Nanoscale Devices - Fundamentals and Applications
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Series II: Mathematics, Physics and Chemistry – Vol. 233
Nanoscale Devices - Fundamentals and Applications edited by
Rudolf Gross Bayerische Akademie der Wissenschaften, Garching, Germany
Anatolie Sidorenko Institute of Electronic Engineering and Industrial Technologies ASM, Kishinev, Moldova
and
Lenar Tagirov Kazan State University, Kazan, Russia
Published in cooperation with NATO Public Diplomacy Division
Proceedings of the NATO Advanced Research Workshop on Nanoscale Devices - Fundamentals and Applications Kishinev, Moldova 18--22 September 2004 A C.I.P. Catalogue record for this book is available from the Library of Congress.
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Table of Contents
Preface.........................................................................................................ix Acknowledgments.......................................................................................xi Contributing Authors ............................................................................... xiii SCIENCE AGAINST NONTRIVIAL THREAT Surface Acoustic Wave Studies for Chemical and Biological Sensors........3 A. Müller, A. Darga, A. Wixforth The Experience of Utilizing the Explosives Detection System on the Basis of Neutron Radiation Analysis Together with X-Ray Units at the Airport “Pulkovo” in St.-Petersburg .................................................15 Y. Olshansky, A. Vishnevkin, A. Sorokin, A. Vikdorovich, A. Golovin, E. Stepanov Thermodynamic Principles of Artificial Nose Based on Supramolecular Receptors.....................................................................23 V. V. Gorbatchuk, M. A. Ziganshin Molecular Detection with Magnetic Labels and Magnetoresistive Sensors .......................................................................................................35 J. Schotter, M. Panhorst, M. Brzeska, P. B. Kamp, A. Becker, A. Pühler, G. Reiss, H. Brueckl WEAK MAGNETIC FIELDS DETECTION TECHNIQUES AND DEVICES Magnetic Tunnel Junctions Based on Half-Metallic Oxides .....................49 R. Gross MEMS Tunable Dielectric Resonator......................................................111 G. Panaitov, R. Ott, N. Klein v
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Ultra-Thin Spin-Valve Structures Grown on the Surface-Reconstructed GaAs Substrate......................................... 123 B. Aktaş, F. Yıldız, O. Yalçın, A. Zerentürk, M. Özdemir, L. R. Tagirov, B. Heinrich, G. Woltersdorf, R. Urban NOVEL IDEAS AND PRINCIPLES OF DEVICES Negative U Molecular Quantum Dot.......................................................137 A. S. Alexandrov Configuring a Bistable Atomic Switch by Repeated Electrochemical Cycling.......................................................................... 153 F.-Q. Xie, Ch. Obermair, Th. Schimmel Realization of an N-Shaped IVC of Nanoscale Metallic Junctions Using the Antiferromagnetic Transition ..................................................163 Yu. G. Naidyuk, K. Gloos, I. K. Yanson
PI-SHIFT EFFECT AND FERROMAGNET/SUPERCONDUCTOR NANOSCALE DEVICES Josephson Effect in Composite Junctions with Ferromagnetic Materials ..................................................................................................173 M. Yu. Kupriyanov, A. A. Golubov, M. Siegel Depairing Currents in Bilayers of Nb/Pd89Ni11 ........................................189 A. Yu. Rusanov, J. Aarts, M. Aprili Superconductor-Ferromagnet Heterostructures .......................................197 A. I. Buzdin, M. Fauré, M. Houzet Superconducting/Ferromagnetic Nanostructures: Spin Fluctuations and Spontaneous Supercurrents ..................................225 M. Aprili, M. L. Della Rocca, T. Kontos COHERENCE EFFECTS IN F/S AND N/F NANOSTRUCTURES Proximity Effect and Interface Transparency in Nb-based S/N and S/F Layered Structures ......................................................................241 C. Attanasio
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Properties of S/N Multilayers with Different Geometrical Symmetry.....251 S. L. Prischepa Andreev Reflection in Ballistic Superconductor-Ferromagnet Contacts....................................................................................................265 L. R. Tagirov, B. P. Vodopyanov Superconductor-Insulator Transition in a PbZSn1-ZTe: In Solid Solution.......................................................................................277 D. V. Shamshur, D. V. Shakura, R. V. Parfeniev, S. A. Nemov
ADVANCED SENSORS OF ELECTROMAGNETIC RADIATION Thermoelectricity of Low-Dimensional Nanostructured Materials .........291 V. G. Kantser Organic Semiconductors – More Efficient Material for Thermoelectric Infrared Detectors .....................................................309 A. Casian, Z. Dashevsky, V. Dusciac, R. Dusciac Submillimeter Radiation–Induced Persistent Photoconductivity in Pb1-xSnxTe(In) ..........................................................................................319 A. E. Kozhanov, D. E. Dolzhenko, I. I. Ivanchik, D. M. Watson, D. R. Khokhlov Quasioptical Terahertz Spectrometer Based on a Josephson Oscillator and a Cold Electron Nanobolometer ....................................... 325 M. Tarasov, L. Kuzmin, E. Stepantsov, A. Kidiyarova-Shevchenko
NOVEL MATERIALS FOR ELECTRONICS Origin of the Resistive Transition Broadening for Superconducting Magnesium Diboride..............................................339 A. S. Sidorenko Aharonov-Bohm Oscillations in Single Bi Nanowires ............................349 D. Gitsu, T. Huber, L. Konopko, A. Nikolaeva Some Application of Nanocarbon Materials for Novel Devices..............355 Z. A. Mansurov
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NANOMATERIALS AND DOMAINS Nanocrystalline Iron-Rare Earth Alloys: Exchange Interactions and Magnetic properties .......................................................................... 371 E. Burzo, C. Djega–Mariadassou The Influence of Applied Field on the Nucleation and Growth of Heteroepitaxial Carbon Films ............................................................. 387 B. Z. Mansurov
Preface
Over the last decade the interest in nanoscale materials and their applications in novel electronic devices has been increasing tremendously. This is caused by the unique properties of nanoscale materials and the outstanding performance of nanoscale devices. The fascinating and often unrivalled properties of nanoscale materials and devices opened new and sometimes unexpected fields of applications. Today, the widespread applications range from the detection of explosives, drugs and fissionable materials to bio- and infrared-sensors, spintronic devices, data storage media, magnetic read heads for computer hard disks, single-electron devices, microwave electronic devices, and many more. This book contains a collection of papers giving insight into the fundamentals and applications of nanoscale devices. The papers have been presented at NATO Advanced Research Workshop on Nanoscale Devices – Fundamentals and Applications (NDFA-2004, ARW 980607) held in Kishinev (Chişinau), Moldova, on September 18-22, 2004. The main focus of the contributions was on the synthesis and characterization of nanoscale magnetic materials, the fundamental physics and materials aspects of solidstate nanostructures, the development of novel device concepts and design principles for nanoscale devices, as well as on applications in electronics with special emphasis on defence against the threat of terrorism. We would like to thank the members of the International Organizing Committee, Sasha Alexandrov, Alexander Andreev, Jochen Mannhart, Thomas Schimmel, and Igor Yanson for their support in putting together the scientific program of the workshop, and all the participants for their invaluable contributions. Rudolf Gross, Anatolie Sidorenko, and Lenar Tagirov The Editors
ix
Acknowledgments
The Editors are grateful to NATO Scientific Affairs Division for financial support of the Advanced Research Workshop on Nanoscale Devices – Fundamentals and Applications, and also for the assistance in preparing the Workshop Proceedings.
xi
Contributing Authors
Aarts J. Kamerlingh Onnes Laboratory, Leiden University, The Netherlands Aktaş B. Gebze Institute of Technology, 41400 Çayırova-Gebze, Turkey Alexandrov A.S. Department of Physics, Loughborough University, Loughborough, United Kingdom Aprili M. Universite Paris 11, CSNSM, CNRS, Orsay, France Attanasio C. Dipartimento di Fisica “E.R. Caianiello” and INFM-Laboratorio Regionale Supermat, Università degli Studi di Salerno, I-84081 Baronissi (Sa), Italy Becker A. Department of Genetics, University of Bielefeld, 33615 Bielefeld, Germany
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Contributing Authors
Brueckl H. ARC Seibersdorf research GmbH, Nano-Systemtechnologien, 1220 Wien, Austria Brzeska M. Department of Physics, University of Bielefeld, 33615 Bielefeld, Germany Burzo E. Faculty of Physics, Babes-Bolyai University, 400084 Cluj-Napoca, Romania Buzdin A.I. Université Bordeaux I, CPMOH, 33400 Talence Cedex, France Casian A. Department of Computers, Informatics and Microelectronics, Technical University of Moldova, MD-2004, Chisinau, Moldova Darga A. Center for NanoScience, University of Munich, 80799 Munich, Germany Dashevsky Z. Department of Materials Engineering, Ben-Gurion University, Beer-Sheva 84105, Israel Djega–Mariadassou C. LCMTR UPR 209 CNRS 2/8, Bat F Rue Henri Dunant 94320 Thiais, France
Contributing Authors
Dolzhenko D.E. Moscow State University, Moscow 119992, Russia Dusciac R. Department of Computers, Informatics and Microelectronics, Technical University of Moldova, R. Dusciac MD-2004, Chisinau, Moldova Dusciac V. Department of Physics, State University of Moldova, MD-2012, Chisinau, Moldova Fauré M. Université Bordeaux I, CPMOH, 33400 Talence Cedex, France Gitsu D. Institute of Electronic Engineering and Industrial Technologies ASM, Kishinev MD-2028, Moldova Gloos K. Nano-Science Center, Niels Bohr Institute fAFG, Universitetsparken 5, DK-2100 Copenhagen, Denmark Golovin A. “Pulkovo” Aviation enterprise, Saint-Petersburg, Russia Golubov A.A. Faculty of Science and Technology, University of Twente, The Netherlands
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Gorbatchuk V.V. Kazan State University, A. M. Butlerov Institute of Chemistry, Kazan 420008, Russia Gross R. Walther-Meißner-Institut, Bayerische Akademie der Wissenschaften, D-85748 Garching, Germany Heinrich B. Simon Fraser University, Burnaby, BC, V5A 1S6, Canada Huber T. Department of Chemistry, Howard University, 525 College St. N.W. Washington, DC 20059, USA Ivanchik I.I. Moscow State University, Moscow 119992, Russia Kamp P.B. Department of Genetics, University of Bielefeld, 33615 Bielefeld, Germany Kantser V.G. International Laboratory of Superconductivity and Solid State Electronics, Academy of Sciences of Moldova, Chişinau, Moldova Khokhlov D.R. Moscow State University, Moscow 119992, Russia
Contributing Authors
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Kidiyarova-Shevchenko A. Chalmers University of Technology, Göteborg SE41296, Sweden Klein N. ISG-2, Research Centre Jülich GmbH, Germany Konopko L. Institute of Electronic Engineering and Industrial Technologies ASM, Kishinev MD-2028, Moldova Kontos T. LPQ-ESPCI, 10 rue Vauquelin, 75005 Paris, France Kozhanov A.E. Moscow State University, Moscow 119992, Russia Kupriyanov M.Yu. D.V. Skobeltsyn Institute of Nuclear Physics, Moscow State University, Moscow, Russia Kuzmin L. Chalmers University of Technology, Göteborg SE41296, Sweden Mansurov B.Z. al-Farabi Kazakh National University, 96A, Tole be Str., 480012, Almaty, Kazakhstan
xviii Contributing Authors
Mansurov Z.A. al-Farabi Kazakh National University, 96A, Tole be Str., 480012, Almaty, Kazakhstan Müller A. Center for NanoScience, University of Munich, 80799 Munich, Germany Naidyuk Yu. G. B. Verkin Institute for Low Temperature Physics and Engineering, National Academy of Sciences of Ukraine, 47 Lenin Ave., 61103, Kharkiv, Ukraine Nemov S.A. State Polytechnical University, St. Petersburg, Russia Nikolaeva A. Institute of Electronic Engineering and Industrial Technologies ASM, Kishinev MD-2028, Moldova Obermair Ch. Institute for Applied Physics, University of Karlsruhe, D-76128 Karlsruhe, Germany Olshansky Y. Scientific & Technical Center “RATEC”, St. Petersburg 193079, Russia Ott R. ISG-2, Research Centre Jülich GmbH, Germany
Contributing Authors
Özdemir M. Gebze Institute of Technology, 41400 Çayırova-Gebze, Turkey Panaitov G. ISG-2, Research Centre Jülich GmbH, Germany Panhorst M. Department of Physics, University of Bielefeld, 33615 Bielefeld, Germany Parfeniev R.V. A.F. Ioffe Physical-Technical Inst., RAS, St. Petersburg, Russia Prischepa S.L. Belarus State University of Informatics and RadioElectronics, P. Brovka 6, Minsk 220013, Belarus Pühler A. Department of Genetics, University of Bielefeld, 33615 Bielefeld, Germany Reiss G. Department of Physics, University of Bielefeld, 33615 Bielefeld, Germany Della Rocca M.L. Dipartimento di Fisica,, Università di Salerno, via S. Allende,84081 Baronissi, Italy Rusanov A.Yu. Kamerlingh Onnes Laboratory, Leiden University, the Netherlands
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Contributing Authors
Schimmel Th. Institute of Nanotechnology, Forschungszentrum Karlsruhe, D-76021 Karlsruhe, and Institute of Applied Physics, Universität Karlsruhe, D-76128 Karlsruhe,
Germany Schotter J. ARC Seibersdorf research GmbH, Nano-Systemtechnologien, 1220 Wien, Austria Shakura D.V. A.F. Ioffe Physical-Technical Inst., RAS, St. Petersburg, Russia Shamshur D.V. A.F. Ioffe Physical-Technical Inst., RAS, St. Petersburg, Russia Sidorenko A.S. Institute of Electronic Engineering and Industrial Technologies ASM, MD-2028 Kishinev, Moldova Siegel M. Institute for Micro and Nanoelectronic Systems, Karlsruhe University, Germany Sorokin A. Scientific & Technical Center “RATEC”, St Petersburg 193079, Russia Stepanov E. “Pulkovo” Aviation enterprise, Saint-Petersburg, Russia
Contributing Authors
Stepantsov E. Institute of Crystallography RAS, Moscow 117333, Russia Tagirov L.R. Kazan State University, Kazan 420008, Russia Tarasov M. Institute of Radio Engineering and Electronics RAS, Moscow 125009, Russia Urban R. Simon Fraser University, Burnaby, G. Woltersdorf BC, V5A 1S6, Canada Vikdorovich A. Scientific & Technical Center “RATEC”, St. Petersburg 193079, Russia Vishnevkin A. Scientific & Technical Center “RATEC”, St. Petersburg 193079, Russia Vodopyanov B.P. Kazan Physico-Technical Institute of RAS, 420029 Kazan, Russia Watson D.M. University of Rochester, Rochester, 14627 NY, USA
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xxii Contributing Authors
Wixforth A. Chair for Experimental Physics I, University of Augsburg, 86159 Augsburg, Germany Woltersdorf G. Simon Fraser University, Burnaby, BC, V5A 1S6, Canada Xie Fang-Qing Institute for Applied Physics, University of Karlsruhe, D-76128 Karlsruhe, Germany Yalçın O. Gebze Institute of Technology, 41400 Çayırova-Gebze, Turkey Yanson I.K. B. Verkin Institute for Low Temperature Physics and Engineering, National Academy of Sciences of Ukraine, 47 Lenin Ave., 61103, Kharkiv, Ukraine Yıldız F. Gebze Institute of Technology, 41400 Çayırova-Gebze, Turkey Zerentürk A. Gebze Institute of Technology, 41400 Çayırova-Gebze, Turkey Ziganshin M.A. Kazan State University, A. M. Butlerov Institute of Chemistry, Kazan 420008, Russia
SCIENCE AGAINST NONTRIVIAL THREAT
Surface Acoustic Wave Studies for Chemical and Biological Sensors
A. Müller1, A. Darga1, A. Wixforth2 1
Center for NanoScience, University of Munich, 80799 Munich, Germany
2
Chair for Experimental Physics I, University of Augsburg, 86159 Augsburg, Germany
Abstract:
Surface Acoustic Waves on piezoelectric substrates are very sensitive to any external modulation of the mechanical and/or electrical boundary conditions at the surface on which they propagate. This makes them a perfect tool for sensor applications. In this manuscript, we demonstrate that a sophisticated transducer design allows for a spatial resolution of the interaction of SAW and local modulation of the electrical and mechanical boundary condition. If such local disturbances of parts of the functionalized sample surface are due to a chemical or optical interaction, a single chip with many different ‘pixels’ can act as a novel type of sensor.
Keywords: surface acoustic waves, biosensors, biochips
Modern sensors are nowadays also meant to act as the ‘interfacing link’ between high performance electronic circuitry and the ‘outside world’. More and more electronic systems are equipped with a whole variety of sensing abilities to make them able to react to and possibly interact with the environment. A good example is the automobile industry. Modern cars have the ability to “sense” their environment and to react accordingly. For instance, the windshield wipers turn on and off automatically depending on whether it is raining or not, the lights are automatically switched on if it gets dark, and the air conditioning system is able to not only adjust the right temperature within the car, it also “smells” the environmental air and reacts by adjusting the outside air supply accordingly. The engine and exhaust system is a very complex feed back mechanism, these days. Many different 3 R. Gross et al. (eds.), Nanoscale Devices - Fundamentals and Applications, 3–13. © 2006 Springer. Printed in the Netherlands.
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parameters of the environment, driving behavior and street conditions are the input for an ‘intelligent’ modern (though at least upper middle class) car. Hence, sensor technology is not only a niche market. In the recent past, unfortunately, we constantly hear about the need for sensors that are able to detect chemical warfare reagents, biological substances, bacteria and other frightening material. Fortunately, however, this kind of sensors still remains a relatively small market. Much more important are those uncountable sensors and smart systems out there, of their ambiguous presence we sometimes not even know. Exactly for this reason, it would be desirable to have a sensor system available, where many different environmental parameters can be determined at the same time. Let us compare it to the sensing abilities of a living organism: Most of them are able to “see”, “hear”, “smell”, “taste”, and “feel”. These five senses have been developed during evolution and – apart from some species living under special environmental conditions – seem to be sufficient to satisfactorily react to the environment. If we try to categorize the senses into the framework of “sensors”, we thus have an optical sensors (eyes), two types of sensors being sensitive to mechanical quantities (tactile sense and ears), and two sensors being sensitive to basically chemical reactions (nose and tongue). All the five sensor systems have in common that they not only consist of a single element but a usually large number of “channels” being able to differentiate different colors, sound frequencies, and chemicals. In this article, we wish to describe the fundamentals of a sensor system that in principle is able to act as a simplified eye, an artificial nose, and even a tactile sensor for smallest forces employing the exact same basis technology in all cases. The sensor is electrically addressable and hence also fulfils the requirements to act as a link between an electronic circuit and the environment. The sensor principle is based on the interaction between surface acoustic waves (SAW) and an externally induced change of the boundary conditions which determine their wave equation. SAW are modes of elastic energy propagating at the surface of a solid. They usually have two components of particle displacement in certain directions with respect to the surface. Two of the simplest modes a called “RayleighWave”, where the wave particle displacement as compared to the unperturbed surface is elliptically polarized with the two axes in the direction parallel to the propagation direction, and the one normal to the surface. Another simple mode is the “Shear-Wave”, where the particle displacement is polarized along the two directions in the plane of the surface [1]. In Fig. 1, we schematically depict a snapshot of a Rayleigh mode. This decay gives the wave the name “surface-wave”. The energy
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flux in such waves is usually confined to a layer of approximately one wavelength thickness, and the decay is more or less exponential.
Fig. 1. Sketch of a Rayleigh-wave at the surface of an elastic solid. Note the decay of the wave amplitude into the depth of the substrate.
In contrast to the case of an isotropic solid, where all properties of a SAW are independent of the choice of the surface and propagation direction of the SAW, for anisotropic solids like semiconductor crystals, one has to include this anisotropy into the description of the SAW itself. For crystals with the lack of inversion symmetry (like the zinc blende lattice as in GaAs), additional effects arise from the piezoelectricity of such materials. Regarding such a piezoelectric crystal and ignoring free charges for a moment, the wave equation for a Rayleigh SAW is usually written in terms of a modified elastic constant c*, taking into account the effect of piezoelectricity [2].
ρ
2 ∂ 2u ∗ ∂ u − =0 ; c ∂t 2 ∂r 2
p2 2 c* = c 1 + ≅ c 1+ K . c ⋅ε
(
)
(1)
Here, p, c, and ε denote the components of the piezoelectric, the elastic, and the dielectric tensor. Usually, these material constants are combined into a single constant K2=p2/cε, describing the amount of piezoelectricity of the respective substrate. The effect of piezoelectricity hence slightly stiffens the substrate, leading to a somewhat higher sound velocity v=v0+∆v/v0, being connected to the bulk coupling coefficient K2 via [2] 2 ∆v K eff K 2 = ≅ 2 2 v0
(2)
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To distinguish between the constant K2 used in eq. 2, and defining the piezoelectric stiffening in bulk material, the index eff is introduced for the effective electromechanical coupling to surface waves. Here, it is interesting to note that eqs. (1) and (2) basically describe the possibility to use SAW as a sensing element. All the quantities defining K2, namely the piezoelectric, the dielectric, and the elastic tensor components can be slightly modified by the interaction with an external source, and hence modify the SAW propagation parameters. Usually, these are the attenuation Γ, and the renormalization ∆v/v0 of the sound velocity. Both quantities can be read out and hence provide the sensor signal.
Fig. 2. Simple SAW delay line. In between the inter-digital SAW transducers, a functionalized and sensitized thin film is responsible for the interaction with an external parameter to be sensed. This interaction is detected by a change of the propagation parameters of the SAW.
Moreover, as the sensitivity usually strongly increases with increasing frequency, SAW are usually regarded to be superior to bulk crystal resonators like quartz micro balances, for example. The reason is that SAW can be excited employing planar metal electrode arrays, whose lateral spacing determines the resonance frequencies. Bulk resonators, on the other hand, rely on thickness vibration modes. Apart from some modern implications like “FBARs”, fabrication processes and reproducibility restrict their application to rather low frequencies [3]. In a SAW, however, only a thin layer of the order of a wavelength is effectively oscillating and hence sensitive to external changes of the boundary conditions. The simplest SAW sensor hence consists of a so-called delay line, where one transducer is used to excite a SAW, and another is used to detect the transmitted SAW after passing a sensitized area in between the
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Fig. 3. The principle of the tapered SAW transducer for sensing purposes. A varying distance between adjacent fingers provides a frequency dependent position of the launched SAW beam. In (b), we depict the frequency response of such a tapered transducer. In this case a split-4 design was used, resulting at four different bandpass regions at odd harmonics of the SAW mode.
two transducers. In Fig. 2, we depict such a simple sensor element. The sensitized area needs to be a functionalized region of the sensor chip, changing some of its properties under the influence of a sensor signal. This could be for instance a change of the conductivity [4] under illumination [5] or accumulation of a reagent, a change of the mass loading the chip, a change of the dielectric properties and alike. Based on this concept, a variety of sensors have already been described and even commercialized. Usually, each “channel” in these cases consists of a single SAW delay line, being more or less sensitive to a single ingredient of the analyte. To gain specificity, at least of order ten different sensors have to be combined to result in reliable, specific analysis of, say, a gas mixture. A more sophisticated SAW sensor scheme relies on the combination of gas
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-60
SAW-attenuation [a.u.]
SAW transmission [dB]
chromatography and a mass sensitive SAW delay line [6]. Here, the specificity of the chromatographic process acts as the different “channels”. The SAW delay line only detects unspecific mass loading of the different ingredients of the analyte mixture, and the time sequence of the sensor signal results in the specific signal.
-80 -100
800
Y
X
820
840
860
880
900
920
Frequency [MHz] X Y
reconstructed position of a laser spot on the sample Fig. 4. Spatially resolved perturbation of the electrical boundary conditions for SAW propagation on a semiconductor thin film. In this case, a laser was used to locally excite free carriers in the film which locally altered the sheet conductivity. This local conductivity change can be monitored by the spatially resolved SAW – thin film interaction as described in the text.
Here, we wish to describe a sensor element, which by a special design of the sound transducers allows for the parallel detection of many different ingredients in a gas mixture at the same chip. We therefore use so-called “tapered” transducers, where the applied high frequency signal is converted into a narrow SAW beam, propagating at different sound paths for different frequencies [7]. The basic idea behind such a “tapered transducer” is shown in Fig. 3, where we show a two-dimensional version of our sensor element. Both sets of transducers each define a bandpass filter, as shown in Fig. 3b. Once an external perturbation of the boundary conditions for SAW propagation is present on part of the active sensor area, a signal in either Γ
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or ∆v/v0 is observed in the respective bandpass, at a specific frequency which can be easily converted into the real space coordinate on the chip.
Fig. 5. SEM micrographs of a zeolite thin film (silicatlite-1), deposited on a sensor chip (top). In the bottom picture, we show the sensor response (SAW phase shift) for different i-butane partial pressures in a carrier gas.
To prove the concept of such a sensing element, we depict in Fig. 4 sensor being sensitive to illumination. This is accomplished be depositing a semiconducting thin film on the piezoelectric chip providing the SAW. Illumination creates free electron and holes in the semiconductor layer, thus increasing its conductivity, locally. This change in conductivity returns a
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SAW signal according to eqs. (1), and (2), respectively, which can be used to reconstruct the position and intensity of the illuminated pattern on the chip (see Fig. 4). Even complex optical images can be reconstructed this way, by employing a tomographic technique [8].
Fig. 6. Sensitivity of eight different functionalized thin films for different gas mixtures. Note that basically each pixel is sensitive to all the three gases, the degree of sensitivity, however, strongly varies [10].
For chemical or biological applications, sensitivity to mass loading is sometimes an appropriate tool to detect specific substances. There are many different approaches for molecular specific capture functionalizations on such sensors [9], all of which have some pros and some cons. The major disadvantage of most of them, however, is the fact that only monolayer mass loading can be detected. Here, we wish to describe a novel type of mass loading functionalization, being molecular specific, and at the same time provide a large mass loading capacity. We functionalize the active surface of our chip by monolayers of nanocrystalline zeolithes with chemically adjustable pore size. In Fig. 5, we show the micrograph of such a thin zeolite layer used for sensor purposes on our spatially resolving SAW chip. In this case, a silicalite-1 system has been used in which the pore size can be adjusted to a diameter of about 0.55 nm. In the lower panel of the figure, we show the response of the sensor for different butane-1 partial pressure in a carrier gas at room temperature. Many different sensitized functionalized thin films are presently under investigation, according to their specificity with respect to different gases. If such sensor “pixels” are deposited within the active area of a two-dimensional spatially resolving SAW sensor chip with two sets of tapered transducers, like the one described in Fig. 4, a very specific and
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highly sensitive sensor for different gas mixtures and/or contents can be devised. For this purpose, an array of differently sensitized pixels (single sensors) is deposited in a checkerboard like manner in between the four senor transducers. The different pixels aught to have a different response for a given gas or gas mixture (see Fig. 6). The spatially resolving SAW chip employing the tapered transducers is the used to read out the accumulated sensor signal for a set of pixels in either a row or a column.
Fig. 7. Spatially resolving SAW sensor employing tapered SAW transducers as described in the text. The active area consists of an array of different sensitized pixels, each having a specific response (numbers on squares) to a given reagent. The SAW can read out the accumulated signal (Σ) of, in this case three pixels at a time.
In Fig. 7, we depict the idea of such a sensor chip with many different pixels. Each pixel exhibits a specific sensitivity for a given gas mixture. This sensitivity results in a SAW sensor signal like attenuation and/or phase change. In the figure, we have denoted the sensor signal for a given gas mixture by the numbers superimposed to the pixels. The read-out SAW signal is then given by the accumulated signals for a single row or column, respectively. In the figure, we have denoted these accumulated signals by the sum sign and the arithmetic sum of the different rows and columns. In Fig. 8, finally, we propose a display technique for such sensors, especially well suited for human inspectors. Humans are very good in pattern recognition, hence we convert the sum signals of figure 7 into a polar diagram, for instance, resulting in an easily recognizable pattern for a given gas mixture. A similar technique had been described in [6]. There, however, a gas chromatograph has been used as a sensor.
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A. Müller, A. Darga, A. Wixforth
Fig. 8. Proposed readout scheme for the sensor depicted in Fig. 7. The accumulated sensor signals (S) are plotted in a polar-type diagram, providing an easily recognizable pattern for a human inspector.
In summary, we have described a highly sensitive sensor scheme for different external parameters. We use surface acoustic waves which can interact with such external parameters, altering the propagation parameters of the SAW. Such parameters may be conductivity changes due to illumination, adsorption, intercalation etc., or more direct measurements like mass loading of the surface. To increase the sensitivity for gas adsorption, we have used functionalized mesoporous zeolite thin films, which can be used as sensor pixels on our chip. The authors thank T. Bein, and J.P. Kotthaus, CENS, Munich for their continuous support and many useful discussions. Financial support of the Bayerische Forschungsstiftung under the program FORNANO, and support of Advalytix AG, Brunnthal, Germany is gratefully acknowledged.
References 1. Auld BA (1973) Acoustic Fields and Waves in Solids. VI, John Wiley & Sons, Toronto 2. Ingebrigtsen K A (1969). J Appl Phys 40:2681; Ingebrigtsen KA Jr (1970). Appl Phys 41:454 3. Farnell GW, Cermak IA, Silvester P, Wang SK (1970). IEEE Transactions on Sonics and Ultrasonics Su27:188 4. Wixforth A, Kotthaus JP, Weimann G (1986). Phys Rev Lett 56:2104 5. Streibl M, Beil F, Wixforth A, Kadow C, Gossard AC (1999). Proc of IEEE International Ultrasonics Symposium 6. Staples EJ (1999) Electronic Nose Simulation of Olfactory Response Containing 500 Orthogonal Sensors in 10 Seconds. Proceedings of the IEEE Ultrasonics Frequency Control and Ferroelectrics Symposium, Lake Tahoe, CA, Oct. 18-21
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7. Streibl M, Govorov AO, Wixforth A, Kotthaus JP, Kadow C, Gossard AC (2000). Physica E 6:255-259 8. Wixforth A (2000) International Journal of High Speed Electronics and Systems. World Scientific Publishing Company 10:1193-1227 9. Ballantine JS, White RM, Martin SJ, Ricco AJ, Zellers ET, Frye GC, Wohltjen H (1996) Acoustic Wave Sensors. Academic Press, NY 10. Wessa T, Küppers S, Rapp M, Reibel J (2000). Sensors and Actuators B Special issue of Prof Göpel in memoriam, 70:203-213
The Experience of Utilizing the Explosives Detection System on the Basis of Neutron Radiation Analysis Together with X-Ray Units at the Airport “Pulkovo” in St.-Petersburg
Y. Olshansky1, A. Vishnevkin1, A. Sorokin1, A. Vikdorovich1, A. Golovin2, E. Stepanov 2 1
Scientific & Technical Center “RATEC”, St Petersburg 193079, Russia
2
“Pulkovo” Aviation enterprise, Saint-Petersburg, Russia
Abstract:
The security system utilizing the explosives detection based on the neutron radiation analysis is described. The system successfully operates in the Pulkovo airport, St. Petersburg, Russia, since spring 2003.
Keywords: explosive detection, thermal-neutron analysis
Because of tragic events during the last twenty years on hijacking and explosion of passenger aircrafts, the Russian Federation laws prescribe rigid security requirements to preflight inspection of aircraft crew members, maintenance staff, air passengers, carry-on baggage, check-in baggage, mail, cargo and on-board load. According to these requirements each airport has to organize 100% preflight inspection of the passenger baggage for the purpose of detection of forbidden objects and substances. This led to significant re-organization of preflight inspection, as many airports of the former Soviet Union had been built long before the new security requirements, and they were not adjusted to operate in such rigid conditions. This article deals with a new approach to organization of the aviation security systems in the Pulkovo airport (St. Petersburg, Russia). Pulkovo is the main airport of the second largest city in the country – St. Petersburg.
15 R. Gross et al. (eds.), Nanoscale Devices - Fundamentals and Applications, 15–21. © 2006 Springer. Printed in the Netherlands.
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Nowadays Pulkovo has two terminals: Pulkovo-1 services domestic flights, in particular to the North Caucasus region, and Pulkovo-2 is fully used for international flights. The rigid security requirements and necessity to inspect 100% of the baggage called for a serious re-evaluation of the available equipment by the aviation security service. The checking lines must meet the security requirements and provide for efficient and convenient inspection. A task of development of fundamentally new checking lines was realized. Special attention was paid to stable detection of explosives, especially plastic explosives, disregarding of their type, form and masking. To realize this purpose it was decided to take a range of administrative and technical measures. First of all it was decided to organize the doorway inspection of all baggage (check-in and carry-on) prior to checking-in. This allows inspecting all baggage of a passenger in his/her presence, and if something suspicious is found in the baggage, the situation can be clarified right on the spot. The actual inspection of a baggage was split into two phases. In the first phase an operator with the help of an X-ray unit carries out standard inspection of the baggage for the presence of forbidden objects. If inside the carry-on or check-in baggage there are suspicious objects, similar by density to explosives, then this baggage is sent to the second phase of inspection. In the second phase the thermal neutron radiation analysis (TNA) detection system is used. It was developed by the Scientific Technical Center RATEC (St. Petersburg, Russia), model EDS-5101 (Explosive Detection System). In both Russia and the United States for the last fifteen years there have been attempts to develop systems for detection of explosives in the air passengers’ baggage on the basis of thermal neutron analysis. The explosives are identified by registration of the gamma radiation of the nitrogen nuclei under irradiation by thermal neutrons (explosives, especially plastic explosives, are characterized by high content of nitrogen). However, so far these attempts have not been resulted in building the systems feasible for a practical use in inspection of the passengers’ baggage and other objects. The main obstacle was a considerable amount of nitrogen in the passengers’ baggage that is not related to explosives. This led to a great number of false alarms. At the same time, X-ray units are aimed for registration of differences in substance density, but this method cannot allow for stable detection of camouflaged explosives, especially plastic sheet explosives. Aviation security specialists generally agree that success in detection of such explosives can be achieved only through a combination of different technologies.
The Experience of Utilizing the Explosives Detection
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In the Pulkovo airport this combination is introduced in two phases of baggage inspection: in the first checking line X-ray units are used, and in the second phase TNA systems EDS-5101 are used. It is important to note that the thermal neutron analysis enables to look into the structure of the baggage substances and to detect the presence or absence of explosives by a noninvasive analysis with a high degree of probability. The EDS-5101 system uses target designation of a suspicious area obtained from an X-ray unit. This allows to inspect not the whole content of the baggage, but only the area where explosives were suspected, and to detect explosives with a high degree of precision and with a low level of false alarms. In addition, new improved algorithms have been used in the EDS-5101 system, which utilize latest developments in physics and mathematical statistics to provide stable separation of signals from nitrogen in explosives on nitrogen in other substances. This decreases the number of false alarms significantly. The likelihood of presence of suspicious objects in the check-in and carry-on baggage is approximately 20–25%, that is, every fourth or fifth baggage item. The time of inspection of one piece of the check-in (or carryon) baggage in an X-ray unit is about 5–7 seconds, the time of inspection of suspicious check-in (or carry-on) baggage with EDS-5101 is about 12–17 seconds. Thus, one explosive detection unit can serve from two to four usual X-ray units, and check the carry-on baggage and suspicious check-in baggage for the presence of explosives without reduction in the inspection line capacity. EDS-5101 can be successfully used for checking of personal computers, photo and video cameras, mobile phones which can used to camouflage explosives. EDS-5101 has a mode of checking the entire carryon baggage, if X-ray units cannot provide a clear picture of a suspicious area against the background. In this case EDS-5101 works practically without target designation, however, it efficiently detects explosives. Special algorithms allow to neutralize the influence of nitrogen not related to explosives, so that it does not practically affect the final result. It is important that the system comes to a conclusion automatically, without an operator. The explosives detection system was tested in the Livermore National Laboratory under operation of specialists from the Transport Security Administration (USA). It received positive assessment both of the quality of operation and the level of safety of such systems in airports. As the testing was carried out without X-rays, it was recommended to carry out repeated trials with an X-ray unit. For comparison, it can be said that the level of the stray radiation from the surface of EDS unit is practically on the level of natural background in the airport owing to special protective material for the case of the equipment. The EDS system received a certificate of the Sanitary Inspection (the Russian requirements to radiation
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safety conform to the international requirements). The system can work with two types of neutron sources: neutron generator and californium isotope Cf-252.
Fig. 1. EDS-5101 developed on the basis of the thermal neutron radiation analysis and aimed for exploiting in airports and other transport objects.
Explosives detection system EDS-5101 has successfully passed tests and received a certificate of the Aviation Security Department of the Transportation Ministry of Russia. A special decree of the Transportation Ministry of Russia recommends for the use of EDS-5101 in airports. The first system EDS-5101 was installed in the checking line of Pulkovo1 airport in spring 2003. The choice of this particular terminal was determined by several reasons: firstly, Pulkovo-1 serves flights on the southern direction, including the North Caucasus region; these flights require special checking and keen attention. Besides, in May-June 2003 St. Petersburg celebrated its 300th anniversary, and an international summit was planned for this time. Many VIP guests were invited, including leaders of
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forty-five states. Pulkovo-1 was the only airport which received the delegations, and where every forty seconds another governmental flight was landing. Unprecedented security measures were taken in the airport. The explosives detection system EDS-5101 was efficiently used in Pulkovo-1 upon receiving of guests, including trials when real explosives were used at nonstandard situations with minimal quantities of explosives. The system operated reliably during the 300th anniversary celebration and the summit meeting.
Fig. 2. Principle of organization of baggage inspection line exploiting the EDS5101 system based on the neutron radiation analysis.
From the very beginning the system was widely used because of many suspicious objects in a hand baggage and check-in baggage of the air passengers. Practically in every flight X-ray units identify objects that could contain explosives. In such cases EDS-5101 efficiently helps operators to check these objects without taking them apart. It is not always possible, for example, in case of notebook accumulators, photo and video camera batteries, etc. At present time specialists of the Scientific and Technical Center RATEC develop an interface that will allow the operator himself to tune the system depending on the object in question; this is needed mainly for checking small-size objects and detecting the minimal quantities of explosives.
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The Pulkovo-1 experience in using checking line with EDS-5101 has demonstrated that introduction of the two-phase inspection allows to separate suspicious baggage from the ordinary baggage immediately. In this case the X-ray unit operator does not loose time on scrutinizing the suspicious object inside carry-on (or check-in) baggage, but sends the suspicious baggage to the second phase of inspection. This enables minimization of the time for inspection of one unit of baggage by the X-ray checking line. In other words, there is no point in rejecting the ordinary Xray systems at the first phase. X-ray is, undoubtedly, not perfect, but with its help a suspicious area can be identified pretty fast, and the final decision can be made later utilizing TNA. An additional advantage is that EDS-5101 can work in a special mode without target designation. In the baggage inspection scheme it is also important to combine systems that are based on different physical principles. Thereupon a combination of thermal neutron analysis and X-ray is an optimal solution to baggage inspection, where each system performs its own task and together they combine a highly reliable, efficient and comparatively inexpensive inspection complex. At the beginning of 2004 RATEC was commissioned by government to develop a special system for detecting small amounts of explosives in cases, handbags, packages and other similar objects. By now RATEC has developed a range of TNA systems for protection of airports, governmental organizations, banks and special facilities from terrorist attacks with use of explosives, radioactive and fissible materials. Systems for checking letters and parcels have been developed, as well as a system for protecting VIP persons. It allows fast and efficient checking of mobile phones, photo and video cameras, dictaphones, notebooks, microphones and other similar objects for presence of explosives, radioactive and fissible materials. For airports with dense traffic flows RATEC currently develops an automatic combined system with both X-ray and TNA systems for inspection of the hand baggage. In this new system the target designation of a suspicious area will be done automatically, which will speed up inspection of one piece of the hand baggage. RATEC staff develops also a TNA system for inclusion in the inspection line for additional checking of a baggage suspicious in the first phase of inspection. Development of such combined systems provides large airports with the state-of-the-art, highly efficient inspection systems.
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Table 1. EDS systems on the basis of TNA – “ready for sale” and under development of RATEC. # Models
Systems application
1 EDS-1101
For inspecting suspicious objects, found in public places
2 EDS-2001
For VIP persons protection. Detection of minimum quantity of explosives
3 EDS-3100 EDS-3101
For offices, State institutions, banks and various special objects
4 EDS-4100
For nuclear power stations with detecting small quantities of special nuclear materials
5 EDS-5101
For airports and other transport objects
6 EDS-6101
For special service with detailed investigation of checked objects
7 EDS-7101
For inspecting check-in baggage of the passengers
8 EDS-8101
For inspecting cargo air and sea containers,
In conclusion, it should be emphasized that the Pulkovo airport experience in organization of checking lines can be widely used in airports with relatively modest traffic flows. This approach does not require serious reconstructions and investments. It includes already available equipment to organize efficient and reliable inspection of carry-on and check-in baggage in the fight against the common enemy of the civilized world – terrorism.
Thermodynamic Principles of Artificial Nose Based on Supramolecular Receptors
V. V. Gorbatchuk, M. A. Ziganshin Kazan State University, A. M. Butlerov Institute of Chemistry, Kazan 420008, Russia
Abstract:
The cooperative effects in the substrate vapor binding by solid receptors and their relevance for structure-property relationships in the odor sensor applications are discussed and reviewed.
Keywords: calixarenes, cyclodextrins, cross-linked hydrophilic polymers, proteins, vapor binding cooperativity, molecular recognition, odor sensors, headspace GC analysis, model sensor systems, structureproperty relationship.
Introduction Application of supramolecular receptors such as calixarenes, cyclodextrins and proteins in the odor recognition devices provides the enhanced vapor binding selectivity [1, 2]. The answer on the question, why a given receptor is more or less selective, is necessary for a design of the effective odor sensors that can substitute for this purpose the living beings. In this review, the effect of cooperativity in the substrate vapor binding by solid receptors on the structure-property relationships of this process is discussed. Cooperativity is one of the major factors defining the receptor selectivity. However, this effect is often underestimated and even neglected in the sensor applications. In this review, we analyze two types of the binding cooperativity. One of them can be observed in binary systems. For another to be revealed, a third component is necessary. These two types of cooperativity are of thermodynamic nature (i.e. can be seen on sorption isotherms) and relate to the well-known cooperative effects of molecular biology and biochemistry. 23 R. Gross et al. (eds.), Nanoscale Devices - Fundamentals and Applications, 23–34. © 2006 Springer. Printed in the Netherlands.
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The first was observed nearly a hundred years ago by Hill [3]. It is a homotropic cooperative binding of oxygen by hemoglobin. The second is a cooperative hydration effect on the enzyme activity [4]. The structural requirements for synthetic supramolecular receptors (hosts) to mimic these types of cooperativity are discussed in this paper. Being cooperative, a substrate-receptor binding has a bundle of related cooperative phenomena. These are the memory of the receptor preparation history, the cooperative effect of the third component, and the temperature effect on the binding threshold. Most of these effects are hard to be seen using sensor devices. So, the data obtained for the model systems with receptor powder, using headspace GC analysis, where these cooperative effects can be controlled, are regarded here together with the corresponding experimental approaches. Because of the complicated interference of cooperative effects, they can have a strong influence on the observed structure-property relationships. Here several simple relationships of this kind are discussed, which were obtained in standard conditions removing the memory effects, so that some objective basis is provided for an aimed molecular design of sensitive materials for “electronic nose” devices being capable to match the best receptors of living nature.
Cooperative Binding in Binary Substrate-Receptor Systems Cooperative binding in the system with two relevant components: substrate vapor and solid receptor, or vaporous guest and solid host, were observed in a lot of works [5-11] for calixarenes and other clathrate forming
Fig. 1. A typical sorption isotherm of guest vapor by solid host in the binary system.
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receptors. The typical shape of guest binding isotherm observed for nitromethane vapor – solid tert-butylthiacalix[4]arene is shown on Fig. 1. This isotherm has a threshold of guest thermodynamic activity, or relative vapor pressure, P/P0, below which no significant binding can be seen. Above this threshold the guest-host binding capacity sharply increases up to a saturation level, which corresponds to the formation of the saturated clathrate. The same shape of sorption isotherms was observed for the binding of oxygen by hemoglobin in water solution [12]. While this cooperative effect for a single hemoglobin molecule in solution is not very simple to explain, mostly because the crystals of this protein, for which the structural X-ray data were obtained, do not perform a significant binding cooperativity [12, 13], for organic hosts like calixarenes a sigmoidal sorption isotherm has rather trivial explanation. In terms of the Gibbs phase rule it is a result of phase transition in solid host phase at the binding of the guest vapor that gives a host-guest inclusion compound, or clathrate [5, 6, 9, 14, 15]. This conclusion was confirmed by the comparison of the powder X-ray diffractograms of the initial host and of the host saturated by guest [6, 16]. A structural illustration of the phase transition at the formation of inclusion compound with solid host is shown on Figure 2.
Fig. 2. A structural illustration of the phase transition at the formation of inclusion compound with solid host.
For description of sigmoidal sorption isotherms the Hill equation can be used: A = SC(P/P0)N / (1 + C(P/P0)N),
(1)
where S – inclusion stoichiometry, C – sorption constant, N – cooperativity constant, A – experimentally determined solid phase composition (mol of guest per mol of host). The fitting of the sigmoidal sorption isotherms with the Hill equation (1) gives two stable solutions: the stoichiometry S and the ratio (lnC)/N. The last value directly relates to the threshold activity of the guest at the half saturation of host A = 0.5S: a0.5S = exp(-(lnC)/N).
(2)
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The Gibbs energy of the guest inclusion can be calculated by the integration of the sorption isotherms having a saturation part: 1
∆Gc = RT ∫ ln( P / PO )dY = RT ln a 0.5 S .
(3)
0
Here, Y = A/S is the host saturation extent. The inclusion free energy ∆Gc is the free energy of transfer of 1 mole of guest from the standard state of pure liquid to the saturated solid phase (inclusion compound). The right part of equation (3) is valid if the ln(P/P0) value is given by equation (1) as a function of Y. The problem is that no sorption isotherms, obtained up-to-date using thin layer of solid receptors on QCM sensors, have a shape shown on Figure 1. The initial part of these isotherms have rather Langmuir [1, 17-19] or the linear [1, 20] shape, Figure 3. For the same toluene vapor – solid tertbutylcalix[6]arene pair the thin layer of this host on QCM sensor gives a BET isotherm [17], while its powder gives a sigmoidal isotherm [21].
Fig. 3. Typical isotherms of the guest vapor sorption by the host thin layer on QCM sensor with linear and BET shape.
The reason for that may be the different history of the host samples. The low-temperature decomposition of clathrates, which occurs on the surface of QCM sensors, may produce the zeolite-like material with empty cavities in the host keeping the packing of clathrate. Such transition was observed using X-ray method [22]. The zeolites with the fixed surface of gas–solid interface have the sorption isotherms with the Langmuir shape like shown on Figure 3 [23]. The host polymorphism as a function of thermal history was also observed in the other studies [24-26]. The sigmoidal isotherms (Figure 1) were observed for the host powder, where the clathrate memory effect was removed by heating [5 ,6 ,9 ,14 ,15].
Thermodynamic Principles of Artificial Nose
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Cooperative Hydration Effect on the Substrate Vapor Binding The molecular design of biomimetic receptors for the sensor applications is often confined to the synthesis of molecularly imprinted polymers (MIP) with the binding sites similar to those of antibodies and enzymes [27]. For a MIP to be a genuine analogue of antibodies, it should have also a biomimetic hydration effect on a substrate binding [28]. Hydration is a crucial factor for the protein receptor properties. It cooperatively enhances the rates of enzymatic reactions in low water conditions [4]. Proteins in contact with some water-organic mixtures show a cooperative increase both in water uptake [29, 30] and uptake of hydrophobic organic components [31, 32] above a certain hydration threshold. Antibodies also need a sufficient hydration to bind antigens [2]. The typical sigmoidal sorption isotherm, where the partition coefficient A/(P/P0) between the pure liquid sorbate and solid protein phase is plotted against the protein hydration, is shown on Figure 4.
Fig. 4. The hydration effect on the binding affinity of cross-linked poly(N-6aminohexylacrylamide) (data from Ref. [28]) and human serum albumin (data from Ref. [31]) for benzene and cyclohexane.
The protein hydration is favorable for the binding of hydrophobic compounds, like benzene, but only above a threshold value of water contents. Above this threshold the protein binding affinity for a sorbate (substrate) goes up to the saturation level that is approximately equal to the
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value observed for the same protein-substrate pair in water solutions [31, 32]. For this favorable hydration effect to be seen, a synthetic polymer should be hydrophilic to let water penetrate inside its bulk phase and have a rather rigid structure preventing the sorption of hydrophobic or large substrates without hydration [28]. The cross-linked poly(N-6aminohexylacrylamide) (PNAHAA), studied in [28], fits rather well to these requirements. The sorption isotherm of benzene on this polymer has almost the same shape as for human serum albumin, Figure 4. Cooperative phenomena in binary protein-water systems were described as a result of protein microheterogeneous structure [33]. The clathrates of water were observed around hydrophobic groups of amino acid residues [34] and bound hydrophobic compounds [35, 36] in protein crystals. Since the same cooperative hydration effect is observed for the binding of hydrophobic compounds by an amorphous non-protein macromolecular material, like PNAHAA [28], and proteins that do not perform significant cooperativity at the binding of organic vapors in absence of water [31, 32], one can conclude that a source of protein hydration cooperativity may be rather the properties of bound water itself than the special protein structure. Water bound by hydrophilic macromolecular receptor contributes much to its binding selectivity. The hydration of PNAHAA increases its selectivity for the pair benzene-cyclohexane up to almost that of liquid water [28]. But this polymer as well as proteins [31, 32] becomes much less selective for the pairs of more hydrophilic sorbates when hydrated. Strong dependence of the substrate binding selectivity on the receptor hydration can be a powerful tool in the applications of the odor recognition devices. A further justification of the clathrate nature of the cooperative hydration effect and the role of hydration in the substrate binding by hydrophilic receptors comes from the sorption studies for beta-cyclodextrin (BCD) [37]. Dry beta-cyclodextrin does not bind monofunctional compounds larger than ethanol. Hence, being hydrophilic, it fits to the above-mentioned requirement for the receptor to have the biomimetic hydration effect. BCD does bind benzene up to hydration of 0.06 g H2O/g BCD and benzene activity P/P0=0.8 as shown in Figure 5. But when BCD is almost completely hydrated, it forms 1:1.3 clathrate with benzene [37] (see Figure 5). The shape of benzene sorption isotherm on BCD hydrated to 0.172 g H2O/g BCD has the same shape as for the guest vapor binding by solid hydrophobic hosts as shown by Figure 1.
Thermodynamic Principles of Artificial Nose
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Fig. 5. The hydration effect on the sorption of benzene by beta-cyclodextrin (BCD). Data taken from ref. [37].
Secondary Cooperative Effects for the Substrate Vapor Binding The main two types of cooperative effects described above define the odor sensing technique based on the substrate-receptor (host-guest) binding with the formation of clathrates, or inclusion compounds, in the binary systems with the homogeneous initial host, where the memory effects are removed, and in ternary systems, where the change of receptor hydration is a dominating factor. Generally, in the sensor applications more complex systems and/or conditions may be used. In these cases, the secondary cooperative effects may be observed, which, in essence, relate to the main two, but may significantly change the substrate-receptor affinity and binding selectivity, or even mask the system cooperativity as it is observed in the systems with memory effects (Figure 3). The cooperative effect of the third component (second guest) on the guest binding by solid host is one of such secondary effects. This effect can reduce the inclusion threshold by the guest activity so that the sorption isotherm acquires a Langmuir shape (Figure 6) due to the addition of a small amount of the third component [8]. This change may occur when
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Fig. 6. The cooperative effect of third component (0.1 mol toluene/mol host) on the sorption of acetonitrile by tert-butylcalix[4]arene at 298 K. Data from ref. [8].
both guests included can coexist in a single crystal [5]. Otherwise, the threshold activity of the guest binding increases a little [8], probably because of the competition between two guests for the binding sites in solid host phase. Formally, the reduction of the guest threshold activity in the presence of a small amount of the third component (Figure 6) corresponds to the increase of the host-guest binding affinity below the host saturation level. Nearly the same effect was observed for hydrated beta-lactoglobulin, which in the presence of 1.2 % of lipids shows more than a double binding affinity for decane and terpenes than the defatted preparation [32]. Lipids behave as included in the protein solid phase, because the combined effect of lipids and hydration perform an apparent synergism. The cooperative effect of the third component may have a profound effect on the structure-energy relationships of the host-guest binding, making the host selectivity in ternary systems with the binary mixtures of guest vapors much different from the value calculated from the binding Gibbs energies in binary “guest vapor – solid host” systems. Another significant secondary cooperative effect is a temperature effect on the receptor hydration threshold. It was observed for beta-lactoglobulin, which shows a reduction of the hydration threshold value on almost 0.1 g H2O/g BLG at relatively small temperature increase from 298 to 309.5 K (Figure 7) [32].
Thermodynamic Principles of Artificial Nose
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Fig. 7. The temperature effect on the hydration threshold of alpha-terpinene sorption by initially dried defatted beta-lactoglobulin. Data taken from Ref. 32.
Besides, the value of the protein hydration threshold depends on the protein hydration history: preliminary hydration pushes the hydration threshold of hydrophobic sorbate (substrate) binding to the higher values, as compared with the value of threshold hydration observed when protein is hydrated in situ – in the presence of the hydrophobic sorbate [31]. Such hydration history or memory effects were observed in studies of enzymatic reactions with enzymes suspended in organic solvents [38]. The described secondary cooperative effects can give a larger variety of structure-affinity relationships for the same set of receptors prepared or used in different conditions, so that a more specific “fingerprint” can be obtained for a given organic component or a complex organic vapor mixture than in the case, when liquid sensitive material is used.
Structure-Energy Relationship The secondary effects, especially the receptor memory of previous treatment and clathrate structure are detrimental for the predictability of the receptor behavior in the odor sensor applications. Moreover, the apparent structure-energy relationship depends much on the choice of substrate (guest) standard state or concentration scale in the estimations of the sensor effect or selectivity. When the substrate vapor with a certain pressure or concentration is chosen as a standard state, the difference in Gibbs energies of the substrate vapor condensation is often a major contribution in the observed sensor selectivity, no much matter what is the nature of the receptor used [39]. In this case, the sensor selectivity for organic vapors follows the selectivity of their solvation in liquid solvents, like
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hexadecane. The last process can be easily quantified in terms of LSER (Linear Solvation Energy Relationship) approach, which may be used for the characterization of unknown vapors by array of polymer-coated sensors [40].
Fig. 8. Nonlinear influence of the guest molecular size on the excessive Gibbs energy of guest inclusion by solid host from the infinitely dilute solution in model liquid solvent.
Still the supramolecular receptors (hosts) exist that are able to bind a very restricted number of guests, when the host memory of the former clathrate structure is removed by heating [7, 10]. This phenomenon corresponds to the essentially nonlinear structure-energy relationship for clathrate formation. A scheme of such relationship for the case, where only guest (substrate) size is important, is given on Figure 8. This scheme describes the guest size effect on the inclusion Gibbs energy ∆Gtrans determined for the standard state: an infinitely dilute liquid solution of the guest in a model solvent having the same energy of pair-wise molecular interactions with the guest as the host cavity interior. This approach allows extracting a pure contribution of supramolecular effect in inclusion Gibbs energy ∆Gc calculated using equation (3) from the sorption data. The minimum on the V-like dependence corresponds to the guest size such that its further infinitely small increase gives the same cavity energy costs in the host phase as in the model solvent. Above this point an ordinary size exclusion effect should be observed. It was found as a linear structureenergy relationship for the binding of organic vapors by 2,2’-bis(9hydroxy-9-fluorenyl)biphenyl [7] and tert-butylthiacalix[4]arene [10] powders with toluene chosen as a model solvent, and the guest molar refraction MRD used as a guest molecular size parameter.
Thermodynamic Principles of Artificial Nose
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Generally, the structure-energy relationships for the guest vapor – solid host binding may be much more complicated than shown on Figure 8, when the host has an essentially different packing in inclusion compounds with different guests and the guest molecular shape is relevant [7, 10]. While a lot of supramolecular hosts can be synthesized with different packing modes, each of them may have its own specific structure-energy relationship, the bright perspective is open for the applications of supramolecular hosts in sensor arrays for intelligent recognition of odors.
Acknowledgments This work was supported by the programs of RFBR (No.03-03-96188 and 05-03-33012), URBR (No. UR.05.01.035) and BRHE (REC007).
References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19.
Grate JW (2000). Chem Rev 100:2627 Setford SJ (2000). Trends Anal Chem 19:330 Edsall JT, Gutfreund H (1983) Biothermodynamics. J Wiley, New York Halling PJ (1994). Enzyme Microb Technol 16 :178 Furusho Y, Aida T (1997). Chem Commun 2205 Dewa T, Endo K, Aoyama Y (1998). J Am Chem Soc 120:8933 Gorbatchuk VV, Tsifarkin AG, Antipin IS, Solomonov BN, Konovalov AI, Seidel J, Baitalov F (2000). J Chem Soc Perkin Trans 2 11:2287 Gorbatchuk VV, Antipin IS, Tsifarkin AG, Solomonov BN, Konovalov AI (1997). Mendeleev Commun p 215 Gorbatchuk VV, Tsifarkin AG, Antipin IS, Solomonov BN, Konovalov AI (1999). J Inclusion Phenom Macrocyclic Chem 35:389 Gorbatchuk VV, Tsifarkin AG, Antipin IS, Solomonov BN, Konovalov AI, Lhotak P, Stibor I (2002). J Phys Chem B 106:5845 Gorbatchuk VV, Savelyeva LS, Ziganshin MA, Antipin IS, Sidorov VA (2004). Russ Chem Bull 53:60 Perutz MF, Wilkinson AJ, Paoli M, Dodson GG (1998). Annu Rev Biophys Biomol Struct 27:1 Bettati S, Mozzarelli A (1997). J Biol Chem 272:32050 Coetzee A., Nassimbeni LR, Achleitner K (1997). Thermochim Acta 298:81 Barbour LJ, Caira MR, Nassimbeni LR (1993). J Chem Soc Perkin Trans 2 :2321 Caira MR, Horne A, Nassimbeni LR, Toda F (1997). J Mater Chem 7:2145 Dickert FL, Schuster O (1995). Microchim Acta 119:55 Dalcanale E, Hartmann J (1995). Sensors and Actuators B 24-25:39 Wang C, Chen F, He X-W, Kang S-Z, You C-C, Liu Y (2001). Analyst 126:1716
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Molecular Detection with Magnetic Labels and Magnetoresistive Sensors
J. Schotter1,2, M. Panhorst2, M. Brzeska2, P. B. Kamp3, A. Becker3, A. Pühler3, G. Reiss2, H. Brueckl1,2 1
ARC Seibersdorf research GmbH, Nano-Systemtechnologien, 1220 Wien, Austria
2
Department of Physics, University of Bielefeld, 33615 Bielefeld, Germany
3
Department of Genetics, University of Bielefeld, 33615 Bielefeld, Germany
Abstract:
For future lab-on-a-chip devices, compact and inexpensive detection units are required that directly translate the abundance of certain biomolecules into an electronic signal. By detecting specifically bound magnetic labels with magnetoresistive sensors, a versatile platform can be designed that fulfils those requirements and even enables on-chip manipulation of biomolecules by suitable magnetic gradient fields. Here, we present sensitive recognition of different types of magnetic labels by magnetoresistive sensors based both on giant magnetoresistance (GMR) and tunneling magnetoresistance (TMR). Hybridization experiments show that our prototype magnetoresistive biosensor can detect complex DNA with a length of one thousand base pairs down to a concentration of 24 pM. A direct comparison of our magnetoresistive and a standard fluorescent detection method clearly shows the advantage and competitiveness of our approach.
Keywords: biosensor, DNA, lab-on-a-chip, microbead, magnetoresistance, GMR, TMR
35 R. Gross et al. (eds.), Nanoscale Devices - Fundamentals and Applications, 35–46. © 2006 Springer. Printed in the Netherlands.
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Introduction Selective and quantitative detection of small amounts of biomolecules plays an important role in the biosciences, in clinical diagnostics or medical research. So far, it is standard procedure to collect the samples on site and send them to specialized laboratories for analysis, which is rather cost intensive and time consuming. Faster and also more reproducible results could be obtained by so-called ‘lab-on-a-chip’ devices, which have received great attention recently [1, 2]. Ideally, these chip-based systems integrate all the necessary steps to detect certain biomolecules from an originally unprocessed specimen (e.g. separation, amplification, chemical modification and detection) into a single easy-to-use portable device. Recent progress in the field of microfluidics [3] suggests that the preparation of biological samples within an integrated microfluidic device is evolving and will become commercially available within the next few years. Concerning the molecular detection unit of future lab-on-a-chip devices, different techniques are currently employed or actively researched. With respect to DNA analysis, they all rely on the principle of detection by hybridization, which allows a highly parallel analysis of thousands of sequences at a time, each of them within a separate specifically functionalized spot of the sensor. As the sample solution is spread across the entire sensor area, target DNA that is complementary to the immobilized probes hybridizes. As the sequence and position of the preassembled probe DNA is known, the type and concentration of the unknown sample is mapped by measuring the abundance of hybridized target DNA at each spot location. The majority of the established detection methods accomplish this task by specifically adding labels to the hybridized target DNA only and collecting the signals from those labels. Concerning today’s semi-integrated micro-arrays, optical methods that employ fluorescent markers are most common [4]. However, due to spatial restrictions for lab-on-a-chip devices, it seems unlikely that they are going to be used extensively for this new market segment. More fit for that purpose are electrochemical detection methods via suitable electroactive labels, which can easily be integrated into chip-based formats and show sensitivity values comparable to optical techniques [5]. Another concept for DNA detection was introduced in 1998 by Baselt et al. [6, 7]. It is based on detecting the stray field of magnetic labels by embedded magnetoresistive sensors. Such a technique, apart from being sensitive, flexible and compatible with standard CMOS fabrication, also has the advantage of directly providing an electronic signal that is suitable
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for automated on-chip analysis. Furthermore, the possibility to attract the magnetic labels by suitable gradient fields also opens up a completely new prospect that is not possible in such a way for any of the competing detection schemes, i.e. the selective on-chip manipulation of desired biomolecules [8]. This option could be used, for example, to reduce incubation times during hybridization, or to test binding forces of adhered molecules [9]. Due to these opportunities, magnetoresistive biosensors represent a very promising alternative detection unit for lab-on-a-chip devices, and since the first publication in 1998, a number of research groups initiated work on that subject [10]. Here, we present some of our key results on magnetoresistive biosensors [11]. Figure 1 displays the different steps involved in DNA detection by our magnetoresistive biosensor. First, samples of probe DNA are spotted onto the sensor surface and are immobilized via epoxy groups embedded into the top polymer layer (part a). Second, the biotin-labeled analyte DNA is added and hybridizes to complementary probe DNA (part b). In the final step, streptavidin-coated magnetic markers are introduced and bind specifically to the biotin of the hybridized analyte DNA (part c). After each step, washing removes unbound DNA or markers. The magnetic stray field of the markers is detected as a resistance change in a magnetoresistive (MR) sensor embedded underneath the probe DNA spot.
Fig. 1. Sketch of the DNA detection process by the magnetoresistive biosensor.
GMR Type Biosensor A magnetoresistive sensor based on the giant magnetoresistance (GMR) effect [12] is developed and optimized for spotter-based microarray applications. It consists of multilayers in the second antiferromagnetic coupling maximum of the following composition: (Si/(Ni80Fe20)1.6nm/[Cu1.9nm/(Ni80Fe20)1.6nm]10/Ta3nm).
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The film stack is sputter-deposited and subsequently patterned by electron beam lithography and Argon ion etching. A thin Ta film on top of the layer stack serves as an adhesion promoter for the final SiO2 protection layer. The inset of Fig. 2 shows an electron micrograph of a typical sensor element. The spiral-shaped line has a width and separation of 1 µm, and it covers a circular area with a diameter of 70 µm. An entire probe DNA spot can thus be covered by a single sensor element. The total resistance of a sensor element depends on layer thickness and multilayer number and amounts to about 12 kΩ in this case. The response characteristic to an inplane magnetic field is also displayed in Fig. 2. Due to the isotropic geometry, the response is the same for every angle of the in-plane field. The relative resistance change is about 7% at a saturation field of roughly 15 kA/m, resulting in a sensitivity to in-plane magnetic fields of 0.5% per kA/m. The prototype sensor consists of 206 individual spiral-shaped elements on a total area of 5×12 mm2. The contact pads and interconnect lines are made of a Ta10nmAu50nmTa10nm sandwich and are patterned using positive photo lithography and lift-off.
Fig. 2. Layout and magnetoresistance response of a GMR-type sensor element.
The entire sensor with the exception of the contact pads is passivated by a 160 nm thick sputter-deposited SiO2-layer. Additionally, a 60 nm thick methacryl-based polymer layer is spin-coated onto the surface from a dioxane solution. It contains epoxy side-groups and allows covalent
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bonding of the probe DNA amino groups to the surface. Furthermore, the polymer acts as an additional protection layer for the underlying sensor. To test the dependence of the sensor signals on the marker coverage, different marker types in varying concentrations are directly spotted on top of separate elements of the same sensor. In a differential setup, the sensor element of interest is measured relative to a reference element, which is not covered by any markers. In order to magnetize the superparamagnetic labels, a magnetic field is applied perpendicular to the plane, which minimizes its direct influence onto the sensor while assuring high magnetic moments of the markers. The in-plane components of the labels’ stray fields are radially symmetric around their center positions and cause local distortions of the magnetization configuration of the underlying magnetoresistive sensor. Figure 3 displays typical data for such a measurement. If the measured element is also free of any markers, almost no change of resistance occurs in any of the two measurement branches while increasing the magnetizing field, and the output signal of the differential amplifier stays constant (gray line). The black line shows the data for a measured element in which 5% of the surface area is covered by Bangs 0.86 µm magnetic markers. With increasing magnetizing field, the induced dipole moment of the microspheres becomes stronger, and the resistance in the upper branch drops. The output signal is symmetric and displays a nearly linear dependence on the magnetizing field, which is expected as a direct consequence of the linear rise of the marker moment at small fields and the linearity of the GMR characteristics.
Fig. 3. Sensor signals with (black line) and without (gray line) magnetic markers in dependence on the perpendicular magnetizing field.
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Fig. 4. Dependence of the sensor output signals on the surface coverage of three different marker types. The lines are linear regressions to the data.
The response of three different types of magnetic markers is investigated at different surface coverages (Fig. 4). For each single measurement, the maximum difference in the output signal is taken and plotted versus the marker coverage of the specific sensor element. The shaded area represents the maximum signal obtained from reference elements with zero marker coverage, which is always less than 40 mV. At coverages not too close to saturation, a linear increase of the sensor signal on the marker coverage is observed, as more and more of the sensor area gets affected by the induced stray fields. Deviations can be attributed to conglomerations of markers in specific regions. In summary, the GMR-based magnetoresistive sensors are able and appropriate to directly determine the density of magnetic markers on the sensor surface. Therefore, just by measuring an electrical resistance, it is possible to conclude the abundance of hybridized DNA strands on the surface of each sensor element, which is demonstrated in the next paragraph.
DNA-detection by the GMR Type Biosensor In order to test the biological sensitivity of the magnetoresistive biosensor, a comparison experiment with standard fluorescent DNA detection is carried out. Different concentrations of double stranded PCR-amplified DNA sequences with a length of 1 kb are used as specific probe, while the unspecific probe consists of double stranded salmon sperm DNA of the same length in a single but much larger concentration (see Table 1).
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Table 1. Overview of the different probe DNA spots used in the comparison experiment.
In the case of the magnetoresistive biosensor, the probe DNA spots are positioned at the rows of the sensor elements, while a polymer coated glass slide (TeleChem SuperClean substrate) is used in the case of fluorescent detection. The optimized spotting solution contains 29% DMSO and the pH of 10 is adjusted by the addition of 1% TEMED. It denaturates the doublestranded DNA under test. Afterwards, the probe DNA amino groups are covalently coupled to the epoxy sites embedded into the polymer. Nonbound probe DNA is removed in a subsequent washing step, and the remaining epoxy-groups are inactivated by incubation in a high molar acetate buffer of pH 5.0 at 55°C. In the next step, single stranded biotin-labeled (5´ and internal) analyte DNA complementary to the specific probe DNA with a concentration of 15 nM is hybridized by incubation in a 35% formamide solution at 42°C for 12 hours. Subsequently, non-hybridized analyte DNA is removed by washing. Only at this stage, as the markers are being added, the magnetoresistive biosensor and the fluorescent chip are treated differently. In the case of the magnetoresistive biosensor, streptavidin-coated Bangs 0.35 µm microspheres are bound to the biotin-labeled analyte DNA in a neutral solution at a mass concentration of 1% for one hour at 37°C. Afterwards, unspecifically bound magnetic markers are washed away. For the fluorescence sample, Cy3 streptavidin markers are coupled to the biotinlabeled analyte DNA, and the fluorescence is measured with a laser scanner. The sensitivity of the laser scanner is adjusted to almost saturation for the highest specific DNA concentration. The average value of 7-8 spots for each specific probe DNA concentration is taken relative to the average fluorescence of the unspecific DNA spots.
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Fig. 5. Sensitivity comparison of the magnetoresistive biosensor and a comparable fluorescent detection experiment.
In case of the magnetoresistive biosensor, sensor elements covered by probe DNA are contacted via Au-wire-bonding. For each sensor element, the maximum output signal at a magnetizing field of 40 kA/m is taken relative to the averaged signal of the unspecific sensor elements. These sensitivity values are plotted together with the fluorescent sensitivities over the probe DNA concentration in Fig. 5. The standard deviations of the signals taken from different probe DNA spots of the same concentration are represented by error bars. Both methods are sensitive to the whole range of probe DNA concentration (i.e. almost over a range of three orders of magnitude). At the high concentration region, both sensor types are saturated, whereas the sensitivity at the lower end is limited by unspecific signals. However, in the case of the magnetic biosensor, the density of unspecifically bound markers within unspecific probe DNA spots is the same as in regions outside of the DNA spots. Contrary to that, there is some additional background signal within the probe DNA spots in the case of fluorescent detection, which decreases the relative sensitivity. Therefore, in this experiment the sensitivity of the magnetic biosensor is superior to the fluorescent detection at low probe DNA concentrations, for example by a factor of 2.7 at 600 pM. Due to the good comparability of the parallel experiments, our magnetoresistive biosensor has proven to be compatible to standard fluorescence in terms of sensitivity.
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TMR Type Biosensors GMR type biosensors are a good starting point, but they do not represent the most sensitive magnetoresistive device possible. Higher signals can be obtained by tunneling magnetoresistance (TMR) type sensors [12], which show a much larger resistance variation over a smaller field range. Therefore, we are also developing a biosensor based on tunneling magnetoresistance. Just like our GMR biosensors, it is designed to be compatible with DNA microarray type applications. Thus, the size of individual sensor elements is chosen to be comparable to typical DNA spot diameters (around 100 µm), so that each sequence is detected by one sensor element. A SEM image of a sensor element of our TMR type biosensor is shown as an inset to Fig. 6. One of its ferromagnetic electrodes (3 nm of Co70Fe30) is magnetically hard due to direct exchange interaction with an underlying 15 nm thick antiferromagnetic Mn83Ir17 film, while the magnetization of the electrode on the opposite side of the 1.6 nm thick Al2O3 tunneling barrier is free to rotate (8 nm of Ni80Fe20). Due to the unidirectional exchange interaction, the in-plane sensor characteristic depends upon the direction of the applied field, which is apparent from the measurements displayed in Fig. 6. Only when the field is applied parallel to the pinning direction, an antiparallel magnetization configuration is achieved and the full TMR amplitude is reached.
Fig. 6. Layout and magnetoresistance response of a TMR-type sensor element.
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Fig. 7. Signal comparison of GMR- and TMR-type biosensors for similar marker coverage.
Since the stray fields of the magnetic markers are radially symmetric in our standard measurement setup, the actual sensor response to magnetic labels can be regarded as the average of its characteristics parallel and perpendicular to the pinning direction. The TMR-type biosensor is passivated by the same protection layer as described above. Afterwards, magnetic labels are directly spotted on top of individual sensor elements in varying concentrations, and, similarly to the method described for GMR-type biosensors, their signals are taken in dependence of a perpendicular magnetizing field. For maximum sensitivity, an additional in-plane bias field is applied in order to set the operational point of the sensor close to the switching field of the free magnetic layer. Figure 7 shows a direct comparison of the resulting signals both for GMR- and TMR-type biosensors. In each case, the respective sensor elements are covered to about 6% by Bangs magnetic microspheres with a mean diameter of 0.86 µm. The reference line displays the signals obtained for a pair of uncovered sensor elements, clearly indicating the absence of any response to the out-of-plane magnetizing field. The measurements for label-covered sensor elements, however, clearly reveal the increasing effect of the marker’s stray fields onto the sensors’ magnetization configurations with rising magnetizing field. In principle, the data of both sensor types share the same features, but the response of the TMR sensor is larger by a
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factor of 3.5 in this case, which reflects its improved sensitivity to in-plane fields. In addition to being more sensitive, TMR type biosensors could also be employed for single molecule detection, which is an integral part of any science investigating the physics and applications of such entities. In this case, the detection of single molecules translates into the detection of single magnetic labels, which is quite possible provided the size of the magnetoresistive sensor is comparable to the magnetic label [13]. Because the sensor elements are easily scaleable to the required dimensions, TMRbased magnetoresistive biosensors are especially fit for that purpose as they also offer the unique possibility of on-chip manipulation of biomolecules by magnetic gradient fields applied to their labels. Currently, we are working on the integration of on-chip manipulation and TMR-based detection of single molecules [8].
Conclusion We have demonstrated the feasibility of GMR-based biosensors for detecting specific complex DNA sequences down to a concentration of 24 pM. At these low concentrations, the GMR-type biosensor has proven to be more sensitive than a comparable standard fluorescent DNA-detection method. Due to the direct availability of an electronic signal and the small size of the required instrumentation, magnetoresistive biosensors are a promising choice for the detection unit of future integrated lab-on-a-chip systems. Furthermore, TMR-based biosensors with superior signal amplitudes have been fabricated for the detection of magnetic labels. As these systems are easily scalable, they could be employed in the regime of single molecule detection. This is especially true when adding on-chip manipulation of single molecules by magnetic gradient fields applied to their labels.
Acknowledgments This work was supported by the German ministry of education and research (BMBF) under grant number 13N7859 and by the Sonderforschungsbereich 613.
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References 1. Kricka LJ (2001). Clinica Chimica Acta 307:219 2. Liu RH, Yang J, Lenigk R, Bonanno J, Grodzinski P (2004). Anal Chem 76:1824 3. Thorsen T, Maerkl SJ, Quake SR (2002). Science 298:580 4. Heller MJ (2002). Annu Rev Biomed Eng 4:129 5. Umek RM, Lin SW, Vielmetter J, Terbrueggen RH, Irvine B, Yu CJ, Kayyem JF, Yowanto H, Blackburn GF, Farkas DH, Chen YP (2001). Journal of Molecular Diagnostics 3:74-84 6. Baselt DR, Lee GU, Natesan M, Metzger SW, Sheehan PE, Colton RJ (1998). Biosensors and Bioelectronics 13:731 7. Rife JC, Miller MM, Sheehan PE, Tamanaha CR, Tondra M, Whitman LJ (2003). Sensors and Actuators A 107:209 8. Brzeska M, Panhorst M, Kamp PB, Schotter J, Reiss G, Puehler A, Becker A, Brueckl H (2004). Journal of Biotechnology 112:25 9. Panhorst M, Kamp PB, Reiss G, Brueckl H (2005). Biosensors and Bioelectronics 20:1685 10. Graham DL, Ferreira HA, Freitas PP (2004). Trends in Biotechnology 22:455 11. Schotter J, Kamp PB, Becker A, Puehler A, Reiss G, Brueckl H (2004). Biosensors and Bioelectronics 19:1149 12. Gregg JF, Petej I, Jouguelet E, Dennis C (2002). J Phys D: Appl Phys 35 :R121 13. Tondra M, Porter M, Lipert RJ (2000). J Vac Sci Technol A 18:1125
WEAK MAGNETIC FIELDS DETECTION TECHNIQUES AND DEVICES
Magnetic Tunnel Junctions Based on Half-Metallic Oxides
Rudolf Gross Walther-Meißner-Institut, Bayerische Akademie der Wissenschaften and Physik-Department, Technische Universität München Walther-Meißner Str. 8, D-85748 Garching, Germany
Abstract:
Magnetic tunnel junctions (MTJs) are key elements of spintronic devices with widespread applications in magnetic data storage and magnetic sensors. They are based on magnetic multilayer structures and their optimization for applications requires the nano-engineering of the interfaces in these multilayers. The key figure of merit of MTJs is the magnitude of the achievable tunneling magneto-resistance (TMR). A promising way for improving TMR values is the use of half-metallic ferromagnets, that is, ferromagnets in which only states of one spin direction are present at the Fermi level. We review the history and present understanding of spin-polarized tunneling in MTJs and their improvement by using half-metallic ferromagnets. Doing so, we give a classification of various types of half-metallic materials. Furthermore, we address the various existing definitions of the quantity spin polarization and the experimental methods for measuring it. Promising half-metallic materials are ferromagnetic oxides. We review the physical properties and present understanding of the most prominent representatives of this materials class and give an overview on recent attempts to fabricate MTJs with high TMR values from these materials. Here, we particularly focus on problems related to the nano-engineering of interfaces.
Keywords: magnetic tunnel junctions, spin polarized tunneling, spintronics, halfmetallic oxides, spin polarization, nano-engineering of interfaces, tunneling magnetoresistance
1
Electronic mail:
[email protected] 49
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Introduction The field of magneto- or spin-electronics has attracted increasing interest over the last decade. This was stimulated both by the interesting fundamental physics issues associated with the transport, manipulation and detection of spins in magnetic and non-magnetic materials and the increasing number of commercial applications of spintronic devices. For example, the high storage capacity of today's hard-disk drives would have been impossible without the development of read heads based on magnetoelectronic concepts. Moreover, at present the feasibility of the Magnetic Random Access Memory (MRAM) is investigated vigorously by several companies. For future applications more complicated three-terminal devices such as spin transistors are envisioned. The common idea behind all these activities is to exploit both the charge and spin degree of freedom of electrons to arrive at spintronic devices with improved or novel properties that may be able to satisfy the continuously growing demands to devices used in our communication and information technology. Most spintronic devices are based on multi-component materials systems consisting of magnetic and non-magnetic materials. Their operation usually strongly depends on the properties of the interfaces in these structures. Therefore, improving the quality and functionality of spintronic devices always requires not only a solid understanding of the underlying physics but also of the involved materials and even more important a controlled nanoengineering of interfaces. Here, we are discussing the relevant physics, materials and nano-engineering issues related to magnetic tunnel junctions with particular emphasis on the use of half-metallic ferromagnets. Spintronic devices already have and will most likely have their largest application potential in magnetic storage and sensors (for recent reviews see [1-11]. For example, for our today's computer industry there is great interest in the possibility of fabricating a non-volatile random access memory which retains its information even after removing power from the device – an ideal memory. The new concept of a nonvolatile Magnetic Random Access Memory (MRAM) has been proposed and will possibly revolutionize semiconductor memory and other spintronic devices such as programmable logic elements in the near future [2, 12-14]. The basic elements for MRAMs [15, 16] as well as many other spintronic devices are micron-sized magnetic tunnel junctions (MTJs), which consist of two ferromagnetic (FM) electrodes sandwiching a thin insulating (I) barrier. Since the discovery of a large tunneling magnetoresistance (TMR) at room temperature in MTJ devices in the early 90's, [17-21] this research field has
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become very active. Moreover, the various physical phenomena which govern the operation of these magnetoresistive devices and the need for suitable materials made the field of magnetic tunnel junctions very attractive both from the basic physics and materials point of view. This stimulated a tremendous research activity in experimental and theoretical physics as well as materials science aiming at the thorough understanding of the electronic, magnetic and magnetotransport properties of MTJs. We emphasize that one of the keys to a successful application of spintronic devices is the ability to control the magnetization direction in ferromagnetic materials. Evidently, this can be realized by applying small magnetic fields. However, this is not an ideal solution for submicron-sized devices due to the large currents and space required for the generation of the control fields. An interesting solution has been proposed by Berger [22] and Slonczewski [23] called spin-transfer torque. This phenomenon, where the flow of a spin-polarized current is transferring angular momentum to a ferromagnet and changes the orientation of its magnetization has been studied both theoretically and experimentally [24-30]. In ferromagnetic semiconductors additional control is possible both by optical means [31-33] and by applying a gate voltage [34-35]. With the development of so-called multiferroic materials it even may be possible to switch the magnetization direction by applying electrical fields. History and foundations of magnetic tunnel junctions Tunneling always has played an important role in understanding spin effects in electrical transport. First experiments on spin-dependent transport have been made using normal metal/ferromagnet/normal metal (N/FM/N) type junctions based on the ferromagnetic semiconductor EuO [36, 37]. When an unpolarized current passes the ferromagnetic semiconductor the current was found to become spin-polarized [38, 39]. In the early 1970s, spin polarized tunneling was studied by Tedrow and Meservey in a series of experiments on ferromagnet/insulator/ superconductor (FM/I/S) type junctions [40-44] followed by the first tunneling experiments on ferromagnet/insulator/ferromagnet (FM/I/FM) type junctions by Jullière in 1975 [45]. A magnetic tunnel junction (MTJ) consists of two ferromagnetic electrodes separated by a thin insulating barrier allowing for spin polarized tunneling between the junction electrodes. The key feature of a MTJ is the fact that the tunneling resistance depends on the relative orientation of the magnetization in the junction electrodes, which can be changed by an external magnetic field. That is, we observe different resistance values Rp and Rap (or conductance values Gp and Gap) for a parallel and anti-parallel
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magnetization orientation. This phenomenon is called tunneling magnetoresistance (TMR) or sometimes junction magnetoresistance (JMR) with the TMR or JMR effect usually defined as
TMR ≡ −
JMR ≡ −
R p − Rap Rp R p − Rap Rap
=
=
Rap − R p Rp Rap − R p Rap
=
=
G p − Gap (1)
Gap
G p − Gap Gp
.
(2)
Of course, TMR in MTJs is just a manifestation of a magnetoresistance that yields a change in electrical resistance in the presence of an applied magnetic field. Historically, the first magnetoresistance effect (beyond the usual positive magnetoresistance observed for every normal metal) was the anisotropic magnetoresistance in bulk ferromagnets dating back to experiments of Lord Kelvin in the 19th century [46]. Although TMR is known since the early experiments of Jullière [45] and Maekawa et al. [47] about 30 years ago, the interest in TMR initially was modest due to the fact that the TMR values obtained in the first experiments were small (only a few percent) and/or could be observed only at low temperatures [48-52]. Certainly, this was related to the difficulties related to the controllable and reproducible fabrication of MTJs. However, with the strong improvement of magnetic thin film heterostructures triggered by the discovery of the giant magnetoresistance effect [53-55], Miyazaki and Tezuka [18] as well as Moodera et al. [17] developed reproducible fabrication processes for MTJs employing smooth and pinhole-free Al2O3 tunneling barriers. For such MTJs for the first time reproducible TMR values above 10% could be observed at room temperature. This stimulated an enormous research activity resulting in a steady improvement of TMR values. Today, for MTJs based on various transition metal alloys TMR values above 50% have been achieved at room temperature [56-60]. Very recently, for Fe/MgO/Fe MTJs with epitaxial or highly oriented MgO barriers TMR values of even above 200% at room temperature have been obtained [61-65] in agreement with theoretical predictions [66, 67]. The physics behind the TMR is spin-dependent tunneling. The tunneling probability and hence the tunneling resistance depends on the relative orientation of the magnetization of the two magnetic layers. The reason for that is simple. Whereas in non-magnetic metallic materials electrons at the Fermi level with opposite spin direction (we denote them spin-up and spindown in the following) have the same Fermi wave vector and subsequently
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the same tunneling probability for spin-up and spin-down electrons, this is no longer the case for ferromagnetic materials. Here, due to the exchange splitting, we usually have different Fermi wavevectors for spin-up and spindown electrons and in turn different tunneling probabilities. Moreover, due to the exchange splitting also the density of states (DOS) giving the density of occupied and open states is usually different for the spin-up and spindown electrons. Since the tunneling current is proportional to the product of the occupied states in one electrode and the open states in the second times the tunneling probability, it is evident that the tunneling current is spin dependent. The spin dependent tunneling of electrons has been discovered by Tedrow and Meservey [40-43]. Using tunnel junctions with a ferromagnetic and a superconducting Al counter-electrode separated by an Al2O3 tunneling barrier, they could use the superconducting electrode to detect the difference of the tunneling current of the spin-up and spin-down electrons (for a review see [44]). Theoretical models Model of Jullière
Already in 1975, Jullière developed a very simple model relating the TMR value of MTJs to the spin polarization P1 and P2 of the two ferromagnetic junction electrodes. This model, which is illustrated in Fig. 1, is based on two simplifying assumptions. First, it is assumed that the spin of the electrons is conserved during tunneling. Then, the tunneling of spin-up and spin-down electrons could be analyzed within a two conduction channel model, where electrons originating from one spin state in the one ferromagnetic junction electrode can tunnel only into empty states of the same spin direction in the other junction electrode. This two-spin channel model goes back to the pioneering work of Mott [68, 69] and was extended later by Campbell [70] and Fert [71, 72]. Secondly, it was assumed that the conductance for a particular spin orientation only depends on the product of the effective DOS in the junction electrodes. That is, the tunneling probability was assumed to be the same for both spin directions. With these assumption the TMR or JMR could be expressed in terms of the spin polarization in the junction electrodes as
54
Rudolf Gross
Fig. 1. Schematic illustration of the tunneling in a FM/I/FM tunnel junction according to the Jullière model for parallel (a) and anti-parallel (b) magnetization orientation. In the lower part the spin-resolved density of states of a ferromagnetic metal is shown, which is exchange split by Eex. The dotted arrows mark the spin conserving tunneling of the spin-up and spin-down electrons.
TMR =
2 P1 P2 1 − P1 P2
JMR =
2 P1 P2 . 1 + P1 P2
(3)
Here, the spin polarization is expressed in terms of the spin-resolved density of states N↑ and N↓, the majority and minority spin in the ferromagnetic electrodes, as
P ≡
N↑ − N↓ = a − (1 − a) = 2a − 1 N↑ + N↓
(4)
with ai ≡ N i↑ /( N i↑ + N i↓ ) = (1 + Pi )/2 being the fraction of majority charge carriers and 1 − ai ≡ N i↓ /( N i↑ + N i↓ ) the fraction of minority charge carriers at the Fermi level in the two ( i = 1,2 ) junction electrodes. Conductance in eqs.(1) and (3) can be expressed as G p ∝ N ↑ ,1 N ↑ ,2 + N ↓ ,1 N ↓ ,2 and Gap ∝ N ↑ ,1 N ↓ ,2 + N ↓ ,1 N ↑ ,2 (compare Fig. 1) to give eq. (3). Evidently, with
the assumptions made the Jullière's model predicts that the spin polarization of the tunneling current is determined solely by the spin polarization of the density of states (DOS) at the Fermi level. Jullière's result can be obtained from a much more general Kubo/Landauer approach by assuming that the wavevector parallel to the tunneling barrier is not conserved (incoherent
Magnetic Tunnel Junctions Based on Half-Metallic Oxides
55
tunneling) [73]. The loss of coherence can be attributed to the amorphous tunneling barrier (e.g. Al2O3) commonly used in magnetic tunnel junctions. Model of Stearns
Of course, Jullière's assumption that the tunneling conductance is proportional to the product of the effective DOS is oversimplifying. In fact the tunneling conductance not only depends on the number of the electrons at the Fermi level, but also on their tunneling probability, which may be different for different electronic states. This is in particular important for the 3d transition metals, where the electronic structure is characterized by dispersive s-bands which are hybridized with more localized d-band. These features were taken into account by Stearns [74], who pointed out that the tunneling probability depends on the effective mass, which is different for different bands. Since the effective mass of the dispersive s-like electrons is much smaller, the tunneling probability of these electrons is much larger and therefore they essentially carry most of the tunneling current. Stearns also pointed out that in many cases the dispersive bands dominating the tunneling current are close to free-electron bands and therefore the DOS of these bands at the Fermi level is proportional to the Fermi wave vector. Then, assuming that the tunneling conductance is proportional to the DOS of these itinerant electrons we can rewrite eq.(4) as
P ≡
k F↑ − k F↓ , k F↑ + k F↓
(5)
where k F↑ and k F↓ are the Fermi wave vectors for the spin-up and spindown electrons. Using band structure calculations, a spin polarization of 45% and 10% were found for Fe and Ni, respectively, in good agreement with experiment. We see that in the model by Stearns the relevant DOS is identified by the Fermi wave vectors of the itinerant electrons. This was an early hint for the fact that the spin dependent tunneling sensitively depends on the electronic structure of the electrode material.
56
Rudolf Gross
2
Fig. 2. Effective spin polarization P plotted versus κ 2 /k F↑ for different values of
P = ( N ↑ − N ↓ )/( N ↑ + N ↓ ) . P is calculated according to eq. (6).
Model of Slonczewski
Although the models by Jullière and Stearns have been quite successful for the interpretation of some experiments, they have several drawbacks that will be discussed in more detail below. The first more accurate theoretical description of MTJs taking into account the effect of the tunneling barrier was made by Slonczewski [75]. He considered the tunneling through a rectangular potential barrier of height V0 modeling the ferromagnetic electrodes by two parabolic bands shifted rigidly against each other by the exchange splitting. By solving Schrödinger's equation for the left and the right electrode as well as for the barrier region for the parallel and antiparallel magnetization orientation and matching the solutions at the interfaces (doing so coherent tunneling, i.e. k║ conservation was assumed), he recovered the result (3), however with the effective spin polarization
P ≡
k F↑ − k F↓ κ 2 − k F↑ k F↓ κ 2 − k F↑ k F↓ P = . k F↑ + k F↓ κ 2 + k F↑ k F↓ κ 2 + k F↑ k F↓
(6)
Here, κ = (2m/h 2 )(V0 − EF ) is the decay constant of the electron wave function in the barrier region. As seen from Fig. 2, P depends on the
Magnetic Tunnel Junctions Based on Half-Metallic Oxides
57
2
barrier height. For high barriers (large κ 2 /k F↑ ), the Jullière result is 2 recovered: P ≅ P . However, for low barriers (small κ 2 /k F↑ ), the effective spin polarization decreases with decreasing V0 and even changes sign. This result clearly shows that the measured spin polarization may not be characteristic for the electronic structure of the electrode material alone but also reflects properties of the tunneling barrier. Since coherent tunneling is assumed in Slonczewski's model, it is appropriate for the description of junctions with epitaxially grown barriers [67]. More advanced models
Recently, it was found that the sign of the spin polarization derived from the measured TMR effect using Jullière's model depends on the material used for the tunneling barrier. Whereas for Co a negative spin polarization was found for MTJs using a SrTiO3 barrier [76, 77], in experiments with Al2O3 barriers for Co and all other transition metals a positive spin polarization is derived [44]. This clearly demonstrated the relevance of the nature of the tunneling barrier. The energy, symmetry and orientation of the unoccupied orbitals in the insulating barrier have a significant influence on the tunneling probability and therefore the observed TMR. Actually, it has been predicted that crystalline tunneling barriers may give rise to much higher TMR values due to a highly spin-dependent decay of certain wave functions with specific transverse momentum in the tunneling barrier [6678]. This clearly shows that the early models such as Jullière model are much to simple, since they do neither include effects due to imperfect barriers nor the detailed electronic structure at the interface between the barrier and the junction electrodes [79-81]. Recent theoretical and experimental work (for reviews see [20, 21, 82, 83]) made quite clear that a quantitative description of spin polarized tunneling is a demanding task, if one has to include all the details on structural and electronic properties of the junction electrodes and the barrier material as well as specific properties of the ferromagnet/insulator interfaces. A discussion of further aspects is given in the section on the tunneling definition of the spin polarization. Decay of TMR with temperature and bias voltage
In most experiments a significant decrease of TMR with increasing temperature is observed. This effect is not predicted by the majority of theoretical models and can be attributed to various mechanisms. First, there
58
Rudolf Gross
may be an increase of inelastic tunneling processes with increasing temperature which are not spin conserving. These processes can be caused by tunneling via magnetic impurity states in the barrier [84-91]. A further process that results in a decrease of TMR with increasing temperature is magnon scattering [90, 92]. Tunneling including the excitation of a magnon can be viewed as an inelastic tunneling process accompanied by spin-flip scattering. Finally, the decrease of the surface magnetization with increasing temperature obviously leads to a reduction of TMR [93-95]. For example, M (T ) ∝ T −3/2 is expected according to Bloch's law due to thermal magnon excitation. The decrease of TMR with increasing voltage can have similar reasons as that due to an increasing temperature. Both for increasing kBT or eV there may be additional inelastic tunneling channels or the excitation of magnons resulting in additional tunneling channels which are not spin conserving [85, 86]. For sufficiently small temperatures and voltages, the contribution of inelastic tunneling processes due to magnon excitations is expected to follow
Gin (T ) ∝ T 3/2
and
Gin (V ) ∝ V 3/2 ,
(7)
whereas the inelastic tunneling current due to multi-step inelastic tunneling via impurity states is expected to follow [86, 87]
Gin (T ) = a1T 4/3 + a2T 5/2 + K Gin (V ) = b1V 4/3 + b2V 5/2 + K
for for
eV >> k BT
(8)
k BT >> eV .
(9)
Here, ai and bi are constants depending on the barrier thickness and the density of defect states. We finally note that a variation of the TMR with varying bias voltage also may be caused by the energy dependence of the density of states. In this case even a sign change of the TMR with varying bias voltage may be obtained as already discussed above [76, 77]. Magnetic tunnel junctions based on half-metals Form the application point of view MTJs with high TMR effect are desirable. Based on the most simple models this can be achieved only with materials having a large spin polarization (the more detailed discussion in the section dealing with the spin polarization will show that actually the so-called
Magnetic Tunnel Junctions Based on Half-Metallic Oxides
59
tunneling spin polarization is the relevant quantity). Until now, MTJs are mostly based on ferromagnetic transition metals and alloys. However, the only partial spin polarization of the charge carriers at the Fermi level in these materials sets an upper limit for the maximum TMR (about 60% at room temperature according to Jullière's model). In order to further improve the TMR of MTJs, materials with a full spin polarization of the charge carries, so called half-metals, are desired. Half-metals are ferromagnets with an unusual band structure in which only states of one spin direction are present at the Fermi level, whereas there is a gap in the density of states for the other spin direction. That is, in half-metals only half of the electrons are conducting. With only one spin band present at the Fermi level, half-metals are 100% spinpolarized. Several classes of potentially half-metallic materials such as the doped manganites, the double perovskites, the Heusler compounds, magnetite, CrO2 or diluted magnetic semiconductors have been proposed as electrode materials for MTJs or as materials for spin injectors [97]. Some of them already have been successfully used for MTJs and TMR values well above 100% have been reported [98]. However, whereas the fabrication techniques for MTJs based on ferromagnetic transition metals is well developed and the magnetic properties of these materials are well understood, this is often not the case for the half-metallic materials listed above. In this book chapter we will discuss the status of the fabrication and the understanding of MTJs based on oxide materials with large spin polarization. We try to address the key factors affecting the magnetotransport properties of such MTJs, in particular the magnitude of the measured TMR effect. In order to do so we first give different definitions of the quantity "spin polarization'', which are relevant for different kinds of experiments, and briefly address methods to measure the spin polarization. We then introduce the relevant oxide materials and present some selected experimental results on MTJs based on these materials. Here, we will in particular address the requirement of nanoengineering of interfaces in multilayer structures used for the implementation of MTJs.
Spin Polarization Discussing spin dependent transport the term spin polarization is often used in a different context. In particular, the degree of spin polarization determined by different experimental methods such as spin-polarized photo emission, spin-dependent tunneling, or Andreev reflection does not
60
Rudolf Gross
necessarily coincide with the definition (4) of the spin polarization. That is, one has to analyze in detail how to extract the spin polarization from a specific experiment. For example, the information obtained from a spindependent tunneling experiment is giving useful information on the degree of spin polarization only if the experimental data can be compared to suitable model predictions. Therefore, in the following subsections we will deepen our knowledge on the spin polarization in ferromagnets. We introduce different definitions of this quantity and discuss, which quantity is actually measured in which experiment. DOS definition of spin polarization The most natural and most often used definition of the spin polarization is the DOS definition PDOS which coincides with eq. (4). In this definition, N↑ and N↓ are the densities of states of the spin-up and spin-down electrons at the Fermi level given by
Nσ
= =
1 (2π ) 1 (2π )3 h 3
∑ ∫δ ( E α
k ,α ,σ
∑ ∫v α
− EF ) d 3k
dS F
kF
,α ,σ
(10)
.
Here, Ek,α,σ and vk,α,σ are the energy and velocity of an electron in band α with spin σ = ↑,↓ and wave vector k. In eq. (10) we have replaced the volume integral in k-space (d³k) by integrals on surfaces of constant energy (dSEdE) by using the relation d³k = dSEdk┴ = dS E
dE dE = dS E . ∇k E hv(k)
Although PDOS can be measured by spin-polarized photo electron spectroscopy [103-105], this definition is of limited meaning for transport experiments, since transport properties are usually not determined by the DOS of the spin-up and spin-down electrons alone. This is of particular importance for materials such as transition metals with "heavy'' (e.g. delectrons) and "light'' (e.g. s-electrons) electrons at the Fermi level. While the DOS often is dominated by the d-electrons, the transport properties are dominated by the s-electrons due to their smaller effective mass [102].
Magnetic Tunnel Junctions Based on Half-Metallic Oxides
61
Transport definition of the spin polarization - the diffusive case We now derive a transport definition of the spin polarization which is more relevant for magnetotransport studies and spintronic devices. We start with the diffusive transport in a metal. Classical Boltzmann transport theory allows us to formally distinguish the current J↑ und J↓ of the spin-up and spin-down electrons and therefore to give a transport definition of the spin polarization as
PJ
J↑ − J↓ . J↑ + J↓
≡
(11)
By using the Boltzmann expression for the current density (diffusive transport) [106]
J
e 2τ (2π )3 h 1
=
∑ ∫ ασ ,
v(k)v(k) dS F ⋅Ε, v(k )
(12)
where Ε is the electrical field and we take into account that several bands with index α can contribute to transport, we obtain for the current density of the two spin directions
Jσ
∝ e 2 〈 Nv 2 〉 σ τ σ
(13)
with
〈 Nv 2 〉 σ
= =
v kασ v kασ 1 dS F 3 ∑∫ (2π ) h α | v kασ |
(14)
1 ∑ vkασ dS F . (2π )3 h α ∫
Note that the second equality only holds for an isotropic material. With this expression we can express the electrical conductivity as
σ
= 〈 Nv 2 〉 ↑ e 2 τ ↑ + 〈 Nv 2 〉 ↓ e 2 τ ↓ .
(15)
2 The comparison with the simple Drude expression σ = ne τ shows, that the
m*
expression 〈 Nv 〉 σ corresponds to (n/m*)σ , that is, to the effective density of electrons of a specific spin direction weighted by 1/m*, i.e. their effective contribution to transport. 2
62
Rudolf Gross
Assuming that the scattering time τ σ is spin independent, we can give the following simplified definition of the transport spin polarization in the diffusive limit [108, 109]:
PJdiff ≡ P
Nv 2
=
〈 Nv 2 〉 ↑ − 〈 Nv 2 〉 ↓ . 〈 Nv 2 〉 ↑ + 〈 Nv 2 〉 ↓
(16)
If, however, the scattering time is spin dependent and/or if there is spin-flip scattering, the expression for the spin polarization becomes much more complicated, since the current density in each spin channel then depends on the properties of both spin channels. We see that in contrast to the DOS definition of the spin polarization, where only Nσ enters, the transport definition of the spin polarization is determined by 〈 Nv 2 〉 σ . This is expected, since transport is not only affected by the DOS, i.e. the number of charge carriers, but also by their velocity v ∝ 1/m * . The definition (16), although interesting from the pedagogical point of view, is however of minor practical relevance, since usually the currents J↑ and J↓ cannot be measured separately in a specific material. In typical experiments the spin dependent transport between a ferromagnetic and a non-magnetic material (e.g. a superconductor, semiconductor or normal metal) is measured. Therefore, we have to discuss the transport across the interface between such materials. In the next subsection, we start with the purely ballistic case, where interface scattering and the mismatch between the Fermi velocities is negligible. Then, this case is extended to more general situations. Transport definition of the spin polarization - the ballistic case We now derive a definition of the spin polarization for the transport across a contact between a ferromagnetic and non-magnetic material. We denote this as the contact definition of spin polarization. Our discussion follows the original work of Sharvin [110], which has been extended by Mazin et al. to allow for an arbitrary Fermi surface [108, 109]. We assume that an electron passing the contact area experiences an acceleration through the electrical field so that its energy has increased by eV after passing the contact. Here, V is the voltage drop across the contact. If in this process the quasi-momentum of the electrons is changed from ħk to ħk', we can immediately state the phase space available for this process at T=0. Since only initial states below EF are occupied, i.e. εk = Ek - EF ≤ 0 and there are
Magnetic Tunnel Junctions Based on Half-Metallic Oxides
63
free final states only above EF, i.e. εk´ ≥ 0, we can write the available phase space by using the Heavyside function θ as
θ (ε k' )θ (−ε k ) = θ (ε k + eV )θ (−ε k ) = eVδ (ε k ) .
(17)
Following the discussion in [108, 109] we now consider the fraction of electrons with a given k, which can reach the contact area per unit time. If the contact plane is perpendicular to the z-direction, this fraction is just vzA with A the contact area and vz the z-component of the electron velocity. The total single spin current density is given by the product of the charge unit, velocity and the number of available states, that is, by
Jσ
=
e (2π ) 3
∑ ∫ α
vkασ , z eVδ (ε kασ )d 3 k = e 2V 〈 N | v z |〉 σ , v z >0
(18)
where we have defined 〈 Nv z 〉 σ completely analogous to eq. (14). Using eq. (10) we can rewrite this expression as
Jσ
=
e 2V
1 dS F e2 SF ,z v = ∑ ∫ kασ , z | v | h (2π ) 2 V , (2π ) 3 h α v > kασ z 0
(19)
where SF,z is the projection of the Fermi surface on the contact plane, which we have assumed perpendicular to the z-direction. Note that for a spherical Fermi surface we just recover the well known Sharvin result. We can compare eq. (19) with the well-known Landauer expression saying that the conductance G = J·A/V of a single conduction channel is just G0 = e²/h. The total conductance is then given by G0 times the number of conduction channels Ncc, which is given by the number of electrons that can pass the contact area. If translational symmetry within the contact plane is not violated, the parallel component k║ of the wave vector is conserved and Ncc is given by the product of the contact area A and the twodimensional momentum density. The latter is just given by SF,z/(2π)², that is, by the product of the projection SF,z of the Fermi surface on the contact plane and the two-dimensional density of states 1/(2π)² in k-space. We obtain for the single spin conductance
Gσ
=
e2 SF ,z e2 N cc ,σ = A = e 2 〈 N | v z |〉 σ A . 2 h (2π ) h
(20)
64
Rudolf Gross
We see that the conductance in the ballistic limit is proportional to 〈 N | vz |〉 , while the conductivity in the diffusive limit is proportional to 〈 Nv 2 〉 . In analogy to eq. (16) we can define the "contact spin polarization'' in the ballistic limit as
PCball ≡ PNv
z
=
〈 Nv z 〉 ↑ − 〈 Nv z 〉 ↓ . 〈 Nv z 〉 ↑ + 〈 Nv z 〉 ↓
(21)
Again, PCball is determined by 〈 Nv z 〉 σ and not by Nσ alone, since transport across the contact is not only determined by the DOS in the two contact electrodes but also by the velocity of the charge carriers. For a spherical Fermi surface the projection SF,z of the Fermi surface on the contact plane is given by πk F2 ,σ resulting in Gσ = e 2 k F2 ,σ A/4πh ∝ k F2 ,σ . This gives PCball = (k F2 ↑ − k F2 ↓ )/( k F2 ↑ + k F2 ↓ ) , in contrast to expressions (5) or (6). We note that ballistic transport between a ferromagnetic metal and a 2D electron gas has also been studied in the context of spin filtering effects [111, 112]. Ballistic point contacts between superconductors and normal metals resp. ferromagnets have been discussed by de Jong and Beenakker [113, 114]. Transport definition of the spin polarization - the extended ballistic case We now extend our discussion of the transport across an interface between a ferromagnetic and a non-magnetic material to the case of a non-ideal interface, where we have an additional barrier and also a mismatch of the Fermi velocities of the two materials. The non-ideal barrier is modeled by a δ-type potential U(z) with a strength determined by the dimensionless parameter Z:
U ( z ) = δ ( z ) ⋅W = δ ( z ) ⋅ ZhvF .
(22)
Here, vF and kF are the Fermi velocity and the Fermi wave vector, respectively. For a vanishing barrier (Z=0) we have an ideal interface (ballistic limit with transmission probability (T=1), whereas for a strong barrier (Z>>1) we have T<<1 similar to a tunnel junction. That is, by introducing the parameter Z we can describe the continuous transition from an ideal contact to contact with small transparency as a tunnel junction.
Magnetic Tunnel Junctions Based on Half-Metallic Oxides
65
In contrast to the case of an ideal interface with Z=0, for the non-ideal interface we now receive a finite reflection probability. That is, the transmission probability T through the contact is 0
Gσ
=
e2 ∑ h k|| ,α
Tk ,α ,σ , ||
(23)
where we have to sum up the transmission probabilities of all available conduction channels. The calculation of the transmission probability depends on the particular model used for the description of the MTJ. Here, we use an extended Blonder-Tinkham-Klapwijk model [116]. The transmission probability Tk ασ is determined by the barrier strength ||
Z and the mismatch of the Fermi velocities v zf in the ferromagnet and v zn in the non-magnetic material. In order to determine the transmission probability we are solving Schrödinger's equation
h2 ∂2 − + U ( z ) Ψ 2 2m * ∂z
=
EΨ
(24)
in the region of the ferromagnet (left) and the non-magnetic material (right) and match the solutions at the interface taking into account the boundary conditions. An electron moving from left to right in the ferromagnet will be reflected/transmitted at the interface to the non-magnetic material with certain probabilities. For the wave function of a spin-up electron we have
1 ik f ⋅r 1 −ik f ⋅r Ψ↑f (r ) = e ↑ + r↑ e ↑ 0 0
(25)
1 ik n ⋅r Ψ↑n (r ) = t↑ e ↑ 0
(26)
with equivalent expressions for a spin-down electron. Here, t↑↓ and r↑↓ are the transmission and reflection coefficients for the spin-up and spin-down electrons at the interface. For Z=0 we have t↑↓ = 1 and r↑↓ = 0. The nomenclature
1 0
means that we are dealing with an electron and not with a
hole. If we would use a superconductor instead of a normal metal, the
66
Rudolf Gross
electron can be Andreev reflected as a hole we have quasiparticles
uk v , k
0 1
and in the superconductor
which represent mixed electron/hole states
[107].
Fig. 3. Sketch of the band structure at the contact between a ferromagnet (left) and a non-magnetic metal (right). In the ferromagnet the lower conduction band edge of the minority spin band is shifted upwards by the exchange splitting Eex. The quantity Γ represents the difference of the lower conduction band edges of the ferromagnet and the non-magnetic metal.
If we assume that only electrons with energies E ≅ EF are contributing to the electrical transport, we have E↑f ≅ E↓f ≅ E n ≅ EF and we can write the absolute values of the Fermi wave vectors as
k Ff ,↑
=
2me EF h2
(27)
k Ff ,↓
=
2me ( EF − Eex ) . h2
(28)
k Fn
=
2me ( EF − Γ) . h2
(29)
Here, Γ is the difference between the lower conduction band edges of the ferromagnet and the non-magnetic metal (see Fig. 3).
Magnetic Tunnel Junctions Based on Half-Metallic Oxides
67
In order to calculate the transmission probability, we assume that the interface is perfectly flat so that we can neglect diffusive scattering. In this case we can assume conservation of k║. For an angle of incident φ of the electrons on the interface we then have sin φ = k||f /kσf and we can write for the wave vectors in z-direction, i.e. perpendicular to the interface (compare Fig. 4)
k zf,σ (φ ) = k Ff ,σ cos φ k zn (φ ) =
(30)
(k Fn ) 2 − [k||f,σ (φ )]2 .
(31)
Applying the boundary conditions at the interface at z=0 [116]
Ψσf ( x, y, z = 0) = Ψσn ( x, y, z = 0)
h 2 ∂Ψσf | − +U Ψσf ( x, y, z = 0) = 2me ∂z z = 0
h 2 ∂Ψσn | +. 2me ∂z z = 0
(32) (33)
and using the wave functions (25) and (26) for both spin directions it is straightforward to calculate the spin dependent transmission and reflection coefficients (see e.g. [148-150]). One obtains
Tσ (φ , Z ) =
v zn vσf , z v zn 2 t . | | = σ 1 n vσf , z f 2 2 (vσ , z + vσ , z ) + (W/h ) 4
(34)
Here we have assumed parabolic bands, i.e. v Ff ,σ = hk Ff ,σ /me and v nF = hk nF /me . We see that the transmission probability depends on the Fermi
velocities and the parameter Z characterizing the barrier strength. Even for the case of an ideal interface (Z=0) the transmission probability may be less than unity due to a possible mismatch of the Fermi velocities. It is evident from Fig. 4 that for large k║,σf the transmission probability will be zero for the chosen situation, since there is no available k║n in the non-magnetic metal. This corresponds to the case of total reflection in optics. Only if the Fermi velocities are equal in both junction electrodes (Eex = Γ = 0), we obtain the well-known Blonder-Tinkham-Klapwijk result T = 1/(Z²+1) and, hence, T = 1 for Z = 0 [116]. To calculate the current density across the interface we have to replace the transmission probability T = 1 in (19) by Tσ (φ , Z ) < 1 and obtain
68
Rudolf Gross
= e 2V
Jσ
1 ∑ (2π ) 3 h α
dS Tσ (φ , Z ) vkfασ , z f F . | v kασ | f vσ , z >0
∫
(35)
Fig. 4. Wave vectors at an interface between a ferromagnet and a non-magnetic metal for spin-up (top) and spin-down (bottom) electrons. Note that due to the conservation of k║ and different Fermi velocities at both sides of the interface there is electron diffraction at the interface.
If we consider the case of a strong barrier, we can neglect in the expression for Tσ (φ , Z ) the first term in the denominator of eq. (34) and obtain
Jσ
∝
1 ∑ (2π ) 3 h α
∫
f vσ , z
>0
(vkfασ , z ) 2
dS F = 〈 N (v zf ) 2 〉 σ . f | v kασ |
(36)
Defining again the spin polarization with this expression for the current density, we get
Magnetic Tunnel Junctions Based on Half-Metallic Oxides
PCbarrier ≡ P
Nv z2
=
〈 Nv z2 〉 ↑ − 〈 Nv z2 〉 ↓ . 〈 Nv z2 〉 ↑ + 〈 Nv z2 〉 ↓
69
(37)
We see that in the tunneling limit a definition of the contact spin polarization is obtained that corresponds to that derived for the diffusive limit. We finally would like to emphasize that it is difficult to give a simple expression for the spin polarization for the case of intermediate barrier strength. Tunneling definition of the spin polarization In many experiments spin polarized tunneling through a thin insulating barrier is used to determine the spin polarization. As already pointed out in introduction, the tunneling conductance is determined by both the DOS in the two junction electrodes and the tunneling probability of the spin-up and spin-down electrons given by the tunneling matrix element |M↑|² and |M↓|², respectively. With ρi↑ = Ni↑ |M↑|² and ρi↓ = Ni↓ |M↓|² with i= 1,2 for the two junction electrodes we obtain the tunneling conductance for the parallel and antiparallel magnetization orientation to
Gp Gap
∝ ρ1↑ ρ 2↑ + ρ1↓ ρ 2↓
(38)
∝ ρ1↑ ρ 2↓ + ρ1↓ ρ 2↑ .
(39)
With the definition (1) of the TMR we obtain
TMR =
2 Ptun ,1 Ptun ,2 1 − Ptun ,1 Ptun ,2
(40)
with
Ptun ,i
=
ρ i↑ − ρ i↓ N i↑ | M ↑ |2 − N i↓ | M ↓ |2 = . ρ i↑ + ρ i↓ N i↑ | M ↑ |2 + N i↓ | M ↓ |2
(41)
We see that (40) reduces to the well-known Jullière expression (3) for |M↑|² = |M↓|², that is, for the same tunneling probability of the spin-up and spindown electrons, which was explicitly assumed in deriving the Jullière result (3). In this case the tunneling spin polarization Ptun is equal to the density of
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Rudolf Gross
state polarization PDOS. However, since the assumption |M↑|² = |M↓|² is not valid in most cases, the experiments based on spin dependent tunneling rather yield the tunneling than the DOS spin polarization. Although there is a huge amount of experimental data on spin dependent tunneling, the interpretation and comparison of the experimental data is complicated, since the measured TMR values are often affected by the following factors: - the crystallographic orientation of the ferromagnetic electrodes [117], - the presence of thin nonmagnetic interface layers [79, 80, 120-126] or surface layers with reduced or changed magnetic properties [127-130], - the reduction of TMR due to inelastic tunneling via localized states in the barrier [85-100] or magnon excitation [92, 93, 118, 119] - the influence of the barrier material on the tunneling probability [60, 76, 131-144], - the influence of the crystallographic orientation of the barrier on the tunneling probability [61-67]. Hence, an accurate description of spin dependent tunneling taking into account all the details of the electronic, structural and magnetic properties of the tunnel junction electrodes and interfaces as well as imperfections of the tunneling barrier is very demanding. For pedagogical reasons we briefly discuss the case of a perfect planar FM/I/FM tunnel junction. Due to periodicity within the plane we have k║ conservation and hence obtain the tunneling conductance per spin channel by summing up over all k║ to (compare (23))
Gσ
e2 h
=
∑T k ||
σ .
k ||
(42)
In order to get Gσ we have to calculate the transmission coefficient Tk||σ which depends again on the particular model. In the most simple case we just can assume a rectangular potential barrier and free electrons in the junction electrodes (free electron model). Then, the transmission coefficient is given by [75, 145]
= 16κ 2
Tk σ ||
Here,
d
is
the
k1,σ k 2,σ e −2κd . 2 2 2 κ + k1,σ κ + k 2,σ 2
thickness
of
the
tunneling
(43) barrier,
ki ,σ = {(2m/h 2 )[( E − Eex ,σ ) − EF ] − k||2 }1/2 is the component of the wave
vector perpendicular to the tunneling barrier (i = 1,2 denotes the two junction electrodes) and κ = {(2m/h 2 )(V0 − EF ) + k||2 }1/2 is the decay
Magnetic Tunnel Junctions Based on Half-Metallic Oxides
71
constant of the electron wave function in the barrier region of height V0. The exchange splitting of the free electron bands is included via Eex,σ, which has opposite sign for the two spin directions. Equation (43) shows that for a thick enough barrier due to the dominating exponential factor only those electrons contribute significantly to the tunneling conductance, which have the smallest κ, i.e. those with k|| ≈ 0. In this limit we recover the expression (6) for the spin polarization derived by Slonczewski. For thinner barriers the spin polarization will depend on the barrier thickness as shown by MacLaren [145]. Although a free electron description is quite intuitive it cannot be used for deriving quantitative results, since the calculations are sensitive on the barrier shape and cannot take into account multi-band effects [146]. Nevertheless we see that even in the very simple free electron model the TMR and the derived spin polarization Ptun are not only determined by the properties of the ferromagnetic junction electrodes but also by the barrier properties. Since the transmission coefficient that enters the Landauer formula (42) is flux conserving, it incorporates a factor of vz, the component of the band velocity perpendicular to the interface. This has been emphasized already above discussing the importance of the band velocity perpendicular to the interface when interpreting spin-dependent transport in the ballistic and diffusive limit. Note that expressions (23) and (42) for the extended ballistic case (strong barrier) and the tunneling case look quite similar. However, one has to keep in mind that in the tunneling case the transmission probability is dominated by the exponential factor exp(-2κd). This results in a selection of states within a narrow tunneling cone, which have vz ≈ vF or equivalently k|| ≈ 0. This is important if one has to deal with single crystalline junction electrodes, since in this case only certain crystallographic directions contribute to the tunneling current. The derived spin polarization then does no longer represent a Fermi average. This is different for the extended diffusive case, where one is averaging over all vz. Nevertheless, in junctions with polycrystalline electrodes the experimental results for the tunneling case and the extended diffusive case will be quite similar, since in the tunneling case the Fermi surface averaging is obtained by the averaging over many grains with different crystallographic orientations. An important aspect in spin dependent tunneling is the chemical bonding at the ferromagnet/insulator interface. It was shown that a change of the interfacial bonding can reduce the spin polarization and even can change its sign [131]. The physics behind this is simple. Suppose, for example, that for a specific interfacial bonding mainly the s states of a 3d transition metal are coupled to those of the insulator. Then the tunneling current is mainly
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determined by the s-states and also the resulting TMR and spin polarization. If in contrast there is a significant coupling of the d-states to those of the insulator, the d electrons due to their high DOS will mainly determine the tunneling current and the resulting spin polarization. As a result, for Co or Ni a different sign of the spin polarization is obtained, if the s- (positive P) or the d-electrons (negative P) are dominating the tunneling current. Today spin dependent tunneling in MTJs is modeled by combining density functional theory within the local spin density approximation (LSDA) for the electronic structure and the Landauer-Büttiker formalism for the conductance [147]. In this approach a multiband description of the electronic structure is used taking into account the electronic states in the ferromagnetic junction electrodes, the interfacial states, the variation of the potential across the tunneling barrier and the evanescent states in the insulator. Recently, Mavropoulos et al. pointed out the relevance of the evanescent gap states in the tunneling barrier [78]. They showed that states of a specific symmetry have a small decay constant κ and therefore will eventually dominate the tunneling current. Recently, Butler et al. [66] showed, that due to this effect in Fe/MgO/Fe tunnel junctions for Fe(100) oriented electrodes the majority spin electrons dominate the conductance resulting in a very high TMR. This was confirmed by Mathon and Umerski [67] and by recent experiments [61, 62]. Spin polarization revisited It is interesting to note that until now the DOS definition of the spin polarization is still commonly used to define the degree of spin polarization although this quantity is completely irrelevant for the magnetotransport properties mostly exploited in devices. Experimental values for PDOS are still derived by interpreting the measured TMR values with the oversimplifying Jullière's model. A likely explanation for this strange situation may be the fact that in tunneling experiments usually the conductance is measured as a function of voltage and then this dependence is used to derive the DOS. Indeed this can be done for superconductor/normal metal contacts. However, this is not possible for ferromagnet/non-magnetic metal junctions, where two different spin channels are compared. Discussing the expressions for the spin polarization derived above we have to make clear that PDOS, PNv, PNv² and Ptun may be completely different in real materials. For example, in transition metals we often can distinguish parts of the Fermi surface that have overwhelming d- or s-character [102].
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73
Those parts with d-character are responsible for the major part of the DOS at the Fermi level given by eq.(10). In contrast, those parts with s-character have small effective mass and high velocity and, hence, dominate
(compare eq. (14)). In the same way, the tunneling probability is much larger for s electrons with their small effective mass, since T ~ exp(-2κd) with κ ∼ (m*)1/2. Hence, it is evident that the differences in PDOS, PNv, PNv² and Ptun are the larger the larger the differences in the Fermi velocities of the d- and s-like electrons. This can be shown by band structure calculations as shown in Fig. 5, where PDOS, PNv, and PNv² are plotted as a function of energy for Fe and Ni [108].
Fig. 5. The different degrees of spin polarization, PDOS, PNv, and PNv² plotted versus energy for Fe (a) and Ni (b) (data according to Ref. [108]).
In general, a large value of PNv² or Ptun can have two different reasons. First, the magnitude of the Fermi surfaces of the spin-up and spin-down electrons can be very different while the Fermi velocities (effective masses) are quite similar. Second, the Fermi surfaces can be quite similar while the Fermi velocities (effective masses) of the two spin channels are very different. In the first case the additional factors v and v² do not significantly change the degree of spin polarization. Furthermore, the tunneling probability is similar for the spin-up and spin-down electrons. That is, we expect PDOS ≈ PNv ≈ PNv² ≈ Ptun. We see that in this case the differently defined values of the spin polarization show only small quantitative differences. However, qualitative differences such as a sign change are absent. A typical example for this case is Fe, for which 〈 N〉 (E ) and 〈 Nv 2 〉 ( E ) have a similar shape. In the second case we have electrons with small and large effective mass at the Fermi level. Then, PDOS is dominated
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Rudolf Gross
by the heavy, whereas PNv² and Ptun are determined by the light electrons. That means, that PDOS and PNv² or Ptun can be very different, since they are determined by different types of electrons. Moreover, PDOS and PNv² or Ptun can have opposite sign. A typical example for this second case is Ni. Although band structure calculations yield PDOS < 0 for Ni, the transport spin polarization has opposite sign for a large energy range, i.e. PNv² > 0. Indeed, for Ni usually a positive spin polarization is derived from transport experiments, where PNv² is measured, which is however much larger than expected from band structure calculations. This indicates that band structure calculations underestimate the effect. This most likely is due to the fact that band structure calculations yield a too weak localization of the d-electrons. Classification of half-metals In this subsection we give a classification of half-metals using the definitions for the spin-polarization given above. Doing so, we will follow Coey's work [151, 152]. Usually, a half-metal is defined as a material which is a normal metal with a well-defined Fermi surface for one spin direction, but shows a gap in the density of states around the Fermi level for the other spin direction [153]. This classical definition is shown in Fig. 6a. Following Coey we denote these half-metals as type-I half-metals. A further classification into IA and IB can be made depending on whether there are less or more than five d electrons so that either the spin-up or spin-down band is only partly filled. In either case there is a spin gap ∆↑ or ∆↓ and a smaller gap ∆sf for spin-flip excitations (see Fig. 6a). It is well known that ferromagnetic transition metals are no half-metals. Although e.g. Co and Ni have a d-band split so that all 3d↑ subbands are filled and only the 3d↓ subbands cross the Fermi level, the presence of an unsplit 4s conduction band results in a finite density of states for both spin directions at the Fermi level. A half-metallic behavior can be generated by shifting upwards the 4s band. In oxides this is achieved by the hybridization of the 4s metal and the 2p oxygen states pushing the 4s band up above EF or shifting down the Fermi level into the d band below the bottom of the 4s band. Typical type-I half-metals are the transition metal oxides CrO2 (IA) and double perovskites such as Sr2FeMoO6 (IB). Another well-known example are the so called Heusler compounds. Here, heavy d elements such as Sb tend to suppress the 3d levels below the 4s band edge by p-d hybridization. For example, in NiMnSb [154] there are spin-up electron with Ni eg character at EF (type IA). In contrast, in the compound Mn2VAl [155] there are Mn t2g spin-down electrons at EF (type IB). It is evident that according to the different definitions of the degree of spin polarization
Magnetic Tunnel Junctions Based on Half-Metallic Oxides
75
given above, for type-I half-metals we have PDOS = PNv = PNv² = Ptun = 1. That is, we have 100% spin-polarization independent of the specific definition of the spin-polarization.
Fig. 6. Schematic sketch of the densities of states for the spin-up and spin-down electrons for the classification of different types of half-metals. For the type-I halfmetals, there is a finite density of states only for the majority, spin-up (not shown) or minority, spin-down electrons (a), whereas there is a gap in the density of states for the other spin direction. Type-II half-metals (b) are equivalent to type-I halfmetals. However, due to narrow bands the states at the Fermi level are localized and transport is due activation across a mobility edge ∆Eµ. Finally, type-III halfmetals (c) have mobil charge carries for one spin direction and localized ones for the opposite. ∆↑,↓ is the representing the gap for a particular spin direction and ∆sf is the gap for spin-flip processes. (d) shows a half-metal as expected from the spinsplit impurity band model [157], which describes the magnetic exchange in a highκ oxide (e.g. ZnO) doped with transition metal ions (e.g. Co) far below the percolation threshold.
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Note that the d-bands may be very narrow. Then, in oxides the charge carriers often form small polarons with large effective mass and small mobility. Conduction is then by polaron hopping. Also, if the Fermi level is close to the bottom or the top of the conduction band, disorder may cause localization below or above a mobility edge. According Coey's classification scheme, type-I half metals with localized charge carriers are denoted as type-II half-metals (Fig. 6b). Again we can further subdivide into type IIA and IIB depending on whether the band is less or more than half filled. Obviously, we have PDOS = PNv = PNv² = Ptun = 1 also for type-II half-metals. A typical candidate for a type- IIB half-metal is magnetite for which the Fermi level is in the t 2↓g band of octahedral iron. The charge carriers of this band are localized to form polarons. The strong localization of charge carriers in oxides brings us to the definition of type-III half-metals (Fig. 6c). In such half-metals we have a finite density of states at the Fermi level for both spin directions. However, whereas the charge carriers for one spin direction are mobile, those for the other are localized and cannot contribute to transport. Then, we have a halfmetal in the sense that we have metallic behavior for one and insulating behavior for the other spin direction, i.e. again only half of the electrons contribute to transport. A typical example for a type- IIIA half-metal is La0.7Sr0.3MnO3 with mobile Mn eg spin-up electrons and localized Mn t2g spin-down electrons at EF. With respect to the different definitions of the degree of spin polarization given above, for type-III half-metals we have PDOS < 0, that is, the DOS spin polarization is less than 100%, it even may be zero. However, since the charge carriers for one spin direction are localized (e.g. v↓ = 0), we have PNv = PNv² = Ptun = 1. That is, the transport spin polarization is 100%. This means that also type-III half-metals have full spin-polarization for the majority of situations, where transport properties are relevant. As the last type we consider diluted ferromagnetic semiconductors. On the one hand, these materials can become ferromagnetic by a polarization of the spins s of the conduction or valance band carriers due to the spin S of localized ion cores by an s·S exchange. This exchange can cause a spin splitting which is larger than the Fermi energy. Examples are (GaMn)As, where the S = 5/2 core spin of the Mn2+ (3d5) ions split the top of the valence band producing spin-down holes [156], and EuO or EuS doped with trivalent rare earth elements. On the other hand, ferromagnetism can arise from the fact that the dopant impurity atoms are sufficiently close to form a narrow impurity band, which is unstable with respect to spin splitting. Possible candidates for this category, where none of the atoms needs to be magnetic, are (LaCa)B6 and ferromagnetic carbon. Another possibility is that the ferromagnetic exchange is mediated by charge
Magnetic Tunnel Junctions Based on Half-Metallic Oxides
77
carriers in a spin-split impurity band [157]. Such spin-split impurity band model can account for ferromagnetism in insulating or conducting high-κ oxides with concentrations of magnetic transition metal ions that lie far below the percolation threshold (Fig. 6d). Measurement of spin polarization The analysis given above makes clear that the spin polarization of a material reported in the literature depends on the method used to measure it. We therefore briefly summarize the various experimental methods sketched in Fig. 7 and discuss the kind of spin polarization that is achieved.
Photoelectron spectroscopy
As illustrated in Fig. 7a, in spin-resolved photoemission experiments [158] the surface of a material is irradiated by photons of energy ħω and the photoelectrons emitted from the surface are analyzed using a spin-sensitive detector (Mott analyzer) [159]. This method directly measures the DOS spin polarization PDOS. However, it has several drawbacks. First, it is a surface sensitive method, since the emission depth of the photoelectrons is
Fig. 7. Illustration of different measuring methods for the spin polarization of ferromagnets. (a) Spin-polarized photoemission from a ferromagnet upon irradiation by photons of energy ħω (b) spin-polarized tunneling in magnetic tunnel junctions, (c) spin-polarized tunneling in superconductor/insulator/ ferromagnet (S/I/FM) tunnel junctions, (d) spin-polarized transport in point contacts between two ferromagnets and (e) spin-polarized transport between a superconductor and a ferromagnet involving Andreev reflection. Hext denotes the applied magnetic field, M the magnetization in the ferromagnet, Itun the tunneling current and I the ballistic or diffusive transport current in the point contacts.
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Rudolf Gross
of the order of 10Ǻ. Therefore, the derived values of the spin polarization are susceptible to surface states, reconstructions, contamination etc. Second, the energy resolution is limited. Finally, in photoemission experiments the spin polarization is usually measured only in a few k-directions and does therefore not represent a Fermi surface average. Superconducting tunnel junctions
The measurement of the spin polarization using a tunnel junction consisting of a ferromagnetic and a superconducting electrode (see Fig. 7c) was pioneered by Meservey and Tedrow [40, 44]. The tunneling conductance of such junction is determined by the convolution of the density of states of the ferromagnet and the superconductor. As illustrated in Fig. 8, the DOS of the superconductor has a sharp peak at the gap energy ∆ and is symmetric for spin-up and spin-down electrons. Applying a magnetic field parallel to the junction electrodes this peak is spin-split by ±µBB. The tunneling conductance for the two spin direction is obtained by convoluting this split DOS by the function aK and (1-a)K, where a ≡ N ↑ /( N ↑ + N ↓ ) and
(1 − a) ≡ N ↓ /( N ↑ + N ↓ ) is the fraction of the spin-up and spin-down electrons at the Fermi level in the ferromagnet and K is the derivative of the Fermi function with respect to energy. Due to the finite spin polarization of the ferromagnet we have a ≠ (1-a) and the conductance versus voltage curves become asymmetric with four peaks at eV = ∆ ± µBB and eV = -∆ ± µBB as shown in Fig. 8. In a first approximation the spin polarization can be evaluated by measuring the conductance values σ1 to σ4 at these peak positions giving [44, 161]
P =
(σ 4 − σ 2 ) − (σ 1 − σ 3 ) = 2a − 1 , (σ 4 − σ 2 ) + (σ 1 − σ 3 )
(44)
where the second equals sign is only correct, if the tunneling probability is the same for the spin-up and spin-down electrons. In a more detailed evaluation the measured conductance spectra have to be fitted to the theoretical prediction using the spin polarization as fitting parameter and taking into account spin-orbit and spin-flip scattering [161]. Note that spinorbit scattering is smearing out the Zeeman-split density of states while magnetic impurities act as pair breakers and result in a reduction of the energy gap. In practice, usually Al is used for the superconducting material, since it has small spin-orbit scattering and its native oxide forms a very good tunneling barrier. The drawback of this method is that it requires very low temperatures (typically below 1 K) so that the spin-
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79
splitting 2µBB of the superconducting DOS peak is larger than kBT. Therefore, it cannot be used to measure the temperature dependence of the spin polarization. Recent experimental values are listed in Table 1.
Fig. 8. (a) Zeeman splitting of the DOS in a superconductor into spin-up (dashed) and spin-down part (dotted) due to an applied magnetic field. (b) Energy derivative K of the Fermi function for a specific voltage multiplied by the fraction a and (1-a) of the spin-up and spin-down electrons at the Fermi level. (c) Spin-up (dashed) and spin-down (dotted) conductivity vs voltage as well as the total conductivity (solid line).
Although early experimental results have been interpreted in terms of the DOS of the ferromagnetic electrodes, of course this method rather yields the tunneling spin polarization Ptun than PDOS. Indeed the experimental values listed in Table 1 show that the spin polarization
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determined according to eq. (44) is always positive for 3d transition metals and alloys in obvious contradiction to band structure calculations [162], which yield negative values for PDOS of Co and Ni. This contradiction was first addressed by Stearns [74], who pointed out that for 3d transition metals the tunneling current is dominated by the 4s electrons due to their small effective mass and in turn large tunneling probability, whereas the DOS is dominated by the 3d electrons. Taking this into account, a positive tunneling spin polarization is obtained for Co and Ni in agreement with the experiment. Table 1. Spin polarization obtained from spin dependent tunneling experiments with Al/AlOx/FM tunnel or Al/SrTiO3/FM (for FM = La2/3Sr1/3MnO3 (LSMO)) junctions using equation (44) (data taken from Refs. [21, 44, 160, 161]).
Point contacts
Ferromagnetic point contacts. In point contacts or nano-constrictions between two ferromagnets usually a contact area is formed that allows for ballistic transport. Therefore, with such configuration the quantity PNv can be measured over a wide temperature range [163]. The drawback of this technique is that in most cases there is a finite barrier strength Z or equivalently a transmission probability Ts < 1 of the contact. Furthermore, usually both Z and the relative orientation of the two ferromagnets are not precisely known. In most experiments this is circumvented by repeated measurements with many contacts to obtain average results. Point contact Andreev-reflection. If one part of the point contact is replaced by a superconductor, the spin polarization of the ferromagnet can be obtained by point contact Andreev spectroscopy [113, 114]. Andreev scattering at a normal metal to superconductor (N/S) interface is caused by the fact that electrons with energies smaller than the superconducting energy gap cannot enter the superconductor due to the lack of available states (see Fig. 9a). Therefore, an electron incident on the NS interface from the N side is retro-reflected as a hole, while at the same time a singlet Cooper pair formed by two electrons with opposite spin and wave vector is
Magnetic Tunnel Junctions Based on Half-Metallic Oxides
81
transferred to the superconductor [164]. That is, in total a charge of 2e is transferred across the NS interface. As a result, the conductance is twice to that of an N/N interface at bias within the gap. Andreev scattering is a phase coherent scattering process and is responsible for the proximity effect at N/S interfaces, where phase correlations are induced in the N material [165, 166]. The situation changes for a ferromagnetic metal with finite spin polarization. If P = 1, the Andreev reflection described above is completely blocked. If an electron with wave vector k and spin-up is incident on the FM/S interface, it requires an electron with wave vector -k and spin-down to form a singlet Cooper pair and to enter the superconductor (formally, this is equivalent to the retro-reflection of a spin-down hole, see Fig. 9b). However, since for P = 1 such electrons are not available, this process is blocked and the conductance is zero for bias within the gap [167]. In order to understand Andreev reflection at a FM/S interface we first consider a FM/N interface. For a clean interface (Z = 0) the conduction at zero bias, G(0), is given by
GFM/N
=
e2 ( N↑ + N↓ ) , h
(45)
that is, the conduction quantum times the number of contributing channels. For a FM/S interface all the (minority) N↓ but only the fraction (N↓/N↑)N↑ of the (majority) N↑ channels contribute to Andreev reflection and transfer the charge 2e across the interface [113, 114]. This results in the conduction
GFM/S
=
e2 h
2 N 2 N↓ + 2 ↓ N↑ = 4 e N↓ , h N↑
(46)
which would be zero for the situation shown in Fig. 9b. Normalizing to GFM/N, we see that the suppression of the conductance at V = 0 and Z = 0 scales linearly with the spin polarization P of the ferromagnet as [108, 113, 114]
GFM/S (0) GFM/N
= 2 (1 − P) ,
(47)
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Fig. 9. Schematic illustration of Andreev scattering at (a) a normal metal/superconductor (N/S) interfaces and (b) a ferromagnet/superconductor (FM/S) interface. For simplicity in (b) a half-metal is assumed. The reflection of a spin-down hole is forbidden in (b) due to lack of available states.
where GFM/N is the normal state conductance and the extracted spin polarization is P = PNv. For large barrier and/or diffusive transport the extracted spin polarization is P = PNv² in the first approximation. However, for this case in general a more detailed analysis is required for exact results [108]. Furthermore, for finite interface scattering (finite barrier strength Z) the analysis of the experimental data is not as simple and one has to determine the spin polarization by fitting the experimental conductance versus voltage curve to model predictions incorporating interface scattering [168]. Note that point contact Andreev reflection probes 〈 Nv z2 〉 averaged over the entire Fermi surface, whereas tunneling through a thick barrier is restricted to a narrow tunneling cone probing Nv z2 only at those selected points of the Fermi surface, where quasimomentum is perpendicular to the interface. However, the experimental data for the spin polarization obtained from both methods are usually very close. The reason for that is the use of polycrystalline materials in the tunneling experiments. Apparently, averaging over individual grains helps to bring the tunneling spinpolarization results close to the Fermi surface averaged point contact Andreev reflection results [169].
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The advantage of the Andreev reflection technique is the detection of spin polarization for a wider range of materials due to the simple sample preparation. A large number of experimental results has been reported [167176]. Certainly, a drawback of the Andreev scattering method is that it can be applied only at temperatures lower than the transition temperature of the superconductor and therefore does not allow to determine the temperature dependence of the spin polarization. Furthermore, care has to be taken in the quantitative analysis of experimental data, which should take into account factors such as an non-spherical Fermi surface [108] as well as the roughness and size of the point contact leading to a diffusive component of the transport [177-179]. Ferromagnetic tunnel junctions
For planar tunnel junctions formed by two ferromagnetic electrodes separated by a thin insulating barrier the spin polarization can be determined by measuring the resistance for parallel and anti-parallel magnetization direction according to eqs. (2) and (3). However, as discussed in the introduction the measured TMR values depend critically on the electronic states at the interface and the choice of oxide barrier (height, thickness, defect states etc.). The quantity measured in such experiment is usually the tunneling spin polarization Ptun given by eq. (41). Furthermore, one has to distinguish between junctions with well oriented electrode and barrier materials and those based on polycrystalline materials. As discussed in the section on the tunneling definition of the spin polarization, due to the dominating exponential factor in the tunneling probability for junctions with thick tunneling barrier only those electrons contribute significantly to the tunneling conductance, which have the smallest κ , i.e. those with k║ ≈ 0. Therefore, tunneling only probes certain crystallographic directions for oriented electrode material. A Fermi surface average is only obtained for polycrystalline electrodes due to the averaging over many grains with different crystallographic orientation. Furthermore, crystalline tunneling barriers can give rise to a much higher tunneling spin polarization due to a highly spin-dependent evanescent decay of the relevant wave functions [61, 62, 78]. That is, for single crystalline or well a oriented electrode material one has to take into account the specific crystallographic orientation of the electrode material with respect to the tunneling direction, whereas for polycrystalline electrodes one gets an average over all possible orientations. In general, using planar MTJs it is difficult to extract reliable information on the intrinsic DOS spin polarization. However, the tunneling spin polarization obtained in experiments based on MTJs is the quantity relevant for a large variety of
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applications and hence provides very valuable information from the practical point of view.
Half-Metallic Ferromagnetic Oxides Half-metallic ferromagnets are interesting for spintronics applications because they provide fully spin-polarized charge carriers at the Fermi level. Of course, from the application point of view not only their full spin polarization is relevant, but also their Curie temperature, which should be well above room temperature. There are several classes of potentially halfmetallic materials such as the doped manganites, the double perovskites, the Heusler compounds, magnetite, CrO2 or diluted magnetic semiconductors (for recent reviews see [97, 152]). Most of them are transition metal oxides, which we discuss here. In transition metal oxides there is a strong hybridization of the outer electron shell (4s/4p) of the transition metal ion with the oxygen 2p-states. This causes a wide s-p-gap, into which the narrow 3d-band tends to fall. A half-metallic oxide is then achieved, if only one of the spin polarized 3d-bands intersects the Fermi level, whereas there is a gap in the DOS for the other spin direction. A similar situation also applies for the Heusler alloys such as NiMnSb [154, 180], PtMnSb or Co2MnSi [181], which are half-metals with high Curie temperatures above about 600 K. Chromium dioxide CrO2 is a ferromagnetic metal with tetragonal rutile structure (space group P42/mmm) with Cr positions at (0,0,0); (½,½,½) and oxygen positions at (±x, ±x,0); (½ ± x, ½ m x, ½) with x = 0.302 (see Fig. 10a). Each oxygen has three chromium neighbors and each Cr4+ (3d²) ion is octahedrally coordinated by oxygen with two short apical bonds (0.189 nm) and four longer equatorial bonds (0.191 nm) [151, 182]. The octahedra are sharing a common edge and form ribbons along the c-axis. The lattice parameters are a = 0.4422 nm and c = 0.2917 nm. Due to the crystal field the five 3d-states are split (~2.5eV) into three t2g and two eg states with the t2g triplet further split into a nonbonding dxy orbital in the equatorial plane of the octahedron and two bonding dyz ± dzx orbitals with respect to the oxygen p-orbital directed perpendicular to the Cr30 triangles [183, 184]. According to Hund's rule the two electrons are occupying two of the t2g states with parallel spin. Ferromagnetism in CrO2 arises from the exchange interaction between the Cr4+ ions. It is mainly caused by a delocalization of one of the t2g spin-up electrons. These
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electrons are hopping from site to site, while they are ferromagnetically coupled to the localized S = 1/2 core spins by the on-site Hund's rule exchange. Band structure calculations [185, 186] have indicated that CrO2 is a type-IA half-metal (see Fig. 6a) with a saturation magnetization of 2µB per formula unit, a spin gap ∆↓ of about 1 eV and a spin-flip gap ∆sf of a few 100 meV (a more detailed analysis shows that there is a finite covalent mixing of the Cr t2g spin-up states and the O 2p states resulting actually in a moment of 2.1µB/f.u. [187, 188]). The Curie temperature of CrO2 is TC ≈ 390 K.
Fig. 10. (a) - (d) Crystal structure, the effect of the crystal field on the one-electron levels, and schematic density of states of some half-metallic oxide materials: (a) CrO2, (b) Sr2FeMoO6, (c) La2/3Sr1/3MnO3, (d) Fe3O4. In (e) and (f) the crystal structures of two ferromagnetic semiconductors are shown together with the schematic density of states: (e) (Ga,Mn)As and (e) (Zn,Mn)O.
The spin polarization of CrO2 has been measured using superconducting tunnel junctions [173, 189, 190] and point contact Andreev spectroscopy [167, 171]. For thin films values as high as 98% have been observed for ballistic contacts at 4 K. However, the spin polarization was found to
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decrease rapidly with increasing temperature and becomes vanishingly small at room temperature [151, 191, 192]. Doped manganites Another class of transition metal oxides with a very rich phase diagram and fascinating physics are the mixed valence manganites with composition La1-xAxMnO3 with A an alkaline earth [193, 194]. These oxides have perovskite type structure where the Mn ions are surrounded by an oxygen octahedra (see Fig. 10c). Due to the crystal field the five 3d-states are split into three t2g and two eg states. In the undoped case (x = 0)) only Mn3+ ions (3d4) are present. Due to the strong on-site Hund's rule exchange the four d electrons are filled in with parallel spin. The half-filled eg band is split by the Jahn-Teller effect making this compound an antiferromagnetic insulator. Doping with divalent alkaline earth ions introduces holes into the eg band. If the hole density is sufficiently high, the holes can delocalize. However, due to the strong on-site Hund's coupling this is only possible, if the itinerant holes are ferromagnetically aligned with the local S = 3/2 core spins of the t2g electrons. That is, hopping provides a ferromagnetic double exchange and hence a ferromagnetic ground state with a maximum Curie temperature between 280 and 380 K for x = 0.3 and A = Ca, Ba, Sr. For the compound La0.7Sr0.3MnO3 with the maximum TC the spin polarization has been discussed controversially, because the expected value sensitively depends on the details of the band structure. From the experimental point of view a high degree of spin polarization has been derived from magnetic tunnel junctions (up to 95% at 4 K [101, 195-197]), superconducting tunnel junctions (72%, [161]), and point contact Andreev spectroscopy (80%, [172, 198]). However, these values have been obtained interpreting experimental data on the basis of the oversimplifying Jullière formula. At present it is still not fully established whether this compound is a type-IA or type-IIIA half-metal. It may be that there are both itinerant eg spin-down states and localized t2g spin-up states at the Fermi level, so that La0.7Sr0.3MnO3 is a type- IIIA half-metal. However, it also may be that the t2g spin-up band is above the Fermi level, so that we have a type-IA halfmetal. For La0.7Ca0.3MnO3 a spin polarization of about 80% at 4 K has been derived from grain boundary tunnel junction experiments [100]. For all studied manganites, the spin polarization has been found to decrease rapidly with increasing temperature and to become vanishingly small at room temperature. For grain boundaries this has been attributed to a strong
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increase of inelastic tunneling processes with increasing temperature [85, 86]. For MTJs the detailed origin has not yet been clarified. Double perovskites Double perovskites of composition A2BB'O6 with A=Ca, Sr, and Ba and BB'= FeMo, CrW, FeRe, or CrRe recently hace attracted considerable attention, since band structure calculations [199, 200] have predicted halfmetallic behavior and furthermore Curie temperatures as high as 420 K for Sr2FeMoO6 [201], 460 K for Sr2CrWO6 [202, 203], and 630 K for Sr2CrReO6 [204, 205] have been observed. In the ideal case, the double perovskites have an ordered NaCl-type array of B and B' cations on octahedral sites (see Fig. 10b). However, in reality there is always a finite amount of antisite defects in the cation order. The magnetic exchange in double perovskites is explained within a generalized double exchange or kinetic energy driven model proposed by Sarma et al. [206] with subsequent extensions and generalizations by Kanamori and Fang [207, 208]. The model can be easily understood considering Sr2FeMoO6 as an example. For the magnetic B ion (Fe3+) Hund's splitting is much larger than the crystal field splitting, and the majority spin t2g↑ and eg↑ bands are filled with five electrons. In contrast, at the 'non-magnetic' B' site (Mo5+) the crystal field splitting between the t2g and eg band is large and Hund's splitting is negligibly small. The Mo 4d t2g band is filled with a single electron. As the majority spin bands at the magnetic site are occupied, kinetic energy gain can only be obtained by hybridization and the hopping of the minority spin electrons from the 'nonmagnetic' site into the empty minority t2g↓ spin band of the magnetic ion. This results in t2g↑ electrons of mixed 3d/4d character at the Fermi level. By shifting electrons from the majority spin band of the 'non-magnetic' site into the minority spin band, the system can gain energy because the spin-down electrons can delocalize. As a result, the charge carriers become strongly polarized, in the extreme case even half-metallic. According to the classification given above the double perovskites are type-I B half-metals. Furthermore, at the 'non-magnetic' site a finite negative spin magnetic moment is induced, which has been detected directly by XMCD measurements [209-211]. This induced spin moment is oriented anti-parallel to the localized moments at the magnetic B site. Therefore, the double perovskites can be viewed as ferrimagnets. Band structure calculation predict a spin gap ∆↑ of about 1 eV and a spin-flip gap ∆sf of several 100 meV. The expected saturation magnetization is given by the difference of the spin moment of the
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localized t2g↑ states at the magnetic site (5µB/f.u. for Fe3+ and 3µB/f.u. for Cr3+) and the induced moment at the non-magnetic site (about -0.3 µB/f.u. both for Mo5+ and W5+). The measured saturation magnetization is always considerably smaller due to antisite defects. Measurements of the magnetoresistance of granular samples and on tunnel junctions with Co counter electrodes indicate a spin polarization above 50% at 5 K, which is decreasing rapidly with increasing temperature [202, 203, 212]. Magnetite Magnetite, Fe3O4, is a ferrimagnet with a very high ordering temperature of 860 K. Magnetite has the so-called cubic inverse spinel structure in which the Fe cations occupy interstices of a face-centered-cubic (fcc) close packed frame of oxygen ions (see Fig. 10d). The moments of Fe3+ are localized at the eight tetrahedral (A) sites and are aligned anti-parallel to those of the Fe3+ and Fe2+ ions equally sharing the 16 octahedral (B) sites [213]. The expected saturation moment is 4µB/f.u. Rapid hopping of electrons between Fe2+ and Fe3+ ions in the B sites results in a room-temperature conductivity of about 200 Ω-1cm-1. Upon cooling, bulk magnetite undergoes a metal to insulator transition (Verway transition) at 120 K, where the electron hopping is frozen and the conductivity decreases by several orders of magnitude. This prevents the study of spin polarization by methods employing superconductors and therefore requiring low temperatures. Although it is generally accepted that the Verwey transition [214-218] is due to the ordering of the Fe2+ and Fe3+ ions, the mechanism governing the transport and magnetic properties of this material above and below the transition is still not well understood. Band structure calculations show that Fe3O4 contains a gap in the majority spin band at the Fermi level, but there is no gap in the minority spin band [219-223]. This suggests that magnetite is a half-metal with the Fermi level in the t2g↓ band of the octahedral iron. However, the small conductivity of magnetite indicates that the carriers are localized forming polarons [224]. That is, magnetite is probably best classified as a type-IIB half-metal. A large negative value of spin polarization of about up to -80% has been measured in spin-polarized photoemission experiments [105, 225228]. Detailed magnetotransport measurements on Fe3O4 have performed only recently [229-232]. For both single crystals [229] and epitaxial films [230] a large negative magnetoresistance of several 10% was found around and below the Verwey transition. Coey et al. [231] have studied the magnetoresistance of Fe3O4 in polycrystalline thin film, powder compact, and single-crystal form above the Verwey transition. A low-field negative
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MR effect has been observed in the polycrystalline thin film and powder compacts, but not in the single crystal. MR effects of (1-3%) have been measured, with the larger values appearing in the powder compacts. It has been suggested that the effect is associated with intergranular transport of spin-polarized electrons, with the MR arising from misaligned magnetization in adjacent ferromagnetic grains that are exchangedecoupled. In general, the MR values reported for polycrystalline samples and tunnel junctions [233-240] are small and show wide spread. At 4 K values of up to -25% have been reported by Hu et al. [233, 234], whereas a value of +43% has been reported by Seneor et al. [235]. However, these values drop to zero at room-temperature. Large MR has sometimes found also in point contacts, from which a spin polarization as large as -72% has been inferred [163]. The latter value agrees reasonably well with those found in spin-polarized photo emission experiments [105]. The fact that for magnetite based MTJs often low TMR values are obtained most likely is related to the problem that controlling the stoichiometry and defect structure at the interface between magnetite and the tunneling barrier is challenging. However, recently magnetite based MTJs with Al2O3 barriers and Ni or Co counter electrodes have been fabricated, which show a room temperature TMR up to more than 20%. Recently, it has been suggested that the large scatter and even differing sign of the TMR effect observed for magnetite is related to disorder at the electrode/barrier interface [241]. In a complex crystal structure such as magnetite the surface can have a number of nonidentical terminations and surface reconstructions with completely different properties regarding spin-dependent transport. For example, it has been shown that an oxygen layer deposited on a magnetic electrode could change the properties of a MTJ drastically including a sign reversal of the spin polarization [242, 243]. Ferromagnetic semiconductors Ferromagnetic semiconductors are interesting regarding the integration of spin electronics with conventional electronics. They should have fully spinpolarized charge carriers and ferromagnetic ordering temperature well above 300 K to allow for room temperature applications. Oxide magnetic semiconductors have been studied intensively in the 1960s and 1970s and came into the focus again recently with the growing interest in spin electronics. The textbook example certainly is EuO with TC = 69 K [37, 244]. In this material the Eu2+ 4f7 level falls into the gap between the oxygen 2p valence band and the Eu 5d/6s conduction band. The latter is occupied by n-type charge carrier due to oxygen deficiency or
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Ln3+ doping. The problem with EuO and related materials such as EuS or EuB6 is to achieve a high Curie temperature. The ferromagnetic cationcation superexchange can be traced back to the ferromagnetic s-f coupling, and the presence of s-f hybridization. However, due to the local nature of the 4f levels it is difficult to find a sufficiently strong exchange mechanism delivering room temperature ferromagnetism [245]. The discovery of ferromagnetism in zinc-blende III-V [246, 247] and IIIV [248, 249] Mn-based compounds strongly stimulated work exploring the interesting combination of quantum structures and magnetism in semiconductors. This was further stimulated by theoretical calculations predicting room-temperature ferromagnetism in transition metal doped IIIV and II-VI semiconductors [260-262]. Recently, ordering temperatures well above 100 K have been achieved with transition metal doped GaAs (see Fig. 10e). Optimizing the deposition conditions allowed to raise TC up to about 180 K for Ga0.95Mn0.05As [35, 247, 250-255]. The charge carriers in this material are p-type in a spin-split 4p (As) valence band. For this material a high spin polarization of 85% has been measured at low temperature recently using Andreev reflection spectroscopy from high transparency Ga0.95Mn0.05As/Ga point contacts [256]. A high spin polarization also has been inferred from the large tunneling magnetoresistance observed in MTJs from this material [257, 258]. This is consistent with band structure calculations that predicted P ≈ 100% [259]. Much excitement has been generated by the observation of Curie temperatures well above room temperature in various transparent oxides such as ZnO [263] (see Fig. 10f), SnO2 [264], TiO2 (anatase) [265], and HfO [266, 267] when doped with a few percent of transition metals such as Co or Mn. Although there is still a controversial debate on these materials, it is likely that the cobalt or manganese levels reside in the gap, with a spinsplit conduction band. They would then be n-type half-metallic semiconductors, like EuO. The magnetic exchange may also be provided by a spin-split impurity band [63, 157, 268]. Unfortunately, doubts persist as to whether these materials are truly uniform and inherently semiconducting. Very recently, several experiments suggested the existence of ferromagnetism in the La0.5Sr0.5TiO3 doped with Co [269, 270]. This has been further confirmed by tunneling magnetoresistance measurements showing a significant spin polarization in Co-doped (La,Sr)Ti1-xCoxO3 films at low temperature [271, 272]. We note that the host material La0.5Sr0.5TiO3 may not be considered as a semiconductor but rather as a strongly correlated metal [273].
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Status There is a large number of ferromagnetic materials that are predicted halfmetallic by band structure calculations. Most of these materials are ferromagnetic oxides. These materials are interesting for studying basic physics aspects of spin-dependent transport and single spin band effects. Furthermore, they provide the basis for applications requiring enhanced spin effects. Among the oxides, chromium dioxide is probably the best studied material. It shows full spin polarization at low temperatures. However, the ferromagnetic ordering temperature (387K) is too low for room-temperature applications. The same is true for the doped manganites with a maximum Curie temperature of about 380 K. In this respect magnetite (TC = 860K) and the double perovskites (TC ≤ 630K) are more promising. However, so far no reliable data exist regarding the spin polarization of the latter materials at room temperature. Moreover, it is still unclear whether or not a high degree of spin polarization can be achieved also at interfaces to insulating barriers as required for MTJs because these materials usually have complicated crystal structures. If these problems can be solved, the half-metallic oxides offer good prospects for applications such as MTJs. Of course, there is hope that new high Curie temperature half-metallic ferromagnets will be discovered. The new family of ferromagnetic oxide semiconductors is highly interesting with respect to the underlying physics. In particular, the clarification of the magnetic exchange in these materials is an important task of future research. However, from the application point of view the Curie temperatures in most cases still are too low and doubts remain whether or not these materials are homogeneous ferromagnets with spinpolarized charge carriers. Future work will show whether these materials can enter spin electronics applications. Finally, we would like to emphasize that ferromagnetic oxide materials are promising with respect to combining half-metals and multiferroics [274276] in epitaxial heterostructures. Such structures are interesting for spintronics, since they may allow to switch magnetization by electric fields or design novel multi-functional materials systems [277, 278]. However, at present this field is not yet well developed and first experiments mostly aim at materials issues such as the heteroepitaxial growth of the relevant materials [279-281].
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Magnetic Tunnel Junctions In this section we discuss the fabrication of MTJs based on half-metallic oxides. In particular we address questions regarding the degradation of the expected tunnel magnetoresistance due to imperfections of the tunneling barrier and the involved interfaces. In our discussion we will focus on MTJs based on the doped manganites and magnetite. Tunnel junctions based on chromium dioxide so far have reached only small TMR values of less than 10% even at at low temperature [190]. Experimental data on junctions based on the double perovskites are hardly available [282]. Fabrication process Magnetic tunnel junctions in most cases are fabricated by first depositing the complete multilayer thin film structure with base electrode (sometimes with additional buffer layer), tunneling barrier, top electrode, and wiring layer. Then the multilayer is patterned ex-situ, e.g. by etching a mesa structure into the multilayer film (see Fig. 11), to achieve the desired junction geometry. This process has the advantage that the multilayer structure can be fabricated in-situ avoiding contamination effects of the sensitive interfaces. For thin film deposition of the oxide heterostructures mostly pulsed laser deposition is used. Details about the laser deposition process including in-situ reflection high energy electron diffraction (RHEED) control of the growth process and laser substrate heating are described elsewhere [283-286]. A critical point for the fabrication of MTJs is the surface roughness of the base electrode. A rough surface usually results in an inhomogeneous thickness of the tunneling barrier and even pinholes.
Fig. 11. (a) Cross-sectional view of the multilayer structure used for planar MTJs. (b) Optical micrograph of a chip containing for test junctions.
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MTJs based on doped manganites Magnetic tunnel junctions based on the doped perovskite manganites have been successfully fabricated using both planar type tunnel junctions as shown in Fig. 11 [98-101, 287-294] as and artificial bicrystal grain boundary junctions fabricated on SrTiO3 bicrystal substrates [8587, 100, 295-299]. As shown by Fig. 12a, a large TMR of up to 300% at 4 K has been achieved for manganite bicrystal junctions with almost ideal switching behavior of the resistance versus applied magnetic field curves. According Jullière's formula, the measured TMR value corresponds to a spin polarization of about 60%. However, as shown in Fig. 12b, the large lowtemperature TMR effect was found to rapidly decrease with increasing T due to a strong increase of the inelastic tunneling current [85-87, 100], which is not spin-conserving. This was attributed to the large density of defect states in the grain boundary barrier, which has been confirmed by noise measurements [300].
Fig. 12. (a) Resistance vs. applied magnetic field curves for a La2/3Ca1/3MnO3 grain boundary junction. The direction of the field sweep is indicated by the arrows (according Ref. [100]). (b) TMR (symbols) plotted versus bias voltage and temperature. The lines show the expected decrease due to the measured increase of the inelastic tunneling current (according Ref. [86]). The inset shows the sample configuration.
Large TMR values up to more than 1800% corresponding to a spin polarization of more than 90% also have been measured for planar type junctions based on doped manganites [98-101, 195]. However, these large values only have been achieved at low temperatures and a rapid decrease
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was found with increasing T resulting in a vanishingly small TMR at roomtemperature. The origin of this decay (inelastic tunneling, magnon excitation, temperature variation of interface magnetization, etc.) has not yet been clarified in detail. Furthermore, a strong voltage dependence of the TMR was found, which in some cases could be attributed to the band structure of doped manganites [195]. Most experiments on manganite based MTJs showed that the TMR and, in turn, the derived value of the spin polarization strongly depend on the chemical composition, the microstructure and the strain state of interfaces and barriers. For La1-xSrxMnO3 (LSMO) based MTJs with SrTiO3 barrier it was found that there exists an instability towards the A-type (layered) antiferromagnetic phase causing a suppression of ferromagnetism by spin canting in the vicinity of the interface as x increases from 0.2 to 0.4. With the proper choice of x at the interface, "atomically regulated'' LSMO/STO/LSMO MTJs could be fabricated, which showed a weaker decrease of TMR with temperature and still a finite TMR value of about 1% at 300 K [301, 302]. For MTJs based on La2/3D1/3MnO3 / SrTiO3 / La2/3D1/3MnO3 (D = Ca or Ba) trilayers large differences in the measured TMR values have been observed, although the structural quality of the layer structure was almost the same for the Ca and Ba doped material (see TEM micrograph in Fig. 13a). Whereas TMR high values above 1000% have been measured for the La2/3Ba1/3MnO3 (LBMO) based MTJs, only small TMR values typically below 30% have been observed for the La2/3Ca1/3MnO3 (LCMO) based MTJs. This could be attributed to the different strain state of the manganite films. Whereas LBMO growth almost strain-free on (001) SrTiO3, LCMO is under tensile strain (-1.05%). During nucleation the LCMO film can accommodate for the epitaxial coherency strain by incorporating less Ca and/or Ca/La site swapping, thereby forming a Ca rich surface layer. During growth of the LCMO base electrode this Ca rich surface layer is swimming on top of the growing film and finally results in a Ca rich interface layer between the LCMO base electrode and the STO barrier. Correspondingly, a Ca deficient LCMO layer is expected at the interface between the STO barrier and the LCMO top electrode. These Ca rich and deficient interface layers have been indeed detected by an EELS analysis (see Fig. 13c) [303]. Both Ca rich and deficient layers are expected to have strongly reduced Curie temperatures (magnetically dead layers) or may even be antiferromagnetic. Note that the Ca enriched and deficient interface layers have negligible influence on the microstructure but are strongly affecting the magnetic properties of the interface. We further note that the tensile biaxial strain itself favors an A-type antiferromagnetic order in LCMO. Therefore, strongly reduced TMR values are expected for the LCMO/STO/LCMO MTJs grown on SrTiO3. In contrast, for MTJs
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based on almost strain free LCMO films grown on a lattice matched NdGaO3 substrate and barrier layer high TMR effects have been observed [99]. In the same way, for LCMO grain boundary junctions large TMR values have been measured as shown in Fig. 12 [100].
Fig. 13. (a) Cross-sectional TEM micrograph of a LBMO/STO/LBMO trilayer structure for MTJs. (b) Illustration of the Ca enriched and deficient layers for a LCMO/STO/LCMO heterostructure, which are absent for a LBMO/STO/LBMO heterostructure. (c) Chemical profile (Ti, La, Ca, and Mn) obtained by EELS along the spatial coordinate perpendicular to the layer structure of a LCMO/STO/LCMO trilayer. In contrast to Ti, La, and Mn the Ca profile is asymmetric with excess Ca at the top of the base electrode and Ca deficiency at the bottom of the counter electrode. The nominal barrier thickness is 3.1 nm (according to [303]).
The two examples discussed for manganite based MTJs clearly show that a controlled engineering of interfaces on a nanometer scale is required not only with respect the microstructure but also with respect to the chemical composition and strain state.
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MTJs based on magnetite Although large TMR values have been reported for junctions based on the doped manganites, these large values are obtained only at low temperature. They rapidly decrease with increasing temperature and become negligible at room temperature. In the same way, for chromium dioxide a large value of the spin polarization (up to 98%) has only been measured at low temperature. However, MTJs with large TMR have not yet be fabricated. Even at low temperature only small values below about 10% have been achieved [190]. Furthermore, both chromium dioxide and the doped manganites have Curie temperatures below 400 K what is insufficient for room temperature applications. This has stimulated work on magnetite, which has a very high Curie temperature of 860 K and is supposed to be a half-metal. There have been several attempts to fabricate magnetite based MTJs, however, in most cases with limited success. As already discussed above, the TMR values reported for various versions of magnetite based MTJs are small and show large spread (even opposite sign) [233240]. Maximum values reported in literature are -25% for Fe3O4/CoCr2O4/ La0.7Sr0.3MnO3 junctions at 4 K [233, 234] and +43% for Co/Al2O3/Fe3O4 junctions [235] also at 4 K. Tunneling magnetoresistance also has been studied for Fe3O4/MgO/Fe3O4 MTJs with maximum TMR values of less than 2%. The largest room-temperature TMR value of 10% has been reported by Matsuda et al. for Fe3O4/Al Ox/CoFe trilayer junctions [240]. The low values and large scatter of achieved TMR values of magnetite based MTJs have been attributed to disorder at the electrode/barrier interface and different surface terminations of the magnetite layer, which are difficult to control during fabrication. We recently have fabricated MTJs based on epitaxial magnetite thin films deposited on (001) MgO single crystal substrates. We used magnetite as the base electrode and Ni or Co for the counter electrode. For the tunneling barrier five materials namely MgO, SrTiO3, NdGaO3, SiO2, and Al2O3-x have been used to study the effect of different barrier materials on the magnetotransport properties of the MTJs. The whole thin film multilayer structure was grown in-situ using pulsed laser deposition (PLD) and electron beam evaporation in an ultra high vacuum system in order to avoid the degradation of interfaces due to ex-situ processing [304].
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Fig. 14. (a) Magnetization versus applied magnetic field measured at room temperature for multilayer films with two different barrier materials. Left panel: Fe3O4/SiO2/Ni. Right panel: TiN/Fe3O4/Al2O3-x/Ni. (b) Resistance versus applied magnetic field as well as measured TMR effect of a Fe3O4/Al2O3-x(2.5nm)/Ni MTJ at T = 320 K.
Best results have been obtained for MTJs with SiO2 and Al2O3-x barriers, which provide magnetically decoupled junction electrodes as necessary for sharp switching. This is shown in Fig. 14a, where we have plotted the magnetization curves of a Fe3O4 /SiO2 /Ni and a Fe3O4/Al2O3-x/Ni trilayers used for the fabrication of MTJs. These samples clearly show a fully decoupled switching of the magnetization of the junction electrodes at all temperatures. Fig. 14b shows the resistance versus applied magnetic field curve as well as the TMR-effect of a Fe3O4/Al2O3-x(2.5nm)/Ni MTJ at T = 320 K. The MTJs with Al2O3-x barrier reproducibly showed a clear positive TMR effect both for Ni and Co counter electrodes. Assuming a negative spin polarization of Co and Ni as predicted by band structure calculation, we can conclude that magnetite also has a negative spin polarization as expected from band structure calculations. Figure 14b shows that the measured R(H) curves have an almost ideal, symmetric switching behavior. Furthermore, a well defined tunneling magnetoresistance was observable in the whole measured temperature range from 150 K to 350 K. Evaluating the true TMR effect of these junction one has to take into account the large series resistance Rs of the magnetite electrodes due to the large resistivity of magnetite. The series resistance Rs is included in the measured resistance for both parallel and anti-parallel magnetization configuration and is entering the expression for the measured TMR as
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Fig. 15. True TMR effect and measured TMR effect plotted versus temperature for a Fe3O4/Al2O3-x(2.5nm)/Ni magnetic tunnel junction.
TMR meas =
( Rap + Rs ) − ( Rp + Rs ) Rp + Rs
=
∆R . Rp + Rs
(48)
In order to obtain the true tunneling magnetoresistance we have to correct for the series resistance Rs and obtain the corrected value as
TMR true =
∆R = TMR meas Rp
Rs 1 + . R p
(49)
Because the measured 4-point resistance in the parallel configuration is Rpmeas = Rp + Rs and hence Rs = Rpmeas − Rp , we can rewrite this equation and obtain
TMR true =
∆R = TMR meas Rp
Rs 1 + . m eas R − R p s
(50)
TMRtrue is plotted in Fig. 15 versus temperature. In contrast to TMRmeas shown in the inset, the corrected values TMRtrue show the expected monotonic decrease with increasing temperature. True TMR values of
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above 20% at room temperature and about 50% at 150 K have been obtained. Slightly larger values are obtained for MTJs with Co counter electrodes not discussed here [305]. Acknowledgements The author wish to thank Th. Brenninger, M. Bibes, A. Boger, J.M.D. Coey, J. Fontcuberta, S. Geprägs, S. Gönnenwein, A. Gupta, D. Kölle, A. Marx, P. Majewski, E. Menzel, K. Nielsen, M. Opel, J.B. Philipp, S. Qureshi, M.S. Ramachandra Rao, and D. Reisinger for valuable and stimulating discussions, disposal of experimental data and technical support. Financial support by the German Science Foundation and the Federal Ministry of Education and Research is gratefully acknowledged.
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254. Sawicki M, Wang KY, Edmonds KW, Campion RP, Staddon CR, Farley NRS, Foxon CT, Papis E, Kaminska E, Pitrowska A, Dietl T, Gallagher BL (2005). Phys Rev B 71:121302 255. Wang KY, K. Edmonds W, Campion RP, Zhao LX, Foxon CT, Gallagher BL (2005). Phys Rev B 72:085201 256. Braden JG, Parker JS, Xiong P, Chun SH, Samarth N (2003). Phys Rev Lett 91:056602 257. Tanaka M, Higo Y (2001). Phys Rev Lett 87:026602 258. Chun SH, Potashnik SJ, Ku KC, Schiffer P, Samarth N (2002). Phys Rev B 66:100408(R) 259. Ogawa T, Shirai M, Suzuki N, Kitagawa I (1999). J Magn Magn Mater 197:428 260. Dietl T, Ohno H, Matsukura F, Cibert J, Ferrand D (2000). Science 287:1019 261. Dietl T, Ohno H, Matsukura F (2001). Phys Rev B 63:195205 262. Dietl T (2002). Semicond Sci Technol 17:377-392 263. Ueda K, Tabata H, Kawai T (2001). Appl Phys Lett 79:988-990 264. Ogale SB, Choudhary RJ, Buban JP, Lofland SE, Shinde SR, Kale SN, Kulkarni VN, Higgins J, Lanci C, Simpson JR, Browning ND, Das Sarma S, Drew HD, Greene RL, Venkatesan T (2003). Phys Rev Lett 91:077205 265. Matsumoto Y, Murakami M, Shono T, Hasegawa T, Fukumura T, Kawasaki M, Ahmet P, Chikyow T, Koshihara S-Y, Koinuma H (2001). Science 291:854 266. Venkatesan M, Fitzgerald CB, Coey JMD (2004). Nature 430:630 267. Coey JMD, Venkatesan M, Stamenov P, C. Fitzgerald B, Dorneles LS (2005). Phys Rev B 72:024450 268. Coey JMD (2005). J Appl Phys 97:10D313 269. Zhao YG, Shinde SR, Ogale SB, Higgins J, Choudhary RJ, Kulkarni VN, Greene RL, Venkatesan T, Lofland SE, Lanci C, Buban JP, Browning ND, Das Sarma S, Millis AJ (2003). Appl Phys Lett 83:2199 270. Qiao PT, Zhao ZH, Zhao YG, Zhang XP, Zhang WY, Ogale SB, Shinde S. R, Venkatesan T, Lofland SE, Lanci C (2004). Thin Solid Films 468:8 271. Herranz G, Ranchal R, Bibes M, Jaffrès H, Jacque E, Maurice J-L, Bouzehouane K, Wyczisk F, Tafra E, Basletic M, Hamzic A, Colliex C, Contour J-P, Barthélémy A, Fert A (2006). Phys Rev Lett 96:027207 272. Herranz G, Basletic M, Bibes M, R Ranchal, Hamzic A, Tafra E, Bouzehouane K, Jacquet E, J. Contour P, Barthélémy A, Fert A (2006). Phys Rev B 73:064403 273. Tokura Y, Taguchi Y, Okada Y, Fujishima Y, Arima T, Kumagai K, Iye Y (1993). Phys Rev Lett 70:2126 274. Hill NA (2000). J Phys Chem B 104:6694-6709 275. Fiebig M (2005). J Phys D 38:R123-R152 276. Hill NA, Rabe KM (1999). Phys Rev B 59:8759 277. Binek C, Doudin B (2005). J Phys: Condens Matter 17:L39-L44 278. Binek C, Hochstrat A, Chen X, Borisov P, Kleemann W, Doudin B (2005). J Appl Phys 97:10C514
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279. Wang J, Neaton JB, Zheng H, Nagarajan V, Ogale SB, Liu B, Viehland D, Vaithyanathan V, Schlom D G, Waghmare UV, Spaldin NA, Rabe KM, Wuttig M, Ramesh R (2003). Science 299:1719 280. Eerenstein W, Morrison FD, Dho J, Blamire MG, Scott JF, Mathur ND (2005). Science 307:1203 281. Béa H, Bibes M, Sirena M, Herranz G, Bouzehouane K, Jacquet E, Fusil S, Paruch P, Dawber M, Contour J-P, Barthélémy A (2006). Appl Phys Lett 88:062502 282. Bibes M, Bouzehouane K, Barthélémy A, Besse M, Fusil S, Bowen M, Seneor P, Carrey J, Cros V, Vaurès A, Contour J-P, Fert A (2003). Appl Phys Lett 83:2629 283. Gross R, Klein J, Wiedenhorst B, Höfener C, Schoop U, Philipp JB, Schonecke M, Herbstritt F, Alff L, Lu Yafeng, Marx A, Schymon S, Thienhaus S, Mader W (2000). In: Pavuna D, Bosovic I (eds) Superconducting and Related Oxides: Physics and Nanoengineering IV. SPIE Conf Proc vol 4058, pp 278-294 284. Klein J, Höfener C, Alff L, Gross R (1999). Supercond Sci Technol 12:1023; see also (2000) J Magn Magn Mater 211:9 285. Reisinger D, Blass B, Klein J, Philipp JB, Schonecke M, Erb A, Alff L, Gross R (2003). Appl Phys A 77:619 286. Reisinger D, Schonecke M, Brenninger T, Opel M, Erb A, Alff L, and Gross R (2003). J Appl Phys 94:1857 287. Lu Yu, Li X W, Gong GQ, Xiao Gang, Gupta A, P Lecoeur, Sun JZ, Wang YY, Dravid VP (1996). Phys Rev B 54:R8357 288. Sun JZ, Gallagher WJ, Duncombe PR, Krusin-Elbaum L, Altman RA, Gupta A, Lu Yu, Gong GQ, Xiao Gang (1996). Appl Phys Lett 69:3266 289. Gupta A, Sun JZ (1999). J Magn Magn Mat 200:24 290. Sun JZ (2001). Physica C 350:215 291. Wong PK, Evetts JE, Blamire MG (2000). Phys Rev B 62:5821 292. Ishii Y, Yamada H, Sato H, Akoh H, Kawasaki M, Tokura Y (2005). Appl Phys Lett 87:022509 293. Jo Moon-Ho, Mathur ND, Evetts JE, Blamire MG (2000). Appl Phys Lett 77:3803 294. Ozkaya D, A. Petford-Long K, Jo Moon-Ho, Blamire MG (2001). J Appl Phys 89:6757 295. Mathur ND, Burnell G, Isaac SP, Jackson TJ, Teo B-S, MacManus-Driscoll J L, Cohen LF, Evetts JE, Blamire MG (1997). Nature 387:266 296. Steinbeck K, Eick T, Kirsch K, O'Donnell K, Steinbeiß E (1997). Appl Phys Lett 71:968 297. Blamire MG, Schneider CW, Hammerl G, Mannhart J (2003). Appl Phys Lett 82:2670 298. Wagenknecht M, Eitel H, Nachtrab T, Philipp JB, Gross R, Kleiner R, Koelle D (2006). Phys Rev Lett 96:047203 299. Gunnarsson R, Hanson M (2006). Phys Rev B 73:014435 300. Philipp JB, Alff L, Marx A, Gross R (2002). Phys Rev B 66:224417 301. Izumi M, Ogimoto Y, Manako T, Kawasaki M, Tokura Y (2002). J Phys Soc Jpn 71:2621
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302. Ogimoto Y, Izumi M, Sawa A, Manako T, Sato H, Akoh H, Kawasaki M, Tokura Y (2003). Jpn J Appl Phys 42:L369 303. Simon J, Walther T, Mader W, Klein J, Reisinger D, Alff L, Gross R (2004). Appl Phys Lett 84:3882 304. Reisinger D, Menzel E, Qureshi S, Boger A, Majewski P, Opel M, Gross R (2006). Unpublished 305. Menzel E, Qureshi S, Boger A, Majewski P, Opel M, Gross R (2006). Unpublished
MEMS Tunable Dielectric Resonator
G. Panaitov1,2 , R. Ott1, N. Klein1 1
ISG-2, Research Centre Jülich GmbH, Germany
2
Institute of Electronic Engineering and Industrial Technologies ASM, MD-2028 Kishinev, Moldova
Abstract:
We present a novel approach for tuning of dielectric resonators by microelectromechanical switches (MEMS). The concept is based on intermodal coupling between the TE01δ mode of a cylindrical dielectric resonator and a planar slotline resonator. The resonance frequency of the resonator can be changed by MEMS switch leading to variation of the coupling between the planar resonator mode and the TE01δ mode. As a consequence, the resonance frequency of the TE01δ mode changes.
Keywords: MEMS, microwave resonator, slotline
Introduction The rapidly growing market of microwave communication systems requires new approaches which may allow to more effectively utilize the available frequency bandwidth. Increasing of cannel capacity can be provided by improvement of the quality of frequency selective elements, like filters and resonators. Moreover, a confident operation of mobile systems under strong interferences requires fast switchable filters and resonators. Due to higher data rate in new 3-generation UMTS (universal mobile telecommunication system) the requirements to the tuning time even faster than for conventional cellular mobile systems. MEMS (microelectromechanical systems) technology has proved to be an appropriate method to attain switching and tuning capability with losses being superior to any other techniques such as varactors or field effect transistors [1]. However, due to the small dimensions of MEMS of the 111 R. Gross et al. (eds.), Nanoscale Devices - Fundamentals and Applications, 111–121. © 2006 Springer. Printed in the Netherlands.
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order of several hundred micrometers, their use is restricted to planar microwave circuits. In order to achieve tuning of a bulky dielectric resonator, coupled modes of planar and dielectric resonator will be investigated here.
Tuning Concept We propose a tuning mechanism, which is based on three main components: a basic dielectric resonator, which determines the main frequency of the resonator; a slotline, as an additional planar microwave resonator; and a MEMS switch, which is used to control the intermodal coupling between dielectric and planar resonator modes. Dielectric resonator We apply dielectric resonators (DRs), which conventionally play an important role for microwave communication systems. Due to the high quality factors, high mechanical stability, and small temperature coefficient of permittivity of advanced microwave ceramics, dielectric resonators are of common use in high-Q filters for base stations and satellite communication, as local oscillators with low phase noise, and integrated satellite receivers [2]. We use a common cylindrical dielectric resonator which operates at TE01δ mode. At high permittivity constant of the dielectric material (ε~30) the 95% of the stored electric energy of the TE01δ mode, as well as a great part of the stored magnetic energy (~ 60%) are located within the cylinder [3]. The resonance frequency of the resonator is strongly dependent on the distribution of the rf electromagnetic fields in the resonator. Any variation of the fields in the resonator volume ∆V leads to change of the resonance frequency ω, according to perturbation principle [3]:
ω −ω 0 ω0
≅
∫∫∫ ∫∫∫
2
∆V
2
V
2
( µ H 0 − ε E0 ) dV 2
( µ H 0 + ε E0 ) dV
,
(1)
where ω0, E0, H0 - frequency and field parameters of non-disturbed resonator. In our experiments we change the frequency of DR resonator by coupling the magnetic field of TE01δ mode to the field of planar slotline resonator.
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Slotline A slotline is a planar transmission line structure, which can be fabricated by etching of a narrow slot in the metallization on one side of the dielectric substrate. If short circuited at one end, it may be considered as a resonator, which frequency depends on the length and the load at the open end of the slotline.
Fig. 1. Slotline as the transmission line circuit loaded at the ends.
Figure 1 presents a slotline as a transmission line loaded at two ends with impedances Z1 and Z2. Resonance parameters of the slotline can be calculated using a transmission line formulation. The impute impedance for the load Z1 at the length l is given by Z in ( l ) = Z 0
Z1 + jZ 0 tan β l , Z 0 + jZ1 tan β l
(2)
where Z0 is the characteristic impedance; β = 2π λ is the wave propagation constant. At the left end of the slotline Zin (- l ) = Z2. If this end is short-circuited, i.e. Z2 = 0, then the equation (2) gives: tan β l = j
Z1
Z0
.
(3)
Next, we consider two cases, when the right end of the slotline is open and closed. For the open end case, i.e. when Z1 = ∞, the ratio between the slotline length and wavelength can be found as following: tan β l = ∞ ⇒ β l = n π
2
⇒ l=λ , 4
(4)
for short slotlines, l < λ. As the length of open end slotline is a quarter of the wavelength this slotline resonator can be considered as a quarter-wave resonator. Similarly, one can show that in the case of shorted end (Z1 =0) holds:
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tan β l = 0 ⇒ β l = nπ
⇒ l=λ . 2
(5)
In this case, the closed end slotline operates as a half-wave resonator. Using a MEMS switch at the right end of the slotline one can control the load of the slotline and, as the consequence, to change the resonant frequency of the slotline. We show below that this behavior can be used for development of a frequency tuning mechanism for the dielectric resonator. MEMS Microelectromechanical switches are known to demonstrate very useful performance at microwave frequencies [1]. Advantages of the MEMS over the conventional semiconductor switches are low losses, low consumption and a higher “off/on” resistance ratio. Figure 2 shows an example of MEMS switches, which we develop for application in the tuneable resonator. A sacrificial layer technique used to fabricate a free-standing Au microbridge structure. The two microstrip lines underneath supposed to be contacted by bending of the microbridge structure due to actuation of the MEMS. A Nb layer on the top of the Au microbridge illustrates a bimetal thermo-expansion actuation mechanism. Nb Au
Si
Fig. 2. Microelectromechanical switch design.
Tuning approach Our concept of a MEMS tunable dielectric resonator is illustrated in Fig. 3. A substrate with four radial slotlines (white strips, Fig. 3) is symmetrically arranged above the cylindrical dielectric resonator of TE01δ mode. The inner end of each slotline is a short, while outer end is loaded by a MEMS switch (black strips). For the case of a MEMS off state, each of 4 slots represent a quarter wave slotline resonator, which possess a strong magnetic field component in radial direction providing strong coupling to
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the resonant fields of the TE01δ mode. Upon closing a single MEMS, the corresponding quarter wave resonator changes into a half-wave resonator. Due to the sign change of magnetic field in the middle of the slotline, the coupling between slotline and TE01δ mode is much weaker. The variation of the coupling between two resonance modes leads to change of the resonance frequency of the assembly.
h
Fig. 3. Schematic of the MEMS tunable resonator. The substrate with 4 slotlines (white strips) is symmetrically arranged above the cylindrical dielectric resonator at distance h. Black strips indicate MEMS position at the end of slotlines.
Due to resonant interaction between two resonance modes the stored electromagnetic energy in the MEMS open state is distributed between the DR and slotlines. This leads to an increase of the effective size of the resonator and, as a consequence, to decrease of resonant frequency of the TE01δ mode. In the MEMS closed state the intermodal coupling is much weaker, the electromagnetic energy is concentrated within a DR and resonance frequency is about of non-disturbed DR. The tuning range of the resonator is determined by the coupling between the TE01δ mode and the quarter wave slotline mode, which can be controlled by a variation of the distance h (Fig. 3). The major advantage of the technique is that in the MEMS closed mode there is almost no coupling between the dielectric resonator and the slotline, i.e. one can expect only small additional losses due to the MEMS structure. In particular, the contribution of contact resistance of the closed MEMS should be rather small.
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Numerical Field Simulations In order to evaluate the expected tuning range and to choose an optimum slotline configuration, numerical field simulations have been performed employing the commercial code “CST Microwave Studio”. For the resonator design shown in Fig. 3, the spatial distribution of the resonant fields was calculated for the capacitive (open) and resistive (closed) state of the MEMS. Fig. 4 illustrates the contours of rf magnetic field in the meridian plane for both two states. The black contours correspond to the higher field amplitude. As expected, the field distribution differs substantially. There is an obvious excitation of the slotlines with open end (Fig. 4a). For the capacitive state the resonance frequency was found to be at f = 1.878 GHz, the quality factor is limited to about of Q = 6.700 due to losses in the magnetic field sections of the slotlines.
a)
z
z
b)
Fig. 4. Magnetic field contours in the meridian plane at: a) quarter-wave mode, MEMS in the capacitive state; b) half-wave mode, MEMS in the resistive state.
For the resistive state, the field is localized primarily inside the dielectric resonator (Fig. 4b), therefore the effects of slotlines and MEMS are not significant. Nevertheless, the quality factor of Q = 21.500 is still slightly lower than that of the non-perturbed dielectric resonator. This is due to a weak excitation of the half-wave slotline mode and corresponding losses in the resistive state. The excitation of the half-wave mode is obviously seen in the Fig. 5b. Here is the distribution of the electric field in the meridian plane of the resonator at half-wave mode. The two maximums within the DR contour indicate that the electric field has an azimuthal symmetry. One notices that the amplitude of the quarterwave field (Fig. 5a) in the slotline area is about one order of magnitude larger than that of the half-wave mode. Figures 4-5 show that both magnetic and
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electric fields are substantially different in two resonator modes. Therefore, the variation of the resonance frequency by switching of MEMS is a result of a complex variation of both fields, according to (1). For the parameters taken in the calculation the tuning range is about 61 MHz, i.e. 3.2%.
a)
b)
Fig. 5. Electric field contours in the meridian plane at: a) quarter-wave mode; b) at half-wave mode.
Experimental Results The principle of tuning by MEMS has been proven in an experimental setup, which is based on a commercial piezoelectric bimorph actuator. The details of the experimental design are shown in Fig. 6. A cylindrical dielectric resonator (εr = 28, BZT ceramic, by TransTech) with TE01δ mode at about f = 1.86 GHz is arranged in a cooper cavity. A substrate with four radial slots was assembled coaxially above the dielectric resonator. As MEMS equivalents, four metallized bimorph piezoelectric actuators control the slotlines states and consequently, the coupling between the DR
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µ
Substrate with slotlines
Actuation Supply
DR
HF Coupling
Fig. 6. Design of the tunable resonator with piezo actuators as the MEMS equivalents.
and the slotlines. The maximum deflection of actuators is about ±330 µm with a quite high blocking force of 2.5 N at its tip. The latter is very important in order to achieve a high contact resistance for the slotline closed state. A compact bipolar ±100 V power supply has been developed in order to actuate the bimorph. In addition, the bimorphs respond quite fast (~1.5 ms switching time) and they are very stable with specified cycling up to 109. The properties of single slotlines have been tested separately in a rectangular cooper cavity. The resonant frequency of the slotlines was found to be correlated with the slots length according to (4). The quality factor of the slotline, depending on the width of the slots, was evaluated to be in the range of 200 ÷ 400 at f ~ 2 GHz. Four slotline disks based on metal thin films deposited on sapphire wafers and bulk copper disks with different slot size have been tested in resonator design (Fig. 4) to achieve an optimum tuning. Table 1 shows the frequency and quality factor (Q) of the resonator depending on the slotline state for four bulk cooper slotlines at two distances above the DR.
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Table 1. Parameters of the resonator with slotlines of 19×0.4×0.5 mm3 size at two distances, h, between slots and dielectric resonator.
Slots state
h = 2.5mm
h = 12mm
4 closed 3 closed
f, MHz 1917.2 1913.3
Q 11500 8500
f, MHz 1856.26 1856.00
Q 25800 25000
2 closed
1909.1
8000
1855.72
25000
1 closed 0 closed
1906.3 1902.9
7000 6000
1855.45 1855.20
26800 26500
For slotlines of 19×0.4×0.5 mm3 size the highest Q was achieved in the case of four slotlines being closed. Upon increasing the number of open slotlines, Q decreases gradually down to a minimum for all slotlines being in the open state. The most interesting result is that the resonant frequency changes step-kind by a discrete almost constant value. The mean frequency step is about 3.5 MHz and 0.25 MHz per actuator switch for two distances. The discrete variation of the frequency by switching of the slotline state means that there is no substantial intermodal interaction between slotlines. This behavior can be used for a digital frequency tuning. Any required tuning range δFn = nδf can be realized by actuation of an appropriate number, n, of tuning units (slots) with a frequency step δf. In order to minimize the number of tuning elements while keeping the required tuning range one can even take slotlines with two different frequency steps δf1 and δf2. For example, in order to realize a tuning within a 5 MHz range with the frequency step δf = 0.2 MHz one needs just 8 slotlines with frequency steps δf1 = 0.2 MHz and δf2 = 1 MHz, instead of 25 slotlines in case of a single frequency step of δf = 0.2 MHz. In order to further increase the Q,-factor of the resonator we have tested broader slotlines (w = 1 mm), which are known to have a higher Q [4]. However, the Q-values for the closed wide slots have been not as high as expected. One possible reason for this might be the additional dielectric losses in the actuator material. These losses may be reduced by a metallization of the entire surface of the actuator. As the main difference, the achieved frequency step per slotline is 7 MHz, which is twice as high as for the 0.4 mm slotlines. Another possibility to change the tuning range is variation of the distance between the DR and slotlines.
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Frequency/switch, MHz
6 5
f, 19x1x0.5mm f, 19x0.4x0.5mm f, 12x1x0.5mm
4 3 2 1 0 2
4
6
8
10
12
h-distance, mm
Fig. 7. Frequency step, δf, as the function of the h-distance between DR and slots of three different sizes: 19×1×0.5 mm3, 19×0.4×0.5 mm3 and 12×1×0.5 mm3.
Figure 7 shows a series of experimental dependences of the frequency as the function of separation between the slotline disk and the dielectric resonator, h, for different slotline size. Thus, depending on the application one can choose an appropriate slot size to achieve a specified frequency tuning step. It is important to note that the figure-of-merit for tuning, κ = Q ⋅ δ f f , is not constant for the different slotline size, and therefore may further be optimized. An example of measurements of resonance parameters as the function of the slotline length is shown in Fig. 8. The measured data illustrate that there is a strong compromise between tuning range and quality factor. The quality factor drops down with the length of the slotline due to additional losses in the slots. Further investigations are directed towards replacing the slotline resonators by other MEMS-tunable planar resonator with higher Q. In any case, a switching of the resonance frequency by about 40 MHz is observed with quality factors being about 7000. This result corresponds to the figure-of-merit values of about 160, which is very challenging for tunable resonators.
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8 12000
11000
f, MHz
Q
4
10000
Quality Factor
Frequency/switch, MHz
6
9000 2
8000 0 10
12
14
16
18
20
Slotline length, mm
Fig. 8. Frequency step and quality factor as the function of slotline length.
Conclusion We proposed a concept of fast tuning of a bulky dielectric resonator coupled to planar slotline resonator. It was shown that MEMS tuning of dielectric resonators is quite challenging and has a great potential to beat the performance of semiconductor based tuning techniques. The discrete variation of the frequency by switching of the slotline state can be used for a digital frequency tuning. Optimization of the planar resonator structures, in order to reach the larger tunability while keeping the highest quality factor, is a subject of our future activity.
References 1. Rebeiz GM, Muldavin JB (2001). Revue HF (Belgium) 2:39-52 2. Klein N (2003) “Microwave Communication Systems: Novel Approaches for Passive Devices”. In: Waser R (ed) Nanoelectronics and Information Technology, Wiley, New York 3. Kajfez D (1986). In: Kajfez D, Guillon P (eds) Dielectric resonators. Artech House, Dedham, MA 4. Rozzi M et al. (1990). IEEE Trans MTT-38:1069-1078
Ultra-Thin Spin-Valve Structures Grown on the Surface-Reconstructed GaAs Substrate
B. Aktaş1, F. Yıldız1, O. Yalçın1, A. Zerentürk1, M. Özdemir1, L.R. Tagirov2, B. Heinrich3, G. Woltersdorf3, R. Urban3 1
Gebze Institute of Technology, 41400 Çayırova-Gebze, Turkey
2
Kazan State University, Kazan 420008, Russia
3
Simon Fraser University, Burnaby, BC, V5A 1S6, Canada
Abstract:
We studied magnetic anisotropies of epitaxial, ultra-thin iron spinvalve structures grown on surface reconstructed GaAs substrates. The ferromagnetic resonance (FMR) technique has been exploited to determine magnetic parameters of the ferromagnetic films in the temperature range 4-300 K. The unusual and unexpected angular dependence of FMR spectra allowed us to build and verify a model describing the magnetic anisotropy of the studied systems. Switching of the principal anisotropy axes of the second layer has been observed. It was attributed to a drastic relaxation of the uniaxial component of anisotropy induced by the surface reconstruction of the substrate. The linear variation of magnetic anisotropy parameters with temperature has been observed. The results on temperature dependence are discussed in terms of thermal expansion induced magneto-elastic anisotropies.
Keywords: ferromagnetic resonance, magnetic anisotropy, ultra-thin film spinvalve
Introduction The magnetic anisotropy of thin films is of crucial importance in spintronic applications. It is well known that ferromagnetic resonance (FMR) is the most sensitive and accurate technique to determine magnetic anisotropy fields of very thin magnetic films [1, 2]. In this paper we study the 123 R. Gross et al. (eds.), Nanoscale Devices - Fundamentals and Applications, 123–134. © 2006 Springer. Printed in the Netherlands.
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magnetic anisotropies in iron bi-layer structures grown on the surfacereconstructed GaAs single-crystalline substrates and demonstrate how surface-induced anisotropy can be used to tailor overall magnetic properties of the studied spin-valve structure. In our experiments we observed unconventional and unexpected FMR spectra. They are interpreted and explained based on the model proposed in this study. Switching of the principal anisotropy axes of the second layer with respect to the first layer has been observed. Consistent fitting of angular and frequency dependencies of the FMR spectra in the temperature range 4-300 K allowed us to determine accurately the cubic, uniaxial and perpendicular components of the magnetic anisotropy, as well as establish directions of easy and hard axes for the magnetization in the layer(s). The origin and temperature dependence of the magnetic anisotropy fields are extensively discussed in terms of the surface-induced anisotropy and lattice mismatch between materials subject to stress induced by differences in thermal expansion coefficients of materials in a contact.
Experimental Result
Samples and general FMR measurement procedure Single and double layers of ultra-thin iron films were prepared by Molecular Beam Epitaxy (MBE) on (4×6) reconstructed GaAs(001) substrates. A brief description of the sample preparation procedure is as follows. The GaAs(001) single-crystalline wafers were sputtered under grazing incidence using 600 eV argon-ion gun to remove native oxides and carbon contaminations. Substrates were rotated around their normal during sputtering. After sputtering the GaAs substrates were annealed at approximately 500°C and monitored by means of reflection high energy electron diffraction (RHEED) until a well-ordered (4×6) reconstruction appeared [3]. The (4×6) reconstruction consists of (1×6) and (4×2) domains with the (1×6) domain being As-rich and the (4×2) domain Ga-rich. The Fe films were further deposited directly on the GaAs(001) substrates at room-temperature from a resistively heated piece of Fe at the base pressure of 1×10-10 Torr. The film thickness was monitored by a quartz crystal microbalance and by means of RHEED intensity oscillations. The deposition rate was adjusted at about one mono-layer
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(ML) per minute. The gold layer was evaporated at room temperature at the deposition rate of about one monolayer per minute. RHEED oscillations were visible for up to 30 atomic layers. Noble metals are known to have long spin-diffusion length that makes them suitable as a spacer-layer in the spin-valve magnetic field sensor applications. Films under study were covered by a 20 ML thick Au(001) layer for protection in ambient conditions. More details of the sample preparation are given in Ref. [3]. Main FMR measurements have been carried out using the commercial Bruker EMX X-band ESR spectrometer equipped by an electromagnet which provides a DC magnetic field up to 16 kG in the horizontal plane. The small amplitude modulation of the field is employed to record fieldderivative absorption signal at the temperature range 4-300 K. An Oxford Instruments continuous helium-gas flow cryostat was used to cool the sample down to the measurement temperatures, and the temperature was controlled by the commercial LakeShore 340 temperature-control system. A goniometer was used to rotate the sample around the rod-like sample holder in the cryostat tube. The sample-holder was always perpendicular to the DC magnetic field and parallel to the microwave magnetic field.
Fig. 1. The sketch of the samples studied in the paper.
The samples were placed on the sample-holder in two different geometries. For the in-plane angular studies the films were attached horizontally on the bottom edge of the sample holder. During rotation the sample normal was kept parallel to the microwave field and the direction of the external DC magnetic field was varied in the sample plane. A sketch of the layer structure of the sample, the coordinate system, as well as the principal vector directions and their angles with respect to the coordinate system are
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shown in Fig. 1. The described geometry is not conventional and gives quite asymmetric absorption curves still at the same resonance field as in the conventional geometry for the in-plane measurements (both DC and microwave magnetic fields always lie in the film plane). Thus, we have been able to study at least the angular dependence of the resonance field and magnetic anisotropies using the FMR data taken from the unconventional in-plane geometry. We have also recorded some FMR data in the conventional, in-plane geometry for some specific crystallographic direction to check the validity of the data obtained in the unconventional geometry. For the out-of-plane measurements the samples were attached to a flat platform precisionally cut at a cheek of the sample holder. Upon rotation of the sample the microwave component of the field remained always in the sample plane, whereas the DC field was rotated from the sample plane towards the film normal to get additional data for accurate determination of the anisotropy fields. In-plane FMR measurements The in-plane FMR measurements have been carried out in the temperature range 4-300 K. Figure 2 shows temperature evolution of the in-plane FMR spectra for the 20Au/40Fe/40Au/15Fe/GaAs(001) sample taken at two angles. As a reference, we used data of our previous measurements on the single-layer sample, 20Au/15Fe/GaAs(001), in which we have established easy and hard directions for the 15ML thick iron layer [4]. The spectra at Figs. 2a and 2b have been recorded at the easy and hard axes for the first, 15ML thick iron layer. Three FMR absorption peaks are clearly visible in the spectra recorded at the angle labelled “easy”. As we do not expect any marked exchange or magnetostatic interaction through the 40ML-thick gold layer, the contribution of the first, 15 ML-thick, iron layer to the multi-component FMR signal can be easily attributed (see labelling in the Figure 2a) by comparison with the measurements on the single-layer sample [4]. The double-peak signal from the second, 40ML thick, iron layer gives a hint that easy and hard axes of the second iron layer are tilted with respect to the principal anisotropy axes of the first layer.
Ultra-Thin Spin-Valve Structures 40Fe
24K 60K 76K 100K 126K 160K 197K 244K 260K 287K
500
138K 180K 213K 244K 15Fe
291K
a) easy
15Fe
1000
1500
Magnetic field (G)
5K 16K 59K 82K 105K
40Fe
5K
FMR signal amplitude (a.u.)
15Fe
40Fe
2000
127
500
b) hard
1000 1500 2000 2500
Magnetic field (G)
Fig. 2. The in-plane FMR spectra of the double-layer sample for two orientations of magnetic field with respect to the crystallographic axes: a) DC field is parallel to the easy axis of the first, 15LM-thick iron layer; b) DC field is parallel to the hard axis of the first layer.
1600
Resonance field (G)
1400
40ML - expt 40ML - theor
15ML - expt 15ML - theor
1200 1000
Fe15 H1 = - 280 G Hu = 560 G ω/γ = 3200 G Meff = 1250 G
800 600
Fe40 H1 = - 460 G Hu = - 150 G ω/γ = 3200 G Meff = 1550 G
400 200 0 0
30
60
90 120 θ (degree)
150
180
Fig. 3. The angular dependence of the in-plane resonance field for the double-layer sample at room temperature: open symbols - experimental data, solid symbols results of the fitting.
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The FMR spectra recorded in the “hard” direction (Fig. 2b) show a fourpeak structure within a certain range of temperatures. Three peaks could be attributed to the hard-axis of the first, 15ML thick, iron layer [4]. Then, the remaining peak in the FMR spectrum comes from the 40ML-thick layer and clearly indicates the easy direction for this layer. Full angular dependence of the FMR spectra at room temperature is displayed in Fig. 3 (experimental points are given by empty symbols). From the figure it is evident that the hard axis (angle for maximal resonance field) of the second, 40ML thick, iron layer is switched on the 90 degrees apart of the hard axis of the first, 15ML thick, iron layer. Let us emphasize here the importance of the FMR measurements at different temperatures (see Fig. 2) for the identification of the angular dependence of the resonance fields in Fig. 3. As far as we know the observation of four FMR signals for double-layer, homogeneous ferromagnetic film structure is unique. Higher-order spinwave modes are not expected because of their shift to negative fields due to very high excitation energies of the short-wave-length standing spin-waves across the thickness of ultrathin films.
The Model and Computer Simulations of the FMR Spectra
Formulation of the model The FMR data are analyzed using the model free energy expansion given as
(
)
(
)
ET = −M ⋅ H + 2π M 02 − K p α 32 + K1 α12α 22 + α 22α 32 + α12α 32 + K uα 32 .
(1)
Here, the first term is the Zeeman energy in the external DC magnetic field, the second term is the demagnetization energy term including the effective perpendicular anisotropy as well, the third term is the cubic anisotropy energy characterized by the parameter K1, and the last term is the uniaxial anisotropy energy. In this equation αi represent directional cosines [5] of the magnetization vector M with respect to the reference axes (see Fig. 1), M0 is the saturation magnetization at the measurement temperature. It should be remembered here that one of the crystallographic axes is always perpendicular to the sample plane, and the remaining two lie in the sample plane. That is why we could combine demagnetization and perpendicular anisotropy terms in a single term (second one) using only the α3 directional cosine. The relative orientation of the reference
Ultra-Thin Spin-Valve Structures
129
axes, a sketch of the sample configuration, and the various vectors relevant in the problem are shown in Fig. 1. The fields for resonance are obtained using the well known equation [6]: 2
1 ∂ 2 ET ω0 ∂ 2 ET ∂ 2 ET 1 1 = − 2 2 2 γ M 0 ∂ θ M 0 sin θ ∂ ϕ M 0 sin θ ∂ θ ∂ ϕ
2
(2)
Here ω0 =2πν is the angular frequency of the ESR spectrometer, γ is the gyro-magnetic ratio for the material of the magnetic film, θ and ϕ are the usual polar and azimuthal angles of the magnetization vector M with respect to the reference system. We do not consider standing spin-wave excitations in the film because the film thickness is too small (20-60 Å). In this condition we expect that the spin waves to be visible far beyond our DC field range from zero to 16 kG. The imaginary component of the dynamic magnetic susceptibility corresponding to the microwave energy absorbed by the sample is given by [1, 7] ∂ 2 ET 2ω = 4π M 0 χ 2 = 4π 2 2 hφ ∂θ γ T2 mφ
2 2 2 4ω 2 ω0 ω − + 4 2 γ γ γ T2
−1
(3)
Here ω =γH is the Larmour frequency of the magnetization in the external DC magnetic field, T2 represents the effective homogeneous relaxation time of the magnetization that contributes to the line width of the FMR signal. We deduce the model parameters as a result of the fitting of the experimental data using the above two equations for computer simulations. Fitting of the FMR data To analyze the data for the in-plane geometry of our measurements, in Eqs. (2) and (3) both polar angles, θ for the magnetization and θH for the external DC magnetic field, are fixed at θ, θH =π/2. The azimuth angle ϕ of the magnetization direction is obtained from the static equilibrium condition for the directions ϕH of the external field varied in the range from zero to π. Then, the set of equations for the in-plane geometry reads:
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H sin(ϕ − ϕ H ) +
1 H1 sin 4ϕ − Hu sin 2ϕ = 0, 2
2
ω0 1 2 = H cos(ϕ − ϕ H ) + 4π M eff + H1 (3 + cos 4ϕ ) − 2 Hu cos ϕ 2 γ × { H cos(ϕ − ϕ H ) + 2 H1 cos 4ϕ − 2 Hu cos 2ϕ}.
(4)
The effective magnetization Meff includes the contribution 2πMeff=2πM0Kp/M0 from the perpendicular anisotropy. Hi = 2K/M0 . Fitting of the full angular dependence for the double-layer sample FMR spectra using the equation set (4) for the signal from each layer (see Fig. 2) is displayed in Fig. 3 by solid symbols. The best-fit parameters are also given in the Figure. The fitting
20Au/40Fe/40Au/15Fe/GaAs(001)
FMR signal amplitude (a.u.)
H||b
253K
expt theor
203K 155K
H||a 100K
65K 5K
0.0
0.5 1.0 1.5 2.0 Magnetic field (kG)
2.5
Fig. 4. Simulated FMR spectra taken as some selected temperatures.
revealed that the hard axis of the second, 40ML-thick iron layer, is switched by 90 degrees with respect to the hard axis of the first, 15MLthick iron layer. The fitting parameters allow us to conclude that origin of
Ultra-Thin Spin-Valve Structures
131
the observed switching is mainly a drastic relaxation and change of sign of the uniaxial component of the magnetic anisotropy in the second layer (~ 150 G as compared with ~ 560 G for the first, 15ML-thick layer). Comparison of the model calculations with the temperature evolution of the experimental spectra for both orientations is given in Fig. 4 by the dash lines. Except for a reversed phase for certain components of the FMR spectrum the agreement between the calculated and the experimental spectra is fairly good. This confirms consistency of the proposed model of anisotropy Eq. (1).
Temperature dependence of the magnetic parameters The temperature dependence of the magnetic parameters is given in Fig. 5. All the parameters are given in magnetic induction units (Gauss). It should 1800 M eff (40ML)
1600
Magnetic parameters (G)
1400
M eff (15ML)
1200 1000 800 600
2K u/M 0 (15ML)
400 20Au/40Fe/40Au/15Fe/GaAs(001)
200 0
2K u/M 0 (40ML)
-200
-400 2K1/M 0 (15ML) -600
2K1/M 0 (40ML)
0
50
100 150 200 250 300
Temperature (K)
Fig. 5. Temperature dependence of the magnetic parameters.
be recalled that the effective magnetization, Meff, includes perpendicular anisotropy (see the paragraph just after Eq. (4)). That is why the values of
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Meff are essentially reduced compared with the bulk magnetization [8]. This means that a very strong, perpendicular to the film plane, surface anisotropy field (about 5 kG) is induced in the first ultra-thin film at room temperature. It is also a general feature that both the uniaxial (along the b axis of the GaAs substrate) and the cubic anisotropy fields are significantly large. That is why the anisotropy energy dominates the Zeeman energy and causes such unusual and surprising double- and triple-line FMR spectra in a single ferromagnetic film. It can be seen from Fig. 5 that all magnetic anisotropy parameters strongly depend on temperature. As the temperature decreases the effective magnetization increases. This is partly due to increase of the saturation magnetization of iron according to Bloch’s law, and partly due to decrease of the easy-axis perpendicular anisotropy. The ferromagnetic transition temperature of bulk iron is about 980 K. This is the reason why even at room temperature the magnetic moment is almost fully saturated. Using the literature data on the temperature dependence of the magnetic moment of iron in the ferromagnetic phase [8] we estimate an increase of the iron magnetization in the range from 300 K to 5 K as about 64 G. The temperature variation is expected ∝ -T3/2 according to the Bloch’s law. Obviously, the observed magnitude (~ 300 G) and almost linear temperature dependence of the effective magnetization do not follow the conventional temperature dependence of saturation magnetization described above. That is why we conclude that the main contribution to the temperature variation of the effective magnetization comes from the temperature dependence of the perpendicular anisotropy. As temperature decreases the perpendicular anisotropy relaxes. The absolute values of both the in-plane uniaxial and the cubic anisotropies increase with decreasing the temperature. The sign of cubic anisotropy parameter is negative, making all of the three principal crystalline axes easy for magnetization. However, the sign of more strong uniaxial anisotropy along the b axis, induced by the surface reconstruction of the substrate, is positive making the b axis a hard direction for magnetization. As can be noticed from Fig. 5 all magnetic anisotropy parameters depend almost linearly on temperature. This implies that there is a common physical reason behind this unified behavior. Taking into account the large differences between the lattice constants of bulk Fe and epitaxial Fe films on the GaAs substrate (see Table) one may attribute the temperature dependence of the anisotropy parameters to the linear magneto-elastic effect.
Ultra-Thin Spin-Valve Structures
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Table. Parameters of the materials. material
thermal expansion (×10-6K-1)
lattice parameters
298 K
523 K
1273 K
a=b=c (Å)
Fe
11.8
15
24
2.867
GaAs
5.73
Au
14.2
14.6
16.7
effect on the iron layer
5.654/2 =2.827
compressive strain
4.078/√2=2.892
tensile strain
Actually, there is about -1.5% misfit between the lattice parameters of Fe, Au and the GaAs substrate. As a result, the Fe film is under a compressive strain. When temperature is lowered down to 4K from room temperature, this stress decreases by about 40%, as can be calculated by using the thermal expansion coefficient of Fe, Au and GaAs crystals, given in the Table. The lattice parameters vary linearly with the temperature, and, in the first approximation, we expect the anisotropy parameters to vary linearly with temperature as a result of different thermal expansion coefficients of materials in the contact and the magneto-elastic coupling.
Conclusion We studied the magnetic anisotropies of epitaxial, ultra-thin iron spin-valve structure grown on the surface reconstructed GaAs substrate. The ferromagnetic resonance (FMR) technique has been explored to determine magnetic parameters of the films in the temperature range 4-300 K. The presence of strong uniaxial anisotropy induced by the surface reconstruction of the substrate has been revealed. The magnitudes and temperature dependencies of the uniaxial, cubic and perpendicular anisotropy fields have been deduced by fitting the experimental data to model predictions. Switching of the principal anisotropy axes of the second iron layer with respect to the first one has been observed. It was attributed to drastic relaxation of the uniaxial component of anisotropy induced by the surface reconstruction of the substrate. The linear variation of magnetic anisotropy parameters with temperature has been observed. The observed temperature dependence is discussed in terms of thermal-expansion induced magneto-elastic contribution to the magnetic anisotropies. The work was supported by Gebze Institute of Technology grant No 03A12-1 and CKP-KSU grant of Russian Ministry of Education and Science.
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References 1. Heinrich B, Cochran JF (1993). Adv Phys 42:523 2. Farle M (1998). Rep Prog Phys 61:755 3. Monchesky TL, Heinrich B, Urban R, Myrtle K, Klaua M, Kirshner J (1999). Phys Rev B 60:10242 4. Aktaş B, Heinrich B, Woltersdorf G, Urban R, Tagirov LR, Yıldız F, Özdoğan K, Özdemir M, Yalçın O (2004). Magnetic anisotropies in the ultra-thin iron films grown on the GaAs substrate. In: Aktaş B, Tagirov LR, Mikailov F (eds). NATO Science Series II: Mathematics, Physics and Chemistry, vol 143. Kluwer Academic Publishers, pp 213-227 5. Gurevich AG, Melkov GA (1996). Magnetic Oscillations and Waves. CRC Press, New York 6. Suhl H (1955). Phys Rev 97:555 7. Aktaş B, Özdemir M (1994). Physica B 193:125 8. Landolt-Börnstein New Series (1986). Numerical Data and Functional Relationships in Science and Technology, vol. III/19A. Springer, Heidelberg
NOVEL IDEAS AND PRINCIPLES OF DEVICES
Negative U Molecular Quantum Dot
A. S. Alexandrov Department of Physics, Loughborough University, Loughborough, United Kingdom
Abstract: While the correlated transport in mesoscopic systems with repulsive electron-electron correlations received considerable attention in the past, and continues to be the focus of intense investigations, much less has been known about a role of attractive correlations in molecular nanowires and quantum dots. Here a negative −U Hubbard model of a d -fold degenerate quantum dot is reviewed. The attractive electron correlations caused by a strong electron-vibron interaction and/or by valence fluctuations in molecules provide a molecular switching effect, when the current-voltage (I-V) characteristics show two branches with high and low current for the same voltage, if the degeneracy of the dot is larger than two. Keywords: attractive correlations, molecular junctions, switching, nanowires, quantum dots
Introduction Molecular electronic junctions are structures in which single molecules or small groups of molecules conduct electrical current between two electrodes [1]. Combine unique I-V characteristics of molecules with the endless possibilities of synthesis offered by chemistry, molecular devices may revolutionize the future of the electronic industry. There is a real progress on that front. A number of laboratories are now testing new types
137 R. Gross et al. (eds.), Nanoscale Devices - Fundamentals and Applications, 137–151. © 2006 Springer. Printed in the Netherlands.
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of reversible switches, as well as fabricating nanowires needed to connect the circuit elements together. Once one has the appropriate switches and wires, the next significant step will be integrating them together into more complex structures that perform useful functions. Hewlett-Packard Research Labs have already constructed a 64-bit molecular memory chip that fits inside a square micron. More recently they fabricated in collaboration with UCLA one kilobit cross-bar molecular memory circuits by nano-imprint lithography [2]. Consequently the actual mechanisms of molecular switching and transport in molecular nanowires are of the highest current experimental and theoretical value [3, 4]. A few experimental studies [5, 6] provide evidence for various molecular switching effects, where the current-voltage (I-V) characteristics show two branches with high and low current for the same voltage below the conventional threshold. This phenomenon can result from a conformational transformation of certain molecules containing a "moving part" like a bypyridinium ring, which changes its position if the voltage is sufficiently high, or from the interaction of metallic leads with the molecules. Such transformations necessarily involve a large displacement of many atoms so this "ionic" switching is rather slow, perhaps operating on a millisecond scale. Remarkably, reversible switching was also observed in simple molecules (organic molecular films [6]). Molecular devices that exhibit intrinsic switching could be the basis of future active elements of molecular electronics. Thus further progress will depend upon finding reversible molecules and understanding intrinsic mechanisms of their switching from low to high current state. Solid-state mesoscopic systems with repulsive electron correlations have received considerable interest in the past and continue to be the focus of intense experimental and theoretical investigations [4]. The Coulomb repulsion suppresses tunneling for certain range of applied voltages, leading to what is commonly known as the Coulomb blockade. Our [7, 8] and other [9] recent studies led to a new insight into a possible bistable current state of molecular circuits due to attractive electron-electron correlations. This mechanism of current-controlled electronic molecular switching requires many-particle attractive correlations, which could appear as a result of strong electron-vibron interactions and/or valence fluctuations in molecular quantum dots (MQDs). Here I describe the switching behaviour of MQD caused by attractive correlations in the framework of a simple negative − U Hubbard model [7].
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Attractive Correlations One mechanism that can produce a negative attractive potential in molecular systems is a strong electron-vibron interaction. It is well known that the lattice deformation strongly affect the interaction between electrons in solids. At large distances electrons repel each other in ionic crystals, but their Coulomb repulsion is substantially reduced due to the ion polarization. However, when a short-range deformation potential and molecular-type (e.g. Jahn-Teller) vibrations are taken into account together with the longrange Fröhlich electron-phonon interaction, they overcome the Coulomb repulsion in complex lattice structures [10]. The resulting interaction becomes attractive at a short distance of about the lattice constant. Strong electron-vibron interactions and attractive correlations are quite feasible also in molecular nanowires and quantum dots (MQD) used as the “transmission lines” and active elements in molecular-scale electronics. There is a wide range of bulk molecular conductors with polaronic carriers formed by the strong electron-phonon interaction. Since the formation of polarons in polyacetylene (PA) was theoretically discussed [11], they were detected optically in PA [12], in conjugated polymers such as polyphenylene, polypyrrole, polythiophene, polyphenylene sulfide [13], Csdoped biphenyl [14], n-doped bithiophene [15], organic-based light emitting diodes [16], and other molecular systems. More recently it has been experimentally demonstrated that the low-bias conductance of molecules is dominated by resonant tunneling through coupled electronic and vibronic levels [17]. Conductance peaks due to electron-vibron interactions has been seen in C 60 [18].
Fig. 1. Localized electrons shift the equilibrium position of the ion ( 3 ). As a result two electrons on neighboring sites 1 and 2 can attract each other.
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Let us elaborate the physical origin of attractive correlations in solids and molecules caused by lattice deformations. We consider a toy model of two electrons localized on the neighboring sites 1,2 and interacting with a single ion vibrating near other site 3 as a three-dimensional harmonic oscillator, Fig. 1. The vibration part of the Hamiltonian in our the model is 2
H ph
1 ∂ ku 2 =− + , 2 M ∂u 2
(1)
where M is the ion mass, k is the spring constant, and u is the ion displacement (here and further we use h = 1 ). Electron potential energies due to the Coulomb interaction with the ion are approximately V1,2 = V0 (1 − u ⋅ e1, 2 /a ),
(2)
where V0 = − Ze 2 /a is the Coulomb energy in a rigid lattice (an analog of the crystal field potential), a is the average distance between electron sites and the ion, and e1,2 are units vectors connecting sites 1, 2 and site 3 , respectively. Hence the Hamiltonian of the model is given by 2
H = Ea (nˆ 1 + nˆ 2) + u ⋅ (f1 nˆ1 + f 2 nˆ 2) −
1 ∂ ku 2 + , 2 M ∂u 2
(3)
where Ea is the atomic level at site 1 and 2 in the rigid lattice, which includes the crystal field, f1, 2 = Ze 2e1, 2 /a 2 is the Coulomb force, and † nˆ 1,2 = c1, 2 c1, 2 are occupation number operators at every site expressed in
terms of the electron annihilation c1, 2 and creation c1†, 2 operators. This Hamiltonian is readily diagonalized by a displacement transformation of the vibronic coordinate u as u = v − ( f1 nˆ1 + f 2 nˆ 2 ) /k .
(4)
The transformed Hamiltonian has no electron-phonon coupling, 2
1 ∂ kv 2 H% = ( Ea − E p )(nˆ 1 + nˆ 2) + V ph nˆ 1nˆ 2 − + , 2 M ∂v 2
(5)
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141
where we used nˆ 12,2 = nˆ 1, 2 because of the Fermi statistics. It describes two small polarons at the atomic level shifted by the polaron level shift E p = f1,22 / 2k , which are entirely decoupled from ion vibrations. The ion vibrates near a new (shifted) equilibrium with the "old" frequency ω = k /M . As a result of the local ion deformation, the total energy of each polaron decreases by E p since a decrease of the electron energy by −2 E p overruns an increase of the deformation energy by E p . Another
important consequence of the lattice deformation caused by two electrons is the appearance of an effective interaction between carriers, V ph , which should be added to their Coulomb repulsion, Vc , V ph = −f1 ⋅ f 2 /k .
(6)
When V ph is negative and large compared with positive Vc the full interaction becomes negative. Our toy model allows us to elaborate its physical origin. If a carrier (electron or hole) acts on an ion with a force f , it displaces the ion by some vector x = f/k . The total energy of the carrierion pair is −f 2 / (2k ) . In case of two carriers interacting with the same ion (see Fig.1) the ion displacement is x = (f1 + f 2 ) /k and the energy is −f12 / (2k ) − f 22 / (2k ) − (f1 ⋅ f 2 ) /k . The last term depends on the scalar product of f1 and f 2 and consequently on the relative positions of the carriers with respect to the ion. If the ion is an isotropic harmonic oscillator, as we assume here, then the following simple rule applies. If the angle φ between f1 and f 2 is less than π/ 2 the polaron-polaron interaction will be attractive, if otherwise it will be repulsive. In general, some ions will provide attraction, and some repulsion between polarons. Applying the displacement transformation, Eq. (4), to a generic “Fröhlich-Coulomb” Hamiltonian, allows us explicitly calculate the effective interaction of small polarons in complex lattices and molecules [19]. An estimate of the short-range attraction (called here the negative Hubbard- U ) yields its value c.a. 0.1 eV or larger in ionic solids. We expect the negative U of the same order of magnitude in carbon-based molecules.
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Negative Hubbard- U Model of MQD Repulsive electron correlations cause the “Coulomb blockade” in the I-V characteristics of quantum dots [20]. However, they cannot cause a switching. We show here that a negative Hubbard U of any origin provides an intrinsic non-retarded current switching of MQD. One mechanism that can provide the negative U in molecular systems is the strong electronvibron interaction as discussed in the previous section. Even if bipolarons are not formed in molecular quantum dots because of a short life-time of carriers on the molecule, the attractive correlations between them still remain. Moreover, attractive short-range correlations are feasible even without electron-phonon interactions. They might be of a pure “chemical” origin, as in the mixed valence compounds [21]. The exact solution of MQD with strong electron-vibron interactions involves not only attractive correlations but also the vibron side-bands [8]. Since main physical conclusions with respect to the switching are qualitatively similar to the results of a much simpler negative Hubbard- U model [7] we limit our consideration to this model. Steady current We employ the Landauer-type expression for the steady current through a region of interacting electrons, derived by Meir and Wingreen [22] as I (V ) = −
e
π
∫
∞
−∞
) d ω [ f1 (ω ) − f 2 (ω )] ImTr Γˆ (ω )G R (ω ) ,
(7)
where f1(2) (ω ) = {exp[(ω + ∆ m eV / 2) /T ] + 1} , T is the temperature, ∆ is −1
the position of the lowest unoccupied molecular orbital (LUMO) with respect to the chemical potential. Γˆ (ω ) depends on the density of states (DOS) in the leads and on the hopping integrals connecting one-particle states in the left (1) and the right (2) leads with the states in MQD, Fig. 2.
Negative U Molecular Quantum Dot
143
Fig. 2. Schematic of energy levels of the molecular quantum dot under bias voltage V . MQD is assumed to be (quasi)4-fold degenerate (d = 4) . Switching occurs in the voltage range V1 < V < V2 ( eV1 = 2(∆− | U |) and eV2 = 2∆ ) due to a lowering of the HOMO-LUMO gap by the attractive electron-electron potential U in the current state.
Formula (7) includes, by means of the Fourier transform of the the molecular retarded Green’s function (GF), Gˆ R (ω ) , the electron-vibron and Coulomb interactions inside MQD and coupling to the leads. Since the leads are metallic, electron-electron and electron-phonon interactions in the leads, and interactions of electrons in the leads with electrons and vibrons in MQD can be neglected. We are interested in the tunneling near the conventional threshold, eV= 2∆ , Fig. 2, within a voltage range about an effective attractive potential | U | . The attractive energy is the difference of two large interactions, the Coulomb repulsion and the phonon mediated attraction, of the order of 1 eV each. Hence, | U | is of the order of a few tens of one eV. We neglect the energy dependence of Γˆ (ω ) ≈ Γ on this scale, and assume that the coupling to the leads is weak, Γ<< | U | . In this case Gˆ R (ω ) does not depend on the leads. Moreover we assume that there is a complete set of one-particle molecular states µ , where Gˆ R (ω ) is diagonal. With these assumptions we can reduce Eq. (7) to
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A. S. Alexandrov ∞
I (V ) = I 0 ∫ d ω [ f1 (ω ) − f 2 (ω ) ] ρ (ω ), −∞
(8)
allowing for a transparent analysis of essential physics of the switching phenomenon. Here I 0 = eΓ and the molecular DOS, ρ (ω ), is given by
ρ (ω ) =−
1
π
∑µ Im Gˆ µ (ω ), R
(9)
R R where Gˆ µ (ω ) is the Fourier transform of Gˆ µ (t ) = −iθ (t ) cµ (t ), cµ† ,
{L,L}
is the anticommutator, cµ (t ) = eiHt cµ e − iHt , θ (t ) = 1 for t > 0 and
zero otherwise. MQD Green’s functions The negative Hubbard- U Hamiltonian includes the attractive electronelectron interaction U in the molecular eigenstate ε µ coupled with the left and right leads by the hopping integrals tα k µ 1 H = ∑ ε µ nˆ µ + U ∑ nˆ µ nˆ µ ′ + ∑ ξα k aα† k aα k + 2 µ ≠ µ′ k ,α µ
∑ (tα µ aα cµ + H .c.). †
k , µ ,α
k
k
(10)
Here aα k and cµ are the annihilation operators in the left ( α = 1) and right ( α = 2) leads, and in the molecule, respectively, nˆ µ = cµ† cµ , ξα k is the energy dispersion in the leads, and U < 0. Similar Hamiltonian was applied to glassy semiconductors [23, 24], high Tc superconductors [10, 25] including doped fullerenes [26], and mixed valence compounds [21]. We use the perturbation theory with respect to the hopping integrals, neglecting any contribution to the current other than tα2k µ , but keeping all orders of the negative Hubbard U . Terms of higher orders in tα k µ cannot change the gross I-V features for any voltage except the narrow transition regime from one branch to another. Applying the equations of motion for
Negative U Molecular Quantum Dot
145
the Heisenberg operators cµ (t ), nˆ µ (t ) and aα k (t ) we obtain a set of coupled equations for the molecular GFs as i
dGµ( N ) (t ) dt
N −1
= δ (t )
∑ ∏ nµ (0)
µ1 ≠ µ2 ≠... µ i =1
i
+ [ε µ + ( N − 1)U ]Gµ( N ) (t ) + UGµ( N +1) (t ),
(11)
where nµ (t ) = cµ† (t )cµ (t ) is the expectation number of electrons on the molecular level, and the N -particle retarded Green’s function is defined as Gµ( N ) (t ) = −iΘ(t )
∑ µ
µ1 ≠
2 ≠... µ
µ
N −1
i =1
c (t )∏ nˆ µi (t ), cµ†
(12)
R for 2N < ∞ and Gµ(1) (t ) ≡ Gˆ µ (t ).
For the sake of analytical transparency we solve this system for a molecule having one d -fold degenerate energy level with ε µ = 0. In this case the set is finite, and it is solved using the Fourier transform. The oneparticle GF is found as d −1
Z r ( n) , r = 0 ω − rU + iδ
Gµ(1) (ω ) = ∑
(13)
where δ = +0 , n = nµ (0), and Z r ( n) =
( d − 1)! n r (1 − n) d −1− r . r!(d − 1 − r )!
This is an exact solution with respect to correlations which satisfies all sum rules. The electron density nµ (t ) obeys the rate equation, which is obtained by using the equations of motion as
dnµ (t ) dt
where
= 2∑ tα k µ ImAα(1)k µ (t ), α ,k
(14)
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A. S. Alexandrov
Aα( Nkµ) (t ) =
∑ µ
µ1 ≠
2 ≠... µ
N −1
cµ† (t )∏ nˆ µi (t )aα k (t ) .
(15)
i =1
These correlation functions should be calculated up to the first order with respect to the hopping integrals tα k µ . In this order, they satisfy the set of coupled equations as i
dAα( Nk µ) (t ) dt
N −1
∑µ ∏ nµ (t ) µ
= tα k µ [nµ (t ) − f (ξα k )]
2 ≠...
i =1
i
+UAα( Nk µ+1) (t ) + [ξα k − ( N − 1)U ] Aα( Nk µ) (t ).
(16)
When we have a finite number of molecular states, the set is finite. One readily solves the set in the stationary case, when nµi (t ) and Aα( Nkµ) (t ) do not depend on time. For the d -fold degenerate energy level, the one-particle correlation function is found as d −1
Aα(1)k µ = [ n − f (ξα k )]tα k µ ∑ r =0
ξα k
Z r ( n) . − rU + iδ
(17)
Substituting it into Eq. (14), we obtain the stationary rate equation for the electron density on the molecule as d −1
∑ ∑ [n − fα (rU )]Z (n) = 0. α r =0
r
(18)
Switching effect Now using Eqs. (13, 8, 9) the current is found as d −1
j = ∑ [ f1 (rU ) − f 2 (rU )]Z r (n),
(19)
r =0
where j = I / (dI 0 ) . Let us consider two-fold, four-fold, and six-fold degenerate molecular level. For d = 2 the kinetic equation is linear in n , and there is only one solution,
Negative U Molecular Quantum Dot
n=
∑α
fα (0)
2 + ∑ α [ fα (0) − fα (U )]
.
147
(20)
The current through a two-fold degenerate molecular dot is found as j=2
f1 (0)[1 − f 2 (U )] − f 2 (0)[1 − f1 (U )] . 2 + ∑ α [ fα (0) − fα (U )]
(21)
There is no current bistability in this case. Moreover, if the temperature is low ( T ∆,| U | ) there is practically no effect of correlations on the current, j ≈ V Θ(e | V | −2∆ ) / | V | . Remarkably, four-fold or higherdegenerate negative U dots reveal a switching effect. In this case, the kinetic equation is nonlinear, allowing for a few solutions. If eV < 2(∆− | U |), the only physically allowed solution of Eq. (18) for d = 4 and d = 6 at zero temperature is n = 0. If 2(∆− | U |) < eV < 2∆ , and T = 0 , the rate equation is reduced to 2n = 1 − (1 − n) d −1 .
(22)
For d = 4 it has two physical roots, n = 0 and n = (3 − 51/ 2 ) / 2 ≈ 0.38 . In this voltage range f1 (0) = f 2 (rU ) = 0, but f1 (U ) = f1 (2U ) = f1 (3U ) = 1 at T = 0. Using the sum rule
∑
d −1 r =0
Z r (n) = 1 and the rate equation (18), the
current is simplified in this voltage range as j = 2n. Hence we obtain two stationary states of the molecule with low (zero at T = 0 ) and high current, I ≈ 0.76 I 0 for the same voltage in the range 2(∆− | U |) < eV < 2∆ , Fig. 3. For d = 6 , the kinetic equation has two physical roots in this voltage range, n = 0 and n ≈ 0.48, which corresponds to I = 0 and I ≈ 0.96 I 0 , respectively. Above the standard threshold, eV > 2∆, where f1 ( rU ) = 1 and f 2 (rU ) = 0 there is only one solution, n = 0.5 with the current I = I 0 . Hence, different from the non-degenerate or double-degenerate MQD, the rate equation (18) for d > 2 has two physical roots in a certain voltage range and the current-voltage characteristics show a hysteretic behavior, Fig. 3. Ermakov [9] also calculated the I-V curves of the 4-fold degenerate dot with the Coulomb and electron-phonon interactions, and found a switching effect in the numerical I-V curves. However, Ref. [9] obtained an unphysical population of each molecular state, n = 1 . Moreover the exact theory [8] showed that the averaging procedure over phonons in the
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A. S. Alexandrov
transformed Hamiltonian used in Ref. [9] cannot be applied in MQD since it misses the vibron-side bands.
Fig. 3. The current-voltage hysteresis loop in the degenerate negative- U model of a molecular quantum dot for two temperatures.
Mean-field approximation and switching We can better understand the origin of the switching phenomenon by taking the limit d >> 1. The physical roots to Eq. (22) are n = 0 and n = 0.5 in this limit with the current I = 0 and I = I 0 , respectively. This is precisely the solution of the problem in the mean-field approximation (MFA), which is a reasonable approximation for d 1. Indeed, using MFA one replaces the exact two-body interaction in the Hamiltonian for a mean-field potential as 1 2
U∑
µ
≠ µ′
nˆ µ nˆ µ ′ ≈ U ∑ µ ≠ µ ′ nˆ µ nµ ′ − 12 U ∑ µ ≠ µ ′ n µ nµ ′
Then the MFA DOS is given by ρ µ (ω ) = δ [ω − U (d − 1)n]. Using the Fermi-Dirac Golden rule the rate equation for n becomes
Negative U Molecular Quantum Dot
dn = −2Γn + Γ ∑ fα [nU (d − 1)]. dt α
149
(23)
For T = 0 there are two stationary solutions of Eq. (23), n = 0 and n = 0.5 in the voltage range 2(∆− | U% |) < eV < 2∆ , and only one solution, n = 0.5 for eV > 2∆, where U% = U (d − 1) / 2. The MFA current is found as j = f1 (2nU% ) − f 2 (2nU% ).
(24)
Combining this equation and the rate equation (23) with dn/dt = 0 , we obtain the I-V characteristic equation as | U% | T (1 − R ) = 1 − × ∆ ∆ (1 + R )sinh ( eV / 2T ) − cosh ( eV / 2T ) , ln j
where 1/ 2
j2 1/ 2 R = [1 − j coth(eV / 2T )] − . 2 sinh (eV / 2T )
(25)
The I /V curves are shown in Fig. 3 for different temperatures and | U% |= 0.4∆. Interestingly, the temperature narrows the voltage range of the hysteresis loop, but the transition from the low (high)-current branch to the high (low)-current branch remains discontinuous. Let us examine the stability of each branch in the framework of the MFA rate equation (23). Introducing small fluctuations of the electron density as n(t ) = n + δ n exp(γ t ) and linearizing Eq. (23) with respect to δ n we find the increment γ ,
γ = −2Γ +
+
∆ − 2n | U% | −eV / 2 | U% | Γ cosh −2 2T 2T
∆ − 2n | U% | + eV / 2 | U% | Γ cosh −2 . 2T 2T
(26)
One can see from this equation that at temperatures T | U | the lowcurrent branch ( n = 0) looses its stability at the threshold V2 = 2∆/e, while
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the high-current branch looses its stability at V1 = 2 (∆− | U% |) /e. In the voltage range V1 < V < V2 both branches are stable, γ ≈ −2Γ < 0. Finally, let us analyze the effect of a splitting of the degenerate molecular level on the bi-stability. The degeneracy could be removed because of JahnTeller distortions and/or the coupling with the leads. We assume that d 1 levels are evenly distributed in a band of a width W. Then the MFA rate equation (23) is modified as dnµ dt
= −2Γnµ + Γ ∑ fα (ε µ + NU ), α
(27)
where N = ∑ µ ′ nµ ′ . For T = 0 in the stationary regime ( dnµ /dt = 0) it has two solutions, N = 0 and N = d / 2, in the voltage range V1 + W < V < V2 with the current j = 0 and j = 1 , respectively. We conclude that the level splitting W <| U | leads to a narrowing of the voltage range of the bistability similar to the temperature narrowing shown in Fig. 3.
Conclusion In conclusion, we have reviewed the negative Hubbard- U model of tunneling through molecular quantum dots (MQD) with attractive correlations. The degenerate MQD with effective attraction between carriers has a hysteretic volatile memory if the degeneracy of the molecular level is larger than two, d > 2. The origin of the bistability is illustrated in Fig. 2. When the current flows through MQD, the "HOMO-LUMO" gap is renormalized down to a lower value due to attractive correlations, so the current-off voltage V1 turns out smaller than the current-on voltage V2 . The hysteretic behavior strongly depends on temperature. The current bistability vanishes above some critical temperature. Among potential candidates for the negative U quantum dot are single C60 molecules ( d = 6 ), where the electron-vibronic coupling proved to be particularly strong [18], or other carbon nano-structures including short nanotubes ( d 1 ) connected to metal electrodes. Other likely candidates are mixed-valence molecular complexes [21].
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References 1. Aviram A, Ratner M (eds) (1998) Molecular Electronics: Science and Technology. Ann NY Acad Sci, New York; Nitzan A, Ratner MA (2003). Science 300:1384 2. Wu W, Jung G-Y, Olynick DL, Straznicky J, Li Z, Li X, Ohlberg DAA, Chen Y, Wang S-Y, Tong WM, Williams RS (2004). Applied Physics A, accepted 3. Heath JR, Stoddart JF, Williams RS, Chandross EA, Weiss PS, Servic R (2004). Science 303:1136 4. Alexandrov AS, Demsar J, Yanson IK (eds) (2004) Molecular Nanowires and Other Quantum Objects. NATO Science Series, Kluwer Acad. Pub., Dordrecht, the Netherlands 5. Collier CP et al. (1999). Science 285:391; Chen J et al. (1999). Science 286:1550; Gittins DI et al. (2000). Nature (London) 408:677 6. Stewart D, Chen Y, Williams RS, unpublished 7. Alexandrov AS, Bratkovsky AM, Williams RS (2003). Phys Rev B 67:075301 8. Alexandrov AS, Bratkovsky AM (2003). Phys Rev B 67:235312 9. Ermakov VN (2000). Physica E 8:99 10. Alexandrov AS (1983). Zh Fiz Khim 57:273 [Russ J Phys Chem 57:167]; (1998) Models and Phenomenology for Conventional and High-temperature Superconductivity. Course CXXXVI of the Intenational School of Physics ‘Enrico Fermi’, G Iadonisi, JR Schrieffer, ML Chiofalo (eds), Amsterdam: IOS Press, p 309 11. Su WP, Schrieffer JR (1980). Proc Natl Acad Sci 77:5626 12. Feldblum A et al (1982). Phys Rev B 26:815 13. Chance RR, Bredas JL, Silbey R (1984). Phys Rev B 29:4491 14. Ramsey MG et al (1990). Phys Rev B 42:5902 15. Steinmuller D, Ramsey MG, and Netzer FP (1993). Phys Rev B 47:13323 16. Swanson LS et al (1993). Synth Metals 55:241 17. Zhitenev NB, Meng H, Bao Z (2002) Phys Rev Lett 88:226801 18. Park J, Pasupathy AN, Goldsmith JI, Chang C, Yaish Y, Retta JR, Rinkoski M, Sethna JP, Abruña HD, McEuen PL, Ralph DC (2000). Nature (London) 417:722 19. Alexandrov AS, Kornilovich PE (1999). Phys Rev Lett 82:807; (2002) J Phys: Condens Matter 14:5337 20. Wingreen NS, Meir Y (1994). Phys Rev B 49:11040, and references therein 21. Wilson JA (2001). J Phys: Cond Matter 13:R945 22. Meir Y, Wingreen NS (1992). Phys Rev Lett 68:2512 23. Anderson PW (1975). Phys Rev Lett 34P:953 24. Street RA, Mott NF (1975). Phys Rev Lett 35: 1293 25. Micnas R, Ranninger J, Robaszkiewicz S (1990). Rev Mod Phys 62:113, and references therein 26. Alexandrov AS, Kabanov VV (1996). Phys Rev B 54:3655
Configuring a Bistable Atomic Switch by Repeated Electrochemical Cycling
F.-Q. Xie1, Ch. Obermair1, Th. Schimmel1,2 1
Institute for Applied Physics, University of Karlsruhe, D-76128 Karlsruhe, Germany 2 Institute of Nanotechnology, Forschungszentrum Karlsruhe, D-76021 Karlsruhe, Germany
Abstract:
Recently, we have reported the stable and reproducible operation of atomic-scale switches, which allow us to open and close an electrical circuit by the controlled reconfiguration of silver atoms within an atomic-scale junction. Here, we investigate the operation of such atomic quantum switches, and we study in more detail the process during which these switches are formed by repeated electrochemical deposition and dissolution. We find that only after repeated deposition/dissolution cycles, a bistable contact is formed on the atomic scale, which allows to switch between a configuration where the contact is closed, the conducting state or “on”-state, and a configuration where the contact is open, the non-conducting state or “off”-state. We demonstrate that before a bistable contact is formed, irregular changes of the contact conductance are observed as a function of the electrochemical cycling process. A sudden transition to regular switching of the contact between a well-defined “on”-state and the non-conducting “off”-state is observed, which indicates the formation of a bistable contact configuration. Conductance quantization at integer multiples of the conductance quantum G0 = 2e2/h (≈ 1/12.9 kΩ) is found at room temperature for the “on”state conductance.
Keywords: Quantized Conductance, Atomic-Scale Point Contact, Electrochemical Deposition, Silver, Atomic Quantum Switch, NanoElectronics, Molecular Electronics
153 R. Gross et al. (eds.), Nanoscale Devices - Fundamentals and Applications, 153–162. © 2006 Springer. Printed in the Netherlands.
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F.-Q. Xie, Ch. Obermair, Th. Schimmel
Introduction Due to their interesting physical properties and technological perspectives, atomic-scale metallic point contacts are an object of intensive investigations by numerous groups [1-12]. As the size of these constrictions is smaller than the scattering length of the conduction electrons, transport through such contacts is ballistic and as the width of the contacts is on the length scale of the electron wavelength, the quantum nature of the electron is directly observable. The conductance is quantized in multiples of 2e2/h, where e is the charge of an electron and h is Planck´s constant. Experimentally, two different approaches are available for the fabrication of these metallic quantum point contacts: mechanically controlled deformation of thin metallic junctions [1-5] and electrochemical fabrication techniques [6-12]. In both cases, conductance quantization was observed experimentally even at room temperature. As the contact in such junctions is ultimately formed by only one or a few individual atoms, it has already been suggested that the controlled movement of the contacting atom(s) could lead to a switch or a relay on the atomic scale. By moving only one atom in and out of position, a quantized electrical current could be switched on and off. Such a development would not only mean that the ultimate limit for the size of a switch would be reached. It would also provide the basic functional unit for the potential future field of quantum electronics. However, building such a device will require reproducible control of the position of individual atoms within the quantum point contact by means of a third electrode, the control electrode or gate electrode. In the past, first experiments demonstrating controlled switching of atomic positions were reported. Eigler et al. [13] showed the switching of the position of a xenon atom between a scanning tunnelling microscope (STM) tip and a nickel sample surface by applying voltage pulses between STM tip and sample. Fuchs and Schimmel [14] demonstrated the switching of atomic positions in a solid surface at room temperatures by applying voltage pulses with the tip of an STM. Another approach was performed recently by Terabe et al. [15], who switched the conductance across a small gap containing a solid electrolyte by voltage pulsing. In these experiments, however, there was no way of controlling the conductance across the atomic-scale junction by means of an independent third electrode, the control electrode or “gate” electrode. Such an independent control electrode, however, is necessary for the fabrication of transistors or relays on the atomic scale.
Configuring a Bistable Atomic Switch
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Recently, we could demonstrate the first implementation of a transistor on the atomic scale [16, 17]. The atomic-scale transistor can be reversibly switched between a quantized conducting on-state and an insulating offstate by applying a control potential relative to a third, independent gate electrode. For this purpose, an atomic-scale point contact is formed by electrochemical deposition of silver within a nanoscale gap between two gold electrodes, which subsequently can be dissolved and re-deposited, thus allowing to open and close the gap. However, it turned out that only after numerous such deposition/dissolution cycles, a contact is formed which reproducibly opens and closes as a function of the voltage applied to the gate electrode. Here, we study the effect of this electrochemical cycling process, and we discuss the mechanisms of formation and operation of the atomic-scale quantum transistor.
Experimental The experimental setup is illustrated in Fig. 1. Atomic-scale silver quantum point contacts were fabricated by electrochemical deposition from aqueous solution. The electrolyte consisted of 1 mM AgNO3 + 0.1 M HNO3 in bidistilled water. For the deposition of the point contacts, a gap was fabricated between two gold electrodes on a glass substrate. On the ends of these gold electrodes which form the gap, silver was deposited electrochemically until the gap was closed. Three types of electrodes are involved in this electrochemical experiment: a) the two gold electrodes, which served both as leads for the atomic-scale point contact and as electrochemical working electrodes, on which the metal is deposited from the electrolyte, b) the counter electrode for the electrochemical deposition current and c) the quasi-reference electrode providing the electrochemical reference potential. The potential difference between the working electrodes and the quasireference electrode is controlled by a “gate voltage” applied to a bipotentiostat. Within the atomic-scale transistor described below, the two gold working electrodes served as “source” and “drain” while the input of the bipotentiostat which controls the potential of the working electrodes relative to the quasi-reference electrode in the following is referred to as the “gate” electrode. By variation of the voltage applied to this “gate” electrode, the conductance between “source” and “drain” could be switched (see below).
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F.-Q. Xie, Ch. Obermair, Th. Schimmel
Fig. 1. Illustration of the experimental setup. Silver quantum point contacts are electrochemically grown within a nanoscale gap between two electrodes deposited on a substrate. After repeated electrochemical deposition/dissolution processes, a bistable contact configuration is formed, and the reproducible switching of the contact between the two Au working electrodes is achieved by means of an independent gate electrode.
The gold working electrodes (thickness approx. 100 nm), were covered with an insulating polymer coating except for the immediate contact area where the electrode ends were next to each other, forming a sub-micron or nanometer-scale gap. Typical gap widths were of the order of 100 nm. Silver wire of 0.25 mm in diameter with 99.9985% purity was used for the counter electrode and the quasi-reference electrode. For conductance measurements, an additional voltage of -12.9 mV was applied between the two gold electrodes. When applying a positive potential at the gate electrode (i.e. gold working electrodes having a negative bias relative to the electrochemical reference electrode), silver islands formed on the two gold electrodes, two islands finally meeting each other by forming an atomic-scale contact. During deposition, the conductance between the two gold electrodes was continuously measured. As soon as a predefined conductance value was reached, the computercontrolled feedback stopped further deposition of silver on the working electrodes.
Configuring a Bistable Atomic Switch
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Results and Discussion After deposition of a silver point contact as described above, an electrochemical cycling process was started in order to configure an atomicscale switch, which allows reproducible bistable switching between an offstate and a well-defined quantized on-state. As soon as an upper threshold (4.9 G0) near the desired conductance value for the on-state (5.0 G0) was exceeded, the gate voltage was changed from a voltage in the deposition regime (+4 mV) to a voltage in the dissolution regime (−36 mV), the voltage being changed at a rate of 10 mV/s. As soon as the conductance dropped below a lower threshold of the source-drain conductance (0.1 G0), the gate voltage was changed back to a voltage within the deposition regime (+4 mV), again at a rate of 10 mV/s. This deposition process was continued until conductance exceeded the upper threshold of 4.9 G0. At this point, a new cycle consisting of dissolution of the contact and subsequent deposition was started. Figure 2b shows the conductance of the silver contact between the two gold working electrodes in units of the conductance quantum G0 as a function of time during this cycling process. Figure 2a gives the corresponding voltage applied to the gate electrode. As seen in Figure 2b, during the first such dissolution/deposition cycles of each freshly-formed contact, conductance values strongly vary from cycle to cycle. In most cases, contact formation resulted in contact conductance values exceeding 20 G0. When dissolving the contact, conductance immediately drops to zero. During some of the cycles, however, deposition leads to the formation of a contact at a significantly lower conductance value. Yet no reproducible response is observed as a function of the applied gate voltage. Not only the conductance observed by closing the gap, but also the time needed for forming and for dissolving a contact varies from cycle to cycle. While contact conductance values near 5 G0 were observed several times, the behaviour of the contact as a result of the applied gate voltage was still erratic in the beginning.
Gate Voltage (mV)
158
F.-Q. Xie, Ch. Obermair, Th. Schimmel
15
(a)
0 -15 -30 0
50
100 150 200 250 300 350
(b)
2
Conductance (2e /h)
20 15 10 5 0 0
50
100 150 200 250 300 350
Time (s) Fig. 2. Configuring a bistable atomic-scale switch by repeated electrochemical cycling. a) Externally applied gate voltage as a function of time. b) Corresponding change in contact conductance. Only after repeated cycling, regular switching is observed as a function of the applied gate voltage (see arrow).
Only after 290 seconds from the beginning of the experiment (see black arrow in Fig. 2b), a sudden transition from an erratic to a regular behaviour of the contact is observed. Beginning at this point, each of the following cycles of the gate voltage in Fig. 2a results in a corresponding opening and closing of the gap in Fig. 2b, the conductance after closing the gap always being 5 G0. A zoom-in into this sequence of regular switching events of
Configuring a Bistable Atomic Switch
159
Fig. 2 is given in Fig. 3. While Fig. 3a gives the potential applied to the gate electrode as a function of time, Fig. 3b gives the corresponding conductance value on the same time scale. Note that each cycle of the gate voltage results in the corresponding switching of the conductance between the “source” and “drain” electrodes: The device now reproducibly operates as an atomic-scale transistor. This sudden transition from an irregular opening and closing of the contact to a bistable switching between zero and a well-defined quantized conductance value was also regularly observed for other electrochemically deposited silver point contacts, the on-state conductance frequently exhibiting values which were integer multiples of the conductance quantum G0 . The fact that quantized conductance is observed at values of a few conductance quanta (e.g. 5 G0 for the data shown above) means that the contact cross-section is still on the atomic scale [18, 19]. The observation that reversible switching is found between two well-defined conductance values indicates that the contact switches between two well-defined configurations on the atomic scale. This could also explain the observed sudden transition between irregular and regular behaviour of the sourcedrain conductance as a function of the gate voltage. As long as no atomicscale bistability is formed, each deposition cycle leads to the electrochemical deposition of a new contact, different contacts having different conductance values. As soon as a bistable contact configuration has formed, the variation of the applied gate voltage could lead to a switching between the two contact configurations even before dissolution or re-deposition of the contact happens. This switching does not necessarily involve electrochemical deposition and dissolution, but could also be induced by changes of local surface forces due to changes of the applied gate voltage: a variation of the gate voltage will lead to changes of the electrochemical double layer which, in turn, will change surface forces and surface tension [20, 21].
F.-Q. Xie, Ch. Obermair, Th. Schimmel
Gate Voltage (mV)
160
15
(a)
0 -15 -30 300
325
350
6
2
Conductance (2e /h)
(b)
4
2
0 300
325
350
Time (s) Fig. 3. Regular switching of a bistable atomic-scale quantum point contact, induced by the applied gate voltage. a) Gate voltage as a function of time. b) Corresponding contact conductance in units of the conductance quantum. Fig. 3 represents a zoom-in into the data of Fig. 2.
Conclusions To conclude, the configuration process of an atomic-scale transistor or relay was studied. The device, which can be controlled by an independent gate electrode, reproducibly operates at room temperature. Depending on the voltage applied to an electrochemical gate, an atomic-scale metallic
Configuring a Bistable Atomic Switch
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quantum point contact is opened or closed. In this way, the source-drain conductance can be switched in a controlled manner between an insulating off-state and a quantized conducting on-state. The device is based on an atomic-scale silver point contact electrochemically deposited between two gold electrodes. It is shown that prior to the reversible operation of the device, electrochemical cycling of the contact is necessary. Our results indicate that during the repeated electrochemical deposition/dissolution cycles, a bistable contact configuration is formed, which subsequently allows reproducible switching between a conducting and a non-conducting contact configuration by means of an applied gate voltage.
Acknowledgements The authors thank B. Bruhn, S. Brendelberger and L. Nittler for experimental support. This work was supported by the Deutsche Forschungsgemeinschaft within the Center for Functional Nanostructures (CFN) and by the Research Award of the State of Baden-Württemberg (Landesforschungspreis).
References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16.
Agraït N, Levy Yeyati A, van Ruitenbeek JM (2003). Phys Rep 377:81 Agraït N, Rodrigo JG, Vieira S (1993). Phys Rev B 47:12345 Pascual JI et al (1993). Phys Rev Lett 71:1852 Krans JM, van Ruitenbeek JM, Fisun VV, Yanson IK, de Jongh LJ (1995). Nature 375:767 Scheer E et al (1998). Nature 394:154 Li CZ, Tao NJ (1998). Appl Phys Lett 72:894 Li CZ, Bogozi A, Huang W, Tao NJ (1999). Nanotechnology 10:221 Morpurgo AF, Marcus CM, Robinson DB (1999). Appl Phys Lett 74:2084 Li CZ, He HX, Tao NJ (2000). Appl Phys Lett 77:3995 Li J, Kanzaki T, Murakoshi K, Nakato Y (2002). Appl Phys Lett 81:123 Elhoussine F, Mátéfi-Tempfli S, Encinas A, Piraux L (2002). Appl Phys Lett 81:1681 Obermair Ch, Kniese R, Xie F-Q, Schimmel Th (2004). In: Alexandrov AS, Demsar J, I. Yanson K (eds) Molecular Nanowires and Other Quantum Objects. Kluwer Academic Publishers, The Netherlands, pp 233-242 Eigler DM, Lutz CP, Rudge WE (1991). Nature 352:600 Fuchs H, Schimmel Th (1991) Adv Mater 3:112 Terabe K, Hasegawa T, Nakayama T, Aono M (2005). Nature 433:47 Xie F-Q, Nittler L, Obermair Ch, Schimmel Th (2004). Phys Rev Lett 93:128303
162 17. 18. 19. 20. 21.
F.-Q. Xie, Ch. Obermair, Th. Schimmel Xie F-Q, Obermair Ch, Schimmel Th (2004). Solid State Communications 132:437 Brandbyge M, Jacobsen KW, Norskov JK (1997). Phys Rev B 55:2637 Cuevas JC, Levy Yeyati A, Martín-Rodero A (1998). Phys Rev Lett 80:1066 Bach CE, Giesen M, Ibach H, Einstein TL (1997). Phys Rev Lett 78:4225 Friesen C, Dimitrov N, Cammarata RC, Sieradzki K (2001). Langmuir 17:807
Realization of an N-Shaped IVC of Nanoscale Metallic Junctions Using the Antiferromagnetic Transition
Yu. G. Naidyuk1, K. Gloos2, I. K. Yanson1 1
B. Verkin Institute for Low Temperature Physics and Engineering, National Academy of Sciences of Ukraine, 47 Lenin Ave., 61103, Kharkiv, Ukraine 2
Nano-Science Center, Niels Bohr Institute fAFG, Universitetsparken 5, DK-2100 Copenhagen, Denmark
Abstract:
We have observed at low temperatures (<8 K) hysteretic I(V) characteristics for sub-µm (∼200 nm) metallic break-junctions based on the heavy-fermion compound UPd2Al3. Degrading the quality of the contacts by in situ increasing the local residual resistivity or temperature rise reduces the hysteresis. We demonstrate that those hysteretic I(V) curves can be reproduced theoretically by assuming the constriction to be in the thermal regime. Our calculations show that such anomalous I(V) curves are due to the sharp increase of ρ(T) of UPd2Al3 near the Neel temperature TN ≈ 14 K. From this point of view each metal with similar ρ(T) should produce similar hysteretic I(V) characteristics. As example we show calculations for the rareearth manganite La0.75Sr0.25MnO3, a system with colossal magnetoresistance. In this way we demonstrate that nano-sized point contacts can be non-linear devices with N-shaped I(V) characteristics, i.e. with negative differential resistance, that could serve like Esaki tunnel diodes or Gunn diodes as amplifiers, generators, and switching units. Their characteristic response time is estimated to be less than 1ns for the investigated contacts.
Key words: point contacts, negative differential resistance, UPd2Al3, La0.75Sr0.25MnO3.
163 R. Gross et al. (eds.), Nanoscale Devices - Fundamentals and Applications, 163–170. © 2006 Springer. Printed in the Netherlands.
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Yu. G. Naidyuk, K. Gloos, I. K. Yanson
Introduction Point-contact (PC) spectroscopy is widely used to study the interaction of conduction electrons with elementary excitations or quasiparticles in conducting solids [1-5]. On the other hand, PC investigations can shed light on peculiarities of the electronic transport in nano-scale devices at ultrahigh current densities. The latter matters for mesoscopic or nanoscale physics, and especially for applied research, where electronic devices have already reached this sub-µm scale. Electron transport in nanostructures can be distinguished by basically three different current regimes, depending on the relationship between the elastic lel and the inelastic lin mean free path of electrons, and the constriction or point-contact diameter d (for a review see [1], Chapter 3). The constrictions are either ballistic lel, lin >> d, diffusive lin << d << lel lin , or thermal lel, lin << d. A priori it is not trivial to determine the specific current regime, which is important to further characterize the nano-object: first, because the nano-sized object needs a well-defined geometry; second, to evaluate the electronic mean free path, especially the inelastic one, can be rather speculative. To investigate the non-linear I(V) curves using PC spectroscopy seems to be the best method to solve this problem.
Experiment and Discussion Here we present experiments on PCs between two pieces of the heavyfermion compound UPd2Al3, using the technique of mechanicallycontrollable break junctions (see Fig. 1, Refs. [1], Chapter 4.1.5 and [6]). For further details of experimental setup and sample preparation see also [7, 8]). UPd2Al3 becomes antiferromagnetic (AFM) at TN ≈ 14 K [9]. We have observed huge non-linearities of the PC resistances and even hysteretic I(V) characteristics below TN (Fig. 2). We derived the contact size and the residual resistivity in the PC region as described in [10]. We found that the very short elastic mean-free path in the constriction lel << d points to at least the diffusive regime of electron transport through the PC. Considering also the small inelastic mean-free path in UPd2Al3, reflected by the steep ρ(T) rise with temperature around the AFM transition (see Fig. 3 inset), we applied the thermal model developed in Refs. [11-15]. In this case the excess electron energy eV dissipates within the constriction,
Realization of an N-Shaped IVC of Nanoscale Metallic Junctions
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increasing the temperature inside the contact when a bias voltage V is applied. As a result I(V) is governed by the resistivity ρ(T) via [14,15]: 1
I (V ) = Vd ∫ 0
dx
ρ (T 1− x 2 )
COUNTER SUPPORT
,
SOLDERING
SAMPLE
BENDING BEAM ISOLATOR
SCREW+PIEZO
Fig. 1. Break-junction set-up sketch.
where the temperature T in the center of the constriction is set by
T2 = T2 bulk + V2/4L and L is the Lorenz number. The calculated according to the above mentioned equation I(V) curve at T = 1 K in Fig. 3 has maximum at around 2.5 mV, resulting in a hysteresis for up- and downward sweeps when the junction is driven by a current source. Figure 3 shows that the theoretical I(V) describe well the experimental data, including the width of the hysteresis, using d = 200 nm and ρ0 = 10 µΩcm, where ρ0 is the additional residual resistance: ρ(T) = ρ0 + ρbulk (T). These are the only two adjustable parameters which have been derived independently from the measured contact resistance R(T) in [10]. This agreement strongly supports our interpretation that local thermal effects at the PC determine the behavior of our UPd2Al3 breakjunction conductivity. The UPd2Al3 junctions presented here are non-linear devices. Their Nshaped I(V) characteristics have a negative differential resistance at very
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Yu. G. Naidyuk, K. Gloos, I. K. Yanson
Current (mA)
2 T=0.1K 3K 5K 7K
1 0 -1
(a) -2 -15
-10
-5
0 5 Voltage (mV)
10
15
Fig. 2. I(V) curves of a UPd2Al3 break junction with RN = 0.66 Ω at the indicated temperatures. Solid (dashed) lines correspond to sweeps with increasing (decreasing) current. The hysteretic loops become smaller when the temperature rises and vanish above ~5 K.
high current densities up to 5×1010 A/m2. Those devices could be applied – in principle – like an Esaki tunnel diode or a Gunn diode [16, 17] as amplifiers, generators, or switching units. Of practical interest is therefore the possible minimum response time. We estimate [10] a thermal relaxation time τ ≈ 100 ps for a d = 100 nm wide contact. This is three orders of magnitude larger than for a standard tunnel diode, but it could be reduced by using smaller contacts as long as they remain in the thermal regime. Obviously UPd2Al3 is not such a unique material for creating N-shaped I(V) curves – each metal with a similar ρ(T) should also produce similar I(V) characteristics. This can be expected for many materials which order magnetically, since their resistivity typically increases steeply when the magnetic order is destroyed by thermal fluctuations. An example for this behavior is the well known rare-earth manganite La0.75Sr0.25MnO3 , a system with colossal magnetoresistance [19, 20] as
Realization of an N-Shaped IVC of Nanoscale Metallic Junctions
167
T=1K T=8K ρ0=10 µΩcm
I (mA)
2
ρ (µΩ cm)
150
0
100 50
TN
T (K)
0 0
0
5
50
10 Voltage (mV)
100
15
Fig. 3. Measured (solid) and calculated (dashed) I(V) curves at two temperatures. The inset shows ρ(T) of the bulk compound [9]. After [10]
shown in Fig. 4. Indeed, the calculated I(V) curves in Fig. 5(a) are N-shaped up to room temperature. Even enhancing disorder by increasing the residual resistivity up to ρ = 1000 µΩcm as in Fig. 5(b) does not suppress completely the maximum in I(V) at low temperatures. 5 TC=342K
ρ (mΩ cm)
4 3 2 1
B=6T 0
0
100
200 T (K)
300
400
Fig. 4. Resistivity ρ(T) of La0.75Sr0.25MnO3 at zero (upper curve) and B = 6 T magnetic field according to [18].
168
Yu. G. Naidyuk, K. Gloos, I. K. Yanson
T=4K
8 (a) 6
100K
4
200K 300K
2
I (mA)
ρadd=0
0
ρadd=0 µΩ cm
8 (b)
T=4K
10
6
50
4
200
2
1000 0
0
20
40
60
80 100 120
V (mV) Fig. 5. (a) I(V) calculated for d = 100 nm at the indicated temperatures, (b) the same calculation of I(V) for different residual resistances at T = 4 K.
On the other hand, the position of the current maxima shifts to the higher voltages with increasing temperature (or with the residual resistivity increase), contrary to what has to be expected for a simple antiferromagnetic transition. Thus it would be very interesting to measure I(V) characteristic or voltage dependent resistance for point contacts with
Realization of an N-Shaped IVC of Nanoscale Metallic Junctions
169
rare-earth manganite compounds experimentally and make a comparison with the analogous calculations in the thermal regime.
Acknowledgments The partial support of the complex program "Nanosystems, nanomaterials and nanotechnologies" of the National Academy of Sciences of Ukraine and Project Φ1-19 of the National Academy of Sciences of Ukraine are acknowledged.
References 1. Naidyuk YuG, Yanson IK (2004) Point-contact spectroscopy. Springer Series in Solid-State Sciences. Vol. 145 Springer, New York 2. Jansen AGM, van Gelder AP, Wyder P (1980). J Phys C 13:6073 3. Yanson IK (1983). Sov J Low Temp Phys 9:343 4. Yanson IK, Shklyarevskii OI (1986). Sov J Low Temp Phys 12:509 5. Duif A, Jansen AGM, Wyder P (1989). J Phys: Condens Matter 1:3157 6. Moreland J, Clark AF, Soulen Jr RJ, Smith JL (1994). Physica B 194-196:1727 7. Naidyuk YuG, Gloos K, Menovsky AA (1997). J Phys: Condens Matter 9:6279 8. Gloos K, Anders FB, Assmus W, Buschinger B, Geibel C, Kim JS, Menovsky AA, Mueller-Reisener R, Nuettgens S, Schank C, Stewart GR, Naidyuk YuG (1998). J Low Temp Phys 110:873 9. Geibel C, Schank C, Thies S, Kitazawa H, Bredl CD, Böohm A, Rau M, Grauel A, Caspary R, Helfrich R, Ahlheim U, Weber G, Steglich F (1991). Z Phys B 84:1 10. Naidyuk YuG, Gloos K, Yanson IK, Sato NK (2004). J Phys: Condens Matter 16:3433 11. Kohlrausch N (1900). Ann Physik (Leipzig) 1:132 12. Verkin BI, Yanson IK, Kulik IO, Shklyarevskii OI, Lysykh AA, Naidyuk YuG (1979). Solid State Commun 30:215 13. Verkin BI, Yanson IK, Kulik IO, Shklyarevskii OI, Lysykh AA, Naidyuk YuG (1980). Izv Akad Nauk SSSR Ser Fiz 44:1330 14. Kulik IO (1984). Phys Lett 106 A:187 15. Kulik IO (1992). Sov J Low Temp Phys 18:302 16. Esaki L (1992). In: Lundquist Stig (ed) Nobel lectures in Physics 1971-1980. World Scientific Publishing Company; (1974) Science 183:1149 17. Price PJ (1992). In: Moss TS, Landsberg PT (eds) Handbook on Semiconductors. Elsevier Science Publishers B.V., Amsterdam Volume I Chapter 12
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18. Mitra J, Raychaudhuri AK, Gayathri N, Mukovskii YaM (2002). Phys Rev B 65:140406 19. Rao CNR, Raveau B (eds) (1998) CMR, Charge Ordering and Related Properties of Manganese Oxides. World Scientific, Singapore 20. Tokura Y (ed) (2000) Colossal Magneto-Resistance Oxides. Gordon and Breach Sciences, Netherlands
PI-SHIFT EFFECT AND FERROMAGNET/SUPERCONDUCTOR NANOSCALE DEVICES
Josephson Effect in Composite Junctions with Ferromagnetic Materials
M.Yu. Kupriyanov1, A.A. Golubov2, M. Siegel3 1
D.V.Skobeltsyn Institute of Nuclear Physics, Moscow State University, Moscow, Russia
2
Faculty of Science and Technology, University of Twente, The Netherlands,
3
Institute for Micro and Nanoelectronic Systems, Karlsruhe University, Germany
Abstract: The proximity effect in the system consisting of a superconducting electrode S with ferromagnetic F and normal N layers on the top of it, is studied theoretically in the framework of the Usadel equations. The results are applied for the calculation of the electronic density of states and the Josephson current in a S(F/N)I(F/N)S tunnel junction. Keywords: proximity effect, Josephson π contacts
Introduction In the past few years there was a noticeable interest in "Unconventional Josephson Junctions" (UJJ). Contrary to already well known "Conventional Josephson Junctions" (CJJ) or 0-Josephson junctions, the current phase relationship I s (ϕ ) in UJJ may have negative critical current (the so-called π- junctions) or even cross the ϕ-axis at a position in between ϕ =0 and ϕ = π (see [1] for a recent review). SFS π- junctions were first realized by Ryazanov et. al. [2]. The existence of UJJ opens the way for fabrication of Josephson structures with engineered properties [3] by putting together UJJ and CJJ into a discrete analogue of a Josephson
173 R. Gross et al. (eds.), Nanoscale Devices - Fundamentals and Applications, 173–187. © 2006 Springer. Printed in the Netherlands.
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junction (DJJ). By changing the shape of I s (ϕ ) of individual junctions in the DJJ it is possible to engineer a junction with desired properties. Even the Josephson structures composed from arrays of 0- and π-junctions possess extraordinary characteristics [4-15]. In particular, the existence of Josephson vortices with unconventional shape of magnetic field distribution was predicted in these structures [7-15]. There is also a possibility of realization of Josephson structures, where the average phase difference ϕ the ground state may have any value in the interval between 0 and π [7-9] (so called "ϕ-junctions"[15]). In the developed theoretical models [10-15] it was supposed that both ϕ and its first derivative are continuous at the boundaries between 0- and πsegments inside a DJJ. This suggestion is not obvious since the transition from 0- and π-segments should be supplemented by a redistribution of a local current near the segment interfaces. In the present work we attack this problem by studying the generic building block of these structures, namely the S(FN)I(FN)S structure presented in Fig.1. In the framework of the Usadel equations [16] we start with the examination of the proximity effect in the S(FN) system and get the analytical solutions describing the oscillatory behavior of superconducting correlations induced into thin ferromagnetic film. These solutions have been used for calculations of the density of states in the S(F/N) proximity system and the lateral spatial variations of the critical current density in the S(FN)I(FN)S Josephson junction. Based on these results we derived the boundary condition for ϕ at the interface of 0- and π- segments in DJJ. This condition was further applied for the determination of the equation for the slow varying part of ϕ. Finally the conditions of experimental realization of "ϕ-junctions" in the discussed system have been formulated.
Model of S(FN)I(FN)S Multilayer We consider S(FN)I(FN)S type structures as shown in Fig. 1, which consists of a superconductor electrode with ferromagnetic and normal layers on top of it. The F/N bilayer is additionally covered by an insulator, I, to form the tunnel barrier and the second superconducting electrode having the same structure.
Josephson Effect in Composite Junctions with Ferromagnetic Materials
175
Fig. 1. Schematic view of S(FN)I(FN)S Josephson junctions
We assume that the S layers are bulk and that the dirty limit conditions are fulfilled in all metals. We further assume that the F metal is a monodomain ferromagnet with a value of the exchange integral equal to H and that the effective constant of electron-phonon interaction is equal to zero in the F and N layers. For simplicity we also assume that interfaces are not magnetically active and can be described by spin independent suppression parameters
γ BF =
R BF ABF
ρFξF
γN =
,γF =
ρSξS R A , γ BN = BN BN , ρFξF ρNξ N
ρSξS R A ρ ξ ,γ = B B , γ = F F , ρNξN B ρNξN ρNξN
where RBF, RBN, RB and ABF, ABN , AB are the resistances and the areas of the SF, SN and NF interfaces respectively; ρS(F,N) is the resistivity of the S(F,N) layer and the coherence lengths are related to the diffusion constants DS(F,N) as ξS(F,N) = (DS(F,N)/2πTC)1/2. This approach is valid for relatively weak ferromagnetic materials when the exchange integral is smaller than 0.1 eV and spin dependent corrections to resistivity can be neglected. Under the above assumptions we can reduce the problem to the solution of the Usadel equations. To simplify it further we assume that γN<<γBN, γF<<γBF so that the rigid boundary conditions FS 1,2 = sin θ S =
∆
ω 2 + ∆2
exp{i χ1,2 }, GS = cos θ S =
ω ω 2 + ∆2
(1)
are valid for the superconductor. Here ∆ and χ are the magnitude and phase of the order parameter in the electrodes, FS1,2 and GS are the Usadel Green's functions We will assume additionally that the transparency of the tunnel barrier is much smaller than the transparency of SN and SF
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interfaces of the structure thus reducing the problem to examination of the proximity effect in S(FN) electrodes.
Proximity Effect in S(F/N) System To study the proximity effect we choose the x axis perpendicular to the plane of the NF-interface and the y axis perpendicular to the plane of the SF- and SN-interfaces with the origin at the NF-interface, as it is shown in Fig.1. The Usadel functions G and F obey the normalization condition Gω2 + Fω F−*ω = 1 , which allows for the so-called θ parameterization. Making use of the boundary conditions [17] at the SF (y=0, -∞<x≤0), SN (y=0, 0≤x<∞) and FN (x=0, 0≤y≤dF, dN) interfaces and demanding the absence of a current across free interfaces, in the limit of small F and N layers thickness (dF,N<<ξF,N) we can reduce the problem to the solution of one dimensional equations
πTC γ BNM cos θ N (∞) 2 ∂ 2 ξ N 2 θ N − sin(θ N − θ N (∞)) = 0, ωγ BNM + πTC cos θ S ∂x
(2)
πTC γ BNM cos θ F (−∞) 2 ∂ 2 ξ F 2 θ F − sin(θ F − θ F (−∞)) = 0, ω~γ BFM + πTC cos θ S ∂x
(3)
where γ BNM = γ BN d N / ξ N , γ BFM = γ BF d F / ξ F , ω~ = ω + iH and πTC sin θ S ∆ θ F , N (m ∞) = arctan ~ , θ S = arctan . ωγ BFM + πTC cos θ S ω
Solution of equations (2), (3) has the form
θ N (0) − θ N (∞)
4
θ N = θ N (∞) + 4 arctan tan
,
(4)
x exp , ζ F
(5)
x exp− ζN
θ F (0) − θ F (−∞)
4
θ F = θ F (−∞) + 4 arctan tan
Josephson Effect in Composite Junctions with Ferromagnetic Materials
177
where the effective decay lengths are
ζN =
ζF =
πTC γ BNM 2
ω γ
2 BNM
ω γ
2 BFM
+ 2πTC ωγ BNM cos θ S + (πTC ) 2
πTC γ BFM 2
+ 2πTC ωγ BFM cos θ S + (πTC ) 2
,
(6)
.
(7)
The integration constants θN(0) and θF(0) in (4),(5) have to be determined from the boundary conditions at the FN interface (x = 0)
ξN
∂ ∂ ∂ θ N = γ F ξ F θ F , γ BF ξ F θ F = sin(θ N − θ F ). ∂x ∂x ∂x
(8)
They can be found analytically in the limit of large transparency of the interface between the F and N metal. Then the condition (8) in the first approximation may be simplified to the continuity of Usadel functions at FN interface θ N (0) = θ F (0) = θ (0) . In this case
θ N (∞ ) θ F (−∞) sin 2 + g sin ζN 2 θ (0) = θ N (∞) + 2 arctan . , g = ζF cos θ N (∞) + g cos θ F (−∞) 2 2
(9)
The developed theory of proximity effect in complex S(F/N) proximity multilayers opens the way for examination of spatially resolved density of states (DoS) in the F/N film. To calculate the DoS we need N (ε ) = Re G (ω → −iε ) . In the SN bilayer ( x → ∞) DoS does not depend on H and is given by [18-20] N N (ε ) = N 0 Re
sign(ε )ε~N d ∆2 − ε 2 ~ , ε~N = ε (1 + γ~ ), γ = γ BN N , 2 2 ~ T π ξ C N ε −∆ N
where N 0 is the total DoS for both spin directions at the Fermi level in the normal state. The DoS has a minigap at ε g < ∆ , and the peaks at ε = ε g and ε = ∆ . The minigap ε g characterizes the strength of superconducting correlations induced into N metal due to the proximity effect.
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In the SF bilayer ( x → −∞) modifications of DoS due to spin splitting of energy levels in F were investigated in [21, 22]. The DoS per spin projection in the F layer has the form (below we will suppose for simplicity that F and N metals differs only due to existence of exchange field H ≠ 0 in the F film)
sgn(ε m H )ε~F ↑,F ↓ N0 N F ↑,F ↓ (ε ) = Re 2 ε~ 2 − ∆2
(10)
F ↑,F ↓
ε F ↑, F ↓ = ε + γ~ (ε m H )
∆2 − ε 2 , πTC
γ~ = γ BF
dF
ξF
= γ BN
dN
ξN
.
Equation (10) demonstrates the energy renormalization due to the exchange field. From (10) it follows that there are two minigaps in the spectrum ε g ↑ and ε g ↓ , and ε g ↓ ≤ ε g ≤ ε g ↑ . Spin resolved DoS for identical transport parameters on the F and N metals at FN interface (x = 0) has the form 2 N F ↑, F ↓ (ε ) N0 = Re
ε%FN =
−iε%N − iε%F ↑, F ↓ + 2ε%FN 2 ∆ 2 − ε%F2 ↑, F ↓ sgn(ε m H ) + ∆ 2 − ε%N2 sgn(ε ) + 2 ∆ 2 − ε%FN
(11) ,
∆ 2 − ε%N2 ∆ 2 − ε%F2 ↑, F ↓ sgn(ε )sgn(ε m H ) − ε%F ↑ε%N − ∆ 2 2
.
It follows from Eq. (11) that, similar to the case of the SF bilayer considered above, the minigap exists if γ~H < πTC . With increasing exchange field the total DoS at ε = 0 becomes nonzero if γ~H ≥ πTC and is given by the simple expression N (0) = N 0
γ~ 2 H 2 − (πTC ) 2 . γ~H
(12)
Figure 2 shows the densities of states calculated far from FN interface both in the SN region (open squares) and the spin resolved DoS in the SF part (open circles and triangular). Shot dash, dash dot and solid lines show
Josephson Effect in Composite Junctions with Ferromagnetic Materials
179
the spin resolved and full DoS at the FN interface. All calculations were done for γ~H = 0.5πTC It is clearly seen from both (11) and the results of numerical calculations (see Fig. 2) that there are four characteristic energies in the spectrum. They are ε g ↑ , ε g , ε g ↓ , and ∆. Here ε g ↓ is the minigap for the spin-down subband in the SF bilayer at x→-∞. It follows from (11) that N F ↓ = 0 at ε ≤ ε g ↓ and becomes nonzero at ε > ε g ↓ , i.e. ε g ↓ is the minigap for the spindown subband in S(FN) at x = 0. However, contrary to the SF case N F ↓ (ε ) has no peak at ε = ε g ↓ but grows continuously from zero value. For the spin-up subband, the minigap in N F ↑ (ε ) is not equal to the gap ε g ↑ in the spin-up subband in the SF bilayer at x→-∞. Instead, the gap value in this subband is determined by ε g , the minigap in the SN bilayer at x→∞. The formal reason is that for energies ε ≥ ε g both the numerator and denominator in (11) become complex thus leading to a nonzero DoS in this energy range. Similar to the spin-down case, N F ↑ (ε ) has no peak at the gap energy ε = ε g , while the peak occurs at ε = ε g ↑ (see Fig. 2). With further increase of energy there is a sharp peak in DoS at ε = ∆ followed by saturation of full DoS at N 0 for ε >> ∆ .
Hγ=0.5πTC
5
SN SF down SF up S(FN) down S(FN) up S(FN) total
N(ε)/N0
4
3
2
1
0 -2.0
-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
2.0
ε/∆ Fig. 1. Densities of states calculated far from the FN interface both in the SN region (open squares) and spin resolved DoS in the SF part (open circles and triangles). Short dash, dash dot and solid lines show the spin resolved and full DoS at FN interface.
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M.Yu. Kupriyanov, A.A. Golubov, M. Siegel
Critical Current of S(F/N)I(F/N)S Tunnel Junctions The theory for the proximity effect in complex S(F/N) proximity multilayers developed above allows for the direct examination of the Josephson effect in the junctions composed of S(F/N) electrodes. To calculate a dc current J across the tunnel barrier in tunnel junctions with complex S(F/N) electrodes we may start with the general expression for the current in θ-parameterization J=
∂ ∂ iπ T ω =∞ sin θω cos θ −*ω θ −*ω − sin θ −*ω cos θω θω , ∑ 2eρ ω =−∞ ∂x ∂x
(13)
and making use of boundary conditions [17] at the isolator I of the S(F/N)I(F/N)S tunnel structure and symmetry relations
cos θ ω = cos θ −*ω , sin θ ω exp{−iχ } = sin θ −*ω exp{iχ } ,
(14)
we arrive at J=
2πT eR N AI
ω =∞
∑ Re[sin θ ω
U
ω = −∞
sin θ ωB ]sin ϕ ,
(15)
where ϕ =(χU – χB) is the phase difference of the order parameters across the structure, RN is a normal junctions resistance, AI is the junction area and the Usadel functions of the upper (U) and the bottom (B) electrodes are defined by equations (4)-(7), (9). In particular, for relatively small transparency of SN and SF interfaces γ~ ≥ TC T the equations (4)-(7), (9), (15) may be linearized resulting in the fully symmetric structure
∑ [
]
∑ [
]
∞ 2 Re θ N ( x) , x ≥ 0 2πT ω =0 J C ( x) = , eR N AI ∞ Re θ F2 ( x) , x ≤ 0 ω =0
(16)
where ∆π TC Re θ = ω (ωγ% + π TC ) 2 N
2
x 1 + 2u exp − ζN
2 x + v exp − , ζ N
(17)
Josephson Effect in Composite Junctions with Ferromagnetic Materials
u=
1+ 1+ q2 2(1 + q 2 )
− 1, v =
181
Hγ~ 1 , − − u q = 1 2 , ωγ~ + πTC (1 + q 2 )
2
ax 2 xa ∆π TC Re θ = 1 − q 2 + 2U exp + V exp , 2 ω (ωγ% + π TC )(1 + q ) ζ N ζ N 2 F
U = ( µ + qη ) cos
V = ( µ 2 − η 2 )sin
2 xb
ζN
a=
xb
ζN
+ ( q µ − η )sin
− 2 µη cos
2 xb
ζN
1+ q2 +1 2
xb
ζN
π TC , ωγ% + π TC
, ζ N = ξN
, µ = a − 1 + qb, η = b − qa + q,
, b=
1+ q2 −1 2
.
It follows fom (16) and (17) that for positive x > 0 the critical current density increases exponentially with x and saturates at the value previously derived for SNINS tunnel structures. For negative x < 0 the critical current density exhibits damping oscillations and tends to the value previously obtained for SFIFS tunnel junctions. For q > 1 the critical current may change sign far from the FN interface. Therefore the expressions (16) and (17) describe the continuous transition from "0" to "π" junction in this structure.
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M.Yu. Kupriyanov, A.A. Golubov, M. Siegel 1
0,10
2 3
0,08
eJC(x)RN/2πTC
0,06
4 0,04
5 6
0,02
0,00
-0,02 -4
-3
-2
-1
0
x/ξF
Fig. 3. Critical current density calculated at T=0.1TC and as a function of the distance from F/N interfaces into F metal for various values of H g%/ pT C = 1; 3; 5; 8; 10; 15 (curves 1-6, respectively).
Hγ=15πTC Hγ=10πTC Hγ=5πTC Hγ=2πTC Hγ=30πTC Hγ=50πTC
e|JC(x)|RN/2πTC
0,1
0,01
1E-3
1E-4
1E-5 -4
-3
-2
-1
0
x/ξF
Fig. 4. Magnitude of critical current density calculated at T=0.1TC and as a function of the distance from F/N interfaces into F metal for various values of H.
Josephson Effect in Composite Junctions with Ferromagnetic Materials
183
At low temperatures this transition is illustrated in Fig. 3 and Fig. 4. At H < πTC / γ~ , there is only the "0" state and J C (x) falls monotonically with x ≤ 0 in the SFIFS part of the structure. At H = πTC / γ~ , the transition to the "π" state occurs and the critical current density tends to zero as x → −∞ . At larger H the oscillation evolves near FN boundary and the amplitude of this oscillation can be of the order of magnitude larger compared to J C ( x → −∞) for H ≥ 5πTC / γ~ . The number of oscillations of J C (x) increases with increasing H.
Boundary Conditions for the Sine-Gordon Equation If the size of the S(FN)I(FN)S tunnel junction in x direction is large or comparable with the Josephson penetration depths λ± of its SFIFS and SNINS parts located at x ≤ 0 and x ≥ 0 , respectively, than the spatial distribution of phase difference ϕ across the structure should obey the socalled sine-Gordon equation:
λ2±
d2 ϕ ± − δ (1 + j ± ( x / ξ ± )) sin ϕ ± = 0 dx 2
(18)
where δ = 1 for x ≥ 0 and δ = ±1 depending on whether we have a 0 or π junction on the left hand side of the structure ( x ≤ 0 ). The functions j± (x / ξ ± ) decay exponentially with x at the distances ξ+, ξ- which are much smaller than the Josephson penetration depths λ±. It follows from our consideration that j + ( x) ∝ 1 , while j − (x) may be at least one order of magnitude larger than unity. Despite of this fact in typical experimental situation ξ ± << λ±
j ± and a solution of Eq. (18) may vary in space only
on a scale much larger ξ ± . Taking this into account we can integrate (18) along x and find the effective boundary conditions which relate the functions ϕ± at ξ± <<|x|<< λ±
ϕ (−0) = ϕ (+0) = ϕ (0),
η =δ
ξ− λ2−
d d ϕ (+0) − ϕ (−0) = η sin ϕ (0) , dx dx
0
∫
−∞
j − ( z )dz −
ξ+ λ2+
∞
∫j 0
+ ( z ) dz
.
(19)
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M.Yu. Kupriyanov, A.A. Golubov, M. Siegel
Below we apply these boundary conditions for the examination the properties of periodic arrays of 0 -π Josephson junctions in S(F/N)I(F/N)S structures.
Periodic Arrays of 0 - Π Josephson Junctions in S(F/N)I(F/N)S Structures Consider now a periodic structure composed of alternating 0- and πJosephson junctions of lengths d0i and dπi respectively. The ground state phase distribution in the array is determined by the equations
λ20
d2 ϕ − sin ϕ = 0, xi − d oi ≤ x ≤ xi , dx 2
(20)
λπ2
d2 ϕ − sin ϕ = 0, xi ≤ x ≤ xi + d πi , dx 2
(21)
where xi is the position of one of FN interfaces. At these interfaces, the phase difference ϕ and its derivative have to satisfy the conditions (19). Suppose now that ξ±<
λ 2eff
Φ =1+
(d 0i + d πi )λ 20 λπ2 d2 2 ψ = Φ sin ψ , λ = eff dx 2 (λπ2 d 0i − λ 20 d πi )
λ2eff η ( x) (d 0i + d πi )
−
(22)
λ2eff d 0i d πi (λ 20 + λπ2 )(λ 20 d π2i + λπ2 d 02i ) cos Ψ . 12(d 0i + d πi ) 2 λ40 λπ4
In the limit λπ = λ 0 = λ , d 0i − d πi << d 0i ≈ d πi , and η(x)=0 this equation transforms into the Buzdin and Koshelev result [15]. Several assumptions were made to derive Eq. (22). First, it was supposed that the scale of ψ (x) variations is larger than d0i, dπi or (d + d πi ) 12( d 0i + d πi ) 2 λ 40 λπ4 d 02i , d π2i << min λ 2eff , 0i , . 2 2 2 2 2 2 η ( x) d 0i d πi (λ 0 + λπ )(λ 0 d πi + λπ d 0i )
(23)
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Second, it was also supposed that the array is periodic with the period d0i + dπi. It means that local size variations of the contiguous items of the array should be smaller than the accuracy of the consideration, namely dπ i − dπ ( i −1) dπ i − dπ ( i +1) d2 d2 dη min 02i , π2i >> , , ( d 0 i + dπ i ) max {d 0i , dπ i }. λ λ d + d d + d dx 0i 0i π πi πi 0
(24)
Generally, the parameter η in (22) is small and can be neglected if d 0i d πi (λ 20 + λπ2 )(λ 20 d π2i + λπ2 d 02i )
λ 40 λπ2 ξ −
≥ 10 2 (d 0 i + d πi ) ,
(25)
or, in other words, if the lengths of the segments d0i, dπi are not too small. It follows from Eq. (22) that if η is not a function of x than the ground state in alternating 0- and π-Josephson array made of S(F/N)I(F/N)S tunnel junctions may have the phase difference ψ0 determined by equation d 0i d πi (λ 20 + λπ2 )(λ 20 d π2i + λπ2 d 02i ) (λπ2 d 0i − λ 20 d πi ) cos ψ = +η . 0 12 (d 0i + d πi )λ 40 λπ4 λ 20 λπ2
(26)
The phase difference ψ0 may take any value in the interval –π ≤ψ0≤ π depending on the properties of 0 and π segments in the array.
Conclusion In conclusion, we have investigated theoretically the proximity effect in the S(F/N) structure and calculated the density of states and the spatial distribution of critical current in the S(F/N)I(F/N)S tunnel junction. We have shown that the density of states near the FN interface depends on four characteristic energies and has a form essentially different from that in the SN or SF bilayers. This fact should be taken into account in the calculation and interpretation of the current voltage characteristics of the S(F/N)I(F/N)S tunnel devices. The critical current density exhibits damped lateral oscillations in SFIFS part of S(F/N)I(F/N)S structure. Moreover, the amplitude of these oscillations may become an order of magnitude larger compared to the bulk value of J C far from the FN interface. This fact modifies the form of
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the boundary conditions which previously have been used at the interfaces of “0” and “π” segments of alternating arrays studied in [10]-[15] and should be taken into account if the sizes of the segments d0i, dπi are small enough (see Eq. (25). The conditions for realization of “ϕ - junctions”, Eqs. (23)-(25), have been formulated.
Acknowledgments This work has been supported by the Russian Ministry of Education and Science and RFBR Grant N 04 0217397-a.
References 1. Golubov AA, Kupriyanov MYu, Il'ichev E (2004). Rev Mod Phys 76:411 2. Ryazanov VV, Oboznov VA, Rusanov AY, Veretennikov AV, Golubov AA Aarts J (2001). Phys Rev Lett 86:2427 3. Kupriyanov MYu, Golubov AA, Khapaev MM, Siegel M (2003) Engineering of Josephson junctions with predetermined properties. ESF Pi-Shift Workshop. Josephson Junctions: Basic Studies and Novel Applications. 16-19 June, Jena, Germany p 18 4. Smilde Ariando HJH, Blank DHA, Gerritsma GJ, Hilgenkamp H, Rogalla H (2002). Phys Rev Lett 88:057004 5. Hilgenkamp H, Smilde Ariando HJH, Blank DHA, Rijnders G, Rogalla H, Kirtley JR, Tsuei CC (2003). Nature (London) 422:50 6. Ryazanov VV, Oboznov VA, Prokofiev AS, Bolginov VV, Feofanov AK (2004). J Low Temp Phys 136:385 7. Mints RG, Kogan VG (1997). Phys Rev B 55:R8682 8. Mints RG (1998). Phys Rev B 57:3221 9. Mints RG, Papiashvili I (2001). Phys Rev B 64:134501 10. Goldobin E, Koelle D, Kleiner R (2002). Phys Rev B 66:100508 11. Goldobin E, Koelle D, Kleiner R (2003). Phys Rev B 67:224515 12. Goldobin E, Sterck A, Gaber T, Koelle D, Kleiner R (2004). Phys Rev Lett 92:057005 13. Zenchuk A, Goldobin E (2004). Phys Rev B 69:024515 14. Aladyshkin AYu, Buzdin AI, Fraerman AA, Mel'nikov AS, Ryzhov DA, Sokolov AV (2003). Phys Rev B 68:184508; (2004) Phys Rev B 69:099903 15. Buzdin A, Koshelev AE (2003). Phys Rev B 67:220504 16. Usadel KD (1970). Phys Rev Lett 25:507 17. Kupriyanov MYu, Lukichev VF (1988). Zh Eksp Teor Phys 94:139 [(1988) Sov Phys JETP 67:1163] 18. McMillan WL (1968). Phys Rev 175:537 19. Golubov AA, Kupriyanov MYu (1988). J Low Temp Phys 70:83
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20. Golubov AA, Kupriyanov MYu (1989). Zh Eksp Teor Fiz 96:1420. 21. Golubov AA, Kupriyanov MYu, Fominov YaV (2002). Pis'ma Zh Eksp Teor Fiz 75:223 [(2002) JETP Lett 75:190] 22. Krivoruchko VN, Koshina EA (2002). Phys Rev B 66:014521
Depairing Currents in Bilayers of Nb/Pd89Ni11
A. Yu. Rusanov1, J. Aarts1, M. Aprili2 1 2
Kamerlingh Onnes Laboratory, Leiden University, the Netherlands Univ Paris 11, CSNSM, CNRS, Orsay, France
Abstract:
In superconductor/ferromagnet (S/F) hybrids the order parameter induced in the F-layer due to the proximity effect is spatially inhomogeneous. This could result in the nonmonotonic behavior of the critical temperature Tc and the depairing current density Jdp as a function of the thickness of the ferromagnet dF. Specifically Tc(dF) and Jdp(dF) may demonstrate a minimum around λex /4, where λex is the period of the spatial oscillation of the order parameter Ψ(x) in the F-layer. We experimentally studied the superconducting properties of Nb/Pd0.89Ni0.11 bilayers structured in 15 µm long strips of different width. Close to 1/4 of the period of Ψ(x), Jdp(dF) shows a minimum, in contrast to the behavior of Tc(dF) where a minimum is absent. We argue that this difference is caused by the higher sensitivity of the depairing current to the order parameter changes in comparison with Tc.
Keywords: proximity effect, superconductivity
SF
hybrids,
spintronics,
inhomogeneous
Introduction The proximity effect between a superconductor (S) and a ferromagnet (F) is in the focus of intensive studies. It is now known that the spatially inhomogeneous superconducting order parameter Ψ(x) induced inside the Flayer in S/F hybrid structures can manifest itself in a number of different ways [1-3]. Among them, one of the interesting phenomena is the nonmonotonous behavior of the critical temperature Tc of S/F multilayers [4] and bilayers [5] as a function of the thickness of the ferromagnet dF. In 189 R. Gross et al. (eds.), Nanoscale Devices - Fundamentals and Applications, 189–196. © 2006 Springer. Printed in the Netherlands.
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bilayers in particular, this effect is difficult to observe [6-8], probably because it is strongly dependent on the interface transparency [5]. Since Tc is basically determined by the largest amplitude of Ψ(x) in the S-part of the bilayer, it quickly regains the bulk value. On the other hand, the depairing current Jdp of the system should be more sensitive to the average of Ψ(x) over the S-layer, and therefore also more sensitive to the lowered Ψ(x) close to the interface [9]. Here we present a first set of measurements of both Tc(dF) and Jdp(dF) on bilayers of Nb/Pd89Ni11, indicating such a difference in behavior.
Experimental Details Bilayers of type s/S/F/N were fabricated by e-gun evaporation in the system described in Refs. [2, 3]. Here, s denotes the substrate consisting of a Si waver covered with 50 nm SiO, S is a Niobium layer with a thickness of 13 nm, F is a layer of Pd89Ni11 with a thickness varying between 0 nm and 12 nm, and N is a thin (5 nm) Ag protection layer. The Niconcentration of the PdNi layers was analyzed by Rutherford Backscattering and determined to be 11 – 12 %. For the depairing current 1.0
2.240 2.238
0.8
2.236 2.234 2.232 2.230
0.4
2.228
T [K]
V [au]
0.6
2.226 0.2
2.224 2.222
0.0
350
400
450
500
550
600
2.220 650
I [µA]
Fig. 1. Typical dependence of voltage V (open squares) and temperature T (open circles) on current I. The measurement was performed on a 1.6 µm wide Nb/Pd89Ni11 bridge of sample 12 (see Table 1).
experiments, samples were structured in the shape of 3 µm wide bridges by e-beam lithography and Ar-ion etching. All samples were liquid-nitrogen cooled during etching in order to avoid undesirable diffusion of materials.
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The structure included the current and voltage leads. In all cases, the distance between the voltage leads was about 15 ± 0.1 µm, while the actual width of the bridge turned out to depend on the details of the lithography process and varied between 1.5 µm and 3.5 µm. Since the width is a crucial parameter, it was determined for each sample separately. The width of the transition from the normal to the superconducting state was about 0.5 K, which is somewhat wider than in the case of single superconductors. Measurements in the normal state yielded an average value of the specific resistance ρF for the Pd89Ni11 films of about 35 µΩ·cm and ρS for the Nb films of about 7.5 µΩcm. The relevant experimental parameters are given in Table 1. Measurements of the depairing current were performed by a pulsed current method as described in ref. [10]. For temperatures close to Tc a small onset of voltage was observed in all samples, probably because of vortex motion. In order to make certain that this effect has no influence on the determination of Idp, the temperature was monitored during every current pulse. Measurable differences were found only very close to Idp, as shown in the Fig. 1. Hence we can conclude that a short current pulse in a combination with a long pause does not cause sample heating and keeps the system in temperature equilibrium until the dissipation related to the normal state occurs.
Results And Discussion Figure 2a shows a set of typical current (I) - voltage (V) characteristics at different temperatures for the sample with dF = 3.5 nm. All show an almost instantaneous jump of the voltage to the value corresponding to the normal state of the sample, which is identified as Jdp. Fig. 2b shows the compilation of all values of Jdp as function of reduced temperature t = T/Tc for bilayers of Nb/Pd89Ni11 with a different thickness dF. Below t = 0.8, the samples with 5.6 nm and 9.5 nm of Pd89Ni11 show a sudden upturn, and differ significantly from other samples in the set. Possibly, the Nb thickness for those two is slightly higher than for the other samples. In order to account for small sample-to-sample variations, all values for Jdp were first normalized by determining the slope S = ∆Jdp/∆t near t = 1. An average slope Sav was defined by the samples with equal slope (3.5 nm, 5.6 nm, 9.5 nm; black dotted line in Fig. 1b). Values for Jdp(t) were then normalized according to Jdpnor(t) = cnorJdp(t), with cnor = Sav / S. The correspondent normalization values cnor for each Jdp curve are presented in Table 1.
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2.0 T=4.29K T=3.84K T=3.48K T=3.17K T=2.0K T=1.86K
Voltage [V]
1.5
1.0
s/Nb/PdNi dF = 3.5 nm
0.5
0.0
0
50
100
150
200
250
300
350
I [µA] 4.0
dF, [nm] 0; 5,0; 6,4; 6,8; 8,2; 9,9;
3.5
10
2
Jdp [10 A/m ]
3.0 2.5 2.0
3,5; 5,6; 6,6; 8,0; 9,5; 11,1;
1.5 1.0 0.5 0.0
0.5
0.6
0.7
0.8
0.9
1.0
t
Fig. 2. (a). Current (I) - Voltage (V) characteristics for a Nb/Pd89Ni11 bilayer with dF = 3.5 nm taken at different temperatures. (b). Experimental values for the depairing current density Jdp(t) as a function of reduced temperature t for Nb/Pd89Ni11 bilayers with different Pd89Ni11 thickness dF. The black dashed line indicates the average slope near t = 1 used for calculation of normalization values, see text.
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Table 1. Experimental values of the critical temperature Tdpc determined by the onset of Ohmic behavior in Jdp(t), the critical temperature Tc determined using a resistance criterion (20% resistance drop), bridgewidth w , the sample resistance R at T = 10 K, the Pd89Ni11 thickness dF , and the normalization constant cnor for the Jdp(t) curves plotted in Fig. 1b.
Sample
Tdpc [K]
Tc [K]
w [µm]
R [Ω]
dF [nm]
cnor
1 2 3 4 5 6 7 8 9 10 11 12
7.46 4.40 4.20 4.50 4.10 4.20 4.00 4.30 4.35 4.35 4.25 4.00
7.46 4.7 4.5 4.65 4.50 4.00 4.28 4.10 4.70 4.60 4.30 4.00
1.6 1.5 1.7 3.4 1.7 2.8 3.2 1.6 1.6 2.3 2.5 1.6
122.7 130.3 46.2 55.8 67.4 33.4 38.1 44.5 72.8 49.6 34.0 54.0
0.0 3.5 5.0 5.6 6.4 6.6 6.8 8.0 8.2 9.5 9.9 11.1
1 1 0.76 1 1.8 1.8 1.5 1.5 1.4 1 1.2 0.9
The normalized values of Jdpnor at t = 0.5 together with Tc as function of dF are plotted in Fig. 3. It is useful to note that Tc(dF) is quite featureless and in particular does not show a dip (although it would have been useful to have one more sample at smaller thickness). Looking at Jdpnor (t = 0.5), and disregarding the two samples with deviating behavior of Jdp(t) (the samples with dF = 5.6 nm and 9.5 nm are indicated as black solid squares in Fig. 3) the asymptotic value for Jdpnor is 2.0 ± 0.4×1010 A/m2. However, the correspondent Jdpnor values at dF = 3.5 nm and 5 nm are 0.5×1010 A/m2 and 0.8×1010 A/m2, significantly lower that the asymptotic value. It appears therefore that Jdpnor (t = 0.5) shows a dip in the region of dF around 40 nm, where Tc(dF) flattens. In order to understand these data, we analyze the behavior of Tc(dF) by using the proximity effect model for S/F bilayers as developed by Fominov et al. [5]. In this model, the parameters used are the following. Firstly there are what can be called the diffusion lengths ξS = (ħDS/(2πkBTcS))1/2 and ξF* = (ħDF/(2πkBTcS))1/2, with DS,F the different confusion constants.
A. Yu. Rusanov, J. Aarts, M. Aprili
12
10
t = 0.5
10
Jdp [10 A/m 2]
10
9 8 7
8
6
6
5 4
4
T [K]
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3 2
2
1
0
0
20
40
60
80
100
0 120
dF [A]
Fig. 3. Depairing current density Jdpnor (open squares) at t = 0.5 and critical temperature Tc (open circles) of Nb/PdNi bilayers as a function of ferromagnet thickness dF. Black solid squares indicate the samples with the deviating values of Jdp. The black solid line shows the fitted Tc(dF) curve. Dashed lines are guides to the eye.
Of course, ξS is nothing else than the superconducting coherence length, and related to the Ginzburg-Landau coherence length by ξS = 2ξGL(0)/π. It can be directly determined by the temperature dependence of the upper critical field, and for our Nb films we use ξS = 8 nm. Note that ξF* is not exactly the same as what is often called the coherence length in the ferromagnet. It does not contain the exchange energy Eex, which in the model is a separate parameter, and therefore does not by itself set the scale for the oscillations of the order parameter. We estimate its value from a Fermi velocity vF for PdNi of 2×105 m/s [11] and a mean free path ℓmfp of the order of the film thickness, around 4 nm. Together with TcS = 7.5 K this yields DF = 2.7×10-4 m2/s and ξF* = 6.6 nm. The other parameters are the specific resistances ρS,F, given before, the bulk transition temperature TcS, and the interface transparency parameter γB.
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The free parameters are therefore Eex and γB, and we find a good fit for the values γB = 0.45 and Eex = 150 K. The fit is shown in Fig. 3. Note that it shows a very slight minimum around 4 nm, emphasizing that the minimum is almost absent even for modest values of γB. From the value for Eex we can make the following estimate. Since Eex >> kBT, the characteristic oscillation length for the inhomogeneous order parameter in the F-layer can be written as λex = 2πξF = 2π(ħDF/Eex)1/2. It follows that λex ≈ 25 nm, and ξF ≈ 3.9 nm. From this, a possible minimum in Tc(dF) could be expected around 0.7λex/4 [5], which is around dF = 4.3 nm. The value for λex can also be compared to the data of Josephson junctions of Nb/Pd89Ni11/I/Nb (with I an insulator), where the 0-π transition should be found at 3λex/8 = 9 nm. The actual transition is found around 7 nm [3], a little bit lower, but still in a reasonable agreement. We conclude therefore that the dip in the behavior of Jdp(dF) at low temperatures is due to the presence of an inhomogeneous order parameter in the F-layer, which is sensed by Jdp but not by Tc.
Conclusion We measured Tc(dF) on Nb/Pd0.81Ni0.11 bilayers together with the depairing current. Close to 1/4 of the period of Ψ(x) in the F-layer Jdp(dF) shows a clear minimum, in contrast to the absence of the correspondent effect in Tc(dF). We believe, that this difference is caused by a higher sensitivity of the depairing current to the order parameter changes than Tc. While the critical temperature senses only the highest value of the ∆ in S, depairing currents are more localized tools and shows the complete behavior of the superconducting order parameter ∆(x) across the whole superconductor.
Acknowledgments This work is part of the research program of the ‘Stichting voor Fundamenteel Onderzoek der Materie (FOM)’, which is financially supported by the ‘Nederlandse Organisatie voor Wetenschappelijk Onderzoek (NWO)’. Support of the ESF program ‘Pi-shift’ is gratefully acknowledged, as well as discussions with A. A. Golubov and Ya. V. Fominov, who also allowed the use of his computer code.
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References 1. Ryazanov VV, Oboznov VA, Rusanov AYu, Veretennikov AV, Golubov AA, Aarts J (2001). Phys Rev Lett 86:2427 2. Kontos T, Aprili M, Lesueur J, Grison X (2001) Phys Rev Lett 86:304 3. Kontos T, Aprili M, Lesueur J, Genet F, Stephanidis B, Boursier R (2002). Phys Rev Lett 89:137007 4. Radović Z, Ledvij M, Dobrosavljević-Grujić L, Buzdin AI, Clem JR (1991). Phys Rev B 4:759 5. Fominov YaV, Chtchelkatchev NM, Golubov AA (2002). Phys Rev B 66:014507 6. Mühge Th, Theis-Bröhl K., Westerholt K, Zabel H, Garif’yanov NN, Goryunov YuV, Garifullin IA, Khaliullin GG (1998). Phys Rev B 57:5071 7. Ryazanov VV, Oboznov VA, Prokof’ev AS, Dubonos SV (2003). JETP Lett 77:43 8. Sidorenko AS, Zdravkov VI, Prepelitsa AA, Helbig C, Luo Y, Gsell S, Schreck M, Klimm S, Horn S, Tagirov LR, Tidecks R (2003). Ann Phys 12:37 9. Geers JME, Hesselberth MBS, Aarts J, Golubov AA (2001). Phys Rev B 64:094506 10. Rusanov AYu, Hesselberth MBS, Aarts J (2004). Phys Rev B 70:024510 11. Dumoulin L, Nedellec P, Chaikin PM (1981). Phys Rev Lett 47:208
Superconductor-Ferromagnet Heterostructures
A. I. Buzdin1, M. Fauré1, M. Houzet 2 1
Université Bordeaux I , CPMOH, 33400 Talence Cedex, France Commissariat à l'Énergie Atomique, DSM, Département de Recherche Fondamentale sur la Matière Condensée, SPSMS, F-38054 Grenoble, France
2
Abstract:
We provide a general description of superconductor/ferromagnet structures and study the evolution of the critical temperature, critical current and magnetization variation with the thickness of the ferromagnetic layer. Special attention is given to the influence of the magnetic scattering on the properties of such hybrid systems.
Keywords: magnetism, superconductivity, magnetic impurities
Introduction Ferromagnetism and singlet superconductivity are two antagonistic orderings. Indeed, a magnetic field can destroy conventional superconductivity via the orbital effect and via the paramagnetic effect and therefore, they usually try to avoid each other. The competition between these two orderings has always been a subject of great interest for both theoretical and experimental physics. The question of their coexistence was first addressed by Ginzburg as soon as 1956 [1]. In fact, it was demonstrated later that a modulated magnetic structure (cryptoferromagnetic state [2]) appears instead of ferromagnetism in a singlet superconductor. As a review of the problem of singlet superconductivity and magnetism coexistence, see [3]. Moreover, Larkin and Ovchinnikov, and Fulde and Ferrell demonstrated in 1964 that the superconductivity of a pure ferromagnetic superconductor may be non uniform at low temperature ([4] and [5]). It is unfortunately not easy to 197 R. Gross et al. (eds.), Nanoscale Devices - Fundamentals and Applications, 197–224. © 2006 Springer. Printed in the Netherlands.
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verify this prediction on experiment because of the incompatibility of ferromagnetism and superconductivity in bulk materials. However, their interplay may be studied when the two orderings are spatially separated, which is obtained in artificially made superconductor/ferromagnet (S/F) structures. These hybrid systems give us the unique possibility to study the properties of superconducting electrons under the influence of a huge exchange field acting on the electron spins. In such systems, Cooper pairs can penetrate into the F layer and induce superconductivity there, which is the so called proximity effect. In addition, it is possible to study the interplay between superconductivity and magnetism in a controlled manner, since varying the layer thicknesses changes the relative strength of the two competing phenomena. Note that almost all the interesting effects related to superconductivity and magnetism interplay in S/F structures occur at a nanoscopic scale. The observation of these effects became possible only a few years ago thanks to recent progress in the preparation of high-quality hybrid layers. The most striking feature of S/F systems is the highly non monotonic behavior of the critical temperature Tc and the critical current I c with the thickness of the ferromagnetic layer. In S/F/S junctions and S/F multilayers, this is related to 0- π transitions, which are studied in the present work. Another interesting manifestation of the proximity effect is the variation of the magnetization in both types of layers. A general review of S/F structures was reported in [6]; see also [7]. Here, we would like to concentrate on the influence of magnetic scattering on the properties of S/F systems. Although the oscillatory behavior of Tc and I c is well known, a noticeable difference between theoretical calculations and experimental results still exists. It could be understood by the introduction of an additional scattering mechanism in theoretical descriptions. Indeed, magnetic impurities, spin-wave or non stoichiometric lattices can play an important role as the spin-flip process has dramatic consequences on superconductivity (on the contrary of non magnetic impurities which have very little impact). More precisely, the pair-breaking effect induced by magnetic impurities leads to the decrease of the decay length of Tc and I c and to the increase of the oscillations period. Note that the question of the spin flip scattering was firstly addressed by Tagirov [8], while Demler et al. studied the spin-orbit scattering role [9]. In the present work, we study the critical temperature and current as well as magnetization in S/F bilayers and multilayers in the framework of the Usadel equations and report on the spin-flip scattering influence.
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Proximity Effect in S/F Systems
Generalized Ginzburg-Landau functional The physics of the proximity effect can be qualitatively described by a standard Ginzburg-Landau functional in Superconductor/Normal (S/N) metal structures (see, for example [10]): r 2 b 4 2 F = a ψ + γ ∇ψ + ψ , 2
(1)
where ψ is the superconducting order parameter, and the coefficient a ∝ (T − Tc ) vanishes at the transition temperature Tc . It should be noted that expression (1) is valid only if the effect of the exchange field h may be neglected. In the F layer, the functional has to be modified. The coefficients a , b , and γ are dependent on h . In particular, the gradient term coefficient γ becomes negative for a relatively large value of h / T > 1 . In that case, it is then necessary to add a higher order derivative term. Finally, the generalized Ginzburg-Landau expansion may be written as following: r 2 η ( h, T ) r 2 2 b ( h , T ) 4 2 ∇ψ + ψ . FG = a( h, T ) ψ + γ (h, T ) ∇ψ + 2 2
(2)
The critical temperature of the second order phase transition into a superconducting state may be found from the solution of the linear equation for the superconducting order parameter aψ − γ∆ψ +
η 2
∆ 2ψ = 0 .
(3)
rr If we seek for a non-uniform solution ψ = ψ 0 exp(iqr ) , the r corresponding critical temperature depends on the wave-vector q and is
given by the expression a = −γ q 2 − η q 4 / 2 . The coefficient a can be written as a = α (T − Tcu (h)) , where Tcu (h) is the critical temperature of the transition into the uniform superconducting state. In a standard situation, the gradient term in the Ginzburg-Landau functional is positive, γ > 0 , and the highest transition temperature coincides with Tcu (h) ; it is realized for the uniform state with q = 0 .
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However, when γ < 0 , the maximum critical temperature corresponds to the finite value of the modulation vector q02 = −γ / η and the corresponding transition temperature into the non-uniform state Tci (h) is given by a = α (Tci − Tcu ) = γ 2 /(2η ) . It is higher than the critical temperature Tcu of the uniform state. Therefore, we see that the non uniform state appearance, called FFLO state, may simply be interpreted as a change of the sign of the gradient term in the Ginzburg-Landau functional.
Damped oscillatory decay of the Cooper pair wave function in ferromagnets To get some idea about the peculiarity of the proximity effect in S/F structures, we may start with the description based on the generalized Ginzburg-Landau functional (2). Such an approach is adequate for a small wave-vector modulation case, otherwise a microscopic theory must be used. This situation corresponds to a very weak ferromagnet with an extremely small exchange field h ≈ Tc . We address the question of the proximity effect for a weak ferromagnet described by the generalized GinzburgLandau functional (2). More precisely, we consider the decay of the order parameter in the normal phase, i.e. at T > Tci assuming that our system is in contact with another superconductor with a higher critical temperature, and the x axis is chosen perpendicular to the interface. The induced superconductivity is weak and we may use the linearized equation for the order parameter (3), which is written for our geometry as aψ − γ
∂ 2ψ η ∂ 4ψ + =0. ∂x 2 2 ∂x 4
(4)
The solutions of this equation in the normal phase are of the type
ψ = ψ 0 exp(kx) , with a complex wave-vector k = k1 + ik2 , and k12 =
γ T − Tci 1+ − 1 , Tci − Tcu 2η
(5)
k22 =
γ T − Tci 1+ + 1 . 2η Tci − Tcu
(6)
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If we choose the gauge with the real order parameter in the superconductor, then the solution for the decaying order parameter in the ferromagnet is also real
ψ ∝ exp ( −k1 x ) cos ( k2 x ) ,
(7)
where the choice of the root for k corresponds to k1 > 0 . So the decay of the order parameter is accompanied by oscillations (Fig. 1b), which is the characteristic feature of the proximity effect in the considered system. Let us compare this behavior with the standard proximity effect [11] described by the linearized Ginzburg-Landau equation for the order parameter aψ − γ
∂ 2ψ =0, ∂x 2
(8)
with γ > 0 . In such a case Tc simply coincides with Tcu and the decaying solution is ψ = ψ 0 exp(− x / ξ n ) where the coherence length ξ n = γ / a (Fig. 1a). The oscillations of the superconducting order parameter in S/F systems may also be understood when considering a Cooper pair picture. Indeed, a Cooper pair is usually formed by two electrons with opposite momenta k F and − k F and opposite spins. The resulting momentum of the Cooper pair k F + (− k F ) = 0 . When a magnetic field is applied, because of the Zeeman’s splitting, the Fermi momentum of the electron with the spin parallel to the field (up) will shift from k F to k1 = k F + δ k , where δ k = µ B h υ F , υ F being the Fermi velocity and h the exchange field in the F layer. Similarly, the Fermi momentum of the down spin electron will shift from − k F to k2 = − k F + δ k (see Fig. 2). Then, the resulting momentum of the Cooper
pair is k1 + k2 = 2δ k ≠ 0 , which implies the space modulation of the superconducting order parameter with a resulting wave-vector 2δ k .
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Fig. 1. Schematic behaviour of the superconducting order parameter near the interface (a) superconductor/normal metal and (b) superconductor/ferromagnet.
Fig. 2. Energy band of a 1D superconductor near the Fermi surface.
Therefore, we can see again that ψ does not only decays into the F layer, but it is also space modulated, which gives the following behaviour of the pair wave function:
ψ ∝ exp ( − x ξ f 1 ) cos ( x ξ f 2 ) ,
(9)
where ξ f 1 ≡ 1/ k1 and ξ f 2 ≡ 1/ k2 are the decaying and oscillations length of the superconducting correlations in the F layer (see Fig. 1(b)) while it only decays in S/N systems.
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Consequences of the superconducting order parameter oscillations in S/F systems The damped oscillatory behavior of the superconducting order parameter in ferromagnets may produce commensurable effects between the period of the order parameter oscillation (given by ξ f ) and the thickness of a F layer. This results in a striking non-monotonic dependence of the superconducting transition temperature on the F layer thickness in S/F multilayers and bilayers. Indeed, for a F layer thickness smaller than ξ f , the pair wave function in the F layer changes a little and the superconducting order parameter in the adjacent S layers must be the same. The phase difference between the superconducting order parameters in the S layers is zero, which is the so-called 0-phase. On the other hand, if the F layer thickness becomes of the order of ξ f , the pair wave function may go trough zero at the center of the F layer providing the state with the opposite sign (or π shift of the phase) of the superconducting order parameter in the adjacent S layers, which is the π phase. The increase of the thickness of the F layers may provoke subsequent transitions from 0- to π -phases, results in a very special dependence of the critical temperature on the F layer thickness. For S/F bilayers, the transitions between 0 and π -phases are impossible. The commensurable effect between ξ f and the F layer thickness nevertheless leads to the non-monotonous dependence of Tc on the F layer thickness due to the commensurability effect between the period of superconducting wave function oscillation and the thickness of the F layer. The first experimental indications on the non-monotonous variation of Tc versus the thickness of the F layer was obtained by Wong et al. [12] for V/Fe superlattices. However, the strong pair-breaking influence of the ferromagnet and the nanoscopic range of the oscillations period complicated the observation of this effect. Advances in thin film processing techniques were therefore crucial for the study of this subtle phenomenon. The predicted oscillatory type dependence of the critical temperature was finally clearly observed in 1995 in Nb/Gd [13] and then in other systems (for more detail, see [6]). Another consequence of the superconducting order parameter modulation is the damped oscillatory behavior of the critical current of a S/F/S junction. A S/F/S sandwich realizes a Josephson junction in which the weak link between the two superconductors is ensured by the ferromagnetic layer.
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The supercurrent I s (ϕ ) flowing across the structure can be expressed as I s (ϕ ) = I c sin(ϕ ) , where I c is the critical current and ϕ stands for the phase difference between the two superconducting layers. A standard junction has at equilibrium I c > 0 and ϕ = 0 , and therefore, no current
exists. It may appear however that I c becomes negative, which implies that the equilibrium phase difference is ϕ = π and the ground state undergoes a π phase shift, namely the π junction. The first unambiguous experimental evidence of the 0- π transition with the temperature variation via critical current measurements was observed by Ryazanov et al. in 2001 [14]. Sellier et al. recently obtained a similar result [15], while Kontos et al. [16] observed the damped oscillations of the critical current as a function of the F layer thickness in Nb/Al/Al 2 O 3 /PdNi/Nb junctions.
Theoretical Framework
Usadel equations Real ferromagnets present rather large exchange fields and the GinzburgLandau functional is not an adequate approach for S/F systems description. A microscopic theory has to be used to theoretically describe the proximity effect in such structures. The most convenient schemes are the use of the Boboliubov-de Gennes equations [10] or the Green's functions in the framework of the quasiclassical Eilenberger [17] or Usadel equations [18]. If the electron scattering mean free path l is small (which is usually the case in S/F systems), the most natural approach is to choose the Usadel equations for the Green's functions averaged over the Fermi surface. The normal Green’s function will be noted Gs ( f ) in the S(F) layer, while the anomalous Green’s function is Fs ( f ) . In the general case, magnetic and spin-orbit scatterings mix up the up and down spins states. Choosing the spin quantization axis along the direction of the exchange field, and introducing the Green functions G1 ~ ψ ↑ψ ↑+ and F1 ~ ψ ↑ψ ↓
( G2 and F2 for the opposite spin orientations), we may
write the nonlinear Usadel equation in the following form
Superconductor-Ferromagnet Heterostructures
1 2 ( F1 ∇ 2G1 ) + ω + ih + + G1 F1 + τ z τ x 1 1 G1 ( F2 − F1 ) − τ x τ so
1 1 + F1 (G2 − G1 ) + τ x τ so
= ∆,
205
(10)
where τ so−1 is the spin-orbit scattering rate while the magnetic scattering rates are τ z−1 = τ 2−1 S z2 / S 2
and τ x−1 = τ 2−1 S x2 / S 2 . The rate τ 2−1 is
proportional to the square of the exchange interaction potential, and we follow the notations of work [19]. For the spatially uniform case, equation (10) naturally gives the same result as the microscopic approach developed in [20, 19]. The ferromagnets that are usually used in S/F heterostructures contain elements with relatively small atomic numbers. Therefore, the spin-orbit scattering may be neglected, and henceforth, τ so−1 = 0 . The influence of the spin-orbit scattering on the critical temperature of S/F bilayers was studied in [9]. In addition, the uniaxial anisotropy strongly suppresses the perpendicular fluctuations of the local exchange field, that is τ x−1 → 0 . In such a case, the Usadel equation is simplified and may be written in the F layer as −
Df 2
(G f ∇ 2 F f − Ff ∇ 2G f ) + (ω + ih + τ m−1G f ) Ff = 0 ,
(11)
where τ m−1 = τ 2−1 S z2 / S 2 may be considered as a phenomenological parameter. Here, there is no spin mixing scattering anymore. Therefore, there is no need to retain the spin indexes 1, 2. In that case, we may use the parametrization of the normal and anomalous Green's functions G = cos Θ( x) and F = sin Θ( x) when the pair potential can be chosen real. For ω > 0 , the Usadel equations are
ω sin Θ s −
Ds ∂ 2 Θ s = ∆ ( x) in the S layer and 2 ∂x 2
(12)
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A. I. Buzdin, M. Fauré, M. Houzet
cos Θ f ω + ih + τm
D f ∂ 2Θ f = 0 in the F layer. sin Θ f − 2 ∂x 2
(13)
Note that the Usadel equations are nonlinear but may be linearized over the pair potential ∆ ( x) near Tc or when the S/F interface has low transparency. Oscillating Cooper pair wave function For a semi-infinite bilayer without magnetic impurities, the decaying solution for Ff is 1+ i Ff ( x, ω > 0) = A exp − x , ξ f
(14)
where ξ f = D f h is the characteristic length of the superconducting correlations decay (with oscillations) in F- layer. In real ferromagnets, the exchange field is very large compared with the superconducting order parameter (h >> Tc ) . Consequently, ξ f is much smaller than the superconducting coherence length ξ s = Ds (2π Tc ) . The constant A is determined by the boundary conditions at the S/F interface. In a ferromagnet, the role of the Cooper pair wave function role is played by ψ : x x cos ξ f ξf
ψ ~ ∑ F ( x, ω ) ~ ∆ exp − ω
.
(15)
Thus, the damping oscillatory behavior of the order parameter is retrieved. It can be seen from this microscopic approach that the decay length ξ f 1 and the oscillation period ξ f 2 are quite the same for a ferromagnet in the dirty limit with no magnetic impurities. In presence of magnetic scattering, the decaying solution has the form Ff ( x, ω > 0 ) = A exp(−(k1 + ik2 ) x) , which implies
(16)
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207
x x cos , ξf1 ξ f 2
ψ ~ ∆ exp − where in the limit h >> Tc k1 =
k1 =
1
ξf 1
ξf
1+ α 2 + α =
1+ α 2 −α =
1
ξf1 1
ξf2
,
(17)
,
(18)
1 . hτ m If the spin-flip scattering time becomes relatively small α >> 1 , i.e. the magnetic impurities concentration is not negligible, the decaying length can become substantially smaller than the oscillating length, see Fig.3. This results in the much stronger decrease of the critical temperature in multilayers and critical current in S/F/S junctions with the increase of the F layer thickness.
with α =
Fig. 3. Schematic evolution of ψ without magnetic scattering (solid line) and with magnetic scattering (dashed line). Note that the oscillations do not disappear in the presence of magnetic scattering, but become very small.
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A. I. Buzdin, M. Fauré, M. Houzet
Oscillatory Superconducting Transition Temperature in S/F Systems
Theoretical description of S/F multilayers We consider a S/F multilayered system with a thickness 2d F of the F layers and 2d S of the S layers, see Fig. 4 (this case is equivalent to a S/F bilayer of thicknesses d F and d S respectively). The critical temperature is determined by the self consistent equation for the superconducting gap: ∆ ln
∆ Tc* + π Tc* ∑ − Fs ( x, ω ) = 0 , Tc ω ω
(19)
where Tc is the bare transition temperature of the superconducting layer in the absence of the proximity effect.
F
F
S
− ds 0
ds
d s + 2d f
x
Fig. 4. Geometry of the studied multilayered system.
Consequently, the anomalous Green’s function in the S layer has to be determined to find the critical temperature. It can be deduced from the anomalous Green’s function in the F layer, and the boundary conditions at the S/F interface [21]:
Superconductor-Ferromagnet Heterostructures
∂FS σ ∂Ff = n ∂x int erface σ s ∂x
, int erface
∂Ff FS ( 0 ) = Ff ( 0 ) − ξ n γ B ∂x
209
(20)
int erface
(21)
with σ n (σ s ) the conductivity of the F(S) layer, ξ n = D f ( 2π Tc ) and
γ B = RBσ n ξ n is related to the S/F resistance per unit area RB . The anomalous Green’s function in the F layer Ff is the solution of the linearized Usadel equation. Therefore, taking into account the symmetry of the system, it can be written as Ff ( x, ω ) = A cosh k ( x − d S − d f ) in the 0 phase and
(22)
Ff ( x, ω ) = A sinh k ( x − d S − d f ) in the π phase where
(23)
k = k1 + ik2 =
1
ξf
i + α and ξ f =
Df h
.
Finally, when considering that the superconducting layer is thin, i.e. d S << ξ S , FS varies a little in the S layer and is β FS ( x, ω ) = F0 1 − ω x 2 , 2
(24)
where F0 is the value of the anomalous Green's function at the center of the S layer: F0 =
∆ . ω + τ s−1
(25)
The parameter τ s−1 a pair-breaking parameter, that plays the same role as the corresponding parameter in the Abrikosov-Gorkov theory of superconductivity with magnetic impurities [22]. Note however that in our case, it can be complex. It is written as
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A. I. Buzdin, M. Fauré, M. Houzet
τ s−,01 (ω > 0 ) = τ 0−1
q tanh(qd% f ) , in the 0 phase γ%q tanh(qd% ) + 1
(26)
q coth(qd% f ) in the π phase. γ% q coth(qd% ) + 1
(27)
f
τ s−,1π (ω > 0 ) = τ 0−1
f
The parameter τ 0−1 =
Ds σ n 1 ξ while γ% = n γ B , q = kξ f and d% f = d f / ξ f . 2d S σ s ξ f ξf
Next, the critical temperature Tc* is directly determined from the following equation 1 T* 1 1 ln c = Ψ − Re Ψ + . * 2 2 2π Tc τ s Tc
(28)
Critical temperature versus F layer thickness In Fig. 5, the critical temperature Tc* has been plotted for two values of the magnetic scattering time for transparent interfaces. It can be deduced that the magnetic scattering decreases the damping length and increases the oscillation period. The decrease of the decay length obviously makes the observation of the oscillations more difficult.
Fig. 5. Influence of the spin-flip scattering on the evolution of the critical temperature: (a) τ 0 = 11/ h and α = 0 ; (b) τ 0 = 11/ h and α = 1/ 2 .
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211
Moreover, the intriguing evolution of Tc* in bilayers with the finite interface transparency must be underlined (see Fig. 6). First, if α = 0 , there is no magnetic scattering and it could intuitively be believed that, the higher the barrier, the less is the influence of the proximity effect and therefore, Tc* (γ% >> 1) > Tc* (γ% ~ 1) . However, it can be seen from Fig. 6 that the critical temperature is a decreasing function of the barrier γ% for a small thickness of the F layer. This counter-intuitive behavior can be qualitatively understood as following. The probability for a Cooper pair to leave the S layer is smaller for a low transparent interface (γ% >> 1) . Nevertheless, the probability for this pair to come back again in the S layer is much higher for a transparent interface. Indeed, when the F layer is thin, the reflection of the Cooper pair at the other interface of the F layer allows the pair to cross again the first interface, which is easier when γ% is small. Consequently, the staying time in the F layer increases with the barrier, and when this time becomes bigger than the coherence time of the Cooper pair, the pair is destroyed, leading to a weakened superconductivity and therefore, the critical temperature decreases with the barrier. On the other hand, if d f is not so small, the Cooper pair is hardly reflected by the external interface of the F layer whatever the value of γ% is and the critical temperature is expected to increase with the barrier.
Fig. 6. Influence of the interface transparency on the evolution of the critical temperature.
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A. I. Buzdin, M. Fauré, M. Houzet
Let us now consider briefly the general case, with spin-orbit and perpendicular spin flip scattering. An additional parameter has to be introduced, namely 1 1
1
1 1
2
α⊥ = ( − ) , h τ x τ so and the parameter α becomes
α = ( + ). h τz τx The Usadel equation (10) can be linearized and the complex pair breaking parameter may be determined. The results are as following. In the 0 phase, q tanh qd% f in expression (26) is replaced by
(
)
q* tanh(q* d% f ) − q tanh(qd% f ) % , q tanh(qd f ) + β 1 + ξ n γ B q* tanh(q*d% f ) β+ 1 + ξ γ q tanh(qd% ) n
B
(29)
f
where q 2 = 2(
ω h
+ α − α ⊥ + i 1 − α ⊥2 ) ,
(30)
and
β = −α ⊥
α ⊥ − i 1 − α ⊥2 1 + 1 − α ⊥2
.
(31)
In the π phase, tanh(qd% f ) is replaced by coth(qd% f ) in expression (29). Therefore, the influence of ‘perpendicular’ spin-flip scattering and spinorbit scattering is quite similar to the influence of ‘parallel’ spin-flip processes, in the sense that it also implies the decrease of the decaying length and the increase of the oscillations period. However, a special situation arises when α ⊥ > 1 . Then, the oscillations of the Cooper pair wave function are completely destroyed. Similar conclusion for spin-orbit mechanism was obtained in [9]. In fact, the influence of the ‘perpendicular’ magnetic scattering is analogous to the spin-orbit scattering. Probably the role of ‘perpendicular’ spin-flip or spin-orbit scatterings is important for the understanding of experimental results
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213
where no oscillations of the critical temperature were detected. Besides, note that the critical temperature oscillations can not disappear when there is only ‘parallel’ spin flip.
Behavior of the critical current The constant I c in the relation I s = I c sin ϕ is negative in the π phase while it is positive in the 0 phase. Thus, the transition from the 0 to π state may be considered as a change of the sign of the critical current, though the experimentally measured critical current is always positive and is equal to I c . The 0- π transition occurs at each minimum of I c . The so called π junction' was first predicted for S/F/S structures by Buzdin et al. in 1982 in the clean limit [23], and later in the more realistic case of the diffusive limit [24]. Although the critical current behavior was a subject of intensive theoretical study, the experimental observation of the π state was difficult to obtain because the characteristic thickness of the F layer corresponding to the crossover from 0 to π state ξ f is rather small. The first experimental evidence was finally reported by Ryazanov et al. in 2001 [14] as a function of the temperature and later by Kontos et al. as a function of the ferromagnetic layer. The experimental data that are now available [14 - 16] on S/F/S junctions can be qualitatively understood in the framework of the existing approach. However, further development of the theory is needed for a more complete description. Below, we consider in more detail the influence of the magnetic scattering on the properties of S/F/S junctions. Experimental hints on the presence of relatively strong spin-flip effects were obtained in [15, 25]. The studied geometry is a S/F/S junction of a thickness 2d f of the F layer and large superconducting electrodes (see Fig. 7). The dirty limit conditions are supposed to be fulfilled. Therefore, the Usadel equations may be used. The supercurrent is determined by the following expression ∞ d % d I s (ϕ ) = ieN (0) D f π TS ∑ Ff F f − F% f Ff , dx dx −∞
(32)
where F% f ( x, h ) = F f* ( x, − h ) , S is the area of the cross section of the junction and N (0) is the electron density of states per one spin projection.
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A. I. Buzdin, M. Fauré, M. Houzet
F S
F S
S F
− d fs 0
dfsf
d s + 2d f
x
Fig. 7. Geometry of the studied S/F/S Josephson junction.
Linearized Usadel equations
In the limit T → Tc , the amplitude of Ff is small and the linearized Usadel equations are valid. Using the rigid boundary conditions, which are valid if σ n ξ s / σ f ξ f << max(γ B ,1) , the critical current may be easily calculated Ic =
eN (0) D f π TS
ξf
∞ ∆2 2q , Re ∑ 2 2 −∞ ω + ∆ sinh(2qd% )(1 + γ% 2 q 2 ) + 2qγ% cosh(2qd% ) f f
(33)
where γ% = γ Bξ n / ξ f , and q = a + ib . Expression (34) takes into account the magnetic scattering and a =
1 + α 2 + α and b =
1 + α 2 − α , with
α = 1/(τ m h) . This formula can be generalized to the case when the interface barriers are different (denoted γ B1 and γ B 2 ). Then, γ% 2 must be replaced by
γ B1γ B 2 (ξ n / ξ f ) 2 and γ% by
γ B1 + γ B 2 ξ n . 2 ξf
Besides, this expression is also valid at all temperatures if γ B >> 1 , with the substitution γ% → γ% / Gs . Gs is the normal Green function in the superconducting electrodes Gs = ω / ω 2 + ∆ 2 . The evolution of the critical current for different values of the barrier transparency is given in Fig. 8.
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215
Fig. 8. Evolution of the critical temperature with the thickness of the F layer for different interface transparencies.
Besides, if γ% >> 1 , the previous expression may be simplified and the critical current becomes Ic =
∆ 2 eN (0) D f π 3TS
ξf
2 Re 2 γ% sinh(2qd% f )q
.
(34)
When there is no magnetic scattering, the transition into the π phase 2∆ (0) h ln occurs in the limit T → 0 at d% cf = , see [26] and [6] for h ∆ (0) further detail. It can be seen from the previous expression that in the absence of magnetic scattering, the exchange field determines the critical thickness. Therefore, when α = 0 , the critical thickness may be much smaller than ξf .
In presence of magnetic scattering, if Tc / h < α < 1, the first minimum is achieved at d% c = 3α . Note that the experimental measuring of the critical f
thickness allows a direct determination of the magnetic scattering.
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A. I. Buzdin, M. Fauré, M. Houzet
When the spin flip scattering controls the system, i. e. α > 1 , then the ψ + nπ , where ψ is defined subsequent 0- π transitions occur at d% cf = 2b a by tanψ = . In that case, the critical thickness d cf is larger than ξ f . b If one interface is completely transparent γ B1 = 0 and the other interface has a large barrier γ B 2 >> 1 , then we obtain the following expression for the critical current near Tc Ic =
∆ 2 eN (0) D f π 3TS
ξf
2 Re . % γ B 2 cosh(2qd f )
(35)
π It vanishes for d% cf = . Thus, a vanishing barrier interface tends to 4b increase the critical thickness, as can be seen from Fig. 8. It can be seen that the critical thickness increases with the increase of the spin flip rate. At the same time, the magnetic scattering strongly increases the damping of I c with the increase of the F layer thickness. Consequently, the magnetic scattering role is quite controversial for the experimental observation of the 0- π transitions. Indeed, even though the increase of the critical thickness leads to an easier observation of the transitions, the decrease of the decay length is on the contrary quite harmful for experiments. Transparent interfaces
For a transparent interface, the linearized Usadel equation can not be used at low temperature and the complete nonlinear equation must be solved. Introducing the dimensionless parameters ω% = ω h , α = 1 (τ m h) and y = x d f , it becomes
−
2 1 ∂ Θf + (ω% + i + α cos Θ f )sin Θ f = 0 in the F layer. 2 ∂y 2
(36)
A S/F/S junction presents two interfaces. In the limit of relatively large thickness of the F layer d f > ξ f , the decay of the Cooper pairs wave function occurs independently near each interface. It can therefore be treated separately enough to consider the behavior of the anomalous
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217
Green’s function near each S/F interface, assuming that the F layer thickness is infinite - one interface. Although Usadel equation (36) is nonlinear, one may find its exact solution. Using the first integral of (36) with the following boundary conditions: ∂Θ f Θ f ( y → ∞) = ∂y
= 0 , ∞
we obtain the following relation g=
where
1 − p 2 sin 2 (Θ 2) + cos(Θ 2) 1 − p 2 sin 2 (Θ 2) − cos(Θ 2)
,
(37)
p 2 = α (ω% + i + α ) .
q = 2(ω% + i + α ) and
The
function
g = g 0 exp(−2qy ) , where the constant g 0 is determined by the continuity of the Green’s functions at the interface. In the case of the rigid boundary conditions, the inverse proximity effect may be neglected. As a result, g0 =
(1 − p 2 )U ( n) (1 − k 2 )U (n) + 1 + 1
2
and U (n) =
∆2 , (ω + Ω) 2
where Ω = ω 2 + ∆ 2 . The anomalous Green’s function at the center of the F layer may be taken as the superposition of the two decaying Ff functions. As a result, the current-phase relation is sinusoidal and the critical current becomes
I c = 64
eN ( 0 ) D f π TS
ξf
∞ U ( n ) q exp −2qd% f , Re ∑ 2 −∞ (1 − p 2 )U ( n ) + 1 + 1
(
)
(38)
where d% f = d f / ξ f . This expression generalizes the corresponding formula from Ref. [24] to the case of finite magnetic scattering. The critical current is proportional to the small factor exp(−2qd% f ) . The terms neglected in our approach are much smaller and are of the order of exp(−4qd% f ) . Therefore, they give a tiny second harmonic term in the current-phase relation.
218
A. I. Buzdin, M. Fauré, M. Houzet
When T → Tc , expression (36) may be simplified and we have Ic =
4π SeN (0) D f ∆ 2
where tanψ =
Tc ξ f
b exp(−2ad% f )sin(2 yd% f + ψ ) , cosψ
(39)
a and q = a + ib . Note that if the magnetic scattering is b
negligible α → 0 , ψ =
π
while if α > 1 ,
π
<ψ <
π
. 4 4 2 The two characteristic lengths, namely the decay length and the oscillations period, appear in expression (39). Accordingly, this formula should be very helpful to fit experimental data. Recent systematic studies of the thickness dependence of the critical current in junctions with Cu x Ni 1− x alloy as an F layer [25] revealed a very
strong variation of I c with the F layer thickness. Indeed, a five orders of magnitude change of the critical current was observed in the thickness interval (12-26) nm. The magnetic scattering effect - inherent to all the ferromagnetic alloys- is probably at the origin of this behavior. Besides, the presence of a rather strong magnetic scattering in Cu x Ni 1− x alloy S/F/S junctions was also noted in [15, 26, 27]. The theoretical fit based on Eq. (39) shows good agreement with the experimental data [28] if the spin-flip scattering is taken into account (see Fig. 9).
π
Fig. 9. Critical current of Cu 0.47 Ni 0.53 junctions as a function of the F layer thickness (Ryazanov et al., 2005). Two 0- π transitions are revealed. For this fit, the choice for the parameters was: α = 1.33 and ξ f = 2.4 nm . The insert shows the temperature mediated 0- π transition for a F layer thickness of 11nm .
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219
In presence of ‘perpendicular’ magnetic scatterings, up and down spins states are mixed and equation (10) has to be used. When the temperature is close to the critical temperature, this expression may be linearized. In that case, the critical current becomes ∞ ∆2 cosh(qd% f ) sinh(qd% f ) 2q * I c = I 0 Re ∑ 2 1 q q β β + + , * % * % −∞ Ω sinh(2qd% f )(1 + β 2 ) cosh( q d ) sinh( q d ) f f
where
I0 =
eN (0) D f π TS
ξf
, Ω 2 = ω 2 + ∆ 2 and β is defined by expression
(31). Note that when α ⊥ → 0 , the previous expression gives the critical current when there is only spin flip in the direction of the exchange field. The oscillations of the critical current (and the 0- π transitions) completely disappear if α ⊥ > 1 . Then, the experimental observation of these oscillations [14 - 16, 26] may be considered as an indication of the weakness of spin-orbit and ‘perpendicular’ spin-flip scatterings effects. Besides, this expression may be written in the same form as expression (39) which is used to fit experimental data. However, the expressions for the corresponding parameters are rather lengthy and therefore are not presented here.
Electronic magnetization variation in S/F systems The mutual influence of ferromagnetism and superconductivity has until now been considered through its consequences on the evolution of the superconducting critical temperature and critical current. Nevertheless, the proximity effect can also manifest itself by a modification of the electronic magnetization. Indeed, the presence of the ferromagnetic magnetization leads to a magnetization onset in the S layer. Similarly, the ferromagnetic layer may present a variation of its magnetization. It should be underlined that this topic has already been investigated in the dirty limit by Krivoruchko et al. [29] and Bergeret et al. [30] and in clean multilayered S/F structures by Halterman and Valls [31].
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A. I. Buzdin, M. Fauré, M. Houzet
Ferromagnet at the contact with a superconductor We consider a S/F system, with a thickness d s of the S layer and an infinite thickness of the F layer. The x axis is chosen to be perpendicular to the layer, with the origin at the vacuum-S layer interface. The magnetization of the F layer is M = MP + Ms ,
(40)
where M p is the magnetization due to the Pauli paramagnetism while M s stems from the superconductivity contribution. M s may be expressed as M s = iN (0)π T ∑ (G f ↑↑ − G f ↓↓ ) , ω
(41)
where G is the normal Green function in the F layer and may be deduced from Ff thanks to the normalization condition G 2f − Ff2 = 1 . For T ~ Tc , the anomalous Green’s function is small, and therefore, G f (ω ) ~ 1 − Ff2 / 2 . Also taking into account that Ff ↓↑ = Ff*↑↓ = F f* , the
magnetization may be presented as M s = iN (0)π T ∑ ( Ff*2 (ω > 0) −Ff2 (ω > 0)) . ω >0
(42)
Note that (42) gives in fact the part of the magnetization related to the presence of superconducting correlations. Since T → Tc , the linearized Usadel equation may be solved to find the anomalous Green’s function. If the interface is supposed to be transparent, i.e. γ B = 0 , calculations give the following final expression of the magnetization Ms = −2 N (0)π T ∆ 2 exp(−2 x / ξ f ) A(ω )sin(2 x / ξ f ) + B(ω ) cos(2 x / ξ f ) , (43)
with A(ω ) = ∑
1
ω (ω + 2τ 0−1 ) ,
((ω + τ ) + (τ ) ) 1 and B (ω ) = ∑ 2τ 0−1 (ω + τ 0−1 ) , −1 2 −1 2 2 ω ((ω + τ 0 ) + (τ 0 ) ) ω
where τ 0−1 =
−1 2 0
−1 2 2 0
Ds σ n 1 is the pair breaking parameter. 2d S σ s ξ f
In the case of very low S/F interface transparency, the magnetization becomes
Superconductor-Ferromagnet Heterostructures
M s = − N (0)π T ∆ 2 exp(−2 / ξ f ) cos(2 x / ξ f )∑ ω
1 (ω + τ / γ% ) 2 γ 2 −1 0
.
221
(44)
A qualitative evolution of the magnetization variation is given in Fig. 10.
Fig. 10. Qualitative oscillating evolution of the magnetization variation in the F layer
Magnetization variation in the S layer in a thin S/F bilayers Let us now consider a thin S/F bilayer (see Fig. 11) with d f << ξ f and d s << ξ s . Vacuum
0
S
F
ds
df
Vacuum
x
Fig. 11. Geometry of the studied system.
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The magnetization in the S layer may be easily determined in this case. For a transparent interface, it reads M s = −8 N (0)π T ∆ 2τ 0−1d% f ∑ ω
(
ω ω 2 + ( 2τ 0−1d% f )
)
2 2
.
(45)
Therefore, the inverse proximity effect leads to the appearance of a negative magnetization in the superconducting layer. Ref. [31] explains this fact quite simply. Namely, although the Cooper pairs inside the F layer do not contribute to a magnetization onset, a Cooper pair with one electron in each layer does. Indeed, the electron in the F layer has its spin parallel to the exchange field (up), while the second electron of the pair has a spin down. Therefore, each of theses Cooper pairs gets a favorite orientation, with the up spin electron in the F layer and the down spin electron in the S layer. That is how a magnetization appears in the S layer, whose direction is the opposite of the one in the F layer.
Conclusion We have investigated the particularities of the proximity effect in S/F multilayered systems in the dirty limit. We demonstrated that the spin-flip scattering may strongly influence the behavior of the critical temperature and critical current. Indeed, it decreases the decay length and increases the oscillations period. Moreover, perpendicular magnetic scatterings can even lead to the complete destruction of the oscillations. It should therefore be taken into account for theoretical fits of experimental data for systems with weak magnetic anisotropy. Besides, another manifestation of the proximity effect in S/F bilayer is the appearance of a negative magnetization in the S layer, while the magnetization of the F layer is being oscillating.
Acknowledgments The authors are grateful to V. Ryazanov for useful discussions and J. Cayssol for critical reading of the manuscript. This work was supported in part by ESF PI Shift Program.
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References 1. Ginzburg VL (1956). Zh Eksp Teor Fiz 31:202 [(1957) Sov Phys JETP 4:153] 2. Anderson PW, Suhl H Phys (1959). Rev 116:898 3. Bulaevskii LN, Buzdin AI, Kulic ML, Panjukov SV (1985). Advances in Physics 34:175 4. Larkin AI, Ovchinnikov YN (1964). Zh Eksp Teor Fiz. 47:1136 [(1965) Sov Phys JETP 20:762] 5. Fulde P, Ferrell RA (1964). Phys Rev B 135:A550 6. Buzdin AI (2005). Rev Mod Phys 77:935 7. Golubov AA, Kuprianov MYu., Il’ichev E (2004). Rev Mod Phys 76:411 8. Tagirov LR (1998). Physica C 307:145 9. Demler EA, Arnold GB, Beasley MR (1997). Phys Rev B 55: 15174 10. De Gennes PG (1966) Superconductivity of Metals and Alloys. Benjamin, New York 11. Deutscher G, De Gennes PG (1969). In: Parks RD (ed) Superconductivity: 1005-1034. Marcel Dekker, New York 12. Wong HK, Jin BY, Yang HQ, Ketterson JB, Hilliard JE (1986). J Low Temp Phys 63:307 13. Jiang JS, Davidovic D, Reich DH, Chien CL (1995). Phys Rev Lett 74:314 14. Ryazanov VV, Oboznov VA, Veretennikov AV, Rusanov AYu, Golubov AA, Aarts J (2001). Phys Rev Lett 86:2427 15. Sellier H, Baraduc C, Lefloch F, Calemczuk R (2003). Phys Rev B 68:054531 16. Kontos T, Aprili M, Lesuer J, Genet F, Stephanidis B, Boursier R (2002). Phys Rev Lett 89:137007 17. Eilenberger G (1968). Z Phys 214:195 18. Usadel L (1970). Phys Rev Lett 25:507 19. Fulde P, Maki K (1966). Phys Rev 141:275 20. Abrikosov AA, Gor’kov LP (1960). Zh Eksp Theor Fiz 39:1781 [(1961) Sov Phys JETP 12:1243] 21. Kuprianov MY, Lukichev VF (1982). Zh Eksp Teor Fiz 94:147 [(1988) Sov Phys JETP 67:1163] 22. Abrikosov AA, Gorkov LP (1963). Methods of Quantum Field Theory in Statistical Physics (English version). Prentice-Hall, New Jersy 23. Buzdin AI, Bulaevskii LN, Panyyukov SV (1982). Pis’ma Zh Eksp Teor Phys 35:147 [(1982) JETP Lett 35:178] 24. Buzdin AI, Kuprianov MY (1991). Pis’ma Zh Eksp Teor Phys 53:308 [(1991) JETP Lett 53:321] 25. Ryazanov VV, Oboznov VA, Prokov’ev AS, Bolginov VV, Feofanov AK (2004). J Low Temp Phys 136:385 26. Buzdin AI (2003). Pis'ma Zh Eksp Teor Fiz 78:1073 [(2003) JETP Lett 78:583]
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27. Cretinon L, Kupta AK, Sellier H, Lefloch F, Fauré M, Buzdin AI, Courtois H (2005). Cond-matt 0502050. 28. Ryazanov VV, Oboznov VA, Bolginov VV, Feofanov AK, Buzdin AI (2005). To be published 29. Krivoruchko VN, Koshina EA (2002). Phys Rev B 66:014521 30. Halterman K, Valls OT (2004). Phys Rev B 69:014517 31. Bergeret FS, Volkov AF, Efetov KB (2004). Phys Rev B 69:174504
Superconducting/Ferromagnetic Nanostructures: Spin Fluctuations and Spontaneous Supercurrents
M. Aprili 1,2, M. L. Della Rocca 2,3, T. Kontos 2 1
LPQ-ESPCI, 10 rue Vauquelin, 75005 Paris, France
2
CSNSM-CNRS, Bât.108 Université Paris-Sud, 91405 Orsay, France.
3
Dipartimento di Fisica,, Università di Salerno, via S. Allende,84081 Baronissi, Italy
Abstract:
We present a quantitative analysis of the superconducting proximity effect at the paramagnetic-ferromagnetic transition. In the ferromagnetic state we have shown than the normal layer for some thickness becomes a phase provider and can be used as a “quantum battery”.
Keywords: proximity effect, superconductivity, hybrid nanostructures.
Introduction Superconducting correlations change when Cooper pairs leave from a superconductor and enter in a strong interacting normal metal. This simple consideration has two main consequences. First, a vanishing superconducting state with new specific properties can be induced in the normal metal. Second, the modifications of the BCS ground state can be used as a probe of the interactions. Thus, on an energy scale comparable or even larger than the superconducting energy gap, interactions can be investigated by spectroscopy of the induced superconducting state. In this paper, we investigate these two points separately. First of all, we shall show that the strong exchange enhancement of spin fluctuations at the paramagnetic-ferromagnetic transition can be directly measured by 225 R. Gross et al. (eds.), Nanoscale Devices - Fundamentals and Applications, 225–237. © 2006 Springer. Printed in the Netherlands.
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proximity effect tunneling spectroscopy (PETS) [1]. Second, the superconducting proximity effect in a ferromagnet results in a oscillating superconducting wave function due to the finite momentum transferred from the exchange energy to the Cooper pairs [2]. This vanishing superconducting state can be used as a “quantum battery”. Specifically, by connecting the ferromagnet on both sides to the same superconductor, the phase difference resulting from the oscillations in the ferromagnetic layer, can generate a spontaneous supercurrent [3].
The proximity effect at the paramagnetic-ferromagnetic transition The electron-spin fluctuation coupling constant and the exchange field at the paramagnetic-ferromagnetic transition are obtained by fitting the tunneling spectra within the theory of proximity effect in the dirty limit [4]. The DOS is measured by planar tunneling spectroscopy in Pd thin films with different Ni concentrations either in the paramagnetic or in ferromagnetic regime. As a comparison, the superconducting density of states of pure Ag, Pt and Pd is also investigated. Minigap in N/S bilayers Spin fluctuations as well as the exchange field are expected to modify the DOS of an SN structure over the whole energy range relevant to superconductivity. This is better understood if one considers the basic mechanism responsible for the proximity effect: the Andreev reflection [5]. When an electron reaches an SN interfaces, it cannot enter into the superconducting region if its energy E is smaller than the energy gap ∆ of the superconductor. It is then backscattered as a hole of energy -E. The electron and the hole are coupled coherently. Their phase relationship ∆φ = 2 Et / h , t being the relevant time scale in the normal region of the structure, controls the quantum interferences in the normal metal. When ∆φ is an integer multiple of 2π , a constructing interference occurs, thereby defining the energy scale of the SN structure. If the N metal consists of a thin layer whose thickness dN is smaller than the superconducting coherence length ξ 0 , the relevant time scale is the dwell time τ in the structure. In this case, the criterium ∆φ = 2π may be fulfilled only for a finite energy Eg, which can be of the same order of magnitude than ∆ . Therefore, no quasiparticle can be found with energy below Eg. This energy scale is commonly referred to as a minigap and
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depends also on the amplitude of the Andreev process which can be altered by imperfect S/N boundaries [6]. In a strong paramagnet, spin fluctuations are known to renormalize the quasiparticle energy [7], just like phonons do in strong coupling superconductors, when their energy is much larger than ∆ . This renormalization factor relates the energy of bare quasiparticles to dressed quasiparticles by E = E (1 + λS ) , λS being the electron-spin fluctuation
λS diverges at the ferromagnetic-paramagnetic transition. As a consequence, the condition ∆φ = 2π becomes 2 Eg (1 + λS )τ / h = 2π and the minigap is expected to provide a direct and very sensitive measurement of λS near the paramagnetic-ferromagnetic coupling constant.
transition. In a ferromagnet the exchange field acts like a spin dependent potential, as Andreev reflexions revert the spin. The condition ∆φ = 2π becomes
2( E + Eex )τ / h = 2π . If Eex > ∆ , which is usually the case, the condition for constructive interferences becomes independent of E and Eg vanishes. The energy dependence of quantum interferences cannot be related anymore to constructives interferences between electrons and holes. The only energy dependence expected is that of the Andreev process itself and the effect of the exchange field is to scale the amplitude of these correlations by a cosine exponentially damped whose phase is just 2 Eexτ / h [8]. We will see that this provides a direct way to determine the exchange field in the ferromagnetic layer. The Usadel equations The Usadel equations provide the formalism to implement the qualitative arguments developed above. The effect of a high Stoner enhancement as well as that of an exchange field are particularly simple to include. In the paramagnetic regime, the S/N hybrid structure is described in the dirty limit by the following equations: hDN ∂θ + i (1 + λsf ) E sin θ + 2Γ AG sin θ cos θ + ∆ N cosθ = 0 , 2 2 ∂x ∆ N = 2π T λN ∑ sin θ . ω
(1)
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∆ N and λN are respectively the diffusion constant and the effective electron-electron coupling constant in the normal slab, λsf is the electron-
spin fluctuations renormalization constant and Γ AG is an AbrikosovGor'kov (AG) pair-breaking term . The pairing angle θ contains all the information about equilibrium properties and the spatially resolved DOS, N(E,x), is found by N(E,x)=Re (cos θ ). In the ferromagnetic regime, the pairing angle depends on the spin σ = ±1 of the quasiparticles, because of the exchange field Eex , eq. (1) reads: hDN ∂θσ + i (1 + λsf ) ( E + σ Eex ) sin θσ + 2Γ AG sin θσ cosθσ 2 2 ∂x +∆ N cos θσ = 0 .
(2)
The spatially resolved DOS N(E,x) is found by N ( E , x) = Re {cos θ + + cos θ − } / 2 . As predicted above, one can see from eqs. (1) and (2) that the exchange field and the electron-spin fluctuation coupling constant both affect directly the energy E of quasiparticles. While spin fluctuations renormalize it, the exchange field acts like a spin dependent potential. Note that, as the Coulomb interaction is expected to be non-negligible, one has to consider a priori a non zero negative gap, ∆ N . Tunnel junction fabrication We measure the density of states by planar tunneling spectroscopy in Counter-Electrode/Insulator/Normal-Metal/Superconductor (CE/I/N/S) junctions. The structures are fabricated in a UHV system of a base pressure of 10-9 Torr. A thick Al layer of 1500 Å is deposited at room temperature on a Si(100) substrate with a SiO buffer of 500 Å. We oxidize the Al layer with an O2 glow discharge. The typical pressure of the plasma is 8×10-2 mbar. Depending on the time of oxidation (several minutes), we get junction resistances ranging from 10 Ω to 10 k Ω , for a typical area of 100 µ m × 100 µ m . The size of the junctions is defined by evaporating two 500 Å SiO layers. A normal metal layer of 50 Å is then evaporated and it is after covered by a Nb layer of 500 Å whose Tc is typically 8.7 K. The 4-probes cross-junction geometry is shown in the inset of fig. 2. The
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thickness are measured with a quartz deposition controller with an accuracy of 1 Å. The normal metal is either Ag, Pt, Pd or Pd1-xNix. Pd is a highly paramagnetic metal and only few percents of Ni impurities are necessary to drive it into the ferromagnetic state. We probe the onset of ferromagnetism by Anomalous Hall Effect (AHE) on Pd1-xNix thin films deposited on a bare substrate. The Hall resistance (i.e. VHall/I) is reported in Fig. 1a for diffrent Ni concentration ranging from 2.4 to 10%, an anomalous component proportional to the magnetization appears for a Ni concentration higher than 2.4%, as observed in bulk Pd1-xNix, while a hysteresis in the Hall resistance appears for 7% (not shown). The Ni concentration is determined by Rutherford Back Scattering spectrometry. In Fig. 1b the magnetoresistance for the highest Ni concetration of Fig. 1a is also reported. The magnetic field is oriented either in the film plane but perpendicular to the current or perpendicular to the film plane. A strong anisotrpy is observed, this anisotropy appears at concetration higher than 2.4% Ni together with the AHE. The hysteresis on the magnetoresistance is not of magnetic origin, it results from fluctuations of the bath temperature. 0.3 0.2
0.4 9.8 %
T=1.3K
Field⊥
0.2
2.4 %
0.0
0.0 -0.2
-0.1
-0.4
-0.2 -0.3
∆R
Rxy(ž)
0.1
-0.4
0.0
0.4
H(T)
Field // -0.4
0.0
0.4
H(T)
Fig. 1. a) Hall resistance for different Ni concentrations. An anomalous Hall component (AHE) appears at 2.5 % Ni. b) The anisotropy of magnetoresistance.
The paramagnetic regime Figure 2 shows the tunnel conductance of CE/I/N/S junctions with pure Ag, Pt, Pd and Pd0.988Ni0.012 as a normal metal. The main feature of the conductance is a minigap, Eg. Note that the minigap decreases when the susceptibility of the metal increases as expected by faster dephasing. In Fig. 2 are also shown the fits (blue lines) of the conductance by the DOS
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found solving the Usadel equations. We used a self consistent code to solve equation (1) together with the usual boundary conditions [9]. It turns out that a finite ∆ N is unnecessary to account for the data and that an upper bound for the coupling constant λN is -0.1.
A finite interface resistance of about 5.10-4 µΩ cm2 was used to account for both the shape and the value of the energy gap for pure Pd. This value is consistent with the one measured in Nb/Pd/Nb Josephson jutions [10]. This finite resistance is supposed not varying when changing material and remains also constant when adding Ni impurities in Pd. From the fits we obtained λsf = 0.05, 2.05, 4.15, 5.3 for Ag, Pt, Pd and Pd0.988Ni0.012, respectively. The electron-spin fluctuation coupling constant increases between Ag and Pt and between Pt and Pd. It increases further when increasing the Ni concentration as expected since the magnetic susceptibility increases below the paramagnetic-ferromagnetic transition consistently with specific heat data. Even though the precise relationship between λsf and the Stoner enhancement is model dependent, a rough estimate of it is given by λsf ~2 (1-1/S)\lnS, where S is the Stoner factor. This yields a Stoner factor of 10 for Pd, 3.3 for Pt and 1 for Ag, in surprisingly good agreement with the known values for these metals.
Fig. 2. The superconducting density of states induced in a normal metal by the proximity effect (red dots). The density of states of pure Nb is also shown. The fits (blue lines) are obtained by solving the Usadel equations self consistently as indicated in the text. In the inset the four probes cross geometry of the tunnel junction is presented.
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The ferromagnetic regime For Ni concentrations higher than 5.5%, the tunnel conductance displays only one characteristic energy, ∆ , of about 1.35 meV, while the shape of the spectra remains roughly the same. This is shown in Fig. 3, where all the spectra have been scaled to the DOS corresponding to 5.5% of Ni. As finite size effects vanished altogether, the proximity bilayer becomes similar to that of a semi-infinite mesoscopic system d F > ξ F . The overall amplitude is smaller than 5% of the background conductance and therefore it is possible to find an analytical expression for the DOS and extract directly the value of the exchange field in the Pd1-xNix layer. Linearizing eq. (2) we find the following DOS: N ( E ) = 1 + [ N 0 ( E ) − 1] exp(−2 Eex / Eth ) cos(2 Eex / Eth ) ,
(3)
where No(E) is the DOS at the Nb/Pd1-xNix boundary. We found Eex = 0.1, 0.5, 2.8, 3.3, 3.9 for 5.5%, 6%, 7%, 9.8%, 11.5% Ni, respectively.
Fig. 3. The proximity effect tunneling spectroscopy in a weak ferromagnet (PdNi) is presented. The spectra are normalized by the PdNi exchange energy. The red line is the best fit using the analytical expression.
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Therefore superconductivity is a very sensitive probe of Eex provided that the size of the magnetic domains is larger than the superproduction coherence length. On other words, the spatial resolution of a «Cooper pair Eex sensor» is given by ξ 0 .
Spontaneous Vortices in Ferromagnetic Josephson Junctions In the ferromagnetic regime for a Ni concentration of about 10% the superconducting wave function oscillates in F on a length scale given by ξ F ≈20 Å. A negative critical current occurs when the ferromagnetic thin layer is coupled with a second superconductor and the superconducting wave function is negative, originating π -coupling [11]. By shorting a ferromagnetic π -junction with a 0-junction, a spontaneous supercurrent sustaining half a quantum vortex occurs. We have detected such a current by measuring the related phase gradient. The ground state of the 0π junction is doubly degenerate, so that either half a quantum vortex or half a quantum antivortex appears. As no coupling is observed at low temperature between the two levels, the junction can be seen as the classical limit of a two level system. It behaves macroscopically as a magnetic nanoparticle of quantized flux, the magnetic anisotropy axis being defined by the junction plane. Device Design The π -junction in a superconducting loop behaves as a phase bias generator producing a spontaneous current and hence a magnetic flux. In the limit 2 π LIc< Φ 0 , the system gains energy by minimizing its magnetic energy against the junction energy. The system maintains a constant phase everywhere and a shift of Φ 0 /2 in the current-phase relationship of the junction is expected. On the other hand, when 2 π LIc >> Φ 0 the system's minimum energy corresponds to that of the junction while maximizing its magnetic energy. A phase gradient is maintained by generating a spontaneous superconducting current, which sustains exactly half a quantum flux. The ground state is degenerate as, the spontaneous supercurrent can circulate clockwise and counterclockwise with exactly the same probability. Applying a small magnetic field can lift the degeneracy and define an easy magnetization direction. The existence of a
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spontaneous supercurrent sustaining half a quantum flux in π -rings has been recently shown in Nb loops interrupted by a ferromagnetic (PdNi) π junction [12]. Analogously, in a highly damped single Josephson junction fabricated with a 0 and a π -region in parallel, a spontaneous half quantum vortex is expected at the 0- π boundary. The way we detect such a spontaneous supercurrent is by measuring the phase gradient by a phase sensitive device, a second Josephson junction. The ferromagnetic 0- π junction (source) and the detection junction (detector) are coupled, as schematized in Fig. 4a, by sharing an electrode. I.e., the top electrode of the detector Josephson junction is simultaneously the bottom electrode of the ferromagnetic one. If half a quantum vortex is spontaneously generated in the ferromagnetic junction, the spontaneous supercurrent sustaining it circulates in the common electrode [Nb2, Fig. 4a] producing a phase variation equal to π /2. A π /4-shift of the detection junction's magnetic diffraction pattern is thus produced.
V+
Nb2SiO
I-
I+
SiO Nb1A l
V-
Detector
I (mA)
5
T = 1.5K B=0
0 -5 -4
-2
0 2 V (mV)
4
Fig. 4. a) Device geometry b) I-V characteristic of the detector junction.
When an external magnetic field is applied, in the hypothesis that the thickness of the common electrode, Nb2, is comparable to the penetration depth, the diffraction pattern of the detection junction is given by: I(B)= I(0)
sin((π /Φ 0)( ks+ k'Φ'+Φ )) , (π /Φ 0)( ks+ k'Φ'+Φ )
(4)
where Φ ' = BDt ' and Φ = BDt are the magnetic fluxes through the ferromagnetic and detection junction respectively, with D the junction width, t and t' the effective barrier thickness. Js is the spontaneous supercurrent density, ks=0.5( µ0 λ 2 )D and k’=( µ0l 2 )/L(D/wdNb2) with µ0
the vacuum permittivity, λ the penetration depth, L the ferromagnetic junction inductance, w the junction length and dNb2 the thickness of the common electrode. The term ksJs generates the shift due to the spontaneous supercurrent contribution, while the term k’ Φ ' reduces the diffraction
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pattern period as a result of the contribution due to the screening current in the ferromagnetic junction. Device Fabrication Samples are fabricated as described above for the single tunnel junctions. First the bottom planar Nb/Al/Al2O3/Nb detection junction is made. A 1000 Å thick Nb [Nb1, Fig. 4(a))] strip is evaporated and backed by 500 Å of Al. Al2O3 oxide layer is achieved by oxygen plasma oxydation during 12 min, completed in a 10 mbar O2 partial pressure during 10 min. The junction area is 0.6×0.8 mm2 (D x w). Then, a 500 Å thick Nb [Nb2, Fig. 4(a)] layer is evaporated perpendicular to the Nb/Al strip to close the junction. This procedure results in a junction critical temperature, Tcj, equal to 8.5 K. Typical junction normal state resistances are of the order of 0.11 Ω and critical current values are of 1-10 mA at 4.2 K. The resulting critical current density is 10-1 A/cm2 leading to a Josephson penetration depth λ j ~1 mm, i.e. larger than the size of the junction (small limit). The IV characteristic of a typical detector is shown in Fig. 4(b). The Nb2 layer acts as both the counterelectrode of the bottom detection junction and the base electrode of the top ferromagnetic 0- π junction. Its thickness is comparable to the Nb penetration depth to insure good coupling between the two junctions. The same procedure is used to prepare the top planar Nb/PdNi/Nb/Al junction. Specifically, after defining the same junction area by evaporating 500 Å thick SiO layers, a PdNi layer was evaporated directly on the Nb layer, without any Al-oxide barrier. This results in a very large critical current and very small junction resistance. An estimate of the critical current density is 104-105 A/cm2, so the Josephson penetration depth, λ jf < 10-2 mm << D, is smaller than the size of the junction. Hence the ferromagnetic junction is in the large limit, with a large screening capability. The maximum degree of misalignment between the top and bottom junction is 100 µ m. A detailed SEM and AFM analysis of devices where the fabrication process has been stopped before closing the ferromagnetic junction with the top Nb/Al bilayer, revealed the presence of inhomogeneities at the junction window edges. SEM images (Fig. 5c) of the PdNi surface show such inhomogeneities in the shape of black bubbles with 2-3 µ m diameter. Fig. 5b shows an AFM picture of the same surface, the line cut indicates the direction along which a thickness profile has been measured (Fig. 5a). Apart from inhomogeneities, the roughness is lower then 100 Å, while picks as high as 500 Å develop on the top of the inhomogeneity itself. This indicates that the PdNi layer is not continuous, thus inducing 0-coupling in
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such zones. Therefore, 0-junction and 0- π junctions are obtained with PdNi layer thickness coresponding to 0-coupling and π - coupling respectively.
a)
QuickTime™ et un décompresseur TIFF (LZW) sont requis pour visionner cette image.
π b)
c)
Fig. 5. a) Roughness profile along the black line in (b). b) AFM picture of an inhomogeneity at the window edge. c) SEM image of one window edge.
Results Residual fields were screened by cryoperm and µ -metal shields. All measurements showed comparable residual fields at 4.2 K of some tenths of mG. Depending on the PdNi layer thickness, the ferromagnetic junction is either 0 or π while a spontaneous half quantum vortex or antivortex is expected only for π -coupling. In Fig. 6a we show the diffraction patterns of a sample for a PdNi thickness of 100 Å corresponding to π -coupling at T = 4.2 K (red data) and T = 2 K (blue data). The magnetic field corresponding to a quantum flux is 300 mG. At T < TcF period reduction is accompanied by a Φ 0 /4 shift in the detector, as expected for a spontaneous half quantum vortex in the ferromagnetic junction. Similar period reductions have been observed for PdNi thickness of 40 Å and 200 Å corresponding to zero coupling together with not shift in the
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detector diffraction pattern [13]. The period reduction at the lowest temperature (2 K) for all the samples was about 15%. This value results from two competing effects: a smaller period is expected because of screening in the ferromagnetic junction as shown in eq. 4, whereas the decrease in the pénétration depth λ (T), at lower temperatures should increase the modulation period. Note that no period reduction or diffraction pattern shift are measured when the source and the detector are decoupled with a thin insulating layer. This indicates that inductive coupling between the two junctions is negligible. Might an external residual field or the magnetic layer itself affect the spontaneous supercurrents? In this regard it is important to stress that the shift is always about Φ 0 /4 for π -coupling and zero for 0 coupling. Since this depends neither on the residual field nor, for 0-coupling, on the thickness of the ferromagnetic layer, any effect of the PdNi magnetic moment on the amplitude of the spontaneous supercurrents seems to be ruled out. Instead, the PdNi magnetic structure breaks the degeneracy of the ground state and polarizes the supercurrents. As a consequence, the sign shift is always the same below the Curie temperature (100 K), indicating the same spontaneous current polarization. When cooling down from room temperature to 2 K, the shift, while reproducible in magnitude, becomes random in sign. This is shown in fig. 3b for 26 cool-downs of two different junctions. φ
0
16
B
dPdNi = 100A
14
percentage (%)
4
4
I (mA)
3
2
(a)
12
10
8
6 4
1
2 0
0
a)
b)
Fig. 6. a) Φ 0 /4 shift in the diffraction pattern of the detector junction above (T = 4.2 K, red dots) and below (T = 1.5 K, blue dots) the ferromagnetic junction critical temperature, Tcf. b) Statistics of the measured shift for 26 different cooling down from room temperature. A positive shift correspond to half a quantum antivortex, a negative shift to half a quantum vortex.
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Conclusion In conclusion, the strong enhancement of the Stoner factor at the paramagnetic/ferromagnetic transition is measured by proximity effect tunneling spectroscopy (PETS). Ferromagnetic Josephson π -junctions behaves as classical spins resulting from spontaneous supercurrents which are revealed by measuring the shift in the magnetic diffraction pattern of a Josephson junction directly coupled to a 0- π junction.
Acknowledgments This work has been supported by the ESF through the “ π -shift” program. We are indebted with E. Reinwald of the University of Regensburg for performing the AFM analysis.
References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13.
Wolf (1986) Principles of electron tunneling spectroscopy. Wiley and sons, New York For a review see Buzdin AI (2005). Rev Mod Phys 77:935 Bulaevskii LN, Kuzii VV, Sobyanin AA (1978). Solid State Comm 25:1053 Kontos T, Aprili M, Lesueur J, Dumoulin L (2004). Phys Rev Lett 93:137001 Andreev AF (1964). Sov Phys-J Exp Theor Phys 19:1228 Belzig W, Bruder C, Schön G (1996). Phys Rev B 53:5727 Daams JM, Mitrovic B, Carbotte JP (1981). Phys Rev Lett 46:65 Kontos T, Aprili M, Lesueur J, Grison X (2001). Phys Rev Lett 86:304 Golubov AA, Kupriyanov Yu (1988). J Low Temp Phys 70:823 Guichard W (2003). Ph.D. thesis, Université Joseph Fourier, Grenoble Kontos T, Aprili M, Lesueur J, Genêt F, Stephanidis B, Boursier R (2002). Phys Rev Lett 89:137007-1 Bauer A, Bentner J, Aprili M, Della Rocca ML, Reinwald M, Wegscheider W, Strunk C (2004). Phys Rev Lett 92:217001 Della Rocca ML et al cond-mat/0501459
COHERENCE EFFECTS IN F/S AND N/F NANOSTRUCTURES
Proximity Effect and Interface Transparency in Nb-based S/N and S/F Layered Structures
C. Attanasio Dipartimento di Fisica “E.R. Caianiello” and INFM-Laboratorio Regionale Supermat, Università degli Studi di Salerno, I-84081 Baronissi (Sa), Italy
Abstract:
The interface transparency T has been investigated for different S/N and S/F layered structures realized by sputtering and by Molecular Beam Epitaxy (MBE). We have chosen Nb as a superconductor (S), Cu, Ag and Pd as a normal metal (N) and PdNi and Fe as ferromagnetic materials (F). The obtained result for T do not seem strongly influenced by fabrication methods but more by intrinsic factors related to the microscopic properties of the two materials.
Keywords: proximity effect, transparency, heterostructures
Introduction The proximity effect between a superconductor (S) and another material (M) has been object of study since the early sixties [1], but only recently an important parameter, the interface transparency T, has been added to the description of this coupling mechanism [2-7]. The quantity T takes into account all the effects that cause electrons to be reflected rather than transmitted at the interfaces, with the consequence that the proximity effect results somehow screened. In principle T can depend on both extrinsic and intrinsic factors such as, for example, interface imperfections, fabrication methods, Fermi velocities and band-structure mismatches [4]. When the non-superconducting material M is magnetic the situation is even more difficult because of the additional splitting of the spin sub-bands and the spin-dependent impurity scattering [8]. For these reasons the study of the interface transparency turns to be important from a theoretical point of 241 R. Gross et al. (eds.), Nanoscale Devices - Fundamentals and Applications, 241–249. © 2006 Springer. Printed in the Netherlands.
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view because it is related to the microscopic properties of the metals. In addition it is also important from a practical point of view because having high quality interfaces it is essential for superconducting devices in which the working mechanism is based on coupling between different materials [9]. In this paper we analyze the topic of the interface transparency T when Nb is placed into a contact with a normal metal N and with a magnetic material F. For N we have chosen Cu, Ag and Pd while for F we have chosen PdNi, which is a weak ferromagnet, and Fe. To investigate the effect of the fabrication method on T some samples have been prepared by sputtering and others by Molecular Beam Epitaxy (MBE). What essentially emerges from this study is that the obtained values for T do not seem strongly dependent from the fabrication methods but more from intrinsic properties of the constituent metals.
Samples Preparation High quality multilayers were grown by MBE and by a dual-source magnetically enhanced dc triode sputtering system on Si(100) substrates kept at room temperature. By using suitable shutters, more than one sample has been fabricated in the same deposition run in order to obtain the same deposition conditions for all the samples of the series: we prepared eight samples in the sputtering and four in the MBE chamber. Nb/Cu systems have been prepared both by sputtering and by MBE while Nb/Ag has been fabricated by MBE [7] and Nb/Pd by sputtering [5]. In the case of S/F systems we have prepared Nb/PdNi by sputtering and Nb/Fe by MBE. The superconducting properties, that is the transition temperature Tc and the upper critical magnetic fields were resistively measured using a standard dc four-probe technique. In the case of Nb/Cu system to better study the effect on T of the deposition technique a detailed structural characterization have been realized by performing low-angle and highangle X-ray reflectivity measurements [7].
Proximity Effect and Interface Transparency
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The S/N Case
Theoretical background The starting point for a complete description of proximity effect in S/N multilayers, valid for arbitrary transparency, was given by Kupriyanov and Lukichev [2] in the framework of Usadel equations (dirty limit). In particular, the model we used to describe the dependence of the critical temperature versus the S layer thickness, Tc(ds), for N/S/N trilayers is based on the Werthamer approximation, valid for not too low temperatures, provided the boundary transparency is sufficiently small [3]. In this limit the system of algebraic equations to determine Tc is: Ω d γ , Ω1tg 1 S = 2ξ S γ b
(1a)
1 Ω 2T T 1 Ψ + 1 CS − Ψ = ln CS , 2TC 2 2 TC
TCS γ , >> γb TC
(1b)
with the identification of the Abrikosov-Gorkov pair-breaking parameter ρ = π Tc Ω12 = π Tc ( γ / γ b ) ( 2ξS / dS ) , where Ψ ( x) is the digamma function and Tcs is the bulk critical temperature of the S layer. These equations contain two parameters γ and γ b defined as
γ =
ρ sξ s , ρ nξ n
γb =
RB
ρ nξ n
,
(2)
where ξs and ξ n are the superconducting and the normal coherence length,
ρs and ρ n are the low-temperature resistivities of S and N, respectively, while RB is the normal-state boundary resistance times its area. The parameter γ is a measure of the strength of the proximity effect between the S and N metals. The parameter γ b , instead, describes the effect of the boundary transparency T, to which is roughly related by T
=
1 1 + γ
. b
(3)
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While γ was determined experimentally by measuring ρs , ρ n , ξs and ξ n ,
γ b (or T) can’t be determined experimentally, because RB is difficult to measure, so it is extracted by a fitting procedure. In the free electron model it is possible to express the interface transparency in terms of the Fermi velocities by [4] T=
4vN vS
[ v N + vS ]
2
,
(4)
where vN,S are the projections of the Fermi velocities of N and S metals on the direction perpendicular to the interface.
Discussion Two different samples typology have been prepared: N/S/N and S/N/S trilayers. The first, with external layers of normal metal with constant thickness, dn, and an internal layer of superconducting material with variable thickness, ds, were used to determine the dependence of Tc on ds. The others, with external layers of superconducting metal with constant ds and an internal layer of normal material with variable dn were instead used to estimate the normal coherence length [5,7]. N/S/N trilayers were also used to determine the Ginzburg-Landau coherence length at zero temperature, ξ (0) , from the slope S= - dHc2/dT|T=Tc of the upper perpendicular magnetic field close to the critical temperature. ξs is, in fact, related to ξ (0) by the relation ξs = 2 ξ (0) / π . What we found is that in all
our systems ξ (0) typically decreases when increasing ds until a saturation value. So for samples with thicker Nb interlayer the perpendicular Hc2(T) reflects the properties of the single Nb film [5,7]. To determine ξ n by S/N/S trilayers we should consider that, if two S layers are separated by a thin N layer, the decay of the superconducting order parameter from both sides overlaps. By increasing the thickness of the N layer the S layers become more and more decoupled until no overlap is left. For this reason the behavior of the Tc(dn) curve will go from a maximum value (related to the critical temperature of the S layer with thickness equal to 2ds) to a limiting value (related to the critical temperature of the S layer with thickness equal to ds). The thickness for which the minimum is reached is called decoupling thickness and can be associated with approximately twice the coherence length (dndc ≈ 2 ξ n )
Proximity Effect and Interface Transparency
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[4, 5, 7]. Finally the S and N resistivities ρs and ρ n have been measured on samples deliberately fabricated [5, 7]. All the above values have been used to reproduce the Tc(ds) for all the sets of fabricated trilayers using equations (1) with T (given by equation (3)) as the only fitting parameter. In Table 1 are reported all the measured quantities for all the systems together with the values obtained from the fitting procedure for the interface transparency. In the last column are reported the theoretical results for T obtained using equation (4) with the following values for the Fermi energies: vNb= 2.73 × 107 cm s-1 [10], vCu= 1.57 × 108 cm s-1 [11], vAg= 1.39 × 108 cm s-1 [11] and vPd= 2.00 × 107 cm s -1 [12]. What we see is that we obtained the higher value for T in tha case of Nb/Pd system for which the values of the Fermi energies are very similar. In addition, in the case of Nb/Cu and Nb/Ag we got very similar values for T as well as for the case of MBE and sputtered Nb/Cu samples in spite of their different interface quality [5,7]. Table 1. Measured values of the electrical resistivities and of the coherence lengths of the N and S materials and of the bulk critical temperature. T is the transparency of the systems obtained by fitting the experimental data of the Tc(ds) curves. Ttheo has been calculated from equation (4). The first (second) line in the table refers to MBE (sputtering) prepared Nb/Cu trilayers. Samples Nb/Cu1 Nb/Cu2 Nb/Ag Nb/Pd
ρ n ( µΩ ρs ( µΩ ξ n (Å)
cm) 1.3 1.8 4.0 5.0
cm) 3.6 4.6 7.3 2.5
260 170 190 60
ξs (Å)
Tcs(K)
64 67 54 64
9.2 8.8 9.2 8.8
T
Ttheo
0.30 0.26 0.33 0.46
0.50 0.50 0.55 0.98
The S/F Case
Theoretical background The experimental investigation of S/F systems, that is the study of interplay of two competing phenomena the superconductivity and the ferromagnetism, started almost forty years ago [13]. More recently the improvement of the deposition techniques has made possible to prepare very thin high-quality ferromagnetic layers and a rich variety of
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phenomena have been predicted [14-16], such as, for example, the nonmonotonic behavior of the superconducting critical temperature Tc in S/F layered structures as a function of the F layer thickness, df [14, 17, 18]. In general the presence of an exchange field Eex in F cause an energy shift between the quasiparticles of the pair entering the ferromagnet and this results in the creation of non-zero momentum Cooper pairs [19]. This implies that the superconducting order parameter does not simply decay in the ferromagnetic material, as happens for normal metals, but also oscillates over a length scale given by ξ f , the coherence length in F, which, in the dirty limit, is given by ξ f = (hDf/2 π Eex )1/2 , where Df is the diffusion coefficient [19]. In the case of S/F systems where F is a strong ferromagnet (Fe, Co, Ni) the exchange energy is typically of the order of 1 eV, resulting in a coherence length ξ f of few Angstroms. In the case of the so-called weak ferromagnets (CuNi [20], PdNi [15, 21]), Eex is in the meV range leading to a ξ f of the order of hundred of Angstroms. The theoretical prediction of the Tc(df) behavior has been done first in the limit of high transparency [22] and then considering the possibility of finite T [23] in the case of strong ferromagnet, where the exchange energy is very large. Recently the theory has been extended to the case of S/F bilayers in a more general case [24] which also applies for the weak ferromagnets for which, again, df is large with respect the interatomic distance.
Discussion Two different sets of samples have been fabricated: F/S and S/F bilayers, using for the ferromagnetic material PdNi (with Ni percentage equal to 10) and Fe. The first set, consisting of an F layer with constant thickness, df, and a Nb layer with variable thickness, ds, were used to determine the dependence of Tc on ds and to estimate the interface transparency of the S/F barrier. The other, with constant ds and variable df were instead used to estimate the coherence length in F and, consequently, the value of the exchange energy Eex. The S and F resistivities, ρs and ρ f , have been measured on samples deliberately fabricated. The Nb coherence length was determined through the dirty limit expression ξs = (hDs/4 π 2 kBTc)1/2. Here Ds is the Nb diffusion coefficient which is related to the low temperature resistivity ρs via the electronic
Proximity Effect and Interface Transparency
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mean free path [25]. All these values have been used to reproduce the Tc(ds) for all the sets of fabricated bilayers using the Fominov theory in the case of Nb/PdNi and the Tagirov calculations for Nb/Fe, leaving γ b as the only fitting parameter. In Table 2 are reported all the measured quantities for all the systems together with the values obtained from the fitting procedure for the parameter γ b . We see that in the case of Nb/PdNi we got a value which is much smaller of that obtained for the case of Nb/Fe as well as for other S/F systems reported in the literature [4, 18, 26]. In addition the value of γ b is of the same order of what reported for the case of Nb/CuNi (for which γ b = 0.3) [24] but a bit higher probably due to the slightly higher value of Eex (or smaller ξ f ) found in our Nb/PdNi system. Table 2. Values of the electrical resistivities and of the coherence lengths of the F and S materials and of the bulk critical temperature. The parameter γ b has been obtained by fitting the experimental data of the Tc(ds) curves. Samples
ρf ( µΩ ρs ( µΩ cm) cm)
ξ f (Å)
ξs (Å)
Tcs(K)
γb
Nb/PdNi
24
6
20
80
9.0
0.7
Nb/Fe
7.5
3.6
7
64
9.1
45
Conclusion In conclusion, we have experimentally studied the interface transparency in Nb-based S/N and S/F layered structures. The obtained results seem more influenced by the intrinsic properties of the two metals more than by the fabrication methods. In the S/N case, as expected by simply theoretical arguments, we get higher transparency for the Nb/Pd system. In the S/F case Nb/PdNi shows a relatively high transparency with respect to the case of combining a superconductor with strong ferromagnets, confirming the fact that ferromagnetic alloys are could be of great interest in the physics of S/F contacts.
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Acknowledgments This work has been made possible thanks to the collaboration of Carla Cirillo, Serghej L. Prischepa, Matteo Salvato and Achille Angrisani Armenio.
References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23.
de Gennes PG (1964). Rev Mod Phys 36:225 Kupriyanov Myu, Lukichev VF (1988). Sov Phys JETP 67:1163 Golubov AA (1994). Proc SPIE 2157:353 Aarts J, Geers JME, Bruck E, Golubov AA, Coehoorn R (1997). Phys Rev B 56:2779 Cirillo C, Prischepa SL, Salvato M, Attanasio C (2004). Eur Phys J B 38:59 Cirillo C, Prischepa SL, Romano A, Salvato M, Attanasio C (2004). Physica C 404:95 Tesauro A, Aurigemma A, Cirillo C, Prischepa SL, Salvato M, Attanasio C, accepted for publication in Supercond Sci Technol de Jong MJM, Beenakker CWJ (1995). Phys Rev Lett 74:1657 Vasko VA, Larkin VA, Kraus PA, Nikolaev KR, Grupp DE, Nordman CA, Goldman AM (1997). Phys Rev Lett 78:1134 Kerchner HR, Cristen DK, Sekula ST (1981). Phys Rev B 24:1200 Ashcroft NW, Mermin ND (1976) Solid State Physics. International Thomsom Publishing, Washington DC Dumoulin L, Nedellec P, Chaikin PM (1981). Phys Rev Lett 47:208 Hauser JJ, Theuerer HC, Werthamer NR (1966). Phys Rev 142:118 Jiang JS, Davidovic D, Reich DH, Chien CL (1995). Phys Rev Lett 74:314 Kontos T, Aprili M, Lesueur J, Grison X (2001). Phys Rev Lett 86:304 Ryazanov VV, Oboznov VA, Rusanov AYu, Veretennikov AV, Golubov AA, Aarts J (2001). Phys Rev Lett 86:2427 Muhge T, Garif’yanov NN, Goryunov YuN, Khaliullin GG, Tagirov LR, Westerholt K, Garifullin IA, Zabel H (1996). Phys Rev Lett 77:1857 Lazar L, Westerholt K, Zabel H, Tagirov LR, Goryunov YuN, Garif’yanov NN, Garifullin IA (2000). Phys Rev B 61:3711 Demler EA, Arnold GB, Beasley MR (1997). Phys Rev B 55:15174 Rusanov A, Boogaard R, Hesselberth M, Sellier H, Aarts J (2002). Physica C 369:300 Cirillo C, Prischepa SL, Salvato M, Attanasio C (2004). Journ Phys Chem Sol, accepted for publication Radovic Z, Ledvji M, Dobrosavljevic-Grujic L, Buzdin AI, Clem JR (1991). Phys Rev B 44:759 Tagirov LR (1998). Physica C 307:145
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24. Fominov YaV, Chtchelkatchev NM, Golubov AA (2002). Phys Rev B 66:014507 25. Broussard PR (1991). Phys Rev B 43:2783 26. Geers JME, Hesselberth MBS, Aarts J, Golubov AA (2001). Phys Rev B 64:094506
Properties of S/N Multilayers with Different Geometrical Symmetry
S. L. Prischepa Belarus State University of Informatics and RadioElectronics, P. Brovka 6, Minsk 220013 Belarus
Abstract:
The influence of finite dimensions of superconducting metallic multilayers on the H-T phase diagram and angular dependences of the upper critical magnetic field Hc2 is studied. It is established experimentally that the geometrical symmetry determines crucially the Hc2 values and their temperature and angular dependencies. For multilayers with the symmetry plane in the center of the superconducting layer and for temperatures close to Tc the values of the parallel critical magnetic fields are larger than for the samples for which the symmetry plane lies in the middle of the normal layer. This reveals the characteristic feature of bi-dimensional behavior in the whole temperature range up to T c . The angular dependencies of the upper critical magnetic field seem to be more sensitive to the presence of the S/N interfaces in the system than the temperature dependencies.
Keywords: S/N multilayer, upper critical magnetic field, proximity effect, dimensional crossover
Introduction Superconductivity in type II superconductors first was studied in infinite samples [1]. Later the peculiarities of superconducting phase nucleation in homogeneous finite samples of different geometry were discussed. In particular, the cases of semi-infinite samples [2-4], slabs and thin films [36] were considered.
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It is reasonably to suppose that the same research program should be also realized for S/X multilayered structures (X denotes the non superconducting material). Until recently, the obtained theoretical results for multilayers have been mostly related to the temperature and angular dependencies of the second upper critical magnetic field, Hc2(Θ,T) [7, 8]. At the same time the results of experimental measurements of Hc2(Θ,T) dependencies differ from theoretical models [9,10] which, however, mostly consider infinite stacks of S and X layers [7]. For example, at present the effect of the surface critical magnetic field, Hc3, on the multilayered structures is not well understood. Firstly, the definition of Hc3 for S/X samples is not known and secondly it is not clear how to calculate this quantity for multilayers. Furthermore, it is not well known in which cases the S/X structure could be considered as (i) infinite, (ii) semi-infinite medium or as (iii) sample with two boundaries. The criterion, at least empirical, for these definitions, is absent in the literature. For these reasons we decided to perform a series of experiments in order to study the influence of finite dimensions on the thermodynamic quantity Hc2 of S/X multilayers as well as its temperature and angular dependencies. We choose X=N, where N stands for a normal (non superconducting) metal. This means that the proximity effect is responsible for the coupling between the different S layers. At this initial stage of our research we restrict ourselves to the two simplest cases: (i) the symmetry plane of the multilayer falls into the center of the superconducting layer and (ii) the symmetry plane falls into the center of the normal layer. In this work we present results of the experimental investigation of (i) the temperature dependence of the upper critical magnetic field paying special attention to the configuration with the applied magnetic field parallel to the film surface and (ii) the angular dependence of the upper critical magnetic field at different temperatures. Both kinds of experiments were performed for samples with the symmetry plane in S or in N layers. We have studied the properties of both Nb/Cu and Nb/Pd multilayers. The choice of Nb for the fabrication of the superconducting multilayers was related to the fact that it has the highest critical temperature among the superconducting elements. The choice of Cu was motivated by the possibility of creating the Nb/Cu multilayers with high quality structural properties [11]. Moreover, the proximity effect is very well studied in this system [12]. On the other hand, we choose Pd because among the normal metals it turns out to be particularly interesting. It is in fact characterized by a large value of the spin susceptibility and in some alloys (i.e. Pd1-xCox, Pd1-xFex) ferromagnetic behavior is obtained even for small values of x [13]. Moreover, the interface of the Nb/Pd system is more transparent compared to the Nb/Cu system [14]. These differences between the two
Properties of S/N Multilayers with Different Geometrical Symmetry
253
studied systems give us the opportunity to generalize the observed symmetry effect to the class of the proximity coupled metallic multilayers.
Fig. 1. The geometrical configuration of the Nb/Pd samples for Nb=9 and Nb=10.
For all investigated samples the top and the bottom layers consist of the normal metal. In this way, taking into account the constant period of the multilayer, the central layer is superconducting in the case of an odd number of bilayers (Nb) and normal for an even Nb. This means that for odd (even) Nb values, taking into account the capping N layer, the symmetry plane of the whole sample falls into the center of the S (N) layer (see Fig. 1). The thickness dS of the S layers is always 200Å for both compositions. In Nb/Pd samples, the thickness dN of the Pd layers is equal to 100Å. In fact, as has been shown recently, the presence of a 100Å thick Pd layer results in a temperature induced dimensional crossover in the Nb/Pd systems [15]. When the Pd layer thickness increases towards 200Å, the system behaves as 2D in the whole temperature range [15]. For the Nb/Cu system we chose dN = dS = 200Å. In fact, for these values a pronounced 2D3D crossover is usually observed on increasing temperature at a crossover temperature T* [12]. We show that changing the number of bilayers, i.e. changing the symmetry of the samples, a drastic change in the dimensionality of the system occurs for T→Tc, with a 2D behavior observed almost up to Tc for samples in which the symmetry plane lies in the center of the S layer. Also we show that, as it was mentioned in [12], the global angular dependence of the Hc2 for S/N layered superconductors is much more complicated than for isolated films [5,16]. Moreover, the measured Hc2(T, Θ = 0) and the Hc2(Θ) dependencies do not correlate to each other
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with the Hc2(Θ) dependences being more sensitive to the nucleation character of the superconducting phase.
Sample Preparation and Experimental Details Pairs of Nb/Pd and Nb/Cu samples with Nb = 9 and 10 were grown on Si(100) substrates at room temperature by using a dual source magnetically enhanced dc triode sputtering system [15]. The deposition rates were 9 Å/s for Nb, 8 Å/s for Pd and 5 Å/s for Cu. Nb/Cu samples were deposited in different deposition runs, while Nb/Pd samples were sputtered simultaneously. A specially designed movable shutter allowed the simultaneous deposition of the two samples with different number of bilayers. The starting position of the sample holder was located between the Pd and Nb guns and two substrates were mounted on it. Nb+1 bilayers were deposited on the substrate positioned closest to the Pd target, while Nb bilayers were deposited on the other substrate. The platform where the samples are mounted can be rotated in a controlled way over 360 degrees using a stepper motor to reach every position. At the beginning of the process the sample holder is sent to the Pd gun to sputter the first layer (the two samples will consist of Nb or Nb+1 Nb/Pd bilayers plus a bottom Pd layer). Then, after Pd deposition, the sample holder is moved, through the zero position, to the Nb gun. This movement from the Nb to the Pd gun is repeated alternately until on both samples Nb bilayers plus the Pd bottom layer have been sputtered. Then, the sample holder is moved from the Pd to the Nb gun in the direction opposite to the zero position to sputter the Nb layer only on one sample. The sample holder has a diameter of 2.5 cm and the distance between the two substrates is almost 2 cm. This allows us to use a shutter close the Nb gun to prevent Nb deposition also on another sample during this last Nb deposition step. After this, the sample holder is slightly moved back and the Nb gun is switched off. When we are sure that the Nb rate is zero (usual waiting time is 1 minute), the sample holder is moved, through the Nb gun, to the zero position, and finally to the Pd gun to sputter the last Pd layer only on this second sample. This last step is possible due to the presence of another shutter close the Pd gun. Finally the sample holder is moved back to the zero position and the deposition process is completed. X-ray reflectivity measurements confirmed the layered structure of the samples with an interface roughness of the order of 10 Å [15]. Transport measurements with a standard four probe technique were performed for both parallel and perpendicular magnetic field orientations. The resistance
Properties of S/N Multilayers with Different Geometrical Symmetry
255
was measured with the accuracy of 10-4 Ohm, while the measured accuracy of the magnetic field was 10-4 T. The samples from each pair were simultaneously mounted in an insert with the possibility to rotate them in the liquid helium bath. The accuracy of the rotation angle was ± 0.1°. The value Θ = 0 corresponds to the magnetic field direction parallel to the film surface. A superconducting Nb-Ti solenoid with Tc = 7.2 K was used to produce the external magnetic field. The Hc2 values were extracted from the R(H) curves measured at the onset of the superconducting transition. The temperature stabilization during the measurements was ± 0.01 K. The transition widths ∆Tc in zero magnetic field were always less than 20 mK, while at parallel fields higher than 2 Tesla their values were less than 300 mK, confirming the high quality of the samples. From Hc2⊥(T) curves we have calculated the values of ξ||(0) which was of the order of 120 Å for all samples. We named the samples using the letter S or N according to whether the symmetry plane lies in the center of S or in the center of N layers, respectively, followed by a letter that indicates the normal metal used (P for Palladium and C for Copper). For example, SP is the Nb/Pd multilayer whose symmetry plane lies in the S layer (Nb=9) while NC is the Nb/Cu sample with the symmetry plane in the center of the N layer (Nb=10). The deposition of the Nb/Pd multilayers in the same run allows us to consider the same Nb as well as the same interface properties in each sample [17]. This hypothesis is confirmed by the fact that for these two samples the same values have been obtained for the resistivity ρ10 ∼ 9 µΩ×cm, the residual resistivity ratio β10 ≈ 1.6 and the coherence length ξ||(0) ≈ 125 Å. The macroscopic parameters of Nb/Cu samples were also very similar. Therefore, we believe that the only difference between each pair of samples is in their symmetry due to their different finite dimensions. The choice of 9 and 10 number of bilayers for the samples studied in this work was based on the result of our previous research [17]. In ref. 17 the effect of the symmetry on the resistive characteristics of Nb/Cu multilayers prepared in the same way as in this work was investigated for Nb in the range 5…12. It was shown that for Nb > 10 the symmetry effect becomes less pronounced due to the smaller influence of the surface effects with increasing the Nb value. In order to demonstrate the validity of the observed phenomena we have also prepared another pair of Nb/Cu samples with Nb equal to 5 and 6. This couple of multilayers have been fabricated in the same deposition run.
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Temperature Dependence of the Upper Critical Magnetic Field In Fig. 2a we present the measured temperature dependencies of parallel and perpendicular magnetic fields for the sample NP. The behavior of Hc2||(T) reveals the conventional 2D-3D crossover for S/N multilayers [7, 9, 10, 12, 15, 18]. In Fig. 2b we present the H-T phase diagram for sample SP, with the symmetry plane in the center of the Nb layer. It is clearly seen that the Hc2||(T) curve is quite different, while the Hc2⊥(T) dependence is very similar to that of NP sample. We did not see the pronounced linear part in the Hc2||(T) dependence as it is usually observed for S/N multilayers in the case of dN ≈ dS ≈ ξS. The Hc2||(T) curve seems to be square root like in the whole temperature range even close to Tc. This is the signature of the 2D behavior.
H C2|| H C2perp
HC2(T)
3
2
1
0
2
3
T(K)
Fig. 2a. H-T phase diagram for sample NP.
4
Properties of S/N Multilayers with Different Geometrical Symmetry
257
3,0
HC2|| HC2perp
2,5
HC2(T)
2,0
1,5
1,0
0,5
0,0
2,0
2,5
3,0
3,5
T(K) Fig. 2b. H-T phase diagram for sample SP.
As already pointed out, both the samples SP and NP were obtained in the same deposition run and therefore represent the same Nb properties and Nb/Pd interface behaviors. The only difference between these two samples is the different number of bilayers and, consequently, their different symmetry. To confirm the effect of the samples’ symmetry on the Hc2||(T) dependencies we have investigated another S/N system. We have fabricated and measured a pair of multilayers with a different N material, namely Nb/Cu, with 10 (NC sample) and 9 (SC sample) bilayers. Taking into account the absence of the large spin susceptibility in Cu compared to Pd [13], the thickness of Cu was larger, dN = 200Å. The results of the Hc2(T) measurements for the NC and SC samples are presented in Fig. 3a and Fig. 3b, respectively [18]. Again the Hc2||(T) dependence for the sample NC is usual for the S/N multilayers revealing a pronounced 3D-2D crossover. The Hc2||(T) dependence for the sample SC is square root like in the whole temperature range up to Tc. In the inset of Fig. 3a we show the H-T plot for the Nb/Cu multilayer with Nb = 6. Furthermore, in the inset of Fig. 3b the H-T phase diagram is presented for another Nb/Cu sample with
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S. L. Prischepa 3,0 2
Hc2(T)
2,5
Hc2(T)
2,0
0
1,5
3
4
5
6
7
T(K)
1,0
Hc2|| Hc2perp
0,5 0,0
2
3
4
5
6
7
T(K) Fig. 3a. H-T phase diagram for sample NC. Inset: the same for sample with Nb=6.
3,0 Hc2(T)
2
2,5 2,0
Hc2(T)
0
3
4
5
6
7
T(K)
1,5
Hc2|| Hc2perp
1,0 0,5 0,0
2
3
4
5
6
T(K) Fig. 3b. H-T phase diagram for sample SC. Inset: the same for sample with Nb=5.
Nb = 5. Also in this case the behavior of Hc2|| versus T depends almost only on the symmetry of the multilayers. Based on the obtained results for different systems and a different number of bilayers, we may conclude that the observed increase of the temperature interval of the 2D nature of samples with the symmetry plane
Properties of S/N Multilayers with Different Geometrical Symmetry
259
in the center of S layer is a general feature of S/N metallic multilayers. According to our opinion, this behavior is mainly due to the effect of finite dimensions on the nucleation of superconductivity. In order to get deeper insight into the problem, we performed angular measurements of the upper critical magnetic field for Nb/Pd and Nb/Cu samples at different temperatures.
Angular Dependence of the Upper Critical Magnetic Field In this section we will present the angular dependences of Hc2 at different temperatures. For sample SP we performed such kind of measurements in the temperature interval 1.91 K ≤ T ≤ 3.61 K (0.52 ≤ t = T/Tc ≤ 0.98). The results of the measurements for two temperatures (3.50 K and 1.91 K) are presented in Fig. 4. As it is clearly seen, there is no significant difference in the shape of the Hc2(Θ) curve for both temperatures. The dashed lines in Fig. 4 correspond to Tinkham’s formula [5] for a 2D thin film. It is seen that the theory explains well the Hc2(Θ) data, especially at small angles (|Θ| < 20°). The present result is typical for this sample. The same behavior was observed also at other temperatures as well as for Nb/Cu multilayers. According to our opinion it confirms the 2D character of superconductivity of S/N multilayer with an odd number of bilayers in the whole temperature range. 30000 25000
T=3.50K T=1.91K 2D
20000
Hc2(Oe)
15000 10000 6000 5000 4000 3000 2000 1000 -20
-10
0
10
20
30
40
50
60
70
80
90
Θ
Fig. 4. Hc2(Θ) for sample SP at two temperatures. The dashed lines are for the Tinkham result.
260
S. L. Prischepa
In Fig. 5 we show the Hc2(Θ) dependences for sample NP at 4 different temperatures: T = 4.19 K (t = 0.99), T = 3.97 K (t = 0.94), T = 3.45 K (t = 0.82), and T = 2.05 K (t = 0.49). As it is clearly seen, the shape of the Hc2(Θ) curves changes significantly with temperature. At T very close to Tc (t = 0.99), the features of 3D behavior is present and the curve is bellshaped. The solid line in this figure corresponds to the 3D LawrenceDoniach result [16]. 800
2500
700
T=4.19K T=3.97K
600
2250
2000 500 1750
400
1500
300
Hc2(Oe)
200
1250
100 1000
14000
27500
T=3.45K T=2.05K 12000
22500
10000 17500
8000 12500
6000 7500
4000 -100
-80
-60
-40
-20
0
20
40
60
80
100
Θ
Fig. 5. Hc2(Θ) for sample NP at 4 temperatures. The dashed lines are for the Tinkham result and the solid line is for the Lawrence-Doniach result.
Properties of S/N Multilayers with Different Geometrical Symmetry
261
Good agreement with the experiment is seen for |Θ| < 20°. This, indeed, corresponds to the linear Hc2(T, Θ = 0) dependence. But at the slightly smaller reduced temperature t = 0.94, at which Hc2(T, Θ = 0) is still a linear function, the measured Hc2(Θ) curve reveals a pronounced cusp. This is in disagreement with the dimensionality derived from the measurement of the temperature dependence of the parallel upper critical magnetic field. Moreover, the experimental data are well described by the thin film limit [5] (dashed line for data of this temperature). In the region around T* (t = 0.82) the Hc2(Θ) curve becomes more complicated showing a sudden increase of the Hc2 values at Θ < 10°. Finally, at low temperatures (t = 0.49), the experimentally measured Hc2(Θ) dependence becomes similar to the Hc2(Θ) dependence of a thin film, however, in the same way as found previously for different multilayers [19], the experimental points fall faster than the Tinkham curve (dashed line). The results presented in this section show the different behavior of the angular dependences of the upper critical field for two different kinds of samples. For multilayers with the symmetry plane in the center of S layer, the Hc2(Θ) curves are well described by the expression for 2D thin films, for which the nucleation position of the superconducting phase is supposed to be in the center of the sample. Moreover, the two-dimensionality of these samples was also confirmed by the results of the Hc2(T) measurements (Figs. 2b, 3b). From this point of view it is reasonable to suppose that for this kind of sample there is only a single superconducting nucleus at the Hc2 value for the whole investigated temperature range in the parallel magnetic field configuration. Moreover, it is likely that this nucleus is located in the middle of the central S layer. At the same time the physical picture for samples with the symmetry plane in the central N layer is more complicated. First of all, at T very close to Tc, where the coherence length is larger than the sample dimensions, the Hc2(Θ) curve is bellshaped with the derivative d Hc2 (Θ)/d Θ |Θ=0=0. This reflects a 3D behavior. Then, still in the temperature region of 3D behavior (according to the Hc2(T, Θ = 0) result), but at slightly smaller temperatures (t = 0.94), the Hc2(Θ) curve has a cusp. Previously, the presence of a cusp in the Hc2(Θ) curve was considered as a prove of two-dimensionality. Our experimental results strongly indicate that for S/N multilayers this is not always the case. At least it is not valid for samples with the symmetry plane within N layer. In the region of the 2D behavior (according to the Hc2(T, Θ = 0) result), where the value of ξ becomes comparable to the multilayer period, the Hc2(Θ) curves become more complicated revealing the probable complex character of the superconducting phase nucleation in these samples. The likely reason of such effects could be related to the
262
S. L. Prischepa
integral surface effects in multilayers. Theoretical work is in progress in order to get a deeper insight into the problem.
Conclusion In conclusion, we have performed a systematic study of the influence of the finite dimensions of the S/N proximity coupled multilayers on the H-T phase diagram and angular dependences of the upper critical magnetic field. Our experiments were performed with two different systems, namely Nb/Cu and Nb/Pd, with different ratios between dS and dN and different Nb values (i.e. symmetry of the samples). The observed Hc2||(T) dependences differ with respect to the position of the geometrical symmetry plane. For samples with the symmetry plane located in the middle of the N layer the measured Hc2||(T) curves were typical for S/N multilayers, presenting the well known 3D-2D crossover at a certain T*
Acknowledgments This work has been partially supported by the national Program "Nanoelectronics" under the grant 01-3109. Valuable discussions with V.N. Kushnir and C. Attanasio are gratefully acknowledged. The experimental work has been done with assistance of C. Cirillo and M. Salvato.
References 1. 2. 3. 4.
Abrikosov AA (1957). Sov Phys JETP 32:1442 Saint-James D, De Gennes PG (1963). Phys Lett 7:306 Saint James D (1965). Phys Lett 16:218 Yamafuji K, Kusayanagi E, Irie F (1966). Phys Lett 21:11
Properties of S/N Multilayers with Different Geometrical Symmetry
263
5. Tinkham M (1963). Phys Rev 129:2413 Tinkham M (1964). Phys Lett 9:217 6. Fink H (1969). Phys Rev 177:732 7. Takahashi S, Tachiki M (1986). Phys Rev B 33:4620 8. Mineev VP (2001). Phys Rev B 65:012508 9. Koperdraad RTW, Lodder A (1996). Phys Rev B 54:515 10. Ciuhu C, Lodder A (2001). Phys Rev B 64:224526 11. Schuller IK (1980). Phys Rev Lett 44:1597 12. Chun CS, Zheng G-G, Vicent JL, Schuller IK (1984). Phys Rev B 29:4915 13. Nieuwenhuys GJ (1975). Adv Phys 24:515 14. Cirillo C, Prischepa SL, Salvato M, Attanasio C (2004). Eur Phys J B 38:59 15. Cirillo C, Attanasio C, Maritato L, Mercaldo LV, Prischepa SL, Salvato M (2003). J Low Temp Phys 130:509 16. Lawrence WE, Doniach S (1970). Proc LT-12:361 17. Kushnir VN, Prischepa SL, Della Rocca ML, Salvato M, Attanasio C (2003). Phys Rev B 68:212505 18. Kushnir VN, Prischepa SL, Cirillo C, Della Rocca ML, Armenio A, Maritato L, Salvato M, Attanasio C (2004). Eur Phys J B 38:59 19. Jin BY, Ketterson JB, NcNiff EJ, Foner S, Schuller IK (1987). J Low Temp Phys 69:39
Andreev Reflection in Ballistic SuperconductorFerromagnet Contacts
L. R. Tagirov1,2, B. P. Vodopyanov2 1 2
Kazan State University, 420008, Kazan, Russia Kazan Physico-Technical Institute of RAS, 420029 Kazan, Russia
Abstract:
On the basis of Eilenberger-type equations of superconductivity for metals with exchange-split conduction band we derive an original expression for the Andreev conductance of superconductorferromagnet point contacts. We give particular formulas for different relations between Fermi momenta of contacting metals, and calculate the Andreev conductance as a function of exchange splitting of the ferromagnet conduction band. From comparison with experimental data on point contact Andreev reflection spectroscopy we estimate the conduction band spin polarization for a series of ferromagnets in contact with superconductors.
Keywords: Quasi-classical theory of superconductivity, Andreev reflection, ferromagnet-superconductor point contacts
Introduction At low temperatures, an electric current from a normal metal enters through a contact in a superconductor by means of Andreev reflection [1]. In the process of Andreev reflection an electron coming to the normal metal/superconductor interface is reflected back into the normal metal as a hole with the opposite spin, and the formed Cooper pair moves through the superconductor transferring a charge 2e . This results in doubling of the differential conductance of a pure N/S nanocontact which has been demonstrated theoretically in Ref. [2] (BTK) based on the solution of the Bogoliubov equations. In Ref. [3] it was emphasized that the Andreev 265 R. Gross et al. (eds.), Nanoscale Devices - Fundamentals and Applications, 265–275. © 2006 Springer. Printed in the Netherlands.
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L. R. Tagirov, B. P. Vodopyanov
reflection in ferromagnet/superconductor (F/S) contacts is suppressed as the spin polarization of the ferromagnet conduction band sets up. The general understanding of this suppression is by a reduction of the efficiency of the Andreev reflection process associated with the decrease of the number of conducting channels in the minority spin-subband (the subband with the lower value of the Fermi momentum). It has been proposed to use the suppression of Andreev reflection in F/S contacts to determine the spin polarization of the conduction band of ferromagnets (Andreev spectroscopy of ferromagnets) [4-9]. Experimental data were interpreted on a phenomenological basis [4, 5, 7]. The BTK theory was generalized and applied to F/S point contacts in the theoretical works [8-13]. However, the general requirement that the Fermi momenta of contacting ferromagnetic and superconducting metals are not the same has not been taken into account. The number of experiments on Andreev spectroscopy of diverse ferromagnets grows [14-20], what demands an adequate theoretical description. In this paper we derive particular formulas for the Andreev conductance of a superconductor in a point contact with different types of ferromagnets. From comparison with experiments on Andreev spectroscopy we estimate the polarizations of the conduction bands of ferromagnetic metals.
Equations of Superconductivity and General Boundary Conditions for Superconductor-Ferromagnet Contacts The equations for the equilibrium thermodynamic Green functions (GF) read: v w 1 w gˆ & (vˆ xj gˆ gˆ vˆ xj1) [ Kˆ gˆ ] 0 wx 2 wȡ w l v& w 1 l l 1 l ] 0 (vˆ xj G G vˆ xj ) [ Kˆ G sgn( pˆ xj ) G wx 2 wȡ sgn( pˆ xj )
(1)
The quantities which enter the above equation are defined as follows: Kˆ
1 1 i vˆ xj2 (iH nW z 'ˆ 6ˆ )vˆ xj2 i ( pˆ xj Wˆ x pˆ xjWˆ x ) 2
> a b @ r
ab r ba
(2)
Andreev Reflection in Ballistic Superconductor-Ferromagnet Contacts
F 6ˆ
S 6ˆ
ic _ u _2
i
³
dp& (2S )
1
2
1
(vˆ x ) 2 gˆ F (vˆ x ) 2
1 1 gˆ S ! S S 2W W
c _ u _2
mp S
S
267
(3)
(4)
l are the quasiclassical GF of incident and where the functions gˆ and G reflected quasiparticles, W D are the Pauli matrices, D is the spin index,
Hn
(2n 1)S T is the Matsubara frequency, ' is the order parameter, pDF
is the Fermi momentum of the ferromagnet spin-subbands, p S is the Fermi momentum of a superconductor, u is the potential of interaction of electrons and impurities, c is the concentration of impurities, W S is the mean free time of electrons in a superconductor, and brackets mean averaging over the solid angle: ! ³v d : / 4S . Hereafter we use the unit system in which = c 1 , so that we do not distinguish momentum and wave number, for example. We assume that the F/S interface coincides with the plane x 0 . Then, r ( x ȡ) , and ȡ ( y z ) is the coordinate within the contact plane, vˆ1x the matrix of the Fermi velocities projections to the x direction, and the other notations are standard. The above Eilenberger-type equations for the metal with the exchangefield-split conduction band had been derived for the first time in [21, 22]. They are valid for arbitrary band splitting and arbitrary spin-dependent mean free paths within the quasiclassic approximation. The system of Eqs. (1) must be supplemented with boundary conditions at the F/S interface (see details in Refs. [22, 23]):
gˆ gˆ Gl Gl S a
d
F a
d
S a
d
F a
d
gˆ a bˆ1 bˆ 2 gˆ a gˆ a bˆ 3 bˆ 4 gˆ a bˆ 3 bˆ 4 gˆ a bˆ1 bˆ 2 gˆ a gˆ a bˆ 3 bˆ 4 gˆ a bˆ1 bˆ 2
The bˆi matrices are given by
(5)
(6)
268
L. R. Tagirov, B. P. Vodopyanov
l s gˆ s G l s gˆ s bˆ1 G l s gˆ s G l s gˆ s bˆ 3 G
l s gˆ s G l s bˆ 2 gˆ s G l s gˆ s G l s , bˆ 4 gˆ s G
(7)
l s ( a ) are the matrices, which are symmetric (s) and where gˆ s ( a ) and G
antisymmetric (a) with respect to the variable p jx : gˆ s ( a )
1 l s(a) ª gˆ r gˆ ¼º G 2¬ !
1 l rG l ¼º ªG 2¬ !
(8)
Note that only the diagonal part of the functions is continuous at the F/S interface (Eq. (5)). These BCs take into account explicitly the spindependence of F/S interface.
General Expression for the Andreev Conductance of F/S Point Contacts We start with the equation for the current I in the linear approximation with respect to the electric field E ( Ex 0 0) . The current is calculated on the ferromagnet side of the contact at xo 0 : Ix
f § w ·¸ ° e2 dH ¨ w W Tr lim ¨ ¸ ® z ³ c ¸ 2 c ¨ 2m r or © wx wx ¹ °¯ f 4S cosh 2 (H 2T )
u ³ dr1G R (H r r1 ) Ex (r1 )W z
½ w A G (H r1 r c ) ¾ wx1 ¿
(9)
Here, G R ( A) is the retarded (advanced) GF, which is obtained from the temperature dependent GFs, Eqs. (1), by substituting H r iG for iH n . After a Fourier transformation with respect to ȡ ȡ ' coordinate, we obtain the ballistic conductance GF S of the F/S point contact:
GF S
f Ae 2 ° dH ³ dp& 2 Tr ®W z ³ 2 16S ° f cosh (H 2T ) 2S ¯
`
l sRW z G l sA G l aRW z G l aA º u ª¬1 gˆ sRW z gˆ sA gˆ aRW z gˆ aA G ¼
(10)
Andreev Reflection in Ballistic Superconductor-Ferromagnet Contacts
269
Now, we have to solve the first of Eqs. (1) with the BCs given by Eqs. (5)-(7). When gˆ is independent of ȡ the solution to Eq. (1) takes the form gˆ j
e
ˆ sgn( pˆ jx ) Kx
ˆ sgn( pˆ jx ) Kx Cˆ j (p Fj )e Cˆ j
(11)
The matrices Cˆ j represent the values of GF gˆ j at large distances from the F/S interface, and for the superconductor we have § g Cˆ 2 ¨ © f
f · ¸ g ¹
1
H n2 '
§ ¨ ¨ 2 ¨¨ ©
Hn i'
i' ·¸ ¸ H n ¸¸¹
(12)
In the ferromagnet gˆ 1 must tend to Cˆ1 W z sgn(H n ) at x o f , and in the superconductor gˆ 2 must tend to Cˆ 2 at x o f . Performing the matrix multiplication in Eq. (11), we find that these conditions are fulfilled if the following relationships are satisfied: ˆ ˆ Cˆ j Cˆ j (p Fj ) C j (p Fj )C j
sgn( p jx )(1) j Cˆ j (p Fj )
(13)
From these relationships, it follows that gˆ js Cˆ j (1) j Cˆ j Cˆ ja
gˆ ja Cˆ ja
(14)
Passing in Eq. (14) to the functions gˆ rs , substituting them into the system of BCs given by Eqs. (6), and solving the resulting equations in the linear approximation with respect to Cˆ ar 1 2[Cˆ aS r Cˆ aF ] , we find, r Cˆ a
f (1 RD RD r DD DD ) 2[1 RD RD (1 RD RD ) g ]
Wx
(15)
Now, from Eq. (10) and making use of Eqs. (5) and (15), we find the Andreev conductance GA of the F/S point contact '
GA u³
GF S (V
dH 4S cosh 2 (H 2T ) 0
Ae 2 ³ 2
dp &
2S
0)
4 ' Dp Dn 2
2
(1 Rp Rn ) 2 ' 4 Rp Rn H 2
(16)
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L. R. Tagirov, B. P. Vodopyanov
The experimental data [4-8] are given in the normalized form. The normalization is, in fact, the conductance at high voltage, V>>ǻ, which is equal to the conductance between a ferromagnet and a normal metal. This conductance can be calculated as follows: GF N (V
Ae 2 §¨© ppF ·¸¹
0)
2
Tcr
³ dT
8S 2 Ae2 §¨ pnF ·¸ © ¹ 8S 2
p
sin(2Tp ) Dp
0
2
(17)
T cr
³ dT sin(2T ) D n
n
n
0
The equations (16), (17) are valid for an arbitrary transmission coefficient DD . They depend on the relationship between the Fermi momenta of the ferromagnet spin-subbands ppF , pnF , and the Fermi momentum of the superconductor, p S . Upon integration on the angles the conservation of the component of the momentum parallel to the interface plane must be obeyed: p__
ppF sin Tp
pnF sin Tn
p S sin T S ,
(18)
which determines the critical angle șcr in equation (17).
Andreev Conductance for Various Relations Between Fermi Momenta of Contacting Metals We analyze three possible relations between the Fermi momenta of contacting metals: 1) p S ppF pnF , 2) ppF p S pnF , and finally, 3) ppF pnF p S .
Case 1), p S ppF pnF : The expression for GA takes the form:
Andreev Reflection in Ballistic Superconductor-Ferromagnet Contacts
2
G
Ae 2 pS '
(1) A
2
1
dx
³ cosh x ' / 2T 2
4S T
0
S /2
(19)
Dp Dn
u ³ dT S sin(2T S )
(1 Rp Rn ) 2 4 Rp Rn x 2
0
271
The corresponding normal state conductance is given by G
(1) F N
(V
0)
Ae 2 pS
2 S 2
8S 2
³ dT
sin(2T S )( Dp Dp ).
S
0
(20)
For particular calculations we use model expressions for the transmission coefficients corresponding to the direct contact between metals: 4 pxn pxS
Dn
( pxn pxS )
4 pxp pxS
Dp
2
( pxp pxS ) 2
.
(21)
With these transmission coefficients GF(1)N can be calculated analytically: 2
G
where, G nF
(1) F N
Ae 2 §¨© p S ·¸¹ §¨ G nF (2 G nF ) ¨ ¨ ¨ ©
6S 2
p S pnF and G pF
(1 G nF ) 2
G pF (2 G pF ) ·¸ ¸ (1 G pF ) 2 ¸¸¹
(22)
p S ppF .
Case 2): ppF p S pnF . The expression for GA takes the form: 2
G
(2) A
Ae 2 §¨© ppF ·¸¹ ' 2
4S T
S /2
u ³ dTp sin(2Tp ) 0
1
dx
³ cosh x ' / 2T 2
0
Dp Dn (1 Rp Rn ) 2 4 Rp Rn x
(23) 2
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L. R. Tagirov, B. P. Vodopyanov
The corresponding normal state conductance is given by G
(2) F N
(V
Ae 2 §¨© ppF ·¸¹
0)
2
S 2
³ dT
8S 2
p
sin(2Tp ) Dp
0
2
Ae2 §¨ p S ·¸ © ¹ 8S 2
(24)
S 2
³ dT
S
sin(2TS ) Dn .
0
It can be evaluated with the model transmission coefficients as follows: 2
G
where G nF
Ae 2 §¨© p S ·¸¹ §¨ G nF (2 G nF )
(2) F N
¨ ¨ ¨ ©
6S 2
p S pnF and G pF
(1 G nF ) 2
(G pF )3 (2 G pF ) ·¸
¸ ¸ ¸ ¹
(1 G pF ) 2
,
(25)
ppF p S .
Case 3), ppF pnF p S . The expression for GA takes the form: 2
G
(3) A
Ae 2 §¨© ppF ·¸¹ '
4S 2T
1
dx
³ cosh x ' / 2T 2
0
S /2
u ³ dTp sin(2Tp ) 0
Dp Dn
(1 Rp Rn ) 2 4 Rp Rn x 2
(26)
The corresponding normal state conductance is given by G
(3) F N
(V
0)
Ae 2 §¨© ppF ·¸¹
2
S 2
³ dT
8S 2 Ae2 §¨ pnF ·¸ © ¹ 8S 2
p
sin(2Tp ) Dp
0
2
(27)
S 2
³ dT
n
sin(2Tn ) Dn .
0
It can be evaluated analytically as follows:
Andreev Reflection in Ballistic Superconductor-Ferromagnet Contacts 2
G
where G nF
(3) F N
Ae 2 §¨© ppF ·¸¹ §¨ G pF (2 G pF ) 6S 2
¨ ¨ ¨ ©
pnF p S , G pF
(1 G pF ) 2 ppF p S
G nF (2 G nF ) ·¸ ¸, G 2 (1 G nF ) 2 ¹¸¸
and G
273
(28)
ppF pnF d 1 . The parameter
ppF pnF is the measure of the ferromagnet conduction band exchange splitting.
G
Discussion of Results Let us now compare our results with the original procedure of polarization estimation proposed in [5]. The authors argue that the normalized conductance measured in their work depends on polarization as GF S Gn 2(1 PI ) (Eqs. (4)–(6) in [5]), where PI ( I n I p ) ( I n I p ) , and Gn GF N is the conductance at high voltages across the contact ( eV ' ). In the course of discussion, the authors silently have made the following two assumptions: Firstly, they identified the current polarization PI with the contact polarization Pc
( N n vnF N p vpF ) ( N n vnF N p vpF ) (1 G 2 ) (1 G 2 ) ,
(29)
where ND and vD are the density of states and the Fermi velocity in the D spin subband of the ferromagnet, respectively. We think that this identification is not quite correct, because it tacitly implies the independence of the total current I n I p through the contact of the spin polarization of the ferromagnet in the normal state. However, it is evident from Eqs. (22), (25) and (28) that GF N essentially depends on G . As a result, the reduced conductance GA (V 0) GF N is a nonlinear function of the contact polarization Pc [23]. Secondly, they do not distinguish the band parameters of the ferromagnet from those of the superconductor in contact. That is, they do not take into account the Fermi-momenta relations of the contacting metals, for which we have considered three distinct cases in the previous section. We believe that our theory allows to estimate more accurately the polarization parameter G of the ferromagnet conduction band by applying the proper case distinguished in the previous section to the particular couple of materials in contact.
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L. R. Tagirov, B. P. Vodopyanov
Conclusion In this paper we present results of our theoretical studies of the Andreev conductance in contacts between superconductors and ferromagnets. On the basis of a microscopic approach we derived an original expression for the Andreev conductance. Our expression takes into account explicitly the spin dependence of the interface transparency and the spin-dependent conservation laws for scattering at the F/S interface. It also distinguishes between the metals being in contact by suitable relations between the Fermi momentum of a superconducting metal and the Fermi momenta of the spinsubbands of the conduction band of the ferromagnetic metal. The work was supported by the Russian ministry of education and science, grant CKP-KSU.
References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18.
Andreev AF (1964). Zh Eksp Teor Fiz 46:1823 [Sov Phys JETP 19:1228] Blonder GE, Tinkham M, Klapwijk TM (1982). Phys Rev B 25:4515 Jongh de MJM, Beenakker CWJ (1995). Phys Rev Lett 74:1657 Upadhyay SK, Palanisami A, Louie RN, Buhrman RA (1998). Phys Rev Lett 81:3247 Soulen RJ, Byers JM, Osofsky MS et al. (1998). Science 282:85; (1999) J. Appl. Phys. 85:4589 Osofsky MS, Nadgorny B, Soulen RJ et al. (1999). J Appl Phys 85:5567 Nadgorny B, Soulen RJ, Osofsky MS et al. (2000). Phys Rev B 61:3788(R) Ji Y, Strijkers GJ, Yang FY, et al. (2001). Phys Rev Lett 86:5585 Strijkers GJ, Ji Y, Yang FY, Chien CL (2001). Phys Rev B 63:104510 Mazin II, Golubov AA, Nadgorny B (2001). J Appl Phys 89:7576 Kashiwaya S, Tanaka Y, Yoshida N, Beasley MR (1999). Phys Rev B 60:3572 Golubov AA (1999). Physica C 326–327:46 Kikuchi K, Imamura H, Takanashi S, Maekawa S (2001). Phys Rev B 65:20508 Osofsky MS, Soulen RJ, Nadgorny BE et al. (2001). Mater Scien Eng 84:49 Nadgorny B, Mazin I, Osofsky M et al. (2001). Phys Rev B 63:184433 Ji Y, Chien CL, Tomioka Y, Tokura Y (2002). Phys Rev B 66:012410 Kant CH, Kurnosikov O, Filip AT et al. (2002). Phys Rev B 66:212403 Nadgorny B, Osofsky MS, Singh DJ et al. (2003). Appl Phys Lett 82:427
Andreev Reflection in Ballistic Superconductor-Ferromagnet Contacts
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19. Raychaudhuri P, Mackenzie AP, Reiner JW, Beasley MR (2003). Phys Rev B 67:020411 20. Auth N, Jakob G, Block T, Felser C (2003). Phys Rev B 68:024403 21. Vodopyanov BP, Tagirov LR (2000). Physica B 284-288:509 22. Vodopyanov BP, Tagirov LR (2003). Pisma Zh Eksp Teor Fiz 77:153 [JETP Letters 77:126] 23. Vodopyanov BP, Tagirov LR (2004). In: B. Aktaú, L.R. Tagirov and F. Mikailov (eds) NATO Science Series II: Mathematics, Physics and Chemistry. Kluwer Academic Publishers v 143, pp 145-167
Superconductor-Insulator Transition in a PbZSn1-ZTe:In Solid Solution
D. V. Shamshur1, D. V. Shakura1, R. V. Parfeniev1, S. A. Nemov2 1
A. F. Ioffe Physical-Technical Inst., RAS, St.Petersburg, Russia
2
State Polytechnical University, St. Petersburg, Russia
Abstract:
We report on the influence of the composition and In content of heavily doped PbzSn1-zTe:In semiconducting compounds on their superconducting characteristics and low temperature conductivity.
Keywords: solid solutions, superconductivity, critical parameters, dielectric states, magnetoresistance, low temperature
Introduction A particular problem of superconductivity (SC) in semiconductor materials based on the AIVBVI compounds is related to the influence of doping with group III elements such as In and Tl, which form impurity states located within the allowed band – in the conduction band for PbTe:In and in the valence band for SnTe:In and PbTe:Tl [1, 2]. The doping of PbTe with In induces an effect known as long-time photoconductivity, while the In doping of SnTe results in the transition to SC at low temperatures with a threshold type increase of the critical temperature Tc with increasing the In content above about 1 at.% [3]. According to [2, 3], SC in SnTe:In is connected with the filling of the In impurity states with a peak value of the density of states (DOS) that exceeds the valence band DOS. The hole mobility is also extremely sensitive to the Fermi level position within the impurity band (resonant impurity states). Furthermore, the finite energy width of the impurity band IB is determined by hybridization of band states with the impurity states [4]. It should be pointed out that SnTe and solid 277 R. Gross et al. (eds.), Nanoscale Devices - Fundamentals and Applications, 277–288. © 2006 Springer. Printed in the Netherlands.
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solutions Sn1-zPbxTe without In doping are also superconductors, however with SC critical parameters which are lower by an order of magnitude [5]. The bulk character of the SC was determined by specific heat measurements on SnTe1+y, SnTe:In and PbSnTe:In [5, 6]. The present work reports the results of a systematic study of the influence of the composition and In content on the SC characteristics as well as on the low temperature conductivity of the In doped compounds of the tin and lead-chalcogenides.
The Samples and Experiment Bulk polycrystalline samples of composition (PbzSn1-z)1-xInxTe with grain sizes d ~ 100 µm were prepared by using the metallocheramic technology with final homogenization by annealing (600°C, 200 h) [7]. The composition of the samples varies from z = 0 to 0.9 at the fixed In content: x = 0.05, 0.16 and 0.20. The electron microprobe analysis did not reveal any trace of a secondary phase in the samples. The samples had the p-type conductivity with a hole concentration in the range p ≅ 5×1021 – 1022 cm-3. The hole mobility Rσ = 10 – 1 cm2/Vs was determined from the Hall and resistivity data. The low mobility values were caused by hole scattering on the In resonant impurity states. The SC transition was determined by four probe measurements of the electrical resistivity ρ in the temperature range between 0.4 K and 4.2 K in 3He and 4He setups and applied magnetic fields up to 12 kOe. The critical parameters Tc and HC2 (the upper critical magnetic field) were determined from the condition ρ = 0.5 ρN, where ρN is the normal-state resistivity. The SC transition width determined from the interval between the resistivity levels 0.1 ρN ÷ 0.9 ρN was equal to 0.1 K ÷ 0.2 K for temperature dependences ρ (T) and 0.5 – 1 kOe for magnetic field dependences ρ (H) (see Fig. 1).
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Fig. 1. The resistance of a (Pb0.5Sn0.5)0.8In0.2Те solid solution as a function of temperature (left) and magnetic field for T < Tc (right).
Experimental Results and Discussion Before discussing our experimental results we would like to comment on the energy position of the impurity band (IB) in the energy band structure of the solid solutions between SnTe:In and PbTe:In (see Fig. 2). The main extrema in the conduction band and valence band are located at the L-points of the Brillouin zone and exhibits an inversion of the band edge positions L6+ and L6- on changing the Pb content as indicated by the zero-gap state at x = 0.65 (T = 4.2 K [1]). The extra maximum in the valence band is situated at the Σ-points of the Brillouin zone. It is separated from the main extremum by ∆εLΣ = 0.3 eV for SnTe at 4.2 K and varies slowly in the solid solutions [1]. The In impurity states in SnTe:In are located on the background of the allowed valence band spectrum. They lie near the edge of the Σ-valence band with a high DOS and shift on the energy axis towards the conduction band edge during isovalent doping with Pb. The width of the IB decreases from Γ ~ 200 meV in SnTe:In to Γ ~ 1 meV in PbTe:In due to a reduction of the interaction between the IB states and VB (L and Σ) states and the lack of the interaction with the conduction band in PbTe [1]. The location of the Fermi level within the IB not only determines the hole mobility but also the SC parameters of the investigated solid solutions.
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Fig. 2. The energy band spectrum of (PbzSn1-z)1-xInxТе alloys.
Figures 3 and 4 show a comparison of the SC parameters in the three series of the samples with different compositions and In doping levels over the investigated composition range (z = 0 ÷ 0.6) when the IB shifts from the Σ-band into the L- valence band. These figures indicate an unusual behavior of the SC parameters in materials with impurity resonance states, which interact strongly with the valence band states and pin the Fermi level of holes. Tc and Hc2(0) have bell-shaped dependences on the composition and both the value and the position of the maximum on the z scale increase with increasing In content.
Fig. 3. Dependence of the critical temperature Tc (left) and the temperature derivative of Hc2 near Tc (right) on the composition.
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Fig. 4. Dependence of the upper critical magnetic field Hc2(0) extrapolated to T = 0 (left) and the DOS at the Fermi level N(0) (right) on the composition. The DOS was calculated using the relation N(0) = 2.84×1014|dHc2/dT |TcρΝ−1 valid for “dirty” superconductors.
The highest SC parameters are obtained for the composition (Pb0.5Sn0.5)0.8In0.2Te. At low In content (5 at.%) a sharp decrease of both Tc and Hc2(0) are observed at z > 0.2. This correlates with the total DOS at the Fermi level N(0) estimated from our experimental data and using the relation for “dirty” superconductors (see Fig. 4). The features in the measured z-dependencies of Tc and dHc2/dT can be understood if one takes into account the rapid shift of the IB with increasing z towards the main valence band edge. As a result, on the left-hand side of the bell shaped curve Tc(z) the IB filling by holes increases with increasing z due to holes transferred from the valence band states to the IB states thereby resulting in a monotonic increase of the SC parameters. On the right-hand side of the curves the rapid decrease of the SC parameters and the destruction of SC for T > 0.4 K are caused by the decrease of the total DOS at the Fermi level due to the IB leaving the z-valence band with high DOS as shown in Fig. 4. On the other hand, it is possible to restore the high SC properties of the system at fixed z by raising the In content, that is, by passing the Fermi level deeper into the z-valence band of the compound. In these samples there is a strong exchange of holes between the impurity states and the Σ-valence band states producing an additional smearing of the IB. Shelankov [8] had suggested a model, which relates the high critical SC parameters in the In doped IV-VI solid solutions to the
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interaction between the quasi-local impurity states and the lattice with a high dielectric constant. It was shown that a mixed-valence model may be used for the description of an energy variation of the local impurity level produced by a lattice distortion. The electron-electron interaction mediated by a virtual shift of the localized energy level is the physical reason for the enhancement of SC. Besides the BCS coupling constant g0, the additional term gimp was introduced in the weak coupling approximation: g = (1 – aimp)g0 + aimpgimp Γ 2 /E2 + Γ 2
(1)
Here, aimp = Nimp(0)/(Nb(0) + Nimp(0)) is the fractional impurity contribution to the total DOS composed of the band value Nb(0) and Nimp = 4π Γ (Cimp./(E2+ Γ 2 )
(2)
Here, Cimp is the concentration of impurities creating the local states,
Γ is the level width, and E is the energy level position relative to EF. Equa-
tion (1) predicts a bell-shaped dependence of Tc on the Fermi level position within the IB. The heavily doped solid solutions with z > 0.5 show a transition from the SC state to a dielectric state at low temperatures (Fig. 5). In the temperature regime T = 40 ÷ 100 K the resistivity versus temperature curves show an activated behavior with an activation energy ∆ ~ 10 meV for z = 0.8. At low temperature in samples with the high Pb content (z = 0.9) the ρ(T) dependence is weaker than an exponential one because a finite conductivity through conduction band states (z = 0.9) appears. In the sample with z = 0.8, for which the Fermi level coincides with the energy gap, a Mott T– 1/4 is obtained down to 0.6 K indicating the presence of variable range hopping conductivity near the Fermi level via impurity states [9]. The conductivity in the overlapping energy region of the IB and valence band states may be less than the hopping conductivity due to the resonant scattering of band holes.
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Fig. 5. Temperature dependence of the resistivity in (PbZSn1-Z)0.84In0.16Te solid solutions with various Pb contents (z = 0.5 – 0.9) and fixed x = 0.16.
Due to a high In concentration in the investigated solid solutions fluctuations in the composition lead to the appearance of SC regions and normal state/SC state (N/S) interfaces in a disordered superconductor - semiconductor mesoscopic system. The SC insertions in the samples reveal themselves in the measured temperature dependences of the resistivity especially in the x = 0.2 series for z > 0.6 (see Fig. 6). When the SC transition in the insertions begins, the resistivity decreases at low temperature and nonlinear current-voltage characteristics are observed (see Fig. 7). We believe that the resistivity decrease in Fig. 7 near the SC onset is associated with the nonlinear current dependence of the normal exponential resistivity in the z = 0.8 sample. The absolute value of the low current resistivity at T = 0.4 K was one order of magnitude less than in the N-state indicating the transition into the SC state. The current-voltage characteristics (IVC) for the samples with lead content z = 0.9 ÷ 0.7 correspond to a transition with decreasing z from isolated SC regions to a structure with weak links between the SC insertions. Around zero bias, the IVC deviated from the ohmic behavior. The curves show an increase of a conductance with decreasing z what is reminiscent a nonzero critical current at V = 0 with a value increasing from
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JC ≅ 10 mkA (z = 0.8) to ~ 100 mkA (z = 0.7) due the increasing size of the links between SC islands in the samples. The weak link critical current was suppressed by low external magnetic fields.
Fig. 6. Temperature dependence of the resistivity in (PbZ Sn1-Z)0.8TeIn0.2 solid solutions with Pb contents z = 0.5- 0.9 and fixed x = 0.2.
Fig. 7. Temperature dependence of the resistivity in Pb0.8Sn0.2)0.8In0.2Te measuredfor different applied currents showing a nonlinear current dependence.
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Fig. 8. Current–voltage characteristics in (PbZ Sn1-Z)0.8In0.2Te (z = 0.7, 0.8, 0.9).
In contrast to the x = 0.2 series the samples with the lower In content (x = 0.16) exhibit on the SC side of the S-I transition (z = 0.6) a threshold type current increase near zero bias similar to that observed for S-I-S junctions (see Fig. 9). Fig. 9 shows the voltage and dV/dI as a function of the bias current in the sample (z = 0.8, x = 0.16) with a residual resistance at low temperatures after the SC transition. A salient feature of the dV/dI curve at T = 1.2 K is a zero bias differential resistivity peak corresponding to the pair tunneling through probable S-I-S interface in the SC-Sm system. The role of SC clusters decreases when the lead content z increases and, in turn, the SC step in the ρ (T) dependence decreases for the sample with z > 0.6. The features of the SC-Sm mesoscopic system are more remarkable in the magnetoresistance (MR) near the critical temperature. In the SC region the MR sharply increases and the negative MR in H > 1 kOe transforms into a positive MR in (Pb0.7Sn0.3)0.84In0.16Te. In this case the low temperature positive MR is sensitive to the tunneling conductance between SC clusters. It changes in the x = 0.16 series from a positive value at z = 0.7 (destruction of S-I-S interfaces) to a negative MR at z = 0.9 (weak localization in the Sm matrix) at the same temperature (see Fig. 10).
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Fig. 9. Voltage-current characteristic (left) and dV/dI as a function of bias current (right) in (Pb0.6Sn0.4)0.84In0.16Te at T = 1.2 K.
Fig. 10. Magnetoresistance as a function of applied magnetic field at T = 1.4 K for (PbZSn1-Z)0.84In0.16Te solid solutions with Pb content z = 0.7, 0.8, 0.9.
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Conclusions The optimal composition and In doping level to achieve the highest SC parameters in solid solutions based on the PbSnTe:In system has been determined. The filling of the In impurity band located against the background of the heavy hole band is responsible for the high SC parameters in the PbSnTe:In system. The higher the concentration of the In doping level the wider the range of the Pb content for which high SC parameters can be obtained in the PbSnTe:In system. A superconducting to insulating state transition was found in the low temperature conductivity of the PbSnTe:In compounds with varying compositions. At low currents we have observed a transition from a SC mesoscopic system to a SC – Sm system with the dielectric state of the semiconductor matrix. The anomalous magnetoresistance in the vicinity of the S – I transition consists of negative and positive parts associated with a transition from a SC mesoscopic system to a semiconductor with the weak localization effect in the normal state for a random fluctuating potential. The studied material is of interest for the development of SC bolometers and nanoscale devices due to observed high sensitivity of the normal and SC properties to temperature variations.
Acknowledgments The authors acknowledge support of the Presidium of RAS grants, RFBR 04-02-16638 grants and “Leading scientific school” – 2200.2003.2 grant.
References 1. Kaidanov VI, Ravich YuI (1985). Sov Phys Usp 28:31 2. Berezin AV, Nemov SA, Parfeniev RV, Shamshur DV (1993). Phys Solid State 35:28 3. Nemov SA, Parfeniev RV, Shamshur DV, Stepien-Damm J (1996). Czechoslovak Journal of Physics, Suppl S2 46:863 4. Nemov SA, Ravich Yu I (1998). Usp Fiz Nauk 169:817 5. Hulm JK, Jones CK (1968). Phys Rev 169:388, Hein RA, Meijr PHE (1969). Phys Rev 179:497 6. Tahar MZ, Popov DI, Nemov S (2003). Physica C 388-389, 581
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7. Chernik IA, Kaidanov VI, Ishutinov EM (1968). Sov Phys – Semiconductors 2:995 8. Shelankov AL (1987). Solid St Commun 62:327 9. Mott NF (1974). Phil Mag 19:835
ADVANCED SENSORS OF ELECTROMAGNETIC RADIATION
Thermoelectricity of Low-Dimensional Nanostructured Materials
V. G. Kantser International Laboratory of Superconductivity and Solid State Electronics, Academy of Sciences of Moldova, Chişinau
Abstract: Efficient thermoelectric technologies of power generation, heat removal and thermal management by refrigeration require solid-state materials and structures with significantly improved thermoelectric characteristics. The achievements in physics of low dimensional structures and the development in the corresponding fabrication techniques has recently reopened the field of thermoelectricity in order to engineer solid-state systems with high thermoelectric performance. Some aspects of thermoelectricity and technology of low dimensional nanostructured materials are highlighted in the present paper. As illustration of the ways to improve the figure of merit and others thermoelectric parameters some recent achievements in the investigation of solid-state nanostructures with quantum wells, wires, and dots are reviewed. Keywords: thermoelectricity, nanostructures, size quantization, nanowires, electron and phonon transport
General Considerations on Thermoelectricity The motion of charges in solid-state materials and structures is accompanied also by energy and heat transport. Therefore, this opens the possibility of solid-state thermoelectric thermal management, heat removal and energy conversion, which transforms heat directly into electricity, or of heat transport, when the energy is transferred from one to the other side 291 R. Gross et al. (eds.), Nanoscale Devices - Fundamentals and Applications, 291–307. © 2006 Springer. Printed in the Netherlands.
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of materials. At present there is growing interest stimulated by application demands in developing of novel thermoelectric materials and structures for efficient solid-state cooling and energy conversion devices. In particular, heat removal and thermal management becomes extremely important for further development of electronic industry as the feature size of the devices decreases and the dissipated power increases. For these purposes it is important to have the possibility of selective spot cooling of chips with high heat flow extraction and decreasing of electric power input. Solid-state power generation and cooling are based on thermoelectric effects, which are known as the Seebeck effect (for power generation) and the Peltier effect (for cooling and heat pumping). The Seebeck effect is associated with the generation of a voltage along a conductor when it is subjected to a temperature difference and this effect is the principle for thermocouples. Charged carriers (electrons or holes) diffuse from the hot side to the cold side, creating an internal electric field that opposes further diffusion. The Seebeck coefficient S is defined as the voltage generated per degree of temperature difference between two points [1] V = − S ∆T
(1)
Conversely, when carriers flow through a material they carry heat and this phenomenon is described by the Peltier effect. The heat current Q is proportional to the charge current J Q= PJ
(2)
and the corresponding coefficient P is the Peltier coefficient , which is connected to the Seebeck coefficient by the Kelvin relation P = ST
(3)
The dc transport of electrical current J and heat J Q are described by J = σ ( E − S ∇T ) , J Q = σ TE − K ' ∇T ,
(4)
where σ is the electrical conductivity and K is the thermal conductivity. Equations (4) under the conditions J = 0 lead to the following expressions for the heat current J Q = STJ − K ∇T
(5)
Thermoelectricity of Low-Dimensional Nanostructured Materials 293
K = K ' (1 − Z 'T ) Z'=
σS 2
Z=
K
σ S2 K'
=
Z' 1 − Z 'T
The last parameter Z is the thermoelectric figure of merit – the central issue of thermoelectricity as well as the power factor P = S 2σ
(6)
Thermal conductivity K enters Z in the denominator of Z because in thermoelectric coolers or power generators the thermoelectric elements also act as the thermal insulation between the hot and the cold sides. On the basis of continuity relation of one dimensional model ∂ρ ∂J + =0 ∂t ∂x
and heat balance equation 0 = ρ J 2 + K ∇ 2T
the temperature distribution along axes x can be obtained [1] T ( x ) = Tc +
x ρJ2 ∆T + x ( L − x) , 2K L
J Q ( x ) = SJT ( x ) − V ( x) =
K ∆T L − ρJ 2 − x , 2 L
ρJ2 x x ( L − x) , ( ρ JL + S ∆T ) + 2K L ∆T = Th − Tc
The last allow to show that thermoelectric refrigerator efficiency and generator efficiency are directly related to the material figure of merit Z . The refrigerator efficiency is described by the ratio.
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The efficiency of refrigerator is defined as η = J Qc P . The rate of heat K ∆T L − ρ J 2 ) removed from the cold source is L 2 divided by the input power, P = J ( ρ JL + S ∆T ) . These quantities depend on the current J . One maximizes the efficiency by varying J . This maximum of efficiency is called the coefficient of performance (COP):
( J Qc = J Q ( 0 ) = SJTc −
COP =
Tcγ − Th , ∆T ( γ + 1)
γ = 1 + ZT , T=
1 (Th + Tc ) 2
The COP value increases by increasing Z and achieves the Carnot limit COP → Tc ∆T in the case when Z → ∞ . A similar analysis can be done for generators. The COP is defined as η = P J Qh , the power divided by the heat flow from hot source J Qh = J Q ( L ) = SJTh −
K ∆T L − ρJ2 L 2
The COP coefficient η is optimized by varying the current and the COP has the form
ηmax =
∆T ( γ − 1)
γ Th + Tc
Again the ideal Carnot efficiency of ∆T Th is obtained in the case when ZT → ∞ . Comparison of COP and efficiency of different cooling and power generation technologies, inclusive thermoelectric one for different ZT values shows [2] ,that solid-state thermoelectric cooler and power generators can become competitive with other energy conversion methods provide values of ZT parameters are larger than 3-4. The present state of established thermoelectric materials and of emerging TE materials are plotted in Fig. 1.
Thermoelectricity of Low-Dimensional Nanostructured Materials 295
Fig. 1. ZT vs T for established thermoelectric materials.
The best room temperature cooling bulk materials available at present are alloys of Bi2Te3 with Sb2Te3 such as Bi0.5Sb1.5Te3, p-type, and Bi2Te3 with Bi2Se3 such as Bi2Te2.7Se0.3 , n-type, with typical ZT values close to one [2]. Thus the major objective in thermoelectric materials engineering and investigation is to increase the figure of merit ZT = σ S2×T/K, i.e. to increase electrical conductivity σ and thermoelectric power S, and to reduce thermal conductivity K.
Basis of Improved Thermoelectric Efficiency in Nanostructured Materials Recently, thermoelectric materials research experienced a resurgence inspired by the development of new concepts and principles to engineer thermoelectric transport in low dimensional nanostructures. Extensive investigations of thermoelectric properties of low-dimensional structures started in the 1990’s after a seminal paper [3] of Hicks and Dresselhaus. In parallel the interests in certain bulk thermoelectric materials such as skutterudites and others have been renewed. The general approach in developing new thermoelectric materials and structures can be succinctly summarized as engineering of electron and phonon transport. Solid-state nanostructured materials are unique in offering the possibilities of tailoring both the quantum-mechanical and transport characteristics of electrons and phonons through the artificial structure potential landscape.
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Being quantum quasi-particles the motion of electrons and phonons in nanostructured materials can develop in the range between two different regimes: i) totally coherent motion (when electrons or phonons spreads in the structure as waves); ii) totally incoherent motion (when either or both of them spreads in the structure as classical particles). The regime of the transport processes in the structures are determined by the correlation between size-scale of the structure potential landscape and three physical length scales of the quasi-particles: mean free path (MFP), phase breaking length (PBL) and the Fermi wavelength (FWL). In the terms of electron parameters high values of ZT request both high mobility and high density of states. This can be realized in anisotropic materials or in multivaley semiconductors with anisotropic carrier characteristics, when it is possible to have a small effective mass in the current flow direction to give a high mobility and large effective masses in the directions perpendicular to the current flow to give a high density of states. Thus, in comparison with the usual electronic transport in traditional low dimensional structures, the thermoelectric structures involve the factor of carrier anisotropy. At the same time such structures are characterized by several groups of carriers in different energy valleys and the possibility of band pocket engineering occur, which together with anisotropy offer a new opportunity to tailor the thermoelectric transport. Hence the above mentioned length scales in thermoelectric solid state structures based on anisotropic and multivalley semiconductors (such as bismuth (Bi) like semimetals, IV-VI narrow gap semiconductors, n-type Si and Ge) MFP, PBL and FWL of the carriers become anisotropic. Therefore, in addition to the issue of thermoelectricity such structures open new possibilities for the investigation of traditional low dimensional transport effects in situations where several groups of carriers with anisotropic physical characteristics are present. In the regime of coherent motion due to quantum size effects in nanostructured materials, such as quantum wells, superlattices, quantum wires, and quantum dots, the energy spectra of electrons and phonons can be manipulated through the variation of the size of the structures. Such low-dimensional nanostructures can be considered to be new materials [2], when a new set of size parameters provides a “new” material. Since the constituent components of nanostructures are well known, the structures are suitable to a certain degree of analysis, prediction and optimization. When the quasi-particle motion is incoherent, it is still possible to utilize classical size effects to tailor the transport properties providing for example a more effective scattering of phonons at the boundaries and interfaces than of the charge carriers. In the context of structure boundaries
Thermoelectricity of Low-Dimensional Nanostructured Materials 297
and interfaces classical size effects can manifest themselves for charge carriers too, leading to a stronger energy dependence of the carrier time relaxation. Thus we can mention the following two physical approaches to improve the thermoelectric efficiency of low dimensional nanostructured materials: I. ZT enhancement due to quantum confinement of carriers - energy states engineering - carrier pocket engineering - band structure anisotropy engineering - concentration of electronic density of states near Fermi energy - semimetal – semiconductor transition - increasing of energy asymmetry of electronic density of states II. ZT increase due to a decrease in the lattice thermal conductivity by phonon engineering - increased phonon – boundary scattering: thickness W ≤ phonon MFP - reduced phonon group velocity due to phonon confinement effect in 2D and 1D structures: L ~ W << MFP -phonon stop-band materials The main aspects of the transformation of the density of states due to the change in the dimensionality of the electron system are illustrated in Fig. 2.
Fig. 2. The density-of-states of electrons in systems of different dimensionality.
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Therefore it is challenging to develop new non-expensive methods of fabrication of Bi2Te3 and PbTe thermoelectric elements as well as to improve significantly its and others materials like bismuth thermoelectric performance, in particular on the basis of approaches for the nanoscale driven enhancement of thermoelectric properties.
Multiple Quantum Well and Superlattice Layered Structures The engineering of electron states in nanostructures in the direction to improve thermoelectric efficiency have started with the consideration of a layered systems with multiple quantum wells [3] analyzing the transport parallel to the layers. A large number of theoretical and experimental papers have demonstrated the enhancement of Z 2 DT in the QW material of n-type structures [4 - 7] due to electron confinement. Being one of the best bulk thermoelectric materials lead telluride was among the first system analyzed in the context of low dimensional thermoelectricity. The effect of dimensionality on the power factor for n-type PbTe/PbEuTe Quantum Well structures is shown in Fig. 4 on the basis of theoretical modeling [6, 7].
Fig. 3. Power factor for (100) and for (111) ones, (S2Φ)111 (dashed lines) vs. well width d at T = 300 K. Curves 1 and 4 are for n = 1019 cm–3; 2 and 5 for n = 5·1018 cm–3; 3 and 6 for n = 1018 cm–3.
Thermoelectricity of Low-Dimensional Nanostructured Materials 299
Fig. 4. Thermoelectric figure of merit Z2DT of PbTe/Pb1-xEuxTe as a function of carrier concentration n for (100) oriented QWs (continuous curve 1) and for (111) ones (dotted curve 1’). Potential barrier height U = 171 meV, x = 0.073, d = 2 nm, T = 300 K (after [4]).
In p-type PbTe/PbEuTe quantum well structures the first experimental demonstration of the principle of low dimensionality in thermoelectricity was obtained. As an illustration in Fig. 5 the first experimental results obtained in [8] for p-type PbTe/PbEuTe quantum well structures can be reproduced taking into account the finite weight of the barrier. 6
ZT2D/ZT3D
5
DA, U = ∞
4
DA, U = 141 meV LO, U = ∞
3 2 1 0 0
20
40
60
80
d (Å)
100
120
140
160
Fig. 5. Thermoelectric figure of merit of p-type PbTe/Pb0.927Eu0.073Te quantum well structures scaled by its bulk value as a function of well width d for two different scattering mechanisms (on acoustical phonons (DA) and optical phonons (LO)).
In the context of p-PbTe wells an other low dimensional approach have been reveal [9] it was suggested that quantum confinement will lead to the
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change of the thermoelectric transport from L-type subbands to Σ -type ones. The latter with higher density of states must be more favorable for the improvement of thermoelectric properties. This is the case for PbTe where we have 12 full valleys (energetic minimas) and 12 full ellipsoids of constant energy in Σ -points of the Brillouin zone. As a result, the density of states must be higher and, respectively, the Seebeck coefficient and the figure of merit must be higher too. GeSi/Si QW structures are also materials utilized in bulk thermoelectricity. The modeling of their thermoelectric properties and the first experimental results indicate that the may have a higher possible figure merit. It has been also shown that in such structures the phonon scattering rate is considerably increased and, as a result, the QW in-plane lattice thermal conductivity is strongly diminished [10]. For the optimization of the thermoelectric properties of QW structures it is very important to take into account the prospects of phonon engineering as well [2].ZA lot of investigations so far has focused on the thermoelectric transport perpendicular to the layer plane of structures. There are several aspects related to the study the transport perpendicular to superlattice layer plane [2]: i) tailoring the density of states using quantum size effects; ii) energy filtering through thermionic transport; iii) reducing thermal conductivity by stopping the propagation of some phonon modes; iv) selective scattering of carriers with high and low energy. The results obtained with the 1 nm / 5 nm Bi2Te3/Sb2Te3 p-type superlattices indicate that the phonon and charge carriers transport can be tuned to improve ZT [11] by the so-called phonon-blocking/electrontransmitting mechanism [12]. In the same paper encouraging results have been also obtained with n-type 1 nm / 5 nm Bi2Te3/Bi2Te2.83Se0.17 n-type superlattices, indicating ZT values higher than one at 300 K. The reason for the less-than impressive ZT value of 1.46 at 300 K in n-type Bi2Te3/Bi2Te2.83Se0.17 superlattices, compared to 2.4 at 300 K in the best ptype Bi2Te3/Sb2Te3 superlattices, is their higher thermal conductivity K.
Thermoelectric Wire Nanostructures General theoretical considerations suggest that, because of their increased quantum confinement effects, one-dimensional quantum wires could have an even larger enhancement in ZT. However, even without quantum
Thermoelectricity of Low-Dimensional Nanostructured Materials 301
confinement effects microwires are of significant importance due to the following aspects: (1) the one-dimensional shape of the microwire should be favourable for some synthetic textures; (2) microwires may be convenient for some tiny-dimensional applications such as natural “thermoelectric wires” taking advantage of their characteristic shape. Several mechanisms are combined to enhance ZT in nanowire arrays: 1. strong carrier quantum confinement in each quantum wire resulting in a δ-function like density of states 2. suppression of phonon scattering 3. strong asymmetry of the energy dependence of the DOS. Due to the very small effective mass of the charge carriers in Bi and PbTe-like semiconductors, the effects of reduced dimensionality can be seen in samples with dimensions of the order of 10 to 50 nm. In addition, it was shown recently [13] that due to the anisotropy of the electron (hole) energy ellipsoids, the quantum size effects in cylindrical Bi- like wires are strongly enhanced. The origin of this effect is related to the peculiarities arising from the wave propagation of anisotropic carriers - like caustics in the geometrical optics of light propagation in ellipsoidal resonators. Recently, the methods and corresponding technological equipment for the fabrication of an array of Bi nanowires have been developed. Arrays with wire diameters of 5-200 nm have been obtained by liquid-phase pressure injection [14], vapour-phase deposition [15] and electrochemical deposition [16]. Various transport investigations have been carried out to highlight classical and quantum size effects [17-19]. As a host template materials porous anodic alumina has been used and even single crystalline nanowires of two orientations (normal to the (202) and (012) lattice planes) were fabricated. Figure 6 shows an image of an anodized alumina template which is filled with bismuth by electrochemical deposition.
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Fig. 6. SEM image of Bi nanowire arrays in the alumina template. The wire diameter is ~ 100 nm.
However, regarding the compatibility of the nanowire array technology with the conventional technology of micro- and optoelectronic materials and from the perspective of wire orientations and their doping the nanowire array fabrication in porous silicon and III-V semiconductors seems to be more attractive. Investigations in this direction started recently. Nanowire arrays based on Bi are the most promising systems for reengineering the transport properties by making use of the quantum confinement effect of the anisotropic carriers and size-induced semimetalsemiconductor phase transitions, which are expected to result in good parameters for thermoelectric applications [19]. In particular, the anisotropy of the carrier pockets in Bi is outlined to be one of the key elements for achieving significantly improved thermoelectric parameters. On the basis of theoretical investigations, the thermoelectric figure of merit Z1DT of Bi nanowires is expected to increase significantly [19, 20] However, the measurements of the thermoelectric properties (such as the Seebeck coefficient and the thermal conductivity as well as a quantitative explanation of these measurements (and even of existing results on electrophysical investigations) are more challenging. Only recently, the first results of the Seebeck coefficient measurements in Bi-wires nanocomposites have been obtained [21]. The thermoelectric power was found to increase considerable in Bi-nanowires with diameters less than 10 nm in alumina matrices. On the other hand the investigations of the Electric Field Effect (EFE) on thermoelectric transport in solid state structures have been started
Thermoelectricity of Low-Dimensional Nanostructured Materials 303
recently [22]. In the context of nanowire structures the first theoretical analysis show that thermoelectric transport and figure of merit can be engineered by radial EFE [23]. For coaxial cylindrical capacitor configuration of PbTe and Bi2Te3 nanowire structures was established that the existence of the transition bipolar-monopolar semiconductor in radial electric field, differences in carrier masses and mobility can essentially modified the thermoelectric properties (Fig. 7).
Seebeck coefficient [µV/K]
500
Sp
400 300
Sn
200 100 0
-300 -200 -100 0
100 200 300
Gate voltage [V] Fig. 7. Dependence of the Seebeck coefficient for electrons Sn (dash line), holes Sp (dash dot line) and bipolar Seebeck coefficient S (solid line) on the gate voltage cylindrical capacitor Bi2Te3 nanowire structures with diameter of 100 nm.
Quantum Dots and Three-Dimensional Nanostructured Thermoelectric Systems The most significant improvement of the thermoelectric figure of merit ZT can be achieved by strongly reducing of the lattice thermal conductivity. The effect of phonon confinement on the lattice thermal conductivity in low-dimensional structures is frequently treated in terms of additional boundary scattering and the decrease of the phonon mean free path. However, there is also another approach. When the phonon modes in the confined structures are modified significantly, the group velocity of these phonons usually decreases and this can strongly influences ZT [22]. Due to the specific properties of quantum dot systems both implications of size
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effects – classical and quantum – on the characteristics of the phonon system and the thermal conductivity have been intensively studied over the last years [24]. In such approach quantum dot systems represent an example of the abovementioned ‘phonon-blocking electron-transmitting’ [12] structure with a great potential for thermoelectric applications. An illustrative result of numerical calculations for a structure that consists of multiple layers of Si with array of Ge quantum dots separated by wetting layers and spacers are shown in Fig. 8. 103
ZTQDC/ZTB
2
10
101 10-0 10-1 10-2 10-3
4
1 -0.3
Fermi Energy (eV) -0.2
-0.1
0.1
0.2
Fig. 8. Figure of merit ZT of Si/Ge quantum dot systems scaled by its bulk value (ZTexp(Si)=0.05 - 0.06 at 300 K) as a function of Fermi energy for two different value of thermal conductivity (15 W m–1 K–1 and 156 W m–1 K–1 respectively).
There the electron and phonon transport modifications due to the space confinement caused by the mismatch in electronic and thermal properties between dot and host materials have been taken into account. The analysis shows that the enhancement of the thermoelectric figure of merit ZT is mostly due to the significant drop in the lattice thermal conductivity caused by the acoustic phonon scattering by quantum dots. To take advantage of both the superlattices and the nanowires to design a better thermoelectric system in [26] another variant of nanostructured quantum dot like systems have been proposed and investigated. The electronic density of states and the thermoelectric properties of superlattice nanowires made of various lead salts (PbS, PbSe, and PbTe) have been investigated as a function of the segment length, wire diameter, crystal orientation along the wire axis, and the length ratio of the constituent
Thermoelectricity of Low-Dimensional Nanostructured Materials 305
nanodots as the superlattice period decreases. The values of ZT are found to be higher than 4 and 6 for 5 nm-diameter PbSe/PbS and PbTe/PbSe superlattice segmented nanowires. The enormous prospects of quantum dot like systems for the improvement of the thermoelectric performance of materials has been confirmed recently by experiments. In particular, a high value of the thermopower has been observed at room temperature in PbTe/PbSexTe1-x quantum dot superlattice structures, which have reached values of ZT = 2 [27]. Investigating the thermoelectric properties of PbTe nanostructures with grain sizes of the order of 30–50 nm in [28, 29] a considerable enhancement in the thermopower relative to bulk PbTe materials has been demonstrated. The magnitude of the enhancement is similar to the abovementioned result for PbTe/PbSexTe1-x quantum dot superlattices. An other nanostructured material like PbTe and the system AgPbmSbTe2+m [30] have been fabricated and shown to have a high thermoelectric figure of merit ZTmax of 2.2 at 800 K. The observed ZT enhancement is explained by the presence of quantum “nanodots” in these materials similar to those found in the PbSe/PbTe MBE–grown thin films quantum dot system [27]
Conclusions The paper reviews some recent investigations dealing with the ways of radical improvement of the thermoelectric figure of merit ZT in low dimensional solid state nanostructures. If the conditions of nanoscale size effects are fulfilled the transport of the charge carriers and phonons can be significantly modified and this can be used to improve the thermoelectric energy conversion efficiency. Recent studies have revealed a lot of new aspects of thermoelectric transport in low dimensional structures, which show promising ways to reach a large increase in ZT values. A number of approaches have been proposed for thermoelectric nanostructures of different dimensionality: quantum wells, superlattices, quantum wires and quantum dot systems. A brief analysis of existing results in the research area of quasi-two dimensional systems is presented, including some experimental data confirming high power factor S2σ and other thermoelectric parameters in the PbTe/PbEuTe and Si/SiGe systems. Recently, superlattices with quasi two-dimensional electron gases have been produced in the Bi2Te3 system
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(the best bulk thermoelectric material), which offers significantly higher Z (more than twice of the bulk value) with the transport across the structure. The report has outlined these new important results. Quasi one-dimensional nanowire array composites are the most promising systems for improving the thermoelectric properties by making use of the quantum confinement effect and phonon transport engineering. Due to the large anisotropy and other electronic peculiarities bismuth is a very attractive material for low-dimensional thermoelectricity, when a semimetal-semiconductor transition appears due to an anisotropy enhanced quantum size effect leading to a large enhancement in the figure of merit. An even more significant improvement of the thermoelectric performance in nanowire systems was revealed by employing a mechanism based on the reduction of thermal diffusivity. Some selected results in this area as well as in the field of electronic properties engineering by the quantum confinement in nanowire systems have been highlighted in the paper. In the last part of the review some aspects of nanostructured bulk materials and quantum dot systems have been discussed. A significant improvement of the thermoelectric figure of merit ZT can be achieved in such systems by a strong reduction of the lattice thermal conductivity as well as by a scattering-induced enhancement of the thermopower, when the nanostructure dimensions become less than the electron mean free path. An important aspect of low dimensional nanostructured thermoelectric materials outlined in the paper is its capability to be implemented into microsystem devices and wafer based microelectronic technologies. They can be used to solve some problems of heat removal and thermal management of nanoscale devices, which become more important with increasing amount of dissipated power.
Acknowledgments This work has been supported by INTAS project N 0184 and National Program of Nanotechnology of R.Moldova .
References 1. Mahan GD (2001) Semiconductors and Semimetals. 71 pp 157-174 2. Chen G, Shakouri A (2002). Transactions of the ASME 124:242 3. Hicks LD, Dresselhaus MS (1993). Phys Rev B 47:12727
Thermoelectricity of Low-Dimensional Nanostructured Materials 307 4. Koga T, Harman TC, Sun X, Dresselhaus MS (1999). Phys Rev B 60:14286 5. Casian A, Dashevsky Z, Kantser V, Scherrer H, Sur I, Sandu A (1999). Proc of Material Research Symposium MRS98, Fall Meeting, 545:99-104 6. Casian A, Sur I, Scherrer H, Dashevsky Z (2000). Phys Rev B 61:15965 7. Casian AZ, Dashevsky Z, Kantser V, Scherrer H, Sur I, Sandu A (2000). Phys Low-Dim Str 5/6:49 8. Hicks LD, Harman TC, Sun X, Dresselhaus MS (1996). Phys Rev B 53:10493 9. Sur I, Casian A, Balandin A, Dashevsky Z, Kantser V, Scherrer H (2003) Proc of 21st Int. Conf. on Thermoelectrics. Long Beach, Piscataway, IEEE, NY pp388-391 10. Sun X, Cronin SB, Liu JL, Wang KL, Koga T, Dresselhaus M, Chen G (1999) Proceedings of Int Conf Thermoelectrics. ICT’99 pp 652–655 11. Venkatasubramanian R, Silvola E, Colpitts T (2001). Nature 413:597 12. Slack G (1995). In: Rowe DM (ed) Boca Raton CRC Handbook of Thermoelectrics. CRC Press pp 407-440 13. Bejenari IM, Kantser VG, Myonov M, Mironov OA, Leadley DR (2004). Semicond Sci Technol 19:106 14. Zhang ZB, Gekhtman D, Dresselhaus MS, Ying JY (1999). Chem Mater 11:1659 15. Heremans J, Trush CM, Lin YM, Cronin S, Zhang Z, Dresselhaus MS, Mansfield JF (2000). Phys Rev B 61:2921 16. Lin K et al (1998). Appl Phys Lett 73:1436 17. Lin K et al (1998). Phys Rev B 58:14681 18. Zhang Z, Sun X, Dresselhaus MS, Ying JY, Heremans J (2000). Phys Rev B 61:4850 19. Lin YM, Sun X, Dresselhaus MS (2000). Phys Rev B 62:4610 20. Sun X, Zhang Z, Dresselhaus MS (1999). Appl Phys Lett 74:4005 21. Heremans J, Thrush CM, Morelli DT, Ming-Cheng Wu (2002). Phys Rev Lett 88:216801 22. Sandomirsky V., Butenko A.V. Levin R., Schlesinger Y (2001) J Appl. Phys. 90:2370. 23. V.Kantser, Z.Dashevsky , H.Scherrer, D.Meglei, M.Dantu (2004). Proc. of 22
24. 25. 26. 27. 28. 29. 30.
Int. Conf. on Thermoelectrics, p. 350 -355. Balandin A and Wang K L (1998). J Appl Phys 84:6149 Khitun A, Wang KL, Chen G (2000). Nanotechnology 11:327 Lin YM, Dresselhaus MS (2003). Phys Rev B 68:075304 Harman TC, Taylor PJ, Walsh MP, LaForge BE (2002). Science 297:2229; Beyer H, Nurnus J, Botner H, Lambrecht A (2002). Appl Phys Lett 80:1216 Heremans JP, Thrush CM, Morelli DT (2004). Phys Rev B 70:115334 Hsu KF, Sim Loo, Fu Guo, Wei Chen, Dyck JS, Uher C, Hogan T, Polychroniadis EK, Mercouri G, Kanatzidis I (2004). Science 303:818
Organic Semiconductors – More Efficient Material for Thermoelectric Infrared Detectors
A. Casian1, Z. Dashevsky2, V. Dusciac3, R. Dusciac1 1
Department of Computers, Informatics and Microelectronics, Technical University of MoldovaMD-2004, Chisinau, Moldova.
2
Department of Materials Engineering, Ben-Gurion University, Beer-Sheva 84105, Israel
3
Department of Physics, State University of Moldova, MD-2012, Chisinau, Moldova
Abstract:
The opportunities for the application of new quasi-one-dimensional organic semiconductors as sensitive elements in nonselective thermoelectric infrared detectors for the long wave length regime of the spectrum are analyzed. It is shown that under certain conditions the effective scattering probability for some carriers is considerably reduced due to the interference of two main electron-phonon interactions. As a result, a significant increase of the thermoelectric power factor, which is the most important parameter of the materials used for infrared detectors, is predicted.
Keywords: Infrared detectors, thermoelectric semiconductors, detector sensitivity.
power
factor,
organic
Introduction In the last years, the demand for detectors of infrared radiation (IR) in the long wave length regime of the spectrum has increased considerably. In this context, the attention of investigators has been focused on the development of new, more sensitive nonselective thermoelectric detectors. As it is known, the photoelectric detectors are sensitive only up to the wave length of the order of 1.3 µm. On the other hand, the earth’s 309 R. Gross et al. (eds.), Nanoscale Devices - Fundamentals and Applications, 309–317. © 2006 Springer. Printed in the Netherlands.
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atmosphere has only several “windows” of sufficient transparency. Such detectors can be applied for the measuring of radiation up to the most far infrared region. An important parameter of the materials used for the sensitive element of such detectors is the thermoelectric power factor, P = σS 2 , were σ is the electrical conductivity and S is the thermopower (Seebeck coefficient). As large as possible values of P are desired for applications. However, the requirements to realize in the same material high electrical conductivity and high values of the thermopower are, in general, contradictory. Therefore, the search and investigation of materials and structures with more complicated energy spectrum giving the possibility to overcome these contradictions and to realize higher values of the thermoelectric power factor are of great importance not only from the theoretical point of view, but also for the prospective practical applications. Recently, it was shown [1] that it is possible to improve the thermoelectric properties of some materials by preparing them in the form of low-dimensional quantum-well (QW) superlattice structures. Several types of such structures have been investigated [2–5]. In n-type PbTe/PbEuTe quantum wells values of P as high as 62–66 µW/cmK2 were measured [2]. This is almost two times higher than the best measured value of 38 µW/cmK2 in the bulk PbTe, a material widely used in thermoelectric applications. In our work [6, 7] it has been shown that the expected values of P for (100) and (111) oriented QWs are P100 = 175 µW/cmK 2 and P111 = 108 µW/cmK 2 , which are, respectively, 4.6 and 2.8 times higher than the above mentioned measured value in bulk PbTe. The application potential of n-type PbTe/PbEuTe QWs in the most rigorous model has been investigated in [8]. A high value of P = 160 µW/cmK 2 , the highest measured up to now, was observed in p-type PbTe/PbEuTe QWs [9], although our calculations show that for certain parameters of QWs and hole concentrations the power factor could achieve usually large values of the order of 250 µW/cmK 2 [10]. However, as it also has been established that the technology for fabrication of such QW structures is very complicated and expensive. Therefore, in spite of the great success in the realization of QW nano-structures, the search and investigation of novel materials with higher thermoelectric power factor continue to remain an important and urgent problem. Lately, organic materials attract more and more attention, since they are less expensive and show a broad spectrum of physical properties [11]. In the present paper we predict that in quasi-one-dimensional organic crystals it is possible to obtain, under certain conditions, an extremely high value
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of the thermoelectric power factor at room temperature. These conditions are determined by the interference and mutual compensation of two major electron-phonon interaction mechanisms in such systems.
Quasi-One-Dimensional Organic Crystals Initially, the interest in the synthesis and investigation of new Q1D organic crystals was connected with the predicted possibility to find high temperature superconductors. Superconductivity in such crystals was indeed discovered, but the achieved critical temperatures are still quite low. Simultaneously, it was established that the Q1D organic crystals have rather diverse and often unusual physical properties, coming mainly from their specific structure. They represent needle-like three-dimensional crystals formed by linear stacks or chains of usually planar molecules (see Ref. 11, Chap. V). Significant π-bonding along the stacks ensures band-like conduction in the stack direction, whereas between the stacks the conduction is hopping-like and is usually negligible. The Q1D conduction takes place in the ion-radical salts of the tetra-thiofulvalenetetracyanoquinodimethane (TTF-TCNQ) type in a number of complexes by charge transfer in crystal polymers. The crystal model considered here has been described in [12, 13]. It must be mentioned that due to rather localized molecular orbitals the tight binding and nearest neighbor approximations are applied for conduction along the chains and the transversal hopping conduction is neglected for simplicity. Two main electron-phonon interaction mechanisms are taken into account simultaneously. One mechanism is deformation potential like and is determined by the variation (caused by intermolecular acoustical vibrations) of the transfer energy W (or transfer integral) of an electron from one molecule to the nearest one along the chain. The coupling constant is proportional to the derivative W ′ of W with respect to the intermolecular distance. The second mechanism is polaron-like and is determined by the variation of the polarization energy of molecules surrounding the conduction electron, due to the same intermolecular vibrations. The coupling constant for this mechanism is proportional to the average polarizability α 0 of the molecule. As it was shown in [12–14], an interference between these interactions can take place. As a result, these interactions considerably compensate each other for some states in the conduction band. For carriers in these states the effective scattering rate is significantly reduced, and the mobility is considerably increased. Increased mobility is favorable for the desired increase of the thermoelectric power factor.
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We are interested in the study of the electrical conductivity σ and thermopower S near room temperature. Usually, in the Q1D conducting crystals there are different phase transitions to other, less conducting or insulating phases. We will assume that the highest critical temperature of these phase transitions is lower than room temperature. In this case, in order to investigate the electronic transport it is possible [12, 15] (i) to neglect in first order approximation the weak interaction between the 1D chains and to consider a one-dimensional crystal model, and (ii) to use the Boltzmann kinetic equation for the 1D model. It was shown in Ref. 12 that near room temperature the scattering of carriers on acoustical phonons can be considered elastic. However, in this case the electron-phonon interactions can fully compensate each other for some states in the conduction band. The mobility of carriers in these states will be limited by the scattering on impurities or defects, which always exist in the crystal. Therefore, it is necessary to take into account the scattering on impurities, as well. The latter, for simplicity, are considered neutral and point-like. The energy of the conduction electron is taken in the usual form
ε ( k ) = 2W (1 − cos ka ) , where k is the projection of the electron wave-vector, and a is the lattice constant along the chains. The sign of W is chosen in such a way that W > 0 for the s-type band, and W < 0 for the p-type one. The dispersion law of longitudinal acoustical phonons is taken in the standard form
ω q = 2υ s a −1 sin qa 2 , where υ s is the sound velocity along the chains and q is the projection of the phonon wave-vector in the chain direction.
Thermoelectric Power Factor The kinetic equation is presented in Ref. 12. It has the form of a Boltzmann equation and in the considered case can be solved exactly. For nondegenerated charge carriers the thermoelectric power factor takes the form P = R12 ( e 2T 2 R0 ) ,
(1)
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where ∆ E −E 1 ( E − EF ) n σ ( E ) exp F Rn = dE , ∫ k0T 0 k0T
(2)
σ ( E ) = e 2υ 2 ( E )τ ( E ) ρ ( E ) ,
(3)
e is the electron charge, k 0 is the Boltzmann constant, EF is the Fermi energy, ∆ = 4W is the conduction band width, υ 2 ( E ) = h −2 a 2 E ( ∆ − E ) is the square of the carrier velocity as a function of charge carrier energy E, τ (E ) is the relaxation time, which has the form of a Lorentzian
τ s, p ( E ) =
hMυ s2W 2 [E ( ∆ − E )]1 2 ⋅ , 2a 2 k 0TW ′2 γ 2 ( E − E 0s , p ) 2 + 4W 2 D 2
D2 =
nim I 2 d 2 Mυ s2 = D02T0 T . 4a 3k 0TW ′2
(4)
(5)
Here, M is the mass of the molecule, γ is a dimensionless parameter which represents the ratio of the amplitudes of two above mentioned interaction mechanisms, E 0s , p = 2W (γ ± 1) γ is the Lorentzian resonance energy, D is a dimensionless parameter which describes the impurity scattering, nim is the linear impurity concentration, I and d characterize the effective height and width of the impurity potential, T0 = 300 K is the room temperature. In (4) the phonon distribution function has been taken in the high T limit, because in these materials the Debye temperature is of the order of 80– 100 K. If γ = 0 and D = 0 , i.e. if only the first interaction mechanism is active, the crystal model coincides with that used in [16]. If γ is small, γ << 1 , the resonance energy E 0s , p is placed into the forbidden band and the interference of above mentioned interactions is weakly pronounced. In this case the usual results for a narrow 1D conduction band are obtained. If γ ≥ 1 , E 0s , p proves to be in the allowed band, and if D << 1 , the electronic states near E 0s , p will be characterized by very high values of the relaxation time. This situation changes essentially the behavior of σ , S and P.
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For nondegenerate carriers the transport integrals R0 and R1 in (1) can be calculated analytically, but the respective expressions are very cumbersome and cannot be presented here. They will serve for the comparison with the approximate or numerical calculations. Obviously, most interesting is the case when the Lorentzian from the relaxation time (4) is placed in the conduction band ( γ > 1 ) and is very sharp ( D << 1 ). In this case R0 and R1 can be calculated approximately by decomposing the Lorentzian in series on D. For the calculation of R0 and R1 one can take only the first term from the decomposition, corresponding to the replacement of the Lorentzian by delta-function. As a result we obtain
R0 = σ 0
E0s , p − EF π (γ 2 − 1)T03 2 exp − k T γ 3 D0T 3 2 0
R1 = ( E 0s , p − E F ) R0 ,
(6)
(7)
where σ 0 depends only on crystal parameters
σ0 =
2re 2 Mυ s2 W
3
πhaW ′2 ( k 0T0 ) 2
.
(8)
After the substitution of (6) and (7) in (1), we obtain 2
2 32 2 s, p k πσ 0 (γ − 1)T0 ( E0 − EF ) P= 0 ⋅ exp − ( E0s , p − EF ) k0T . γ 3 D0T 3 2 ( k0T ) 2 e
(9)
It is seen that P can take high values, because D0 appears in the denominator and D0 << 1 . However, P also depends strongly on the difference ( E0s , p − EF ). It is evident from (9) that the optimal value, which corresponds to the maximum of P, is E0s , p − EF = 2k0T . On the other hand, for the non-degenerate case it is necessary that EF is at least equal to − 3k 0T0 . Therefore (9) is not valid for the determination of Pmax , but still can serve for the evaluation of P near the maximum value. Numerical calculations following the exact expression (1) are necessary in order to determine P for a larger interval of the parameters. Unfortunately, not all
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parameters in (1) are known for any Q1D organic crystal. We shall consider a hypothetical crystal with parameters close to those of TTF chains (p-type band) in the TTF-TCNQ crystals: M = 3.7 × 105 me ( me is the mass of the electron),
W = 0.075 eV ,
W ′ = 0.2 eVÅ -1 , υ s = 2 × 105 cm/s ,
a = 12.3 Å , b = 3.82 Å , c = 18.47 Å (b is the direction of chains), r = 2 ac . Let’s put γ = 1.5 and E F = −3k 0T0 , then E 0p = 1.9k 0T0 . From
(9) we obtain P = 7.1 ⋅ D0−1 µW/cmK 2 . For D02 = 10 −3 we will then have P = 225 µW/cmK 2 , a very promising result.
300 1
P, µW/cm⋅K2
200
100
2 3
0 0 1 2 3 4 5 6 7 8 9 10
γ Fig. 1. Thermoelectric power factor P as a function of the parameter γ. D02 values used for the various curves: 1 – 10-3; 2 – 10-2; 3 – 10-1.
We do not know the polarizability α 0 of the TTF molecule. But in order to obtain γ = 1.5 it is necessary to have α 0 = 8.5Å 3 , which is not a very high value. Note, that the calculated value [17] of α 0 for the TTF molecule in the TTF-TCNQ crystal is of the order of 16Å 3 . This value of α 0 would correspond to γ = 2.8 and, respectively, to P = 58.5 µW/cmK 2 for D02 = 10 −3 . This is also a promising result. At the same time, it is seen that P decreases quickly, when γ grows. Estimations show that in order to
achieve D02 ~ 10 −3 the crystal purity must be high, but quite reasonable.
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The results of the numerical calculation of P as a function of γ for a ptype Q1D crystal with the above mentioned parameters and different values of α 0 and D0 are shown in Fig. 1. It is seen that P has a maximum of 240 µW/cmK 2 near γ = 1.2 , and P = 210 µW/cmK 2 for γ = 1.5 , a value very close to one calculated using the approximate expression (9).
Conclusions The thermoelectric power factor P has been modeled for a series of quasi one-dimensional organic semiconductors with the goal to find more efficient materials for the sensitive elements of thermoelectric infrared detectors. It is found that in some of such crystals, in which the interference of two main electron-phonon interaction mechanisms is well pronounced, the value of P can be increased by almost a factor of ten compared to the respective value in the best PbTe material.
Acknowledgments The work was supported by INTAS-01-0184 project.
References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.
Dresselhaus MS et al (1999) Proc 18th Intern Conf on Thermoel. Baltimore, USA (IEEE, Piscataway, NJ) p 92 Harman TC, Spears DL, Manfra MJ (1996). J Electron Mater 25:1121 Beyer H et al (2002). Appl Phys Lett 80:1216 Mironov OA et al (1997). Thin Solid Films 294:182 Harman TC, Taylor PJ, Walsh MP, LaForge BE (2002). Science 297:2229 Casian A, Sur I, Scherrer H, Dashevsky Z (2000). Phys Rev B 61:15965 Casian A, Dashevsky Z, Kantser V, Scherrer H, Sur I, Sandu A (2000). Phys Low-Dim Struct 5/6:49 Sur I, Casian A, Balandin A (2004). Phys Rev B 69:035306 Harman TC, Spears DL, Calawa DR, Groves SH, Walsh MP (1997) Proc of 16th Int Conf on Thermoelectricity. Dresden, Germany p 416 Sur I, Casian A, Balandin AA, Dashevsky Z, Kantser V, Scherrer H (2003) Proc of 21st Intern Conf on Thermoel. Long Beach, USA (IEEE, Piscataway, NJ):288
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Submillimeter Radiation–Induced Persistent Photoconductivity in Pb1-xSnxTe(In)
A. E. Kozhanov 1, D. E. Dolzhenko 1, I. I. Ivanchik 1, D. M. Watson 2, D. R. Khokhlov 1 1
Moscow State University, Moscow 119992, Russia
2
University of Rochester, Rochester, 14627 NY, USA
Abstract: Persistent photoconductivity in a Pb0.75Sn0.25Te(In) alloy initiated by monochromatic submillimeter-range radiation at wavelengths of 176 and 241 µm was observed at helium temperatures. This photoconductivity is shown to be associated with the optical excitation of metastable impurity states. Keywords: persistent photoconductivity, impurity states, submillimeter radiation
Introduction Most of the presently used high-sensitivity non-thermal detectors intended for use in the far infrared range are based on doped silicon and germanium. The longest wavelength corresponding to the photoelectric threshold in such radiation detectors that has been reached in uniaxially strained Ge: Ga was λr = 220 µm [1]. A viable alternative to the silicon- and germanium-based radiation detectors is offered by lead telluride based narrow-gap semiconductors. Doping lead telluride and its solid solutions by group III elements gives rise to effects such as Fermi level stabilization and persistent photoconductivity, which are not characteristic of the starting material [2]. In particular, the Fermi level in Pb1−xSnxTe(In) alloys with a tin content of 319 R. Gross et al. (eds.), Nanoscale Devices - Fundamentals and Applications, 319–324. © 2006 Springer. Printed in the Netherlands.
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0.22<x<0.28 stabilizes within the band gap to produce semi-insulating states of the semiconductor at low temperatures. Because the effect of Fermi level stabilization brings about homogenization of the electrophysical parameters of the material and because the characteristic energy parameters of the alloy, such as the band gap width and the impurity-state activation energy, are of the order of a few tens of meV, the possibility of employing these semiconductors as radiation detectors in the far IR range appears extremely attractive. This possibility was realized [3], and it was found that the sensitivity parameters of a Pb0.75Sn0.25Te(In)-based IR radiometers would substantially exceed those of their counterparts based on doped silicon and germanium. However, the key question of the spectral response of the Pb1–xSnxTe(In)based radiation detector, in particular, of the red cutoff of the photoeffect in this material, has remained unexplored. The phenomenon of persistent photoconductivity observed in Pb1–xSnxTe(In) at low temperatures results in a buildup of nonequilibrium carriers in the allowed band under background radiation present in each standard spectrometer, thus making standard spectral measurements impossible. An instrument that has no background illumination at all but provides the possibility of illuminating a sample with radiation of a fixed wavelength and calibrated intensity is required.
Fig. 1. Setup for measuring photoconductivity spectra of Pb1–xSnxTe(In) [4]: 1blackbody, 2 - entrance window, 3 - “nitrogen” filter, 4 - “helium” filter, 5 interference filter, 6 - rotating disk with filters, 7 - sample, and 8 - helium bath.
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Such an instrument was built and described in [4] (see Fig. 1). A sample was fixed to the bottom of a helium bath in evacuated space. The background radiation was eliminated by means of screens cooled to liquidhelium and liquid-nitrogen temperatures. Blackbody radiation with a temperature of 77 or 300 K impinged on the sample through the entrance window and a series of cooled filters. Special filters, which were maintained at the liquid-nitrogen or liquid-helium temperature, transmitted only the part of the radiation corresponding to the spectral interval under study. The diaphragm in the “helium” screen made it possible to calibrate the radiation flux striking the sample. Finally, a narrow radiation line was isolated by means of an interference filter mounted on a rotating disk within the helium screen. It was demonstrated in [4], that the radiation with wavelengths of 90 and 116 µm gives rise to strong persistent photoresponse in Pb0.75Sn0.25Te(In) at the liquid helium temperature. An important point mentioned in [4] was that the radiation energy quantum corresponding to the used wavelengths is smaller than the thermal activation energy of the ground impurity state that pins the Fermi level. Therefore the persistent photoconductivity was defined in this case by the excitation of metastable, but not the local ground states. However, the interpretation of the results mentioned above has remained an open issue. Indeed, the thermal activation energy Ea was calculated from an analysis of the temperature dependence of the resistivity by using the relation ρ~exp(Ea/2kT) rather than ρ~exp(Ea/kT), as is usually accepted when dealing with impurity states. The calculations made by using the first of the above relations were based on a study of the character of the pressure-induced motion of the impurity level [5]. However, this substantiation is of an indirect nature. If the activation energy of an impurity level is calculated using the second relation, the value of Ea will be one-half of that obtained from the first relation; i.e., it will correspond to a wavelength of 140 µm. Therefore, the conclusion that the metastable impurity states provide a major contribution to the photoresponse at wavelengths of 90 and 116 µm will be invalid. In this paper, we report on the detection of a photoresponse in a Pb0.75Sn0.25Te(In) film at wavelengths of 176 and 241 µm.
Experiment Pb0.75Sn0.25Te(In) films were grown through molecular-beam epitaxy on a BaF2 substrate. The thermal activation energy of the impurity ground state calculated from the relation ρ ~ exp(Ea/2kT) was 20 meV. The experiment
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was carried out in the setup shown schematically in Fig. 1. The temperature of the helium screen after filling in the liquid helium reached a stationary level in ~30 min. The disk with the filters was initially set in the position where the blackbody radiation impinged on the metal shutter. Since it was fixed to the bottom of the helium bath, the sample was cooled faster than the screen and the disk. Therefore, the sample was initially illuminated by the background radiation of the screen and the disk that had not yet cooled down. This background illumination gave rise to the generation of longlived nonequilibrium carriers in the sample. To transfer the sample to the unperturbed state, it was warmed with a heater located close to the sample, after the screen cooling. The increase in the sample temperature to 30 K and subsequent cooling to liquid-helium temperature transferred the electronic system of the sample to a close-to-ground state, with practically all the carriers localized. After this, the disk was rotated such that the radiation with the wavelength determined by the disk filter was directed onto the sample. This rotation usually lasted 3 to 4 s.
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We studied the kinetics of the current increase through the sample for various voltages across the sample and various blackbody temperatures. In Fig. 2 we have plotted the experimental data obtained for a sample voltage of 10 mV and a blackbody temperature of 300 K. A noticeable photoresponse was detected at both wavelengths of the radiation striking the sample. Because our sensitive measuring equipment made it possible to record currents of up to 0.25 µA, we could reliably measure the kinetics of the increase in the photocurrent only at low voltages across the sample, U<40 mV. At higher voltages, the photocurrent grew so fast that the amplifier became overloaded in a time comparable to the time required to rotate the filter disk, i.e., within a few seconds.
Discussion The following features in the photoconductivity may be of interest here. First of all, the current rise observed after switching on the illumination follows a strongly nonlinear kinetics. Switching off the illumination triggers a fast decay of the photocurrent, with subsequent slow relaxation to the dark level. If, however, the illumination is switched on again a short time after its removal, the photocurrent rises very fast (in a time comparable to the “fast” relaxation time) to the value recorded just before the illumination removal, after which the previous, relatively slow dynamics of the increase in the photocurrent sets in again. The fast and the slow processes are apparently of essentially different natures. Another feature may also be noteworthy. The energies of photons corresponding to radiation wavelengths of 176 and 241 µm are substantially less than the thermal activation energy of the impurity ground state, even if this energy is calculated using the relation ρ~exp(Ea/kT). Thus, the results obtained in this study provide direct evidence for the persistent photoconductivity in Pb1–xSnxTe(In) originating from photoexcitation of metastable impurity states. The threshold energy for optical excitation of these states is very low. The wavelength of the corresponding photon is longer than at least 241 µm, which, as far as we know, is the largest value of λr for non-thermal radiation detectors. The photoconductivity cutoff of the materials studied here apparently lies at substantially longer wavelengths. One cannot rule out the possibility that the operating range of Pb1–xSnxTe(In)-based photodetectors extends over the whole submillimeter range.
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Acknowledgments This study was supported in part by the Russian Foundation for Basic Research (project nos. 04-02-16497, 05-02-16657), INTAS (project no. 2001-0184), and a NATO Collaborative Linkage Grant.
References 1. Haller EE, Hueschen MR, Richards PL (1979). Appl Phys Lett 34:495 2. Volkov BA, Ryabova LI, Khokhlov DR (2002). Usp Fiz Nauk 172:875 [(2002) Phys Usp 45:819] 3. Chesnokov SN, Dolzhenko DE, Ivanchik II, and Khokhlov DR (1994). Infrared Phys 35:23 4. Khokhlov DR, Ivanchik II, Raines SN, et al. (2000). Appl Phys Lett 76:2835 5. Akimov BA, Zlomanov VP, Ryabova LI, et al. (1979). Fiz Tekh Poluprovodn (Leningrad) 13:1293 [(1979) Sov Phys Semicond 13:759]
Quasioptical Terahertz Spectrometer Based on a Josephson Oscillator and a Cold Electron Nanobolometer
M. Tarasov 1, L. Kuzmin 2, E. Stepantsov,3, A. Kidiyarova-Shevchenko2 1
Institute of Radio Engineering and Electronics RAS, Moscow 125009, Russia
2
Chalmers University of Technology, Göteborg SE41296, Sweden
3
Institute of Crystallography RAS, Moscow 117333, Russia
Abstract: We have developed a low temperature transmission spectrometer operating in a wide range of frequencies from 100 GHz to 1.7 THz. The spectrometer has utilized the unique properties of high-Tc superconducting Josephson junctions and the wideband response of sensitive Cold-Electron Bolometers (CEB). The voltage response of the CEB integrated with log-periodic and double-dipole antennas, has been measured using an oscillator consisting of high-Tc Josephson junction integrated on a separate substrate with a log-periodic antenna. Superconducting Josephson junctions with high characteristic voltages (IcRn larger than 4 mV at 4.2 K) are fabricated by depositing YBa2Cu3O7-x on miscut sapphire bi-crystal substrates, where the tilting axis is along the grain boundary. The cold electron bolometer having a superconductor-insulator-normal metal-insulatorsuperconductor (SINIS) structure was 200 nm wide, 10 µm long, and terminating tunnel junctions were 200x300 nm² area. The response of the bolometer with a double dipole antenna has resonance shape with maximum corresponding to the designed central frequency of 300 GHz. A voltage response of the bolometer up to 4·108 V/W corresponds to a noise equivalent power of the bolometer of 1.2·10-17 W/Hz1/2. Our measurements demonstrate that the Josephson junction is overheated by the transport current up to 3 K at 1 mV bias when it is placed on a millikelvin stage. A high-Tc Josephson junction operated at temperatures below 2 K has the advantage of a high IcRn product that enhances the oscillation frequency to above 2 THz. The 325 R. Gross et al. (eds.), Nanoscale Devices - Fundamentals and Applications, 325–335. © 2006 Springer. Printed in the Netherlands.
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M. Tarasov, L. Kuzmin, E. Stepantsov, A. Kidiyarova-Shevchenko resolution of the spectrometer is determined by the linewidth of Josephson oscillations and for this temperature is below 1 GHz. Further developments of such a device are possible by using a carbon nanotube.
Keywords: proximity effect, superconductivity, hybrid nanostructures.
Cryogenic Detectors It has become increasingly evident in the last few years that superconducting devices will play a major role in radiation detection and the characterization of the electromagnetic spectrum in terahertz frequency range. The critical difference between detection at terahertz frequencies and detection at shorter wavelengths lies in the low photon energies (1-10 meV). The ambient background thermal noise almost always dominates narrow-band signals requiring cryogenic cooling. The use of superconducting sensors has been further promoted through the recent development of ultra-low temperature coolers which no longer need liquid cryogens. Superconducting devices offer the prospect of fundamental thermodynamic or quantum limited sensitivity, spectral sensitive detection with ability to measure photons individually, and the ability to manufacture large arrays with modern thin-film technology. Ultimate NEP of a bolometer (see [1]) is determined by fundamental thermodynamics and is essentially the same for all types NEP2=4kbT2G, where kb is Boltzmann’s constant, T the electron temperature, and G the thermal conductance. The main limitation for practical detector NEP is determined by a background power load P0 that gives NEP2=2P0kbTe [2]. At present the state-of-the-art practical bolometers demonstrate NEP below 10-17 W/Hz1/2 at 0.1 K. The main generic types of cold direct detectors are as follows: The Transition Edge Sensor (TES) is the most acknowledged type of superconducting bolometer. It consists of a thin superconducting film that can change its resistance under the incoming radiation power. Depending on the frequency range, such a bolometer can be integrated with THz band planar antenna, or attached to an absorber film that interacts with the optical or X-ray radiation. For X-ray detectors at 100 mK one of the best NEP=3.10-18 W/Hz1/2 was demonstrated in [3]. The dynamic range of TES detectors is strictly limited by dc heating power applied before real operation. Any attempt to increase the dynamic range lead to additional
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heating with unavoidable degradation of the sensitivity. Practical TES sensors sacrifice sensitivity to avoid saturation. Superconducting Tunnel Junctions (STJ) of the SuperconductorInsulator-Superconductor (SIS) type, as well as Superconductor-InsulatorNormal metal (SIN) type can be used from microwave to X-ray wavebands. They are currently being developed as photon counting detectors. STJ convert the incident radiation energy into a population of excited charges whose number is proportional to the deposited energy and to the inverse of the superconducting gap. The superconducting electrode in which this conversion takes place serves as absorber. By measuring the tunnel current it is possible to estimate the incoming energy and frequency [4]. Due to the finite leakage current and quantum efficiency about unity, the best NEP is about 10-16 W/Hz1/2. The Superconducting Hot Electron Bolometer (SHEB) is mainly used in HEB mixers as a relatively fast (up to 10 GHz) power meter of interfered signal and LO waves, and in general it is the same type of power detector, because at signal and LO frequencies it is too slow and can’t multiply these components. It has a very low thermal capacitance and a large thermal conductance, and in this way it is optimized for speed, but not for sensitivity. This type of sensor can be optimized for direct detection, a socalled Hot Electron Direct Detector (HEDD). The theoretical estimations of NEP below 10-20 W/Hz1/2 [5] seems to be very optimistic. Taking into account the background power load, HEDD NEP should be limited at the same thermodynamic level of above 10-18 W/Hz1/2. The Normal metal Hot Electron Bolometer with Andreev mirrors (ANHEB) was proposed by M. Nahum and P.L. Richards [6] and consists of a thin normal-metal strip between superconducting electrodes. Low electron-phonon interaction at low temperatures together with Andreev reflection at the boundary of a normal metal and a superconductor prevent heat leakage from hot electrons to phonons and to the electrodes. A superconductor-insulator-normal metal (SIN) junction attached to the normal metal strip is used for temperature sensing. The best NEP=5.10-18 W/Hz1/2 was achieved at 100 mK [7]. The SINIS normal metal cold electron bolometer (CCNHEB or CEB) was proposed in [8] and experimentally demonstrated in [9]. As in all previous cases, the responsivity and noise equivalent power (NEP) of the bolometer are mainly determined by its electron temperature. To improve CCNHEB performance it was suggested using direct electron cooling of the absorber by a superconductor-insulator-normal metal (SIN) tunnel
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junction [10]. The effect of electron cooling was demonstrated in [11]. The CEB is essentially a nanorefrigerator that cools the electrons within a thin metal film by extracting the hottest electrons through SIN junctions. This effect is similar to that used in a thermoelectric cooler (Peltier effect). In contrast to TES, an unavoidable dc heating for electrothermal feedback is replaced by a deep electron cooling, removing all incoming power from the absorber to the next stage. Thereby the electron temperature is maintained at the minimum level below the phonon temperature independently on the relatively high power load. The refrigeration effect allows this detector to operate with high sensitivity under high power load. The response time is determined by the tunneling time of electrons which can be very fast (∼10ns). The NISIN hot electron bolometer [12] is completely complementary to the SINIS bolometer and clarifies the difference between hot electron and cold electron bolometers. In the NISIN case a photon assisted tunneling through the barrier heats the middle S electrode. Hot electrons are injected into a superconductor; they reduce the energy gap, which in turn increases the current through the junction. The Kinetic Inductance Bolometer (KID) is based on the fast change of the kinetic inductance in a superconducting strip when a pair breaking process reduces the superfluid density [13]. The basic principle of operation of the KID is to measure the resonant frequency of a thin-film superconducting resonator, operating at about 5 GHz. The frequency shift is detected by monitoring the transmission or reflection phase, and this shift is proportional to the energy of the absorbed photon. According to [14] the NEP is determined by the quasiparticle generation-recombination noise and at T~1K it can be as small as 10-19 W/Hz1/2, again without accounting for the background power load. Among these six generic superconducting bolometers the majority is operated at an electron temperature that is equal or above the bath or phonon temperature. Only the CEB works at a reduced electron temperature. Moreover, the electron cooling allows extracting the heating power, which leads to an increase of the saturation level. As a result the effect of the background power load is not as severe as for the rest of bolometer types.
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Samples Layout and Fabrication A general view on the cold electron bolometer with capacitive coupling (CCNHEB) chip is presented in Fig. 1. One can see in the center a broadband log-periodic antenna for the frequency range 0.1-2 THz, and double-dipole antennas for 300 and 600 GHz to the left and to the right from the center. Besides above and below the central antenna there are two structures with additional SIN junctions for studies of electron cooling in SINIS structures. The first step of sample fabrication was thermal evaporation of 60 nm Au for fabrication of the normal metal traps and contact pads. The pattern for the traps and the pads were formed using photolithography. The next step was the fabrication of the tunnel junctions and the absorber. The structures were patterned by e-beam lithography and the metals were thermally evaporated using the shadow evaporation technique. The Al (superconductor) was evaporated at an angle of about 60° up to a thickness of 65 nm and oxidized at a pressure of 10-1 mbar for 2 minutes. A Cr/Cu (1:1) absorber of a total thickness of 75 nm was then evaporated directly perpendicular to the substrate. The cooling junctions have a normal state resistance RN equal to 0.86 kΩ, while the two inner junctions have RN equal to 5.3 kΩ. The inner junctions have a simple crosstype geometry, where a section of the normal metal absorber overlaps the thin Al electrodes. The area of overlap, which determines to the area of each of the tunnel junction, is equal to 0.2 x 0.3 µm2. The structure of the outer junctions is such that the ends of the normal metal absorber overlap with a corner of each of the Al electrodes, which have a much larger area compared to the middle Al electrode. The area of each of these junctions is 0.55 x 0.82 µm2. The purpose of the larger area Al electrode is to provide more space for quasiparticle diffusion compared to the middle Al electrode with simple cross-type geometry. In the described structure, the two outer and inner junctions have the RN equal to 0.85 kΩ and 5.4 kΩ, respectively. The volume of the absorber was 0.18 µm3. A bias cooling current is applied through the outer junctions and the absorber. These tunnel junctions act as the cooling junctions, and therefore serve to decrease the electron temperature of the absorber. To determine the electron temperature, the voltage across the inner junctions is measured. A small current bias is applied to these junctions. The bias has to be optimal to obtain the maximum linear voltage response on temperature, and yet not too large so as to disturb the cooling process in the absorber.
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High critical temperature Josephson junctions on tilted bicrystal sapphire substrates were fabricated in YBaCuO epitaxial films with the c-axis inclined in <100> direction by angle 14o+14o. Films 250 nm thick were deposited by pulsed laser ablation on tilted sapphire bicrystal substrates covered by a CeO2 buffer layer. The critical temperature of the film was Tc=89 K and ∆Tc=1.5 K. Bicrystal Josephson junctions with a width ranging from 1.5 to 6 µm demonstrated a characteristic voltage IcRn of over 4 mV at a temperature of 4.2 K. This makes them promising candidates as oscillators for Terahertz frequency band applications.
Fig. 1. Optical micrograph of the central part of the CEB chip.
Power and Temperature Responses of the Bolometer We measured the temperature response of the bolometers at temperature down to 260 mK. The dc response was measured at upper and lower structures with four SIN junctions. Two external junctions were used as thermometers and two internal as heaters. The highest measured value of the voltage response to temperature variations is over 1.6 mV/K and the largest current response about 37 nA/K for a 10 kΩ junction and 55 nA/K for a 6 kΩ junction. It was possible to apply a dc power to the central pair of junctions and measure the response of the outer pair of SIN junctions for these samples with four SIN junctions. We observed the largest voltage response of
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400 V/µW for a 70 kΩ junction and 550 A/W for a 10 kΩ junction. The obtained values of current and voltage responses can be converted to the natural figure of merit for the sensitivity of the bolometer, namely the Noise Equivalent Power (NEP): NEP=In/Si or NEP=Vn/Sv . Here, In is the current noise, Vn is the voltage noise, Si=dI/dP is the current response, Sv=dV/dP is the voltage response of the bolometer. Taking the voltage noise of a room-temperature preamplifier of about 3 nV/Hz1/2 one can obtain the technical TNEP value to TNEP=1.25.10-17 W/Hz1/2 Using measured values of the temperature response and the power response one can also obtain the thermal conductivity of the bolometer.
GV =
∂P ∂V / ∂T = = 0.8 ⋅ 10−11W / K ∂T ∂V / ∂P
Now we can calculate the thermodynamic NEP arising from the electronphonon interaction NEPep2=4kT2G in which the thermal conductivity G=5ΣνT4=10-11 W/K, ν is the absorber volume. This results in a thermodynamical noise equivalent power NEPTD=1.4.10-18 W/Hz1/2 and, if we compare with the thermal conductivity in the voltage bias mode, it corresponds to a NEPV=1.3.10-18 W/Hz1/2.
Irradiation of Bolometer by a Josephson Junction To increase the output microwave power from the Josephson junction and increase the oscillation frequency it is necessary to increase the critical current of the Josephson junction. Placing the Josephson junction on the He4 stage prevents the sample from overheating by the relatively high power absorbed by the Josephson junction. For example, if we take a junction with 10 kΩ normal resistance and an oscillation frequency of 300 GHz, this results in an absorbed power over 0.2 µW. At 1 THz it is already 2.5 µW.
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The layout of the Josephson sample was with similar log-periodic antennas, and the critical current was over 500 µA at 2 K. As a result the IcRn product exceeds 5 mV for non-hysteretic junctions and such oscillators can in principle operate at frequencies above 2.5 THz. The experimental curves shown in Fig. 2 have been measured by bolometers integrated with double-dipole and log-periodic antennas. They reveal that there is a clear maximum at the design frequency of 300 GHz for DDA and a smooth spectrum for LPA. The response at higher bias voltages for the Josephson oscillator is presented in Fig. 3. The highest maximum corresponds to an oscillation frequency of 1.75 THz. For measurements at frequencies below 600 GHz we use also a low Tc Josephson oscillator with resistively shunted Nb SIS tunnel junction. It is integrated with the log-periodic antenna designed for 0.2-2 THz. Samples were fabricated by HYPRES 3 µm Nb process (for details see www.hypres.com). The linewidth of Josephson oscillations was measured by irradiating such junction by a backward wave oscillator and monitoring the detector response by a lock-in amplifier. The selective detector response in Fig. 4 shows the voltage distance between bipolar maxima below 1 µV that corresponds to the Josephson oscillations linewidth below 0.5 GHz.
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Conclusion We demonstrated the first experimental response of a normal metal cold electron bolometer at frequencies up to 1.7 THz. The noise equivalent power of the bolometer is 1.3.10-17 W/Hz1/2. We use an electrically tunable high critical temperature Josephson quasi-optical oscillator as a source of radiation in the range 0.2-2 THz and a shunted Nb SIS Josephson junction for frequencies below 600 GHz. The combination of a Terahertz-band Josephson junction and a high-sensitive hot electron bolometer opens the possibility to develop a quasioptical cryogenic transmission spectrometer with a resolution below 1 GHz. Such cryogenic spectrometer can be used for low-temperature spectral evaluation of biological and chemical samples. Cold electron bolometers can be used for remote atmosphere monitoring for pollution detection, etc.
Acknowledgments This work has been supported by Swedish agencies VR, STINT, and by INTAS-01-686.
References 1. Richards PL (1994). J Appl Phys 76:1 2. Kuzmin L (2004). Proc. 15th Int Symp on Space Terahertz Technology, Northampton, April 27-29 3. Irwin KD, Hilton GC, Wollman DA, Martinis J (1996). Appl Phys Lett 69:1945 4. Peacock A, Verhoeve P, Rando N, et al (1997). J Appl Phys 81:7641 5. Gershenson ME, Gong D, Sato T, Karasik BS, Sergeev AV (2001). Appl Phys Lett 79:2049 6. Nahum M, Richards PL, Mears CA (1993). IEEE Trans Appl Supercond 3 :2124 7. Chouvaev D, Kuzmin L (2001). Physica C 352 :128 8. Kuzmin L (2000). Physica B 284-288 :2129 9. Tarasov M, Fominsky M, Kalabukhov A, Kuzmin L (2002). JETP Lett 76 :507 10. Kuzmin L, Devyatov I, Golubev D (1998). Proc. SPIE 3465:193 11. Nahum M, Eiles TM, Martinis JM (1994) Appl Phys Lett 65 :3123
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12. Barends R, Gao JR, Klapwijk TM (2004) 6-th Eur. Workshop on Low Temp Electronics (WOLTE-6). 23-25 June, ESTEC, Noordwijk, The Netherlands, pp 25-31 13. Grossman E, McDonald D, Sauvageau J (1991). IEEE Trans Magn 27 :2677 14. Sergeev AV, Mitin VV, Karasik BS (2002). Appl Phys Lett 80 :817
NOVEL MATERIALS FOR ELECTRONICS
Origin of the Resistive Transition Broadening for Superconducting Magnesium Diboride
A. S. Sidorenko Institute of Electronic Engineering and Industrial Technologies ASM, MD-2028 Kishinev, Moldova
Abstract:
The origin of the superconducting transition broadening for the novel superconductor MgB2 is investigated. The dominant role of twodimensional fluctuations and thermally activated flux flow in the vicinity of the critical temperature is found to be responsible for the resistivity of MgB2 near the superconducting transition. The reasons of the observed extraordinary strong magnetic field dependence of the activation energy of the flux motion are discussed.
Keywords: superconducting transition broadening, fluctuations, flux flow
Introduction The discovery of superconductivity in MgB2 [1] with the highest critical temperature, Tc = 39 K, for a simple intermetallic compound triggered various expectations of big-scale technical applications of this novel material. This superconductor with a large Ginzburg-Landau parameter κ ≈ 26, a big magnetic penetration length λ(0) = 140-180 nm and short coherence lengths, ξc (0) = 2.3 nm, ξab (0) = 6.8 nm has a rather high critical current density, jc ~ 1.6×107 A/cm2 at 15 K [2], what makes magnesium diboride a very attractive candidate as a high current conductor. On the other hand, the broadening of the superconducting transition, as found in resistivity measurements, could limit the applicability of MgB2. Obviously, it is important to clarify the mechanisms responsible for the appearance of resistivity.
339 R. Gross et al. (eds.), Nanoscale Devices - Fundamentals and Applications, 339–348. © 2006 Springer. Printed in the Netherlands.
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Superconducting transition broadening may have different reasons. It may be caused by an inhomogeneous microstructure of polycrystalline samples with additional phases having different Tc values. Furthermore, thermodynamical fluctuations play an important role in the vicinity of the superconducting transition, especially for low-dimensional and layered superconductors with a short coherence length and a high Tc, as in the case of MgB2. Finally, the thermally activated flow of flux quanta leads to dissipation and results in a broadening of the transition. The first reason of superconducting transition broadening can be eliminated by improving the technological process of sample preparation. The other two mechanisms are of fundamental nature and, therefore, are more interesting for experimentalists and theorists.
Superconducting Fluctuations The fluctuations governing the superconducting transition broadening have been investigated for high quality homogeneous MgB2 films [3] and single crystals [4]. It was shown, that there exists an intrinsic transition width of the resistive transition of a superconductor caused by thermodynamic fluctuations of the order parameter. This minimal width, ∆Tc, is given by the Ginzburg criterion [5], ∆Tc = GiTc Gi = [2µ0 kBTc /Bc2(0)ξ03]2,
(1)
where Gi is the Ginzburg parameter, Bc(0) the critical flux density at T = 0 K, ξ0 = h vF / 2π kBTc, µ0 = 4π ×10-7 Vs/Am, kB the Boltzmann constant, and vF the Fermi velocity. The value of Gi is extremely small for pure 3D conventional superconductors like bulk Pb and Al (G ~ 10-13). However, it increases by many orders of magnitude for dirty and low dimensional systems making the fluctuation effects observable experimentally [6]. For layered superconductors with a small coherence length ξ0, the normalized intrinsic width ∆Tc / Tc may be much larger, up to Gi ~ 10-2 for high-Tc superconductors like YBa2Cu3O7-x [7]. Due to the layered character of MgB2 with its small coherence length it is expected that fluctuation effects resulting in a broadening of the resistive transition can be directly detected. We performed measurements of the upper critical field and the resistive transition for MgB2 films grown on MgO and sapphire single crystalline substrates by dc -magnetron sputtering. The x-ray diffraction study revealed a (101)-textured structure of the films. SEM-characterization of the samples demonstrated a polycrystalline very smooth and homogeneous
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morphology of the films with ~ 100 nm crystallites size. The resistive transitions R(T) in constant external magnetic fields were measured by a conventional four-probe method. The observed linear temperature dependence of the upper critical field demonstrates a 3D character of superconductivity in our samples below Tc as discussed in [3]. In contrast, resistivity measurements yield a temperature dependence of the fluctuation conductivity above Tc which agrees with the Aslamazov-Larkin (AL) theory [8] for a 2D superconductor in the weak fluctuation region. Our experimental results indicate a quasi two-dimensional origin of the nucleation of superconductivity in MgB2.
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T, K
Fig. 1. Resisitive transition broadening for a MgB2 film (thickness d = 5.6 µm, substrate MgO). Fluctuations manifest themselves in the region marked as “a” in the R(T) curve.
Fig. 1 shows the resistive transition R(T) at zero magnetic field for one of the investigated MgB2 samples. The transition width ∆Tc is about 0.7 K (0.1Rn-0.9Rn criterion), demonstrating a high homogeneity of the film. The fluctuation or excess conductivity, σ´ ≡ 1/R(T) – 1/Rn, strongly depends on the superconductor dimensionality D. For a two-dimensional superconductor (D = 2) in the weak fluctuation region the excess conductivity σ´ ~ (T / TcAL – 1)(D-4)/2 is expected to be inversely proportional to the temperature [σ´(T)]-1 = (Rn /τAL)(T - TcAL)/ TcAL, where τAL = (Rn e2) /16 h , and Rn is the sheet resistance of the film [8].
(2)
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Fig. 2. Linear behaviour of the fluctuation conductivity on 1/T in the fluctuation region, i.e. the upper part of the resistive transition, marked as “a” in Fig.1. The solid line corresponds to the AL-theory for a 2D-superconductor, crossing the Taxis at TcAL (marked by the arrow).
Figure 2 presents the results of Fig. 1 in the (σn / σ´) vs T plot, indicating that the resistive behavior of the MgB2 film is really caused by superconducting fluctuations according to eq. (2). The fit of the experimental results in Fig. 2 by equation (2) gave us the values of Rn and TcAL. According to eq. (2), the intersection of the linear fit with the T-axis gives the critical temperature TcAL = 36.77 K, which is higher than the midpoint value of Tc0.5Rn = 35.67 K as the result of the critical temperature shift due to fluctuations. The procedure of the precise determination of the normal state resistance, Rn, is shown in Fig. 3. From the transition width of the film ∆Tc = 0.7 K we obtain the experimental value of the Ginzburg parameter, G = ∆Tc / Tc = 0.02, which is comparable with the value Gi = 10- 2 for high- Tc superconductors.
Origin of the Resistive Transitions Broadening
T = 36.0K T = 37.0K T = 40.0K
0,05
-1
1/R (Ohm )
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0,04 0,03 0,02
0
5
10 15 -1 1/Bperp (Tesla )
20
Fig. 3. Precise determination of the normal-state resistance of the measured sample. The intersection of the 1/R vs 1/B curves, measured at different temperatures, with the 1/R-axis gives the value Rn = 42.185 Ohm.
Thermally Assisted Flux Flow (TAFF) A further mechanism causing a broadening of the superconducting transition of MgB2 in the lower part of the R(T) curve (marked as “b” in Fig.1) was found. The resistivity in that region is governed by the flux-flow processes. In a type-II superconductor like magnesium diboride, in the mixed state the flux lines are fixed at “pinning centers” formed by defects or impurities. The main mechanism of flux creep resulting in a broadening of the resistive transition in an applied magnetic field, is the thermally activated motion of flux lines over the energy barrier, U0, of the pinning center [9]. The layered structure of MgB2 is expected to influence the magnetic flux penetration and motion leading to a resistive transition broadening similar to the case of high-Tc superconductors and artificially multilayered systems [10, 11]. Moreover, magnesium diboride exhibits an exceptional magnetic behavior with dendritic magnetic instabilities for vortex penetration [12] and “noise-like” jumps of the magnetization in an applied magnetic field [13], which should influence the resistive behavior of this novel superconducting material.
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80
ρ ( µΩ cm )
60
40
7
5
4
3
2
1
0
20
0 20
25
30
35
T (K) Fig. 4. Resistive transitions ρ(T) for a 400 nm thick MgB2 film on a sapphire substrate measured at different values of the magnetic field perpendicular to the film plane: curves 0 to 7 correspond to B = 0, 1, 2, 3, 4, 5, 7 Tesla.
Figure 4 shows the resistive transitions ρ (T) at several magnetic fields B, applied perpendicular to the MgB2 film plane for one of the investigated samples grown on a sapphire substrate. The transition width is about 0.3 K in zero and low magnetic fields but increases up to ~ 2 K for high fields. Usually, for layered superconductors the broadening of the lower parts of the resistive transition, ρ (T) < 1 % ρ n ( ρ n is the resistivity in the normal state just above the transition), in a magnetic field is interpreted in terms of the dissipation of energy caused by the motion of vortices. This interpretation is based on the fact that in the low-resistance region the creep of vortices represents a thermally activated process so that the ρ (T ) dependences are described by the Arrhenius equation
ρ (T , B) = ρ0 exp [ −U 0 / kBT ] .
(1)
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B7=7T
10
B6=5T
ρ ( µΩcm )
B5=4T B4=3T B3=2T
0,1
B2=1T B1=0T
1E-3
1E-5 0,8
1,0
1,2
1,4
1,6
Tc / T Fig. 5. Arrhenius plot ρ(T)B=const presents the data of Fig. 4 as ln(ρ) vs (Tc / T). From the slope of the linear parts of the curves the values of the activation energy U0 are obtained.
Here, U0 is the energy barrier that has to be overcome by thermal activation and ρ0 is a field-independent pre-exponential factor. Investigations of highTc superconductors and artificial multilayers showed that the activation energy exhibits weak power-law dependences on the applied magnetic field, U0(B)~B-n with an exponent n ~ 1 [10, 11]. Since eq. (1) with a temperature-independent U0 is used, the values of U0 should be deduced only from the limited temperature intervals below Tc , in which the data of the Arrhenius plot of ρ (T ) yield straight lines. The straight-line behavior over five orders of magnitude of the resistance really indicates that the resistive behavior of the lower parts of the resistive transition is caused by the TAFF-process as described by the Arrhenius law, Eq. (1). The best fit of the experimental data for ρ (T ) B=const yields values of the activation energy, ranging from U0/kB = 10000 K in low magnetic field down to 300 K in the high field region, as shown in Fig. 6. Compared to the power law U0(B)~B-n usually observed for other layered systems, MgB2 shows a much stronger field dependence of the activation energy in the high magnetic field region (B > 1 T). This is clearly demonstrated by Fig. 6, where data for the investigated MgB2 sample are shown together with a typical U0(B)
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dependence for a high-Tc superconductor (data for a Bi-Sr-Ca-Cu-O sample, taken from Palstra et al. [14]).
U0/kB (K)
10000
MgB2 on Al2O3 BiSrCaCuO
100 0,1
1
10
B (T) Fig. 6. Dependence of the activation energy U0 / kB on applied magnetic field for the investigated 400 nm thick MgB2 film deposited on sapphire (solid squares). For comparison, the weak power-law dependence U0 ~ B-n, n = 1/6 and n = 1/3 in two linear parts of the U0(B) dependence, observed for a high-Tc superconductor (solid circles, data for Bi-Sr-Ca-Cu-O sample [14]) is given.
A possible reason for the unusually strong magnetic field dependence of the activation energy of the TAFF process in MgB2 may be the appearance of thermo-magnetic instabilities, leading to a complex flux dynamics, such as the dendritic flux instability found recently for c-axis textured MgB2 films in a magnetic field perpendicular to the film plane. The magnetooptical measurements demonstrate a “fractal-like“ structure of the flux penetration with increasing amount of the flux-dendrite density with increasing applied magnetic field [12]. More generally, thermal instabilities, analyzed by M. Tinkham [15], lead to disastrous consequences for superconducting magnets and cables because the material rapidly heats up due to the dissipation of energy caused by the flux creep. Therefore, for thermal stability of a superconductor, an increase in local heat production needs an increased outflow of heat to the surrounding material which has to respond to the increased dissipation due to flux motion.
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Conclusion From the reported experimental results one may conclude that the resistive transitions broadening for good quality MgB2 films has a fundamental reason: fluctuations in the upper part of the resistive transition and fluxflow processes for the lower part. The rapid decrease of the activation energy for MgB2 in the magnetic field region B > 1T reflects a dramatic loss of the current carrying capabilities of this superconductor due to the weakening of the flux-line pinning with increasing magnetic field. The unusual flux creep behavior of magnesium diboride needs further investigation, especially in view of future technical applications, an increase of the flux line pinning and a thermal stabilization of wires and cables are necessary for electrical transport with high current density.
Acknowledgments I am very grateful to Jan Klamut, Andrei Varlamov, Paul Ziemann, Jan Aarts, Serghej Prischepa, Armin Reller for useful and stimulating discussions, Thomas Schimmel, Thomas Koch, Achim Wixforth, Siegfried Horn, Reinhard Tidecks, Valery Ryazanov and Lenar Tagirov for long-term fruitful collaboration, as well as to the A. v. Humboldt Foundation for the donation of “Coolpower-4.2GM” and “PLN-106” refrigerators. This work was partially supported by BMBF, Project Nr. MDA02/002, and by the International Laboratory of Strong Magnetic Fields and Low Temperatures (Wroclaw).
References 1. Namagatsu J, Nagakawa N, Muranaka T, Zenitani Y, Akimitsu J (2001). Nature 410:63 2. Kim HJ, WN Kang, Choi EM, Kim MS, Kim KHP (2001). Lee SI, Phys Rev Lett 87:087002 3. Sidorenko A, Tagirov L, Rossolenko A, Ryazanov V, Klemm M, Tidecks R (2002) Evidence fort wo-dimensional superconductivity in MgB2. Europhys Lett 59:272-276 4. Masui T, Lee S, Tajima S (2003). Physica C 383:299 5. Schmidt VV (2000) Introduction to the Physics of Superconductors. Ryazanov VV and Feigelman MF (eds) 2nd Russian edition, MCNMO Publishers, Moskow. Chapter III, §19 6. Fogel NYa, Sidorenko AS, Rybalchenko LF, Dmitrenko IM (1979). Sov. Phys JETP 50:120
348 7. 8. 9. 10. 11. 12. 13. 14. 15.
A. S. Sidorenko Kapitulnik A (1988). Physica C153-155:520 Aslamazov LG and Larkin AI (1968). Phys Lett 26A:238 Yeshurn Y, Malozemoff AP (1988). Phys Rev Lett 60:2202 Fogel NY, Cherkasova VG, Koretzkaya OA, Sidorenko AS (1997). Phys Rev B 55:85 Graybeal JM, Beasley MR (1986). Phys Rev Lett 56:173 Johansen TH, Bazilevich M, Shantsev DV, Goa PE, Galperin YM, Kang WN, Kim HJ, Choi EM, Kim MS, Lee SI (2002). Europhys Lett 59:599 Jin S, Mavoori H, Bower C, Dover RB (2001). Nature 411:563 Palstra TTM, Batlogg B, Schneemeyer LF, Waszczak JV (1988). Phys Rev Lett. 61:1662 Tinkham M (1996) Introduction to Superconductivity. 2nd ed, McGraw Hill, New York
Aharonov-Bohm Oscillations in Single Bi Nanowires
D. Gitsu1, T. Huber2, L. Konopko1,3 , A. Nikolaeva1,3 1
Institute of Electronic Engineering and Industrial Technologies ASM, Kishinev MD-2028, Moldova
2
Department of Chemistry, Howard University, 525 College St. N.W. Washington, DC 20059
3
International Laboratory of High Magnetic Fields and Low Temperatures, Wroclaw, Poland
Abstract:
We report on the study of the magnetoresistance (MR) of 55 nm diameter single bismuth nanowire at a temperature of 1.5 K. For a wide range of magnetic fields and different sample orientations the MR exhibit field-periodic modulations. We have observed 3 periods of MR oscillations; one of them is consistent with theoretical predictions for the Aharonov-Bohm oscillations in disordered cylinders. Two others may be connected with Dingle’s predictions for oscillations resulting from quantization of the electron energy spectrum.
Key words: magnetoresistance, Aharonov-Bohm oscillations, bismuth
Introduction The semimetal Bi has a highly anisotropic Fermi surface, low carrier density, and long carrier mean free path. Since for Bi the charge carriers have a small effective mass, quantum confinement effects in Bi are more clearly manifested and can be observed in nanowires of larger diameter than those of any other material. It was theoretically shown that when charge carriers pass through a cylindrical conductor aligned with a 349 R. Gross et al. (eds.), Nanoscale Devices - Fundamentals and Applications, 349–353. © 2006 Springer. Printed in the Netherlands.
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magnetic field, they reveal their wavelike nature in the appearance of the resistance oscillations as a function of the applied magnetic field. A particular type of quantum-size effects in normal conductors with long carrier mean free path is the solid state analogue of the Aharonov-Bohm effect (AB) [1] which exists in cylindrical wires as MR oscillations with a period ∆B that is proportional to Φ0/S, where Φ0=h/e is the flux quantum and S is the cross-sectional area of the wire. The longitudinal magnetoresistance (LMR) oscillations of the AB-type, which are equidistant in magnetic field with period ∆B=Φ0/S and decrease in amplitude with increasing field due to electrons undergoing continuous grazing incidence reflections at the wire wall, have been observed at 4.2 K in single nanowires grown by the Ulitovsky technique with diameter 0.2
Experiment and Discussion The single nanowire samples with diameter 55 nm were prepared by the high frequency liquid phase casting in a glass capillary using an improved Ulitovsky technique [10] and represented cylindrical single crystals with the (1011) orientation along the wire axis. All measurements were
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performed at the High Magnetic Field Laboratory (Wroclaw, Poland) in a superconducting solenoid generating magnetic fields up to 14 T at temperatures between 1.5 K and 2.1 K.
o
θ=90
o
θ=48
o
dR/dB, a.u.
θ=32
o
θ=0
0
2
4
6
8
10
12
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B, T
Fig. 1. Magnetic field derivative of the resistance, dR/dB, for a single Bi wire (d=55 nm) with glass coating vs. magnetic field at T=1.5 K. θ is an angle between the direction of the applied magnetic field and the wire axis, θ=0 corresponds to the longitudinal MR and θ=90o to the transverse MR.
The sample was measured in a two-axis rotator in which the sample holder was rotated by means of a worm drive and a twisted fiber connected to a shaft of a step motor. Together with MR data we measured the first derivative of the MR with respect to the applied magnetic field using modulation technique (a small AC magnetic field of amplitude ∆B=7.5x10-5 T and a frequency of 13.7 Hz was produced by a special superconducting modulation coil). A 5210 lock-in amplifier was used for demodulation of the MR signal. Figure 1 shows the derivative dR/dB of a 55-nm Bi nanowire sample for various orientations between the wire axis and the magnetic field. We have observed oscillations in the derivative of the MR signal at various orientations of the sample. The observed oscillations are a superposition of three frequencies. The distinct character of the beats for longitudinal MR oscillations made it possible to determine their periods with sufficient
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accuracy. For θ=0 (longitudinal MR) we found ∆1=∆2=0.74 T, and ∆3= 1.21 T. Then, as the angle θ is increased, the two periods ∆1 and ∆2 split up. The period ∆1 is in good agreement with ∆B=Φ0/2S obtained for the wire diameter d=60 nm, which very close to the actual wire diameter. The observed oscillations (∆1) have an oscillation period corresponding to half the normal Aharonov-Bohm period. Moreover, the observed variation of the oscillation period as a function of the angle between the magnetic field and the wire axis (see Fig. 2) is in satisfactory agreement with the theoretical dependence ∆1(θ)=∆1(θ=0)/cosθ. The same angle dependence was obtained in [3-4] for a Bi wire with d=0.56 µm, however, with the normal Aharonov-Bohm period (see inset of Fig. 2). The carrier mean free path in our sample is estimated to be longer than 1 µm but a period ∆B=Φ0/2S was predicted [8] for disordered cylindrical samples with short mean free path.
2,5 0,4
- ∆1 - ∆2
0,3
- ∆3
∆1, kOe
2,0
0,2
∆, T
H||J
1,5 0
20
40
60
θ, Deg
1,0
B⊥J B||J 0,5 0
20
40
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80
θ, Deg
Fig. 2. Dependence of the three oscillation periods ∆1, ∆2 and ∆3 obtained from the MR data of a 55-nm Bi nanowire on the angle between the direction of the applied magnetic field and the wire axis. The solid line shows the ∆1(θ=0)/cosθ law. A similar dependence for the oscillation period ∆1 obtained for a Bi nanowire with d=0.56 µm [10] is shown in the insert.
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Due to the Ulitovski technology the interface between the Bi-core and the glass surrounding may contain considerable trapped charge which can produce a narrow (~ 6 nm) space-charge region. The carriers of this region have shorter mean free path because of surface roughness scattering. Two other periods ∆2, ∆3 observed in our experiments, to our opinion are the Dingle oscillations from the Bi core.
Acknowledgments This work was supported by the Civilian Research Development Fund under Grant CGP # MO-E1-2603-SI-04.
References 1. Aharonov Y, Bohm D (1959). Phys Rev 115:485 2. Brandt B, Gitsu DV, Nikolaeva AA, Ponomarev YaG (1977). Zh Exp Teor Fiz 72:2332 [(1977) Sov Phys JETP 45:1226] 3. Brandt NB, Gitsu DV, Nikolaeva AA, Ponomarev YaG (1982). Lect Notes Phys 152:415 4. Brandt NB, Bogachek EN, Gitsu DV, Gogadze GA, Kulik IO, Nikolaeva AA, Ponomarev YaG (1982). Fiz Nizk Temp 8:718 [(1982) Sov J Low Temp Phys 8:358] 5. Brandt NB. Gitsu DV, Dolma VA, Ponomarev YaG (1987). Zh Exp Teor Fiz 92:913 6. Huber TE, Celestine K, Graf MJ (2003). Phys Rev B 67:245317 7. Dingle RB (1952). Proc Roy Soc London Ser A 212:47 8. Al’tshuler BL, Aronov AG, Spivak BZ (1981). Zh Exp Teor Fiz Pisma 33:101 [(1981) JETP Lett 33:94] 9. Sharvin DYu, Sharvin YuV (1981) JETP Lett 34:272 10. Brandt NB, Gitsu DV, Joscher AM, Kotrubenko BP, Nikolaeva AA (1976). Prib Tekh Eksp 3:256
Some Applications of Nanocarbon Materials for Novel Devices
Z. A. Mansurov al-Farabi Kazakh National University, 96A, Tole be Str., 480012, Almaty, Kazakhstan
Abstract:
The results of the synthesis and study of nanocarbon materials, namely the structure and morphology of carbon formation, the sorption of gold by carbonized apricot stones, the mechano-chemical encapsulation of quartz particles by metal-carbon films, the production of carbon-graphite cathodes for chemical sources of current, the development of multifunctional catalysts for oil hydrofining, the carbonization of vegetative raw material, and the development of carbon containing refractories are presented.
Key words: nanocarbon materials, sorbent, carbonization
Introduction The last years are characterized by the intensive development of investigations on the formation of carbon nanotubes. It was found that Fe, Ni, Co, their oxides as well as the alloys of these elements are the most effective catalysts for carbon formation. As a result of a catalytic reaction carbon is formed and is deposited on dispersed metallic particles. These deposits have specific forms and properties which allow us to consider them as perspective ultra-dispersed systems used in different areas of chemistry. Over the last years, some experimental materials point at specific developments which indicate a successful application of metalcarbon compositions as adsorbents, catalysts carriers and catalysts for a number of chemical reactions [1]. 355 R. Gross et al. (eds.), Nanoscale Devices - Fundamentals and Applications, 355–368. © 2006 Springer. Printed in the Netherlands.
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At present, the laboratories of the Chemical Physics Department and Combustion Problems Institute of al-Farabi Kazakh National University carry out investigations on obtaining carbonized materials on the basis of local clays and wastes of the mining industry: chromite and bauxite sludges, agricultural wastes (walnut shells and grape stones) and using them in different applied aspects. The obtained adsorption-catalytic systems find practical applications in oil purification from sulphur containing compounds, purification of water from organic compounds and heavy metal ions, purification of air from SO2 and in obtaining improved types of refractories, such as carriers of catalysts of hydrocarbon transformation reactions [2-4].
The Structure and Morphology of Carbon Formations It is possible to obtain a more suitable surface with a larger number of active centers than present in the initial samples by carbonization. Figure 1 shows electron microscopy micrographs of carbonized chromite slime and clay treated with Co salt. It is seen that formation of carbonaceous deposits results in not only chemical (formation of metal carbide and carry-over of carbide to carbon fiber), but also physical processes – carry-over of particles to carbon mass. Pyrolysis on the surface of the catalyst results in the formation of carbon deposits or the so-called catalytic carbon, the amount of which is 5-10 times larger than the initial one. A developed specific surface is related to the morphology of carbon deposits in the form of fibers, tubes and clusters with the diameter of 1500-3000 Å. It should be noted that the growth of carbon fibers (tubes) in general is complicated and sometimes has a branched character, i.e. one metal particle initiates the growth of several carbon fibers in different directions, the so-called "octopus" effect [2]. Simultaneous participation of carbon and metals in the formation of carbon fibers makes it possible to develop new nanostructured composition materials with complex of widely varying properties.
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Fig. 1. Scanning electron microscopy micrograph of chromium sludge and Cocatalyst carbonized at T = 1073 K.
Sorption of Gold by Carbonized Apricot Stones [5] At the present level of development of metal manufacturing, during the processing of the complex composition a large amount of the noble metals is lost. That is why the questions connected to the extraction of noble metals from industrial solutions by sorption on carbonic sorbents still remains relevant. The necessity of cheap sorbents satisfying industrial requirements regarding sorption capacity, thermostability, controllable size and pore structure etc., leads to the development of new methods for obtaining suitable sorbents. Carbonization of apricot stones (CAS) was carried out by a method described in [2]. Carbonized sorbents based on apricot stones have the codes CAS-1, CAS-2, and CAS-3. The sorption of gold (III) was studied by electrochemical methods using a P-5848 potentiostate by means of recording current vs. time, J-t, curves at constant potential corresponding to the extreme current of gold electroreduction (III). Carbonized sorbents based on vegetative raw material concurrently possess ion-exchanging and reducing properties. Stationary potentials of
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CAS were measured to reveal their reducing properties. The sorbents of the CAS series have relatively low oxidative-reducing potential that depending on the CAS mark change between 0.20 V and 0.25 V. This implies that the CAS sorbents possess reducing properties, which are probably determined by the presence on the surface of carbonized sorbents of reducing groups – carboxylic, phenolic, hydroxyl, amine. The relative amount of these groups depends on the conditions of carbonization. At carbonization, in the beginning predominantly aromatic compounds are formed, which with increasing carbonization temperature are condensed into polycyclic aromatic compounds (PAH) and in fullerene-like compounds. Thus, carbonized sorbents are not only have ion-exchanging but also oxidativereducing properties. The measured stationary (real) potential of the muriatic medium [AuCl4] is 0.76 V. The difference of potentials between the gold-oxidant and the sorbent-reducing agent ranges form 0.51 to 0.56 V. A potential difference of 0.24 V is required in practice for completely passing any oxidativereducing reaction. It follows from these data that there is a real opportunity for the reduction of gold (III) to the metallic state. This opportunity is proved by magnitudes of CAS sorbent potentials after sorption of gold (III) on it. The reducing of gold (III) to the metallic state was proven by optical microscopy images (see Fig. 2).
Fig. 2. Micrograph of a sorbent CAS-2 + Au(III).
Extraction of metallic gold occurs non-uniformly over the whole surface of the CAS particles, and in separate areas where the growth of gold crystals occurs. Hence, the process of metallic gold extraction and sorbent reducing groups’ oxidation is electrochemical, i.e. there are cathodic and anodic areas. Cathode areas are formed in the initial moment of sorption where further the reducing of gold (III) occurs.
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As well-known in chemistry the method of initial areas was used for the determination of the magnitude of gold sorption velocity. Kinetic curves of gold sorption on sorbents CAS-1, CAS-2, and CAS-3 in a solution of 0.1n HCl have been obtained by an electrochemical method. Sorbent CAS-2 was found to have maximum velocity of gold sorption as it is seen from Fig. 3. Complete sorption of gold (III) on this sorbent occurs after 8 minutes, whereas the time of complete sorption of gold (III) on CAS-1 is 11 minutes, and on CAS-3 even 16 minutes. That is, the velocity of gold (III) sorption decreases in the sequence CAS-2 > CAS-1 > CAS-3. Based on this data our further research was focusing on the sorbent CAS-2. Complete sorption of gold (III) occurs after 8 minutes in the range of studied concentrations of gold (III) (8.88-3.5 mg/L) independent of its concentration. At the same time, the steepness of the J-t-curve of gold (III) was found to decrease from solution and consequently its sorption extends with increasing gold (III) concentration. The increase of gold (III) sorption velocity occurs in the presence of copper salt (II) and iron (III) what is connected to the catalytic effect of gold (III) reducing. Complete sorption of gold (III) occurs after 5 minutes in the case of the presence of copper (II) ions, and after 1.5 minutes in the case of iron (III) ions, whereas it takes 8 minutes in absence of these admixtures.
Fig. 3. Kinetic curves of gold (III) sorption on various sorbents in 0.1n HCl solution. Sorbent mass – 0.2 g, particle diameter – 0.5 mm, gold concentration – 17.75 mg/L. Solution volume – 20 mL.
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Such increase of the gold (III) sorption velocity by means of the sorbent CAS-2 can be explained by a catalytic effect: the sorbent CAS-2 reduces iron (III) to iron (II), and copper (II) to copper (I). The ions of iron (II) and copper (I) are formed at the surface of the sorbent and at once react with gold (III) ions in oxidative-reducing interaction. As the reduction of gold (III) by iron (II) and copper (I) ions occurs in the surface layer of the sorbent, the metallic gold is extracted on this surface. The process of gold (III) sorption in the presence of iron (II) salts practically can be represented as following: CAS-Red + Fe(III) → CAS + Fe(II) + Ox. Au(II) + Fe(II) → Au0 + Fe(III) In the presence of copper (II) salts: CAS-Red + Cu(II) → CAS + Cu(I) + Ox Au(III) + Cu(I) → Au0 + Cu(II) The easiness of gold (III) sorption by iron (II) is widely used in analytical methods of gold determination. Thus, our study of gold (III) sorption on the sorbents CAS-1, CAS-2, CAS-3 in a muriatic medium has shown that the sorbent CAS-2 has the strongest sorption ability. As long as gold (III) sorption occurs it it reduced to the metallic state. The stationary oxidative-reducing potentials of the sorbents have been measured. The potentials ranged between 0.20 and 0.25 V. This demonstrates the possibility of oxidative-reducing interaction with gold (III). The sorption velocity is decreased in the sequence CAS-2 > CAS-1 > CAS-3. When studying the influence of Au (III) concentration, sorbent mass and particle size on the sorption process it was found that the gold (III) sorption velocity on the sorbents CAS is directly proportional to the gold (III) concentration and the sorbent mass, whereas it is inversely proportional to the particle diameter. It is shown that superior amounts (in 100-800 times) of the metal ions Fe(III), Cu(II), Cd(II), Zn(II), Hg(II), Pt(IV), and Fe(III) do not impede to gold (III) sorption on the sorbent CAS-2.
Mechanochemical Encapsulation of Quartz Particles in Metalcarbon Films [6-7] When one aims at obtaining composition nanomaterials, usually special attention is paid to carbon- and silicon containing systems due to the peculiarities of their properties and a wide range of applications. Surface
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nanostructures play a significant role in such objects as highly disperse adsorbents, catalysts, filling agents of composition materials as well as in the materials for electronic industry. Mechanochemical synthesis has several advantages compared to other methods in the production of highly disperse nanostructured materials based on silicon containing materials. As it was shown [6-7], it is possible to obtain quartz-based highly disperse materials by mechanochemical synthesis, which exhibit simultaneously interesting magnetic, dielectric and electric properties depending on the regimes of treatment and the conditions of subsequent application. In order to clarify why quartz-based highly disperse materials display the abovementioned properties, structural investigations of the synthesized material were carried out. EPR-spectroscopy has shown the presence of para- and ferromagnetic centers in quartz particles after mechano-chemical treatment. The presence of ferromagnetic part of the quartz EPR spectrum indicates the presence of Fe-containing clusters on the surface of the particles. The formation of Fe compounds connected with the quartz surface was proven by Mössbauer spectroscopy. The applied methods of our structural research have shown that structural and phase transformation occurred on the surface of the quartz particles during the mechano-chemical treatment. The detected compounds are composites of a modified quartz surface and polymers formed on its surface. The structure of the modified quartz particle and its transformation in the process of mechano-chemical treatment is vividly shown by the results of scanning electron microscopy studies. After grinding in a mill without modifying the organic additives, the surface layer of the particle becomes partially amorphous and the ultradisperse iron being ground is introduced into the surface from the walls of the milling vessel and balls. These structural changes were analyzed in detail in [6-7]. The main volume part of such particles remains crystalline. This fact has been verified by the results of electron microdiffraction (Fig. 4a). When using a dispersing organic additive to the quartz during grinding in a centrifugal-planetary mill, the structure of the particles undergoes considerable changes. The surface layer is a dense or porous multilayer system of a different structure. The electron microdiffraction analysis shows the process of transformation of the quartz particles (see Fig. 4a) into a composition material containing particles with a partially amorphous and modified structure (see Fig. 4b) up to the formation of dense carbonaceons structures on the quartz surface (see Fig. 4c).
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Fig. 4. Electron microscopic pictures and electron diffraction of modified particles of quartz.
It is found that there is a certain regularity in the structural transformations taking place on the surface of a quartz particle with the carbon of the organic additives during the treatment of the material in a mechanical reactor: from thin dense films with the introduced ultradisperse particles of iron to porous multilayer structures with the formations of different configurations grafted to the surface of the particles. Films, fibers or nanotubes (see Figs. 4a-c) are formed on the surface depending on the kind of the carbon containing modifier being used (alcohols, polystyrene) and the time of treatment. Ultradisperse iron introduced into the surface of the quartz particles catalyzes the growth of carbon nanotubes on the particle surface. After mechano-chemical treatment, the quartz particles become encapsulated by a thin coating film with a thickness ranging between 10 and 50 nm, which is strongly bound to the surface of the particle. Carbon enters into the composition and structure of this cover layer. The carbon content can be determined by an adsorptionweight method. This carbon was defined as in content of a homopolymer (i.e. in not bound state with surface). Its content varies from 1.5 to 3.5%. When making an activated coal input of 5%, after mechano-chemical treatment 3.03% of the carbon was found to be in a chemically bound state. When using butanol and polystyrene as modifiers, the amount of carbon was equal to 1.76 and 2.49%, respectively. Thus, the structure of the quartz particle surface layer after a mechanochemical treatment may be interpreted as a nanostructural formation containing carbon, silicon, and iron and having a complex structure encapsulating a particle. Finally, a composition material is formed consisting of a carbon surface layer with introduced inclusions of iron, a transition amorphous layer on the basis of silicon and a crystalline quartz base.
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Such structural rearrangements of the surface layers of the quartz particles being dispersed result in a change of the physical properties of the material. The measurement of the specific resistance of a quartz powder after a mechanochemical treatment showed a decrease by more than an order of magnitude. The modified quartz powder after a mechanochemical treatment also exhibits magnetic properties (Table 1). Table 1. The change of specific resistance and magnetic permeability of quartz powder depending on the time of weathering. Grinding was carried out during 5 minutes.
Material Quartz Quartz + butanol Quartz + polystyrene
specific resistance, ρ×106 , Ω×m time of weathering τst, 24 hours 0 30 1.1 6 6. 15.0 2.6 3.5
magnetic permeability time of weathering τst, 24 hours 0 30 2.0 2.0 4.0 3.0 29.0 27.0
Both the electric and magnetic characteristics of mechano-chemically treated quartz are found to change with time, i.e. the material "ages". However, if such "aged" material is placed into an electromagnetic field of definite intensity, its ferromagnetism and conducting characteristics are recovered. This most likely is the result of the interaction of a piezoelectric quartz nucleus with the carbon nanostructural film containing ultradisperse inclusions of iron.
Production of Carbongraphite Cathodes for Chemical Sources of Current [8-9] A coalgraphite layer was grown on metal current taps with the aim to develop an optimal method for production of coalgraphite cathodes for lithium chemical sources of current. The obtained electrodes were tested for the discharge of pure nickel screens, electrodes of fibrous carbon materials and electrodes produced according to conventional technology using polyethylene as a binder. It is shown that the obtained carbonized material is not inferior in its characteristics to a traditional one.
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In order to measure electrochemical characteristics of cathodes made according to the proposed method, we created a model of a chemical source of current with a lithium anode and an electrolyte containing 1.6 mole/L of lithium chloraluminate in thionyl-chloride (the electrolyte composition corresponds to that used in commercial samples for this purpose). The electrodes were tested in a galvanostatic regime using two-sided polarization of the cathode. The latter was wrapped in a separator of nonwoven glass fibre, placed into a slot cell between two lithium electrodes in the electrolyte, and the electrodes of the unit were tightly compressed. After assembling the unit, under galvanostatic conditions, polarization of the model was carried out by a current corresponding to current density on the electrodes of 25 mA/cm2. The discharge of the model was carried out until a sharp decrease of the cell voltage appeared. Our investigations showed that the above mentioned decrease in voltage corresponds to the filling of pores in the carbon material with electrolyte reduction products. The major reduction product is lithium chloride which is slightly soluble under these conditions. For comparison, the tests on the discharge of pure initial nickel screens, electrodes of fibrous carbon materials using different materials as electrodes made according to conventional technology using polyethylene as a binder were carried out. The discharge of the nickel screen at a current density of 6 mA/cm2 showed almost an instant passivation of the electrode by the reaction products that result in a quick decrease of voltage. The discharge of a soot electrode with a polyethylene binder lasted 30 minutes at current density of 12 mA/cm2. The test showed that the discharge characteristics of the models of current sources of the lithium-thionylchloride practically did not depend on which of the metals the fibrous carbon was deposited. Discharge curves were obtained for samples with coated layers of fibrous carbon in three different regimes (see Fig. 5). The mass of the deposited coating varied from 0.01 to 0.05 g. It was shown that the time of the electrode discharge before a sharp decrease of voltage on the cell directly depended on the amount of the coated carbon material and the time of the layer growth. At the discharge in thionylchloride solution with current density of 6 mA/cm2 the time of discharge reaches 4 minutes.
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Fig. 5. Discharge characteristics depending on cathode material (obtained by carbonization at 750°C), j = 20 mA/cm2: 1 - carbonized iron net; 2 - cathode by the standard technology.
Multifunctional Catalysts of Oil Hydrofining [10] Multifunctional catalysts of hydrofining were developed by modifying alumocobaltmolybdenum catalyst with hardened zeolites. Catalysts were prepared by the conventional impregnation method. After moulding, the catalysts had been dried at 393-420 K (4 hours) and calcinated at 823 K during 5 hours. Then, the catalysts were exposed to the process of carbonization to obtain active forms of carbon as a carrier. Activation of the catalysts was carried out in the process of sulphurizing by free sulphur first at 393-423 K, at a hydrogen pressure of 0.7-1.0 MPa during 3 hours, then sulphurizing was continued at 473 K, at a pressure of 2.5 MPa for 2.5-3 hours. The catalysts were investigated by different physico-chemical methods. Specific surface and porosity were determined by the BET method. The types of adsorbed forms of hydrogen and activation energy were determined with the help of thermo-programmed desorption of hydrogen. The texture of catalysts was determined with the help of electron microscopy. The reaction products were analyzed by highly effective gasliquid chromatography with a capillary column on the chromatograph of "Perkin Elmer" company. Chromatograms were computed by "Total Chrom Arneld" software. A detailed hydrocarbon analysis was carried out
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within the framework of "DHA Win" program. The content of sulphur in the samples was determined by the X-ray fluorescence analyzer "Oxford Lab-X-3500". The activity of catalysts was tested in a laboratory flowing plant in the processes of hydrofining gasoline and diesel fractions of oil at a pressure ranging form 2.0-4.5 MPa, temperature 593-673 K, and volume rate of raw material feed 1-4 hour-1. The characteristics of the commercial catalyst GO70 were taken under equal conditions in order to compare the catalytic activity. Butyl-mercaptan and thiophene diluted in n-decane were used as a model compound. The isomerization activity was tested in the reactions of cracking of hexane and decane. The developed catalysts showed higher indices than commercial catalyst GO-70. With the increase of temperature from 593 to 673 K the degree of hydrodesulphurization increases from 75.3 to 98.2%, and isomerization degree increases from 18.3 to 41.7%. According to the data of the detailed hydrocarbon analysis the estimated octane number increased from 70.83 to 83.48 (see Table 2). Obtaining of active forms of carbon on catalysts results in the formation of more highly dispersed systems having correspondingly a higher catalytic activity in hydrofining and hydroisomerization processes. The study of the surface and porosity of catalysts by the BET method showed the increase of the specific surface of catalysts from 80 to 240 m2/g. Catalysts, mainly, have pores with a size less than 50 nm. Electron microscopy studies showed carbon coated catalysts to be highly disperse. The metal particle size was found to vary within 20-40 nm. Table 2. Effect of the influence of over-carbonized industrial CoMo - catalyst on the characteristics of straight-run benzine, concentration of carbon - 10%. octane level initial straight-run 70.83 benzene (SB) SB after overcarbonized catalyst
83.48
boilingpoint °C
concentration paraffin
(mass.%) naphthenes
concentratio n of Scompound
88.1
21.42
12.95
0.12
36.06
1252
16.05
0.001
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Carbonization of Vegetative Raw Material [4] The raw material on the basis of the treatment of agricultural products belongs to quickly renewable sources and is ecologically more friendly. When carbonizing (pyrolysis in an inert medium) samples of walnut shells (WS), apricot (AS) and grape stones (GS) the main mass loss takes place in the temperatures range 200-500°C. At 500°C during 1 hour, about 50% of mass is lost and, finally, at 950°C the loss reaches approximately 75%. The specific surface of the samples is reaching 830 m2/g and was determined by the method of thermal desorption of argon. Electron microscopy analysis showed a change in morphology and structure of the carbon containing sorbents depending on the temperature and time of carbonization. The mechanism of sorption of heavy metal ions by carbonized sorbents was stated by physico-chemical methods of analysis. Carbonized sorbents were found to be effective for adsorption of heavy metal ions, organic compounds and sulphur dioxide.
Carbon Containing Refractories [11] Introduction of chromite sludge increases the density and mechanical strength in comparison with the refractory "Furnon-3XP", slightly increasing refractoriness. Formation of fibrous carbon and carbides of metals accounts for the improvement of physico-chemical indices of carbon refractories on the basis of clay and chromite sludge. The obtained refractory materials have a high slag-resistance that allows forecasting their use in metallurgical processes of precious metals production. On quartz basis, by a mechano-chemical treatment powder material with new properties was obtained. After mechano-chemical treatment quartz was found to become ferromagnetic. The magnetic permeability of quartz reaches µ = 30 depending on the chosen carbon containing modifier and the regime of mechanical treatment. Electroconductivity of the obtained material increases by 2-3 orders of magnitude. A coalgraphite layer was grown on metal current taps with the aim to develop an optimal method for production of coalgraphite cathodes for lithium chemical sources of current. The obtained electrodes were tested for the discharge of pure nickel nets, electrodes of fibrous carbon materials and electrodes produced according to conventional technology using polyethylene as a binder. It is shown that the obtained carbonized material is not inferior in its characteristics to a traditional one.
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Poly-functional carbon containing catalysts which may simultaneously carry out hydroforming, hydro-desulphurization and hydroisomerization of petrol and diesel fractions were synthesized. The developed catalysts allow obtaining petrol of low-sulphur content and diesel fuel with improved properties.
References 1. Buyanov RA, Chesnokov VV (1995) Regularities of catalytic formation of carbon threads in the process of synthesis of new composition materials. Chemistry in the interests of sustainable development 3:177-186 2. Mansurov ZA (2000) Overcarbonized adsorptive-catalytic systems. Eurasian Chemico-Technological Journal 2:59-68 3. Mansurova RM (2001) Physicochemical properties of synthesis of carbon containing compositions. Monograph, Almaty XXI century 4. Mansurov ZA, Mansurova RM, Ualieva PS, Zhilybaeva NK (2002) Obtaining and propetities of sorbents from vegetative raw materials. Chemistry in the interests of stable development 10:339-346 5. Mansurov ZA, Mansurova RM, Zakharov VA, Besarabova IM, Zhylybaeva NK, Nikolaeva AF (2004) Sorption of Gold by carbonized sorbents on the basis of vegetative raw materials. “Carbon 2004”, - Providence, Rhode Island, USA, July 11-16 6. Mofa NN, Chervyakova OV, Ketegenov TA, Mansurov ZA (2003) Magnetic sorbets obtained by mechanochemical treatment of quartz containing mixtures. Chemistry in the interests of stable development 11:755-761 7. Mansurov ZA, Mofa NN, Shabanova TA (2004) Synthesis of powderlike materials with particles encapsulated in nanostructurized carbon containing films. “Carbon 2004”, - Providence, Rhode Island, USA, July 11-16 8. Mansurov ZA. (2003) Nanocarbon materials. Vestnik KazNU (Chemical series) 30:28-31 9. Mansurov ZA, Biisembayev M., Bakenov Zh, Wakihara M (2004) Novel carbon materials for battery applications. “Carbon 2004”, - Providence, Rhode Island, USA, July 11-16 10. Mansurov ZA (2003) Some Problems of the Development of Physics and Chemistry of Carbon Materials. Eurasian Chemico-Technological Journal 5:1-6. 11. Mansurov ZA, Dilmukhambetov EE, Ismailov MB, Fomenko SM, Vonghai IM (2001) New refractory materials on the basis of SHS technology. La Chimica e l′industria 83:1-6
NANOMATERIALS AND DOMAINS
Nanocrystalline Iron-Rare Earth Alloys: Exchange Interactions and Magnetic properties
E. Burzo1, C. Djega–Mariadassou2 1
Faculty of Physics, Babes-Bolyai University 400084 Cluj-Napoca, Romania 2
LCMTR UPR 209 CNRS 2/8 Bat F Rue Henri Dunant 94320 Thiais, France
Abstract:
The exchange interactions in R2Fe17 compounds are analyzed. The contributions to R5d band polarizations are estimated. Then, the physical properties of nanocrystalline Sm(Fe,Si)9 alloys are studied in correlation with possible technical applications. Finally, the magnetic properties of the Nd5Fe66.5-xCr10MxB18.5 nanocomposite system with M = V or Nb are analyzed. Keywords: rare earth compounds, magnetic properties
Introduction Nanocrystalline alloys have very interesting physical properties with a large variety of useful technical applications [1]. The systems based on rare-earth (R)-transition metals (M) are particularly interesting as materials for permanent magnets. In this article we focus on R2Fe17 and R2Fe14Bbased compounds. A characteristic feature of both systems is their low Curie temperature TC. For R2Fe17 compounds the highest TC value was shown to be 477 K for R = Gd [2]. Somewhat higher Curie temperatures, around 600 K, were found in R2Fe14B system [3]. The above values are about half of the Curie temperature of pure iron, although the iron content in the above compounds is about 90%. Furthermore, in both systems the iron moments are close to that of pure iron. Due to the low TC values and planar anisotropies, the R2Fe17 compounds have been considered as not 371 R. Gross et al. (eds.), Nanoscale Devices - Fundamentals and Applications, 371–385. © 2006 Springer. Printed in the Netherlands.
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useful for technical applications such as permanent magnets. At present, the Nd2Fe14B is the basic component for high energy permanent magnets [3]. The low Curie points in both R2Fe17 and R2Fe14B type compounds were attributed to the strong distance dependence of the exchange interactions [4]. The R2Fe17 compounds crystallize either in hexagonal or orthorhombic crystal structures. In both structures there are four different iron sites. In contrast, the R2Fe14B compounds have a tetragonal structure. In this lattice structure there are six crystallographically different iron sites, two R positions and one boron site. The distances between iron atoms, dFeFe, in the above systems cover a large range of values. The interactions between iron atoms situated at distances dFeFe ≤ 2.45 Å are negative, whereas the exchange interactions between Fe atoms situated at larger distances than the above value are positive. There are no negative exchange interactions, since for the particular sites they are smaller than the positive ones. Nevertheless, a considerable magnetic energy is stored in them. As a result, there is a strong decrease of the Curie temperatures in both systems. The exchange interactions in rare-earth-transition metal compounds are rather complex. The exchange interactions of R-Fe type can be described by Campbell’s model [5]. In this model the 4f electrons of the rare-earth polarize their 5d bands and there are short-range 5d-3d exchange interactions with neighbouring iron atoms. The local 4f-5d exchange interactions are positive and the 4f and 5d moments will align parallel. An additional contribution to 5d band polarization resulting from 5d-3d short range interactions is also expected. The Fe(3d)-Fe(3d) exchange interactions are short range. In this paper we analyze the exchange interactions in R2Fe17 and R2Fe14B compounds. Then, we present the physical properties of Sm-Fe-Si-C nanocrystalline alloys with the ratio R/(Fe,Si) close to 2/17 having possible technical applications. In the last part, we report on the physical properties of Nd-Fe-Cr-M-B nanocomposite magnets with M=V or Nb based on Nd2Fe14B type and α-Fe.
Experimental and Numerical Methods The Sm2Fe17-xSix and SmFe9-ySiy nanocrystalline alloys were obtained by high energy ball milling of Sm-Fe pre-alloyed powders and balanced quantity of silicon and samarium powders. The samples were heat treated in vacuum at temperatures between 650oC and 1150oC. Carbonation of the samples was achieved by reaction of the alloy powder with appropriate amount of C14H10 powder [6].
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The Nd5Fe66.5-xCr10MxB18.5 alloys with M=V or Nb were obtained by arc melting the constituents under purified argon atmosphere. Then, flakes were prepared by melt-spinning the alloys onto a cooper wheel in argon atmosphere. The amorphous precursors of nanocrystalline alloys were heat treated at 650oC from 0.5 to 10 min. The X-ray analyses of Sm-Fe-Si alloys were performed with a Bruckertype diffractometer. The crystal structures and lattice site coordinates were obtained by Rietveld analysis by using the FULLPROF computer code in the assumption of Thompson-Cox-Hastings line profile, which permitted the refinement of the coexisting phases and also to take into account the broadening of the diffraction lines induced by grain size and strain effects. The mean grain sizes in Nd-Fe-Cr-M-B nanocomposites were determined from the broadening of X-ray line by using the Scherrer formula [7, 8]. The distribution functions of the grain sizes were also determined. Magnetic measurements were performed in the temperature range 4.2800 K and fields up to 9 T. Band structure calculations were performed by using the ab initio tight binding linear muffin tin orbital method in the atomic sphere approximation (TB-LMTO-ASA) [9]. In the framework of the local density approximation (LDA), the total electronic potential is the sum of external, Coulomb and exchange correlation energies [10]. The functional form of the exchange correlation energy, used in the present work, was the free electron gas parameterization of Von Barth and Hedin [11]. Relativistic effects were included. The 4f states were treated as part of core. These electrons are not part of the band structure, but the polarization of the 4f densities was calculated self–consistently.
Exchange Interactions in R2Fe17 and R2Fe14B Type Compounds As mentioned already above, the exchange interactions between R and Fe atoms can be described as of 4f-5d-3d type, while those between transition metals are of short-range 3d-3d type. According to the Néel-Slater curve, for the rhombohedral R 3m type structure of the R2Fe17 compounds, the interactions between Fe atoms situated in dumbbell 6c sites are strongly negative. The exchange interactions between Fe(9d)-Fe(18f) are weakly negative [12]. The exchange interactions between iron atoms situated at larger distances are positive and impose a parallel alignment of the iron moments. Similar behaviour can be seen in the hexagonal P63/mmm type structure of R2Fe17 compounds.
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2.50
4f 12j
MFe(µB/atom)
2.25
12k 2.00
6g 1.75
Y 1.50
0
Ho 5
Tb 10
Gd 15
2
(9J-1) J(J+1) Fig. 1. The iron moments as function of De Gennes factor for hexagonal R2Fe17 heavy rare-earths and yttrium compounds.
The dependences of the iron moments determined from band structure calculations as function of De Gennes factor, G, are given in Fig. 1 for the case of hexagonal R2Fe17 compounds, where R is a heavy rare-earth or yttrium. The iron moments follow linear dependences MFe=MFe(0)+αG, where α = 4⋅10-2µB for all type of sites and G = (gJ-1)2J(J+1) [13]. From these data it can be seen that a fraction of (5-7) % of the iron moments is induced by 4f-5d-3d exchange interactions when replacing Y by Gd. The sequence of the decreasing the Fe moments is 4f > 12j > 12k > 6g. The iron magnetic moments are sensitive to their local environments. We note that in the above sequence the 4f and 12j are inverted, when only the spin contributions are considered in the computing method. The R5d band polarizations for R2M17 (M = Fe, Co, Ni) as well as for R2Fe14B (R= Gd,Y) compounds follow also linear dependences as a function of the De Gennes factor, M5d = M5d(0)+βG (see Fig 2). The M5d polarizations are translated to higher values as the magnetizations of the transition metal sublattices are higher. The slopes β ≅ 1×10-2 µB are nearly the same for all R2M17 systems. For the P63/mmc type structure of R2Fe17 compounds, where two R sites are present (2b and 2d), their M5d values are only slightly different and the mean value was only plotted in Fig. 2. A higher difference between the M5d band polarizations at R4f and R4g sites can be seen in the R2Fe14B compounds. The observed differences can be
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correlated with their different local environments, with the R(4g) site having a higher number of boron atoms in the first coordination shell.
M5d(µB/atom)
0.75 R2Fe17 (R3m) R2Fe17 (P63/mmc) R2Co17 (R3m)
0.50
RFe7B0.5
R2Ni17 (P63/mmc)
0.25
0.00
0
5
10
2
15
(9J-1) J(J+1)
20
Fig. 2. The computed 5d band polarizations for R2M17 (M=Fe, Co, Ni) heavy-rare earths compounds and GdFe7B0.5. The Y 4d band polarizations in Y2Ni17, Y2Fe17 and YFe7B0.5 are also plotted.
The R5d band polarizations for R2M17 (M = Fe,Co,Ni) as well as for R2Fe14B (R= Gd,Y) compounds follow also linear dependences as a function of the De Gennes factor, M5d = M5d(0)+βG (see Fig 2). The M5d polarizations are translated to higher values as the magnetizations of the transition metal sublattices are higher. The slopes β ≅ 1×10-2 µB are nearly the same for all R2M17 systems. For the P63/mmc type structure of R2Fe17 compounds, where two R sites are present (2b and 2d), their M5d values are only slightly different and in Fig. 2 only the mean value was plotted. A higher difference between the M5d band polarizations at the R4f and R4g sites can be seen in the R2Fe14B compounds. The observed differences can be correlated with their different local environments, with the R(4g) site having a higher number of boron atoms in the first coordination shell. The analysis of the data in Fig. 2 shows the presence of two contributions to the R5d band polarization. The βG contribution is due to local 4f-5d exchange and is the same for a given R atom. The M5d(0) values obtained by extrapolation of the M5d vs G dependences to G = 0 are the same as those induced on the 4d band by 4d-3d short range exchange interactions in the Y2M17 compounds. Thus, this contribution can be only ascribed to the presence of 5d-3d short range exchange interactions resulting from hybridization effects.
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The effect of short-range exchange interactions on the R5d band polarization can be analyzed starting from Hamiltonian which takes into account both 3d-5d and 5d-5d exchange interactions [14].
H = −2 J 3d −5d S5 d (0)∑ S3di (0) − 2 J 5 d −5d S5 d (0)∑ S5 dj , i
j
(1)
where J3d-5d and J5d-5d are the exchange parameters characterizing the 3d-5d and 5d-5d interactions with i an j the nearest neighbors Fe and R atoms, respectively, and S5d(0) and S3d(0) are the spin values characterizing the systems with G = 0. Co
M5d(0)(µΒ/atom)
0.5
R3m
P63/mmc
0.4
Fe YFe7B0.5
0.3
Ni
0.2 0.1 0.0
0
5
10
15
Md(µΒ/RM8.5 or RFe7B0.5)
20
Fig. 3. The M5d(0) contributions to 5d band polarizations as a function of the transition metal moments in the RM8.5 (M=Fe, Co, Ni) rare earth compounds as well as the mean value of M4d in YM7B0.5.
The relation (1) can be analyzed in the molecular field approximation. The effect of 5d-3d and 5d-5d exchange interactions is equivalent to an internal field, Hexch, acting on the R atom. This induces an additional polarization to that resulting from 4f-5d local exchange, similar to that evidenced on 3d band in rare-earth transition metal compounds [15]. The internal field is Hexch = N5d-3dM3d + N5d-5dM5d where N5d-3d and N5d-5d are the molecular field coefficients describing the R5d-Fe3d and R5d-R5d exchange interactions. The R5d-R5d exchange interactions can be neglected as compared to R5d-Fe3d, since the former are very small [14]. Thus, it the relevant result is that Hexch is proportional to Md, where Md is the total 3d magnetization per formula unit. Previously [15], we showed that above a critical field, which determines the appearance of 3d magnetic moments, M3d is proportional to the exchange field, Hexch. Supposing that this relation is valuable for the 5d band, it follows that M5d(0) = αMFe. The
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M5d(0) values determined in RM8.5 (M = Fe, Co, Ni) and the mean value obtained for YFe7B0.5 compounds are plotted in Fig. 3 as a function of the total transition metal magnetization corresponding to one R atom. There is a linear dependence, in agreement with the above conclusions. The determined slope is α = 0.028 ± 0.004. Assuming that the induced 5d band polarization by short range 5d-3d exchange interactions is the same as that induced in the Fe 3d band when changing the exchange interactions due to substitution of a nonmagnetic rare earth by a magnetic one it is found that M5d(0) = (18×102)-1 Hexch, where M5d(0) is given in Bohr magnetons and Hexch in T . Supposing that N5d-3d ≅ NR-Fe we can estimate also the induced 5d polarization, M5d(0). For NFe-R ≅ 30, as found in R2Fe17 compounds [16], we obtained a value α = 0.011 somewhat smaller than that determined from Fig. 3. This shows that the R5d band is more sensitive to exchange interactions than the Fe3d band, and is comparable to that found for the 3d band.
Nanocrystalline Sm-Fe-Si-C Alloys As function of the thermal treatment process both stable and metastable solid solutions can be formed in the Sm-Fe-Si system close to the composition 2/17. Unlike in the cobalt rare-earth phase diagram, no presence of a hexagonal CaCu5 type structure was found in R-Fe system. In the RCo5 type compounds a deviation from 1/5 stoichiometry was shown. The alloys were described by the formula R1-sCo5-2s, where s rare-earth atoms are substituted by s dumbbells pairs of cobalt [17, 18]. The presence of metastable R1-sFe5+2s phases was also reported [19]. For s = 0.22, a TbCu7 type structure can be invoked, while for s = 0.36 ÷0.38 a 1/9 stoichiometry was found with the alloys having P6/mmm type structure. If s = 0.33, a single R atom out of three is substituted for by one dumbbell pair and the stoichiometry is 2/17. If the dumbbell pairs are randomly distributed, the structure remains hexagonal and is of P6/mmm type. This structure is closely related to CaCu5 one (see Table 1). In the theoretical Sm1-sFe5+2s system, the 3g sites occupation is not affected, while for s = 0.22 the 2c site of the CaCu5-type structure would transform partially into Fe6l. The 2e sites are gradually occupied by iron. The Sm1-sFe5+2s system exhibit experimentally only the metastable P6/mmm phase with s = 0.36-0.38 consistent with SmFe9-ySiy after annealing at 650-850oC. These alloys are nanocrystalline with grain sizes varying from 22 to 28 nm. The 2/17 stoichiometry is approached when increasing the annealing temperature compared to the above mentioned. The P6/mmm type structure changes to a rhombohedral R 3m type through an ordering
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process of atoms. Thus, the 2e and 6l sites transform into 6c and 18h sites, respectively. The 3g site splits from one third into 9d positions and two third into the 18h sites in the rhombohedral 2/17 phase. The R1a site gives rise to the R6c site in the R 3m type structure. Table 1. Atom occupancies and positions in the R1-sM5+2s alloys having P6/mmm space group [6]. Atom
s=0 1/5 1Sm(1a) 1-s 1 2Fe(2c) 2(1-3s) 2 6Fe(6l) 6s 0 3Fe(3g) 3 3 2Fe(2e) 2s 0
s = 0.22 TbCu7 type [20] 0.78 2 0 3 0.44
s = 0.33 2/17 0.66 0 2 3 0.66
s = 0.36 1/9 0.64 0 2 3 0.72
In hexagonal R2Fe17 based compounds the substitution of one Fe atom by Si decreases the R(5d) band polarization by ≅ 0.04 µB. This can be correlated with the diminution of short range R(5d)-Fe(3d) exchange interactions due to dilution effects. 5.0
c(Å)
12.45 12.40
Sm 2Fe17-xSixCz
SmFe9-ySiyC z
a(Å)
8.65 8.60
4.2
8.55 8.50 0.0
4.9
4.4
8.70
a(Å)
c(Å)
12.50
0.4
0.8
1.2
1.6
2.0
x
0.00
0.25
0.50
y
0.75
1.00
Fig. 4. Composition dependences of lattice parameters in Sm2Fe17-xSixCz z = 0 (•), 2(*) and Sm2Fe9-ySiyCz z = 0 (•), 1(*) alloys.
The composition dependences of the lattice parameters for Sm2Fe17-xSixCz (z=0, 2) and SmFe9-ySiyCz (z=0, 1) alloys are shown in Fig. 4. A rhombohedral type structure having R 3m space group was found for Sm2Fe17-xSix compounds with x ≤ 2.0. The metastable SmFe9-ySiy solid solutions are formed up to y = 1.04 and crystallize in hexagonal P63/mmm type structure. The a- and c-lattice parameters decrease when replacing iron by silicon in Sm2Fe17-xSix system. The silicon atoms are located in 18h sites. The same behaviour was shown for Sm2Fe17-xSixC2
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carbonated samples, although a rather high increase of the cell parameters was shown after carbonation. In the case of hexagonal P63/mmm alloys, in which the SmFe9-ySiy and SmFe9-ySiyC systems crystallize, the c lattice parameters decrease while the a-parameters increase slightly. We note that as in the 2/17 compounds the carbonation leads to a high increase of the cell parameters. Domain sizes of 22 nm were found for the SmFe9-ySiyC alloy with y = 0.25. They decrease slightly to 18 nm for y = 1, according to the role played by silicon on the nanostructure.
Fig. 5. Composition dependences of the Curie temperatures in Sm2Fe17-xSixCz (z=0, 2) and SmFe9-ySiyCz (z= 0, 1) alloys.
10
Γ
8
6
2/17 1/9
4
2 450
500
550
TC(K)
Fig. 6. The dependence of the Γ values on the Curie temperatures.
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The composition dependences of the Curie temperatures, Tc, are given in Fig. 5. The Tc values of noncarbonated 1/9 and 2/17 samples increase gradually showing the same trend, although the Tc values are higher in the 1/9 phase. The increase of the Curie temperature on replacing iron by silicon can be attributed to the reduction of the antiferromagnetic exchange interactions relative to a slight increase of the Fe-Fe distances concomitant with the filling of the d band by the p silicon electrons implying a shift to strong ferromagnetic behavior. For example, in the R 3m type structure of Sm2Fe17 compounds, Si replaces Fe in the 18h sites, while the distances Fe6c-6c and Fe9d-18f increase but remain just below 2.45 Å. The Curie temperatures in carbonated samples are sensitively higher than those in noncarbonated. Due to the increase of the lattice parameters by the presence of interstitial carbon, the distances between iron atoms are larger. This leads to a decrease or even cancellation of the contributions of the iron pair to the negative exchange interactions. When substituting Fe by Si, in carbonated samples, a decrease of the TC value is expected due to dilution effects as well as hybridization effects of Fe3d, Si3p and C3p bands in agreement with experimental observations. The correlation between the Curie temperatures and the volume variations can be analyzed by using the Γ =
1 dTC d ln Tc = parameter kBTc dp d ln v
[21, 22]. Here, κ denotes the compressibility and v is the volume of the cell. A linear dependence of the Γ values on the Curie temperatures was reported starting from a model, which considers the 3d electrons as having mainly a localized behavior [21, 23]. In this model the Γ value is given by: 2 5 d ln Jeff 5 kBN0 g I Γ= + 2 + Tc d lnv 8 S(S +1) J 2 I , 3 eff b
(2)
where No is the Avogadro number, g is the Landé factor, I is the effective intra-atomic exchange integral, which is reduced from its bare value Ib, and Jeff is the effective exchange coupling parameter. Linear Γ vs Tc dependences were found and described by the relation Γ = a - bTC with a = 34.5 and b = 0.054 K-1 for the 2/17 compounds and a = 47.5 and b = 0.085 K-1for 1/9 alloys. These values are close to those determined in Y2Fe17C(N)x based compounds, where the values a = 38 and b = -0.06 K-1 d ln J eff were reported [23]. The γ = values are 16.4 and 22.9 for the 2/17 d lnν
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and 1/9 systems, respectively, as compared to γ = 16 obtained in the Y2Fe17C(N)x based compounds. The dlnJeff/dlnν values were also estimated from magnetic data by using a molecular field approximation. These values are by about 25 – 30 % smaller compared to those determined from Fig. 6. The differences may be attributed to the approximations used in estimating the Jeff values, particularly mean values of the iron moments and for exchange interactions or a mean field approximation used in analyzing the magnetic data. The data from Fig. 6 suggests that iron atoms show mainly a localized behavior both in the 2/17 and 1/9 systems. As a result of carbonation, the Sm2Fe17-xSixC2 and SmFe9-ySiyC systems have uniaxial anisotropy. In SmFe9-ySiyC nanocrystalline alloys, high coercivities were obtained in samples annealed between 700 and 800oC [6] (see Fig. 7). The coercive fields decrease when decreasing or increasing the annealing temperature outside the above mentioned temperature range. A coercive field of µoHC = 1.3 T was found in case of the sample having y = 0.50 and grain sizes around 22 nm. The highest coercive field µoHC = 1.5 T was obtained for a SmFe8.75Si0.25C alloy. Thus, these alloys have potential technical applications. The high coercivities of mechanically alloyed and carbonated samples originate from the presence of the 1/9 metastable phase.
16
Coercivity (kOe)
14
0.25 0.5 0.75 1
12 10 8 6 4 2 0 600
700
800 900 1000 1100 Annealing temperature (°C)
1200
Fig. 7. The dependence of coercive fields in SmFe9-ySiyC alloys on the annealing temperature.
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Nanocrystalline Nd-Fe-B Based Alloys The reduction of the Nd concentration in Nd-Fe-B alloys, as compared to 2:14:1 stoichiometry, provides less expensive permanent magnets, and thus is of interest for technical applications. With respect to their microstructure, the alloys are constituted from a matrix of a magnetically hard phase Nd2Fe14B and αFe and/or Fe3B particles on the grain boundaries. The increase in the remanence due to the nanoscale structure is further enhanced by the presence of a soft magnetic phase, which has a higher saturation induction than that of the Nd2Fe14B phase [3, 24]. Previously, it has been shown that the addition of chromium in Nd-Fe-B nanocomposites increases the coercivity, while the Curie temperatures and the induction decrease [3]. In order to analyze the influence of other substituting elements on the magnetic properties, particularly the coercivity of nanostructure alloys, we studied the Nd5Fe66.5-xCr10MxB18.5 with M = V or Nb system. The Nd5Fe66.5-xCr10MxB18.5, as melt spin ribbons, with M = V or Nb are amorphous. The distribution of grain sizes is more homogeneous when the vanadium content increase. The thermal treatment at 650oC leads to the crystallization of the alloys. For short annealing time, the presence of α-Fe was found. The mean dimensions of α-Fe crystallites increase rapidly for an annealing time ta up to 1 min, at 650oC and then a nearly linear variation was observed as shown in Fig. 8. For ta = 10 min the mean grain size of αFe is d = 17 nm.
Fig. 8. The α-Fe crystallite sizes as a function of the annealing time at 650o C for the Nd5Fe66.5Cr10Nb2B18.5 alloy.
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70
5
60 50
Hc
3
40 30
2
0
20
Mr
1
M (emu/g)
Hc(kOe)
4
10 0
0.5
2 ta (min)
5
10
0
Fig. 9. Remanence and coercivity for Nd5Fe66.5Cr10Nb2B18.5 samples as a function of the annealing time at 650o C.
Due to the small crystallite dimensions, the as quenched samples are superparamagnetic. The hard magnetic properties were developed after crystallization of as quenched samples. Both the remanent induction and the coercive field increase gradually with annealing time as shown by Fig. 9. For samples thermally treated at 650oC during 10 min, a ratio Br/Bs = 0.7 was determined. The coercive field exceeds 0.36 MA/m. The demagnetizing curves for the Nd5Fe66.5Cr10Nb2B18.5 sample annealed at 650oC for 2 and 10 min, respectively, are plotted in Fig. 10. These are typical for an exchange spring behavior as expected in nanocomposite magnets, when the particle sizes of α-Fe approach the domain size of the hard Nd-Fe-B phase. The exchange interactions would suppress the reversible rotation of the magnetization in the soft α-Fe particles and there is a significant increase of the coercivity. Although the Nd5Fe66.5-xCr10MxB18.5 samples contain high boron content no evidence for the presence of a Fe3B phase was found. It was shown previously [25] that for x > 5 at % Cr the main soft magnetic phase is α-Fe. The presence of chromium as well as Nb and V decreases the saturation magnetization of the alloys. Although the Br/Bs ratio is higher than 0.7, the remanent induction is decreased as compared to Nd-Fe-B magnets.
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50
30 a
20
B (G.cm3/g)
40
b
10 −0.6
−0.4
−0.2 µoH (T)
0
Fig. 10. The second quadrant hysteresis loops for Nd5Fe66.5Cr10Nb2B18.5 alloys annealed for 2 min (a) and 10 min (b) at 650oC.
A high value of the remanence ratio is due to exchange coupling between the magnetic moments at the interface of the hard and soft magnetic phase nanograins. As a consequence of this coupling, there is a large degree of reversibility in the demagnetization behavior [26]. The better magnetic properties are obtained when the mean grain size of α-Fe is around 20 nm.
Conclusions The exchange interactions in R2Fe17 and R2Fe14B compounds are well described by a 4f-5d-3d model. The 5d band polarization is due to both local 4f-5d as well as 5d-3d short range exchange interactions. The analysis of the volume effects on the Curie temperatures show that iron has mainly a localized behavior. Stable Sm2Fe17-xSixCz and metastable SmFe9-ySiyC2 nanocrystalline alloys have been obtained. A transition from a metastable to stable state through an ordering process of atoms was seen on increasing the annealing temperature. High coercivities were obtained in SmFe9-ySiyC alloys thermally treated in the temperature range between 700oC and 800oC. The Nd5Fe66.5-xCr10MxB18.5 nanocomposites with M = V or Nb are constituted from hard magnetic phase based on Nd2Fe14B and α-Fe. A high Br/Bs = 0.7 ratio as well as high coercive fields were found for melt spinning samples after a thermal treatment at 650oC for 10 min. The α-Fe dimensions are of ≅ 20 nm.
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References 1. Lu K (1996). Mat Sci Eng R16:161 2. Burzo E, Chelkovski A and Kirchmayr NR (1992) Landold Börnstein Handbook. Springer Verlag Vol. III 3. Burzo E (1998). Rep Prog Phys 61:1099 4. Givord D, Lemaire R (1979). IEEE Trans Magn 10:109 5. Campbell IA (1972). J Phys F: Metal Phys 2:L117 6. Bessais L, Djega-Mariadassou C, Nandra A, Appay MD, Burzo E (2004). Phys Rev B 69:064402 7. Carr GE, Davies HA, Buckler RA (1988). Mat Sci Eng 99:147 8. Burzo E, Chiriac H, Ersen O, Pop V (1999). In: Materiaux pour l’Electrotechnique. Vol. 1, Politechnical University Bucharest 9. Anderson OK (1975). Phys Rev B 12:3060; Anderson OK, Jepsen O (1984). Phys Rev Lett 53:2571 10. Jones RO, Gunnarson O (1989). Rev Mod Phys 61:689 11. von Barth U, Hedin L, (1972). J Phys C: Solid State Phys 5:1629 12. Li ZW, Morrish AH (1997). Phys Rev B 55:3670 13. Burzo E, Vlaic P, J Magn Magn Mat (in press) 14. Burzo E, Chiuzbaian SG, Neumann M, Valeanu M, Chioncel L (2002). J Appl Phys 92:7362 15. Burzo E (1974). Solid State Commun 14:1295; (1981) J Less Common Met 77:251 16. Burzo E, Lazar DP, Valeanu M (1976). Proc. 12th Rare-Earth Research Conference, Colorado p 104 17. Buschow KHJ, Van der Goot AS (1968). J Less Common Met 14:323 18. Givord D, Laforest J, Schweizer J, Tasset F (1979). J Appl Phys 50:2008 19. Djega-Mariadassou C, Bessais L, Nandra A, Burzo E (2003). Phys Rev B 68:024406 20. Villards P, Calvert LD (1991) Pearson's Handbook of Crystallographic Data for Intermetallic Phases. ASM International 21. Jaakkola S, Parviainen S, Penttila S (1975). J Phys: Metal Phys 5:543 22. Brouha M, Buschow KHJ, (1973). J Appl Phys 64:1813 23. Plugaru N, Valeanu M, Burzo E (1994). IEEE Trans Magn 30:663 24. Davies HA, Manaf A, Zhang PZ (1993). J Mater Eng Perform 2:579 25. Hirosawa S, Kanekiyo H (1993). Proc. 13th Int. Workshop Rare Earth and Their Applications, p.87 26. Kneller EF, Hawig R (1991). IEEE Trans Magn 27:3588
The Influence of Applied Fields on the Nucleation and Growth of Heteroepitaxial Carbon Films
B. Z. Mansurov al-Farabi Kazakh National University, 96A, Tole be Str., 480012, Almaty, Kazakhstan
Abstract:
On the basis of a literature review and the comparison of physical and chemical properties of materials and calculations, it is shown that copper, saturated with hydrogen, is an appropriate substrate material for the heteroepitaxial growth of diamond films. Our estimates have shown that it is possible to create conditions for preferential oriented growth of diamond films by changing the magnitude and configuration of applied magnetic fields. On the basis of theoretical calculations the technological installation for the growth of carbon films by a method of differential magnetron sputtering has been developed and designed. The basic technological parameters of the installation are presented. Also the mathematical algorithm describing the deposition process of carbon films on a copper buffer layer is offered.
Keywords: carbon films, applied fields, differential magnetron sputtering
Introduction The significant efforts devoted to the synthesis of diamonds were motivated by the unique combination of properties of this material. They provide the chance to produce electronic devices both in discrete and integrated form with very high speed and power at a high operation temperature range, at high integration density, and increased mechanical strength and reliability. Various techniques such as plasma-enhanced chemical vapour deposition (CVD), ion-assisted deposition, and hot filament CVD have
387 R. Gross et al. (eds.), Nanoscale Devices - Fundamentals and Applications, 387–399. © 2006 Springer. Printed in the Netherlands.
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been developed for obtaining diamond films and are widely used today [16]. However, the characteristics achieved thus far for electronic devices are not good enough for practical applications. The majority of the deposition processes for CVD crystallization occur under conditions where the massive chaotic crystallization of diamond hinders the growth of high quality crystalline material. Therefore, the search for deposition systems, in which the rate of the backwards process would be noticeably higher and, besides, would allow the continuous control of film growth is an urgent problem. The solution of this problem will determine the further development of the diamond synthesis technology.
Evaluation of Applied Field Effects on the Nucleation of Heteroepitaxial Carbon Films It is generally known that specific problems such as the simultaneous nucleation of non-diamond structures, the prevention of a flawless stable growth of oriented diamond films, the control of growth rates, as well as the selective elimination of highly defective areas and non-diamond structural modifications of carbon films are characteristic for the synthesis of diamond. These problems may be solved by suitable control of applied fields and other control parameters (electrostatic, magnetic, optical, temperature, field of elastic deformations) of certain limiting symmetry group. According to the principles of crystal growth the symmetry of these control fields should correspond to the point and space group symmetry group of diamond and not to that of non-diamond modifications of carbon films. Evaluation of applied field effects on the nucleation The nucleus formation and growth rate of a crystalline film are determined rather by the differences in the magnitude of thermodynamic functions under the various phase conditions than by the magnitude itself. However, the differences are substantially less than the absolute magnitudes. Thus, even insignificant variations of the thermodynamic functions may result in applied field effects on the process of nucleation, as well as the rate and direction of the crystalline structure growth. The energy of formation of a nucleus (Gn) is determined by a chemical potential (µ) of a substance, by the surface energy (σ) and, in general, by further non-considered parameters defining effects of applied fields and
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their spatial symmetry: Gn =µ +σ +WE +WH +WC +Whν +WSE +WSH
The additional terms take into account the following contributions: WE energy of the electric field, WH - energy of the magnetic field, WC - energy of the field of elastic deformations, Whν - energy of the optical influence, WSE and WSH - space symmetry of the applied fields (electric and magnetic fields). Thus, it is possible to create conditions for preferential nucleation and growth of a diamond film by changing the magnitude and the shape of the applied fields. Effects of the elastic deformation field The most convenient substrates for the growth of diamond films would be single crystalline silicon wafers with a diamond-like structure due to their availability, low cost and perspectives for further use in semiconductor device fabrication. In Fig. 1 a linear model of the growth of a monolayer carbon film on a silicon substrate is shown. Because of the difference of the lattice cell parameters of Si (5.43 Å) and C (3.57 Å) [7] elastic deformations of the lattices occur. In the given model the deformation forces acting on the carbon atoms are equal to twice the deformation forces acting on the silicon atoms, FC = 2FSi. That is, the effect of the whole silicon substrate is replaced by the effect of two silicon layers on one layer of carbon.
Fig. 1. Linear model of a monolayer carbon film grown on silicon.
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The deformation force acting on the atoms of carbon and silicon can be written as: FC = CC
εc a
, FSi = CSi
ε Si b
, CC
εc a
= 2CSi
ε Si b
, b - a = ε c + ε Si ,
where CC and CSi are the elastic constants of carbon and silicon, εC and εSi the relative displacements, and a and b the lattice parameters of diamond and silicon, respectively. The energy of the elastic deformation of the diamond lattice is given by 2
ε ε U C = C11c c + C12c c a a
with UC = 0.569×1029,
2
eV or UC = 0.322eV, 1/atom. m3
The energy of the elastic deformation of the silicon lattice is given by 2
ε ε U Si = C11Si Si +C12Si Si b a
2
eV or USi = 2.247eV, 1/atom. The estimated value m3 of the deformation energy of the silicon and diamond lattices (UC + USi) is
with USi = 1.16 ×1029,
then equal to
KT = UC + USi = 2.57 eV, what corresponds to a temperature e
of about 30000 K required to create such deformations. That demonstrates the impossibility of epitaxial growth of crystalline diamond films on silicon. Thus, to obtain high quality diamond films a buffer layer with the following properties is required: 1. The buffer layer should have a diamond-like structure. 2. The lattice parameters of the buffer layer should be well matched to those of diamond. 3. The thermal expansion coefficient of the buffer layer should be close to that of diamond to minimize thermal stresses between the substrate and the diamond films. The analysis of the physical and chemical properties of chemical elements shows that copper saturated with hydrogen is a suitable materials choice acting as a buffer layer for epitaxial growth of diamond films. In Fig. 2 the
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copper (111) surface is shown. The crystallographic calculations of tetrapore and octapore volumes in the copper lattice proved that in copper films grown in a hydrogen atmosphere the tetrapores will be filled-in with atomic hydrogen, and octapores with molecular hydrogen. The estimated value of the space between the atomic hydrogen atom in tetrapore and the carbon atom deposited above tetrapore is L = 0.87 Å. This is sufficient for C–H bond creation, because the typical C-H bond length is 1.09 Å [7].
Fig. 2. Distribution of tetrapores and octapores on the (111) surface of a copper lattice.
Since copper does not create compounds with carbon, the atomic hydrogen in the tetrapores of the copper lattice is expected to act as a center of crystallization of carbon. As a result the difference between the lattice parameters of substrate and the growing diamond film becomes considerably smaller. The distance between the carbon atoms on a (111) surface is aD = 2.517 Å for diamond and aG = 2.84 Å for graphite [7]. The distance between tetrapores on the copper (111) surface is aCu = 2.56 Å, i.e. the difference is: ∆ 1=
2.56 − 2.517 × 100 % = 1.7% 2.56
for diamond
∆ 2=
2.56 − 2.84 × 100% = 10.94% 2.56
for graphite.
Thus, the distribution of tetrapores on the copper (111) surface and, respectively, of atomic hydrogen located in them, create suitable conditions for the preferential formation of tetragonal carbon bonds.
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Estimation of electrostatic field effects on the energy conditions for diamond and graphite The estimation of the effects of electrostatic and magnetostatic fields on the energy state of diamond and graphite nuclei located in these fields, is completed by the limited version of the simulation program ELCUT (see http://www.tor.ru/elcut). On Fig. 3 the schematic model of the installation is presented.
Fig. 3. Schematic model of the installation: 1 - quartz; 2 - anode; 3 - nuclei; 4 substrate-cathode.
Under the above-mentioned conditions of electrode mounting, the energy of the electrostatic field is concentrated in the center of the substrate. Putting a diamond and graphite nuclei of various shapes in the center of the substrate, the following results were obtained. In the Fig. 4 the characteristic distribution of the energy density of the electrostatic field is shown. In the Table 1 the density of the electrostatic field energy in the volume limited by nuclei of diamond, graphite and in the same volume without any nuclei in the vacuum is presented.
Fig. 4. Distribution of the energy density of the electrostatic field.
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As shown in Table 1, for a columnar growth the energy density in volume in comparison with vacuum increases both for graphite and for diamond. For a platelet type growth parallel to the surface the energy density decreases in both cases, however, it decreases stronger for diamond nuclei. Thus, the growth along the substrate surface is the most favourable for diamond. The columnar growth is unfavourable both for graphite and diamond. Table 1. Density of the electrostatic field energy in the volume limited by nuclei of diamond, graphite and in the same volume without any nuclei in the vacuum. Material Vakuum Graphite Diamond Vakuum Graphite Diamond
Shape
density of electrostatic field energy, ×10-10 J/m3 2.667 2.684 5.807 2.504 2.501 1.288
Estimation of magnetic field effects on the energy conditions for diamond and graphite Similar estimations have been made also for magnetic fields. The calculations showed that a constant magnetic field does not have a significant influence on the energetics of the state of the nuclei due to the tiny difference of the magnetic susceptibilities for various carbon modifications (for diamond χ = -1.726.10-12, for graphite χz = -51.665.1012 , χx = -0.906.10-12 [7]). However, it has significant influence on the carbon ions deposited on the substrate. As shown in ref. [8], the hardness of carbon films may be considerably increased, if a magnetic field is applied in the deposition process. Not only the strength of the applied magnetic field has influence on the magnitude of thermodynamic functions but also the symmetry of the field with respect to the structure of crystalline film and its nuclei. The copper structure saturated with hydrogen determines the symmetry of surface forces which is appropriate for a diamond structure. The distribution of electrodes and their shapes should therefore form the symmetry of diamond structure.
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Method of Differential Magnetron Sputtering The pressure of the saturated carbon vapours at a temperature of 500 K is P = 10-35 Torr. In contrast, the typical background pressure in the deposition chamber in the process of plasma-chemical vapour deposition is P ≈ 10-5 Torr. This pressure should be lower than the pressure of the carbon vapour (a molecular stream) by 2 to 3 order of magnitude, i.e. PC ≈ 10-2-10-3 Torr. Then, mass chaotic crystallization occurs at such (1032) oversaturation. For the artificial reduction of the carbon oversaturation the essentially new technique of the differential magnetron sputtering, which is based on the balancing of streams of carbon from two magnetrons, is offered. This technique will allow to solve the basic problem of the gas-phase synthesis of diamond - the problem of irreversibility of the diamond growth process. It will enable to grow diamond and diamond-like films with instrument quality. Moreover, such tool enables the improvement of recipes for epitaxial film growth in the presence of passivating gases with methods of hydrocarbon decomposition, and also technological methods used for the doping of film with well defined doping concentration. On the basis of our theoretical calculations the technological apparatus for the growth of carbon films by the method of differential magnetron sputtering has been developed and designed. Figure 5 show a crosssectional view of the deposition apparatus. Two magnetrons (1) are located symmetrically with respect to the copper anode (6). The experimental study of the influence of the symmetry of external fields on the nucleation and growth of the carbon films is carried out with three forms of anodes, which have various axes of symmetry (3-fold, 4-fold, ∞). Laser irradiation of the substrate comes from a source (7) located outside of the chamber by means of the window VUP-5M (Vacuum Universal Post) and the mirror (9). At the first stage, for depositing a copper buffer layer, copper atoms are sputtered from the target (8) by Ar ions. On the second magnetron the substrate (3) is located. The electrostatic field determines the energy of the surface bombardment of a nucleus and a film, and, hence, it determines the temperature of an adsorption layer and the speed at which the film is etched away. The heaters (2) set the substrate temperature. The deposition process of copper has to be performed in a mixed argon hydrogen atmosphere. At the second stage there is a change from the copper target to a graphite one and then the carbon deposition is carried out in an argon atmosphere. Change of the voltage applied between the target and the anode as well as between the substrate and the anode controls the streams of carbon onto the substrate and from it.
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Fig. 5. Cross-sectional view of the apparatus used for the growth of carbon films by the method of differential magnetron sputtering: 1 - magnets: 2 - heaters: 3 - the substrate - cathode: 4, 5 - a quartz: 6 - the anode: 7 - light source: 8 - cathode target: 9 - a mirror.
Calculation of the differential magnetron sputtering system parameters In order to define the optimum modes for the growth of carbon films on a copper buffer layer the following calculations have been carried out: 1.
Calculation of the analytical dependences of technological parameters
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for the deposition process of a copper buffer layer in the atmosphere of argon - hydrogen: • Dependence of the sputtering coefficient of copper on the applied anode – target voltage • Dependence of the sputtering coefficient of copper on the concentration of hydrogen in an argon – hydrogen gas mixture • Dependences of the sputtering and deposition velocities of copper on the applied voltage and the concentration of hydrogen 2. Calculation of the analytical dependences of the technological parameters of the deposition process of a carbon film in an argon atmosphere: • Dependence of the sputtering coefficient of graphite on the anode – target voltage • Dependence of the sputtering and deposition velocities of carbon on the applied voltage and the discharge current density. The results of the calculations are presented in Figures 6 and 7:
Fig. 6. Dependence of the sputtering coefficient of graphite on the target – anode voltage.
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Fig. 7. Dependence of the deposition velocity of a carbon film on the discharge current and the applied voltage.
The algorithm of deposition process of carbon films on a copper buffer layer developed on the basis of the obtained analytical dependences of deposition parameters is presented in the scheme shown in Fig. 8.
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Fig. 8. The scheme of the deposition process.
Conclusion We have shown that the conditions for the preferential nucleation and growth of a diamond film might be created through changing the magnitude and shape of applied fields. It was proven that to obtain high quality diamond films it is necessary to create an intermediate buffer layer with diamond-like physical-chemical properties. It was found that copper saturated with hydrogen is an appropriate material for the buffer layer with atomic hydrogen in tetrapores of the lattice acting as crystallisation centers for the diamond film. The distribution of tetrapores on the copper (111) surface and, respectively, of the atomic hydrogen located in them, creates the conditions for preferential formation of tetragonal carbon bonds. We developed a mathematical algorithm for modelling the deposition process on the basis of theoretical calculations carried out earlier. This allows us to offer a theoretical model for the essentially new technological method of differential magnetron deposition of carbon films.
References 1. May PW (2000). Phil Trans R Soc Lond A358:473-495 2. Martorell IA, Partlow WD, Young RM, Schreurs JJ, Saunders HE (1999). Diamond and Related Materials 8:29-36
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3. Cui JB, Shang NG, Liao Y, Li JQ, Fang RC (1998). Thin Solid Films 334:156160 4. Takeyasu Saito, Masanori Kameta, Katsuki Kusakabe et al. (1998). Journal of Crystal Growth 191:723-733 5. Bartsch K, Waidmann S et al. (2000). Thin Solid Films 377-378:188-192 6. Mansurov BZ, Aknazarov SKh, Grinev VP, Lezbaev BT (1997) International Symposium “Chemistry of Flame Front”. Proceeding, Almaty, Kazakhstan pp 144-149 7. Samsonov GV (ed) (1965) Physical – Chemical Properties of Elements. Naukova Dumka, Kiev 8. Hou QR, Gao J (1998). Applied Physics A: Materials Science & Processing A67:417-420 9. Mansurov BZ, Kalykova G, Myasnikova N, Mikhailov L (2001). Eurasian Chemico-Technological Journal 3:211-214
