, as function of the D–- center coordinates, is anisotropic one; moreover, value of anisotropy essentially depends on characteristic sizes of quantum disks. It is necessary to point out, that the feature of geometric and potential confinement of quantum disk is appeared in essential
Optical Properties of Quantum Dots dependence for edge of the impurity absorption band quantum disks:
( ω )th = 2
2
U0
( 2m L ) ∗
2
( ω )th
3 + 4 (ξ1,1 )
2
617 from characteristic sizes of
(18m R ) . ∗
∗2 0
K(Ȧ), sm-1 0.5
0.4 2
0.3 1
0.2
0.1
0.03
0.06
0.08
0.1 =Z , eV
Figure 11. Spectral dependence for the light impurity absorption coefficient in quasi-zero-dimension structure with disk-shaped QD (based on InSb) under U = 0.15 eV, L = 15 nm, for different values of quantum disk radius R0 : 1 – R0 = 32 nm, 2 R0 = 65 nm.
In case of QD with shape of ellipsoid of revolution the impurity absorption coefficient
( ) K NQD (ω ) s
with longitudinal light polarization (in relation to the non-spherical QD axis)
eλ s , with account of the QD sizes dispersion, can be determined by (s)
K NQD (ω ) =
3 2
(
)
2π N 0 (s) 2 δ E δ duP u M ( ) QD,λ ∑ ∑ m,0 ∫ nρ ,0,2n +1 + Eλ − ω , (90) I 0 m nρ ,n1 0
618
V.D. Krevchik, M.B. Semenov and R.V. Zaitsev
where
N0
– the QD concentration in dielectric matrix;
δ m, 0
– Kroneker symbol;
δ (x ) –
Dirac delta – function. From
equation
β = Ed / ( ω2 )
L ∗ = L∗ u ,
L∗ = L / a d ,
where
from
formula
we can obtain
ω2 ( u ) =
where
and
Ed , βu
(91)
∗ ⎞ ∗ β = L∗ / ⎛⎜ 4 U 20 = U 20 / Ed . ⎟ ; U 20
⎝
⎠
Then, with account of (91), expression (90) for the impurity absorption coefficient
( s) K NQD
(ω ) can be represented as
(s)
K NQD (ω ) =
3 2
2π N 0 u (s) 2 δ duP u M × ( ) QD,λ ∑ ∑ m,0 ∫ 2 I 0 m nρ ,n1 Ed X − η 0
(
(
)
)
⎛ 2n + 3/ 2 + w 2n + 1 ⎞ 1 ρ ⎜ ×δ −u⎟, ⎜ ⎟ β X −η 2 ⎜ ⎟ ⎝ ⎠
(
where
X = ω / Ed
)
(92)
– photon energy in units of effective Bohr energy;
∗ ∗ ∗ w = L∗ U10 U 20 R0∗ ; ; U10 = U10 / Ed ; R0∗ = 2 R0 / ad . After integrating in (1.3.39) we will have
(
)
22 n1 +3 Δ nρ ,n1 + w 2 + 1 4
) Γ (Δ
(s)
K NQD (ω ) = K 0 S β −1 X −1 X − η 2
×
(
( 2n1 + 1)!
−1 N1
∑
N2
nρ =0 n1 =0
2
nρ ,n1
(
S ∑ P u ( )nρ ,n1
)
)
− w 2 + 3 4 Γ 2 ( n1 + 3/ 2 )
(
⎡−Γ ⎣ (1 2 − w ) ⎤⎦ Γ Δ nρ ,n1 + w 2 + 5 4
)
×
Optical Properties of Quantum Dots
)( (
(
619
))
) (
× ⎡ Δ nρ ,n1 + w 2 + 1 4 Ψ Δ nρ ,n1 + w 2 + 5 4 − Ψ Δ nρ ,n1 − w 2 + 3 4 − 1⎤ ⎢⎣ ⎥⎦
×
(
2n1 + 3/ 2 + (2nρ + 1) w + Δ nρ ,n1
( n + (nρ + 1/ 2)w + Δ 1
nρ ,n1
+ 1/ 4
)
−1
×
2
) ( n + (nρ + 1/ 2)w + Δ 2
1
nρ ,n1
+ 5/ 4
)
2 , (93)
where
K 0 S = 2πλ02α ∗ad2 N 0 ; N1 = [C1 ]
) )
( (
– integer part for the expression
C1 = 3 β X − η 2 − 1 / ( 4 w ) − 1/ 2 ; N2 = [C2 ]
C2
C1
value
– integer part for the expression
S ∗ ⎞ Δnρ ,n1 = βη 2u( ) nρ ,n1 2; β = L∗ ⎛⎜ 4 U 20 ⎟;
value
(
))
S u ( )nρ ,n1 = 2n1 + 3 2 + w 2nρ + 1
(
⎝
( β ( X − η )); n 2
⎠
1 - oscillator quantum
number in radial direction of QD. Change of
η2
on
−η ′2
in (93) gives expression for absorption coefficient under
photo-ionization of D–- centers with
Eλ > 0 .
In case of QD with shape of ellipsoid of revolution the impurity absorption coefficient
() K NQD (ω ) t
eλ t
with transversal light polarization (in relation to the non-spherical QD axis)
can be represented as
(
(t )
K NQD (ω ) = K0t wβ −1 X −1 X − η 2
×
×
(
)
−1 N3
N4
(
(t ) ∑ P u nρ ,n1
∑
nρ =0 n1 =0
) (
)
( 2n1 )!( nρ + 1) × 2 2n1 ! 2 n ( 1)
βη 2 u ( t )nρ ,n1 2 + w 2 + 1 4 Γ βη 2 u ( t )nρ ,n1 2 − w 2 + 3 4 2
(
2 (t ) ⎡−Γ ⎣ (1 2 − w ) ⎤⎦ Γ βη u nρ ,n1 2 + w 2 + 5 4
( ( n + (n
) 2 + 1/ 4 )
t 2n1 + 1/ 2 + (2nρ + 1)2 w + βη 2u ( ) nρ ,n1 1
2 (t )
ρ + 1/ 2) w + βη u
nρ ,n1
2 2
×
)
)×
620
V.D. Krevchik, M.B. Semenov and R.V. Zaitsev
(
t × n1 + (nρ + 3/ 2) w + βη 2u ( )nρ ,n1 + 1/ 4
( (
) ( −2
βη 2u ( t )nρ ,n1 2 + w 2 + 1 4
)
−1
))
) (
t × ⎢⎡ Ψ Δ nρ ,n1 + w 2 + 5 4 − Ψ βη 2u ( ) nρ ,n1 2 − w 2 + 3 4 − 1⎥⎤ ⎣ ⎦
×
−1
, (94)
where
K 0t = 23 π 2λ02α ∗ad2 N 0 ; α ∗ - the fine structure constant with account of the
static dielectric permeability for material of QD;
N3 = [C3 ]
– integer part for the
КNQD(ω), 1.2·104
1·104
8·103
6·103
4·103
2·103
0.05
0.1
0.15
0.2
0.25
0.3
0.35
Figure 12. Spectral dependence for the light impurity absorption coefficient
ω , eV
K NQD (ω )
by
structure with QD, which have shape of ellipsoid of rotation, based on InSb, which are synthesized in a transparent borosilicate glass, in case of longitudinal (curve 1) and transversal (curve 2), in relation to the QD – axis, light polarization, under
Ei = 10 −3 eV
U10 = U 20 = 0.2 eV , N0 = 1015 cm−3 .
,
R0 = 105 nm , L = 35 nm ,
Optical Properties of Quantum Dots
621
) ) C4 = ( ( 3β ( X − η 2 ) − 1) − ( nρ + 1) w ) C S u ( ) n ,n = ( 2n1 + 1 2 + 2 w ( nρ + 1) ) ( β ( X − η 2 ) ) C3
expression
value ,
part for the expression
ρ
( (
C3 = 3β X − η 2 − 1 / ( 4 w ) − 1 ; N4 = [C4 ] 4
value ,
4;
.
1
Change of
– integer
η2
on
−η ′2
in (94) gives expression for the impurity absorption
coefficient under photo-ionization of D–- centers with
Eλ > 0 .
Fig. 12 give spectral dependencies of the light impurity absorption coefficients, which are calculated on formulae (93) and (94) for case of QD, based on InSb. As we can see from fig. 10, in quasi-0D – structure with QD, which have shape of the ellipsoid of revolution, there is dichroism of impurity absorption (compare curves 1 and 2 on fig. 12). This dichroism is connected with change of selection rules for magnetic quantum number in radial direction and oscillator quantum number in z- direction of QD. Hence, dichroism of the light impurity absorption in quasi-0D-structure can be used for the QD (with shape of the ellipsoid of revolution) identification.
References [1] Arakawa Y., Yariv A. // IEEE J. Quantum. Electron. - 1986. - v. 22. - P. 1887. [2] Weisbuch C., Vinter B. Quantum Semiconductor Structures. - Academic Press, INC, 1991. [3] Someya T., Akiyama H., Sakaki H. // Phys. Rev. Lett. - 1996. - v. 76. - P. 2965. [4] Weigscheider W., Pfeiffer L. N., Dignam M. M., Pinczuk A., West K. W., McCall S. L., Hull R. // Phys. Rev. Lett. - 1993. - v. 71. - P. 4071. [5] Rytova N.S. // Doklady Akad. Nauk (USSR). - 1965. - v. 163, № 5. - P. 1118. [6] Mulyarov E.A., Tikhodeev S.G. // Zhur. Eksp. Teor. Fiz. (Russian JETP).- 1997. - v. 111. - P. 274. [7] Keldysh L. V. // Phys. Stat. Sol. (a). - 1997. - v. 164. - P. 3. [8] Galkin N.G., Margulis V.A., Shorokhov A.V. // Fiz. Tverdogo Tela (Russian). - 2001. - v. 43, № 3. - P. 511. [9] Belyavskii V.I., Kopaev Yu.V., Kornyakov N.V. // Uspekhi Fiz. Nauk (Russian). - 1996. - v. 166, № 4. - P. 447. [10] Krevchik V. D., Grunin A. B., Aringazin A. K., Semenov M. B. // Hadronic Journal. 2002. - v. 25, № 1. - P. 23. [11] Krevchik V. D., Grunin A. B., Aringazin A. K., Semenov M. B. // Hadronic Journal. 2002. - v. 25, № 1. - P. 69. [12] Krevchik V. D., Grunin A. B., Semenov M. B. // Izvestiya Vuzov. Fizika (Russian). 2002. - № 5. - P. 69. [13] Krevchik V. D., Grunin A. B., Zaitsev R.V. // Fizika I Tekhn. Polupr. (Russian). - 2002. v. 36, № 10. - P. 1225.
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[14] Huant S., Najda S. P. // Phys. Rev. Lett. - 1990. - v. 65, №12. - P. 1486. [15] Krevchik V. D., Zaitsev R.V. // Fiz. Tverdogo Tela (Russian). - 2001. - v. 43, № 3. - P. 504. [16] Landau L.D., Lifshits E.M. Quantum mechanics (nonrelativistic theory). Vol. 3. - М.: “Nauka” publ., 1989. [17] Nikiforov A.F., Uvarov V.B. Special functions for mathematical physics. - М.: “Nauka” publ., 1978. [18] Beytmen G., Erdeyi A., Highest transcendential functions. Vol. 1, 2. - М.: “Nauka” publ., 1973. [19] Vargin V.V. The coloured glass production. - М., 1940. [20] I.M. Lifshits, V.V. Slezov. // Zurnal Eksp. Teor. Fiz. (Russian, JETP). 1958. - – v. 35 (N 8). – P. 479. [21] Kulish N.R., Kunets V.P., Lisitsa M.P. // Fiz. Tverdogo Tela (Russian). - 1997. - v .39, № 10. - P. 1865. [22] Krevchik V.D., Grunin A.B., Aringazin A.K., Semenov M.B., Kalinin E.N., Mayorov V.G., Marko A.A., Yashin S.V. Magneto-optics of quantum wires with D- - centers // Hadronic Journal – 2003. - vol. 26, N 1. - p. 31-56, http://arXiv.org/abs/condmat/0303478 . [23] Ledentsov N.N., Ustinov V.M., Shchukin V.A., Kopyev P.S. Alferov Zh.I., Bimberg D. // FTP – 1998. - V.32. - №4. – P. 385. [24] Kokurin I.A., Margulis V.A., Shorokhov A.V.// Bulletine of the Povolgsky region universities (section “Natural sciences”, Physics). – 2003. - № 6(9). – P. 96. [25] Zhukov A.E., Kovis A.R., Ustinov V.M. // FTP. – 1999. – V. 33. - №5. – P. 1395. [26] Krevchik V.D., Grunin A.B., Marko A.A.// FTP. – 2006. – V.40 - №4. – P. 433. [27] Krevchik V.D., Grunin A.B., Evstiffev V.V.// FTP. – 2006. – V.40 - №6. – P. 136.
In: Quantum Dots: Research, Technology and Applications ISBN: 978-1-60456-930-8 Editor: Randolf W. Knoss, pp. 623-649 © 2008 Nova Science Publishers, Inc.
Chapter 17
POST-GROWTH ENERGY BANDGAP TUNING OF INAS/INGAAS/INP QUANTUM DOT STRUCTURES: INTERMIXING OF QUANTUM DOT STRUCTURES Tang Xiaohong and Yin Zongyou Photonics Research Center, School of Electrical and Electronic Engineering Nanyang Technological University, Singapore 639798
Abstract Semiconductor quantum dots (QDs) have become a topic of intensive research due to much interest in the fundamental physics of three dimensional (3D) quantum confinement, together with the novel device functionality that they can provide. For example, a QDs based semiconductor laser shows much lower threshold current density and lower temperature sensitivity of the threshold current, etc. Post-growth energy bandgap tuning of semiconductor QD structures is very important for monolithic photonic integration of QDs based passive and active optoelectronic devices. In this chapter, post-growth thermal annealing intermixing of InAs/InGaAs/InP quantum dots has been investigated in detail. The energy bandgap tuning of InAs/InGaAs/InP QD structures through the thermal annealing intermixing under a wide temperature range is studied. To increase the energy bandgap tuning, argon (Ar) plasma exposure enhanced intermixing of the InAs/InGaAs QD structure has been investigated. The energy bandgap blue shift of InAs/InGaAs/InP QD structure through the Ar plasma enhanced intermixing achieves 159 meV. By using a SiO2 mask layer, selective intermixing of an InAs/InGaAs/InP QD structures has been studied. The largest intermixing selectivity of the same wafer reaches 77 meV. Three different energy bandgap tuning across an InAs/InGaAs/InP QD wafer has been achieved using the post-growth selective Ar plasma enhanced intermixing. This large postgrowth selective bandgap tuning of the QD structures paves a way for monolithic integration of QDs based passive and active devices.
Introduction Stranski-Krastanow (SK) self-assembled semiconductor quantum dots (QDs) grown by molecular beam epitaxy (MBE) or metalorganic chemical vapor deposition (MOCVD) have
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Tang Xiaohong and Yin Zongyou
attracted intensive studies in recent years. Unique physical properties of QDs make them very promising for novel optoelectronic devices applications [1, 2]. For example, semiconductor quantum dot (QD) lasers have attracted much attention because they have been predicted that their threshold current density, temperature sensitivity, efficiency, etc, will be improved dramatically when compared with that of semiconductor quantum well lasers due to reduction of dimensionality of the carriers movement in QD structures [3]. Based on GaAs substrate, semiconductor QD lasers with self-assembled InAs/GaAs QD structures as the active region have showed very low threshold current of 6.7 mA under continuous-wave operation at roomtemperature [4]. InP substrate based QDs have shown an impressive wide wavelength range emission: InAs/InP and InAsSb/InP QDs based lasers have realized lasing at 1.516 µm and 2.05µm, respectively [5]. InAs/GaAs QD mid-infrared photodetectors operated at high temperature were also reported by A.D. Stiff et al. [6]. Quantum computing based on the orbital state of electronic motion in QDs has been theoretically investigated [7]. To develop a high performance optical communication system, integration of different functional optoelectronic devices, e.g. semiconductor lasers, high speed modulators, amplifiers, low-loss waveguides, photodetectors, etc., onto same chip is very important. QD based optoelectronics devices have the advantages in densely packed device arrays and monolithic integration of different devices with low-power electronics onto the same substrate [8-11]. This makes it urgent needed to be capable of controlling the tailoring of material energy bandgap for QD structures. Post-growth bandgap tuning based on the material intermixing offers simplicity and flexibility as compared to the epitaxial regrowth technique [12] and the selective area epitaxy technique [13]. Quantum dot intermixing (QDI), which can enlarge the energy bandgap of the QDs material by the QDs/barrier material interdiffusion, has attracted a lot of researches for postgrowth tuning the energy bandgap of the QD structures [14,15,16,17,18,19]. The reported QDI techniques so far include in-situ annealing during QDs growth [14], post-growth rapid thermal annealing (RTA) of QDs [15], impurity-free vacancy disordering [16], laser-induced intermixing [17], and heavy/light ion implantation for intermixing [18, 19]. Among these techniques, ion implantation and laser-induced intermixing have sufficient lateral resolution. But, the thick cap layer of a QD laser structure limits the effectiveness of laser-induced intermixing due to the small light penetration depth [20]. On the other hand, ion-implantation intermixing can be used for much thicker layer structures by using higher energy ions. However, intermixing effect from heavy ion implantation is limited due to the low-density point defects formed [20]; while for light ion implantation, implantation energy of ∼ 100 keV is required to obtain sufficient intermixing effect [19, 20 ,21]. Plasma-induced QDI is attractive since i) the low-energy (with hundreds eV) ions generated in a plasma chamber minimize the degradation of QD optical quality with comparable bandgap tuning; ii) short processing time (typically from several minutes to 10 minutes); iii) relatively simple and low-cost plasma etch facilities; iv) temperature independent of plasma process. Initial plasma-induced intermixing in quantum well (QW) structure was done using H2-plasma generated by reactive ion etcher (RIE) [22], but this technique was limited in the intermixing. Intermixing in QW structures induced by Ar plasma generated in an inductively coupled plasma (ICP) RIE system has shown large energy bandgap tuning capacity [23,24]. In this chapter, post-growth intermixing of InAs/InGaAs/InP quantum dots has been investigated in three aspects. The energy bandgap property of InAs/InGaAs/InP QD structures
Post-growth Energy Bandgap Tuning…
625
after the intermixing induced by the thermal annealing within the whole temperature range is studied firstly. Then, Argon (Ar) plasma exposure enhanced intermixing in the QD structures was investigated. Based on the Ar-plasma exposure and thermal annealing technologies, the selective intermixing of QD structures across one wafer was also developed by adding one simple SiO2 mask deposition process.
Thermal Annealing Induced QD Intermixing The interdiffusion in InAs/InGaAs QD structures under the proper thermal annealing occurs between the group III Indium and Gallium atoms due to the presence of the concentration/strain gradient across the interface, as shown in Figure 1. After the material interdiffusion between InAs QDs and InGaAs barriers in InAs/InGaAs/InP QD structures, the decreased/increased Indium/Gallium content in InAs dots increases the transition energy in the QDs layer, thus blueshifting the dots’ energy bandgap as the dot-line curve in Figure 1. The interdiffusion thus the energy bandgap shift extent is related to the lattice mismatch strain between InAs(QDs) and InGaAs(barrier), composition of InGaAs barrier and the annealing temperature, etc. In this section, the effects of annealing temperature in the whole range of 400 oC - 800 oC on the PL emission properties of InAs/InGaAs/InP QDs with different dot morphologies and barrier structures are investigated in detail. InGaAs
InAs
InGaAs
Ga In
After intermixing
Conduction-band profile
Before intermixing z
Figure 1. Schematic diagram for the intermixing upon thermal annealing.
Experiment In this study, two groups of InAs/InGaAs/InP QD structures are used. The detailed structurelayers of the dots are compared in Table 1. One group of QDs, samples (a), (b) and (c), differs only in the dot morphology, including dot size, dot separation induced by dot density; the other group, samples (c), (c1), (c2), (c3) and(c4), varies only in the composition of dots’ topbarrier layer. For all the QD samples, the InAs QDs are grown with two-step mode on the compressively strained graded In0.53-0.72Ga0.47-0.28As/InP matrix by MOCVD. The MO-sources
626
Tang Xiaohong and Yin Zongyou
used in the growths are trimethylindium (TMIn), trimethylgallium (TMGa) and tertiarybutylarsine (TBA) and tertiarybutylphosphine (TBP). Purified N2 is used as the carrier gas. After growing a 0.2μm InP buffer on InP (100) substrate, around 30 nm graded In0.53o 0.72Ga0.47-0.28As matrix layer was grown at 600 C under 100 mbar reactor pressure before starting the growth of InAs QDs. The reactor pressure was changed to 20 mbar for growing the InAs dot layer. The detailed two-step growth conditions for the dots growth are described in the Ref [25]. Table 1. The layer structures of InAs/InGaAs/InP QDs.
Reflector Heating lamp InP proximity cap P overpressure
Quartz Window
QD sample
Susceptor Quartz pin Gas inlet (N2)
Thermocouple
Figure 2. A schematic diagram of the RTP machine with the capping configuration for QDI process.
In the thermal annealing process, all the QD samples were annealed in the Jetstar Rapid Thermal Processor (RTP) in N2 atmosphere to prevent contamination. A schematic diagram of a typical RTP system using tungsten halogen lamps is shown in Figure 2. The RTP consists of four major components: energy source, process chamber, temperature measurement apparatus and temperature control part. The thermal process is carried out in the single-wafer reactor of the RTP system. The process chamber is usually made of quartz, silicon carbide or stainless steel, and has quartz windows for the optical radiation to illuminate the wafer. A measurement system is placed in a control loop to set wafer temperature. The wafer temperature in an RTP system can be measured with a noncontact optical pyrometer or a thermocouple. The temperature was controlled by a thermocouple. One fresh piece of InP proximity cap was used to provide P over-pressure environment and prevent the sample surface from P outdiffusion during the annealing process. The annealing temperature below
Post-growth Energy Bandgap Tuning…
627
850 oC is chosen to minimize thermal induced damage on the dots of QD samples. QD morphology is characterized by using an atomic force microscope (AFM) and transmission electron microscopy (TEM). Photoluminescence (PL) spectra for the as-grown/annealed dots were measured at 77 K. In the PL measurement, a 488 nm Argon laser was used as the exciting source and a cooled PbS photodetector was used to detect the PL signals. A lock-in amplifier was employed to amplify the PL signal collected before sending it to a computer for processing.
Results and Discussion QD Morphology Effect In Figure 3, the typical AFM and TEM images of the uncapped and capped dots, respectively, for the three QD samples with different dot morphologies in terms of dot size, dot density and dot separation are shown. Dot density based on AFM images for the samples (a), (b) and (c) is 0.2×1010 cm-2, 1.2×1010 cm-2 and 2.5×1010 cm-2, respectively. Dot average size based on TEM images for the samples (a), (b) and (c) is 35nm@9nm (Diameter@heigt), 33nm@8nm, and 21nm@5nm, respectively. The calculated size dispersion in terms of dot size standard deviation [26] for the three samples is smaller than 2. As shown in the PL spectra of as-grown samples (a), (b) and (c) in Figure 4-(A), when the dot density decreases, such as from (c) to (a), the low-energy state filling effect [27, 28] results in the higher energy PL emission. Accordingly, for samples (a) and (b), there are three and two emission peaks after multiple Gaussian fit. The PL peak position, linewidth and peak intensity for the three samples are summarized in Figure 5. As observed, the energy-level separation between the ground and the 1st excited energy levels is 20 meV for sample (b). For sample (a), the separation between ground and 1st excited energy levels is 17 meV. The separation between the splitting energy levels is mainly determined by the dot lateral size [29, 30]. The reduced emission energy separation from the sample of (b) to (a) is due to the increased dot size.
(I) AFM
(II) TEM 20 nm
(a)
(b)
(c)
Figure 3. (I) Top-view 1×1µm2 AFM images and (II) typical cross-section TEM images for samples (a), (b) and (c), respectively.
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Tang Xiaohong and Yin Zongyou
PL intensity
PL intensity (a.u.)
(a)
(a)
Temperature
(b)
(c)
30K 60K 80K 100K 0.50 0.55 0.60 0.65
(B)
0.50 0.55 0.60 0.65
0.50 0.55 0.60 0.65
(b)
(a)
(c) 0.50
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Figure 4. (A) 77-K PL spectra and (B) Temperature PL spectra for the as-grown QD samples (a), (b) and (c); (C) PL spectra for the three samples upon thermal annealing at 400, 450, 500, 550, 600, 650, 700, 750 and 800 oC, respectively, for 60 seconds.
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Besides, as shown in temperature-PL spectra in Figure 4-(B), when the PL measurement temperature increases, the multi-transitions from low and high energy levels gradually become illegible. This is due to the high temperature induced thermal effect on the electron/hole carriers. After thermal annealing on the three samples in the temperature of 400 ∼ 800 oC, the PL peak redshifts before the normal material-interdiffusion induced blueshift, and correspondingly, the PL intensity decreases in the redshift region while recovers to increase in the blueshift region, as shown in Figure 5-(C). Too high annealing temperature, > 750 oC, starts to damage the QDs, thus lowering down the PL intensity again. It is noted that the PL peak redshift is the same (8 meV) for the ground energy level of the three samples, but increased for the higher transition energy levels in sample (b) and (a). The observed PL linewidth of the three samples goes with broadening in the whole annealing temperature range. The literatures also reported the redshift at relatively low annealing temperature in the GaAs/InGaP multiple QW [31] and InGaAsP/InP single QW structures [32]. However, the redshift was attributed to the dominance of group-III sublattice interdiffusion at low annealing temperature. This cannot explain our observed results since the InAs QDs are sandwiched between InxGa1-xAs barrier layers. No matter what extent intermixing between the group-III species, the InAs QDs will blueshift since the Indium atom of InAs QDs will be replaced by the Gallium atom from InGaAs barriers after the material interdiffusion. So, we attribute the dots’ bandgap redshift upon annealed at relatively low temperature to the dots’ size microincrement after the In/As atom self-diffusion in the QDs&top-barrier interface. In selfassembling the QDs, the grown-in defects exist in the dot/barrier interface region due to the large lattice mismatch between the dots and their barriers [33]. For InGaAs/InAs/InGaAs/InP QD structures studied here, the strain-induced grown-in defects are mainly In or As atom vacancies [34, 35]. Vacancy defects migration plays different roles in the thermal annealing process at different annealing temperatures. At low annealing temperature, the thermal energy is low. The material diffusion is dominated by the atom’s activation energy. When the thermal energy increases under high annealing temperature, the material diffusion relies on the composition/strain-gradient [36]. Under low-temperature annealing, the diffusion of each type of atom species (group-V or -III) is self-diffused through its own sublattice; and the activation energy for atom diffusion follows the sequence of In < Ga < As atoms [37]. Some In, Ga and As atoms from top-barrier InGaAs will diffuse into the vacancy positions in the dot&barrier interface region. However, in group III atoms, Ga atom has higher activation energy than In atom, so the In vacancies will be filled by In atoms under the self-diffusion limitation. Correspondingly, the As vacancies will be filled by As atoms. As a result, the effective height of dots will increase, thus redshifting the PL emission. After calculation by using 8 kp theory model [38] for the studied QDs emitting around 2.1 μm (∼0.58 eV), 8 meV transition energy reduction could be generated after only 0.5 nanometer, i.e. a couple of monolayers, increase in dots’ height. With the annealing temperature increase, the high thermal energy weakens atoms’ activation energy difference in the annealing. As a result, composition/strain-gradient induced In-Ga atoms’ interdiffusion dominates the annealing process, thus producing the blueshift. The dots’ blueshift after annealing is dependant on the dot size and dot separation. As reported previously, the main intermixing in QD structures comes from dots’ vertical direction due to the small aspect ratio of the dots [39, 20]. Sample (a), (b) and (c) has the dot
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height of 9 nm, 8 nm and 5 nm, respectively, but the observed average PL peak blueshift of (a), (b) and (c) after annealed at 700 ∼ 750 oC is 25 meV, 41 meV and 13 meV, respectively. Sample (b) with slightly lower dot height blueshift in PL peak further compared with sample (a), which indicates the intermixing is not just determined by the dot height. This is attributed to the dot density induced different dot-dot separation distance. The relative size of unit dot cell, i.e. the average dot-dot distance divided by the QD diameter, is calculated to be 5.2, 2.6 and 2.5, respectively, for sample (a), (b) and (c), respectively. Compared with (b), the dots in sample (a) have much wider space separation. The confinement material of InGaAs next to (not above or below) the QD is compressively strained perpendicular to the growth direction due to the larger lattice constant of InAs dot material [40]. When the dot separation reduces by increasing the dot density, the strain between confinement-layer and dots increases, resulting in an enhanced driving force for material intermixing upon thermal annealing. This will increase the net blueshift of the QDs upon annealing. As a result, the total blueshift of (b) is larger than that of (a). It’s noted that the annealing temperature triggering the PL peak shift from redshift to blueshift, referred as threshold temperature (TRS), is 650 oC, 600 oC and 670 o C, respectively, for sample (a), (b) and (c). The blueshift extent of the dots is proportional to 1/TRS under the same other annealing conditions. PL linewidth, full-width of half maximum (FWHM), of the ground energy level transition for the as-grown samples (a), (b) and (c) is as narrow as around 20 meV. The abnormal linewidth broadening upon annealing will be discussed below. Besides, the noticeable phenomenon observed is that for the very low density QDs of sample (a), the linewidth broadens much faster with the annealing temperature than the higher density QDs of (b) and (c). This further confirms the dependence of dots’ blueshift on the dot separation upon annealing. For the dots with very wide dot separation, their blueshift reduces compared with dots with narrow separation as discussed above. This results in different blueshift rates for the dots distributed with different separations, thus different PL linewidth broadening rate. The dot separation dispersion can be calculated based on the AFM images in Figure 3. For sample (c), the shortest and longest dot-dot separation is around 33 nm and 73 nm, that for (b) is 58 nm and 110 nm; and for (a) is 73 nm and 294 nm. The separation dispersion is then calculated by dividing the maximum separation difference with dot’s average diameter. The received separation dispersion is 6.3, 1.6 and 1.9, respectively, for samples (a), (b) and (c). The dot-dot separation dispersion of (a) is 2.3 times broader than that of samples (b) and (c). So, sample (a)’s PL peak blueshift rates differ mostly and then its PL FWHM goes broader faster than samples (b) and (c) as shown in Figure 5-(B). For all the emissions at different energy levels for the three samples, the PL intensity becomes low in redshift region and increases in the blueshift region. This is because the vacancy filling is not a complete material intermixing process, which does not reduce the total vacancy quantity and may even produce new vacancy-interstitial pairs due to un-complete migration at low temperature. So, the total nonradiative centers were increased which degraded the PL emission intensity. After increasing annealing temperature, the normal material interdiffusion reduces all kinds of nonradiative recombination centers in the QD structure, thus increasing the PL intensity.
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Top Barrier-Layer Effect To study the effect of top barrier-layer’s composition on the dots’ PL properties upon thermal annealing, three more samples (c1), (c2) and (c3) were prepared based on sample (c). The only difference between these samples lies in the cap layer compositions: In0.72-0.53Ga0.280.47As for (c), In0.53Ga0.47As for (c1), In0.33-0.53Ga0.63-0.47As for (c2), and GaAs/In0.53Ga0.47As for (c3), respectively. The PL curves for the four samples are plotted in Figure 6-(A). The PL peak positions and PL linewidth as the function of the annealing temperature are summarized in Figure 6-(B). The PL peak maximum redshift as to the molar fraction of ‘InAs’ content in InGaAs top barrier is plotted in Figure 7-(A). In Figure 7-(B), the ratio of FWHM of the annealed dots vs that of as-grown dots, the ratio of the maximum PL integrated-intensity (IInt) of the annealed QDs (700 ∼ 750 oC) vs that of the as-grown ones, and the ratio of the maximum PL peak-intensity (PInt) vs that of the as-grown ones are all plotted as the function of the absolute FWHM of as-grown QDs. Threshold annealing temperature TRS, calculated diffusion length (Ld), composition-gradient and composition@strain-gradient between InAs QDs and InGaAs top barrier is plotted in Figure 8 as the function of dots’ average blueshift in 700 - 750 oC. Based on Figure 6-(A), the QD’s bandgap energy (PL peak position) can be effectively adjusted through controlling the composition of the top-barrier layer of the dots. The higher the Gallium content in InGaAs material, the higher bandgap of the InGaAs, such as from (c) to (c3), which ultimately increases the bandgap of the QDs. Based on the annealed dots’ PL results as shown in Figure 6-(B), the PL redshift extent relation to the InGaAs composition is
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Figure 7. (A) PL peak maximum redshift as the function of ‘InAs’ content fraction in InGaAs topbarrier layer; (B) Ratios of FWHMAnnealed/FWHMAs-grown, (IIntAnnealed -IIntAs-grown)/IIntAs-grown and (PIntAnnealed-PIntAs-grown)/PIntAs-grown are all plotted as the function of the absolute FWHM of as-grown QDs.
summarized in Figure 7-(A). It is obvious the absolute redshift is proportional to the molar fraction of ‘InAs’ content in the InGaAs top barrier. Especially for sample (c3) with capping a thin GaAs (∼10nm) on InAs dots before growing the top In0.53Ga0.47As layer. There is no redshift in the whole annealing temperatures. Also, we checked the dots capped with only InP, there is still no redshift (not shown here). So, the redshift is only determined by the ‘InAs’ content of top-barrier layer. Since there is only Arsenic content in GaAs and Indium InP, the self-diffusion limited vacancy filling cannot contribute to the InAs size increase for the dots directly capped by GaAs or InP material. Besides, seen from Figure 7-(B), PL linewidth goes broad (narrow) when the FWHM of the as-grown QDs is small (large). The FWHM broadening upon annealing is opposite to that generally observed in the traditional PL linewidth narrowing [41, 42], where the dots are with broad size dispersion so with broad PL linewidth. The PL linewidth broadening was also observed in Ref. [28], where it is attributed to the highly uniform dots formed. To study the real reason behind, we compared the change between PL IInt, PInt and FWHM of the annealed QDs with regard to that of the as-grown ones. PL IInt indicates the total number of rediative emissive dots, PL PInt is proportional to the most dot number with the same size@strain, and PL linewidth reflects the dot size@strain distribution. As shown in Figure 7-(B), in the FWHM narrowing region for samples (c1) and (c3), the ratio of PL IInt of annealed QDs vs that of the as-grown dots is smaller than the ratio in PL PInt case. This shows the dots size@strain dispersion was narrowed after annealing and more emissive dots with the same (peak position) size/strain increase the PL PInt faster. So, for the dots initially with broad FWHM, the size@strain gradient induced intermixing dominates in the annealing, which narrow the linewidth. However, for the highly uniform dots with narrow FWHM, such as for samples (c) and (c2) with FWHM of 18.5 ∼ 20 meV, the dot size@strain gradient is low and the linewidth goes broad upon annealing. Based on Figure 7-(B), the ratio of PL IInt of the annealed QDs with regard to that of the as-grown dots is larger than the ratio in PL PInt change for samples (c) and (c2). It is reasonable to deduce
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that the annealing makes some initial nonradiative dots emit radiatively at non-peak positions and the size@strain of these dots are divergent from that of the dots emitting at the peak position. The PL linewidth sharpness is mainly determined by the QD height dispersion [43]. Narrow linewidth of sample (c)/(c2) indicates their QD height is in high uniformity. However, the dot diameter is not uniform as shown in the AFM/TEM images in Figure 3-(c). Different size induces different strain relaxation in the dots, thus different lattice mismatch between dots and barrier layer. So, the grown-in defects in the dots will be not uniform distributed thus with a concentration gradient between the dots. Such defect concentration gradient will act as the main driving force in the material interdiffusion for the dots with uniform height but non-uniform diameter upon thermal annealing. The dots with more grownin defects go with the higher degree intermixing under the same annealing conditions. This will result in different PL peak blueshift rates, thus broadening the PL FWHM. The dispersing of dot’s initial uniform height upon annealing was further confirmed by the lowering of PL PInt of annealed dots compared with that of as-grown dots in sample (c). So, the traditional thermal annealing for improving the PL performance does not always work for the QD structures, which is dependant on the initial PL properties of the as-grown dots.
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Diffusion length (Ld) is used to characterize the material interdiffusion after annealing. The Ld is calculated by using the approximated sech(x) function as the following [44]:
L ΔE e 0→hh 0 = 1 − sec h( β d ) , E e 0→hh 0 R
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where β is the blueshift rate coefficient which is dependant on the dot size. R represents the size of the QD, which is the radius of a spherical QD of the same volume, given as
R = 3 4VQD /3π , τ is the annealing time. The relation of composition-gradient (δC), (δC)x, and δC×(Strain)x with considering dots&top-barrier lattice mismatch to dots’ average blueshift (δW) will be compared with the Ld ∼ δW curve. The power index, x, in (δC)x and δC×(Strain)x will be determined by linearly fitting to the Ld curve. δC is defined as the difference in InAs molar fraction of dots&top-barrier. As shown in Figure 8, the Ld does not vary linearly with δC, but vary linearly well with (δC)1/3 or δC×(Strain)1/2. This shows, for the QD structures with the only difference in top barrier, the annealing-induced material intermixing can be well described by using the δC parameter only without including the parameter of strain. This is reasonable because the dots&top-barrier strain is fully determined by the composition of dots’ top barrier since all the dots are the same, which are grown on the same InGaAs/InP matrix. Besides, the threshold annealing temperature TRS is further demonstrated to be inversely proportional to dots’ blueshift as shown in Figure 8. A blueshift rate of -9.3×10-2 meV/oC versus the TRS is figured out for the InxGa1-xAs/InAs/InGaAs/InP QD structures with different top barriers. These results are useful in directing the design of QDs’ bandgap tuning by post-growth QDs intermixing.
Argon Plasma Enhanced QD Intermixing When the QD sample is exposed to ICP Ar plasma for a certain duration of time, the point defects will be generated in the surface layers of the sample. These surface point defects diffuse down to the QD active region of the sample during the high temperature thermal annealing process, and subsequently affect the material interdiffusion between the dots and dots’ barriers. This will produce the effect on the energy bandgap tailoring of the QD structure.
Experiment The plasma system used in this work is Plasmalab System100 machine from Oxford Instruments Plasma Technology as shown in Figure 9. Plasma is an ionized gas with equal numbers of free negative and positive charges. The positive charge is mostly of singly ionized neutrals from which single electron has been stripped. The majority of negatively charged particles are usually free electrons. In this system, the plasma is created by electromagnetic induction when an alternating high frequency current in the radio frequency (RF) inductive coil causes a circulating current to flow in a low-pressure plasma. The plasma can be initiated by the induction coil or by the power applied to another system electrode, usually RF power to the substrate electrode. The 13.56 MHz radio frequency (RF) and ICP power supply can provide independent control on ion bombardment energy and ion current density with the operation power up to 500 W and 3000 W, respectively. To create a high current in the RF coil in this nonresonant design without high reflected power, an RF tuning network with automatic matching is mounted close to the coil. The ICP power is supplied via a coaxial Ntype 50 Ω connector on the matching unit.
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Quartz/alumina tube
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The matching unit contains motor-driven vacuum capacitors and a directional detection unit, which acts to make the load impedance presented to the RF generator as close as possible to 50 Ω. During plasma exposure, the chamber base pressure is maintained at 5 ⋅ 10-5 Torr. The ICP chamber is equipped with a water circulator to maintain the chuck (cathode) temperature at 60 oC. However, it should be noted that the actual sample surface temperature during plasma exposure is higher than the reading from the table temperature since the ions produced by plasma is very hot. Based on the ion energy (30 eV ∼ 1 keV) in plasma chamber, an average energy of 500 eV is used to approximatively calculate the plasma temperature. The calculated plasma temperature is around 5.8 million of Kelvin. Such hot plasma induced thermal effect during the ICP process also affects the diffusion process in the QD structures. The ICP system is equipped with a back side He-cooled electrode with quartz as a wafer susceptor. The QD samples were placed on the silicon substrate to provide a fairly large (168 W/mK) heat conduction. Ar ion current density increases nearly linearly with the ICP power increase by fixing the RF power. The working pressure affect slightly the ion current density generated in chamber. High ion density in the order of 1016 m-3 can be generated in such ICP chamber. In the experiment of this work, the ICP and RF power was set at 500W and 480W, respectively, and the Ar gas flow rate was set at 100 sccm with the process pressure set at 60 mTorr. In order to investigate Ar-plasma exposure effects on QD intermixing, InP/In0.53Ga0.47As/InAs/In0.53Ga0.47As/InP QD sample was prepared with growing a thick, ∼ 700 nm, InP top layer for Ar-plasma exposure. The AFM image of the uncapped QDs of the sample is shown in Figure 10. The standard deviation of dot’s diameter/height is around 1 nm, so the dots formed are with good size uniformity. Dot density of the samples are around 1.0×1010 cm-2. The point defects, which affect QD intermixing upon annealing, will be generated in the surface InP layer in the QD samples after plasma exposure. Effects of the
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intermixing under different Ar-plasma exposure and annealing conditions were evaluated by studying the changes in the samples’ PL peak position, intensity and linewidth before and after the ICP treatment or followed by annealing. Argon plasma exposure
InP cap In0.53Ga0.47As InAs QD In0.53Ga0.47As InP Figure 10. The schematic QD structure with the corresponding top-view 1×1 μm2 AFM image.
Results and Discussion Plasma-Exposure Effect In order to study the plasma effect on dots’ PL properties, the QD sample was exposed to ICP Ar plasma with different time but without the RTA treatment. A reference InP/In0.53Ga0.47As/InAs/In0.53Ga0.47As/InP QD sample was deposited with 200 nm SiO2 cover layer before the 90s-ICP plasma exposure, so that the sample’s epilayer would not be exposed to the plasma during the ICP process. All low-temperature PL spectra were shown in Figure 11, and the PL property change trend as the function of exposure time is summarized in Figure 12. It is observed that after the ICP Ar-plasma exposure, PL peak of the reference sample (Dot line curve in Figure 11) is the same as that of the as-grown sample. This shows the 200 nm SiO2 mask can effectively prevent the epilayer from the plasma bombardment. The interesting phenomenon observed is that, for samples without the SiO2 mask layer, the PL spectrum exhibits a blueshift right after the Ar-plasma exposure. This is different from that of QW structure, where the PL emission blueshift occurs only after subsequent thermal annealing. For the sample studied, the blueshift of the QDs’ PL emission increases with the Ar plasma exposure time up to 90 s and then saturates. At the same time, the PL intensity of the QDs increases with the exposure time up to 90 s, and it decreases when exposed to Ar plasma longer. This is attributed to the material sputtering of the samples during the Ar plasma exposure. Based on the top-view AFM images in Figure 13, the sample surfaces become rough after exposure to ICP plasma, a result of surface sputtering during the ICP process. The sputtering rate can be deduced from the surface roughness and the Z-range measured by AFM after different ICP exposure times. The surface RMS (Root-mean-square) roughness and the Z-range versus the exposure time are also shown in Figure 13-(B). Z-range of the sample increases to a maximum of 535 nm with the ICP exposure time increased to 90 s and then decreases when further exposed to the plasma. The decrease in the sputtering with
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longer plasma exposure is because the plasma sputtering tends to remove the surface hills faster, thus reducing the RMS and Z range of the surface.
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The results show that PL emission of the QD sample blueshifts after sample’s exposure to plasma. From our previous study, we have measured the sputtering rate of InP material with the same experimental conditions. It is about 100 nm/min, which is significantly slower for InGaAs or InGaAsP materials. The surface roughness measured by AFM in Figure 13 indicates the roughening rate (proportional to the sputtering rate) of sample’s InP cap layer is around 300-400 nm/min which is higher than the sputtering rate received before. This is because the top InP layer of the studied QD sample was grown at the relatively low temperature. Therefore, the cap layer was removed during ICP process at a higher etching rate. The removal of the cap layer explains the increase of the sample’s PL intensity. At the same time, because of the removal of the cap layer, the plasma-generated defects at the surface are close to the QDs. After sufficient plasma exposure time, some of the defects can diffuse into the QDs, thereby inducing the QD/barrier material intermixing under the high cation concentration gradient between the QD material (InAs) and its surrounding material (InGaAs). These effects contribute to a blueshift of the QDs’ PL emission after ICP Arplasma exposure. After long time exposure to plasma, the whole InP cap layer is removed.
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The etch rate of InGaAs barrier layer of the sputtering is slower. At the same time, the number of defects generated is saturated [24]. Therefore, the sample’s PL blueshift saturates after 120 s plasma exposure. The reduction of the sample’s PL intensity after long time plasma exposure shows that the thin upper InGaAs barrier layer is insufficient to confine the carriers in the QD, and at the same time, some of the QDs may even be removed by the sputtering.
Large Bandgap Tuning To study the plasma-exposure effect on the bandgap shift extent upon thermal annealing, we annealed InP/In0.53Ga0.47As/InAs/In0.53Ga0.47As/InP QD sample at 580 - 620 oC for 60 s after their exposure to the plasma with different time. The 77-K PL curves for the samples annealed at 620 oC are shown in Figure 14. Changes in the sample’s (i) PL peak shift, (ii) FWHM of the PL spectra, and (iii) PL peak emission intensity, as a function of the plasma exposure time for the three groups of samples annealed at different temperatures are plotted in Figure 15. Thermal annealing effects of the as-grown samples are also plotted in Figure 15 with plasma exposure time of zero second. As the reference, the PL peak shift, PL linewidth and peak intensity as the function of exposure time of the samples after their only plasma exposure based on the PL curves in Figure 11 are also shown in Figure 15. It shows that with plasma exposure of less than 90 s, the sample’s PL blueshift and emission intensity increase further after RTA treatment while the FWHM of their PL spectra reduces. The longer the plasma exposure time, the greater these changes received. When the plasma exposure time exceeds 90 s, the samples’ emission blueshift saturates and PL intensity drops dramatically, while their linewidth becomes broader. For the as-grown samples, i.e. without plasma exposure, RTA treatment makes their PL blueshifts and emission intensities changes in small range. 4
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Exposure time (s) Figure 15. (i) The PL peak wavelength, (ii) the FWHM linewidth, and (iii) the relative PL peak intensity of InAs(QD)/In0.53Ga0.47As/InP QD samples after their ICP plasma exposure with different time followed by RTA for 60 seconds at different temperatures are plotted as a function of the Arplasma exposure time. Data points for the as-grown samples after only RTA treatment are circled for notice.
The RTA-treatment enhanced PL blueshift of the samples after Ar-plasma exposure less than 90 s is due to the larger plasma-generated defect density in the interdiffusion between the QDs and their barrier layer during the RTA annealing. The increase in the samples’ PL intensity and the narrowing of their PL linewidth is because (i) thermal annealing increases
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the defects’ mobility and homogenizes their distribution in the QD structure, thereby homogenizing the intermixing between the QDs and their barrier materials; (ii) thermalannealing induced QD/barrier interdiffusion enhances the mobility of the atoms in QD structure, hence averaging the local compositional/strain fluctuation in the dots [45]. However, if the Ar plasma exposure time is too long, over 120 s, the plasma sputtering partially removes the InGaAs upper barrier layer, which reduces the uniformity of InGaAs upper barrier. This causes the non-uniform emission from the QDs and broadens the emission linewidth. The reduced barrier confinement from upper InGaAs barrier layer would also weaken the PL emission intensity. Similar degradation of the PL emission after too long plasma exposure has also been observed in bulk material [46] and QW structure [24]. To investigate the high temperature annealing effect on the QD’s emission shift range, we annealed the as-grown InP/In0.53Ga0.47As/InAs/In0.53Ga0.47As/InP QD sample and the sample after ICP Ar-plasma exposure for 90 s together at 720 oC for 60 s. To study the maximum bandgap tunability of this technology, we also annealed the sample at 780 oC after ICP Arplasma exposure for 90 s. Figure 16 shows the PL curves of these samples. Very large PL emission blueshift of 394 nm when compared with that of the as-grown QDs is obtained from the QD sample under RTA annealed at 780 oC after 90 s ICP
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Ar-plasma exposure. The corresponding energy bandgap increase of the QD sample after ICP+RTA treatment in this study is ΔEg = 159 meV. Such bandgap tuning range based on low energy Ar-plasma exposure is much larger than the reported maximum bandgap tuning range of 126 meV [47] and 120 meV [19], respectively, based on using high energy P-ion and H-ion implantation both for the In∼0.55Ga∼0.45As/GaAs QD structures. This comparison on QD’s bandgap tuning capacity between the different intermixing technologies is sound since the reported In∼0.55Ga∼0.45As/GaAs QD structures have a comparable group-III atom
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concentration gradient, which mainly determines the intermixing degree in terms of QD structures, with the InAs/In0.53Ga0.47As QD structure studied in this work. In addition, it is observed in this study that the PL peak blueshift after Ar-plasma exposure followed by thermal annealing at 720 oC is 330 nm larger than that of the sample only going through the thermal annealing. This further confirms the Ar-plasma-exposure generated mobile point defects in the sample play a great role in further widening the QD’s energy bandgap during the thermal process afterwards. Compared with that of the as-grown sample, the PL intensity and linewidth of the QD sample after 90s-ICP followed by RTA at 720 oC ∼ 780 oC increases by ∼ 2.5 times and narrows by ∼ 46%, respectively.
Selective Intermixing A post-growth multi-bandgap tuning technique for QD structures is important for monolithic photonic integration of passive and active QD based multifunctional optoelectronic devices. Different band engineering technologies of spatial selective intermixing for the QD structures across a wafer, including titanium-dioxide (TiO2) interdiffusion-suppression [16], laser radiation [17], neutral ion-implantation [47] and silica-sputtering [48], etc, have been investigated. In this part, based on the Ar-plasma enhanced intermixing in tuning the QDs’ energy bandgap, spatial selectivity of the intermixing by employing a SiO2 mask on the wafers during the intermixing has been investigated in detail for the InAs/InGaAs/InP QD structures.
Experiment The InP/In0.53Ga0.47As/InAs/In0.53Ga0.47As/InP QD structures top-capped with a 700 nm InP cap-layer were still used for this study. The intermixing of the QD samples was carried out by exposing them to Ar plasma and followed by thermal annealing. To investigate the spatial selectivity of the intermixing, portion of the sample was deposited with a 200-nm SiO2 mask layer by plasma enhanced chemical vapor deposition (PECVD). Then, the samples were exposed to ICP Ar plasma in an ICP chamber. The ICP/RF power, Ar gas flow rate inside the chamber and the process pressure settings in this experiment are the same as before. After the Ar plasma exposure, the samples were annealed in a RTA chamber with atmospheric pressure of nitrogen ambient. Samples were covered with InP wafers to prevent the surface Argon plasma exposure PECVD Thermal annealing SiO2 mask
Sample
Sample
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Figure 17. Process flow in the selective intermixing study for InAs/InGaAs/InP QD structure.
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evaporation during the RTA process as mentioned previously. The whole process line is schematically shown in Figure 17. The energy bandgap tuning of the QD samples after the intermixing was still investigated by using the PL measurement.
Results and Discussion Plasma Exposure Dependence First, we studied the selective intermixing by exposing the QD samples to plasma with different time and followed by the same RTA annealing at 620 oC for 60 s. Two groups of samples are prepared before the plasma-exposure and the following annealing steps as shown in Figure 17: samples (i) not masked by the SiO2 layer during the plasma exposure and (ii) covered with the SiO2-mask layer. The 77-K PL spectra of group (i) samples have been shown in Figure 14, and that of group (ii) are shown in Figure 18. The (i)/(ii)-samples’ peak emission blue shift and linewidth as a function of the Ar plasma exposure time during the intermixing are summarized in Figure 19-(a) and Figure 19-(b), respectively. Large blue shift of the peak emission has been observed from the samples without the SiO2 mask after the intermixing as discussed previously. However, a small blue shift of the SiO2 masked sample is received after the intermixing and the blue shift does not change much with the plasma exposure time. The blue shift of the sample after the Ar plasma exposure is the same as that of the sample only did the RTA annealing without the Ar plasma exposure. It shows that the 200-nm SiO2 mask layer effectively obstructs the Ar plasma from exposing to the epilayer surface of the sample. The blue shift of the SiO2 masked sample after the intermixing is only because of the RTA annealing.
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As shown in Figure 19-(a), the difference in the bandgap blue shift (or spatial selectivity of the intermixing) between the un-masked and SiO2 masked region of sample can be continuously tuned from 0 to 77 meV by increasing the plasma-exposure time from 0 to 90 s and RTA annealing at 620 oC.
Annealing Temperature Dependence To study the temperature dependence of the selective intermixing, samples (SiO2 masked and un-masked) were annealed at different temperatures ranging from 580 ∼ 780 oC after exposed to the ICP Ar plasma for 90 s. Figure 20 shows the PL spectra of (iii) un-masked samples, (iv) SiO2 masked samples annealed at different temperature in the intermixing. For reference, we removed one sample’s (sample (v)) SiO2 mask after the ICP plasma exposure and RTA annealed it together with sample (iii) and (iv). Figure 20-(v) shows the PL spectra of the sample (v) annealed at different temperature. Figure 21 shows the blue shifts of the PL emission peaks and the linewidths of the three samples after going through the intermixing with different annealing temperatures. The results show that for sample (iii), large blue shifts have been received after the intermixing, while its PL peak intensity increases and the linewidth decreases with the ICP Ar plasma exposure up to 90 s, which shows that the ICP plasma dose not damage the InAs QDs’ crystal quality up to 90 s plasma exposure.
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Figure 20. 77-K PL spectra of the (iii) un-masked, (iv) masked and (v) mask removed samples after the intermixing. The samples were exposed to ICP Ar plasma for 90 seconds followed by RTA for 60 seconds and annealed at different temperatures: A: 580 oC, B: 620 oC, C: 650 oC, D: 680 oC, E: 720 oC, F: 750 oC and G: 780 oC.
It is noticed that for the un-masked sample, the sample’s PL peak blue shift increases with the annealing temperature during the intermixing almost linearly. However, for the SiO2 masked sample (iv), the PL peak blue shift increases slowly with the RTA annealing temperature in the intermixing when the annealing temperature is below 650 oC. When the annealing temperature increases from 650 oC to 750 oC, the blue shift of the SiO2 masked sample increases much faster after the intermixing. At low RTA annealing temperature, < 650 o C, the difference of the blueshift between the SiO2 masked and unmasked samples after the intermixing is very large, ~200 nm. When the RTA annealing temperature is above 650 oC, the difference of the blue shift between the masked and un-masked samples after the intermixing becomes smaller when annealed at higher temperature. The blue shift difference between the samples is only 10 nm when annealed at 780 oC during the intermixing. These show that the selectivity of intermixing between the SiO2 masked and un-masked samples is dependant on the annealing temperature. It shows that the lower annealing temperature, the wider intermixing selectivity, which is good for device fabrications. Too high annealing temperature, > 800 oC, will degrades the 3-dimentional carriers’ confinement of the QD structures and even destroy the highly-strained QDs [48, 49].
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Figure 21. (A) PL peak blue shifts and (B) ratio of PL spectra FWHM of the (iii) un-masked sample, (iv) masked sample and (v) mask removed sample as a function of the RTA annealing temperature. All the samples were exposed to the ICP Ar plasma for 90 seconds for the intermixing.
As shown in Figure 21-(A), for the reference sample without the SiO2 mask layer on top during the RTA annealing, its PL peak blue shift after the intermixing increases almost linearly with the annealing temperature, while for sample (v) with the SiO2 on top during the RTA annealing, the blue shift increases much faster when annealed at the temperature above 650 oC. This is attributed to the strain generated between the SiO2 mask and the InP cap layer of the sample during the thermal annealing. Since the thermal expansion coefficient of InP (∼4.6×10-6 oC-1) is more than 8 times larger than that for SiO2 (∼0.52×10-6 oC-1), large strain is generated in the InP cap layer during the thermal anneal. This thermal strain leads to the generation of the group III vacancies in the sublattices of InP [50]. At the same time, the diffusion of the defects into the InAs QDs is enhanced by the strain, which promotes the QD intermixing of the sample as the annealing temperature is increased [17]. Figure 21-(A) also shows when the samples are annealed in the temperature range from 675 oC to 725 oC during the intermixing, three different bandgap tunings across a wafer can be obtained by adding the SiO2-mask removal between plasma exposure and thermal annealing steps in the intermixing. The different blue shift between un-masked (iii) and masked (iv) and mask removed (v) samples reaches as wide as around 50 nm (∼ 20 meV) at annealing temperature of 700 oC in the intermixing. The difference in the intermixing between the un-masked sample (iii) and masked sample (iv) or between the masked sample (iv) and masked removed sample (v) can be modulated by changing the annealing temperature, which achieves the multiple bandgap changes across a wafer through the intermixing. This is important in realizing multi-functional monolithic integration circuits, which generally requires different energy bandgaps across a wafer for different function devices, e.g. the emitter, modulator and detectors, etc. Besides, in the multi-emission tuning
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region, the PL linewidths for the samples (iii), (iv) and (v) are all smaller than that of their asgrown samples. This shows the uniformity of composition/strain in the dots is improved, which advantages improving the spatial resolution across sample’s adjacent regions under different intermixing treatments. Moreover, the spatial resolution in the intermixed QDs is dependent on the lateral diffusion distance/uniformity of the point defects induced by the plasma conditions (plasma energy, density and time), annealing conditions (annealing temperature and time), and mask conditions (SiO2 film thickness and uniformity, etc. This needs further systematic study in future.
Conclusion In this chapter, thermal annealing on QDs’ energy bandgap tuning was studied. It was observed that annealing at low temperatures induces the redshift which is attributed to the dots size micro-increase mainly determined by the ‘InAs’ content in InGaAs top barrier. After increase annealing temperature, composition/strain-gradient induced material interdiffusion dominates and produces the blueshift. The blueshift is dependant on dot separation distribution and top-barrier InGaAs composition. The Ar-plasma exposure on QDs’ energy bandgap tuning was investigated. It was found that the dots’ bandgap could be tuned in an obviously larger range by exposing the sample to Ar-plasma before the thermal annealing compared with by using the thermal annealing only. The enhanced QD intermixing is attributed to the point defects generated near sample’s surface region after its exposure to the plasma. By combining the plasma exposure and SiO2 mask techniques, the spatial selective intermixing has been obtained through controlling the plasma exposure time or the annealing temperature. Based on the selective intermixing, multi-wavelengths across one wafer with 50nm wavelength separation were achieved. Such selective bandgap tuning of the QD structures across one wafer paves a way for monolithic integration based on the passive and active QDs devices.
References [1] Bimberg, D.; Grundmann, M.; Ledentsov, N. N. Quantum Dot Heterostructures, Wiley, Chichester, 1999; chapter 4. [2] Grundmann, M.; Christen, J.; Ledentsov, N. N.; Bohrer, J.; Bimberg, D.; Ruvimov, S. S.; Werner, P.; Richter, U.; Gosele, U.; Heydenreich, J.; Ustinov, V. M.; Egorov, A. Y.; Zhukov, A. E.; Kop’ev, P. S.; Alferov, Zh. I. Phys. Rev. Lett. 1995, vol. 74, 4043-4046. [3] Arakawa, Y.; Sakaki, H. Appl. Phys. Lett. 1982, vol. 40, 939-941. [4] Tatebayashi, J.; Hatori, N.; Kakuma, H.; Ebe, H.; Sudo, H.; Kuramata, A.; Nakata, Y.; Sugawara, M.; Arakawa, Y. Electron. Lett. 2003, vol. 39, 1130-1131. [5] Qiu, Y. M.; Uhl, D.; Keo, S. Appl. Phys. Lett. 2004, vol. 84, 263-265. [6] Stiff, A. D.; Krishna, S.; Bhattacharya, P.; Kennerly, S. W. IEEE J. Quantum Electron. 2001, vol. 37, 1412-1419. [7] Kral, K.; Zdenek, P.; Khas, Z. IEEE Transactions Nanotechnol. 2004, vol. 3, 17-25. [8] Yariv, A. Appl. Phys. Lett. 1988, vol. 53, 1033-1035.
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[9] Chakravarty, S.; Bhattacharya, P.; Mi, Z. IEEE Photon. Tech. Lett., 2006, vol. 18, 26652667. [10] Emary, C.; Sham, L. J. J. Phys.: Condens. Matter 2007, vol. 19, 056203-1-10. [11] Ozaki, N.; Takata, Y.; Ohkouchi, S.; Sugimoto, Y.; Nakamura, Y.; Ikeda, N.; Asakawa, K. J. Cryst.Growth 2007, vol. 301, 771-775. [12] Nishikawa, T.; Kubo, M.; Sasai, Y. Appl. Phys. Lett. 1996, vol. 68, 3428-3430. [13] Tanbun-ek, T.; Sciortino, P. F.; Sergent, A. M.; Wecht, K. W.; Wisk, P.; Chen, Y. K.; Bethea, C. G.; Sputz, S. K. IEEE Photon. Technol. Lett. 1995, vol. 7, 1019-1021. [14] Fafard, S.; Wasilewski, Z. R.; Allen, C. N.; Picard, D.; Spanner, M.; McCaffrey J. P.; Piva, P. G. Phys. Rev. B, 1999, vol. 59, 15368-15373. [15] Girard, J. F.; Dion, C.; Desjardins, P.; Allen, C. N.; Poole, P. J.; Raymond, S. Appl. Phys. Lett. 2004, vol. 84, 3382-3384. [16] Fu, L.; Lever, P.; Tan, H. H.; Jagadish, C.; Reece, P.; Gal, M. Appl. Phys. Lett. 2003, vol. 82, 2613-2615. [17] Dubowski, J. J.; Allen, C. N.; Fafard, S. Appl. Phys. Lett. 2000, vol. 77, 3583-3585. [18] Surkova, T.; Patane, A.; Eaves, L.; Main, P. C.; Henini, M.; Polimeni, A.; Knights, A. P.; Jeynes, C. J. Appl. Phys. 2001, vol. 89, 6044-6047. [19] Lever, P.; Tan, H. H.; Jagadish, C.; Reece, P.; Gal, M. Appl. Phys. Lett., 2003, vol. 82, 2053-2055. [20] Ji, Y. L.; Lu, W.; Chen, G. B.; Chen, X. S.; Wang, Q. J. Appl. Phys. 2003, vol. 93, 12081211. [21] Aimez, V.; Beauvais, J.; Beerens, J.; Morris, D.; Lim, H. S.; Ooi, B. S. IEEE J. Select. Topics Quantum Electron. 2002, vol. 8, 870-879. [22] Ooi, B. S.; Bryce, A. C.; Marsh, J. H. Electron. Lett. 1995, vol. 31, 449-451. [23] Djie, H. S.; Mei, T.; Arokiaraj, J. Appl. Phys. Lett. 2003, vol. 83, 60-62. [24] Djie, H. S.; Mei, T.; Arokiaraj, J.; Sookdhis, C.; Yu, S. F.; Ang, L. K.; Tang, X. H. IEEE J. Quantum Electron 2004, vol. 40, 166-174. [25] Yin, Z. Y.; Tang, X. H.; Deny, S.; Zhao, J. H. Nanotechnology 2006, vol. 17, 16461650. [26] Maeda, Y. Phys. Rev. B, 1995, vol. 51, 1658-1670. [27] Chang, W. H.; Hsu, T. M.; Tsai, K. F.; Nee, T. E.; Chyi, J. I.; Yeh, N. T.; Jpn. J. Appl. Phys. 1999, vol. 38, 554-557. [28] Yang, T.; Tatebayashi, J.; Aoki, K.; Nishioka, M.; Arakawa, Y. Appl. Phys. Lett. 2007, vol. 90, 111912-1-3. [29] Schmidt, K. H.; Medeiros-Ribeiro, G.; Garcia, J.; Petroff, P. M. Appl. Phys. Lett. 1997, vol. 70, 1727-1729. [30] Hospodkova, A.; Krapek, V.; Kuldova, K.; Humlicek, J.; Hulicius, E.; Oswald, J.; Pangrac, J.; Zeman, J. Physica E 2007, vol. 36, 106-113. [31] Francis, C.; Bradley, M. A.; Boucaud, P.; Julien, F. H.; Razeghi, M. Appl. Phys. Lett. 1993, vol. 62, 178-180. [32] Nie, D.; Mei, T.; Tang, X. H.; Chin, M. K.; Djie, H. S.; Wang, Y. X. J. Appl. Phys. 2006, vol. 100, 046103-1-3. [33] Walther, C.; Bollmann, J.; Kissel, H.; Kirmse, H.; Neumann, W.; Masselink, W. T. Appl. Phys. Lett.. 2000, vol. 76, 2916-2918. [34] Park, C. J.; Kim, H. B.; Lee, Y. H.; Kim, D. Y.; Kang, T. W.; Hong, C. Y.; Cho, H. Y.; Kim, M. D. J. Cryst. Growth, 2001, vol. 227-228, 1057-1061.
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[35] Semenova, G. N.; Venger, Y. F.; Valakh, M. Y.; Sadofyev, Y. G.; Korsunska, N. O.; Strelchuk, V. V.; Borkovska, L. V.; Papusha, V. P.; Vuychik, M. V. J. Phys.: Condens. Matter, 2002, vol. 14, 13375-13380. [36] Fleming, R. M.; McWhan, D. B.; Gossard, A. C.; Wiegmann, W.; Logan, R. A. J. Appl. Phys. 1980, vol. 51(1), 357-363. [37] Goldstein, B. Phys. Rev., 1961, vol. 121, 1305-1311. [38] Yin, Z. Y.; Tang, X. H.; Liu, W.; Zhang, D. H.; Du, A. Y. J. Appl. Phys. 2006, vol. 100, 033109-1-5. [39] Dubowski, J. J.; Allen, C. N.; Fafard, S. Appl. Phys. Lett. 2000, vol. 77, 3583-3585. [40] Schumann, O.; Birner, S.; Baudach, M.; Geelhaar, L.; Eisele, H.; Ivanova, L.; Timm, R.; Lenz, A.; Becker, S. K.; Povolotskyi, M.; Dahne, M.; Abstreiter, G.; Riechert, H. Phys. Rev. B, 2005, vol. 71, 245316-1-10. [41] Chia, C. K.; Chua, S. J.; Tripathy, S.; Dong, J. R. Appl. Phys. Lett. 2005, vol. 86, 051905-1-3. [42] Barik, S.; Tan, H. H.; Jagadish, C. Appl. Phys. Lett. 2007, vol. 90, 093106-1-3. [43] Nathalie, P.; M. Denis, P.; Loic, F-F.; Rene, C. Phys. Rev. B, 2000, vol. 62, 5092-5099 . [44] Gunawan, O.; Djie, H. S.; Ooi, B. S. Phys. Rev. B 2005, vol. 71, 205319-1-10. [45] Gao, Q.; Tan, H. H.; Fu, L.; Jagadish. C. Appl. Phys. Lett. 2004, vol. 84, 4950-4952. [46] Lootens, D.; Van Daele, P.; Demeester, P.; Clauws, P. J. Appl. Phys. 1991, vol. 70, 221224. [47] Djie, H. S.; Ooi, B. S.; Aimes, V. Appl. Phys. Lett. 2005, vol. 87, 261102-1-3. [48] Bhattacharyya, D.; Saher helmy, A.; Bryce, A. C.; Avrutin, E. A.; Marsh, J. H.; J. Appl. Phys. 2000, vol. 88, 4619-4622. [49] Kosogov, A. O.; Werner, P.; Gosele, U.; Ledentsov, N. N.; Bimberg, D.; Ustinov, V. M.; Egorov, A. Yu.; Zhukov, A. E.; Kop'v, P. S.; Bert, N. A.; Alferov, Zh. I. Appl. Phys. Lett. 1996, vol. 69, 3072-3074. [50] Kim, H. S.; Park, J. W.; Oh, D. K.; Oh, K. R.; K, S. J.; Choi, In-H. Semicond. Sci. Technol., 2000, vol. 15, 1005-1009.
In: Quantum Dots: Research, Technology and Applications ISBN: 978-1-60456-930-8 Editor: Randolf W. Knoss, pp. 651-668 © 2008 Nova Science Publishers, Inc.
Chapter 18
APPLICATION OF QUANTUM DOTS IN ORGANIC MEMORY DEVICES: A BRIEF OVERVIEW Kaushik Mallicka and Michael J. Witcombb a
Advanced Materials Division, Mintek, Private Bag X3015, Randburg, 2125, South Africa b Electron Microscope Unit, University of the Witwatersrand, Private Bag 3, WITS, 2050, South Africa
Abstract Quantum dots (QDs) are nanosized regions capable of restricting a single electron, or a few electrons, to the region in three dimensions and in which the electrons no longer occupy band-like energy states, but rather discrete energy states just as they would in an atom. Quantum mechanical phenomena result from this, hence the term quantum confinement. Originally, QDs were grown from semiconductors such as cadmium selenide or cadmium telluride. Since then, however, the synthesis of QDs from nearly every semiconductor and from many metals and insulators has been reported. Quantum dots of semiconductors and metals are currently the focus of intense research. Their electrical, optical, and magnetic properties are different from those of the bulk systems being more like those from molecular-like clusters in which a large number of atoms are on or near the surface. Apart from unique physical properties, QDs also exhibit interesting applications. With their advantage of size, they are ideal for data storage or memory applications to provide high-density memory elements. Potential applications of nonvolatile flash memory devices utilizing QDs have resulted in extensive efforts being made to form QDs, acting as both charging and discharging islands, by a variety of methods. Semiconductor or metallic QDs incorporated within organic or polymeric materials have demonstrated a memory effect when subjected to an electrical bias voltage. Memory phenomenon in QDs arise from their electrical bistability, which is triggered by charge confinement via a suitable voltage pulse. These materials have shown potential applications in digital information storage because of their good stability, flexibility and fast response speed. Organic electrical bistable materials are those that exhibit two kinds of different stable conductive states by applying appropriate voltages. The materials can be switched from low conductive state (“0” or OFF state) to high conductive state (“1” or ON states) by applying an activation voltage. This process is called ‘write’. The high conductive state can remain stable without a bias voltage, and can be read back at a lower voltage. The reverse process is realized
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Kaushik Mallick and Michael J Witcomb by applying a reverse bias when the conductive status changes from a high conductive state to a low conductive state, this being termed ‘erase’. Materials functionalized with ‘erase’ and ‘write’ can be used as RAM (random access memory) and Flash memory. Some materials are write-once-read-many times (WORM), which can be used as ROM (read-only-memory) devices. For commercial use of data storage, devices should satisfy a number of requirements, such as, room temperature operation, low activation voltage to save energy, high ON/OFF ratio, short response time, long retention time and durability. This chapter reviews the recent progress of memory devices exhibiting electric bistability, such devices being based on composites containing quantum dots of semiconductors or metals embedded in organic macromolecular materials.
1. Gold Nanoparticles in Memory Devices Gold nanoparticles are the most stable of particles, and they present fascinating aspects with respect to materials science such as the behavior of the individual particles with regard to their size-related electronic, magnetic and optical properties (quantum size effect), and their applications in catalysis, biology and electronics. A hybrid Si-organic memory device has been reported by Kolliopoulou et al. [1], which incorporated gold nanoparticles that were deposited at room temperature by chemical selfassembly processing and which were utilized as charge storage elements. The nonvolatile, electrically-erasable, programmable read-only memory device had a structure very similar to a metal-oxide-semiconductor (MOS) transistor. The source, drain, and channel as well as a thin SiO2 layer on the channel were fabricated using conventional Si technology. A layer of gold nanoparticles of average size 5 nm (Figure 1), an insulating organic cadmium arachidate Langmuir-Blodget film and Al gate electrode were subsequently deposited on top of the infrastructure. The hybrid memory device exhibited non-volatile characteristics at low operation voltages and did not show any sign of decay of its characteristics over eleven hours at ambient conditions. The work opened the way for extreme miniaturization and the threedimensional integration of memory devices.
Figure 1. TEM image of the gold nanoparticles. Reprinted with permission from S. Kolliopoulou et al. [1]. Copyright 2003, American Institute of Physics.
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Subsequently, Ouyang et al. [2] reported on the utilization of a gold nanoparticle based polymer thin film for the fabrication of non-volatile memory devices. The device had a simple structure with an organic film sandwiched between two aluminium electrodes. The organic film was formed by spin-coating a 1, 2-dichlorobenzene solution of 1-dodecanethiolprotected gold nanoparticles (Au-DT NPs), 8-hydroxyquinoline (8HQ) and polystyrene (PS). The current–voltage (I–V) curves of the Al/Au-DT+8HQ+PS/Al device showed a very low current of about 10–11 A at 1 V in vacuum (Figure 2). An electrical transition was found to take place at 2.8 V with an abrupt current increase from 10–11 A to 10–6 A (curve A). The device showed good stability in this high conductivity state during the subsequent voltage scan (curve B). The high conductivity state could be returned to the low conductivity state by applying a negative bias, curve C, where the current suddenly dropped to 10–10 A at -1.8 V. After the device had returned to the low conductivity state, it could be switched back to the high conductivity state by simply applying a higher bias in either polarity. The switching time was ≥25 ns. Irrespective of whether the device was tested under a nitrogen atmosphere or in air, it exhibited a similar electrical behaviour. Switching between the high and low conductivity states of the device was undertaken numerous times, the device repeatedly performed write, read and erased cycles. This demonstrated the potential of the twodimensional device for non-volatile memory purposes. In addition, the device is capable of incorporating vertical integration.
Figure 2. Current–voltage curve for a device of structure Al/Au–DT+8HQ+PS/Al. A, B and C represent the first, second and third bias scans respectively. The arrows indicate the voltage-scanning direction. Reprinted by permission from Macmillan Publishers Ltd: Nature Materials, J. Ouyang et al. [2], copyright 2004.
A non-volatile plastic digital memory device fabricated from a polyaniline-gold composite has been reported by Tseng et al. [3]. The ~1 nm diameter polyaniline nanofiber 30 nm diameter gold nanoparticle based device exhibited a very interesting bistable electrical
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behavior (Figure 3). As the potential was increased to +3 V, an abrupt increase in current was observed. This changed the device from a low conductivity (10-7 amps) OFF state to a high conductivity (10-4 amps) ON state (Figure 3, curve A). The device was stable in the ON state when the potential was lowered back to 0 V (Figure 3, curve B). The high conductivity of the ON state could be changed back to the OFF state by applying a reverse bias of -5 V. The device was then stable in the OFF state until +3 V was applied at which point it reverted to the ON state (Figure 3, curve C). If the potential was raised above +3 V, then a region of negative differential resistance (NDR) was observed. However, the NDR effect was not observed to have any effect on the performance of the device. Only after several days was a slight decrease in conductivity of the ON state detected. ON-OFF switching times of less than 25 ns were measured for the device.
Figure 3. Current-voltage characteristics of the polyaniline nanofiber/gold nanoparticle device. The potential was scanned from (A) 0 to +4 V, (B) +4 to 0 V, and (C) 0 to +4 V. Between +3 and +4 V, a region of negative differential resistance (NDR) was observed. Reprinted with permission from R.J. Tseng et al. [3]. Copyright 2005 American Chemical Society.
The switching mechanism was attributed to an electric field-induced charge transfer from the polyaniline nanofibers to the gold nanoparticles. Under a sufficient electric field, electrons that reside on the imine nitrogen of the polyaniline gain enough energy and migrate towards and onto the gold nanoparticles (Figure 4). As a consequence, the gold nanoparticles become more negatively charged, whereas the polyaniline nanofibers become more positively charged. The conductivity of the polyaniline nanofiber-gold nanoparticle composite was found to increase dramatically after the electric-field-induced charge transfer, in accordance with the transition from the OFF to the ON state. It was noted that if the gold particles had a diameter greater than 20 nm, the more metallic nature of the larger gold particles dominated the switching so that the device could only be switched on once.
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Figure 4. Schematic structure of a polyaniline nanofiber-gold nanoparticle after the application of +3 V. Reprinted with permission from R.J. Tseng et al. [3]. Copyright 2005 American Chemical Society.
Prakash and co-workers [4] have demonstrated electrical bistability in a non-volatile polymer memory device having an active layer consisting of conjugated poly (3hexylthiophene) and 2.8 nm average sized gold nanoparticles capped with 1-dodecanethiol sandwiched between two metal electrodes. The device exhibited a high stability in both the conductivity states and showed a switching behavior even at temperatures down to 240 K. Above a threshold voltage the device, which was in a low conductivity state, exhibited an increase in conductivity by more than three orders of magnitude. The device could be returned to the low conductivity state by applying a voltage in the reverse direction. The electronic transition is due to an electric-field-induced charge transfer between gold nanoparticles and poly (3-hexylthiophene), the latter acting both as the matrix and active component of the device. When the external electric field was high enough, electrons on the HOMO of poly (3-hexylthiophene) may gain enough energy and tunnel through the 1dodecanethiol into the core of the gold nanoparticles (Figure 5). Consequently, the gold nanoparticles become negatively charged while the poly (3-hexylthiophene) is positively charged. The stable negative charge on a gold nanoparticle results from the insulator nature of the 1-dodecanethiol shell. The effect of the charge transfer on the electronic structure of poly (3-hexylthiophene) is similar to the chemical oxidation of a conducting polymer when conductivity increases after oxidation. The switching of the device from the ‘ON’ to the ‘OFF’ state results from the electrons returning from the gold nanoparticles to the poly (3hexylthiophene). Charge transport through poly (3-hexylthiophene) is by a process of charge hopping through the polymer film. The effect of the Coulomb interaction between the negative charge on the gold nanoparticles and the positive charge on the poly (3hexylthiophene) chain was observed as a result of the high charge mobility in the polymer film. The device exhibited excellent stability in both the conductivity states and could be cycled between the two states for thousands of times, fluctuations in the OFF current only occurring after 1500 cycles. The device exhibited strong potential towards its application as a fast, stable, low-cost, high storage density nonvolatile electronic memory. An electrical memory effect in a single layer device made of 11-marcaptoundecanoic caped gold nanoparticles doped poly (N-vinylcarbazole) composite thin film has been reported by Lin et al. [5]. The device was a two terminal structure with the doped polymer film sandwiched between an indium tin oxide bottom electrode and an aluminium top electrode. The device performed write-read-erase-read cycles more than 2.5×105 times at ambient temperature without any significant degradation.
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Figure 5. Energy diagram of the core of a gold nanoparticle, 1-dodecanethiol (DT), and poly-(3hexylthiophene) (P3HT). The two dots on the HOMO of P3HT represent two electrons, E indicates the direction of the electric field, and the arrow from the electrons on the HOMO of P3HT indicates the electron transfer from P3HT to the core of gold nanoparticle. Reprinted with permission from A. Prakash et al. [4]. Copyright 2006, American Institute of Physics.
An electrical bistability device based on a metal-insulator-metal sandwiched structure has been described by Song et al. [6] in which poly-(N-vinylcarbazole) (PVK) mixed with gold nanoparticles served as the active layer between TaN and Al electrodes. An ON/OFF current ratio as high as 105 at room temperature was achieved. Very little degradation in current density for both the ON and OFF states was observed, the device being found to be stable in both these states from ambient to around 70 0C. The polymer serves a multi-role mode as matrix for gold nanoparticles, electron donor to the metal nanoparticles and as the path for charge carrier transport. When a high electric field was applied to the device, the electricfield-induced charge transfer complex between the polymer and nanoparticles will be formed with the polymer being positively charged and the particles negatively charged. The distortion due to the presence of a charge tends to change the electronic states in the vicinity of the charge such that the HOMO energy level shifts upward and the LUMO energy level shifts downward [7]. This change will decrease the band-gap and increase the conductivity of the organic material. Since the charge transfer complex strongly depends on the electric field, a reverse bias can cause the charge transfer complex to return to its original state. Kim et al. [8] have designed a nano-floating gate memory utilizing a monolayer of vertically aligned gold nanoparticles embedded in a dielectric polymide film. The device was fabricated by sandwiching a 3.4 nm thick gold film between two polyimide (PI) precursor layers. The film was in the form of uniform sized nanoparticles which coalesced into 10 nm sized, well dispersed nanoparticles after baking the system at 400oC for 1 hour at ~ 10-3 Pa (Figure 6). The layers were grown on p-Si substrates. The advantage of polyimide is that it has good thermal stability and chemical endurance. Capacitance-voltage (C-V) measurements of the Al/PI/Au nanoparticles/PI/p-type Si structure at 300 K showed that the monolayer of Au nanoparticles functioned as a floating gate in the metal-insulator-semiconductor (MIS) type capacitor. The device exhibited a capacitance hysteresis of 3.4 V at an applied voltage of 6 V. The monolayer of Au
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nanoparticles embedded in the PI film thus exhibited a well behaved memory effect which could be potentially utilized in next generation of flash memories.
Figure 6. Cross-sectional TEM image of the PI/Au nanoparticles/PI sample. Reprinted with permission from J.H. Kim et al. [8]. Copyright 2007, American Institute of Physics.
A new organic memory system using pentacene as the active organic semiconductor layer and citrate-stabilized gold nanoparticles as the charge storage elements has been reported by Leong et al. [9]. The organic memory device comprised a metal-pentacene-insulator-silicon (MPIS) structure (Figure 7).
Figure 7. Schematic illustration of an organic memory device utilizing citrate-stabilized Au nanoparticles. The silicon substrate was used as the bottom gate electrode. Reprinted with permission from W.L. Leong et al. [9]. Copyright 2007, American Institute of Physics.
The device was fabricated on a degenerately doped n-type silicon wafer, used as the bottom gate electrode, while a 4.5 nm thermally grown silicon dioxide layer was utilized as the top electrode. The substrate surface was functionalized with a self-assembled 0.9 nm monolayer of 3-aminopropyl-triethoxysilane (APTES), after which a thin layer of citrate stabilized gold nanoparticles (Figure 8) were decorated on it followed by a 45 nm thick active layer of pentacene and finally a top gold electrode. Double sweeping C-V and conductance - voltage (G-V) curves of the device obtained in the frequency range 50 kHz – 1 MHz exhibited a clockwise C-V hysteresis and almost constant FWHM of the conductance peaks in G-V. Since interface traps are minimal in the high frequency range, it could be concluded that the charge trapping effect originated from the Au nanoparticles rather than from interface traps. The use of functionalized Au nanoparticles as nanotraps, and the simplicity in design and processing implied from this
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work that the fabrication of integrated memory devices in low-cost plastic electronics applications is indeed possible.
Figure 8. TEM images of citrate-stabilized Au nanoparticles of size of 3–7 nm]. Reprinted with permission from W.L. Leong et al. [9]. Copyright 2007, American Institute of Physics.
2. Organic Memory Devices with Metal Oxide Nanoparticles Metal-oxide nanoparticles embedded in a polymer layer have attractive characteristics with regards to high density, good uniformity, single layer controllability, and feasibility for nanofloating gate memory devices. The bistable effects of semiconductor cuprous oxide (Cu2O) nanoparticles fully embedded in a spin coated polyimide (PI) matrix have been investigated by Jung and coworkers [10]. The nanoparticles were chemically self-assembled, relatively uniformily distributed and separated, and fully embedded in the matrix. The size of the Cu2O nanoparticles varied between 3 and 5 nm (Figure 9) while their surface density was estimated to be 2.7x1012 particles cm-2.
Figure 9. Transmission electron microscopy images of Cu2O nanoparticles embedded in a polyimide layer. Reprinted with permission from J.H. Jung et al. [10]. Copyright 2006, American Institute of Physics.
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The I-V curves taken at 300 K for the [Al (100 nm) / PI (20 nm )/ Cu2O nanoparticles / PI (20 nm) / Al/glass] device were measured under forward and reverse bias conditions (-30 to +30 V) and showed an electrical hysteresis behavior. The bistable behavior was found to be asymmetric in nature. This was believed to be due to either the non-uniformity of distribution of the Cu2O nanocrystals embedded in the PI layer or from the existence of interface traps between the PI layer and the Al electrodes. The device was found to have a much higher hysteresis and thus a significantly enhanced information storage capability than Cu2O-free samples thus indicating potential applications in next-generation non-volatile flash memory devices. The memory effects of semiconductor ZnO nanoparticles embedded in a polyimide (PI) matrix have been investigated and reported by Kim and co-workers [11]. The PI precursor was prepared by dissolving p-phenylene biphenyltetracarboximide-type polyamic acid (PI2610d, Dupont) in N-methyl-2-pyrrolidone, the solution then being spin-coated onto p-Si substrates. Subsequently, a 10 nm thick Zn film was deposited followed by another PI layer. The device was then cured at 350oC for 2 hours in a nitrogen atmosphere. The first PI layer acted as the tunneling barrier while the second acted as the insulating layer. TEM studies (Figure 10) confirmed the formation of 4 - 6 nm diameter ZnO nanocrystals in the PI layer.
Figure 10. Plan-view bright-field transmission electron microscopy image of the ZnO nanoparticles embedded in a polyimide layer. Reprinted with permission from J.H. Jung et al. [11]. Copyright 2006, American Institute of Physics.
C-V measurements carried out at 300 K on the Al/PI/ZnO nanocrystals/PI/p-Si structure (Figure 11) indicated a MIS behavior with a flatband voltage shift due to the existence of the self-assembled ZnO nanocrystals since no hysteresis was measured when the nanoparticles were omitted. This was taken as indicative of charge trapping, storing and emission of electrons in the ZnO nanoparticle sites resulting from quantum confinement effects. Koo et al. [12] have reported on a floating gate memory device in which In2O3 nanoparticles acted as charge storage regions embedded in a BPDA-PDA polyimide gate insulator layer. Self-assembled In2O3 nanoparticles were formed within the polyimide gate insulator matrix as a result of chemical reactions between indium ions and the polymer precursor during the curing process at 400oC for 1 hour. The average diameter and distribution density of the In2O3 particle were 7 nm and 6x1011 cm−2 respectively. It was found that post-annealing at 400oC for 30 minutes in hydrogen diluted to 3% in nitrogen ambient resulted in considerable improvement in the memory window, increasing from 2.6 to
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4.4 V, and the retention characteristics. These improvements were believed to be associated with the reduction of interface and bulk traps in the stacked insulator layer.
Figure 11. Capacitance-voltage curve for Al/polyimide/ZnO nanoparticles/polyimide/p-Si(100) structure. Reprinted with permission from J.H. Jung et al. [11]. Copyright 2006, American Institute of Physics.
3. Memory Effects Based on Semiconductor Nanoparticles Research on semiconducting nanoparticles has begun to focus on different applications in order to meet the challenges and advancement of technology. Apart from several optoelectronic devices for photonic applications [13,14], such nanomaterials have demonstrated applications as electrically bistable devices and memory elements [15-19]. Devices based on the CdSe nanoparticles exhibit a high ON/OFF ratio and demonstrate ROM and RAM applications. Sahu and co-workers [20] has deposited layer by layer films based on mercaptoacetic acid (MAA) and poly (diallyl-dimethyl ammonium chloride) (PDDA)-stabilized CdSe quantum dots onto indium tin oxide coated glass substrates by alternate cycles of adsorption of MAA (anionic)- and PDDA (cationic)-capped QD particles from their dispersed solutions via an electrostatic adsorption process. Thus the nature of the surface charge changes for the next layer adsorption. The number of bilayers of CdSe was determined by the number of dipping sequences and yielded thin films controllable in thickness at the nanometer scale. The thickness of a single bilayer film was 5 nm. Electronic absorption and photoluminescence spectra of MAA- and PDDA-capped CdSe nanoparticles in dispersed solution are shown in
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Figure 12. The inset shows a TEM image of MAA-capped CdSe QDs revealing particles of less than 5 nm diameter. Calculated diameters of the MAA- and PDAA-capped nanoparticles matched well being 3 and 4.2 nm respectively.
Figure 12. Electronic absorption and photoluminescence spectra of MAA- and PDDA-capped CdSe nanoparticles in the dispersed solution. The PL spectra are shown in broken lines. The inset shows a TEM image of MAA-capped nanoparticles [20]. Reprinted with permission from S. Sahu et al. [20]. Copyright 2007, American Institute of Physics.
I-V curves obtained from 10- and 20- bilayer devices demonstrated electrical bistability. This was explained on the basis of charge confinement in the nanoparticles. The devices based on these semiconductor QDs exhibited high ON/OFF ratios, up to 2000, and demonstrated potential ROM and RAM memory applications. An electrical multi-stability effect has been observed by Portney et al. [21] for a single layer device fabricated from a hybrid virus-semiconducting QD (CdSe core - ZnS shell) assembled via a bottom-up approach from an icosahedral-mutant-virus template (CPMVT184C). The hybrid was embedded in a polyvinyl alcohol (PVA) matrix. To illustrate the potential of this single layer hybrid device as a functioning memory element, a sequence of simple write-read-erase cycles was performed (Figure 13). The device was programmed with write, read, erase, and read pulses of 5, 1, −3, and 1 V, respectively. The time width of the pulses was 16 ms. The corresponding currents to the different pulses were recorded and showed two distinct high and low conductance states (i.e., “1” and “0”). The write and erase pulses are denoted as “W” and “E” in Figure 13. These cycles could reach about 100 times with one to two orders of magnitude difference in the high/low conductance states. These states were found to be repeatable and non-volatile. The memory effect produced by this hybrid system was shown both from the I-V sweeps and cycle curves. These results thus represent a new dimension and material combination for the fabrication of non-volatile memory systems. They also depict a unique charge interaction behavior between organic and inorganic components resulting in a multi-level stability which is desired for the memory applications.
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Figure 13. Pulsed voltage cycles used to demonstrate a functional memory element [21]. Reprinted with permission from N.G. Portney et al. [21]. Copyright 2007, American Institute of Physics.
Figure 14. Capacitance-voltage curve (1 MHz C-V) for a Au/(CdSe/ZnS nanoparticles embedded in the MEH-PPV layer)/ITO/glass device. Reprinted with permission from F. Li et al. [22]. Copyright 2007, American Institute of Physics.
Liu et al. [22] have reported on the memory effects of a capacitor consisting of a blend of core-shell-type (CdSe core-ZnS shell) nanoparticles and a conducting polymer poly [2-methoxy5-(2-ethylhexyloxy)-1,4-phenylene-vinylene] (MEH-PPV) sandwiched between a metal electrode (Au or Al) and an indium tin oxide (ITO) coated glass. The latter acted as the device substrate. The embedded nanoparticles had a concentration of 1.5 wt% in the hybrid layer and were uniformily distributed. The diameter of the core/shell-type CdSe/ZnS nanoparticles was ~15 nm, while the ZnS shell thickness was ~ 0.5 nm. A C-V curve obtained at room temperature (Figure 14) for the Au/hybrid layer/ITO/glass structure showed a MIS behavior
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with charge trapping, storing, and emission regions through the existence of sites occupied by carriers. The presence of such sites was attributed to charging and discharging of the carriers in the CdSe/ZnS nanoparticles. No hysteresis was measured for an identical device, but containing no nanoparticles. Figure 14 could thus be attributed to carriers trapped in the embedded CdSe/ZnS nanoparticles, which is indicative of a memory effect.
Figure 15. Capacitance-voltage curves for an Al/[CdSe/ZnS nanoparticles embedded in the MEH-PPV layer]/ITO/ glass device under (A) positive and (B) negative voltages. Reprinted with permission from F. Li et al. [22]. Copyright 2007, American Institute of Physics.
Dramatically different C-V characteristics under positive and negative bias voltages were observed for Al/hybrid layer/ITO coated glass devices, Figure 15. The C-V curve shows a symmetric character, and an obvious hysteresis can be observed on both sides of the curve, indicative of the occupancy of CdSe/ZnS nanoparticles by charged carriers for both positive and negative bias voltages. Therefore, while the hysteresis appearing in Figure 15 indicates a memory effect for this device, it must have a different operating mechanism from that of the Au/hybrid layer/ITO structure. A dipolar carrier trapping model has been proposed by the authors to explain the symmetric behavior of the C-V characteristics shown in Figure 15. Both holes and electrons are believed to be involved in the charging and discharging processes of the carriers for the Al/hybrid layer/ITO coated glass structure, as distinct from the proposed
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trapping process of unipolar hole carriers for the Au/hybrid layer/ITO coated glass structure. This discrepancy originates from the blockage of the electrons in the Au device as a result of the relatively large energy barrier between the work function of the Au electrode and the lowest unoccupied molecular orbit (LUMO) level of the MEH-PPV layer. These results indicate that memory devices based on a spin-coated MEH-PPV polymer layer containing the CdSe/ZnS nanoparticles hold promise for potential applications in future non-volatile flash memory devices.
4. Use of Platinum Nanoparticles in Memory Applications Nanostructured viruses are attractive for use as templates for ordering quantum dots to make self-assembled building blocks for next-generation electronic devices. Important developments have been made in the synthesis of bio-nanostructures with nanocrystals, including protein shelled viruses modified by metallic [23-26] or semiconducting nanoparticles [21,27,28] A new memory device based on a hybrid system composed of tobacco mosaic virus (TMV) conjugated with platinum nanoparticles (TMV–Pt) has recently been developed and reported by Tseng et al. [29]. The device was fabricated through a solution process with a hybrid bio-inorganic composite layer in a PVA matrix sandwiched between two aluminum electrodes. The Pt nanoparticles of average size ~10 nm were found to be quite uniformily distributed on the virus surface (Figure 16), there being on average of roughly 16 particles per virus. From TEM and AFM images, the nanowire diameter was about 30 nm.
Figure 16. Transmission electron microscope image of TMV–Pt conjugate and size distribution of the Pt nanoparticles. (A) TEM image of a 300 nm long TMV nanowire conjugated with Pt nanoparticles. The Pt nanoparticles have an average size of ~10 nm and are uniformly attached to the surface of the virus wire. (B) Histogram showing the size distribution of the nanoparticles, which measured between 7 and 15 nm. Reprinted by permission from Macmillan Publishers Ltd: Nature Nanotechnology [29], copyright 2006.
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The composite system revealed electrical bistability depending on the voltage-controlled conductance states. A comparative study of a device fabricated with the same concentration and thickness, but with only Pt nanoparticles exhibited the conduction current close to the ON state whereas a similarly prepared TMV-only device showed the current in the OFF state. Neither device exhibited any memory effect. The unique memory effect therefore must arise from the combination of the TMV and Pt nanoparticles. The function of the TMV is not only as a support for the nanoparticles, but due to the presence of the RNA core with its rich aromatic rings, such as guanine, it can act as a charge donor. The proteins on the surface of the TMV virus separate the RNA and the Pt nanoparticles and therefore act as an energy barrier. The mechanism of the memory device, including the sudden jump of the current in the I–V scan (Figure 17), is likely to be due to a charge transfer from the RNA to the Pt nanoparticles under the high electric field. Once the charge has been transferred, it will be trapped in the nanoparticles and stabilized by the coat proteins. Since this is an electrical field effect, when sufficient numbers of sites have been established, a sudden jump in the current accompanying the charge tunneling through the nanoparticles is observed. Charge transfer and charge traps in nanoparticles must therefore be responsible for the conductance switching behaviour and the memory effect. The concept of conjugating nanoparticles with biomolecules thus opens up new possibilities for making functional electronic devices using biomaterial systems.
Figure 17. When a voltage scan from 0 to 6 V is applied, conductance switching is observed on the TMV–Pt, but the TMV-only configuration shows no switching transition. Reprinted by permission from Macmillan Publishers Ltd: Nature Nanotechnology [29], copyright 2006.
5. Application of Bimetallic Systems to Nonvolatile Memories Kim et al. [30] have extended use of the quantum confinement by nanoparticles to bimetallic systems for the fabrication of the organic memory devices. Self-assembled Ni1−xFex nanoparticles were embedded into a polyimide (PI) matrix. Basically a 5 nm thick Ni0.8Si0.2
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layer was deposited onto a 40 nm thick PI layer that had been spin-coated onto a n-Si (100) substrate. Subsequently another 40 nm thick PI layer was deposited on top and then the complete structure was cured at 400oC for 1 hour at 10-3 Pa. The first precursor PI layer acted as the tunneling barrier while the second PI layer functioned as the insulating gate layer. TEM images showed from both planar, Figure 18, and cross-sectional views that the self-assembled 4-6 nm diameter Ni1−xFex nanocrystals had been created only within the PI layer.
Figure 18. Plan-view, bright-field TEM image of the Ni1−xFex nanoparticles embedded in the polyimide layer. Reprinted with permission from J.H. Kim et al. [30]. Copyright 2005, American Institute of Physics.
Figure 19. C-V curve (1 MHz) for an Al/polyimide/ Ni(1−x)-Fex nanocrystals /polyimide/n-Si(100) structure. Reprinted with permission from J.H. Kim et al. [30]. Copyright 2005, American Institute of Physics.
The C-V curve (Figure 19) taken at 300 K for the Al/PI/Ni1−xFex nanocrystals/PI/nSi(100) structure showed a MIS behavior with the charge trap regions similar to MIS memories with floating gates based on nanocrystalline Si [31,32]. The flatband voltage shift of the C-V curve was about 2V, which is enough to capture electrons within the nanoparticles. Similar C-V curves for samples without Ni1−xFex nanocrystals under the same measurement
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conditions showed no hysteresis. The clockwise hysteresis, indicative of the existence of sites occupied by electrons, was thus attributed to the quantum confinement effect of the nanocrystals. These results thus indicate that Ni1−xFex nanocrystals embedded in a PI layer can act as floating gates and therefore such a device holds promise for potential applications in next generation non-volatile, single electron flash memories.
6. Conclusions The use of hybrid materials, such as organic and nanoparticle composites, which have been shown to be able to yield good ON/OFF ratios, low read/write voltages, short response times down into the nanosecond range, can provide a simplified manufacturing process that can give low-cost, flexible, stackable, high density, light-weight devices that have an active device area approaching the nanoscale. Such quantum dot composites including organicmetal nanocluster, organic-metal oxide nanocluster, and organic-bimetallic nanocluster systems have been reviewed briefly here on the basis of the techniques used for their synthesis and properties. While in their infancy compared to the more mature semiconductor technologies, organic-nanoparticle composites show outstanding promise with respect to digital nonvolatile memory applications. The electrical bistable effect is a fascinating phenomenon covering physics, chemistry, materials science and engineering. Many possible applications for systems showing such an effect have been explored; different fabrication methods and device structures have been studied; more potential applications and devices will be proposed. While it must be acknowledged that there are many questions yet to be answered such as how is bistability influenced by the size and density of the nanoparticles, the thickness of the embedding material around each particle, where are the charges stored in the particles etc., the future of organic bistable device research looks to be exciting, rewarding and productive with potentially strong commercial applications in the field of information technology.
Acknowledgements K. Mallick acknowledges financial support from Project AuTEK and the Nanotechnology Innovation Centre (NIC), Mintek, South Africa.
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INDEX A absorption coefficient, xiv, xv, 129, 130, 131, 133, 143, 159, 335, 346, 351, 530, 532, 533, 577, 578, 589, 591, 592, 593, 596, 597, 599, 601, 602, 603, 616, 617, 618, 619, 620, 621 absorption spectra, xiv, 90, 131, 132, 136, 139, 141, 142, 156, 193, 246, 577, 604 absorption spectroscopy, 328 AC, 50, 327 acceptor, 87, 151, 248, 250, 328 acceptors, 52 access, 49, 450 accidental, 566 accounting, 140, 173, 192, 378, 390, 396, 400, 599 accuracy, xiii, 16, 23, 24, 25, 38, 220, 348, 351, 352, 383, 385, 386, 428, 430, 485 acetate, 405 acetic acid, 255 acetone, 402 achievement, 268, 269, 547 acid, 120, 246, 247, 251, 257, 258, 334, 354, 404, 415, 659, 660 acidic, 120, 250 acidity, 120 acoustic, 341, 515 acoustical, 341, 356 acrylic acid, 254, 404 actin, 247 activation, ix, xvi, 52, 54, 55, 56, 69, 70, 74, 76, 77, 89, 97, 98, 100, 110, 150, 152, 244, 364, 629, 651 activation energy, 52, 54, 55, 56, 69, 152, 364, 629 adaptation, 258 adenomas, 423 adhesion, 119, 253
adiabatic, 190, 191, 524, 538 adsorption, 121, 247, 258, 660 aerospace, 211, 268 agent, 257 agents, 244, 252 aggregation, 121 agonist, 258 air, 124, 129, 364, 415, 416, 653 AJ, 328 alcohol, 402 algorithm, 18, 21, 22, 23, 55, 284, 335, 349 alkaline, 119, 120, 122 alkaline media, 122 alkalinity, 119, 121 alloys, 268, 474 alternative, 67, 89, 170, 246, 279, 373, 539 alters, 497, 508 aluminium, 208, 653, 655 aluminum, 523, 664 amine, 405 amino, 246, 252, 254 amino acid, 246, 254 ammonia, 119, 120, 121 ammonium, 405, 660 ammonium chloride, 660 amorphous, 123, 136, 151, 152, 334, 337, 345, 347, 350, 351, 352, 354, 358, 359, 360, 361, 362, 363, 364, 366, 369 amplitude, xv, 18, 54, 59, 72, 73, 78, 85, 86, 87, 88, 100, 190, 191, 283, 381, 382, 383, 384, 386, 387, 388, 389, 470, 471, 476, 477, 481, 549, 552, 554, 557, 559, 566, 571, 578, 607, 612 analog, 254, 347, 455 angiogenesis, 251, 253 angiotensin, 251 angiotensin II, 251 angular momentum, 4, 5, 6, 11, 12, 24, 25, 26, 28, 29, 31, 32, 33, 117, 125, 126, 138, 140, 302, 303, 373, 374, 375, 497, 509
670 anion, 139 anions, 120 anisotropic, xi, 6, 15, 299, 305, 318, 323, 324, 325, 329, 498, 529, 616 anisotropy, 128, 236, 318, 325, 584, 608, 610, 612, 616 Annealing, 80, 84, 85, 360, 625, 644 annihilation, 156 anomalous, xi, 331, 334, 337, 338, 339, 346, 347, 505, 524 antagonist, 258 antibody, 244, 249, 250, 251, 416, 420 antibonding, 539 antigen, 249, 250, 251, 252 antisense, 415 antiviral, 249 appendix, 306 aqueous solution, 246 AR, 181, 328 arginine, 251 argon, xv, 147, 149, 208, 623 argument, 211, 590, 597 aromatic, 665 aromatic rings, 665 arsenide, 93 artificial, 2, 13, 48, 118, 494, 523, 538 artificial atoms, 2, 48, 118, 494 aspect ratio, 219, 222, 232, 234, 236, 237, 272, 292, 296, 303, 305, 308, 318, 323, 629 assignment, 139, 141 assumptions, 13, 55, 347, 429, 430, 485 asymmetry, 310, 314, 399, 486, 539, 603 asymptotic, 8, 9, 186, 433, 434 asymptotically, 439 atmosphere, 124, 129, 147, 149, 337, 356, 359, 361, 362, 363, 364, 626, 653, 659 atmospheric pressure, 642 atomic force, 58, 226, 401, 405, 406, 627 atomic force microscope, 58, 627 atomic force microscopy, 226, 401, 405 Atomic Force Microscopy, 269 atomic force microscopy (AFM), 226, 405 atoms, vii, xvi, 2, 38, 67, 81, 85, 110, 117, 118, 213, 214, 278, 279, 334, 346, 372, 373, 429, 430, 439, 444, 451, 453, 471, 484, 494, 546, 572, 625, 629, 641, 651 atrial natriuretic peptide, 422 attachment, 249, 472, 550 attention, x, xiii, 16, 17, 25, 126, 136, 137, 267, 268, 315, 333, 356, 364, 371, 383, 430, 493, 494, 505, 530, 578, 624 Au nanoparticles, 656, 657, 658 averaging, 18, 21, 55, 62, 375, 641
Index axons, 422
B backscattered, 204 bacterial, 251 baking, 656 band gap, ix, xii, 81, 109, 115, 116, 130, 131, 132, 134, 135, 136, 137, 140, 149, 150, 151, 156, 157, 246, 268, 332, 333, 356, 357, 367, 499, 509, 522 bandgap, x, xv, 48, 157, 160, 211, 212, 214, 217, 220, 231, 238, 267, 268, 269, 290, 396, 411, 623, 624, 625, 629, 631, 634, 639, 641, 642, 643, 644, 646, 647 bandwidth, ix, x, 169, 174, 178, 182, 183, 184, 191, 192, 196, 198, 200, 203, 212, 213, 237, 238 barrier, 52, 62, 63, 70, 80, 81, 85, 89, 100, 135, 139, 145, 148, 151, 152, 172, 173, 174, 221, 258, 278, 279, 290, 343, 357, 374, 495, 497, 516, 523, 524, 530, 533, 534, 535, 538, 624, 625, 629, 630, 631, 632, 633, 634, 638, 640, 647, 659, 664, 665, 666 barriers, 151, 218, 220, 332, 341, 343, 356, 357, 381, 523, 625, 629, 634 basis set, 16, 20, 26 beams, 283 behavior, vii, viii, xii, 1, 2, 11, 14, 29, 32, 38, 39, 110, 112, 319, 362, 393, 394, 395, 399, 401, 406, 408, 409, 410, 433, 434, 438, 446, 455, 459, 463, 466, 474, 483, 484, 486, 557, 559, 560, 561, 562, 567, 570, 652, 654, 655, 659, 662, 663, 664, 667 Beijing, 299, 493 benchmark, 39 bending, 51, 93, 281, 289 benefits, 217 Bessel, xi, 9, 117, 299, 302, 326, 605, 607 bias, viii, xvi, 24, 47, 48, 50, 51, 52, 54, 61, 62, 63, 67, 68, 69, 71, 72, 73, 77, 84, 85, 87, 89, 91, 92, 97, 98, 99, 100, 101, 102, 183, 184, 186, 188, 190, 191, 505, 651, 653, 654, 656, 659, 663 biaxial, 234, 237, 295, 497 biexciton, 397 bifurcation, xiii, 427, 429, 430, 442, 444, 447, 448, 451, 472, 475, 484, 486 billiards, 450, 455, 482, 573, 574 binding, xiv, xv, 50, 52, 76, 243, 247, 251, 252, 253, 300, 328, 333, 348, 402, 404, 428, 435, 436, 437, 460, 461, 496, 516, 524, 527, 530, 550, 551, 556, 560, 574, 577, 578, 579, 580,
Index 581, 584, 585, 586, 587, 593, 594, 603, 604, 605, 606, 608, 610, 611, 612, 613, 616 binding energies, 50, 328, 527, 530 binding energy, xiv, xv, 52, 76, 333, 496, 530, 577, 578, 584, 585, 586, 593, 603, 604, 605, 606, 608, 610, 611, 612, 613, 616 bioavailability, 243 biochemical, x, 243, 244 biological, x, xii, 243, 244, 245, 247, 248, 249, 251, 255, 256, 259, 268, 307, 393, 404, 414, 424, 425 biological activity, 247 biological processes, 259 biologically, 247, 256, 258, 259 biology, 423, 652 biomaterial, 665 biomedical, x, 203, 205, 208 biomedical applications, 205, 208 biomolecules, 424 Biopharmaceuticals, 261 biotin, 248, 254, 255, 403, 404, 415, 422 birth, 19, 20, 415 bismuth, 208 black, 234, 384, 423, 470, 480, 481, 482, 507, 568 blood, 251, 258 blot, 415, 422 blueshift, 80, 81, 88, 524, 629, 630, 631, 633, 634, 636, 638, 639, 640, 641, 642, 645, 647 body weight, 415 Bohr, xi, 14, 27, 133, 245, 305, 311, 331, 332, 382, 581, 584, 585, 589, 605, 607, 610, 611, 613, 616, 618 Boltzmann constant, 52, 65, 208 Boltzmann distribution, 307 bonding, 403, 524, 539 bonds, 278, 346, 359, 360, 361, 362 bone, 251, 252 bone marrow, 251, 252 borate, 247 borderline, 485 borosilicate glass, 590, 591, 592, 593, 597, 598, 601, 602, 603, 620 Bose, 22, 23, 24, 338 Bose-Einstein, 338 bottleneck, 57, 76, 103 bottom-up, 217, 238, 661 boundary conditions, 139, 374, 497, 547 boundary value problem, 381 bounds, 401 bovine, 415 brain, 258 branching, 19, 20, 21, 22
671 breakdown, 135 breast, 251, 422 broad spectrum, x, 55, 169, 170, 190, 196 broadband, x, 196, 203, 204, 205, 206, 207, 209, 211, 212, 217, 234, 237, 238 Broadband, v, 203, 204, 207 buffer, 119, 120, 121, 226, 246, 247, 258, 269, 284, 359, 364, 404, 626 building blocks, 143, 664 bulbs, 207 bulk materials, 38, 140, 311 burning, 181, 182, 185
C cadmium, viii, xvi, 109, 118, 119, 120, 121, 123, 124, 129, 130, 131, 132, 144, 145, 148, 153, 154, 156, 245, 246, 251, 258, 398, 651, 652 cancer, 251, 252, 253, 422 cancer cells, 252 candidates, 244, 259 capacitance, viii, 3, 47, 49, 50, 51, 52, 53, 54, 55, 59, 61, 62, 67, 71, 72, 73, 89, 95, 97, 507, 656 capacity, 3, 393, 624, 641 carbide, 626 carboxyl, 405 carboxylic, 254 carboxylic acids, 254 carcinogen, 258 catalysis, 652 cathode, 635 cation, 572, 638 cations, 120 cavities, 196, 429, 430, 440, 450, 478, 479, 481 cell, x, 111, 123, 125, 139, 220, 221, 243, 244, 247, 248, 249, 250, 251, 253, 257, 258, 259, 343, 414, 422, 423, 497, 517, 524, 534, 536, 630 cell culture, 249, 250 cell invasion, 250 cerebellum, 250 cervical, 251 cervical cancer, 251 chalcogenides, 121 channels, 430, 432, 435, 450, 452, 456, 478, 479, 486, 575 charge density, 14, 51 charge trapping, 657, 659, 663 charged particle, 634 chemical agents, 358 chemical composition, viii, xi, 109, 120, 121, 134, 140, 148, 154, 331, 334, 366 chemical degradation, 244, 245
672 chemical deposition, 119, 120, 121, 122, 136 chemical etching, 359, 407 chemical interaction, 400 chemical oxidation, 655 chemical properties, 166, 357 chemical reactions, 333, 659 chemical stability, 358 chemical structures, 253 chemical vapor deposition, 332, 623, 642 chemical vapour, 196 chemical vapour deposition, 196 chemistry, x, 24, 115, 133, 243, 244, 253, 257, 264, 299, 667 chemotherapeutic agent, 252 chicken, 249 chiral, 447, 451 chloride, 119, 247, 404, 415 CHO cells, 254 cholera, 249 chromium, 208 chromosomes, 424 circular dichroism, 317 circularly polarized light, 372, 375, 390 circulation, 17 cladding, 172 cladding layer, 172 classes, 112 classical, x, 14, 17, 33, 34, 35, 119, 267, 270, 281, 296, 514 classified, 157, 212, 227, 496, 562 cleaning, 405 cleavage, 248 clinical, 249, 251 clinical trial, 251 clinical trials, 251 closure, 275 clustering, 75, 448 clusters, xvi, 48, 56, 138, 334, 335, 349, 350, 354, 356, 604, 651 coagulation, 121 coatings, 172, 181, 209, 247, 248, 357 coding, 422, 424 coherence, x, xii, 196, 203, 204, 205, 206, 208, 209, 235, 371, 372, 373, 375, 377, 378, 382, 386, 431, 474, 485, 486 coil, 204, 634 colloidal particles, 121 colors, 245, 249, 250, 420 combined effect, 232 commercial, xvi, 652, 667 communication, 182, 624 communication systems, 182 community, 170, 244
Index compatibility, 244 compensation, 190 competition, 182, 185 competitor, 35 complementary, 49, 57, 402, 404 complementary DNA, 404 complexity, 14, 220, 235, 244, 402 complications, 222 components, 3, 52, 54, 55, 63, 78, 91, 96, 110, 127, 140, 235, 247, 249, 269, 286, 306, 357, 376, 377, 379, 380, 381, 382, 383, 384, 385, 386, 387, 388, 389, 400, 401, 410, 411, 457, 463, 464, 466, 497, 517, 526, 626, 662 composite, 34, 349, 357, 539, 653, 654, 655, 664, 665 composites, xvi, 652, 667 composition, ix, 48, 85, 148, 169, 170, 172, 191, 192, 208, 215, 216, 236, 246, 292, 297, 324, 334, 346, 347, 348, 351, 353, 357, 367, 494, 495, 625, 629, 630, 631, 633, 634, 647 compositions, 237, 335, 357, 505, 630 compound semiconductors, 292, 300 compounds, 125, 126, 127, 243, 253, 256, 258, 268 compression, 183, 184, 185, 200, 278, 508 computation, 16, 174, 222, 329, 372, 400, 435, 494, 514, 516, 565 computer, 3, 28, 424, 514, 627 computers, 514 Computers, 45 computing, 371, 624 concentration, xi, 50, 52, 54, 56, 57, 60, 61, 62, 63, 64, 65, 66, 72, 75, 78, 90, 92, 94, 96, 98, 119, 120, 121, 149, 150, 151, 156, 157, 158, 160, 161, 215, 247, 292, 299, 315, 317, 318, 320, 345, 348, 349, 350, 402, 415, 416, 533, 589, 615, 618, 625, 633, 638, 641, 663, 665 concrete, 429, 438, 452, 453 condensation, 356 condensed matter, 143 conductance, 431, 547, 548, 550, 552, 566, 567, 568, 569, 570, 573, 657, 661, 665 conduction, 61, 68, 70, 85, 89, 90, 95, 97, 125, 128, 129, 130, 131, 137, 139, 149, 150, 151, 156, 157, 160, 173, 210, 211, 220, 223, 245, 304, 311, 315, 319, 374, 394, 395, 396, 497, 498, 499, 509, 529, 531, 635, 665 conductive, xvi, 651 conductivity, ix, 109, 149, 150, 151, 153, 158, 163, 357, 653, 654, 655, 656 conductor, 2, 17, 37, 38, 40, 184 confidence, 56
Index configuration, 18, 34, 97, 98, 115, 234, 235, 277, 278, 281, 294, 345, 375, 457, 626, 665 confinement, vii, xii, xiv, xv, xvi, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 16, 24, 27, 29, 37, 40, 60, 82, 83, 85, 115, 132, 134, 135, 136, 139, 141, 143, 148, 155, 172, 176, 179, 182, 192, 210, 218, 223, 224, 237, 245, 312, 315, 332, 337, 354, 393, 497, 508, 517, 534, 536, 541, 574, 577, 578, 579, 587, 603, 604, 605, 606, 607, 608, 612, 616, 630, 641, 645, 651, 661 conjugation, 18, 248, 249, 254, 257 connectivity, 148 consensus, 80, 474 conservation, 373, 375, 547 constituent materials, 508 construction, 126, 404, 407, 485 contamination, 626 continuing, 204 continuity, 406, 547, 553 control, vii, xiv, 1, 3, 67, 76, 78, 79, 89, 119, 120, 170, 235, 246, 279, 335, 345, 357, 371, 372, 373, 374, 393, 416, 420, 430, 450, 453, 457, 485, 486, 514, 577, 578, 626, 634 controlled, vii, 2, 3, 22, 76, 121, 136, 300, 357, 362, 429, 431, 454, 458, 474, 478, 494, 514, 523, 578, 626, 665 convergence, 33, 177 convex, 440, 478, 479, 481 cooling, 152, 373, 515 copolymer, 254 copyright, iv, 653, 665 core-shell, 662 correction factors, 37 correlation, xi, 4, 14, 24, 25, 27, 28, 29, 30, 33, 34, 35, 38, 39, 123, 131, 132, 134, 135, 331, 339, 341, 350, 410, 481, 508, 575, 578 correlation analysis, 131, 132 correlations, viii, 1, 3, 4, 25, 30, 31, 34, 40, 445, 566 Coulomb, xiv, 3, 4, 9, 14, 16, 22, 25, 27, 28, 29, 30, 31, 32, 33, 34, 35, 39, 50, 89, 134, 135, 139, 187, 200, 495, 508, 515, 524, 539, 545, 546, 565, 566, 568, 570, 575, 655 Coulomb interaction, 16, 27, 134, 135, 139, 187, 495, 508, 515, 524, 539, 546, 566, 568, 570, 655 couples, 187, 318, 375 coupling, xii, xiii, xiv, 28, 29, 30, 31, 32, 35, 118, 125, 126, 136, 140, 143, 187, 188, 189, 209, 252, 254, 300, 302, 303, 304, 318, 323, 328, 376, 393, 394, 395, 399, 400, 401, 402, 405, 408, 410, 411, 427, 428, 429, 430, 431, 432,
673 433, 435, 436, 437, 438, 442, 443, 444, 445, 446, 447, 448, 449, 450, 452, 453, 454, 455, 456, 457, 458, 459, 460, 461, 462, 463, 467, 468, 469, 471, 473, 474, 479, 480, 481, 482, 483, 484, 485, 486, 511, 524, 533, 534, 536, 545, 550, 551, 552, 553, 554, 555, 556, 559, 561, 562, 564 coupling constants, 460, 468, 552, 553, 556, 559 coverage, 56, 58, 74, 208, 216, 226, 227, 230, 231, 238, 410 covering, 48, 192, 214, 236, 268, 347, 667 cracking, 367 CRC, 165 credit, 40 critical analysis, 172 critical value, 277, 447, 457, 463, 464, 516, 523, 568 cross-sectional, 205, 231, 232, 236, 270, 272, 292, 666 crosstalk, xii, 393, 394, 399, 401, 406, 408, 409, 410 cross-talk, 402 crystal, vii, viii, 1, 14, 15, 35, 36, 38, 109, 110, 111, 112, 114, 117, 121, 122, 123, 124, 125, 127, 133, 134, 135, 136, 138, 143, 148, 155, 207, 208, 275, 278, 284, 285, 300, 302, 306, 307, 310, 311, 325, 351, 367, 515, 644 crystal growth, 121, 125, 134, 148, 155 crystal lattice, 110, 111, 114, 285 crystal phases, 14, 36 crystal structure, 122, 123, 125, 143, 306, 311, 325 crystal structures, 122 crystalline, viii, xi, 14, 15, 35, 36, 109, 139, 267, 268, 283, 290, 333, 336, 345, 351, 354, 359, 360, 369 crystallinity, 270 crystallites, 122, 364, 592, 593, 601, 602, 603 crystallization, xii, 215, 332, 357, 358, 363, 365, 366, 367, 370 crystallographic, viii, 109, 118, 122, 363, 365 crystals, 119, 125, 244, 274, 282, 285, 364 Curie temperature, xi, 299, 300, 315, 321, 322, 323, 324, 325 curing, 403, 660 curing process, 660 current ratio, 656 cycles, 237, 400, 401, 402, 410, 466, 653, 655, 660, 661, 662 cyclotron, xiv, 4, 5, 14, 24, 312, 577, 579, 590, 603 cysteine, 246 cystine, 258
674
Index
cytokines, 247 cytometry, 244, 251, 424 cytosol, 259 cytotoxic, 258 cytotoxicity, 258
D damping, 187 dark conductivity, 151, 158 data analysis, 56 data set, 409 de Broglie, 143, 494 death, 19, 20 Debye, viii, 52, 62, 65, 97, 109, 122, 124 decay, xiii, 54, 55, 162, 300, 375, 377, 382, 383, 385, 386, 390, 395, 396, 427, 428, 429, 430, 434, 439, 440, 442, 451, 452, 454, 456, 457, 458, 469, 470, 471, 472, 478, 485, 486, 573, 574, 589, 614, 652 decomposition, xi, 232, 267, 268, 270, 336 deconvolution, 54, 56, 81 decoupling, 452 defects, 56, 57, 60, 68, 71, 74, 75, 79, 80, 85, 86, 87, 88, 97, 269, 277, 278, 337, 353, 354, 355, 356, 475, 629, 633, 634, 638, 646 definition, xi, 114, 185, 189, 211, 299, 349, 451, 454, 455 deformation, 170, 173, 277, 286, 289, 526 degenerate, 3, 4, 6, 20, 125, 127, 128, 318, 324, 374, 375, 376, 382, 497, 507, 511 degradation, 80, 215, 217, 247, 624, 641, 655, 656 degree, xi, 126, 134, 144, 150, 236, 268, 280, 281, 292, 294, 296, 371, 444, 455, 459, 473, 475, 477, 483, 538, 539, 633, 641 degrees of freedom, 16, 134, 514 delivery, 244, 248 delta, 55, 80, 92, 115, 211, 332, 374, 431, 547, 580, 589, 590, 597, 618 Delta, 349 demand, 269 dendrimers, 247 dendrite, 364 density functional theory, 17 deoxyribonucleic acid, 402 dependant, 245, 247, 629, 633, 634, 645, 647 dephasing time, 176 deposition, viii, xi, 109, 118, 119, 120, 121, 122, 134, 136, 148, 277, 290, 331, 333, 335, 337, 339, 343, 353, 354, 356, 357, 367, 394, 400, 404, 405, 406, 625 derivatives, 284
desorption, 226 detection, x, 54, 57, 126, 136, 148, 243, 244, 248, 249, 250, 251, 259, 268, 373, 374, 385, 390, 407, 416, 417, 418, 420, 421, 422, 424, 635 deviation, 131, 190, 191, 271, 343, 349, 534 DI, 402, 404, 405 Diabetes, 416 diagnostic, 244, 250, 251 diamagnetism, 5 diamond, 125 Diamond, 298 dielectric, xii, 4, 51, 65, 72, 134, 135, 214, 305, 315, 332, 333, 335, 343, 345, 348, 352, 353, 356, 357, 362, 367, 369, 393, 399, 408, 578, 581, 588, 589, 604, 609, 615, 618, 620, 656 dielectric constant, 4, 72, 134, 135, 305, 333, 348, 357, 362 dielectric materials, 393 dielectric permeability, 581, 588, 609, 620 dielectric permittivity, 65, 352 dielectrics, 333, 357 differentiation, 221, 284, 383, 386 diffraction, vi, xii, 122, 123, 124, 274, 275, 282, 283, 288, 393, 394, 402, 403, 406, 408, 409, 410, 411 diffusion, 17, 19, 20, 21, 22, 38, 174, 192, 198, 332, 334, 347, 356, 475, 629, 631, 632, 633, 635, 646, 647 diffusion process, 347, 635 diffusion time, 192 dimensionality, 12, 28, 29, 40, 48, 125, 197, 547, 624 diminishing returns, 402 diode laser, 206, 208 diodes, ix, x, 50, 95, 97, 169, 170, 172, 194, 196, 199, 200, 201, 203, 207, 208, 300, 328 dipole, xv, 89, 129, 131, 373, 376, 377, 382, 383, 385, 386, 397, 505, 515, 516, 577, 578, 588, 594, 609, 616 dipole moment, 131, 373, 382, 397, 505, 516 dipole moments, 373 direct observation, 73 Dirichlet boundary conditions, 546 discontinuity, 52, 524, 538 discreteness, 2 discretization, 180, 382 dislocation, 56, 74, 86, 231, 233, 269, 274, 275, 277, 278, 279, 282, 285, 286, 287, 288, 289, 290, 294, 296, 354 dislocations, x, 74, 217, 229, 267, 268, 269, 274, 278, 281, 282, 285, 286, 287, 288, 289, 296, 354 disorder, 144, 150, 475
Index dispersion, xiv, xv, 111, 113, 125, 130, 131, 157, 170, 185, 339, 345, 354, 356, 577, 578, 589, 590, 596, 607, 608, 614, 616, 617, 627, 630, 632 displacement, 18, 207, 258, 278, 284, 377, 381, 590, 603 dissociation, xi, 121, 268, 279, 290, 331, 333 dissociation temperature, 268, 290 dissolved oxygen, 119 distortions, 289 distribution, viii, 22, 48, 51, 55, 62, 71, 72, 75, 78, 80, 90, 93, 97, 101, 119, 144, 151, 170, 171, 172, 174, 176, 177, 181, 185, 190, 193, 197, 209, 217, 222, 237, 259, 269, 271, 272, 285, 286, 290, 291, 294, 296, 382, 384, 395, 399, 400, 405, 414, 421, 422, 456, 457, 496, 505, 516, 523, 528, 529, 632, 641, 647, 659, 660, 664 distribution function, 144, 151 divergence, xv, 209, 399, 555, 561, 578, 605 diversity, 373 division, 204, 206, 207, 458 DNA, 247, 394, 402, 403, 404, 405, 410, 415, 424 dominance, 570, 629 donor, 151, 250, 656, 665 donors, 259 dopamine, 255, 257, 258 dopant, 2, 208, 245 dopants, 208 doped, 50, 51, 57, 72, 76, 89, 94, 207, 208, 245, 300, 315, 316, 317, 320, 321, 328, 329, 374, 655, 657 doping, 50, 51, 54, 57, 65, 72, 74, 78, 81, 94, 98, 192, 198, 317, 325, 374 drug delivery, 252 drug discovery, x, 243, 258, 259 drugs, 243, 244, 247, 259 dry, 403, 404, 405 drying, 403, 404, 405, 415, 416 durability, xvi, 652 duration, 86, 381, 382, 383, 386, 389, 395, 403, 634 dyes, 245, 248, 249
E E.coli, 251 economic, 268 efficacy, 251 eigenenergy, 553, 573, 574 eigenvalue, xiv, 11, 25, 26, 440, 447, 448, 449, 450, 457, 462, 470, 475, 545, 559
675 eigenvalues, xiii, xiv, 11, 26, 27, 111, 117, 427, 429, 430, 433, 434, 438, 439, 442, 443, 446, 447, 448, 450, 453, 455, 456, 457, 458, 461, 462, 463, 464, 465, 466, 468, 470, 476, 477, 478, 480, 485, 545, 547, 551, 553, 556, 557, 560, 561, 563, 569, 571 Eigenvalues, 439 eigenvector, 443, 463, 549 elastic constants, 286 elasticity, 524, 538 electric current, 146 electric field, xi, 51, 52, 62, 67, 70, 72, 85, 90, 100, 103, 299, 305, 309, 313, 314, 323, 325, 328, 386, 494, 505, 506, 507, 521, 522, 523, 654, 655, 656, 665 electrical, viii, ix, xi, xvi, 47, 48, 49, 57, 76, 79, 80, 81, 89, 90, 102, 109, 116, 134, 145, 146, 147, 148, 149, 150, 151, 152, 153, 154, 156, 157, 170, 171, 217, 331, 335, 345, 356, 357, 374, 523, 651, 653, 655, 656, 659, 661, 665, 667 electrical conductivity, 145, 149, 150 electrical properties, 76, 81, 89, 170, 171, 217, 335, 523 electrical resistance, ix, 109, 145, 146, 147, 148, 149, 150, 151, 152, 153, 154, 156, 157 electrochemical, 3 electrodeposition, 122 electrodes, vii, 2, 144, 145, 153, 653, 655, 656, 659, 664 electroluminescence, 357 electromagnetic, 153, 159, 163, 204, 374, 390, 514, 588, 634 electromagnetic wave, 374, 390, 588 electron beam, 332, 335 electron charge, 2, 70, 134, 151, 371, 516, 579 electron density, 514 electron diffraction, 340, 343, 363 electron gas, vii, 1, 76, 565, 574 electron microscopy, xi, 297, 331, 334, 343, 356, 425, 659 electron state, 68, 128, 193, 316, 318, 319, 320, 322, 356, 372, 375, 471, 497, 534, 539, 566, 567, 568, 579, 584, 603 electronic structure, viii, xi, xiv, 16, 38, 47, 48, 50, 57, 76, 85, 116, 137, 197, 198, 237, 299, 300, 302, 305, 325, 328, 372, 493, 494, 496, 508, 523, 539, 540, 655 electronic systems, 2, 17, 38, 39 electronics, 48, 126, 371, 393, 604, 624, 652, 658 electrostatic, iv, 9, 10, 57, 65, 247, 523, 660 electrostatic interactions, 247 EM, 204, 280, 282, 288, 414, 421
676 embryo, 422 embryonic, 422 emission source, 207 emitters, ix, 169, 170, 268 encapsulated, 245 encoding, 422, 424 endogenous, 253 endoplasmic reticulum, 421 endothelial cell, 251, 253 endothelial cells, 251, 253 endothelium, 251 endurance, 656 energy density, 335, 526 energy emission, 177 energy level splitting, 311 energy transfer, 250, 257, 394, 399, 400, 401, 408, 410 energy-momentum, 211 engineering, ix, xii, 2, 48, 110, 170, 212, 217, 231, 332, 356, 358, 367, 484, 642, 667 England, 195 enlargement, 136, 268 entanglement, 538, 539 entropy, 197 envelope, xi, xiii, 139, 220, 299, 300, 302, 303, 328, 374, 381, 493, 509, 515, 524, 529, 530, 539 environment, xiii, xv, 246, 372, 424, 427, 428, 429, 430, 432, 438, 443, 444, 445, 452, 453, 454, 455, 456, 458, 459, 462, 467, 485, 486, 578, 626 environmental, 514 enzymatic, 244, 248 enzymatic activity, 248 enzyme, 259 enzymes, 244, 248 epidermal, 251 epidermal growth factor, 251 epitaxial growth, 276, 294 epitaxy, 367, 514, 517, 624 equilibrium, ix, 18, 50, 52, 89, 110, 120, 121, 148, 149, 153, 156, 157, 158, 160, 162, 163, 177, 197, 246, 281, 379, 380, 383, 387, 395, 398, 604 equilibrium state, 52, 383 equipment, 52, 407 erbium, 208 Escherichia coli, 251 ester, 195, 247, 252 estradiol, 415 estrogen, 422, 423 etching, 343, 344, 358, 359, 360, 361, 362, 517, 638
Index ethanol, 257 ethylene, 404 Euler, 608 European, 194 evaporation, 290, 332, 333, 335, 336, 355, 363, 643 evidence, viii, 47, 49, 57, 74, 92, 102, 174, 259, 336, 367, 373 evolution, xii, 38, 56, 88, 110, 127, 133, 191, 327, 376, 380, 382, 383, 385, 386, 387, 388, 389, 405, 457, 464, 470, 480, 514, 515, 546, 563, 570, 572, 604 excitation, ix, xii, 49, 52, 58, 68, 70, 71, 73, 76, 78, 81, 86, 89, 93, 94, 102, 103, 110, 142, 149, 171, 198, 213, 232, 234, 245, 246, 354, 355, 356, 371, 372, 373, 374, 375, 380, 382, 383, 385, 386, 388, 390, 393, 395, 398, 416, 420, 428, 434, 453, 458, 459, 517 exciton, 134, 141, 174, 176, 192, 193, 194, 211, 245, 300, 327, 332, 372, 373, 374, 375, 382, 395, 397, 399, 411, 496, 508, 527, 530, 538 exotic, 14, 15 expansions, 12 experimental condition, 119, 123, 136, 148, 638 experimental design, 115 expert, iv experts, vii, 133 explosive, 126, 249 exponential, 19, 20, 21, 51, 54, 55, 56, 91, 149, 162, 383, 390, 399, 408 exposure, xii, xv, 252, 258, 413, 414, 418, 420, 421, 623, 625, 635, 636, 637, 638, 639, 640, 641, 642, 643, 644, 646, 647 Exposure, 636, 643 external magnetic fields, vii, 1 extinction, 245, 398 extracellular, 251 extravasation, 248 extrinsic, 150 eye, 29, 32, 34, 200 eyes, 208, 224, 225, 234
F fabricate, 2, 268, 493, 517, 533 fabrication, xii, 2, 93, 162, 268, 269, 282, 290, 315, 357, 393, 394, 397, 398, 401, 402, 405, 410, 411, 514, 515, 653, 658, 662, 666, 667 failure, 275, 549 family, 252, 268, 292 fast processes, 428 faults, 74 feedback, 170, 185, 200, 209
Index Fermi, 13, 14, 15, 23, 35, 39, 52, 60, 61, 62, 64, 65, 67, 68, 71, 72, 74, 95, 97, 101, 151, 152, 177, 190, 210, 211, 325, 396, 397, 398, 474, 515, 534 Fermi energy, 52, 68, 71, 152 Fermi level, 60, 61, 62, 64, 67, 72, 95, 97, 101, 151, 152, 177, 210, 211, 325, 396, 397 Fermi liquid, 13, 14, 15, 35, 39, 474 Fermi-Dirac, 60 fermions, 23 ferromagnetic, 315, 323, 324 ferromagnetism, xi, 299, 321, 322, 323, 324, 325, 329 Feynman, 381, 391 fiber, x, 203, 204, 205, 207, 208, 209, 239, 393, 407, 409, 410 fiber Bragg grating, 205 fibers, 235, 407 fibronectin, 253, 422 fidelity, xii, 371, 373, 374, 383 film, ix, xi, 56, 109, 118, 119, 124, 125, 129, 131, 134, 136, 141, 145, 146, 147, 153, 155, 156, 157, 331, 333, 334, 335, 336, 337, 343, 344, 346, 347, 348, 349, 350, 351, 352, 353, 354, 355, 356, 357, 358, 359, 360, 361, 362, 363, 364, 366, 367, 647, 652, 653, 655, 656, 657, 659, 661 film thickness, 146, 350, 647 films, viii, xi, 109, 115, 118, 119, 120, 122, 123, 127, 129, 131, 132, 133, 134, 135, 136, 137, 138, 140, 141, 142, 144, 145, 148, 149, 153, 154, 331, 332, 333, 334, 335, 336, 337, 338, 341, 343, 345, 346, 347, 348, 351, 352, 353, 354, 355, 356, 357, 358, 360, 361, 362, 363, 364, 365, 366, 367, 369, 370, 660 filters, 54 financial support, 238, 667 fine tuning, 134, 407 finite differences, 435, 550 fish, 190 flatness, 192, 508 flexibility, xvi, 405, 624, 651 floating, 656, 658, 659, 667 flow, 244, 249, 251, 349, 350, 351, 358, 399, 400, 424, 634, 635, 642 flow rate, 635, 642 fluctuations, ix, 22, 65, 74, 169, 192, 232, 374, 407, 655 fluorescein isothiocyanate, 418 fluorescein isothiocyanate (FITC), 418 fluorescence, xii, 248, 252, 255, 259, 404, 405, 406, 407, 414, 420, 424 fluorescence in situ hybridization, xii, 414
677 fluorescent markers, 244 fluorophores, 244, 245, 259, 418, 420 focusing, 39 folding, 252, 339, 340 food, 250 Ford, 259, 260 formamide, 415 Fourier, 56, 111, 125, 126, 274, 275, 281, 282, 283, 294, 295 FP, 23, 24 France, 267 free energy, 321, 322, 324 free radical, 257 free radicals, 257 freedom, 126, 170, 371 freedoms, 325 freezing, xiv, 16, 62, 66, 577 fulfillment, 590 full width half maximum, 176 functional analysis, 423 functional memory, 662 funds, 325 fusion, 249, 250 fusion proteins, 250 FV, 328 FWHM, 62, 227, 229, 230, 399, 630, 631, 632, 639, 640, 644, 646, 657
G G protein, 254 gain threshold, 402 gallium, 93 gas, 207, 279, 333, 336, 337, 343, 349, 350, 351, 353, 360, 363, 403, 405, 626, 634, 635, 642 gas diffusion, 279 gas phase, 403 gases, 348 gauge, 4, 5, 25, 508, 579 Gaussian, 7, 15, 24, 25, 28, 33, 34, 36, 39, 55, 58, 81, 123, 151, 176, 206, 283, 627 gene, 243, 248, 259 gene therapy, 259 generalization, 36, 39 generation, xi, xii, 126, 149, 153, 156, 157, 158, 160, 208, 257, 267, 281, 296, 371, 372, 376, 390, 411, 646, 659, 664 generators, 378, 380, 381 germanium, xi, 331, 332, 333, 334, 335, 336, 337, 338, 339, 346, 347, 356, 357, 361, 362, 363, 364, 370 glass, 119, 123, 207, 357, 367, 415, 622, 659, 660, 662, 663
678
Index
glasses, 332, 335, 357 glassy films, 358 glia, 250 glial, 250 glioblastoma, 251 gloves, 415 glucose, 252 glycerol, 247 glycine, 251 glycol, 247 God, 357 gold, 394, 409, 652, 653, 654, 655, 656, 657 gold nanoparticles, 652, 653, 654, 655, 656, 657 government, iv G-protein, 254 grain, 151 graph, 549 gratings, 205 Green's function, 21, 22 grids, 560 Ground state, 33, 34, 540 ground state energy, 11, 18, 20, 21, 23, 27, 29, 30, 32, 33, 34, 35, 500, 502, 506, 541, 585, 586 groups, 93, 102, 127, 136, 139, 170, 174, 176, 181, 182, 249, 259, 268, 347, 403, 405, 452, 474, 625, 639, 643 growth factor, 251 growth hormone, 414, 422, 423, 425 growth rate, 246, 279, 334, 336, 338 growth temperature, 58, 74, 94, 226, 227, 229, 230, 231, 238, 268, 270 guanine, 665 guidance, 49 gyroscope, 204
H H1, 537 H2, 333, 537, 624 Hall resistance, 574 halogen, 207, 626 Hamiltonian, xiv, 4, 5, 10, 11, 18, 19, 20, 23, 24, 25, 26, 27, 29, 127, 138, 140, 220, 222, 300, 301, 302, 303, 304, 306, 315, 373, 376, 377, 378, 380, 381, 429, 432, 433, 435, 436, 437, 443, 445, 447, 450, 452, 456, 459, 460, 461, 462, 463, 464, 466, 467, 468, 469, 470, 476, 480, 498, 499, 505, 508, 509, 510, 511, 512, 515, 518, 523, 525, 526, 527, 530, 534, 545, 547, 550, 551, 552, 553, 554, 555, 556, 557, 558, 560, 562, 563, 565, 568, 569, 571, 572, 575, 579, 580, 587, 588, 594, 596, 609
handling, 415 harmonics, 140 heat, 3, 157, 252, 347, 407, 635 heating, 152, 335 heating rate, 335 helicity, 385, 387, 388 helium, 81, 94, 333 Helmholtz equation, 546, 547, 550 Hermitian operator, 430, 438, 453, 454 heterogeneous, 119, 121, 363, 367 heterostructures, xi, xii, 48, 103, 195, 268, 269, 274, 282, 292, 327, 332, 345, 349, 350, 353, 367, 517, 572 high resolution, 205, 274, 280, 281, 282, 283, 285, 288, 294, 296, 345 high temperature, 57, 62, 98, 142, 268, 321, 333, 354, 363, 624, 629, 634, 641 high-density memory, xvi, 651 Hilbert, 167, 550 Hilbert space, 550 histochemical, 420 histogram, 231, 232, 233, 234, 271, 290, 343 Holland, 390 HOMO, 125, 127, 655, 656 homogeneity, 269, 394, 517 homogeneous, 14, 79, 121, 176, 187, 188, 197, 358, 359, 361, 362, 366 homogenous, 176, 179, 187 Honda, 41 hormone, 414, 415, 422, 423, 425 hormones, 421, 423 host, vii, 1, 48, 49, 103, 249, 250 House, 263 human, vii, 250, 251, 252, 254, 422, 424 human estrogen receptor, 250 humans, 258 hybrid, xiv, 577, 578, 592, 603, 652, 661, 663, 664, 667 hybridization, xiii, 402, 403, 404, 414, 415, 416, 418, 424, 425, 578, 587 hydro, 254 hydrofluoric acid, 407 hydrogen, 78, 138, 660 hydrolysis, 120 hydrophilic, 254 hydrophobic, 254 hydroxide, 121 hydroxyl, 403, 405 hydroxyl groups, 405 hydroxylation, 403 hyperbolic, 381 hyperfine interaction, 373, 375, 382 hypothalamus, 422
Index hypothesis, 93, 174, 181, 192, 293, 375, 387 hysteresis, 322, 335, 357, 656, 657, 659, 663, 667 hysteresis loop, 322
I icosahedral, 661 identification, 69, 91, 93, 101, 118, 122, 414, 423, 450, 621 identity, 554 III-nitrides, x, 267 illumination, 163, 244, 246, 259 image analysis, 424 images, xii, 222, 227, 228, 229, 232, 236, 237, 250, 257, 274, 275, 281, 283, 284, 285, 286, 288, 294, 300, 345, 363, 366, 405, 414, 417, 418, 419, 420, 421, 627, 630, 633, 636, 638, 658, 659, 664, 666 imaging, x, xii, 58, 203, 205, 208, 244, 245, 247, 251, 252, 253, 254, 255, 259, 406, 407, 413, 414, 421, 423, 424 imaging systems, 259 imaging techniques, 205 immersion, 404, 405, 415 immunoassays, 249 immunocytochemistry, 423 immunoglobulin, 416, 420 immunohistochemical, 414, 416 immunohistochemistry, xii, 414, 423 implementation, 18, 23, 146, 372, 373 impurities, 74, 120, 148, 315, 333, 473, 474, 569, 573, 604 in situ, 58, 94, 367, 414, 422, 423, 424, 517 in situ hybridization, 414, 422, 423, 424 in vitro, 248, 259 in vivo, 244, 248, 251, 254, 258, 259 inactive, 56 incandescent, 207 incidence, 334 inclusion, 22, 192, 222, 296, 511 independence, 139 indexing, 111 indication, 25, 97, 129 indices, 111, 401 indium, 524, 655, 659, 660, 662 indium tin oxide, 655, 660, 662 indium tin oxide (ITO), 662 individuality, 424 induction, 579, 590, 591, 593, 597, 598, 601, 602, 634 industry, 211 inelastic, 439 inequality, 445
679 inert, 147, 149, 247, 259, 356 infancy, 667 infection, 249, 250 infections, 249 infinite, 6, 7, 8, 9, 13, 20, 32, 34, 39, 139, 170, 218, 220, 341, 435, 436, 437, 457, 461, 469, 473, 476, 555 information processing, 300, 328, 373 information technology, 667 infrared, ix, 48, 169, 268, 533, 624 infrastructure, 652 inherited, x, 203, 211, 234 inhibition, 251 inhibitors, 251 inhomogeneity, x, 177, 203, 211, 217, 224, 234 initial state, 383, 386, 390 injection, 145, 170, 174, 182, 183, 187, 188, 190, 191, 194, 196, 197, 198, 210, 213, 332, 374, 415 injury, iv Innovation, 667 inorganic, 420, 662, 664 InP, vi, xv, 170, 172, 174, 178, 195, 200, 231, 278, 327, 328, 623, 624, 625, 626, 629, 632, 634, 635, 636, 637, 638, 639, 640, 641, 642, 646 insight, x, xii, 24, 38, 48, 76, 91, 103, 156, 219, 243, 259, 393, 430 inspection, 95, 136 instability, 295 instruments, 368 insulators, xvi, 651 insulator-semiconductor, 50, 656 integrated circuits, 393 integration, x, xv, xvi, 9, 19, 56, 221, 222, 243, 269, 387, 411, 551, 590, 615, 623, 624, 642, 646, 647, 652 integrin, 251, 252, 253 interactions, x, 9, 16, 17, 136, 243, 244, 255, 259, 373, 475, 539 interface, x, 57, 76, 79, 81, 85, 88, 102, 231, 267, 276, 277, 282, 285, 286, 288, 289, 294, 295, 296, 335, 337, 339, 343, 354, 355, 356, 357, 367, 399, 528, 625, 629, 657, 659, 660 interface energy, 276 interference, x, xiv, 143, 203, 204, 205, 273, 336, 447, 472, 478, 545, 546, 575 internalization, 252, 257 interpretation, 20, 21, 89, 92, 96, 128, 137, 151, 367, 375, 451 interstitial, 110, 333, 630 interstitials, 74, 75
680
Index
interval, ix, 96, 101, 110, 111, 112, 122, 149, 158, 160, 161, 162, 204, 383, 385, 386, 404, 438, 455, 458 intramuscularly, 415 intrinsic, ix, x, 49, 57, 60, 67, 68, 69, 70, 73, 76, 80, 84, 85, 87, 91, 93, 95, 97, 100, 101, 102, 103, 109, 123, 126, 129, 134, 148, 150, 152, 153, 203, 211, 267, 268, 474 intron, 422 invasive, 251 inversion, 236, 382, 384 Investigations, v, 47 ion bombardment, 269, 634 ion implantation, 214, 332, 624, 641 ionic, 121 ionization, 67, 148, 149, 150, 619, 621 ionization energy, 150 ions, 119, 120, 121, 300, 315, 321, 324, 325, 332, 349, 373, 377, 382, 514, 624, 635, 659 IR spectra, 358, 359, 360, 361 irradiation, 214 island, 48, 56, 57, 231, 233, 289, 374, 494, 496 island formation, 57, 494 isolation, 148 isothermal, 49, 56 isotropic, 4, 6, 125, 128, 137, 138, 139, 286, 304, 342, 376, 497 I-V curves, 659, 661
J Jefferson, 43 Jordan, 429, 447 Josephson junction, 514 Jung, 242, 490, 658, 659, 660, 668 justification, x, 5, 8, 203, 375, 386
K kidneys, 258 kinetic energy, 19, 21, 33, 135, 304 kinetics, 158, 160, 162 King, 260
L labeling, 117, 251, 257, 307, 415, 420, 424, 425 Langmuir, 261, 412, 652 language, 117 Laplace transformation, 55 laser, ix, xii, xv, 58, 81, 94, 169, 170, 172, 173, 174, 175, 176, 178, 181, 182, 183, 184, 185,
186, 187, 188, 190, 191, 193, 196, 197, 199, 200, 201, 205, 208, 209, 214, 215, 227, 244, 334, 349, 354, 355, 356, 373, 374, 375, 387, 395, 407, 413, 414, 423, 424, 425, 429, 430, 439, 444, 451, 453, 471, 546, 578, 623, 624, 627, 642 laser pointer, 209 laser radiation, 642 lasers, ix, xii, 2, 80, 169, 170, 171, 172, 178, 181, 182, 184, 185, 186, 187, 192, 194, 195, 196, 197, 198, 199, 200, 206, 207, 208, 211, 217, 268, 334, 337, 354, 393, 408, 578, 624 lattice, viii, 15, 36, 47, 48, 56, 57, 70, 74, 75, 78, 79, 87, 90, 102, 110, 111, 114, 125, 135, 144, 157, 170, 232, 268, 272, 273, 277, 278, 279, 280, 281, 282, 284, 285, 292, 295, 296, 357, 367, 373, 375, 377, 382, 435, 436, 437, 494, 517, 523, 526, 551, 560, 625, 629, 630, 633, 634 lattice parameters, 272, 273, 367, 494, 517, 526 lattices, 285, 367 law, 158, 207, 292, 393 lead, vii, xi, xii, xiii, 1, 22, 40, 110, 121, 126, 131, 148, 154, 170, 181, 185, 190, 193, 217, 243, 248, 252, 257, 268, 283, 331, 332, 352, 359, 367, 394, 399, 411, 427, 429, 435, 436, 437, 450, 461, 478, 484, 508 leakage, 135, 141, 409 lens, 171, 172, 222, 223, 224, 374, 524 lenses, 407, 498 leukemic, 251 Lie algebra, 378, 380, 381 lifetime, 156, 159, 162, 175, 184, 355, 375, 430, 431, 473, 474, 475, 508, 515, 547 ligand, 246, 247, 248, 255, 256, 258 ligands, 246, 247, 248, 251, 253, 254, 255, 256, 258 light emitting diode, 208, 209 light scattering, 121 likelihood, 245 limitation, 22, 40, 54, 98, 616, 629 limitations, 6, 8, 192, 199, 244, 246, 421 linear, ix, xi, 11, 25, 26, 51, 70, 110, 115, 116, 123, 131, 132, 134, 136, 137, 141, 145, 150, 151, 152, 158, 160, 161, 162, 163, 164, 208, 209, 254, 282, 285, 292, 299, 301, 307, 321, 325, 357, 387, 395, 439, 460, 549 linear dependence, 131, 302 linear function, 141, 152, 162, 321, 460 liquid nitrogen, 227 liquid phase, 15 Listeria monocytogenes, 250
Index literature, viii, x, 37, 69, 109, 112, 118, 121, 122, 127, 129, 133, 134, 135, 139, 140, 143, 151, 156, 171, 173, 184, 244, 245, 258, 267, 269, 276, 278, 340, 358, 429, 430, 440, 474, 515 lithography, 403, 405 liver, 252, 258 localization, xiii, 35, 135, 148, 154, 211, 274, 341, 372, 414, 415, 421, 422, 424, 425, 505, 508, 524, 555 location, 64, 67, 72, 103, 274, 278, 294, 373, 397 logging, 408 long period, 89, 259 longevity, x, 243 low molecular weight, 254 low power, 191, 207, 245 low temperatures, 62, 65, 92, 334, 354, 357, 363, 366, 375, 473, 508, 517, 647 low-density, 624 low-power, 624 low-temperature, xii, 75, 81, 91, 270, 291, 332, 367, 629, 636 luminescence, 76, 90, 103, 215, 300, 404 luminescence efficiency, 76, 103, 215 lying, xi, 138, 142, 267, 377, 382, 383, 430, 440, 441, 442, 445, 446, 469, 478 lymph, 248, 252 lymph node, 248, 252 lymphocytes, 424 lymphoid, 250 lymphoid tissue, 250 lymphoma, 424 lysine, 246, 252
M magnetic field, vii, xi, xiii, xiv, 1, 2, 3, 4, 5, 6, 13, 14, 15, 16, 17, 23, 24, 25, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 112, 126, 299, 300, 305, 306, 309, 310, 311, 312, 313, 317, 318, 319, 320, 321, 322, 325, 328, 399, 493, 494, 507, 508, 538, 553, 577, 578, 579, 580, 581, 584, 585, 586, 587, 589, 590, 591, 592, 593, 594, 596, 599, 603, 613 magnetic properties, xvi, 651 magnetic resonance, 205, 373 magnetic resonance imaging, 205 magnetization, 321, 322, 323, 324, 325 magnetron, 332 maintenance, 206 malignant, 249 malignant cells, 249 manganese, 329 Manganese, 300, 329
681 manifold, 463 manipulation, 372, 373, 514 manufacturer, 398, 421 manufacturing, 667 many-body problem, 453 mapping, 248, 283, 295, 465, 466, 467, 559 market, 269 Markovian, 387 mask, xv, 623, 625, 636, 642, 643, 644, 645, 646, 647 mass spectrometry, 327 master equation, xii, 181, 199, 371, 373, 378, 379 materials science, 652, 667 mathematical, 55, 131, 147, 351, 388, 429, 430, 431, 440, 459, 622 Mathematical Methods, 42 Maxwell equations, xii Maxwell's equations, 382, 383 MB, 328 MBE, 57, 69, 74, 75, 76, 78, 79, 94, 143, 170, 623 mean-field theory, 321, 325 meanings, 134 measurement, 49, 50, 52, 57, 59, 60, 61, 62, 63, 66, 69, 87, 91, 92, 94, 96, 97, 146, 147, 157, 184, 185, 204, 289, 323, 336, 351, 375, 626, 629, 643, 667 mechanical, iv, xvi, 2, 3, 245, 270, 335, 450, 570, 651 mechanics, 369, 473, 622 media, 117, 351 median, 401 medicinal, x, 243, 244 melanoma, 251, 252 melting, 215, 332 membranes, 343, 344 memory, xii, xvi, 3, 76, 103, 332, 343, 357, 367, 651, 652, 653, 655, 656, 657, 658, 659, 660, 661, 662, 663, 664, 665, 666, 667 mercury, 334, 354, 355 mesoscopic, 37, 126, 458, 483, 488, 547, 573, 574 messenger ribonucleic acid, 423 messenger RNA, 422 metabolic, 245 metal nanoparticles, 656 metal organic chemical vapor deposition, 231 metal oxide, 667 metal-oxide-semiconductor, 652 metals, xvi, 474, 574, 575, 651, 652 metaphase, 424 metastasis, 251 metastatic, 248, 252
682 metric, 5 Metropolis algorithm, 18, 19 mice, 252 microarray, 249 microelectronics, 122, 126, 325 microfabrication, 2, 3 micrometer, 110, 333 microscope, 119, 171, 335, 363, 366, 402, 407, 424, 664 microscopy, xi, xii, 250, 331, 373, 404, 413, 414, 417, 418, 423, 424, 425, 517 microtubules, 424 microwave, 429, 430, 439, 440, 448, 450, 481, 484, 486, 570 microwaves, 539 migration, 75, 227, 229, 251, 629, 630 miniaturization, 110, 652 mining, 178 mirror, 205, 407, 450 misfit dislocations, x, 231, 267, 277, 281, 282, 285, 286, 288, 290, 294, 296 mixing, 120, 128, 138, 140, 142, 143, 237, 373, 374, 394, 496, 497, 504, 505, 507, 508, 517, 520, 523, 528, 538, 539, 541 mobility, 76, 150, 151, 268, 269, 295, 357, 641, 655 model system, 13, 466, 569 modeling, xii, 14, 196, 393, 394, 399, 402, 408, 410 models, 3, 6, 35, 37, 39, 133, 171, 182, 194, 199, 276, 277, 283, 296, 352, 373, 390, 397, 430, 508, 523, 538, 556, 575 modulation, ix, 62, 76, 95, 169, 170, 178, 182, 183, 184, 186, 188, 190, 192, 197, 198, 199, 200, 328, 374, 381, 393, 578 modules, 470, 471 modulus, 483 moisture, 347 molar ratio, 334, 349 mole, 334, 363 molecular beam, 57, 122, 226, 374, 623 molecular beam epitaxy, 57, 122, 226, 374, 623 molecular orbitals, 137, 539 molecular-beam, 514 molecules, xii, 2, 38, 244, 247, 355, 404, 414, 420, 421, 494, 523, 538, 546, 665 momentum, 11, 24, 25, 113, 126, 138, 140, 176, 302, 303, 306, 326, 327, 341, 374 monochromatic light, 122 monochromator, 58, 153, 354 monolayer, 170, 216, 226, 227, 230, 231, 238, 276, 281, 402, 403, 404, 405, 524, 525, 529, 656, 657
Index monolayers, 48, 74, 277, 294, 629 monolithic, xv, xvi, 623, 624, 642, 646, 647 monotone, 616 Monte Carlo, 17, 18, 19, 21, 38, 45, 394, 395, 400, 402, 410, 411 Monte Carlo method, 17, 21, 38 morphological, 269, 290, 292, 296 morphology, 226, 230, 238, 269, 270, 279, 290, 295, 517, 625, 627 MOS, xii, 332, 335, 343, 357, 367, 652 mosaic, 283, 664 Moscow, 166, 369 motion, vii, 1, 2, 3, 26, 27, 115, 134, 376, 380, 432, 448, 565, 569, 624 motivation, 3, 9 mouse, 251 movement, 289, 295, 312, 624 multiples, 383, 386, 588 multiplexing, 204, 206, 207, 245, 258 multiplicity, 249 muscle, 249, 422 muscle tissue, 249, 422 mutant, 661 myosin, 422
N nanoclusters, xi, 128, 136, 138, 331, 332, 334, 335, 347, 351, 352, 357, 369 nanocrystal, xi, 117, 127, 134, 135, 136, 138, 139, 140, 144, 148, 151, 327, 328, 331, 338, 356, 374, 424, 425 nanocrystalline, 136, 145, 151, 667 nanocrystals, xi, xii, 135, 140, 142, 143, 148, 151, 155, 244, 246, 315, 327, 328, 329, 331, 332, 336, 337, 340, 356, 367, 413, 414, 424, 425, 578, 659, 664, 666, 667 nanodots, 315 nanoelectronics, 332, 369 nanofabrication, 3 nanofibers, 654 nanomaterials, 660 nanometer, 110, 138, 244, 332, 420, 629, 661 nanometer scale, 420, 661 nanometers, xii, 135, 170, 341, 393, 399, 402 nanoparticles, 148, 327, 332, 341, 347, 394, 398, 400, 402, 404, 405, 406, 652, 654, 655, 656, 657, 658, 659, 660, 661, 662, 663, 664, 665, 666, 667 nanoribbons, 394 nanoscale structures, 37 nanoscience, 2, 3
Index nanostructures, x, xi, 2, 126, 267, 269, 270, 279, 290, 297, 311, 315, 331, 332, 508, 514, 613, 664 nanosystems, 3 nanowires, 315, 332 nation, 382 natural, 134, 170, 374, 395, 451, 454, 458, 475, 485, 589, 615 neglect, 72, 123, 174, 515 network, x, 206, 267, 277, 282, 283, 285, 286, 287, 288, 289, 294, 295, 296, 634 neurotransmitters, 247 Newton, 239, 571 Newton’s law, 113, 114 next generation, 657, 667 nitride, x, 267, 362 nitrides, 268, 279, 333 nitrogen, 80, 362, 403, 404, 405, 642, 653, 654, 659, 660 Nobel Prize, 48 nodes, 23, 286, 296, 496, 550 noise, 2, 55, 197, 204, 208, 244, 249, 283, 286, 407, 440, 479 non-destructive, 345 nonequilibrium, 161, 215 non-invasive, 205 nonlinear, 289, 347, 373, 374, 375, 467, 468, 551, 578 non-linear, 116 non-linear, 408 nonlinear dynamics, 374 non-uniform, xv, 51, 578, 633, 641, 659 non-uniformity, 659 normal, 151, 251, 257, 271, 278, 334, 363, 415, 416, 420, 422, 423, 454, 547, 629, 630 normal distribution, 271 normalization, 5, 386, 443, 444, 454 normalization constant, 5 novel materials, 2 n-type, 657 nuclear, vii, 2, 247, 373, 375, 377, 382, 416, 428, 429, 431, 439, 440, 441, 451, 458, 459, 574 nuclear many-body problem, 431 nuclear spins, 373, 382 nucleation, x, 58, 94, 119, 121, 227, 267, 276, 278, 279, 295, 296, 347, 363, 367 nuclei, xiii, 247, 375, 427, 428, 429, 430, 439, 458, 459, 484, 485 nucleolus, 424 nucleons, 428, 429, 459 nucleus, 2, 121, 259, 428, 429, 438, 459 numerical analysis, 199, 584, 612
683
O observations, 85, 87, 139, 421, 423 OCT, 205, 206, 235, 415 Ohmic, 58, 94 oligomer, 246 Oncology, 264 one dimension, 115 online, 261, 558, 568, 570 oocytes, 257 openness, 550 operator, xiii, 11, 19, 24, 117, 138, 302, 303, 306, 318, 377, 381, 427, 428, 429, 430, 431, 432, 433, 435, 437, 438, 440, 443, 444, 449, 450, 451, 453, 455, 456, 458, 460, 465, 467, 473, 474, 475, 485, 486, 551, 552, 553, 565, 579, 581, 588, 606 opioid, 254 optical communications, ix, 169 optical fiber, x, 203, 204, 205 optical gain, 190, 197, 198 optical parameters, 348 optical properties, viii, ix, xi, xiii, xv, 47, 48, 118, 125, 126, 129, 134, 136, 141, 192, 226, 230, 232, 234, 238, 282, 300, 305, 328, 331, 335, 336, 345, 346, 349, 352, 356, 367, 369, 493, 494, 523, 524, 578, 604, 652 optical pulses, 372 optical systems, 514 optical transmission, 393 optics, 90, 200, 411, 578, 587, 622 optimization, x, 28, 79, 103, 192, 203, 212, 222, 224, 237, 245, 407 optoelectronic, xv, 48, 195, 268, 269, 279, 282, 290, 307, 493, 623, 624, 642, 660 opto-electronic, xii opto-electronic, 332 opto-electronic, 357 opto-electronic, 367 opto-electronic, 578 opto-electronic, 604 optoelectronic devices, xv, 48, 268, 269, 282, 290, 307, 493, 623, 624, 642, 660 optoelectronic properties, 268, 279 optoelectronics, xi, 48, 116, 269, 325, 331, 332, 357, 624 orbit, 118, 126, 127, 128, 136, 138, 139, 140, 142, 143, 144, 303, 318, 323, 526, 664 organelles, xii, 413, 414, 421, 423, 424 organic, xvi, 196, 244, 245, 246, 420, 651, 652, 653, 656, 657, 662, 666, 667 organic solvent, 246 organic solvents, 246
684
Index
organization, 117, 143 orientation, xi, xii, 40, 126, 267, 270, 272, 273, 274, 280, 281, 283, 284, 286, 292, 296, 311, 324, 371, 372, 373, 374, 375, 383, 384, 385, 386, 388, 390 orthogonality, 444, 554 oscillation, xv, 62, 514, 516, 578, 616 oscillations, 187, 200, 524, 547, 573, 587, 592, 603, 616 oscillator, xv, 4, 6, 11, 12, 14, 16, 27, 37, 116, 577, 578, 579, 580, 585, 586, 587, 603, 604, 605, 609, 614, 616, 619, 621 osmium, 423 ovarian, 251 oxidation, xi, 119, 331, 334, 335, 347, 356, 655 oxidative, 257 oxidative damage, 257 oxide, xi, 119, 121, 246, 331, 334, 337, 338, 339, 346, 347, 357, 362, 364, 370, 394, 658 oxide nanoparticles, 658 oxide thickness, 357 oxides, 333, 363 oxygen, 119, 358, 403, 405 oxygen plasma, 403, 405 oxytocin, 422
P PA, 297 PACS, 545 paper, 115, 118, 136, 151, 300, 306, 375, 390, 414, 430, 439, 471, 547 parabolic, xiv, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 16, 24, 130, 131, 157, 495, 505, 507, 508, 517, 577, 593 paramagnetic, 248, 324 particle collisions, 396 particle density, 19, 20 particles, 18, 19, 22, 23, 34, 38, 110, 121, 139, 245, 350, 394, 399, 400, 401, 402, 404, 420, 432, 493, 560, 652, 654, 656, 658, 660, 664, 667 passive, xv, xvi, 206, 623, 642, 647 pathogens, 251 pathophysiological, 414, 423 pathways, 244 patterning, 402, 406 pears, 428 pedagogical, 39 penalty, 197 peptide, 244, 251, 252, 253, 254, 258, 423
performance, x, 48, 80, 103, 191, 192, 197, 199, 203, 212, 231, 239, 268, 394, 408, 411, 624, 633, 654 periodic, 110, 111, 113, 114, 115, 205, 395, 497, 499 periodicity, 110, 111, 393 permit, x, 243, 244, 257, 259, 374, 375 permittivity, xi, 51, 72, 331, 357 perturbation, 11, 376, 456, 473, 495, 496, 505, 556 perturbation theory, 11, 496, 505, 556 perturbations, 457 P-glycoprotein, 251 pH, 119, 120, 122, 247, 415 pH values, 122 phage, 253 phase diagram, 14, 516 phase shifts, 470 phase space, 19, 23 phase transformation, 357, 363 phonon, 48, 57, 76, 99, 103, 339, 341, 356, 377, 382, 495, 515, 524 phonons, 129, 131, 197, 337, 354, 356, 515 phosphate, 404, 415 photobleaching, 244, 245, 249, 409, 420 photoconductivity, ix, 91, 110, 118, 119, 148, 151, 153, 154, 156, 157, 158, 162, 163 photodetectors, 624 photoelectrical, ix, 109, 118, 119, 120, 145, 153, 156 photoexcitation, 145, 158 photoluminescence, viii, xi, xii, 47, 49, 74, 80, 81, 89, 136, 142, 213, 226, 268, 327, 331, 332, 334, 336, 371, 375, 382, 383, 385, 386, 387, 388, 389, 390, 508, 661 photoluminescence spectra, 81, 213, 661 photon, 129, 130, 131, 156, 157, 170, 171, 175, 176, 179, 180, 181, 184, 185, 187, 196, 209, 245, 246, 346, 347, 374, 383, 394, 395, 396, 397, 399, 589, 590, 591, 597, 598, 609, 616, 618 photonic, xii, xv, 181, 333, 393, 406, 411, 623, 642, 660 photonic crystals, 333, 394 photons, ix, 156, 169, 171, 175, 176, 177, 179, 181, 186, 188, 190, 356, 373, 394, 395, 399 physical properties, xi, xvi, 3, 37, 244, 299, 300, 325, 448, 624, 651 physicochemical, 117 physicochemical properties, 117 physics, vii, xv, 2, 4, 24, 34, 40, 48, 93, 103, 114, 115, 116, 126, 133, 172, 174, 192, 245, 297,
Index 299, 300, 372, 428, 430, 474, 547, 622, 623, 667 physiology, 258 piezoelectric, 279, 367 Piezoelectric effect, 496 piezoelectricity, 222 pituitary, 414, 415, 421, 422, 423, 425 pituitary gland, 415, 422, 423 PL emission, 58, 80, 87, 625, 627, 629, 630, 636, 638, 641, 644 PL spectrum, 58, 94, 354, 355, 636 planar, 286, 288, 292, 332, 335, 364, 365, 666 Planck constant, 70 plane waves, 497, 528 plasma, xv, 179, 186, 190, 268, 332, 357, 403, 421, 623, 624, 625, 634, 635, 636, 637, 638, 639, 640, 641, 642, 643, 644, 645, 646, 647 plasma membrane, 421 plastic, xi, 268, 280, 281, 292, 294, 296, 423, 653, 658 platforms, 244, 258 platinum, 56, 664 play, 4, 37, 67, 81, 84, 190, 445, 450, 460, 523, 642 point defects, viii, 47, 57, 74, 75, 76, 78, 79, 80, 86, 215, 624, 634, 635, 642, 647 Poisson, 16 Poisson equation, 16 polarity, 653 polarizability, 348 polarization, xi, xii, xv, 135, 234, 235, 236, 237, 238, 279, 299, 300, 307, 309, 310, 323, 324, 325, 373, 375, 381, 383, 390, 539, 578, 587, 588, 589, 590, 591, 592, 593, 594, 596, 597, 598, 599, 601, 602, 603, 609, 617, 619, 620 polarized, xi, xii, 13, 15, 16, 36, 39, 234, 299, 300, 306, 313, 327, 328, 371, 372, 374, 375, 381, 382, 383, 384, 385, 386, 387, 388, 389, 390, 507, 575 polarized light, 300, 328, 375 polyaniline, 653, 654, 655 polycrystalline, 123, 151, 363, 365 poly-crystalline, 360 poly-crystalline, 361 polyester, 119 polyethylene, 247 polyimide, 656, 658, 659, 660, 666, 667 polymer, 246, 251, 405, 653, 655, 656, 658, 659, 662, 664 polymer film, 655 polymeric materials, xvi, 651 polymers, 247 polymethylmethacrylate, 402
685 polynomial, 16, 37 polynomials, 5, 15, 580, 605 polystyrene, 653 polyvinyl alcohol, 661 polyvinylpyrrolidone, 415 poor, 13, 39, 54, 207, 243, 268, 270, 279, 517 population, ix, 19, 20, 21, 22, 169, 171, 175, 179, 181, 188, 373, 375, 377, 378, 379, 380, 382, 383, 384, 385, 386, 387, 388, 389, 400, 451, 471, 474, 568, 578 pores, 343 porous, 332, 355 potential energy, 4, 9, 21, 541 powder, 122 power, 163, 170, 174, 179, 181, 182, 184, 187, 188, 190, 191, 192, 193, 196, 197, 204, 205, 206, 208, 209, 213, 214, 215, 217, 232, 234, 268, 332, 354, 355, 395, 397, 398, 399, 400, 403, 407, 408, 410, 411, 514, 634, 635, 642 powers, 408, 409 precipitation, 121, 136, 334 prediction, 76 preparation, iv, 120, 367, 373 pressure, 204, 268, 328, 333, 335, 336, 626, 634, 635, 642 probability, xiii, 18, 21, 22, 23, 48, 92, 176, 177, 193, 195, 211, 356, 375, 428, 451, 453, 469, 480, 508, 557, 558, 560, 561, 562, 570 probability density function, 18 probability distribution, 18, 22, 23, 375 probe, 50, 67, 142, 145, 146, 148, 198, 208, 243, 383, 407, 408, 415, 416, 420, 423, 517 procedures, 30, 55, 56, 97, 402 production, 110, 423, 539, 622 program, 238, 404, 408 projector, 155 prolactin, 414, 422, 423, 424, 425 promote, 209, 250 propagation, 179, 372, 373, 376, 381, 390, 393, 394, 399, 401, 402, 408, 409, 436, 552, 553 property, iv, xi, 48, 116, 118, 126, 170, 299, 303, 325, 334, 349, 393, 420, 473, 474, 624, 636 proportionality, 162 prostate, 249, 251 prostate cancer, 251 protection, 407 protein, xii, 247, 249, 250, 252, 255, 404, 413, 414, 415, 416, 417, 418, 419, 420, 421, 422, 423, 424, 664 protein synthesis, 252, 414, 421 proteinase, 415, 416 proteins, xiii, 247, 249, 250, 252, 414, 418, 421, 422, 665
686
Index
proteolytic enzyme, 248 protocol, 415 protocols, 244 proximal, 117 pseudo, 123 pseudomorphic growth, 279, 286 p-type, 656 public, 56 public domain, 56 pulse, viii, xii, xvi, 48, 52, 67, 71, 76, 86, 91, 97, 98, 100, 101, 191, 239, 334, 341, 356, 371, 373, 374, 381, 382, 383, 384, 385, 386, 387, 388, 389, 390, 395, 514, 651 pulses, viii, 48, 99, 103, 170, 373, 374, 661 pumping, 208, 334, 337, 373, 375, 395, 398, 410 purification, 244 Purkinje cells, 250 pyramidal, 197, 222, 223, 496 pyrophosphate, 415
Q quantitative estimation, 170 quantization, xiv, 118, 126, 127, 128, 133, 134, 136, 138, 139, 140, 342, 347, 372, 374, 495, 505, 577, 578, 584, 586, 587, 603, 616 quantum chaos, 448 quantum computers, 485, 514, 578 quantum computing, 538 quantum confinement, x, xv, xvi, 38, 48, 49, 52, 60, 74, 116, 133, 134, 140, 143, 245, 246, 267, 269, 300, 315, 319, 321, 325, 341, 348, 354, 523, 541, 623, 651, 659, 666, 667 quantum gates, 516 quantum Hall effect, viii, 4, 42, 126 quantum mechanics, vii, 1, 7, 17, 110, 117, 375, 376, 449, 458, 485, 486 quantum phenomena, vii, 4, 13 quantum state, 85, 373, 514, 515, 516 quantum structure, 574 quantum well, ix, xi, 12, 48, 62, 89, 115, 116, 143, 144, 169, 170, 171, 172, 181, 182, 183, 184, 185, 190, 198, 210, 211, 311, 327, 331, 341, 357, 374, 474, 475, 494, 495, 498, 509, 517, 524, 538, 624 quantum yields, 244, 245, 246, 259 quartz, 333, 335, 353, 363, 365, 366, 626, 635 quasiparticle, 134 qubit, xiii, 371, 373, 493, 494, 514, 516 qubits, 300, 328, 373, 514 quinone, 257
R Rabi oscillations, xii, 371, 373, 386, 390 radiation, 122, 153, 154, 159, 163, 207, 208, 211, 269, 339, 373, 385, 386, 387, 530, 556, 557, 626 radical, 359 radio, 204, 634 Raman, xi, 136, 327, 331, 332, 334, 336, 337, 338, 339, 340, 341, 342, 343, 345, 347, 348, 349, 350, 353, 354, 356, 367, 373 Raman scattering, xi, 327, 331, 332, 334, 345 Raman spectra, 334, 337, 338, 339, 341, 348, 349, 353, 354 random, xvi, 18, 20, 21, 144, 309, 325, 382, 400, 401, 480, 652 random access, xvi, 652 random configuration, 382 random walk, 18, 20, 21 rapid prototyping, 394 rat, 258, 414, 415, 416, 417, 418, 422, 423, 425 rats, 415, 422 reactant, 155 reactants, 148 reaction time, 403, 405 reactive ion, 624 reading, 194, 635 reagent, 121, 348, 405 real numbers, 454 real time, 54 reasoning, 139 receptors, 244, 247, 251, 254, 424 reciprocal relationships, 270 recombination, 48, 59, 153, 157, 158, 160, 161, 163, 170, 171, 174, 175, 177, 184, 199, 211, 354, 355, 394, 397, 398, 410, 508, 630 recombination processes, 158, 161 reconstruction, 273 recovery, 199 red shift, ix, 109, 129, 132, 133, 138, 140, 142, 144 redistribution, 52, 86, 431, 478, 479, 486 redox, 119 redshift, 524, 629, 630, 631, 632, 647 reduction, 56, 80, 81, 82, 110, 119, 183, 184, 193, 204, 223, 224, 227, 237, 247, 258, 289, 337, 347, 402, 444, 451, 455, 479, 481, 624, 629, 639, 660 reference system, 285 reflection, 58, 172, 209, 226, 417, 418, 472, 477 reflection high-energy electron diffraction, 58, 226 reflectivity, 181, 205
Index refractive index, 178, 179, 180, 185, 186, 187, 188, 189, 190, 205, 348, 351, 384, 394, 396, 399, 531 refractive index variation, 178, 179, 187, 188, 189, 190 refractive indices, 347, 381 regression, 251 regrowth, 424, 624 regular, 269, 285, 440, 484 Reimann, 41, 43, 44 relationship, xiii, 63, 279, 281, 380, 399, 408, 414, 421 relationships, 378 relative size, 630 relaxation process, xii, 156, 158, 160, 161, 162, 164, 174, 371, 377, 379, 390 relaxation processes, xii, 156, 158, 160, 164, 371, 377, 379, 390 relaxation rate, 185, 187, 382 relaxation time, ix, 110, 157, 162, 182, 183, 187, 190, 372, 373, 375, 379, 383, 396 relaxation times, 190, 372, 379, 383 relevance, 126 renal, 259 renormalization, 483 rent, 485 replication, 278, 375 research, vii, ix, xii, xiii, xiv, xv, xvi, 2, 40, 116, 126, 127, 169, 170, 172, 194, 204, 244, 300, 358, 414, 493, 578, 623, 651, 667 researchers, 244, 333, 494, 496 reservoir, 50, 67, 95, 171, 172, 173, 174, 176, 460, 469, 573 reservoirs, 460, 461, 468, 469 residues, 429 resin, 423 resistance, 145, 146, 147, 148, 149, 153, 154, 155, 654 resistivity, 118, 146 resolution, viii, x, xi, 47, 49, 54, 56, 57, 65, 76, 91, 93, 94, 97, 100, 205, 243, 246, 283, 284, 295, 296, 331, 334, 425, 624, 647 resonator, 450, 546 respiratory, 249 respiratory syncytial virus, 249 response time, xvi, 652, 667 retention, xvi, 652, 660 reticulum, 252 rings, 481, 498, 499, 500, 501, 502, 503, 504, 547, 573 risk, 191 rods, 300, 306, 307, 308, 309, 327, 328
687 room temperature, viii, xi, xvi, 47, 50, 58, 59, 74, 76, 81, 131, 132, 148, 177, 181, 200, 226, 227, 231, 232, 299, 321, 323, 331, 332, 333, 337, 363, 415, 416, 429, 532, 533, 652, 656, 663 room-temperature, 73, 76, 81, 85, 624 roughness, 636, 638 routines, 284 RSV infection, 249, 250
S Sagnac effect, 204 saline, 415 salmon, 415 Salmonella, 251 salt, 247 salts, 404 sampling, 22, 179, 180, 270, 272 sapphire, x, 208, 267, 268, 269, 333, 335, 336, 337, 339, 357 saturation, 193, 194, 197, 209, 268, 307, 310, 312, 333, 383, 398, 401, 474, 475 scalability, 373 scalable, 373 scalar, 137 scaling, 2 Scanning electron, 171 scanning electron microscopy, 226 scatter, 121 scattering, 12, 178, 208, 334, 337, 339, 354, 428, 429, 430, 431, 432, 433, 434, 437, 444, 445, 447, 452, 453, 454, 455, 456, 458, 459, 460, 465, 473, 475, 477, 478, 479, 480, 485, 486, 515, 524, 548, 549, 550, 552, 554, 555, 556, 557, 559, 561, 570, 571, 574, 575 Schottky, viii, 48, 50, 52, 54, 58, 59, 64, 72, 81, 89, 94, 95, 97, 103 Schottky barrier, 89, 94, 97 Schrodinger equation, 605 Schrödinger equation, 110, 117 science, 54, 116, 117, 126 scientific, 3, 122, 268 scientific community, 268 scientists, 40, 48 SE, 380 search, 25, 597 Seattle, 393 secretion, 423 segregation, 524 selecting, 97 selectivity, xv, 40, 55, 248, 258, 405, 623, 642, 644, 645 selenium, 119, 120, 258
688 self-assembling, 629 self-assembly, x, xiii, 203, 400, 402, 403, 410, 493, 494, 517, 518, 521, 652 self-ordering, 269 SEM, 170, 171, 226, 227, 228, 229, 231, 232, 233, 238, 402, 404, 405 semicircle, 450 semiconductor lasers, ix, 48, 169, 172, 181, 196, 198, 199, 624 semiconductors, viii, xi, xvi, 47, 51, 52, 56, 67, 74, 115, 122, 125, 126, 127, 133, 139, 142, 143, 149, 150, 151, 157, 166, 198, 268, 300, 302, 304, 306, 329, 331, 332, 369, 372, 523, 651, 652 sensing, ix, 169, 172 sensitivity, vii, xv, 1, 40, 204, 207, 208, 235, 248, 249, 250, 402, 538, 604, 613, 623, 624 sensors, x, 190, 203, 204, 268 separation, 29, 50, 53, 91, 93, 174, 178, 190, 192, 193, 245, 394, 399, 400, 406, 409, 410, 505, 524, 538, 605, 625, 627, 629, 630, 647 series, 26, 56, 57, 74, 80, 98, 103, 115, 127, 136, 138, 140, 143, 250, 336, 482, 539, 555, 582 serotonin, 255 serum, 415 serum albumin, 415 services, iv shear, 497 shock, 252 short-range, 539, 587, 609 shoulder, 91, 337 Siemens, 574 sign, 19, 20, 23, 145, 182, 283, 312, 320, 450, 463, 466, 497, 614, 652 signals, xi, xii, 57, 58, 59, 72, 74, 82, 87, 97, 227, 331, 337, 355, 414, 415, 417, 418, 419, 420, 421, 578, 627 signal-to-noise ratio, 207 signs, 190 silane, 403 silica, 335, 642 silicate, 357 silicon, vii, xi, 56, 181, 247, 331, 332, 333, 339, 354, 357, 362, 363, 370, 394, 402, 626, 635, 657 silicon dioxide, 247, 354, 657 silver, 145, 146, 153, 394, 407 similarity, 117, 223 simulation, ix, xii, xv, 17, 23, 25, 37, 38, 40, 169, 172, 173, 192, 198, 375, 382, 383, 384, 386, 393, 394, 399, 400, 401, 410, 578, 604 simulations, xii, 30, 181, 184, 187, 201, 278, 371, 382, 383, 386, 390, 395, 402
Index singular, 463, 465, 549, 552, 557, 567 SiO2 films, 349, 355, 358, 363 sites, 36, 227, 279, 403, 404, 435, 436, 437, 469, 483, 484, 551, 560, 659, 663, 665, 667 Slater determinants, 15, 36 social, 268 sodium, 119, 120, 247, 404, 415 software, 55, 56, 173 solar, ix, 110, 162 solar cell, ix, 110, 162 solar cells, ix, 110, 162 solid phase, xi, 38, 331, 333 solid state, 110, 111, 117, 149, 245, 578 solubility, 120, 121, 246 solutions, x, xiii, 110, 117, 120, 203, 238, 357, 399, 415, 427, 439, 454, 455, 462, 485, 497, 508, 546, 547, 550, 552, 660 solvent, 246 somatostatin, 422 spatial, x, xiv, xv, 49, 57, 65, 71, 79, 110, 115, 133, 134, 135, 143, 181, 209, 243, 283, 374, 382, 384, 401, 414, 421, 577, 578, 603, 608, 642, 644, 647 spatial anisotropy, xiv, xv, 577, 578, 603, 608 spatial frequency, 283 spatial location, 79 species, 119, 246, 629 specific heat, 252 specificity, 248, 254 spectroscopy, viii, xi, 3, 47, 49, 50, 52, 54, 57, 61, 67, 89, 103, 130, 142, 311, 327, 331, 334, 337, 345, 499, 507, 517 speed, xvi, 170, 196, 198, 200, 609, 624, 651 speed of light, 609 sperm, 415 spheres, 122, 302, 303, 309, 311, 312, 316, 317, 319, 325, 328, 329 spin, vii, xii, 1, 4, 13, 14, 15, 16, 17, 24, 25, 28, 36, 37, 38, 39, 40, 118, 125, 126, 127, 128, 136, 137, 138, 139, 140, 142, 143, 144, 176, 300, 302, 303, 313, 315, 318, 323, 324, 328, 329, 371, 372, 373, 374, 375, 376, 377, 378, 379, 380, 382, 383, 384, 385, 386, 387, 388, 389, 390, 396, 402, 509, 514, 526, 575, 653, 658, 659, 664, 666 spin dynamics, 373, 375, 376 spleen, 252 sputtering, 332, 636, 638, 641, 642 stability, xvi, 14, 119, 120, 121, 238, 247, 258, 259, 357, 382, 651, 653, 655, 661, 662 stabilization, 14, 453, 603 stabilize, 14, 15 stages, 227, 279, 407
Index stainless steel, 626 standard deviation, 58, 271, 290, 408, 627, 635 standard model, 110 standards, 416 staphylococcal, 249 Stark effect, xi, xiii, 299, 305, 314, 325, 328, 493, 494, 505 stars, 471 statistical analysis, 55 statistics, 270, 271 steady state, 54, 72, 396, 411 stochastic, 17, 119 stoichiometry, 148 storage, xvi, 3, 651, 652, 655, 657, 659 strains, 78 strategies, 247 strength, viii, xiii, 1, 3, 5, 14, 27, 29, 33, 34, 35, 39, 79, 116, 126, 305, 309, 314, 317, 325, 427, 428, 430, 437, 442, 443, 446, 447, 448, 449, 451, 452, 453, 456, 457, 460, 462, 463, 479, 480, 481, 482, 485, 516 streptavidin, 247, 249, 250, 251, 253, 255, 403, 404, 415, 416 stress, 23, 78, 279, 286, 287, 289, 294, 295, 497 stress fields, 286 stretching, 214, 354 strong interaction, 14, 38, 39, 304, 539 structural changes, 356 structural modifications, 358 students, 194 subgroups, 176, 181, 182 substances, 145, 358 substitution, 281, 416, 420, 549, 581 substrates, x, 57, 94, 119, 123, 143, 267, 268, 269, 281, 333, 335, 354, 357, 364, 656, 659, 660 sucrose, 415 sugars, 247 sulfate, 404, 415 Sun, 106, 211, 217, 238, 240, 262, 263, 328, 490 superconductor, 514 superiority, 514 superlattice, xiv, 143, 237, 493, 494, 508, 523, 529, 533, 534, 535, 536, 537 superlattices, 115, 528, 530, 534, 535, 536, 537, 546 superposition, xiv, 112, 115, 285, 457, 514, 549, 561, 562 supply, 57, 407, 634 suppression, 386, 406, 642 surface area, 147 surface chemistry, xii, 246, 393, 404 surface diffusion, 334
689 surface energy, 276, 279 surface layer, 634 surface modification, 244, 247, 248, 255, 258, 259 surface region, 647 surface roughness, 636, 638 surface treatment, 406 surgery, 248, 252 susceptibility, 249 switching, 397, 407, 653, 654, 655, 665 symbols, 150, 224, 225, 531 symmetry, xiii, 14, 78, 79, 117, 127, 137, 190, 285, 286, 301, 302, 305, 311, 320, 374, 378, 428, 450, 452, 462, 471, 497, 510, 511, 512, 538, 539, 553, 554, 569 synthesis, viii, xiii, xvi, 109, 118, 120, 122, 134, 136, 246, 327, 358, 364, 414, 421, 422, 651, 664, 667 synthetic, 118, 120, 134, 422 systematic, 16, 17, 49, 118, 647
T targets, 244, 245, 248, 249, 253, 256 T-cell, 251, 259 T-cells, 251 technological, vii, 1, 2, 3, 110, 117, 332, 351 technology, vii, 2, 116, 332, 335, 343, 356, 357, 370, 393, 514, 515, 641, 652, 660 teflon, 450 telecommunications, x, 267 temperature annealing, 140 temperature dependence, ix, 65, 85, 87, 109, 147, 149, 150, 152, 321, 355, 364, 365, 474, 644 temperature gradient, 343, 351 temporal, x, 203, 209, 243, 374, 390 tensile, 286 terminals, 403 test data, 409 therapeutic, 244, 253 therapeutic agents, 253 therapeutics, 251 therapy, 252, 257 thermal activation, 69, 74, 87, 98 thermal activation energy, 69, 87 thermal energy, 77, 89, 193, 629 thermal equilibrium, 52, 177, 193 thermal evaporation, 122 thermal expansion, 268, 646 thermal oxidation, 354 thermal stability, 237, 656 thermal treatment, ix, 109, 129, 133, 138, 148, 154
690 thermalization, 174, 178, 183, 190 thermodynamic, 120 thermodynamic stability, 120 thin film, viii, 109, 116, 118, 119, 120, 121, 122, 123, 124, 127, 128, 129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142, 143, 145, 146, 147, 148, 149, 150, 151, 152, 153, 154, 156, 157, 158, 159, 161, 162, 163, 164, 653, 655, 661 thin films, viii, 109, 116, 118, 119, 120, 121, 122, 123, 125, 127, 128, 129, 130, 131, 133, 134, 135, 136, 139, 140, 141, 142, 143, 145, 147, 148, 151, 153, 154, 156, 157, 159, 162, 163, 661 thinking, 89 three-dimensional, xii, 48, 56, 110, 116, 125, 132, 140, 143, 144, 211, 220, 226, 227, 273, 413, 414, 421, 422, 424, 494, 505, 584, 652 three-dimensional model, 220, 273 three-dimensional reconstruction, 424 threshold, xii, xv, 2, 7, 8, 12, 13, 48, 72, 74, 80, 170, 178, 181, 182, 184, 185, 186, 187, 190, 193, 196, 198, 200, 208, 393, 397, 398, 410, 411, 439, 440, 441, 476, 547, 554, 623, 624, 630, 634, 655 thresholds, xiii, 71, 427, 438, 439, 442, 458, 485 Tikhonov regularization method, 55, 56 time frame, 247 tin, 119 TiO2, 642 tissue, 244, 248, 249, 250, 251, 415, 416, 424 titanium, 208, 642 title, 118, 136 TM, 234, 235, 236, 237 tobacco, 664 Tokyo, 413, 415 tolerance, 211 toluene, 404, 405 topological, 449, 450, 465 topology, xiii, 427 torque, 379, 380 toxic, 257 toxicity, 243, 244, 257, 258 toxin, 249 toxins, 249, 258 tracking, 248 trajectory, 348 trans, 143, 377, 427, 428, 478, 481, 486 transfer, 76, 93, 145, 332, 343, 374, 375, 377, 378, 382, 383, 386, 399, 400, 401, 654, 655, 656, 665 transformation, viii, 1, 94, 303, 353, 356, 360, 361, 450, 553
Index transformation matrix, 450 transformations, 154, 361 transistor, 3, 76, 357, 652 transistors, vii, 2, 268, 335 transition temperature, 122 translational, 110, 134, 274, 292 transmission electron microscopy, x, 57, 80, 222, 267, 345, 627, 659 Transmission Electron Microscopy, 269, 297 Transmission Electron Microscopy (TEM), 269 transparent, 272, 335, 350, 354, 456, 550, 552, 578, 604, 609, 620 transport, xiv, 16, 41, 118, 126, 145, 149, 151, 192, 193, 269, 357, 373, 435, 437, 488, 508, 545, 549, 550, 554, 555, 563, 565, 571, 574, 575, 604, 655, 656 transport phenomena, 126 traps, 52, 57, 66, 67, 69, 72, 74, 76, 86, 97, 269, 282, 343, 357, 367, 657, 659, 660, 665 travel, 474 trend, 150, 151, 227, 393, 409, 411, 475, 506, 533, 541, 636 trial, 15, 17, 18, 22, 23, 24, 25, 28, 29, 30, 32, 39 tubular, 335, 353 tumor, 248, 251, 252, 253 tumor cells, 251 tumor growth, 251 tumors, 251 tungsten, 207, 626 tunneling, 3, 13, 52, 57, 62, 63, 68, 69, 70, 76, 84, 85, 89, 92, 94, 98, 99, 100, 102, 103, 141, 324, 357, 523, 552, 572, 575, 659, 665, 666 two-dimensional, vii, 56, 76, 116, 227, 277, 414, 421, 585, 653 two-dimensional (2D), 56, 227
U ultrasound, 205 ultraviolet, 205, 268 uncertainty, 31 uniform, ix, 5, 51, 78, 169, 171, 174, 191, 193, 211, 347, 357, 379, 400, 514, 632, 656 uniformity, 374, 633, 635, 641, 647, 658 universities, 622 users, 206
V vacancies, 74, 75, 110, 629, 646 vacuum, 134, 335, 353, 367, 402, 403, 515, 609, 635, 653
Index validity, 142, 375, 386 vapor, xi, 331, 333, 334, 335, 337, 348, 349, 354 variability, 358 variable, xiv, 118, 126, 138, 147, 258, 409, 411, 452, 545, 546, 560, 564, 573 variables, 17, 25, 400, 408, 605 variation, xiv, 52, 54, 79, 80, 95, 97, 100, 111, 116, 178, 179, 180, 182, 185, 186, 187, 188, 189, 190, 205, 208, 219, 222, 227, 232, 269, 294, 351, 358, 497, 505, 520, 545, 556, 564, 570, 571 vascular, 253 vascular cell adhesion molecule, 253 vasculature, 251 vasopressin, 422 velocity, 22, 112, 180, 268 vertical integration, 653 vibration, 15, 66, 338, 339, 340, 354, 515 vibrational, 40, 439 viral, 249 viral infection, 249 virus, 661, 664, 665 viruses, 247, 664 visible, 69, 84, 141, 170, 246, 274, 275, 276, 288, 294, 335, 354, 355, 356, 404, 553 visual, 129, 136 visualization, xiii, 285, 421, 424, 425 voids, 333 Volmer-Weber, 276
W water, 120, 246, 327, 355, 359, 363, 402, 404, 405, 635 water-soluble, 327 Watson, 260 wave number, 548, 562 wave propagation, xii, 371, 373, 381 wave vector, 125, 129, 342, 382, 461, 550, 588, 594, 609 waveguide, xii, xiv, 172, 177, 179, 191, 192, 193, 194, 195, 234, 235, 393, 394, 395, 399, 400, 401, 402, 404, 405, 406, 407, 408, 409, 410, 411, 545, 546, 572, 573
691 wavelengths, 58, 181, 183, 192, 206, 208, 268, 397, 406, 407, 647 wavepacket, 112, 113 weak interaction, 14, 38, 539 wealth, 2 wells, 8, 13, 89, 115, 144, 198, 498 western blot, 249, 250 wetting, 48, 56, 57, 58, 59, 75, 79, 81, 82, 83, 85, 87, 88, 94, 95, 171, 172, 173, 198, 277, 293, 494, 508, 524 Wigner molecule, 14, 15, 34, 35, 36, 38 windows, 67, 70, 626 wires, xi, 115, 299, 305, 315, 328, 331, 461, 469, 474, 483, 494, 498, 501, 502, 514, 546, 547, 550, 551, 552, 553, 554, 555, 556, 557, 559, 560, 561, 562, 564, 566, 569, 571, 572, 574, 622 workers, 122, 655, 658, 659, 660 working conditions, 185, 188
X xenograft, 251 X-ray, 122, 123, 132 X-ray diffraction, 122, 123, 132 xylene, 402
Y yeast, 415 yield, 17, 50, 58, 159, 247, 347, 405, 497, 667 ytterbium, 208 yttrium, 208
Z zero-dimensional structures, xiv, xv, 577, 578, 604 zinc, 125, 126, 127, 128, 140, 141, 142, 143, 245, 247, 258, 302, 304, 306, 324, 327, 398
