Quantum dots
v
Preface
Studies of the basic properties of low-dimensional electron systems that are realized mostly in semiconductor structures, have been recently in the forefront of research in condensed matter physics. Rapid developments in fabricating high-quality, disorder-free systems have led to increasing attention on electron correlation effects rather than the disorder-dominated single-particle effects which used to be the mainstay of research in mesoscopic systems. Here we review the experimental and theoretical developments primarily on the electronic properties of quantum dots. In this book we have made an attempt to systematically follow the original published work from which one can perhaps build an understanding of these fascinating objects. The review is supplemented by a collection of reprints of published papers cited in the text as [R1], [R2], ... together with their usual reference numbers that appear in the list of references. The selection of reprinted articles entirely reflects my own choice and is certainly incomplete. However, I have made every effort to provide an exhaustive list of references at the end of the review. Although, we should keep in mind however, that in a fast developing field like quantum dots, the term "exhaustive" has a short shelf life! A major goal for me while preparing the book has been to collect the rich and diverse properties of these fascinating low-dimensional systems and present them in a palatable format to the beginners in the field. However, I also hope that experts will find the choice of topics a useful record of the present status of the field. The review is an outgrowth of a very productive collaboration for well over a decade with the group at the Department of Theoretical Physics, University of Oulu, Finland and Peter Maksym from the University of Leicester, UK. I would like to thank Pekka Pietil/iinen, Veikko Halonen, and Karri Niemel/i for their continuing collaboration with me in the field of low-dimensional electron systems. I would also like to thank Jiirgen Weis and Thomas Schmidt (MPI-Stuttgart) for their input on transport spectroscopy of quantum dots, and Peter Maksym for very useful discussions. Excellent support from the Max-Planck Institutes (Stuttgart and Dresden) are gratefully acknowledged. I would
vi
T. Chakraborty
like to express my gratitude to Professor Peter Fulde for his continuing support and encouragement. I thank Professor Peter de Chatel and Egbert van Wezenbeek of Elsevier for their interest in the project and for many valuable suggestions. My thanks to Paul Houle for critically reading the entire manuscript and to Peter Maksym for making valuable comments about the choice of topics and reprints. Last, but certainly not the least, a word of appreciation for my family (Kaberi and Rebecca), who has been accompanying me to all parts of the world, enduring all imaginable (and unimaginable) languages, climates, etc. Their patience and understanding made this work possible. This book is written for them.
Tapash Chakraborty Dresden~ June 1999
Quantum dots
vii
A short description of the book
The book is about the electronic and optical properties of two low-dimensional systems: quantum dots and quantum antidots. It consists of two parts. The first part is a selfcontained monograph. This part describes in detail the theoretical and experimental background for exploration of electronic states of the quantum-confined systems. Starting from the single-electron picture of the system, the book describes various experimental methods that provide important informations about those systems. Turning to manyelectron systems, theoretical developments are described in detail and their experimental consequences are also discussed. The field has witnessed an almost explosive growth and some of the future directions of explorations are highlighted toward the end of the monograph. The subject matter of the book is dealt with in such a manner that it is accessible to the beginners but it should also be useful for expert researchers as a comprehensive review of most of the developments in the field. The list of references is also fairly complete (upto the time of publication). This book also contains 37 reprinted articles which are selected to provide a first-hand picture of the overall developments in the field. The early papers are arranged to portray the developments somewhat chronologically. More recent papers are supposed to be fair representative of all the various directions current investigations are leading to.
Quantum dots
1
Introduction
Mesoscopic system, a world which lies between the microscopic world of atoms and molecules and the macroscopic world that surrounds us in our everyday life, has been the center of great attention in recent years. The length scale pertinent to these systems, often called nanostructures, is between I0 - 1000A , and we have learned a great deal only recently, about their electronic and optical properties. The advent of submicron technology has ushered in the era of low-dimensional systems in condensed matter physics. Within the last few years, advances in microfabrication techniques have allowed researchers to create unique quantum confinement and thereby opened up a new realm of fundamental physical ideas [i-5], as well as the nanostructure devices with dominant quantum mechanical effects [2-8]. This happens particularly, when electrons are confined to length scales smaller than the electron wavelength (a few tens of nanometers). Theoretical and experimental researches on the many-electron properties of the mesoscopic systems are a challenging endeavor because of their complexity and their manifestations in several surprising phenomena. While single electron properties are no less interesting, almost all mesoscopic systems, unless tailored to have only one electron at a time (like the electron turnstile [9]), have more than one electron present. As we shall see below, the collective effects are profound in the optical and electronic properties of the systems considered here. This is particularly true for devices where the mean free path of the electrons exceeds the size of the device (the ballistic regime). While electron-correlation effects are clearly of great interest primarily for advances in our basic knowledge of these systems, their consequences are also important for future practical applications. The driving force behind much of the research on mesoscopic systems is the expectation that the miniaturization will lead to new type of electronic (and optoelectronic) devices
2
T. Chakraborty
Figure 1 The evolution (shown schematically) of a (a) three-dimensional electron systems to (b) two-, (c) one- and (d) zero-dimensional systems. Also shown are the corresponding density of states (schematic).
much more advanced in their performance than what the existing devices [4, 7, 10-12], specifically being much faster and dissipatating less heat. Here we shall not discuss those aspects of technological applications. Rather we shall try to review our understanding of the underlying physics of these systems. The fabrication of these devices will not be dealt with here (except for a brief description in some cases), but can be found in several books and reviews in the literature [3, 6, 7, 12]. In Fig. 1, we show schematically how a three-dimensional electron system evolve gradually into a zero-dimensional system. In the case of the three-dimensional (3D) gas in the bulk with effective mass m*, we have a free motion of elctrons in all three directions with the corresponding energy h2
E -
+
2
2
+ kz),
where kz,y,z are the wavevectors in all three directions. The energy spectrum is therefore continuous and the corresponding density of states (DOS) is that of a bulk system, Ds(E) o( E89 [Fig. l(a)]. In a two-dimensional system, on the other hand, the electron motion in the z-direction is quantized into discrete electric subbands. The motion is, however, still free in the xy-plane:
Quantum dots
3 h2
i = 1, 2, 3,..., and the 3D-DOS is strongly modified in this case near the quantization energies, showing step-function like behavior [Fig. l(b)] [12]. Additional lateral confinement of the electron motion leads to the one-dimensional system E-
h2 2 2m*ky+E x+E~,
j = 1, 2, 3 , . . . where the DOS is highly peaked and its modification from the 2D-DOS takes place at all relevant energies [12] [Fig. 1(c)]. Finally, when the electron motion is confined in all directions one gets a zero-dimensional system
E-
k Eyi + E~ + E z,
k = 1, 2, 3 , . . . where the energy spectrum is discrete and the DOS is a series of f-function peaks [Fig. l(d)]. These are quantum dots (QDs) - the subject of this book. These manmade objects have lateral widths in the range of a few hundred to about ten nm, where the smallest ones are the self-assembled systems (Sect. 3.3). The thickness of quantum dots created in GaAs/A1GaAs heterostructures is ~., 1 0 - 20 nm. Self-assembled quantum dots are only a few nm high. In general, a system is strictly two-dimensional only if the lowest two-dimensional subband is occupied [13, 14]. This is the same as the condition that the Fermi energy lies far below the second subband, or stated differently, as the condition that the thickness of the electron plane is much less than the average separation of the electrons. Similar conditions also hold for one-dimensional systems [14]. Low-dimensional electron systems are, therefore, low dimensional only in the dynamical sense. Finally, we add that experimental evidence of the atomic-like 6-function density of states in nm-scale quantum dots has indeed been reported recently [15][R28], [16]. Similar discrete electronic states were also observed in metal quantum dots [17] where tunneling transistors containing single nm-scale A1 particles were made and discrete spectra of energy levels observed via current-voltage measurements. In Chap. 2, we survey the properties of quantum dots. The energy spectrum of a single electron confined in a parabolic dot and subjected to an external magnetic field was first investigated theoretically almost seventy years ago. The interest on this model system today is the realization of that ideal calculation in today's state of the art low-dimensional semiconductor nanostructures. These are discussed in detail in Sect. 2.1.
4
T. Chakraborty
The driving forces behind most quantum dot research are ingenious experiments designed to explore the novel properties of the dots. In Sect. 2.3, we describe the results of conventional capacitance spectroscopy, which were the among the first few experiments on QDs, and more recent work on single-electron capacitance spectroscopy. We describe in detail the results of those experiments pointing out those aspects of the results which are now understood and those which remain to be explained. Major developments in exploring miniature devices have taken place in optical spectroscopy and we have discussed those in detail in Sect. 2.3. In Sect. 2.4, we present a brief account of transport experiments in quantum dots. This is a vast field and our aim here is to focus primarily on the spectroscopic aspects of transport measurements with or without an external magnetic field. Two important topics of transport spectroscopy are discussed: single-electron charging and diamond diagrams. Vertical tunneling in QDs which provides useful information about few-electron systems and is also described. Our theoretical understanding of impurity-free parabolic QDs are presented in Sect. 2.5. We introduce the technical details required to evaluate the many-electron properties of quantum dots. The effects of impurities on the electronic properties are discussed in Sect. 2.6. The exciton spectra of a quantum dot, derived from experimental investigations and theoretical results on an exciton in a parabolic dot, are described in Sect. 2.7. Various other topics, such as tilted-field effects, spin blockade and properties of coupled dots are also discussed in Chap. 2, as well as a discussion of the properties of QDs whose shapes are not circular, viz., elliptical and stadium shaped dots. In Chap. 3, we describe some novel systems closely related to those described in Chap. 2. We describe commensurability oscillations in antidots in detail together with their possible application in the search for quantum phenomena in a half-filled Landau level. Novel quantum-confined systems such as quantum corrals are briefly described in Sect. 3.2. Finally, one of the most intensely studied systems in recent years, self-assembled quantum dots are discussed briefly in Sect. 3.3. There is considerable technological interest in this system for application in optoelectronic devices that would lead to thresholdless lasers with high critical temperatures. Chapter 4 concludes the topic of quantum dots by listing a few directions of current developments. The reprinted articles are meant to supplement the survey by providing a first-hand information about the topics discussed in the review. The papers of the initial periods of research are arranged according to the stages of development. Because of the rapid pace, selection of papers provide only a sampling of current developments. In this review, we shall focus entirely on the zero-dimensional systems and not discuss electron correlation effects in two- or one-dimensional electron systems. One important effect of electron correlations in two dimensions is the fractional q u a n t u m Hall effect
Quantum dots
5
(FQHE) [18, 19] - the subject of the physics Nobel prize in 1998, that has been reviewed earlier in the literature [20, 21]. A few books are already published on quantum dots, one popular [22] and other two for experts [23, 24]. Ref. [24] describes in detail the process of growing the nanostructures with special emphasis on self-organization processes and its application in quantum dot lasers. Earlier experimental techniques to create the nanostructures are available in [6]. Description of quantum dot properties can also be found in several recent publications [25-29].
Quantum dots
7
Quantum dots
Quantum dots, popularly known as "artificial atoms" i, where the confinement potential replaces the potential of the nucleus [I], [30][R7], are fascinating objects. On one hand, these systems are thought to have vast potential for future technological applications, such as possible applications in memory chips [i0], quantum computation [31-36], quantum cryptography [37], in room-temperature quantum-dot lasers [38], and so on. But the fundamental physical concepts we have learned from these systems are no less enticing. We shall discuss many of those basic concepts in this review. Some examples of those concepts are: magic numbers in the ground state angular momentum, the spin singlettriplet transition, the so-called generalized Kohn theorem [i], [30][R7], [39-41], and its implications, shell structure, single-electron charging, diamond diagrams, etc. In the last few years we have witnessed a profusion of new results and ideas in quantumconfined zero-dimensional electron systems. Experimental advances in fabricating quantum dots and precise measurements of various electronic and optical properties have generated an exciting situation both for the theoreticians and experimentalists. As we shall demonstrate in this review, there have been several interesting developments where the theoretical predictions and experimental surprises have resulted in deeper understanding of these systems. In our review of the properties of quantum dots we shall mostly concern ourselves about the case where an external perpendicular magnetic field i Interestingly, as far as we know, this popular name was introduced in the literature by Maksym and Chakraborty [I], [30][R7]. One other appropriate name "designer atoms" was introduced by Reed [4]. There are, however, significant differences between quantum dots (QDs) and real atoms: QDs are larger than atoms and number of electrons in the dot can be independent of the size of the dot.
8
T. Chakraborty
0.3
el--
PS
~--o0 I
/
o 1
/ ~ e O ~1
~
i - -
/
i I
oO
i '
iI -
i
0.2
o 1
--
. . . . . .
- . . . . . . . . .
,
e .
....
1! . l l - - .
e 0 . o 0
.
=
=
=
.
.
,"
i /
II
II
/ .....
.
.
.
.
.
-
o 1
, ___,...
=
=
:
_
-
-
-
-
-
-
- - - _. . . .
-
-e 0
. . . .
,
,
- - - - - - : : : : : - _- -_- 2 -
el
ii
ii
"
,'
. . . . .
,'--
'
_
el oO
, , " ,-,
--
,"
,,;,
.-" ""
e0
/
,/ r ," ,,,,"
i i /
ol
/--eO
i
i i
i
i1
iI
' - . . . . . .
----.-/
i1 /
----' " , ,,
/
o 0
iI
II
,, /
"
i I
n l
0.I
/
iI
,
----~ -/
.
11/ ,,~ ;,;
----
i
---"
--
=--
.
,/'/
'
,,
,' ,'
e 0
i
'
,
.---" /
o0
-'--=:::::~
ol
. . . . . . . ".
.
.
.
.
.
.
.
.
.
.
.
.
el
". - .
.
.
.
e0
.
e0 (a)
(b)
(c)
(d)
(e)
F i g u r e 2 Energy levels of two electrons in a long, narrow box (Ly = 10Lx). The energy spacings are measured relative to the ground state energy scaled by Rs i.e., AE/Rs for (a) non-interacting electrons Ly = 20a0, Eg = 1.905Rs, Rs = 105 x Re, Re is the effective Rydberg and ao is the Bohr radius. In the case of interacting electrons the parameters, Ly/ao, Eg/Rs, and Rs/Re are respectively, (b) 20, 1.907, 105, (c) 200, 1.923, 103; (d) 2000, 1.987, 10; and (e) 2 x 104, 2.314, 0.1. The parity and total spin of each state are also indicated [45] [R4].
is present [30] [R7], [42]. However, our discussion of the electron correlation effects would be incomplete if we did not mention the important work done on q u a n t u m dots in the absence of a magnetic field One of the first reports of three-dimensional quantum confinement in semiconductor nanostructures suitable for measurement of excitation spectra of q u a n t u m dots was by Reed et a l [43][R1], [44] They studied vertical transport in q u a n t u m dot structures realized by etching narrow columns into heterostructures. W h e n the lateral dimension was made sufficiently small, i e , when full three-dimensional confinement was achieved,
Q u a n t u m dots
9
the measured current-voltage characteristics showed a series of peak structures which were attributed to resonant tunneling through the discrete zero-dimensional states in the dot. Electron correlation effects in quantum dots in the absence of a magnetic field were first studied theoretically by Bryant [45][R4]. He considered a two-electron system in a rectangular box with hard-wall potentials and studied the interplay of kinetic and interaction energies as a function of the size of the box. In the infinite-barrier model, the kinetic energy matrix elements scale as 1/L 2 where L is the linear dimension of the quantum-well box. While the interaction matrix elements, scale as 1/L when L is changed without changing the box shape. For small L, the electrons behave as independent, uncorrelated particles because the Coulomb interaction is insignificant compared to single-particle level spacings. However, when L increases the interactions become important and the level spacings change. Electron correlations help electrons form a Wigner crystal. The signature of this state in a confined system is the degeneracy of the levels. In a long, narrow box the evolution of the states into the levels with degeneracies of the Wigner lattice happens for L _> 0.1 #m (Fig. 2). These results show that there is a continuous evolution of the energy level structure, from the single-particle-like states in the limit of a small dot, to a level structure in larger dots where electron-electron interactions are dominant. We should point out here that this was the very first report on the importance of electron correlations in mesoscopic systems. Macucci et al. [46] used density functional theory to investigate the electronic structure of quantum dots in the absence of a magnetic field. For the exchange-correlation terms they used the polynomial representations given in the calculations of Tanatar and Ceperley [47] of a two-dimensional electron gas. The numerical studies of Macucci et al. were done mainly for the chemical potential and the differential capacitance of the dot as a function of the dot size and the number of electrons. It was found that there is a gradual transition from the state where the quantum effects are dominant (in very small dots where the quantization energy dominates over the Coulomb energy) to an almost classical capacitor-like behavior of large dots. Tarucha et al. [48-50], [51][R33] analyzed the electronic states of a few-electron vertical quantum dot 2. Their vertical dots were disks with diameter about 10 times the thickness. The lateral potential had a cylindrical symmetry with a soft boundary profile that can be approximated by a harmonic potential. These authors noticed that at zero magnetic field, the Coulomb oscillation is irregular in period reflecting a shell structure associated with a two-dimensional harmonic potential. At low fields, they observed antiparallel filling of spin-degenerate states. Close to zero magnetic field, they noticed the 2 For details on vertical dots and the work of Tarucha et al., see Sect. 2.4.6.
10
T. Chakraborty
filling of states with parallel spins in accordance with Hund's rule. According to this rule degenerate states in a shell are filled first with parallel spins up to a point where the shell is half filled. Half-filled shells correspond to a maximum spin state t h a t has relatively low energy due to exchange interactions. Observation of the shell filling has prompted several theoretical investigations [52-56] to find the underlying physical reasons for that effect. In the absence of an external magnetic field, Fujito et al. [57] calculated the total energy, chemical potential, capacitance, and conductance peak positions for anisotropic parabolic quantum dots. Here the Hamiltonian is written (in effective atomic units) N
%t =
N
i<j
i=1 1
1 2 ~Vi + - ~ 2 (x~ + y~) + -~zZ~
=
t(ri)
-1
~t(ri)+Erij 2
1
with an anisotropic parabolic potential with the cylindrical symmetry of characteristic _1
_!
frequencies ~x and ~z corresponding to oscillator lengths fx = ~x 2 and gz - ~z 2 respectively. The many-electron ground states were obtained by means of an unrestricted HartreeFock method. The ground state was found to be ferromagnetic up to 12 electrons in large dots (gx - 49.5 nm). For small dots (gx = 7.425 nm), one-electron levels calculated self-consistently are occupied in accordance with Hund's rule. The differential capacitance c
(N) -
-
-
where p ( N ) - E ( N ) - E ( N - 1) is the chemical potential and E(N) the total energy of the ground state of N electrons in a dot, shows a characteristic oscillation with electron numbers. Local maxima or minima of capacitance reflect the shell structure. Shell structure was also seen in the chemical potential and conductance peaks versus the gate voltage. The latter was calculated as follows: a tunneling current flows through the dot between two leads with a small bias voltage when the chemical potential of the dot aligns with that in the leads. The conductance peaks appear at the gate voltages V
(N) -
where ~L is the chemical potential of the leads. More on these topics will be discussed in Sect. 2.4.1. In an interesting paper, Kumar et al. [58][R8] considered a model of a single quantum dot of 300 nm square area. They solved the SchrSdinger equation (using Hartree approximation, i.e., ignoring exchange and correlation effects) and Poisson equations selfconsistently. In the absence of an external magnetic field they found that although the
Quantum
dots
11
0.35
1
1
5 0 0 nm DOT
0.30
0.25 :::k
0.20
0.15
L
0.15
_
1
0.20
. . . . . .
1
0.25
1
0.30
-0.35
x (~ m) Figure 3 Lateral potential contours of a quantum dot in a plane below the GaAs/AIGaAs interface. The innermost contour is 15 meV below the Fermi level, which is indicated in the figure [58][R8].
dot geometry was square, the lateral potential had nearly circular symmetry (the angular momentum was approximately a good quantum number). Further, the effective size of the quantum dot (with a diameter of ~ I00 nm) was considerably smaller than the geometrical size of the structure. The calculated quantum dot potential by Kumar et al. is shown in Fig. 3 in a plane 8 nm below the interface of a square, gated GaAs/AiGaAs clot similar to what was experimentally investigated by Hansen et al. [59][R2]. Clearly, the potential contours are nearly circular, particularly at low energies. These authors studied the evolution of energy levels in the presence of an external perpendicular magnetic field. The calculated energy levels for seven electrons per dot showed good qualitative agreement with the results for states in a two-dimensional harmonic oscillator potential in a magnetic field, as discussed below.
T. Chakraborty
12
2.1
One-electron
systems
The problem of a single ideally two-dimensional electron in a circular dot, confined by a parabolic potential grnl*a;02r2 (m* is the electron effective mass) in the presence of an external magnetic field was solved more than half a century ago by Fock [60][R5] (and later by Darwin [61][R6]). It is interesting to note that the same problem (but for zero confinement potential) was studied two years after Fock's work by Landau [62] leading to the term Landau levels. We shall call the energy levels derived below as Fock-Darwin levels (FDL)[1].
2.1.I Energy spectrum Following the classic work of Fock [60] [R5] the Hamiltonian is written as 7-{--1( 2m*
e ) 2 ira, p - -A + g w02 (x 2 + y2). c
(-1By, l~Bx, 0)
If we choose the symmetric gauge vector potential, A = the Schr6dinger equation is written as 2m* h 2 { -O~r 02~2 + - r1 0- ~rr
002
1
-iz~
e B2r 2
h2 ( di f l df ~ -~ r dr
12 ) r2f
+
Defining the cyclotron frequency c~c ! ~
Sin* c 2
\
i +~m
,~2or2- E)r /
The radial part of the equation is then
2m*
g0 -
and e = -[e[,
2m*cieBhO~ O0
-~. r 1 .2 0 2 . ~ .}
+ Let, ~ - ~ f ( r ) e VzTr
(2.1)
ebb ~2m* l cf 8m,c 2 + ~
lm*w2or2-E
e B / m * c - h/m*g~,
f=0.
and the magnetic length,
, the radial equation is simplified to
h2
Let us define b -
ft'-i' f, (1 +
12f) -Jr-[.E
2 c) 2 89 4c%/c~
177~*
+ 40202) r 2 +
89
and a new independent variable
] S=0.
(2 .2)
Quantum dots
13
X --
m*wcbr2 _ ar 2 = br2 2h 2~ "
Then the radial equation Eq. (2.2) transforms into
x f , + f~
li f ( E - 4z + hwcb
1 1~1 4z + ~ f-O.
11 A further simplification of the radial equation follows if we define 3 - E/hwcb + -~-~,
z f " + f' + / 3 - - g z - ~
f --O.
(2.3)
The single-electron eigenenergies are then obtained from the solution of Eq. E~l
=
lhwc[b(2n+lll+l)-l]
=
(2~ + IZl + 1)(X1 h 2
-
(2~ + IZI + 1 ) h f t - llhwc,
2
h2
1 _1
(2.3) [20]
hWcl
(2.4)
1 2 where ~2 = (XWc + w~). Here the two quantum numbers are, n = 0, 1,..., the radial quantum number, and 1 = 0, • the azimuthal quantum number. The energy spectrum is displayed in Fig. 4 for hw0 = 4 meV. Some interesting properties of the energy spectrum are immediately obvious: At B = O, Wc = 0 and
E~z - (2~ + IZl + 1)hwo. Similarly, for large B, Wc >> wo, n--0,
l_>0,
E-
1 ~hw~
n-l,
l>_0,
E-~
3 ~a2c '
...,
which means that, in the absence of a confinement potential, energies of the positive 1 states are independent of l, but in its presence they increase with I. States with angular momentum 1 < 0 have much higher energies. At low values of the magnetic field such that the magnetic length is larger than or comparable to the size of the confinement potential, there is a hybridization of Landau levels with the levels that arise from spatial confinement. With increasing magnetic field, as the magnetic length becomes much smaller than the radius of the confinement potential, free-electron behavior (Landau-type levels) prevails over spatial confinement. Therefore, one sees a gradual transition from spatial to magnetic quantization that depends on the relative size of the quantum dots as compared to the magnetic length. As an example, if the electrons are confined in a 100~ dot, Landau levels would form in a magnetic field which is above 40 tesla.
14
T. Chakraborty
60 50
,-1)
0,-3)
(3.-5)
/(1,2)
40
A
E LLI
--~ ~h~ c
--~h~ c
=-----(1,0) -~(0,,-1)
30
,(0,10) ,(0,9)
.(0.8)
20
/(0,7)
-.--(O,e)
10
---~89 c
(~,2[0'4) 0.31 t0 1)
o:o)
0 0
4
8
12
16
20
B(T) F i g u r e 4 Single-electron energy levels (the Fock-Darwin levels) in a parabolic dot as a function of the magnetic field. The levels are indicated by their quantum numbers (n, l). The confinement energy is taken to be ha~0 = 4 meV.
The calculated low-energy excitations of the Fock-Darwin diagram (Fig. 4) were well reproduced in the magnetotunneling experiments on double-barrier resonant-tunneling structures [63-65]. Here one employs a strongly asymmetric double-barrier heterostructure 3 where the thick barrier is the emitter and the thin one is the collector [64]. Figure 5 sketches the condition for resonant transport via the single-electron states which is also the basic idea behind the experimental results described below. Here the crucial quantity to understand the experimental results is the addition energy, p ( N ) , the energy required to add one extra electron to a N - 1-electron dot. To begin, at zero bias (V = 0) the q u a n t u m dot contains no electrons because the ground state energy E(1) of the first electron [E(1) = #(1)] lies above the chemical potential of the emitter ch and collector contacts #E,c. Whenever the addition energy PE exceeds # i ( I ) - c~eV where c~ is the voltage to energy conversion coefficient [63-65] and p~(1) = E~(1), single-electron states are available for tunneling from the emitter to the dot and one observes a step A I = 2 F E F c / ( 2 r E + r c ) ~ 2FE, (FE << Fc) in the measured current. Here, FE (Fc) is the tunneling rate through the emitter (collector). The bias position of these steps, 3 F o r m o r e d e t a i l s a b o u t m e a s u r e m e n t s o n v e r t i c a l q u a n t u m d o t s , see Sect. 2.4.6.
Quantum dots
15
~i(1)
2~1(1)
~i (1) ~. LI.E 12
~1(1) ~E
'LtE
~c
__ V:O
V:Vo(1 ) VO(1)
Figure 5 The energy diagram illustrating the transport spectroscopy of single-particle spectrum of a quantum dot [64]. E(1) = [ # i ( 1 ) - #~h] /ec~, provide a direct experimental probe to the single-particle spectrum including the ground state. The gray-scale plot of Fig. 6 shows the differential conductance G = d I / d V as a function of bias voltage and magnetic field (derived from the I - V data) obtained by Schmidt [64]. Some of the low-lying Fock-Darwin states are clearly discernible from the figure. It is quite remarkable how the energy spectrum of an ideal quantum-mechanical model system derived by Fock some seventy years ago, is made visible in semiconductor nanostructures.
2.1.2. Single-electron states To determine the single-electron wavefunction we return to equation Eq. (2.3) X R## JF R ! Jr-
(
fl_
12) R = 0
1 -~x---~a
whose solution is [20]
(Zl+~)~]= I 1
1
Rnl(r)-- aNl+lZl l]!
21Zln.
exp (-r2/4a~) r jzl S ( - ~ ,
r2)
I~1 + l, 2--~H 9
Here we have used the notations
~H
-
a.
-
acb= (C~c 2 +4CJo2) 89 = m*wH
1
go
v~
v~'
.
JLA
c
(2.5)
16
T. Chakraborty
F i g u r e 6 Gray-scale plot of differential conductance G = dI/dV as a function of magnetic field and bias voltage. The location of the lowest L a n d a u level energy ( 1 hCOc)/eOZ and the second L a n d a u level ( 3haJc)/ec~ are also indicated [64].
Q u a n t u m dots
17
and .7- is the confluent hypergeometric function. If we use the relation
br2) = (n n,l,+}t,)!L~I( -~o br2) ' f ( -n, ll[+l,-~o we can write the radial function as
RnI(r)
=
=
go (~)
1+IZl
[
so,
2,t,n,
Ill'
"
21~leo2(~+lzl)! ~xp - ~ 0
exp
-
-~o
\ e0
r
-
(n +- lII)iL~I \ 2 g g J
L~l\2e~
'
where L~ I are associated Laguerre polynomials. The single-electron wavefunction is then O)
27rgo2 21Zl(n + I/l)[
exp
4eo2/\ eo
L~I
(2.6)
We shall make use of this form of the single-electron wavefunctions on several occasions below.
2.2
Dipole
matrix
elements
In what follows, we assume the wavelength of the incident light to be so large (compared to the size of the quantum dot) that its spatial variation can be neglected within the extent of the electronic wavefunction, i.e., the plane wave describing the radiation field can be approximated as e ik'r ~ 1. In the case of quantum dots this corresponds to far infrared radiation. In this approximation the transition probability from an initial state [p} to a final state Iq) is proportional to the quantity [1], [30][R7], [66], [67][R9], [68][R10], [69, 70], [71][R19]
dipole
approximation
2[- E I(plrlq) "e(~)12" OL
The polarization vectors e (~) are the unit vectors i and j in case of
linear polarization
e(+) = v/-~(i 1 i iJ) in case of
circular polarization.
If the incident light is unpolarized we have
or
18
T. Chakraborty Z
-
I(q[xlp}l 2 + I(qlylp}l 2
=
1_ [[(qlreiOlp)[2 +l(qlre_iolp}[2]. 2
We define the single-particle matrix elements dA'A
IreiO I /~}
--
<~'
=
27rSl+l,t,
(2.7)
r2Rn,z,(r)Rnz(r) dr
which hold for circularly symmetric systems (such as parabolic quantum dots) where the quantum number A stands collectively for the principal quantum number n and the angular momentum quantum number 1. Then the single-particle matrix elements of the operators x and y can be written as 1
XA'A
-
(a'l~l~) - ~ [d~,x + d ~ , ]
YA',~
-
(~'lyl~) - ~ [d~,~ - d ~ , ] .
1
In occupation representation the dipole operators corresponding to the configuration space operators x and y are
X
-
E x),),,a~a),,
r
=
ZY~'4a" AA'
It is clear from the above expressions that only transitions with A1 = 1~ - 1 = +1 are possible. If a photon to be absorbed is polarized in the x-direction, only the operator X is to be used, and for photons polarized in the y-direction, the operator Y is used [71][R19], [72], [73] [R24], [74, 75].
2.3
Basic
properties"
Experiments
Experimental information about the electronic properties of quantum dots is primarily from single-electron capacitance spectroscopy [76][R21], [77, 78], gated resonant tunneling devices [79], conventional capacitance studies of dot arrays [59][R2], [80-85], transport spectroscopy [86][R3], [87][R13], [88][R14], [89], [90][R22], [91-94], [95][R23], [96], far-infrared (FIR) magnetospectroscopy [97][Rll], [98], [99][R12], [100-102], and very
Quantum dots
19
Figure 7 (a) Schematic diagram of a quantum dot capacitor. (b) Scanning-electron micrograph of quantum dot (300 nm across) arrays [82].
recently, from Raman spectroscopy [103,104]. In conventional capacitance studies, an oscillatory structure in the measured capacitance was attributed to the discrete energy levels of a quantum dot. In the presence of a perpendicular magnetic field, Zeeman bifurcation of the energy levels of a quantum dot was also observed. This splitting is believed to occur due to the interplay between competing spatial and magnetic quantization. Capacitance spectroscopy has been widely used to study the density of states of lowdimensional electron systems [81]. A schematic view of a quantum dot capacitor is shown in Fig. 7 (a). Here the lateral confinement is provided by depletion of high-mobility twodimensional electron gas. The 30 nm GaAs cap layer is etched away to deplete the underlying A1GaAs. The electrons are then confined in a quantum dot at the heterojunction. A scanning-electron micrograph of quantum dots is also shown in Fig. 7 (b). The n + GaAs is one electrode of the quantum capacitor and the Ni/Au metal on the top is the other. The differential capacitance (at T = 0) of a confined electron system is determined largely by the thermodynamic density of states:
C = dQ
dn d#
= q-Jp~-d-v~ Ds(EF) dV
where dQ is the infinitesimal charge induced by a change in voltage dV, q is the electronic charge, n is number of carriers, # is the chemical potential and Ds = dn/d# is the density of states (DOS) at the Fermi energy EF. A change in the gate voltage changes # and therefore in quantum dot experiments, we can sweep the Fermi energy through the zero-dimensional density of states. The measured capacitance (more precisely, the first derivative of the capacitance vs the gate voltage) reveals structures related to the
T. Chakraborty
20
I
rn dots
T=O.7K f=1OkHz
30Onto dots
>
-0.40
P
E
-0.20
0.00
0.20
0.40
vG (Volts)
Figure 8 Capacitance spectra for three different quantum dots. The size dependence of the spatial quantization is clearly visible in the spectra [82]. zero-dimensional quantum levels. Figure 8 shows capacitance results for three quantum dots of different sizes. The wellresolved oscillations in the derivative of the capacitance reflect zero-dimensional quantization. The period of the oscillations increases with increasing dot size. Figure 9 shows the gate voltage derivative of the capacitance as a function of the gate voltage at different magnetic fields. The peaks at positive gate voltages show complicated behavior even at very low fields (B < 0.2 tesla). However, as the field is increased to about 1 tesla we can clearly distinguish the peak shifts and splitting of many of the peaks. At higher fields, some peaks increase in size and other weaker peaks move between them. The stronger peaks are the precursors of Landau levels which the smaller peaks join as the field increases and become more degenerate. We plot the peak positions as a function of magnetic field in Fig. 9 (b). The stronger peaks that become Landau levels at high magnetic fields are plotted as solid dots while the weak peaks are shown by open dots. The oscillatory structure in the capacitance seen in Fig. 9 (a) can be attributed to the discrete energy levels of the dot indicating that the quantization has been achieved in all directions. The peak splitting is believed to occur due to the interplay between competing spatial and magnetic quantization 4. 4 Silsbee and Ashoori [84] have presented a different interpretation of these results, where they a t t r i b u t e
Quantum dots
21
~t
T:O'7K ~
250
-
!
188 ~o -
~^
I o
8
..n~ o ~".....~ ~" o
UO~ ~-_. ~,,oo ~"
~
ID
-
9
9
9
oo
., 9
o
e
""o"~. " o OOo
o ~ 0
9
U)
(....
~.
63~-~ ~eo foe
~ ~
9
000
~.,&lg~
>o)
'eeD ! ;""
0
~e . ' ~
0
e,i,.~
o0
00 OoOe " 9 e ~ o o 9 O000e O 0 o e e o oo9 o ee ooO~176 e ee e ee o 0
0L~ _.~~,
_
<1: v >
~"
o
9
0
23
~t
;
9
9
-
"lD
-125 i ~ ' ' ~ " o 9 "" imeeeseol o o o ~ 1 7 6 1o7o6o o
~
oo oeoe
iJ
-188
_ I
I
I
......1
~a~NOIONB O O 0 9 9 9 0 0 9 9 9 9 e 0
I ......
-0.30 -0.20 -0.10 0.00 0.10 0.20 0.30
(a) Figure
Vg (V) 9
;meeieeo o 9 e e o c
oO
B=0
-250 0.0
(b)
_I 0.4
J -J 0.8 1.2
|. 1.6
2.0
B (T)
(a) C a p a c i t a n c e s p e c t r a for 300 n m q u a n t u m d o t s at various m a g n e t i c fields.
(b) Gate voltage position of the peaks. Filled and open circles correspond to strong and weak peaks respectively [59][R2]. Another interesting result in capacitance spectroscopy is the observation by Hansen et al. [83] of fractionally quantized states, similar to the fractional quantum Hall effect in a two-dimensional electron system [19-21]. For dots containing about 30 electrons in a very high magnetic field, the derivative of the capacitance vs the gate voltage shows downward cusps at 51 and 52 filling factors 5 [Fig. 10 (a)]. The temperature dependence of the minima [Fig. 10 (b)] at these two filling factors is also consistent with that of the fractional quantum Hall states.
2.3. I. Single-electron capacitance spectroscopy The electronic ground state in a parabolic confinement potential described above has indeed been observed in an ingenious experiment reported by Ashoori et al. [76][R21], [77, 78]. The method involved in this experiment is known as single-electron capacitance spectroscopy, and allows direct measurement of the energy levels of a he-electron dot as a the peak structure of the capacitance spectra as due to successive electron transfer where the peak spacing is determined by the charging energy (see also, Ref. [85]). 5 Defined as tJ = (h/eB)N/A, where A is the area and N the number of electrons in the dot.
T. Chakraborty
22
(a) ~
(b)
l
T=O.7K
.
B=30T
~ v =1/3
=1/3
v=2/3
'
I~\\i
v=2/3
<.% 19T
~
-0.3
-0.1
~
0.1
!i
0.3
0.5
Vg (Volts)
0.7
.....
-0.3
l,
-0.1
I
0.1
0.3
0.5
,_
0.7
Vg (Volts)
Figure 10 (a) Derivative of the capacitance vs the gate voltage for 3000 .~ dots. Only the lowest spin-split Landau level is occupied at the gate voltage shown here. (b) Temperature dependence of the fractional states at B = 30 tesla [83].
function of the magnetic field. The basic configuration of the sample used by Ashoori et al. is shown in Fig. 11 (a). The capacitance was measured between an electrode on top of the QD (the gate) and a conducting layer under the dot that is separated from the dot by a thin tunnel barrier. When the dc gate voltage on the top electrode is varied the Fermi level in the bottom electrode can coincide with the Fermi energy of the dot. That would result in electron tunneling through the thin A1GaAs barrier, as indicated in Fig. 11 (a). Charge modulation in the QD induces a capacitance signal on the gate because of its close proximity to the dot. The capacitance as a function of the gate voltage was found to exhibit a series of uniformly spaced peaks, with separation decreasing with increasing electron number [Fig. 11 (b)]. The peaks are due to the addition of single electrons to the QD. The remarkable aspect of the experiment by Ashoori et al. [76][R21], [77, 78] is that they probed the addition spectrum starting with the very first electron in the dot. The results were plotted by Ashoori et al. [76][R21], [77, 78] as a 2D gray-scale plot where the vertical axis is the voltage, the energy scale of electron addition. The capacitance is highest in the bright areas. The bright lines trace the addition energies as a function of the magnetic field B. The ground state energies of systems with up to 35 electrons are presented in the plot of capacitance peaks in Fig. 12. The magnetic field
Q u a n t u m dots
23
b l o c k i n g b a r r i e r i + * § § ++.§ + * 9 + * + § quantumwell -=i ~ ~i ........." G a A s tunnel barrie~ ___ ~ " . : IAIGaAs spacer ~e Undoped GaAs ~ . / J . / J J . / . 7 _ ~ J J J / / / ' . / / / / . ~ / ; .
(b)' O9 t-"
~
i
~ i ,,,,, 0
o ._1 w
6'3
-390
-360
-330
-300
-270
-240
V(mV) Figure 11 (a) Schematic diagram of the sample. (b) Capacitance data as a function of the gate voltage for a quantum dot. The bottom trace is the signal at the dot in 90 ~ lagging phase [76] [R21].
dependence of the lowest energy state in Fig. 12 (a) is smooth and can be well described by the Fock-Darwin energy for hw0 = 5.4 meV. Further, the high-field asymptote of this curve also follows the dashed line (plot of ~1 hwc) as expected. The ground state energy curve for the two-electron system shows a pronounced bump and change of slope around 1.5 tesla. This was interpreted by Ashoori et al. as an indication of spin singlet-triplet transition. It is interesting to note that a similar transition for He, predicted at an exceedingly high field of 4 x 105 tesla in the vicinity of white dwarfs and pulsars [105] still remains to be observed. Artificial atoms with weak binding and small electron mass are therefore, in certain aspects, more useful tools than real atoms! The b u m p progresses and shifts monotonically to higher fields as the electron number increases, thereby producing a clear ripple through the data set. We should emphasize at this point that some earlier theoretical work predicted this type of spin transition [106][R15], [107][R16]. Such a spin transition in a two-electron q u a n t u m dot was also observed experimentally by Su et al. [108] via single-electron tunneling [for more on these topics, see Sect. 2.4.6]. Ashoori et al. also noticed a distinct loss of intensity as
24
T. Chakraborty
F i g u r e 12 Images of sample capacitance as a function of both magnetic field and energy. The dashed line shows hc~c/2. The number along the traces indicate the electron number [(a) 1-6, (b) 6-35] in the dot [76][R21].
the m a g n e t i c field was increased, which resulted from a decreased t u n n e l i n g rate. T h e reason for this is not entirely clear. G r o u n d s t a t e energies of the dot containing ne = 6 - 35 electrons are shown in Fig. 12 (b). Before we go into the details of these results, let us first take a look at Fig. 13. It is clear from the t h e o r e t i c a l Fock-Darwin states calculated for the confinement energy of hco0 = 1.12 m e V t h a t the g r o u n d s t a t e energy of the 35-th electron should oscillate as the levels cross with increasing m a g n e t i c field. T h e oscillations cease at ~ 2 tesla. If we define the L a n d a u level filling factor t, as the Landau-level o c c u p a n c y at the dot center, the position of the last level crossing in this figure can be identified as y = 2, i.e., two electrons per flux q u a n t u m .
Quantum
dots
25
I~ I~ 9^ V
' ~
~
~
~162
:
:
'
:
%,"_~'-:,~:~.':""" .".." :"" .".".: : :':':':'." ." .:.:.:.:.' ".',',',','," "r: ~',:,: "" /:: ." "" " ".' --".-"":'.".'.'.'.:.:.: " " ' " " " ' " " " . :. . ". . :. . ". . ." : : " :" ..'.'.'.;:.:. " ' " ' " """ "
;
:':-..'.'."""-'-'..'.'.'-'.'-'-:':':':':" :.:.;.~-: , , :
:.
"~" [-::: .:; "; " ":' . :' . '::: ::::::: :::::::::::. :::' .:: :~: -~: :~. :~: .i: .--:: :.:.:.~.:.': , . ::..-."'.: ,: :t"""!::::":." ":~':, [:'i,:~:~'..;:';: "-~" :~',,: "" . . .;-:.: . .. :" .'.':.':':': :..:':.':...';
5
S (T) Figure 13 The Fock-Darwin s t a t e s of a q u a n t u m d o t w i t h hw0 - 1.12 m e V ( d o t t e d c u r v e s ) . T h e b o l d s o l i d c u r v e s h o w s t h e m a g n e t i c field e v o l u t i o n o f t h e 3 5 - t h e l e c t r o n .
T h u s far, we have discussed only the case of n o n - i n t e r a c t i n g electrons. A simple way to i n c o r p o r a t e the interaction b e t w e e n electrons is to follow the c o n s t a n t - i n t e r a c t i o n (CI) model 6. This a p p r o x i m a t i o n is c o m m o n [87][R13], [110] in studies of the C o u l o m b blockade, where the total energy is w r i t t e n as ne
Z
+
(2.S)
i=1
with U as the inter-electron interaction a n d r chemical p o t e n t i a l is t h e n evaluated from #e -~ E ( n e ) -
is the energy of the i-th electron. T h e
E(T~e - 1) = (he - 1) V ~- ~ne
where s~o is the energy of the n~-th electron. Going back to Fig. 12 (b), we notice the d e v e l o p m e n t of the ~ = 2 positions (white triangles). For n~ > 10, the ~ = 2 positions for successive electrons agrees well with the CI model with h~0 = 1.1 meV. At large n~, the t u n n e l i n g rates are a t t e n u a t e d a r o u n d y = 2. Ashoori et al. s p e c u l a t e d t h a t , as the electrons in the dot center are in a q u a n t u m Hall state, the incompressibility of t h a t (a h a l l m a r k of the q u a n t u m Hall state) s t a t e causes suppression of the tunneling. 6 For a critical analysis of this approximation, see Ref. [109].
T. Chakraborty
26
Ashoori et al. [111,112] also noted that large dots (lithographic diameter larger than 0.4#m) containing few (~40) electrons reveal some more puzzling features. Instead of being well separated in gate voltage as in the results described thus far, electron additions in these dots are grouped in bunches. With increasing electron number, the bunching pattern develops into a periodic pairing of electron addition peaks in gate voltage, with every fifth peak paired. When the electron number is increased above about 80, the pairing pattern disappears and the usual periodic Coulomb blockade pattern appears. Application of a magnetic field reinstates the pairing phenomenon, where a larger magnetic field Bpairing(rte) is required to create pairing when the dot contains larger number of electrons. Finally, the bunches cease to appear as the magnetic field strength is increased so that the filling factor is at u = 1. These authors surmised that the bunches are caused by electron localization within the QD. The cause of localization giving rise to periodic bunches are also attributed to interactions. It is interesting to note that, for classical particles confined in a parabolic potential [113-115], the ground state corresponds to localization of the particles to shells with shell filling somewhat similar to that observed in experiments above. However, along that line of reasoning the pairing mechanism still remains to be explained.
2.3.2.
Optical transitions
In the experimental studies described in this section, the quantum dot structures were created either by etching techniques or field-effect confinement (Fig. 14). The samples were prepared from modulation-doped A1GaAs/GaAs heterostructures. For the deepmesa-etched quantum dots, an array of photoresist dots (with a period of a = 1000 nm both in x- and y-direction) was created by a holographic double exposure. The rectangular 200 nm deep grooves were then etched all the way into the active GaAs layer [Fig. 14 (a)]. Field-effect confined quantum dots were prepared by starting from a modulation-doped GaAs-heterojunction. Electrons were laterally confined by a gate voltage applied to a NiCr-gate. A strong negative gate voltage depletes the carriers leaving isolated electron islands (quantum dots). Details about the fabrication of these dots are available in the literature [2], [97][Rll], [98], [99][R12], [101,116,117]. We have already discussed in Sect. 2.2 the expression for dipole matrix elements. More specifically, transitions from an electronic state ~nt to a state Cn'Z' are governed by the the transition amplitude
dnz,n,Z, = (O~zlre+i~162
(2.9)
with associated oscillator strength f~z,~'z' -- (2m*/h)mnz,~,z,
Id~z,~,z, Ie
(2.1o)
Q u a n t u m dots
27
.... n-15aAIAs
GaAIAs s p a c e r OD-Systern
+ + + + +
~
-- - ' - ~
/////
'"
"- J
GaAs
.
_
_
(b)
--
5ateelectrode Phoforesisf
GaAsHeferosfrucfure QuQntumdof
Figure 14 Sketch of the (a) deep-mesa-etched quantum dots and (b) field-effect confined dots. where a&~z,n,z, - ( E n , l , - E n l ) / h is the transition frequency. For the single-electron states in a parabolic quantum dot, Eq. (2.9) can be written in closed form [see, Eq. (2.8)]
dnl,n'l'
(2 )2 --
(~nn'(~l ' l + 1 ,
fft* 02 0
1
1
[7t
+ Ill
+
1] ~ 1__
--
5n,,n+lS1,,l+
1
leading to the selection rules, A1 - l ' - 1 corresponding energies are
2h (1 - St,o) rn*wo - •
A E + - hft +
+ 111
as derived above and An -- 0, 1. The
lh~
c
(2.11)
where the + ( - ) sign corresponds to left (right) circular polarization. As the magnetic field increases, A E + approaches the cyclotron energy ha~c, but A E _ decreases. Typical experimental results for these resonance positions are shown in Fig. 15). Liu et al. [70] investigated experimentally the quantized energy levels and the allowed optical transitions of a quantum dot in a magnetic field. They observed the upper branch
T. Chakraborty
28
n 16 0
..-13
i
20
~ "
30
.t
,
>
01 o
20 v
i s
i lo
i 15
/
E s
lO ,~
T-.4K _.. z%Vg=8K
,,
0
I
I.
I
1.0
2.0
3.0
4.0
B(T) F i g u r e 15 Zeeman splitting of the resonance position of quantum dots. The solid lines are obtained from Eq. (2.11). Independence of the quantization energy on gate voltage and electron number is demonstrated as inset [97][Rll].
of the allowed transitions in a parabolic potential discussed above. These authors also did a theoretical analysis of the energy levels and the dipole-allowed optical transitions and found that the experimental results agree with the calculated transition energies for hcJ0 = 1.5 and 2.8 meV. The FIR spectra observed by Demel et al. [99][R12] for q u a n t u m dot arrays with 210 and 25 electrons per dot are shown in Fig. 16. With increasing magnetic field, the resonance splits into two resonances, as expected. An interesting observation here is the anticrossing behavior of the upper mode at low magnetic fields. The appearance of a resonant anticrossing in the energy levels is believed to be due primarily to a nonlocal interaction in a single dot which is important in low dimensions. Meurer et al. [101] and Meurer [102] pointed out that FIR spectroscopy not only provides the resonant frequency but also the absorption strength, a measure of the electron population per dot. For a parabolic potential at B = 0, all absorption occurs at a single resonant frequency, as demonstrated by the early result in Fig. 15 and more recent results in Fig. 17. However, as the number of electrons in the dot increases, the absorption strength increases proportionately. Figure 18 shows the F I R transmission spectra and the integrated absorption strength vs the gate voltage in a field-effect q u a n t u m dot array with a periodicity of 200 nm. Interestingly, the absorption at low electron occupancy
Q u a n t u m dots
29
(a)
2OO
150 ! o
iO0
N ~ 210 R~16Onm
3 5O
(b)
200
150 A
! 10o Nffi25 R - 100rim
3 5o
0
2
4
6
8
10
12
14
16
B (T) F i g u r e 16 The magnetic field dispersion of resonant absorption in quantum dot structures with 210 and 25 electrons. The solid lines are obtained from Eq. (2.11) [99][R12].
increases step wise with the gate voltage, i.e., most of the l0 s dots in the array change their charge by one electron at the same gate voltage. Meurer et al. explained this unique behavior as due to the large Coulomb energy t h a t one needs to add a n o t h e r electron to a dot. In Fig. 18 one notices t h a t in order to increase the n u m b e r of electrons from ne = 2 to ne = 3, the voltage must increase by AVg = 30 inV. This value corresponds to a capacitance of C = e/AVg = 5.3 • 10 - i s F and a Coulomb charging energy, Ec = e2/2C, of 15 meV. Self-consistent Hartree calculations for q u a n t u m dot systems with nearly the same dimensions as in this experiment [58][R8] predicted t h a t the voltage intervals required to add an additional electron is AVg = 15 mV, the same as what was estimated by Meurer et al. from their experiments. In the F I R spectra of q u a n t u m dots another type of excitation mode associated with the harmonics 2coc, 3a:c,... has been observed resembling the Bernstein modes [118] seen
T. Chakraborty
30
9
i
!
+-- '
+-
+-. . . ,=. ~ o
12 e - / d o t
25eldot
....
60
60
.,.,.o" ..,.o" , 9
9
i~,9
,. l l ' l ' .O0""
E
40
.00 .+ 9
40
....
....
::~"
O
,
"""o~
....
8~
9 ..... o~ .......
20
20
9
I
0
,
I
'
:
2
!
,
IL
3
4
L
5
0
I
2
4
3
5
B(T)
B (T)
F i g u r e 17 Magnetic field dispersion of the resonant frequency in a quantum dot array with 12 and 25 electrons per dot. The lines are derived from Eq. (2.11) [116]. _+
1.00
'
1
i
"1
,
i
9
J ~
'
0.99
?
,'
i
,
I
'
I
,
l
(b)
//
9 v (v, %
i
X=llSlam
),=ll8la
, x
-,-j
i i I I
%
0.98 - - 0 . 7 3 5 . . . . . ~ / -0.730 . . . . . . . . ~ " - - ' /
I I I!
t._.,
0.97
I
i
-0.720 . . . . . . . . . . . -~-/ 0.96
Ill
-0 7~s . . . . . . . . i . . . . -" " S.;S
'
6.00
Vg
'
6.()S
(v)
'
6.10
~ :
-0.76
-0174,
-0.72
-0170
'
B (T)
F i g u r e 18 (a) FIR transmission at a fixed laser frequency of 10.4 meV and at various gate voltages for a quantum dot array of period 200 nm. (b) Integrated absorption strength vs the gate voltage for a series of spectra. The stepwise increase indicates the incremental occupation of the dots with one, two and three electrons per dot [101].
earlier in three-dimensional [119] and the two-dimensional electron systems [120,121]. In the latter case, the explanation for the existence of this mode is t h a t the dynamic spatial modulation of the charge density of a plasmon breaks the isotropy of the system and causes an interaction resulting in an anticrossing when the m a g n e t o p l a s m o n dispersion crosses harmonics nwc (n = 2, 3 , . . . ) of the cyclotron frequency. Experimental evidence of the Bernstein modes in an array of large dots [116,117] (with a period of 2>m, and about 2400 electrons) is shown in Fig. 19. The i m p o r t a n t message of the optical spectroscopy so far is t h a t in F I R spectroscopic measurements on parabolic q u a n t u m dot structures the measured resonant frequency is independent of electron number within the experimental error. These resonances are
Q u a n t u m dots
31
160-
,
,
,
,-"
,
-
.._, 120 E o 80 a'40 O0
2
4
7080/[ .......... ~ ._.
60
9r
50
g
30
'
6 B(T) "'
Y
8
+',
~'
1012 '
"
20
1000
1
2
3
4
5
B(T)
F i g u r e 19 Magnetic field dispersion of the various excitations in an array of large quantum dots. The same experimental results are shown in the top and bottom panels and in the inset with different scales. The Burnstein type of anticrossing is shown in the inset [117].
related to single-particle transition energies in a bare confinement potential [97][Rll], [98], [99][R12], [100-102]. As we shall see below, an intriguing theoretical explanation of this important result is available. For q u a n t u m dots with tailored deviations from parabolic confinement (more like a hard-wall confinement), Bollweg et al. I122] noticed that the co_ modes display oscillatory behavior [Fig. 20]. Here the experimental resonance frequencies are normalized to a
V//co~ 121
calculated dispersion coc_~_ + ~co~ - ~coc which displays no oscillations. The experimental results, in contrast, show two oscillating periods. The m a x i m a and minima are dependent on the filling factor. The maxima of frequency appears at fully occupied Landau levels, while the minima occur at half-filled Landau levels. These authors attribute these oscillations to the formation of compressible and incompressible strips at the edges of the dots. Darnhofer et al. [12a] however presented a different explanation of the observed oscillations in the co_ modes. They performed a self-consistent calculation of the equilibrium state and the FIR absorption spectrum of electrons in a q u a n t u m dot with nonparabolic
T. Chakraborty
32
180 160 140
, 1.08
E= 80
1.06
~" 60
~,, 1.04
6
40
1.02
4
20
1.00
~
0.98
........
0
0
2
4
6 B (m)
8
10
0.0
b
,o
100
ot ar y
,
~
0.1
-
8 >
IK:)N~=7.7.101 lcm-Z 2 ~I~N_1=7.0 101lcm "2 .
0.2 B1 (T "1)
I
.
~""
0
0.3
F i g u r e 2 0 O b s e r v e d dispersion of the resonant frequencies in a dot array. T h e c a l c u l a t e d dispersion for aJ+ and a~_ (with a~0 - 86 cm - 1 ) are also shown for c o m p a r i s o n . T h e co~ m o d e is the acoustic m o d e of the double-layer dot array. (b) O b s e r v e d r e s o n a n c e frequencies a~_ normalized to the c a l c u l a t e d dispersion a~ca (left scale) versus B - 1 . T h e m o t i o n of individual electrons in the dot for the a~_ m o d e is shown as inset [122].
confinement potential. They used the local-density approximation to study the ground state properties, ignoring spin degree of freedom of the electrons. At high magnetic fields, they found that the radial extension of the electron density depend on the filling factor. Under dipole excitation the electronic system probes the radius-dependent curvature of the confinement potential, leading to small filling factor dependent oscillation in the co_ mode. The results agree well with the experimental results of Bollweg et al. [122]. Lorke et al. [124][R20] employed FIR spectroscopy to study the coupling of adjacent quantum dots created in GaAs-heterojunctions where the coupling of the dots can be tuned by a gate voltage. The resonance peaks as a function of magnetic field strength are shown for the isolated dots in Fig. 21. In the region of coupling, four modes are observed at high magnetic fields (B _> 2.5 tesla). The topmost mode is a magnetoplasmon resonance, and the one adjacent to it is a confined cyclotron resonance. The lower branch is an additional edge mode, at frequencies lower than that observed for an isolated dot, which was identified as a charge moving along a boundary of about twice the length of the perimeter of a single dot. A charge moving along a peanut-shaped orbit enclosing two dots, as shown in the inset of Fig. 21 (b), seems to explain the additional edge mode. These results represent a transition from an isolated dot to a electron grid, analogous to the transition from atoms to molecules.
Quantum dots
33
I00
I
80 60
Vg=-3.
I
I
.
.
.
.
.
.
I
IV
--
O 0/.~
-
4O ~
-
2O
Ca) ,.~
=
>
0
t20 100
c~
so
~-
V =-29V 9
50 40 20
.=.
lob)
! 0
- "--o-o.-o-o__
i 1
l 2 B
I 3
I 4
._
(T)
Figure 21 Resonance position for (a) isolated and (b) coupled quantum dots. The solid lines are obtained from Eq. (2.11). The inset shows the relevant trajectories for the upper(A) and lower- (B) frequency edge modes. The dashed line corresponds to the dispersion for type B edge mode [124][R20].
Before closing this section, we should mention that collective response of two-dimensional macroscopic electron disks was studied earlier by Allen et al. [125] by measuring the absorptance of radiation transmitted normal to the surface. In the presence of an external magnetic field a split in the resonance was observed similar to what is observed in few-electron quantum dots. Finally, Kern et al. [100] prepared very large quantum dots containing 600 electrons by deep mesa etching, and measured an energy spacing of 1 meV. FIR spectroscopy of these systems showed, in addition to the two usual modes, other weak modes and anticrossing behavior. Dahl et al. [126] studied the microwave response in elliptical quantum dots. In this geometry, the plasma resonance shows a large gap at zero magnetic field, where a circular dot shows degenerate dipole modes. Lifting of degeneracy is the major effect of the anisotropic dots. A polarization-dependent higher harmonic of the fundamental
T. Chakraborty
34
Figure 22 The gate geometry with the various gates to create the quantum dot and the constrictions [135].
low-frequency mode was also observed. Our theoretical understanding of these effects will be discussed below (Sect. 2.11.1).
2.4
Transport spectroscopy
Excellent articles and reviews on transport phenomena in mesoscopic systems exist, e.g., in Refs. [127-134, 92]. Here we will briefly discuss some interesting aspects of transport spectroscopy in a mesoscopic system primarily to demonstrate that, just like the optical and capacitance spectroscopies, electrical transport measurements can provide information about the energy spectrum of a 0D system. Quantum dots coupled to two electron reservoirs by weak tunnel junctions allow one to study electron transport through the dots. Tunneling of electrons through the dots is influenced by quantum confinement as well as the charging effects. As described below, addition of an electron in the QD costs a finite charging energy and that allows one to control the number of electrons in the dot which can be changed by one at a time. Detailed reports on the transport results and their qualitative explanation can be found in Refs. [130,135,136,96]. In what follows, we briefly touch upon some of the major experimental works. In Fig. 22, we show one example of QDs created for lateral (parallel to the surface) tunneling. Here the QD is created by depleting the 2DEG, which lies 100 nm below the surface of a GaAs/A1GaAs heterostructure, underneath the gate structure. Electrons are depleted by a large negative voltage ( - 4 0 0 meV) to gates F, C,
Q u a n t u m dots
35
(b) Single-electron Tunneling
(a) Coulomb Blockade ,,,
,_,ge(N,l)
Y;["
~1
"J
. . . . . .
'
-pe(N) J
ge(N)
7 .... ,f3
eV;s
.........
Figure 23 Schematic picture of (a) Coulomb blockade (CB) [#e < ~1, ~r < ~e(N nI- 1)], and (b) Single-electron tunneling (SET) [#1 > #e(g + 1) > #r]- The source-drain voltage across the sample VDS = ( # l - #r)/e is very small [eVDs << #e(N + 1) - #~(N)].
1, and 2 forming a QD of diameter ~ 800 nm. Electron transport occurs through the quantum point contact (QPC) constrictions induced by gates 1-F and 2-F. The charging energy for such dots is estimated to be ~ 0.6 meV, and therefore, the charging energy is the major energy scale at temperatures below 4K.
2.4.1. Charging effects The discreteness of electron charge plays an important role in transport through confined regions that are weakly coupled to the leads. In a QD, charge of a single electron becomes important when the capacitance C of the dot to the surroundings is small. This leads to a charging energy of a single electron, Ec = e2/2C that exceeds the thermal energy kBT. As a result, current through the dot is a discrete transport of single charges rather than a continuous flow of electrons. Charging effects have been studied earlier in classical systems such as granular rims, small metal tunnel junctions, etc [137], but in what follows, we focus entirely on charging effects in QDs. The potential energy landscape of the Q D induced by the gates is shown in Fig. 23. In the two reservoirs at left and at the right of the dot, the states are occupied up to the electro-chemical potentials 7 #I and #r. The energy separation between the 0D states in the Q D where N electrons are localized, is E N + I - EN, which is related to the change in the electro-chemical potential 2
/~e
----
# e ( N + 1) - pc(N) --- ~ X + l
7 D e f i n e d as t h e s u m of F e r m i e n e r g y E F ( N )
-
EN
- -
EN + --~.
a n d t h e e l e c t r o - s t a t i c p o t e n t i a l eqoN.
(2.12)
36
T. Chakraborty
(a)
N+2
~
(b)
Nd~I N-1
i
l
l
' .
,,
w v
d
A e~
rm+qrm+ .....
:" ",
|
,
:
:2 e
*'C
(c)
AVg
"-
Vg
Figure 24 (a) Conductance, (b) number of electrons and (c) the electrostatic energy e~ of the dot as a function of the gate voltage Vg. Depending upon the gate voltages, the energy gap Ape can lead to a blockade in electron tunneling in and out of the dot. This is depicted schematically in Fig. 23 (a). Here electrons can not tunnel out of the dot because #1 and Pr are higher than the highest occupied level pc(N) in the QD. Electrons cannot tunnel into the dot because the resulting electro-chemical potential pe(N + 1 ) is at a higher energy level than the electro-chemical potentials of the reservoirs. Therefore, at T = 0, electron transport is b l o c k e d - the Coulomb blockade (CB), when p~ (N) < pl, Pr < P~ (N + 1). As shown in Fig. 23 (b), a change in the gate voltage (which in turn, changes the electrostatic potential ~N), pe(N + 1) can be made to line up between p~ and Pr [P~ > p~(N + 1) > Pr]. It is then possible for an electron to tunnel from the left 2DEC reservoir into the dot [pl > p~(N + 1)]. The electro-chemical potential in the dot then increases by Ape, and so does the electrostatic potential e~N+l - - e ~ N = e 2 / C [Eq. (2.12)]. Since p e ( N + 1) > #r, one electron can now tunnel from the dot to the right reservoir. Then the electro-chemical potential drops back to p~(N). A new electron can now tunnel into the dot and the cycle repeats. This process is called the single electron transport (SETR). The conductance of the QD oscillates between zero (CB) and non-zero (SETR) values as one sweeps the gate voltage. These are the Coulomb oscillations [Fig. 24 (a)]. In
Quantum dots
37
VGS VDS dd/D~s . ~ JITopSJ I[3
(a)
Vsl
777"
(b)
3 __
'
'
'
'
I
. . . .
Back
Gate
~X . . . .
\
I )
I
' 1 pm
V
'
/
(
"'
g O 0
0
I
J
I
I
I
-5
I
Gate
I
I
I
I
0
I
I
I
I
5
10
Voltage VBS(V)
Figure 25 Cross-section of a quantum dot sample (schematic). (b) Conductance versus the gate voltage. The six gates on top of the 2DEG used to define the quantum dot and the two tunneling barriers is shown as inset [138].
the CB regime, the number of electrons in the dot remains fixed at the minimum of the conductance [Fig. 24 (b)]. At the conductance peak, this number oscillates by one electron and e~ oscillates by e 2 / C [Fig. 24 (c)]. Between two conductance maxima, e~ changes by E N + 1 - E N + e 2 / C. Figure 25 (a) depicts the system used by Weis et al. [94], [95][R23], [96,138] to investigate transport spectroscopy in QDs. The dots were made by creating a negatively biased split-gate on top of a standard GaAs/A1GaAs heterostructure. Here the 2DEG lies 86 nm beneath the surface; a negative gate voltage of about Vg = - 0 . 7 V depletes the 2DEG under the gates, and as a result, a disk of electrons of diameter 350 nm containing about 10 electrons is formed. The tunneling barriers were tuned by changing the top-gate voltages, while the back-gate on the reverse side of the GaAs substrate was used to change the electrostatic potential of the QD. Fig. 25 (b) is a typical result for the conductance of the QD as a function of back-
T. Chakraborty
38
gate voltage VBs. The observed periodic series of sharp peaks is largely explained by the qualitative picture of Coulomb blockade oscillations presented above. Similar results were also reported earlier by McEuen et al. [87][R13] and Johnson et al. [90][R22].
2.4.2.
The diamond diagrams
Diamond digrams, or charging diagrams provide important information about tunneling spectroscopy of ground and excited states of quantum dots [90] [R22], [91, 94], [95] [R23], [96,138]. The diagrams are generated by plotting the differential conductance dI/dVDs as a function of the back-gate voltage VBs for a range (-3 mV to +3 mV) of sourcedrain voltages VDS Fig. 26 (a). In this linear gray-scale plot, white regions correspond to dIDs/dVDs below -0.1#S and the dark regions correspond to that above 2#S ( S = l / O h m ) . The main structures visible in Fig. 26 (a) are plotted schematically in Fig. 26 (b). In the limit liDS --~ 0, one observes conductance resonance, as described above. Increasing IVDsl, the range of VBS where transport through the QD occurs, is broadened linearly with IVDsl, enclosing in between the diamond-shaped CB regions. Along the VDS ~ 0 axis, the electron number N changes to N + 1 when the adjacent diamond-shaped zero-current region touch, as indicated in Fig. 26 (b). The transport regime - the light gray regions of Fig. 26 (b) - can be identified in the VDS -- Vcs plane by extrapolating the boundaries between transport and blockade regimes. Within the SET regimes, additional transport channels through the QD [dark gray regions in Fig. 26 (b)] are also visible at finite VDs. During transport as liDS --~ 0, the QD changes between the ground states of, say, N and N + 1 electron system. At finite VDS, excited states for both N and N + 1 electron systems are also accessible, thereby providing new tunneling channels. This can happen in two ways: either an excited state of the (N + 1)-electron system becomes accessible to put the (N + 1)-th electron to the N-electron dot, or the QD is left in an excited state of the N-electron system, and the (N + 1)-th electron leaves the dot. Within the SET regime, one also observes negative differential conductance [white regions in Fig. 26 (a) and dashed lines in Fig. 26 (b)]. Those peaks shift parallel to the boundaries of transport and of the CB regime (upper edge of a diamond). It has been suggested (see Sect. 2.9) that spin selection rules are responsible for these features.
2.4.3. Magnetic field effects In order to extract the magnetic field dependence of N-electron energy levels in a QD, McEuen et al. [87][R13], [88][R14], [89] investigated transport spectroscopy in the presence of an external magnetic field perpendicular to the plane. Theys started with a 2DEG
Q u a n t u m dots
39
F i g u r e 26 (a) Differential conductance dIDs/dVDs as a function of gate voltage VBS for a range of values of VDS. (b) Sketch of main structures visible in (a). Here light gray areas are regions of single-electron tunneling, i.e. the regimes where the number of electrons can energetically fluctuate between for instance N and N + 1. Dark gray areas are the regimes where the number of electrons can energetically fluctuate between, for instance, N - 1, N and N + 1. Dashed lines indicate the regime of negative differential conductances and the dotted lines correspond to suppressed conductance [138].
in G a A s / A 1 G a A s heterostructure using electrostatic gates to confine and adjust the density of the 2DEG. A negative voltage applied to a lithographically-patterned split upper gate creates the QD. A positive bias applied to a lower gate adjusts the electron density. The conductance peak positions in Vg, as observed in their experiments [Fig. 27 (a), inset] are a direct measure of the addition energy - the energy to add an electron to the dot" Vg - #e(N)/ec~ (o~ ~ 0.4 is constant). The evolution of the position of a particular conductance peak with increasing field is shown in Fig. 27 (a). From the peak positions, these authors constructed the level spectra [87][R13], [88][R14], [89] which are shown in Fig. 27 for 3 >_ u >_ 2 (b) and for u _< 2 (c). These spectra were obtained by subtracting a constant [AVg - 1.175 mV in (b) and AVg -- 1.35 in (b)] between successive peak position curves in the experiment. McEuen et al. noted t h a t Coulomb interaction dominate the spectra and simple noninteracting models fail to account for the level crossings observed in the spectra. On the other hand, a self-consistent model t h a t includes both Landau-level
T. Chakraborty
40
316.5
,:.
i-l'~ ~
"
'
>
2 ~- 2(20EG) J
,.r'," $& i,o, L
314.5
.
.
.
1.0
.
~ = 4 (2. DEG) .
.
' . . . . . . .
1.5 2.0 B (tesla)
-;
2.5
3.0
0.30
0.5
--
(c)'
0.25
0.4 ;>
0.20
0.3
0.15 >~ 0 . 2 ~5
0.10
0.1
0.05
0
1.5
1.7
1.9 2.1 B (tesla)
0 2.3 2.6
2.7
2.8 B (tesla)
2.9
F i g u r e 27 (a) Position of the conductance peak in back-gate voltage Vg as a function of the magnetic field. The measured filling factor of the 2DEG is also indicated. Inset: Conductance versus Vg at B - 2.5 tesla. Also shown are the measured addition energies in the regions, (b) 3 _> u > 2 and (c) ~, < 2 [88][R14].
quantization and the proper Coulomb interaction agrees well with the observed spectra. In contrast to the experiment of McEuen et al. where the magnetic field is directed perpendicular to the plane of the dot, Weis et al. [95] [R=a], [96, ~3s] performed experiments with magnetic field directed parallel to the current direction (parallel to the electron plane). The result (Fig. 28) indicates that the q u a n t u m numbers of the ground state energy of the QD changes with an increasing magnetic field. This leads to the observed amplitude modulation of the conductance peaks in Fig. 28.
2.4.4. Electron turnstiles Making use of the Coulomb oscillations described above, Kouwenhoven et al. [9] realized a turnstile operation for electrons in a QD. This is a class of device which can transfer
Q u a n t u m dots
41
O"
-15
A
I-v
0
rn
15 -5
0
5
10
VB(V) F i g u r e 28 Conductance versus the gate voltage for different values of the magnetic field B between - 15 tesla and 15 tesla [138].
individual electrons around a circuit at a well-defined rate. The process of clocking electrons through a QD one by one at a well-defined rate is shown schematically as follows [Fig. 29]" (a) Initially, both barriers are high and hence the probability of an electron tunneling from left contact into the QD is negligibly small. (b) The left-hand barrier is lowered: one electron can now tunnel into the QD. Once it has, the charging energy increases the chemical potential of the QD above that of the left-hand contact, making it energetically unfavorable for another electron to tunnel into the dot (the Coulomb blockade). (c) After the left-barrier is raised the electron is
trapped in the QD.
(d) W h e n the right-hand barrier is lowered, the trapped electron can tunnel into the right-hand contact. After the dot is discharged and the right-hand barrier is raised again, we are back to step (a) and the system is ready to repeat the process. If the frequency at which the
T. Chakraborty
42 (b)
(a)
~
_N+I
--~O ~tr
e~s
.............
(e
(d) ----_
F i g u r e 29 The four cycles of the turnstile process required to clock a single electron.
electrons are clocked through the QD is a~, then the current flowing is I = ecz, when only one electron passes through the dot during each cycle. Experimental results of Kouwenhoven et al. [9] are shown in Fig. 30, where the I - V curve for an oscillating barrier turnstile excited at a frequency of 10MHz is shown. There are clear current plateaus at values given by In = necz, where n is an integer. Increasing the source-drain bias voltage, one can make more than one electron to tunnel in each cycle. Kouwenhoven et al. obtained current plateaus corresponding to controlled transport of up to seven electrons per cycle. In principle, the turnstile can be used as a current standard just as the q u a n t u m Hall effect is used as a resistance standard [139]. One important criterion for such metrological applications is the accuracy of current quantization. One fundamental limitation to the accuracy is a leakage current that arises from macroscopic q u a n t u m t u n n e l i n g this allows an electron to move from source to drain via a virtual tunnel path through forbidden states in the dot. Such a leakage path can, however, be suppressed byreplacing the tunnel barrier with multiple tunnel junctions. On a fundamental level, one could perhaps speculate that this device, under suitable conditions, could be used to transport fractionally-charged Laughlin quasiparticles [140]. Single-electron t r a n s p o r t in the fractional q u a n t u m Hall regime is still an uncharted territory [141].
Q u a n t u m dots
43
.
.
.
.
.
.
.
lol
.
.
.
.
.
.
.
.
.
.
.
I 1 t I
.<
-5
-10
...... -4
_
-2
.
.
,
.
.
0
.
.
.
L
1,
2
4
-
-
V (mV) F i g u r e 30
2.4.5.
Current-voltage curve for a turnstile at a frequency of 10MHz [9].
Photon-assisted
tunneling
Tunneling in the presence of an external microwave field, photon-assisted tunneling (PAT), has received increasing attention in recent years [142-148]. PAT allows investigation of time-dependent tunneling phenomena related to 0D levels. By absorbing or emitting photons from the high-frequency signal during tunneling, electrons can reach the normally inaccessible energy states. In this process, electrons overcome the Coulomb gap and tunnel from the left reservoir in a QD by absorbing discrete photons of energy hu from the microwaves field, see Fig. 31 (a). If the subsequent tunnel process is from the QD to the right reservoir then PAT contributes to the current. The electron-turnstile device described above, also produces frequency-dependent currents, but the photon energy is much too small at MHz frequencies to be energetically important. Enhancement of tunneling has indeed been observed experimentally when the photon energy corresponds to the energy difference between the incoming electron and an available state of the QD into which it tunnels. The PAT current is observed as a shoulder on the Coulomb oscillation current peaks Fig. 31 (b). PAT can thus be used as an additional spectroscopic tool to investigate the energy levels of a QD. It is expected that this phenomenon can also help the development of highly sensitive microwave detectors [149,150].
T. Chakraborty
44
(a)
(b)
~i
,
Wt~ 0
:: .
,.~ ~
i ,,,,iii ,
-
~
_
~.~_
1
~.._
,,,
--~ I
-';', ,,,,
-
5
0
~
,,
-32
-31 -30 GATE VOLTAGE (mV)
-2~
t
F i g u r e 31 (a) Schematic illustration of the PAT process. The solid lines are the occupied levels in the QD, while the dotted lines are unoccupied levels. (b) Observed current versus gate voltage for a QD irradiated by microwave at three frequencies. The solid lines show results for increasing power and the dashed lines are the results for current without microwaves [145].
2.4.6.
Vert ical tunneling
In a vertical QD device the current flows vertically with respect to the heterostructure layers. In these systems vertical confinement of electrons are provided by the various heterostructure layers. The lateral confinement is provided in part by lithographically etching out a pillar in a double-barrier heterostructure. For the purpose of single-electron transport studies, these devices have the advantage over the lateral QD devices because here the contribution to the transport current begins already with the first electron in the dot. In contrast to the planar QD devices which have tunable tunnel barriers t h a t are only a few meV high and ~100 nm long, vertical QDs have essentially fixed tunnel barriers t h a t are typically high (a few hundred meV) and thin (typically ~ 1 0 n m ) . In the case of lateral QDs the tunnel barrier increases with decreasing electron numbers in the dot. This means t h a t even in the absence of a Coulomb barrier, one gets a unmeasurably small current [92]. Vertical dot structure is therefore the best way to investigate the properties of a few electron system via electrical measurements. An example of a vertical QD device used for conductance spectroscopy is sketched in Fig. 32. Details of the fabrication process of such structures can be found in [92,151]. The most straightforward measurement one performs with these devices is the simple I - V curve, which exhibits non-ohmic features with fine structures related to the energy spectrum of the QD. In t h a t respect, here one does a type of conductance spectroscopy
Quantum dots
45
V
!i
iii
li!!
GaAs:Si 5 nm 10 nm
bb= 6,7,8,9 nm
li!l
AIGaAs GaAs AIGaAs GaAs:Si
Figure 32 Cross section of a double-barrier tunneling device used for single-electron spectroscopy. The dashed lines correspond to the depletion layer which confines the electrons to the center of the pillar [64].
because tunneling is enhanced when an available energy level of the QD aligns itself with the Fermi level of one of the contacts. If the barriers are asymmetric and we inject the electrons from the transparent side, electrons accumulate in the QD [63, 65,152-155], and the steps in the I - V curves provide a measure of the addition spectrum. When, on the other hand, electrons are injected through the less-transparent barrier, charge does not accumulate in the dot and structures in the I - V curve correspond to true single-electron spectrum. S u e t al. [63,152] investigated magnetotunneling in double-barrier resonant tunneling nanometer device in the single-electron charging regime. Their I - V results are depicted in Fig. 33 for the case when the collector barrier is less transparent than the emitter barrier and an applied magnetic field parallel to the tunneling direction, i.e., perpendicular
46
T. Chakraborty
"7.5
"i
d. ~ T ----.---f"
180
200
220
24.0
260
v (my) Figure 33 magnetotunneling I - V curves, offeset vertically by 15 pA. Each current step corresponds to an increase of electron in the well by unity [63].
to the barriers. Several steps are observed in the I - V curves which reflect single-electron charging of the dot. As mentioned earlier, each step in the "staircase" corresponds to an increase of the number of electrons by unity, starting from zero. The change in energy required to add an N - t h electron in the dot is (AE)N - EN - E x - 1 , where EN is the total energy of the many-electron state in the dot. This corresponds to the voltage VN at which the Nth step appears. Therefore, A V N -- (VN+I -- VN) corresponds to ( A E ) N + I -- (AE)N = EN+I -- 2EN + F-,N-1. Hence, a step width A V N represents a change in energy needed to add the (N + 1)-th electron in the dot. The magnetic field dependence of the voltage extent AV1 - c~ ( E 2 - 2El) and AV2 - c~ ( E 3 - 2E2 + El), where c~ is the voltage to energy conversion coefficient [63] are shown in Fig. 34. Observation of cusps in AV1 and A V2 at the same value of magnetic field was interpreted by S u e t al. as indication of spin singlet-triplet transition of the two-electron state confined in the QD. Using the device structure sketched in Fig. 32 similar results for spin transitions were
Quantum dots
47
20
I
!
i"
i
"!
I
'"1
I ..... ~ ....... 1
i '
T
"';'
ooooOO~
'
oo
15
E
%
10
+#~
<~
++++++++++++
o AV1 t
+ AV2 ~
0
1
2
i
I
4-
i
I
6
i
1
1
8
10
I
I
9
12
I
/
14
B (tesla) Figure 34 The voltage extent of the current steps which corresponds to charging of the well by one extra electron versus the magnetic field [63].
reported by Schmidt et al. [64,65]. These authors employed single-electron tunneling spectroscopy to investigate the many-particle ground state, in contrast to the singleparticle spectrum of Sect. 2.1.1. observed by Schmidt et al. in a similar type of device. In the present case, however, the thin barrier is the emitter and the thick barrier is the collector [64]. The I - V staircase as a function of the magnetic field oriented parallel to the current direction and a gray-scale plot of the differential conductance as a function of bias voltage and the applied magnetic field are shown in Fig. 35. The lowest curve in Fig. 35 (b) corresponds to the ground state energy of a single electron in the QD. The nonmonotonic behavior of the other curves is attributed to transitions in the ground state energy between states of different angular momentum and spin. In this respect, the results correspond to the capacitance measurements of Ashoori et al. [76] [R21]. Finally, vertical quantum dots containing a tunable number of electrons starting from zero were employed by Tarucha et al. [48-50], [51][R33] to extract information about the many-particle ground states via single-electron transport, as already mentioned in the beginning of this chapter. Their vertical QD is a sub-micrometer pillar fabricated in an In/A1/GaAs double barrier heterostructure. Here the lateral confinement originates from side-wall depletion that is controlled by a "side gate" surrounding the structure, The number of electrons in the dot was controlled by varying the voltage on this gate.
T. Chakraborty
48
-0.8
-o.8 "--
-0.4
-0.2 0.0
-40
-50
-60
-70
v (mV) -60 .
-50
9
.
,
~
.,~
~
-40 i
=
-30
.... 0
(b) 9
,
2
9
4,
9
6,
''
8,
9
"/ 10
9
1 ,2
9
1'~4
9
1 i6
B (Tesla)
F i g u r e 35 (a) Magnetic field dependence of the I - V staircase. The curves are plotted with a vertical offset (step 0.2 tesla). (b) Magnetic field dependence of the differential conductance in a gray-scale plot [64].
The cross section of the q u a n t u m dot device where Coulomb oscillations at zero magnetic field was observed by Tarucha et al. is shown as inset in Fig. 36. We have already indicated in Sect. 2.1.2 that the single-particle states (Fock-Darwin states) of a circularly symmetric dot are degenerate. This level degeneracy and consecutive filling of each set of degenerate states causes the shell structure for 2, 6, 12, 2 0 , . . . electrons. Additionally, parallel filling of electrons amongst half-filled degenerate states occur in a shell at electrons numbers ne = 4, 9, 1 6 , . . . due to an exchange effect (Hund's first rule). The addition energy, AEc = #(he + 1 ) - #(rte), where the electrochemical potential for
Q u a n t u m dots
49
(~"'/" ~
2O Vd - 150 IJV
~
50mK
"~9,~.~~---.L~
/(,
<
.... H
II:
oL _
4
"
Dot
I///~DBH 1//4--- Side Gate
'
9
161
0
-1.5
-1.0
-0.5
Gate Voltage (V)
Figure 36 Observed Coulomb oscillations in the current vs gate voltage at B - 0 tesla in a gated quantum dot device by Tarucha et al. [50], shown as inset.
an he-electron dot corresponds to the position in energy of the current peak in Fig. 36 can now be readily obtained. In fact, the addition energy corresponds to the spacing between the N + 1-th and N - t h current peaks. Plotted in Fig. 3T, it has large maxima for N - 2,6 and 12 and also relatively large maxima for N = 4,9 and 16 [49, 50]. Interestingly, breaking of the circular symmetry was found to destroy this shell structure and will be discussed in Sect. 2.11. Figure 38 (b) depicts the position of current peaks in the presence of an external magnetic field. Positions of the first three peaks show monotonic behavior with respect to the applied field, while the others show oscillatory behavior with B that increases with the electron number. In general, current peaks shift in pairs with B. The spacing between the peaks is roughly constant for odd N and vary strongly with B when N is even. Most of these features are explained in the framework of single-particle FockDarwin states and the constant-interaction model [50] discussed in Sect. 2.3.1. For spinless electrons, each state is twofold degenerate. The degeneracies at B = 0 are lifted as B > 0 (Sect. 2.1.1). The corresponding addition energy for even N is strongly magnetic field dependent. On the other hand, addition energy for odd N is determined only by the
T. Chakraborty
50 I '
'
I
I
I
N=2
6
-
<
00
5
I
I
10
15
.......
I
20
Electron number N
Figure 37 Addition energy vs electron number derived from the Coulomb oscillations in Fig. 36. effect of electron-electron interaction that is responsible for lifting the spin degeneracy. This should naturally lead to a pairing of the current peaks. The good agreement between Fig. 38 (a) and Fig. 38 (b) is an indication that the CI model works reasonably well in the few-electron regime. However, the model must break down when exchange interaction becomes important, and is seen to be the case for filling of electrons in the second shell. Magnetic field dependence of the third, fourth, fifth and sixth current peaks, i.e., peaks that belong to the second shell, is displayed in Fig. 39. Here one clearly sees pairing of third and fourth peaks and fifth and sixth peaks for B > 0.4 tesla. By following the evolution of the respective pairs with magnetic field, Tarucha et al. identified the quantum numbers (n, l) = (0, 1) with antiparallel spins for the low-lying pairs and ( 0 , - 1) with antiparallel spins for the higher pair. Interestingly, this pairing is rearranged for B < 0.4 tesla. In this field range the third and fifth peaks, and fourth and sixth peaks are paired. Therefore, as B is increased from 0 tesla and exceeds 0.4 tesla, the forth electron undergoes an angular momentum transition from 1 = - 1 to 1, while the fifth electron undergoes a reverse transition, i.e., from l = 1 to - 1 . Recently, Eto [173] has reported results for the magnetic field dependence of addition spectra using exact diagonalization scheme that closely duplicates the data in Fig. 39.
Q u a n t u m dots
51 4
I'
(a)
50
I"
'
I
-
(b)
i
> 40
E 3O ,-
..... ~ 1
-.
' -
L
~c 20
o
.......
-
-4
-
10 I~ " ~ I ,,,
0
I
,
~
--
-NS 0
I
2 B (Tesla)
3
o
B (Tesla)
Figure 38 (a) Energies of a parabolic dot in the constant interaction model (b) Gate voltage positions of current peaks as a function of the magnetic field [50].
2.5
Many-electron
systems
Before we discuss the properties of interacting electron systems, let us briefly mention some results for non-interacting many-electron systems. Geerinckx et al. [69] studied theoretically, the electronic states, energy levels and dipole-allowed optical transitions for six non-interacting electrons in a quantum dot in the presence of a perpendicular magnetic field. They compared the results for hard-wall confinement with those for a parabolic confinement potential. In the hard-wall case it was found t h a t there are several transitions with different energies allowed. This contrasts sharply with the parabolic case in which only two transitions are allowed. Among the many transitions in the hard-wall case, however, only a small number of transitions have sufficient oscillator strength to be observable. As the magnetic field is increased, the resonant frequency approaches the two-dimensional result much faster than that for the parabolic confinement potential. Almost sixty years after the pioneering work of Fock [60][R5] on a single electron in a parabolic confinement potential and in an external magnetic field, Maksym and Chakraborty [30][R7] first introduced the electron-electron interaction in that system.
T. Chakraborty
52
-1.2
> v
O
c~.13 m
0 >
O
-1.4
0
0.5
1.0 1.5 B (Tesla)
2.0
Figure 39 Behavior of the third-to-sixth current peaks in a magnetic field. Spin configurations for electrons in the second shell and the angular momentum quantum numbers are indicated in the figure [50].
In order to calculate the interacting electron states, we assume that the magnetic field is strong enough to keep them spin polarized. This assumption helps us to unambiguously study the interplay between confinement and interaction [30][R7]. (The effect of spin will be discussed later.) In the spin polarized electron system the Hamiltonian is ne
E
ne
1
9 2E 2 1 e2 ~ (p~ + ~A~) ~ + ~-~ ~0 ~ + 2-J - 2m* i=l i=1 "
1
Ir~ - rjl
(2.13)
where e is the background dielectric constant. We ignore the neutralizing positive background present in real systems. For an infinite system this cancels the divergence caused by the Coulomb repulsion but for a single dot the matrix elements of the Coulomb interaction are finite. For a periodic array of dots with a large spacing the background
Quantum dots
53
cancellation merely shifts the energy levels for the single dot by a constant. In the following we present explicitly the interaction matrix elements for quantum dots that we have to evaluate numerically to obtain the electronic properties [30][R7], [106] [R15 of interacting parabolic quantum dots.
2.5.1.
Interaction Matrix elements
In order to derive the interaction matrix elements needed for the exact diagonalization scheme, we begin with the single-electron wave functions derived in Sect. 2.1.2
~ , ( x , 0) = where x =
~/b ~
2~eg (~ + Ill)!
e-il~
(2.14)
r and
(~ + IIL)! ~ ( ~ _ ~)!(lll + ~)!~! X 2t~
L~t(x~) = Z ( - I ) t~--0
The wavefunction can be formulated in a more convenient form, n
@nl(X, O) -- Cl (nl) E C2(7Z/, N)e-ilOe--}X2x2~+[l[ t~--O where the coefficients are b n! 27rg2 (n ~-l/l)!
Cl(nl) --
;
(~ + Ill)!
C2(nl, t~) - (-1) ~ ( n - ~)!(Igl + ~>!~!
For the interaction potential we write the Coulomb interaction in the form e2
V([rl-r21) =
~lrl - r21
e 2 J" 27r ik.(rl-r2)dk. (27r)26 ~-~e
(2.15)
The interaction matrix elements are then e 2 47r2g4 [ b Anln2nan4 = 511+12,13+14 g---~ b2 L27cg~" Ill21314 6--
hi!
!
]2
n4!
•
2~e~ ( ~
+ II~l)!
2~e0~ ( ~
+ Izzl)!
27cg~ ( n 4 -f-1/41)!
!
]2
T. Chakraborty
54 nl n2 n3 ~4 1 X E E E E [/'~1 -3L /'~4 -4-- 2 ( ~c1=0 ~2=0 ~c3=0 ~4=0
(--1) ~1+~4
(Ttl
IiI + I/4]- k)]'. [~2--t-
/'63-t-
1 ~(I/21 + ll31- k)]' .
-~-l/~l)!(n4-I-1/41)!
~1!~4!
(~1 -- ~Cl)!(l/ll-I-t~l)!(Tt4-- ~4)!(I/4l-I-/'~4)!
(_1)~+~3
(~2 Jr IZ=I)!(~3+ IZzl)!
(Tt2 -- t~2)!(I/21-I-/~2)!(Tt3-- ~3)!(1/31-I-~3)! /%2!/~3! 1 t~l -~-t~4-}-~(lll I+ IZ~I-- k) 1 [~1 -t- ~4 -t- ~(I/11 + 114I+ k)]' 1 . . [~1 -I-~4 -I-g(l/ll-I-IZ~I-k)- ~]'(kJr ~)' s--0
E
-4- ~(IZ=I + Ilzl + ] [~ + ~z + ~(IZ=l Jr [/31- k)- t].,(k -4-t),. [~2 nL ~3 1
E t=0
(-1) s+t r ( k + s + t + 3 ) 8!t!
1
2 k+s+t+l
'
where F(x) is the G a m m a function. The formula given above works very well for small values of the Fock-Darwin level index (NFD < 5), defined as NFD = n + ( I l l - 1)/2,
(2.16)
but even for moderately high values one runs into numerical problems. The difficulty lies in the severe roundoff errors one accumulates due to cancellations between successive terms of alternating sign. The literature contains some other ways to evaluate the Coulomb integral where we can avoid the numerical problem alluded above. One such approach is due to Stone et al. [156] which is based on an integral representation for 1/r 1 rl-r21 and the Taylor expansion of
e 2u2rl"r2
=
.
2
Zoo due-u2(rl-r2)2
In the lowest Landau level, the single-particle states
are r
_
1
r "~ e -
r 2/4 e imO
v/27r2mm! and the Coulomb matrix elements (in units of e2/t~0, where t~ - hc/eB) are
r
(2.17)
Tt,~) -- / dr I dr2 0 ~+k t (r1)r ~-k(r2)r2---[(~m(rl)~n(r2) lrI 1 _
Q u a n t u m dots
55 (2m + 2n + 1)!! O<2
• E
(n + p)!(m + k + p)!
p!(k+p)!
p=0
(2k + 4p - 1)!! 2k4P(2p + k + n + m + 1)!" (2.18)
This expression has the advantage that all terms in the series in Eq. (2.18) are positive and therefore the roundoff error discussed above is avoided. The convergence is, however, very slow.
Many-electron spectra
2.5.2.
The eigenstates of the system described by the Hamiltonian Eq. (2.13) are eigenstates of the total angular momentum, which is conserved by the electron-electron interaction. They can be classified by a q u a n t u m number J, which is the sum of single-electron 1 values. The states are calculated by numerically diagonalizing the many-electron Hamiltonian. In the following, we provide some technical details regarding the calculation of the eigenstates.
Basis states- The problem was formulated using second quantization. Let us recall that the single-particle states are a~, l O} -- ~Pnl(r) -where
2
a
n!
~-(~ + Ill)!
exp ( - i l O - (c~r)2/2) (ar)IL~ j ( ( a r ) 2 ) ,
~ .~,/h] ~
- [(L~o~ + ~c/4) . In order to construct the basis states, the number of electrons ne, total angular moment u m J = ~ 1 and the m a x i m u m of Fock-Darwin level indices Ntot = ~ NFD in the dot are to be fixed first. As the angular m o m e n t u m increases the number of states belonging to a certain Fock-Darwin level also increases. For example, for four electrons and J = 6 and total FDL index Ntot = 0 or Ntot = 1, the basis consists of twelve states
a~--t%1,lla~?%2,12a~_y~3,13a~n4,1410 ), where q u a n t u m numbers (nl, ll)(1%2,/2)(Tt3,/3)(Tt4,/4) have values for the lowest FDL
(Nto~ =0) (0,0) (0, 1 ) ( 0 , 2 ) ( 0 , 3 )
T. Chakraborty
56 and for the second F D L (Ntot = 1)"
(0,0) (0,0) (0,0) (0,0) (0,0) (0,0)
(0,I)(0,2)(0,3) (0,I)(0,3)(1,2) (0,I)(0,4)(I,I) (0,I)(0,5)(I,0) (0,2)(0,3)(I,i) (0,2)(0,4)(1,0)
(0,-I) (0,-I) (0,-I) (0,-I) (0,I)
(0,0)(0,i)(0,6) (0,0)(0,2)(0,5) (0,0)(0,3)(0,4) (0,I)(0,2)(0,4) (0,2)(0,3)(i,0)
To give an idea of the size of Hamiltonian matrix one needs to deal with: for four electrons restricted to two lowest Fock-Darwin levels when the total angular m o m e n t u m is J = 22, there are 422 basis states and 24346 off-diagonal nonzero m a t r i x elements. Similarly, if we have five electrons at B = 7.5 tesla in two lowest Fock-Darwin levels the ground state occurs when J = 30 and we have 1669 basis states and more t h a n 140 000 off-diagonal nonzero m a t r i x elements in the Hamiltonian matrix. The rapid increase of the n u m b e r of matrix elements is a m a j o r problem t h a t severely restricts the n u m b e r of electrons t h a t can be studied in a q u a n t u m dot using the numerical approach. How do Fock-Darwin levels enter into the computations of the energy eigenstates? Let us begin with the one-particle energy:
Enl
-
(2~ + IZl + 1)(h2co~/4 + h2co02)89-llhcoc
=
(2NFD + 1 + 1)B - 1 C ,
where we denote B - ( h W c2/ 4 + 2 h2co02)89and C _ 1 hcoc. If we assume a basis state of ne non-interacting electrons, the sum of the single-particle energies is then Etot = 2NtotB + Ltot(B - C) + n e B where Ntot - }--]~i~l IVYD is the total Fock-Darwin level index of the m a n y - b o d y state and J - ntot = }-]~=1 ne li is the total orbital angular m o m e n t u m . As described earlier, the Fock-Darwin levels are degenerate when coc >> coo, i.e., at high magnetic fields. If we now restrict ourselves to Ntot = 0 and choose a certain value for Ltot, then we can construct basis states such t h a t the system is in the lowest Fock-Darwin level. If we consider the case of Ntot = 1, then the system is in the second level and so on. Introducing the Coulomb interaction between electrons, degeneracy of the Fock-Darwin levels is lifted and energy bands are formed [30][R7]. We should note that, Ntot is not conserved when inter-electron interactions are switched on, but in order to restrict the basis we need to limit the m a x i m u m value of Ntot.
Quantum dots
57
ne=3 B:10T ~-~40
M
>.
30
2O
u I
I
I
I
I
I
I
l
1
ne-3 B:2T /
~-~40
>
M
30
q
2O 1
I
I
I
1
1,
1
l
0
1
2
3
4
5
6
7
,,
I
8
J Figure quantum
40 Energy levels as a function of Y for three electrons in a GaAs dot for magnetic field B - 2 tesla and B - i0 tesla [30][R7].
parabolic
Energy spectra- The energy levels of a parabolic quantum dot are shown in Fig. 40 for three electrons and in Fig. 41 for four electrons. They were first calculated by Maksym and Chakraborty [30][R7] using the parameters appropriate to GaAs and w0 = 4 meV. The energies are plotted relative to what would be the lowest Landau level, that is, the constant of h (1~w c2 + w02)89per electron is not included. Here the total energies are plotted against Y at magnetic fields representative of low- and high-field behavior. As seen in these figures, there are always two sets of broadened levels separated by a gap. In the limit of zero confinement these would be the lowest two Landau levels. The general trend is that the energies increase with J because the single-electron energies increase with 1. The main difference between low- and high-field behavior is the ground state angular momentum. At B = 2 tesla, the ground state appears at the lowest available J, that is, the smallest angular momentum compatible with placing all the electrons in NFD = 0
T. Chakraborty
58
70 >
r
60
pa
50
=4 B = 8 T
_ n
-
-
_-__:
iRBm
40
30 70
m I
I
ne =4
1
I
1
-7.
l
1
B-2T
60
~
5O
m
~
~--~m"~
m
I
4O 30
B
' 0
'2
'
'
'
'
4
6
8
10
-
'
12
' 14
J Figure 41 Energy levels as a function of J for four electrons in a GaAs parabolic quantum dot for B - 2 tesla and B - 8 tesla [30][R7].
states. As an example, the three electron system has the ground state at Y = 3. For the non-interacting system, the ground state would have the lowest available J provided B is so high that only NFD = 0 is relevant. The interaction, however, causes the ground state J to increase with B. This effect is caused by the interplay of the single-electron energies and the interaction energy. In the following, we consider a simple picture where only the NFD ---=0 states are taken into account. Then the single-electron contribution to the energy (relative to the lowest
Landau level) is simply h ( 8 8 + a~02)1 - ~a~ 1 g. The contribution from the interaction is determined by numerical diagonalization of the Hamiltonian. In Fig. 42, we show these two contributions together with their sum. The single-electron contribution increases linearly with g because electrons in high angular m o m e n t u m states see a higher confining potential. On the other hand, the interaction contribution decreases because electrons with higher angular momenta move in orbitals of larger radii, thereby reducing their Coulomb energy. The net result is that the total energy as a function of
Q u a n t u m dots
59
ne-3 B - I O T no Landau level -mixing
... / /o.. ,.I
9 - qj/O %%
2O -
.'*\ 9
\./
i""
9
\.," 9 'S
o-o
.'"
total
o=o
E
/\
./"
-
\
UJ
/e 9
0%
>-
10 -
\
.'" -,/~..,.'_..
~176
/
of
~'~
single eleclron
,,"/ "" "'! ""/" " "I" " -
/
.""
interoction
/
I
0
.........
10
1
I
20
3 Figure 42 Contributions to the total energy as a function of J. Arrows indicate the steps in the interaction energy [30][R7].
J has a minimum. At low B this happens at the lowest available Y because the singleelectron energy increases steeply with Y. At high fields, the increase is much weaker so the minimum occurs at a higher J value.
Magic numbers- One very important result here is that the ground state of electrons in a magnetic field occurs only at certain magic values [30][R7] of angular m o m e n t u m (and also spin, to be discussed later) which are dependent on the number of electrons. At these magic J values, which satisfy the relation 1 (no - 1) + jn~ J - ~n~ where j is an integer, there are basis states in which electrons are kept apart very effectively. The ground state always occurs at one of these J values and the competition between interaction and confinement determines the optimum J. For ne _< 5 the basis states have all the electrons in a compact cluster in the zeroth Landau level. T h a t is,
60
T. Chakraborty
all the occupied single-electron orbitals have n = 0 and are adjacent in angular momentum space. For ne > 5, when (he - 1)-fold symmetry occurs, the basis states have one electron with 1 = 0 and the remaining electrons in a compact cluster. It was shown in Ref. [157][R17], [158,159] that the Coulomb energy of the compact cluster states is reduced by an exchange contributio'n whose magnitude is very large. As a result, the total energy is reduced for these favored values of J. Similar observations were also made by other authors [160], [161][R18]. R ~ n ~t al. [lao] pointed out that the quantummechanical symmetry (Pauli principle and rotational invariance) plays a crucial role in determining the states and hence the magic numbers. These authors also noticed that the magic numbers are insensitive to details of the dynamics of the system. To summarize, the magic numbers occur because the magnetic field compresses the wavefunction of the system and increases the Coulomb energy. At certain critical fields the system can reduce the energy by making a transition to a new ground state which has a larger lateral extent and a higher angular momentum. As the magnetic field is increased we see a series of abrupt changes in total angular m o m e n t u m and system size. The selected values of angular momentum can be explained in terms of the symmetry of the minimum of the combined confinement and interaction potential [109,158,159]. Imamura et al. [162] studied the quantum states of vertically coupled dots in a strong magnetic field. They found that electron correlations in the double dot lead to a series of angular m o m e n t u m magic numbers which are different from those of a single dot. These results correspond to ground states dominated by the interlayer electron correlation. These authors proposed that the magic numbers can be investigated experimentally in vertically coupled dots. The generalized Kohn theorem, to be discussed below, does not hold for two vertically coupled dots with different confining potentials. I m a m u r a et al. surmised that the jump of angular momentum from one magic number to another should show up as discontinuities in the FIR absorption energies of the double-dot versus the magnetic field. Ruan and Cheung [163] studied a system of vertically coupled parabolic QDs, each containing two electrons. The electrons interact via the Coulomb potential. Numerical diagonalization of the Hamiltonian for this coupled system revealed an extra sequence of ground states as a function of increasing magnetic field not expected in uncoupled dots. As discussed above, the study of the interacting electron states in q u a n t u m dots revealed a wealth of very useful information. Not surprisingly then, these systems have attracted a large number of workers and as a consequence a large number of publications [164-177] on variations of the work initiated by Maksym and Chakraborty exist in the literature.
Generalized Kohn theorem- The magneto-optical results discussed in Sect. 2.3.2 revealed
Quantum dots
61
that the FIR excitation energies are independent of electron number. That finding is, of course, quite surprising. It means that the electron-electron interactions described above do not influence the spectra at all. It turns out that the experimental results demonstrate, albeit in a different situation, a variation of the original Kohn theorem [178]. This theorem states that, in a translationally invariant electron gas, the cyclotron resonance is unaffected by electron-electron interactions. Note that the parabolic confinement potential has the unique property that the Hamiltonian can be written as 1
~ - ~--~ (P -~-QA) 2 -~- 89 ~2-]~2 -+-~'~rel where P - E~'~I Pi and R = y]j rj/ne are the center-of-mass (CM) coordinates, Q nee and M = rn*ne. The last term is a function of only the relative coordinates and contains all the effects of the interaction. As a consequence, the wavefunctions are simply ~b(R)~(rij) and the eigenenergies are Ent + Erd. Here we should point out that the CM energy is identical to the single-electron energy Enl Eq. (2.4) because of the fact that e/m* = Q / M . The dipole operator 7-{'= e E E . r j J
e -iwt
= ~ E . R e -i~t
(2.19)
where E is the applied electric field, is expressed solely in terms of the CM coordinates. It follows that FIR radiation excites the CM but does not affect the relative motion [1], [30][R7], [39-41]. The interaction effects can only be probed by either deliberately engineering the dots so that the CM and relative motions are coupled or measuring the thermodynamic properties of the electrons. This important result for quantum dots by Maksym and Chakraborty has been called the generalized Kohn theorem in later publications. There have been a few theoretical studies on the effect of non-parabolicity and consequent coupling of the CM and relative motions. Deviations from the parabolic confinement potential were studied first by Gudmundsson and Gerhardts [179]. They found that a circular symmetric correction, like c< r 4 to the parabolic confinement explains the occurrence of a higher mode observed by Demel et al. [99][R12]. In order to explain the observed anticrossing, they considered the confinement potential with square symmetry, like cv ( x 4 + y4). Pfannkuche and Gerhardts [180] studied numerically the magneto-optical response to the FIR radiation of quantum dots containing two electrons (quantum dot helium). In order to study the possible deviations from the parabolic confinement, they used 7(r) =
1 9 (.042(a~ "4 -Jr-bx2y 2) ~frt
T. Chakraborty
62 20 14 18 12 16 l0 14
8 12 6 10 4 8 2 6 0 0
1
2
3
B (T)
4
5
0
1
2
3
4
5
B (T)
Figure 43 The low-lying energy values for (a) non-interacting and (b) interacting quantum-dot helium as a function of the magnetic field. The dotted curve in (a) is four-fold degenerate. In the interacting system these degeneracies are lifted [181].
where W4, a and b are constants. They concluded that even small deviations from the strictly parabolic case cause rich structure in the FIR spectra. They also observed that the dominant features of the collective excitations are still those of a single-particle. Anticrossing is also observed in their calculated spectra. The particularly simple system of two quantum-confined electrons (quantum-dot helium) has received a lot of a t t e n t i o n (mostly for theoretical studies) because of the relative simplicity of the calculations involved [176,177, 181-184]. When the confinement potential is parabolic, the energy spectra (or part of it) for the non-interacting and interacting two-electron cases are shown in Fig. 43 (a) and Fig. 43 (b) respectively. Obviously, the Coulomb interaction "destroys most of the clear structures immanent in the non-interacting spectrum" [181]. In the non-interacting system, the energy levels tend to bunch up in groups, thereby forming Landau levels in the limit of high magnetic fields. The interaction, on the other hand, causes the energy levels to spread apart from each other. In the non-interacting system, the energy of states in the same Fock-Darwin level increases with angular momenturn (for a given magnetic field). That is not so in the interacting system. If we compare two states of the same Fock-Darwin level with adjacent angular momenta, the difference of the energies in the non-interacting system decreases as ~ 1/B. The difference between their Coulomb energy however increases as v ~ (at least at high fields). Therefore, above a certain value of the magnetic field, the state with higher angular m o m e n t u m becomes lower in energy. As in the larger systems [30][R7], this feature influences the ground state of the quantum dot helium. The importance of correlations and the accuracy of the
Q u a n t u m dots
(a)
63
"''"~"",,,,
0.3
l
|
(b) 0.2
exact
i---
''l
,~
~
-, .....
exact m
H~
.......
HE
.......
0.15
0.250.2 ....
x
........ , . . . . . . .
01
0.15 0.1
005 0.05 t
0 0
1
2
3
4
5
6
7
0
I
2
3
X
4
5
6
7
8
x
F i g u r e 44 Ground state pair-correlation functions for the exact and Hartree-Fock (a) L - 0 states and (b) L - 1 states as a function of dimensionless variable The magnetic field is 1 tesla and a H = 12.79 nm [181].
x = r/aH.
Hartree-Fock approximations has also been investigated in this system [182]. Calculation of the ground state pair correlation function
g(r)-Tra~lZ(~(r-ri+rj) I where the angular brackets denote the ground state expectation value and ag defined in (2.6) reveals the importance of mixing between single-electron states of opposite angular m o m e n t u m in the L = 0 state. For q u a n t u m - d o t helium with a parabolic confinement potential, the pair correlation function is simplified because then each state is a product of center-of-mass and relative part of the wavefunction. Numerical results for the pair-correlation functions are shown in Fig. 44 for L = 0, 1. In the case of L = 1, the results of Hartree-Fock (HF) and exact diagonalization agree quite well, but they differ considerably in the case of L = 0. This can be understood as follows: In the HF state, both electrons are in L = 0 states and each electron state has one of its m a x i m a at the origin. As a result, there is a high probability t h a t both electrons are close to the origin in each of the product states t h a t form the HF state, producing a peak at g(0). On the other hand, the exact L = 0 state includes products of single particle states where electrons have opposite angular momenta. Since these states have non-zero values of L they do not have a m a x i m u m at the origin. The electrons are able to avoid each other and hence a peak in is away from the origin.
g(r)
Analytic solutions of QD models- Although
the problem of a interacting two-dimensional
T. Chakraborty
64
electron gas in a parabolic confinement potential and a perpendicular external magnetic field can be solved in various numerical methods, an exact analytic solution of the problem for a realistic interaction, of course, is far beyond anyone's reach. The singleelectron problem was solved analytically by Fock, as described in Sect. 2.1. Interestingly, for model interelectron interactions or at certain combination of magnetic and confinement potential strength, the interacting many-electron quantum dot model can be solved analytically. For example, for a model interaction v (ri, rj) -- 2V0 - g1 ?Tt* F 2 I r i - rjl 2
(2.20)
where 170 and F are positive parameters which can be chosen to model different types of dots, Johnson and Payne [165] obtained exact analytic expressions for the energy spectrum as a function of particle number and magnetic field. For the choice of interparticle interaction Eq. (2.20) the Hamiltonian for ne interacting electrons in a parabolic quantum dot 7-{- 2m* 1 E[
1 9 w02 E Pi + eAi] 2 + ~m
i
C
i
]rl2 + E
v (ri , ry) -- g *# B B E
i<j
si,~
i
where g* is the effective g-factor, {Si,z} are the spin components along the z axis, #B is the Bohr magneton and p~ = (p~,x,p~,y) and A = (A~,x,A~,v) are the momentum and vector potential associated with i-th particle, can be diagonalized exactly by introducing the center-of-mass and relative mode ladder operators
A+
-
1 [4n~m.hf~l
a~
--
4nem*hPo
{m*f~ (X =t=iY) =1=i ( P x T iPy)}
1
{m'F0 (xij =]=iyij) :7 i (piy,x T ipij,y)}
and similar operators B • b+ associated with opposite angular momentum. Here F0 = It22- neF 2] 89 In addition, the following transformations were used" R - ( X , Y ) = I Y~i ri rij - (xij yij) - r i - rj and the corresponding momentum operators, P (Px, Pv) = ~ p~ and p~j = (p~j,x, p~j,y) = p ~ - Pj. For a fully spin polarized system, n e
~
,
the eigenstates of the total Hamiltonian ~ are generated by operating with A +, B +, a +, and b+ on the zero-point wavefunction whose unnormalized spatial part ff~0 = 10/ is
~0 - exp
2h where Z = X - iY and
Zij
Iz jr 2
- - n e m * a iZl 2
-- Xij
--
2n~h
i<j
iyij, with the corresponding energy
Quantum dots
65
E 0 - - h~'~ -~- ( ~ e -- 1 ) ~ t F 0 + Tte(Tte -- 1 ) V 0 -
ne(g*m*/4me)hWc. 1
For large magnetic field and strong confinement (~ > n~ F) the ground state is ]L} : Ha+]O} = H ~ o i<j
i<j
with the corresponding energy [165]
1 EL = Eo + -~ne(ne - 1)h
(F0 - 1~c) .
Taut [183] obtained analytic solutions for two electrons in a parabolic quantum dot in a magnetic field. The results are, however, available only at certain values of magnetic and confinement potential strength. Analytic expressions for energy levels and magnetization of a two-electron parabolic dot in a magnetic field were also derived by Dineykhan and Nazmitdinov [184]. As for interelectron interactions, these authors used the Coulomb potentials. For an inverse-square form of the interparticle interaction (~r -2 the twoelectron dot system is exactly solvable for arbitrary values of the magnetic field [166, 167]. The 2N-dimensional quantum problem of N particles (e.g., electrons) with inversesquare interaction, parabolic confinement and an external magnetic field is shown [168] to reduce exactly to a (2N - 4)-dimensional problem independent of magnetic field and confinement potential strength, for which Johnson and Quiroga [168] evaluated an exact set of relative mode excitations. Though interesting for being exact, the results are of limited use because of their choice of interactions and other constraints. Finally, the problem of N particles of effective mass m* interacting via a logarithmic potential (i.e., satisfying Poisson's equation) in a medium of dielectric constant e confined by a harmonic field was solved exactly by Pino [185] in Thomas-Fermi approximation in two dimensions. The chemical potential, total energy and differential conductance were calculated and analyzed for various limiting cases in that approximation.
Other important issues of QD m o d e l s - Wagner, Chaplik and Merkt [186] have proposed that, despite the generalized Kohn theorem in a parabolic dot, internal electronic structure can still be analyzed in the FIR spectroscopy, via the quadrupole interaction of FIR radiation with quantum dots. Electron correlation effects and the FIR absorption in a square-well quantum dot were also studied by various authors [74, 75,187]. Ugajin investigated the absorption spectra and the absorption coefficient of two electrons interacting via the Coulomb interaction and confined in the a square-well dot
Yc~
Y) --
-V0 0
if Ix[ < L / 2 a n d otherwise
[y[ < L / 2 (2.21)
66
T. Chakraborty
for various values of the strength of the Coulomb interaction between the electrons. He found that there are many different types of absorption, some induced by the Coulomb interaction. As the size of the dot was increased the intensity of a few of the absorptions was enhanced, possibly by electron correlations. Although the few-electron calculations described above, are successful in exploring the electron states, they are limited to only a few electrons because of the size of the Hamiltonian matrix, which, as discussed above, grows rapidly with the number of electrons in the dot. In an interesting paper, Bolton [188] presented a systematic approach to calculate the optimum J astrow wavefunction for a quantum dot containing ne ~ 10 or more electrons. For a parabolic confinement potential, he showed that the ne-particle problem (rte >_ 3) can be reduced to a three-body problem, by introducing special derivative operators which act on rij = ri - rj as if they were independent coordinates {rij} i < j. The resulting three-body problem is then solved variationally using the Laughlin [140] function to obtain the optimum pair functions and the ground state energies for upto 10 electrons in a quantum dot. Bolton also reported a quantum Monte-Carlo calculation [189] of the ground state properties of 1-10 electrons in parabolic quantum dots. The generalized Kohn theorem is not valid for quantum dots containing holes (instead of electrons) [190-193] because of strong mixing between the valence bands. These systerns therefore exhibit an observable coupling between the CM and internal motion. In the case of a single hole in a quantum dot, the coupling of heavy-hole (HH) and lighthole (LH) states results in significant lowering of the aJ+ modes [190]. It also induces transitions with energies comparable to the separation between the lowest HH and the lowest LH subband. For many holes in the quantum dot [191], the F I R response exhibits strong deviations from the usual case that is governed by the generalized Kohn theorein. The energy levels of one and two holes in parabolic quantum dots in the absence of any external magnetic field, have been calculated recently [192]. T h a t work was later extended by these authors to the case of a perpendicular magnetic field. The energy levels in the single-hole case show strong anicrossings as a result of valence band mixing. The Coulomb interaction between two holes results in strong correlation effects. As the applied magnetic field is increased the total angular m o m e n t u m of the ground state increases in order to minimize the Coulomb repulsion. We have established that the ground state of n~-electron quantum dots in a magnetic field occurs only at certain magic values of the total angular momentum, and that transitions from one magic value to another should occur as the magnetic field is increased. We also found out that this cannot be probed by infrared spectroscopy because F I R radiation couples to the CM motion and hence is insensitive to the interaction when the confinement potential is parabolic. How should we then observe the effect of inter-electrons? Maksym and Chakraborty [30][R7], [106][R15] suggested that thermodynamic quantities such as, electronic heat capacity and magnetization might be sensitive to interelectron
Quantum dots
67
interaction. These quantities are, in principle, measurable, e.9., they have already been measured in a two-dimensional electron gas [194-197]. Maksym and Chakraborty found that the calculated field dependence of these quantities is oscillatory with discontinuities that occur when the ground state angular momentum changes, a behavior which may be important for observation of the magic angular momenta.
2.5.3. Electronic heat capacity The magnetic field dependence of the electronic heat capacity Cv in a quantum dot was studied by Maksym and Chakraborty [30][R7] from the temperature derivative of the mean energy. The results are shown in Fig. 45 (excluding the Zeeman contribution which is a small, slowly varying background). These calculations do not include the Landau level mixing, which simplifies the problem but do not sacrifice any essential physics. The results for interacting electrons (solid lines) are clearly different than those for noninteracting electrons (dotted lines). In the former case, Cv oscillates as a function of the magnetic field and has minima that are associated with crossovers from one ground state J value to another. The oscillations in C~ are a many-body effect, unlike the low-field oscillations in C~ for a 2DEG [197]. Their origin is best understood by considering the results for T = 1 K: at this tempere~ture the dominant contribution to C~ comes from two competing ground states. This causes the doublet structure around the crossovers and is understood in terms of the magnetic field dependence of the gap between the corresponding ground states. Far away from the crossover the gap is large and hence C~ is small. Similarly, it is small exactly at a crossover because the gap is then zero. However, on either side of a crossover the gap is nonzero but small. As a result, Cv is nonzero because neither the probability of a thermal excitation nor the heat absorbed in one are vanishingly small. The oscillatory heat capacity should reveal more structures when the ground state spin depends on the magnetic field.
2.5.4. Magnet i:zation Once the many-body eigenvalues and eigenstates are available, as described in the preceding section, the magnetization can be calculated by differentiating the eigenvalues with respect to the magnetic field B. The results of such a calculation for a parabolic quantum dot are shown in Fig. 46. The top panel of each figure gives the magnetization as a function of B, calculated with and without interaction for (a) three electrons and (b) four electrons. The remaining panels show the ground state total angular momentum quantum number (J) and the ground state spin (S). All results are for GaAs quantum dots with hco0 = 4 meV. The calculations were done with the m a x i m u m value of N
T. Chakraborty
68
0.3 ne = 4
10.2 ~
3K
T :
~ > 0.2 E
.
r-
0.2 ~ -'--
"~ ~'0.1 E
.
.
.
.
L-:-.................
.
_ ..................
.
2
~ . - ~
T=IK
"~,_J_.=_2_2_]J
r-;-JS--J 2-j~_" 2.;o ., .
o
.
.
t
.
Io.1
_
,
,
,
I
J
T =IK
.
.
.
-
j:6
l
.
J ~2
j-g t. . . . . . . . .
-j
0
0
4
,
_ne:3
_~i_6 2
3K
~-. . . . . 0.1
'
0
ne:4
T =
,
~-
"~o. 1
ne : 3
6
8
10 12 14 16 18 20 B(T)
'
2
4
6
8
-
10 12 14 16 18 20 B(T)
F i g u r e 45 Heat capacity C. as a function of magnetic field for three and four electrons in a q u a n t u m dot. T h e dashed lines in the figure indicates the g r o u n d s t a t e Y [30][R7].
taken to be I; that is, one electron was allowed to have N > 0 and the other electrons had N - 0. This truncation is surprisingly accurate, even at low magnetic fields. The absolute value of the magnetization is insensitive to the upper value of the N sum and the only effect of increasing it is that the position of the steps change. This is illustrated in the inset of Fig. 46(a), where the results of allowing the upper limit of the N sum to rise to 2 are shown. Physically, the steps correspond to the changes of the ground state g or both J and S, as can be seen by comparing the three panels of the figure. The magnetization of non-interacting electrons has no step because the lowest two single-electron levels are unaffected by level crossings as the field is increased. Hence systems of up to four non-interacting electrons in the lowest spin state stay in the same angular momentum state throughout the field range so the magnetization curve is smooth. All the steps in this case are a consequence of the interaction. For five or more noninteracting electrons the magnetization would be affected by negative I levels crossing positive l levels. However, the position of these crossings would be drastically affected by the interaction. In addition, there are relatively few of them when the electron number is small (for five electrons in the lowest spin state there is only one) and they tend to occur at low field. In contrast the steps due to the interaction occur at a regular sequence of J values throughout the field range.
Quantum dots
69
(a)
(b) '
1
#.
0 9
-
0
"
-2.5
-2
-2 ~-~.--.J.-.
-3
-'~
----
-4
10
16
8
12
/
6 4
.
.
.
.
2 0
r./)
.....
J- . . . .
0 2 1 0
1 0
5
10
B (T)
15
0
5
10
16
B (T)
Figure 46 Magnetization of a parabolic quantum dot containing (a) three electrons (N = 1), the ground state angular momentum 3 and spin S, and (b) four electrons per dot. The dash-dot lines correspond to the non-interacting system results [106][R15].
2.5.5. Spin transitions Spin transitions in a quantum dot with increasing magnetic field were also investigated by Maksym and Chakraborty [106][R15] for three and four electrons in a dot. The effects of spin were found to be important at fields B < 10 T. The system is spin polarized at higher fields. One effect of spin is that it causes extra steps in the magnetization. Each spin state has its own sequence of special J values, and each of those J values correspond to a possible ground state at that spin. Such behavior is a direct consequence of the interelectron interaction. Electron correlations are known to play an important role in ordinary electron systems such as the Hubbard model. The effect of spin we have just described is, however, a unique manifestation of electron correlations in high magnetic fields. A somewhat simplified version of this result was found by Wagner et al. [107][R16], who studied the magnetization and spin transitions in a two-electron dot. They observed
70
T. Chakraborty
jumps in the ground state angular momenta that are accompanied by jumps in the total spin, which in the two-electron case is the "spin-singlet - spin-triplet" transitions. Ugajin [198] studied the effect of an external electric field, i.e., the Stark effect, on the spin-singlet-spin-triplet transitions of a two-electron system in a quantum dot. He calculated the energy levels and the absorption coefficient of two electrons confined in a square well dot [74, 75] in the presence of an external electric and magnetic field. Application of an electric field causes level repulsion and makes spin transitions unfavorable. Like the heat capacity and magnetization, there are other physical quantities which show oscillations as a function of increasing magnetic field. One example is the magnetoluminescence energy due to recombination in a quantum dot from an acceptor in the plane of the dot [169]. The magnetic field dependence of the quantity is also oscillatory and undergoes discontinuous jumps as the ground state changes from one angular momentum to another. Pfannkuche et al. [181] performed detailed calculations of the ground state properties of quantum-dot helium (energies, pair-correlation functions, densities, etc.) in a magnetic field, using Hartree, Hartree-Fock, and the exact diagonalization scheme. The results of the Hartree approximation show strong deviations from those of exact diagonalization because of an unphysical self-energy term in the former approach, which is exactly canceled by the exchange (or Fock) term. The HF results for the triplet state agree well with the exact results, showing the importance of the exchange interaction. The exact results for the singlet state, however, differ significantly from the HF results. This difference is clearly due to correlations, neglected in the HF approximation. Finally, in a typical quantum dot nanostructure, such as that of Ashoori et al. [76], tunneling of electrons between the dot and the surrounding electrodes is controlled by an applied gate voltage. The effect of the gate electrodes on electron-electron interaction within the QD has been investigated by Maksym et al. [199,200]. Given the small size of these nanostructures, one reasonably expects that induced charges on the gate electrodes used to define the nanostructures could have an appreciable screening effects on the interelectron interactions in the dot. For the specific case of a dot in a parallel plate capacitor, the screened interaction at realsitic dot-plate separations (~ 800 ~), was found to have a dramatic effect on the energy level spectrum, the electrochemical potential, and on transitions of the total angular momentum and spin of the ground state with magnetic field.
2.5.6. Quantum-Hall dots One particular aspect of quantum dots has received considerable attention lately, viz., the electronic properties in the very strong magnetic field limit such that only the lowest
Quantum dots
71
Landau level is occupied - quantum Hall dots [201-206]. Unfortunately, many of those results are primarily of academic interest and are somewhat removed from real systems as discussed below. Yang et al. [201] studied the addition spectrum (the energy to add one electron to a dot) for quantum dots with 5-6 electrons, using the exact diagonalization method. As discussed in Sect. 2.1, parameters that determine the ground state in a quantum dot, once the spin degree of freedom is included, are the confinement energy, the magnetic field and the Land6 g-factor. This is because the single-electron energy depends on two frequencies: ha0 and hac. But in the high-field limit, the single-electron energy to lowest order in B depends only on -y - ~/~c. Yang et al. used 7 as an independent parameter. This means that their results are only valid at extremely high fields and most of the phase diagram is therefore, not very useful for application to real dots [207]. When the magnetic field is strong enough that only the lowest Landau level is filled by the electrons the system enters the fractional quantum Hall effect regime [20, 21] and conductance peaks in the ~ = 1/2p + 1 state are suppressed algebraically in the largene limit. This was predicted by Kinaret et al. [203-205], who studied a parabolic QD containing up to eight electrons interacting via a potential of the type: V(r) cx 1/r 2. The tunneling probability, when an electron tunnels into the QD containing ne electrons, is proportional to the square of the single-particle matrix element
M : (n~ + llc~Ine> for adding an electron in an angular momentum tion~
state c~ between ground state wavefunc-
l~e> ~nd I~o + 11.
In the integer quantum Hall effect regime, both these states are Slater determinants made up of angular momentum states in the lowest Landau level (for a circular dot). The overlap matrix element is therefore unity. Due to nontrivial correlations in the fractional quantum Hall state, these authors find
M ~ n e (;-1)/4 at u = 1 / 2 p + 1. Palacios et al. [206] performed exact diagonalization of the interacting Hamiltonian in a parabolic dot for up to five electrons. They obtained results for #(ne) with n = 1 (Landau level index) for high fields and n = 2 below 1 tesla. They noticed significant deviations from the results of Yang et al. [201] where the calculations were restricted to n = 0, even at large values of the magnetic field. We mentioned earlier that Ashoori et al. observed a reduction of tunneling rates for large ne around u = 2, where y is the Landau level occupation [76][R21]. Palacios et al. [206] explained that observation as due to electron correlations in the ground state. At low temperatures, the tunneling rates of single electrons are proportional to the spectral function [204,206,208,209]
T. Chakraborty
72
1 _~..~.........................2 L 2 , 2 - _ ~
L
~
-
r
~ . . . .~. .
~
ne = 4
_
.
.
~
~
~
~
-
-
~
~
~
~=
....
ll e = .~
.
.
.
~f . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
'.
.
.
.
.
.
'
l
0
Ill
.
.
.
.
"'" ::====: ..........
--'1 :: ........
=
.... .............
:
=:==IW ._=
ne = 2
....
-'~
.
~
.._
0 -
.
~.. .~ = ...............................................
...............................................................
. . . . . . . . . . . . . .
"
~ .
-[
_
~
~
-
2
I
i
1
m
2
,
! 6
t
I-~
.I
]rl
8
B (Tesla) Figure
47
T h e s p e c t r a l w e i g h t A ( n e ) for ne -- 2 -
A(ne,~)-
E
I ((I)~elc~l(I)~e--1)12
5 as a f u n c t i o n o f m a g n e t i c field [206].
(aJ - E~(ne) + Eo(n~ - 1))
r/,i
(I)~e-1 is the ( h e - 1)-electron many-body ground state wavefunction. In the linear regime, tunneling rates depend on the spectral weight [206,209] i.e., A(ne) = A(ne, #(n~)) and is shown in Fig. 47 for n~ = 2 - 5 as a function of magnetic field/3. When electrons form a compact droplet with ~, = 2 and minimum ISzl, A(n~) ~ 1. W i t h increasing magnetic field, spin correlations in the ground state reduce A(n~). It is proposed that 1 the spectral weight can even be zero due to spin selection rule I S z ( n e ) - S z ( n e - 1)I > g. The experimental work of Klein et al. [210] showed the evolution of the chemical potential #no of a dot of 30 electrons, in a magnetic field. These authors reported a divergent spin susceptibility for which a quantitative description was given using the Hartree-Fock theory. The accuracy of their theoretical analysis is, however, questionable: they used a single Landau level HF approach at fields around 2-3 tesla which will introduce some error. The important issue of how the calculational errors compare with experimental errors is not clear. The effect of electron correlations in this context was investigated by Ahn et al. [211]. They calculated the ground state energy and chemical potential of a droplet containing a few electrons via an exact treatment of electron-electron interactions with Landau level mixing included. The many-electron states in the lowest Landau w
h
e
r
e
Quantum dots
73
B=IT ..~176~ - - ~
...-;'..,.,~ E (meV)
i 2
1 3
I 4
I 5
(a)
{b)
L Figure 48 Energy spectrum of a single dot and the dot-pair (a) without and (b) with inter-dot Coulomb coupling. level and in the symmetric gauge was multiplied with a Jastrow correlation factor, so that, "electron correlations are effectively included through the mixing of higher Landau level" [211]. These authors also found bumps in the chemical potential as the magnetic field is varied. This resulted in a ripple in the # - B diagram as observed experimentally. Finally, Pfannkuche and Ulloa [212] studied the overlap matrix element of ne and ( h e - 1) electrons states of a parabolic dot, which governs the tunneling rate of an electron passing through the dot. They noticed that, as in FIR spectroscopy [Eq. (2.19)], in transport spectroscopy measurements the excitation spectra are dominated by the CM modes. Working with a 3-electron dot they found that electron correlations strongly suppress most other transitions involving excitations of the internal degree of freedom. The CM motion, therefore, dominates the transport resonances.
2.5.7.
Coupled quantum d o t s
In a quantum dot pair coupled in series, the circular symmetry is broken and, as a consequence, radiation couples to the internal motion of the electrons. Such a system was studied by us some time ago [71][R19], [213]. The dot-pair is coupled only via the Coulomb interaction (tunneling of the electrons between the dots is not allowed).
T. Chakraborty
74
We consider the Hamiltonian for the Coulomb-coupled dot-pair in a parabolic confinement to be of the form ~-{
=
7-{o --
']"{0 -Jr- ~-{ee
~.
E a=1,2
[ 1 ~
_-
e
e )2 lm, 2] P a , i - -A(ra,i)c + g Wo2(r~,i- R~)
"
e2 ~-~ee
(
1
E=
,2 i r
e2
+_ye
1 ~,,3 .
lr l i _ r 2 j I
Here the sums over i and j run over the number of electrons in a single dot and R~ is the position of the center of the c~-th dot. No attempt is made to include any singleparticle interaction between the dots, i.e., the electrons feel the other dot only through the Coulomb interaction ~ee. We show in Fig. 48 how the Coulomb interaction between the two dots couples the excitations with L = 2 (CM) and L = 5 (CM) of an individual dot. The lower mode of the dipole transition is caused by the transition to L = 4 (CM) level which has no other level to couple to. The coupling of the CM and relative motions causes interesting structures in the dipole transition energies as a function of magnetic field (Fig. 49). As explained above, the lower mode of the transition energies for electron-dot pair is always close to the single-particle mode. On the other hand, the upper mode is seen to exhibit interesting anticrossing behavior due to the coupling described above. Theoretical work on the correlation effects in coupled quantum dots and F I R absorption in those systems has also been reported recently [214,215]. Here a two electrons system was considered in dots which were assumed to be wells with finite barriers (squarewell quantum dots). A square is divided into two in which the spacing plays the role of tunneling barrier. FIR spectra was calculated and the absorption coefficients were found to be strongly influenced by the separation between the dots. In fact, with increasing separation between the dots or increasing the barrier heights, the quantum states of two electrons in coupled quantum dots were found to change from uncorrelated to highly correlated states [215]. F I R absorption reveals the transition from uncorrelated to correlated Wigner lattice states to Mott insulating states as the separation between the dots increases [74]. Kempa et al. [216] studied a two-dimensional square array of dots modeled as point dipoles. They noted that if dots that contain a large number of electrons are brought sufficiently close, phase transitions occur in which the dot system polarizes spontaneously. The phase transition is of second order and can lead to both ferroelectric and antiferroelectric arrangements; the latter phase is the most stable.
Quantum dots
75
d :100 nm 000 ~
__
o0o0O~ 9 ....o o o O V J
~
--
~,,.,,~
o o o o o ~ " ."~ . . . ~
E (meV)
--
~ ,,' ' ~ ' ~ - ~ ~ " ~ '~Y ""~ ~-o
."
.[
I 0.0
0.5
o o o
~
0
I
,l,
1.0
1.5
2.0
B(T)
Figure 49 Dipole transition energies and intensities of a three-electron per dot pair. Dot separation is 100 nm and the confinement energy is hwo = 2.5 meV. Solid lines are oneparticle transition energies. Diameters of the open points are proportional to the intensity of the transition. Palacios and Hawrylak [217] studied correlated few-electron states in vertical doubledot systems. A critical distance between the dots was calculated where the interdot correlations are found to be important and minimum-isospin (or quantum dot index) ground states appear. Similar studies were also reported by Oh et al. [218] who investigated the electronic structure and the optical properties of vertically coupled QDs in magnetic fields. They performed exact diagonalization of the Hamiltonian matrix for one and two electrons in a coupled dot characterized by a parabolic potential in the zy-plane. In the growth direction they used the vertical potential consisting of two square wells of equal width (150 .~), a barrier width of 50A, and two buffer layers with thickness of 350 ~. The barrier height of 147 meV was chosen to represent the GaAs system. They found that interdot and inter-electron interactions strongly affect the ground state, which can, in principle, be observed via optical spectroscopy. For vertically polarized light, the calculation of resonant energies revealed blueshift and sharp drops with increasing magnetic field that was attributed to the electron_electron interactions.
T. Chakraborty
76
2O
~" > 15 10
5 20
~" 15
lO
5 0
2
4
6
8
10
B (W) F i g u r e 50 Lowest single-electron energy levels vs magnetic field of a parabolic quantum dot containing a repulsive scatterer (170 - 10 meV, d = 5 nm) located (a) at the center of the dot and (b) at 5 nm away from the center. In figure (a) some values of the angular momentum quantum number are indicated.
2.6
Electron-impurity systems
While most theoretical studies of q u a n t u m dots involve impurity-free dots, Halonen et al. [73][R24], [219] studied the effects of a repulsive scatterer in a q u a n t u m dot. Repulsive scattering centers in q u a n t u m dots have profound effects on the energy s p e c t r u m of a q u a n t u m dot in a magnetic field. The s y m m e t r y - b r e a k i n g electron-impurity potential introduces additional structures due to level repulsions, lifting of degeneracy etc. in the otherwise well u n d e r s t o o d s p e c t r u m of an impurity-free parabolic q u a n t u m dot discussed above. Halonen et al. analyzed those structures and extracted some simple rules t h a t determine when the level repulsions are supposed to occur. It should be pointed out t h a t t r a n s p o r t properties of q u a n t u m dots with an impurity whose s t r e n g t h can be controlled independently are under active investigations [220] and energy spectra like the ones presented here can be observed in experiments such as single-electron capacitance We begin with the s t a n d a r d model in which electrons of effective mass m* are confined
Q u a n t u m dots
77
within the z = 0 plane by a parabolic potential and are subjected to a perpendicular magnetic field B. In the presence of a symmetry-breaking impurity potential, the manyelectron Hamiltonian is written as 7-{ 1 ne ~e ne e2 1 E (~Az) 2 1 * 2E 2 E Vimp 1 E -- 277~* i----1 Pi Jr- -- " -Jr- ~7~'b &O i=1 ri + i=1 (ri) + ~ e ~#j I r ~ - r~l (2.22) where ne is the number of electrons in the system and e is the background dielectric constant. We also use the symmetric gauge vector potential, A = g1 ( - B y , Bx, 0). The impurities are modeled by a Gaussian potential V imp (r) -- V0e -(r-R)e/de,
where V0 is the potential strength, d is proportional to the width of the impurity potential (the full width at half maximum ~ 1.67d), and R is the position of the impurity. We apply the exact diagonalization method by constructing a basis using single-particle wavefunctions of a perfect parabolic quantum dot ~nz (r)
--
Ce-il~
where C is the normalization constant, a - v/h/(rn*t2), ft - v/cz02 + a~c2/4, and L~(x) is the associated Laguerre polynomial. The quantum number 1 = 0, +1,-t-2,... is the orbital angular m o m e n t u m quantum number and n = 0, 1, 2 , . . . is the radial quantum number. In the actual calculations electron spins are taken into account but the Zeeman energy is ignored. The impurity potential V imp (r) introduces an interaction between the single-particle states ~nt (r) with matrix elements
f
(~'~1/1 (r) V imp (r)Fn212 ( r ) d r =
V0e-i(/1-/2)O
•
(d/a) 2 (R/a) k e--(R/a)2/[l+(d/a) 2]
nl ,
n2 ,
(~1 + Ill I)! (n2 + 121)!
~= - 1 ) ~ (n2 + • ~ ( 3! \ n2 /3=0 ~+/3+(IZl I+ I/2I-k)/2
p=O
• s~O = s,
1/2~
c~!
o~---0
IZ21)(c~ +/3 ~3
(_1)~ ( n l +
(llll
,, ~1 -c~
+ Iz21- k)/2)!
k +p
-s
II~l)
[l+(d/a)2] s+p+k+l '
T. Chakraborty
78
'
22
I
'
I
'
I
'
I
'
--
16
22
>
20
Q)
18
16 0
1
2
3
4
5
B (T)
F i g u r e 51 Lowest energy levels of a two-electron parabolic quantum dot containing a repulsive scatterer (V0 - 10 meV and d = 5 nm) located (a) at the center of the dot and (b) at 5 nm away from the center.
where k - [11 -12] and the position of the impurity R is represented in polar coordinates
(R,e).
2.6.1. Energy spectrum Some single-electron energy levels of a parabolic q u a n t u m dot with an impurity at the center are shown in Fig. 50 (a). The material p a r a m e t e r s chosen for the numerical results t h a t follow are a p p r o p r i a t e for GaAs q u a n t u m dots, e = 13, electron effective mass m* = 0.067me, and ha0 = 4 meV. There are several features in the s p e c t r u m which distinguish it from t h a t of an impurity-free parabolic dot (Fig. 4). First of all, the ground state has different angular m o m e n t u m at different magnetic fields. Further, the impurity potential mixes energy levels t h a t have the same angular m o m e n t u m but different principal q u a n t u m number. The degeneracy at B = 0 is partially lifted, and different Fock-Darwin bands are clearly visible. The level spacings are also different from those in impurity-free parabolic q u a n t u m dots.
Q u a n t u m dots
79
T h e energy levels of a parabolic q u a n t u m dot with an i m p u r i t y which is near b u t not exactly at the center are shown in Fig. 50(b). Here the angular m o m e n t u m is not a g o o d q u a n t u m n u m b e r and all degeneracies at B = 0 tesla are lifted. T h e r e is no a b r u p t change in the g r o u n d s t a t e as the m a g n e t i c field is changed. Anticrossings in the lowest FockD a r w i n b a n d are clearly visible. T h e r e also are several anticrossings in the higher energy levels. C o m p a r i n g these results with those for an impurity-free parabolic dot (Fig. 4) we find t h a t the s t r e n g t h of the energy level repulsion (anticrossing) d e p e n d s strongly on the difference in the q u a n t u m n u m b e r s n a n d 1 of the a p p r o p r i a t e states. T h e level repulsion is s t r o n g e s t for states t h a t have the same value of n a n d t h a t differ as little as possible in I. C o m p a r i n g Fig. 50 (a) a n d Fig. 50 (b) note t h a t level repulsion is s t r o n g for the levels in the lowest F o c k - D a r w i n band. In Fig. 50(a) the 1 = 0 state (i.e., the g r o u n d state at /? = 0 tesla) crosses some of the lowest energy levels of the lowest b a n d as the m a g n e t i c field is increased. B u t as soon as the circular s y m m e t r y is broken by m o v i n g the i m p u r i t y away from the center [Fig. 50(b)] the states in the lowest F o c k - D a r w i n b a n d are mixed. This results in s t r o n g anticrossing such t h a t only hints of the original 1 = 0 energy level can be seen at higher m a g n e t i c fields b e t w e e n energies 10 and 15 meV. T h e track of the original I = 0 energy level becomes m u c h clearer as it crosses the o t h e r levels with higher value of l, i.e., as the m a g n e t i c field is further increased. A strong anticrossing effect is also seen at B = 0 tesla b e t w e e n states with n = 0, 1 = - 1 a n d n = 0, 1 = 1, i.e., A n = 0 and A1 = 2. In Fig. 50(a) these two states are d e g e n e r a t e at B = 0 tesla with energy eigenvalue a b o u t 8.6 meV. In Fig. 50(b) this d e g e n e r a c y is clearly lifted due to the b r o k e n circular symmetry. A n o t h e r s t r o n g anticrossing can be seen at B = 1.4 tesla near the energy value of 10.5 meV. This anticrossing c o r r e s p o n d s to the crossing of the energy levels with n = 0, l = - 1 a n d n = 0, 1 = 2 of Fig. 50(a). Here A1 = 3 and the level repulsion is clearly weaker t h a n the one b e t w e e n the states with n = 0, 1 = - 1 a n d n = 0, 1 = 1 where Al = 2. T h e r e is also an equally s t r o n g anticrossing at higher energy near E = 19 m e V a n d B = 1.4 tesla. This anticrossing results from the level repulsion b e t w e e n states with n = 1, 1 = - 1 and n = 1, l = 2, i.e., here also A n = 0 and A1 = 3. A l t h o u g h there seem to be m a n y level crossings in Fig. 50(b) these crossings are actually anticrossings. Because the s t r e n g t h of the level repulsion d e p e n d s strongly on the difference in n and 1 the gap b e t w e e n m a n y of the energy levels is too small to be seen in the Fig. 50(b). T h e fact t h a t there are no crossings of the energy levels m e a n s also t h a t there is no conserved q u a n t i t y other t h a n the energy. In t h a t sense the s y s t e m seems to be chaotic. As the i m p u r i t y is m o v e d further away from the center of the dot, interactions b e t w e e n the states of the impurity-free dot first increase resulting in s t r o n g e r anticrossing effects. B u t w h e n the i m p u r i t y is far e n o u g h its effects are reduced a n d the energy levels begin to resemble the levels of an impurity-free parabolic q u a n t u m dot.
80
T. Chakraborty
The energy levels of a quantum dot containing two interacting electrons and the impurity at the center are shown in Fig. 51 (a). Clearly, the spin singlet-triplet transition is moved from about 2.5 tesla (impurity-free case) to about 1.5 tesla due to the presence of the scatterer. Similar results for systems with the impurity is moved away from the center are shown in Fig. 51 (b). As expected, there is no degeneracy at B = 0 tesla. Clear energy level repulsion can also be seen in this case. We have done a detailed analysis of how the energy levels shown in Fig. 51 (a) change as the impurity moves away from the center of the dot. Just like the single-electron energy levels (Fig. 50) we also find angular momentum selection rules that govern the strength of the level repulsion. If the angular momentum of a pair of crossing energy levels of Fig. 51 (a) differs by two there is a large anticrossing of the corresponding levels in Fig. 51 (b). If the difference is some other even number then the anticrossing is weaker but still not insignificant. However if the difference of the angular momentum q u a n t u m numbers is an odd integer then the level repulsion is almost negligible. Since we are dealing here with two mutually interacting electrons with opposite spin it is evident that the change from the one-electron case in the rules governing the strength of the level repulsion is due to the Coulomb force. We have done similar calculations for three- and four-electron dots. These calculations support, for electron numbers higher than one, a simple rule that if there is an even (odd) number of electrons, the level repulsion is strong for states that correspond to the states in a corresponding circularly symmetric system (i.e., impurity is at the dot center) whose angular momentums differ by an even (odd) number.
2.6.2. Optical absorption spectrum Figure 52 shows optical absorption of a quantum dot containing one, two, and three electrons and one impurity near but not at the center of the dot [72], [73][R24], [219]. The point here is that any increase of the number of electrons will result in a decrease of the effect of the impurity on the optical absorption. In the one-electron dot the degeneracy of the absorption modes at B = 0 tesla is clearly lifted. But as the number of electrons is increased the level repulsion decreases substantially. Also the clear structures seen in the one- and two-electron results have almost all disappeared in the three-electron case; the optical absorption intensity is more or less scattered around the original modes of an impurity-free quantum dot. Figure 53 shows the optical absorption of a two-electron quantum dot containing one impurity which is near but not exactly at the center of the dot. The absorbed light is now linearly polarized. The main result is that polarization affects absorption only at low magnetic fields. If the electric field of the electromagnetic radiation points along the axis
Quantum dots
81
ne=l
I e
.o O00
. oooOO
00'
OOOOOOoooo0000 _
>" 6
2
_
1:;
n,=2
oOOO ~.~176
..1
~,! ............ I ..........
I
ne=3
oe6
.6;
s at'ee~
_"~176176176176 ~-.-J --0
J
......1
1
2
........... !_ 3
_
s (T)
Figure 52 Dipole-allowed optical absorption energies and intensities of a parabolic quantum dot containing a repulsive scatterer (V0 = 4 meV and d = 5 nm) located 5 nm away from the center of the dot. Areas of the filled circles are proportional to the calculated absorption intensities. The number of electrons in the dot is indicated inside the figures.
that goes through the center of the dot and the impurity, (say, the z axis), the absorption is strongest for the so-called bulk mode (the upper mode) at low magnetic fields [Fig. 53 (a)]. If the absorbed radiation is polarized perpendicular to that axis, i.e., along y axis, the absorption is strongest for the edge mode (the lower mode) at low fields [Fig. 53 (b)]. As the magnetic field is increased the difference on the absorption intensity between these two polarization directions rapidly disappears. One explanation to this effect could be that the electron density near the impurity decreases as the magnetic field is increased. This is because the electron density moves away from the center towards the edges of the dot as the magnetic field is increased. Finally, in Fig. 54 we show optical absorption of a quantum dot containing five impurities and one, two, and three electrons. Clear similarities with the one-impurity results (Fig. 52) can be seen here. The primary difference is that the degeneracy at B = 0 is not as strongly lifted, i.e., the system is effectively more circular symmetric than the one impurity case. Here also an increase of the number of electrons reduces the effects of impurities.
T. Chakraborty
82
I
I
I
I
(a)
~--
>
v
9 ;
.
6 4
.oo~O *.~176 o
<1
e
O
!
e
9 ~~ e O 0 0 0 0 0 0 _
I
I
i
e
(b)
6
9 O 0
-
9 gO0
; .
O
O ~ ~
O0*ooe6
: oeeeeooo. .
2
t
I
0
1
,
I
I
2
3
B (T)
F i g u r e 53 Dipole-allowed optical absorption energies and intensities of a parabolic twoelectron quantum dot containing a repulsive scatterer (V0 = 4 meV and d = 5 nm) located 5 nm away from the center of the dot at x axis. The absorbed electromagnetic radiation is linearly polarized with the electric field pointing (a) along the z axis and (b) along the y axis.
Thus far, we have discussed about systems with V0 being small (2-10 meV). Let us now consider the case of a repulsive Gaussian potential with V0 = 32 meV placed at the center of the dot. In t h a t case electrons are confined in a wide ring. B o t h the effective radius and width of this ring are taken to be about 20 nm for a single electron. Figure 55 shows electromagnetic absorption energies and intensities of t h a t system with one t h r o u g h three electrons as a function of the magnetic field. One-electron results reveal four distinct modes. The strongest of the upper two modes can be i n t e r p r e t e d as a bulk m a g n e t o p l a s m o n mode according to its asymptotic behavior, i.e., its energy approaches hwc as the magnetic field is increased. The origin of the discontinuities near 5 and 8 tesla can be traced back to the fact t h a t the potential forming the ring, in our case, is highly asymmetric: We have a steep Gaussian potential near the center of the dot and the outer edge is formed by a soft parabolic potential. For a s y m m e t r i c potential we expect the bulk m a g n e t o p l a s m o n mode is a s m o o t h function of the magnetic field. The situation is somewhat similar to the experimental work by Dahl et al. [221], who investigated m a g n e t o p l a s m a excitations of a 2DEG confined in a ring [which can alternatively be considered as a disk with a repulsive s c a t t e r e r - also known as an antidot
Q u a n t u m dots
83
- o io ..........i ..........
_ _,, . . . ' "
-OiOO
OOOO
-
ne =- 1
_
wO00000000000oo_ . : >" 6
i
..
08
0
~176
99 99 S!
he'--2
OO0 0000000000
2
L_._
__l.~l~__..n e 0
,,,,,
. t;,
~t 9 -#~b 0
O
~
S $1~
he=3
on
t~176 1
2
3
B (T) F i g u r e 54 Dipole-allowed optical absorption energies and intensities of a parabolic QD containing five repulsive scatterers. The locations of the scatterers, which are denoted as open circles and with parameters V0 = 2 meV and d = 5 nm, are shown as inset. Areas of filled circles are proportional to the calculated absorption intensities. The number of electrons in the dot is also indicated.
(see chapt. 3) at its center (inset of Fig. 56)]. T h e y employed millimeter-wave spectroscopy and their observed resonance frequencies as a function of applied m a g n e t i c field are displayed in Fig. 56. Several distinct modes are clearly visible in the data, two of which are seen to split into doublets at finite m a g n e t i c fields. These modes are labeled by con,+, n = 0, 1, 2 , . . . , where n is the n u m b e r of modes in the radial direction and the second, a n g u l a r index can in general assume the values • but was found to be restricted to m = +1 in the present case. Due to circular s y m m e t r y , con+ = con_ a t / ? = 0. In the ring geometry, the upper b r a n c h of the lowest doublet, the coo+ m o d e exhibits a positive dispersion near B = 0, as one would expect for the u p p e r branch of a circular disk. But for B > 0.25 tesla, the coo+ b r a n c h decreases with increasing B, while its oscillator s t r e n g t h decreases rapidly. These are the characteristic signatures of edge m a g n e t o p l a s m o n s ( E M P ) . For larger B, Dahl et al. used an a p p r o x i m a t e formula for dispersion in a disk [222], coemp CX:in d/do + c o n s t a n t , where d is the disk d i a m e t e r and do depends on the sample p a r a m e t e r s . T h e solid line in
T. Chakraborty
84
In e
I
9
16
... $
12
9~
I
1
=
9
~ 9
9
9
0
~
v
>
16 12
_
ne
--
8
o~
~
o
,-'-:7 -
O8~
~176
ooO~176176 OO0
- -
o'"
2
9
....
""" i
he--3
4 l-ill'coO
o
~
0
2
-
OOOo0 ~ 9
4
6
o
9
8
-J
10
B (T) F i g u r e 55 Absorption energies and intensities of a QD including a strong repulsive scatterer (wide ring) with one, two, and three electrons as a function of the magnetic field. The areas of filled circles are proportional to the calculated absorption intensity.
Fig. 56 is for d = 12#m and the dashed line is for d = 50pm. These two values correspond to inner and outer diameters of the rings respectively. Comparing the calculated results with the data, these authors concluded that the coo+ mode corresponds to the E M P localized at the inner boundary of the ring, while the coo- mode corresponds to E M P at the outer edge. Coming back to our theoretical results for a wide ring, if we ignore the discontinuities in Fig. 55 discussed above, the two upper modes of the one-electron spectrum behave clearly the same way as the experimental results of Dahl et al. [221]. However, the two lower modes behave differently (in the one-electron case) from those experimental results. The lower modes, i.e., the edge magnetoplasmon modes, reveal a periodic structure similar to the case of a parabolic ring [72], [73] [R24]. T h a t is however true only for the one-electron
Q u a n t u m dots
85
l F i g u r e 56 Frequencies of magnetoplasma resonances for an an array of wide rings. The lines are the dispersions discussed in the text [~21]. I
system. When the number of electrons in the system is increased the periodic structure of the edge modes (the two lowest modes) starts t o disappear. This is, of course, due to the electron-electron interaction. The Coulomb il lteraction is important in wide rings. It should be emphasized that because the spin deg tee of freedom is also included in these calculations, the difference between the one- an( two-electron results is entirely due to the Coulomb force. The lowest mode (which is also the strongest) behaves (even for only three electrons) much the same way as does the lowest mode in ;he experiment [221] (where the system consists of the order of one million electrons). It is interesting to note that this mode is also similar to the observed magnetoplasma resol lance in antidot arrays [223][R37] In the high field regime, the upper mode observed in antidot systems is also qualitatively reproduced in the quantum dots. s For more about this experiment, see Sect. 3.1.1.
86
T. Chakraborty
Figure 57 Capacitance vs voltage for self-organized dots and rings. The ring structures are shown as inset [225].
Hawrylak [224] has studied the effect of a tunable artificial impurity in quantum dots in a magnetic field. The impurity was found to induce electronic spin and charge transitions which modify the chemical potential of the dot and therefore can be observed in resonant tunneling experiments. Our theoretical work presented in Fig. 55 received strong support from recent experiments on quantum rings. With the help of capacitance and FIR spectroscopy, Lorke et al. [225,226] investigated the electronic structure of self-organized quantum rings. In the process of creating self-organized InAs islands (see Sect. 3.3), these authors have been able to change the shape of the islands [227] and create ring-like structures (Fig. 57) of which the inner diameter is ~ 30 nm and the outer diameter ~ 80 nm [225]. Capacitance spectroscopy on these systems revealed structures (Fig. 57) that were distincty different from those of quantum dots. Those structures were interpreted as due to filling of first and second electron level. The FIR response of quantum rings containing two electrons as shown in Fig. 58, exhibit behavior remarkably analogous to those of Fig. 55 (in particular the upper modes), predicted theoretically [73][R24]. Creation of these self-assembled disorder-free quantum rings represents a major development in low-dimensional struc-
87
Q u a n t u m dots
35
I-
II
I
i
"l
'
I
'
i
~ . 3O > 0)
E 25 20 cO
15
(1) 0
c 10 C O
~0
5 0
ne=2 -
0
---
.........
,I
|
,
i
,_i,
2
8 4 magneti c field
I
10 (T)
!
12
14
F i g u r e 58 FIR resonance position of self-as',sembled rings versus the applied magnetic field for two-electrons per ring [225].
tures where many new and exciting physics driv( ~n by confinement, electron-correlations, and the influence of an external applied field ca~1 be explored.
2.7
Exciton spectrum /
Theoretical work on the excitons in a q u a n t u m dot in magnetic fields has been reported by Halonen et al. [228] [R25]. Earlier work by Bryant on excitons [229] and biexcitons [230] in q u a n t u m boxes (in the absence of a magnetic fiel( 1) demonstrated the competing effects of q u a n t u m confinement and Coulomb-induced elecl :ron-hole correlations. More recently, excitons and biexcitons have also been studied in s( miconductor nanocrystallites [231-233]. Measurements of the exciton binding energy in t he presence of a magnetic field has been reported in q u a n t u m wells [234-236] and quantu m wires [237]. In our work on excitons in a quantum dot im magnetic fields [228][R25] the model Hamiltonian for a two-dimensional hydrogenic eXciton" in a parabolic confinement potential and a static external magnetic field is /
- ~e +~J+~-h where the electron, hole, and electron-hole terms are
T. Chakraborty
88
[
1 -ihV~me
e ]2
-Ae c
1 2 2 At- ~ 71~eCUe T e 2
1 [--ih~7h + -eAh] 2rnh c e2 1 ~-h
I~ 2 2 -+- -~ 't t t h ~ h l'h
= 6
Ire -- rhl"
Here e is the background dielectric constant. Let us now introduce the center of mass (CM) and relative coordinates R = ~ (mere + ? T t h r h ) , r - - r e -- rh, and the usual notations, M - rne + mh, # -- m e m h / M , and 7 = (mh -- m~)/M. We also choose the symmetric gauge vector potentials for electrons and holes as A~ = g1B x ( r e - rh) and A h - - s B 1 x ( r e - rh). The Hamiltonian can then be rewritten as ~'{ -- ~-I~CM -Jr- ~'{rel -~- ~-{x,
where the various terms are ~CM --
']-~rel
h2 _~ 1M 2 M V~M + ~
h2
ihe B
--2--fiVr~o~+ 2-~,~ 1 +3#
ihe
~cB.r
e2B 2 4#2c2
_4_
1
1 (meWe2q-mhW~)
R2
.r • Vre, ]
(?TZhad2+ ?Tte(.d~) 7.2
x VOM + ~ ( ~ - - ~ )
e2 1 r
R.r.
The CM t e r m is the Hamiltonian of a well-known two-dimensionM h a r m o n i c oscillator with energy s p e c t r u m
ECM --
2nCM +/CM + 1) hWCM
1 ('%~+'~h~)
1
~CM --
n C M ~ 0. The other term ~-/rel is the Hamiltonian of a two-dimensional charged particle in a magnetic field and in parabolic and Coulomb potentials. T h e radial part of the SchrSdinger equation
with
Q u a n t u m dots
89
80 ---
>o
(o.~1~/.~,~
~, I
60 4o
UJ 20
I ~
~
(o.oi(o.oi (a) 1
,
I
,
8o
oo ~
I
40
_
W
0
10
20
30
B (T) F i g u r e 59 The ground state and low-lying excitation energies of a light-hole (rnh = 0.09rn) exciton as a function of the magnetic field (tesla). The confinement potential energy is taken to be 15 meV for both particles.
R" + - 1 Rl r
-
4h2c2 + Mh2
R-0
can be solved numerically. All interactions between CM and relative motions are included in the cross t e r m ~ x . The results for the lowest energy levels of a light hole (rnh = 0.09rn) exciton as a function of the magnetic field is shown in Fig. 59 without (a) and with (b) the cross t e r m ~ x added to the Hamiltonian. We plot only those levels t h a t have the total angular m o m e n t u m L = 0. Although the effect of the cross t e r m is much larger for the excited states t h a n for the ground state, one can identify at least some of the lowest levels using the non-interacting energy levels [Fig. 59 (a)]. Due to the confinement, the relative angular m o m e n t u m of the exciton can have different angular m o m e n t u m values to get L = 0. In Fig. 59 (a) the energy levels for l r e ] = 0 (solid lines) and l r e ] = +1 (dashed
T. Chakraborty
90
oo 9 9 9 e ~ o.O ~ 9 coco ~176 . 08 *, , oOOOOO~176176176 C O o 9 1 4 9 1 4 9 1 7969 9 9
80-
O 0 0 0 o .
o*
9
eOOOoee..o~ OOO4000OOO
60-
o o
00 B
.o
9
oo~176176176176176176176
eoeoo 9
9~ uJ
4o- OOO000OOOoooo00000oOOOOOOOOO0
20
oooOOOOOO4D
-
OOOOOOOOoooOOOoOOOO 0
[
1
0
5
I "
10
t
I
T
15 B (T)
20
25
30
Figure 60 Optical absorption energies and intensities of a light-hole exciton as a function of the magnetic field (tesla). Diameters of the filled points are proportional to the calculated intensity of the absorption. lines) are shown. For each energy level of the relative motion there is a spectrum of the CM levels separated from each other by the amount of the confinement potential energy. To illustrate this point more clearly, we have labeled each level by its (exact) quantum numbers: nON ,/CM, nrel, and lrel. Addition of the cross term ~• results in anticrossings in the energy spectrum [Fig. 59 (b)]. Moreover, when the magnetic field is increased some energy levels begin to form the first and second Landau levels. The intensity of optical absorption is calculated from (~(r)) - ~
ij
c*cj R~f~,lfe, (0) R L, zL (0) 51~e~,lL5%M,~JCMflSM,l~M,
which gives the probability of finding the electron and hole at the same position. The numerical results are presented in Fig. 60, where a rich antierossing structure of the optically active energy levels is found to be still present. Photoluminescence experiments on quantum dots provide important information about the states of excitons [238], [239] [R26] localized in the quantum-confined low-dimensional systems. Sharp peaks (due to recombination of an exciton from discrete energy levels) in the photoluminescence (PL) spectra are usually attributed to localization of excitons in quantum dots. Zrenner et al. [239][R26], [240,241] reported results on optical spectroscopy on single quantum dots - termed "natural quantum dots". These dots originate from well width fluctuations in a narrow quantum well (Fig. 61). In a narrow well, wellwidth fluctuations of several monolayers (MLs) result in lateral potential variations which can be quite sizable and excitons can be localized in regions with locally enhanced well
Quantum dots
91
Figure 61 Interface roughness in a quantum well on an atomic length scale. Excitons are localized in regions with locally enhanced well w i d t h - the natural quantum dot [241]. width [240,241]. For a GaAs quantum well, a local variation of the well width from 10 ML to 12 ML results in local reduction of the exciton energy by 43 meV [240]. Optical spectroscopy on single quantum dots are possible by spatially resolved PL spectroscopy and resonant charge injection in an electric-field-tunable coupled quantum well structure. Excellent details of the experiments can be found in Refs. [240,241]. A very interesting result by Zrenner et al. [239][R26] was the demonstration of the existence of zero-dimensional states in coupled quantum well structures. Because the dots are asymmetric with strong confinement in the z direction and weak confinement in the xy-plane, their response to the parallel (BII) and pendicular magnetic field (B• should be very different. In their magneto-optical measurements, those expectations were precisely realized. The positions of the observed emission lines as a function of BII and B• are shown in Fig. 62 (a) and Fig. 62 (b). Clearly, there is negligible influence of BII on the position of the lines, but for B• one observes complicated level shifts, splittings and anticrossings. In fact, the 3 or 4 emission lines on the low-energy part of the spectrum, show very similar behavior predicted theoretically (Fig. 60) and discussed above [228][R25]. There are a few other interesting investigations of the optical properties of quantum dots reported in the literature [242,243], [244][R27], [245] which provide information about the single-particle states of electrons and holes. Bayer et al. [242,243] reported results on low intensity magnetoluminescence spectroscopy. They studied electronic transitions corresponding to the ground state and low-lying excited states of GaAs/InGaAs modulated barrier quantum dots. From scanning electron spectroscopy, they concluded that the barrier region is cylindrical. The magnetic field was applied along the growth direction, and therefore the orbital angular momentum around the cylindrical axis was a good quantum number. As the lateral and vertical confinement potentials of these dots, in contrast to a parabolic confinement, are abrupt and rather shallow, the problem is non-separable in the radial coordinate r and the coordinate in the z-direction. These authors calculated the electron-hole states using finite-difference method [242]. The results are shown in Fig. 63 together with the experimentally observed results. Comparison of the theoretical and experimental results clearly show that at zero field the ground state
T. Chakraborty
92 1732 BII
GaAs/AlAs
30A/40A
~" 1730 T = 4.2K
........................ >,, ~1728 i
~
1726
1724
(a)
,. . . . . . . . . . . . . . . . . . . . . . . . . . . .
1732
I730
E~ 1728
~a 1726
1724
1722
0
2
4
6
8
10
12
B (r)
Figure
62
Position of the emission lines as a function of (a) Bil and (b) B •
[239][R26].
is nondegenerate with zero angular momentum, and the first excited electron and hole states are doubly degenerate with angular momenta m + I. At a non-zero magnetic field, the first excited states split according to two orientations of the angular momenta, and a higher state (m = -2) becomes localized in the dots. The magnetic field dependence of the calculated transition energies compare well with the experimental results. Similar results were also obtained by these authors for the deep etched InP/InOaAs/InP quantum dot structures [243]. Rinaldi et al. [244][R27], [245] also reported observation of the Zeeman effect in parabolic InGaAs/GaAs quantum dots in magneto-luminescence experiments. Their photoluminescence spectra show splitting of interband transitions corresponding to quantum dot states with n + Iml < 5 induced by an axial magnetic field. Here, n and m are the principal and azimuthal quantum numbers respectively. The splitting is due to lifting of degeneracy of the excited states (Zeeman effect), similar to what was discussed in the FIR experiments. The authors concluded that the magneto-optical properties of strongly
Q u a n t u m dots
93 '
i
,
1
9
|
9
!
!
9
,
55nm 1.4e
54nm
i
,
9~ ,
~/
9
['
9
~
~
I
9
r
('
DO T ~ A
DOTS /
41nm
DOTS
/ ""
m-+1
,,~1.45 m--2 /
,,"
m= : --22 1
" , : , , . . 1.44
~176176176 v ~
mnnwm~ 0
2
4.
6
80
5
,""
m=O
I
10
0
2
4
6
8
B (T) F i g u r e 63 Experimental results on magneto-photoluminescence as a function of magnetic field for modulated barrier dots of three different diameters. Theoretical results (solid lines) are discussed in the text.
confined quantum dots reflect the single-particle states rather than excitonic effects. The effect of magnetic and electric fields on excitons in parabolic quantum dots was investigated by Jaziri and Bennaceur [246]. They calculated the energy and oscillator strength of a Is exciton in a parabolic quantum dot in the presence of parallel electric and magnetic fields. They found that application of an electric field results in a spatial separation of carriers leading to a decrease in the exciton energy and the oscillator strength. Application of a magnetic field leads to additional confinement that in turn leads to additional exciton energy and oscillator strength. For narrow dots (R0 < I00 .~, /~0 ~- V/~/P(-~)0, OgO : W e --- ~2h), electric and magnetic fields have little effect on the properties of the excitons. Kulakovskii et al. [247] reported magnetophotoluminescence spectroscopy of multiexciton complexes consisting of two and three excitons confined in InGaAs/GaAs quantum dots with lateral dimensions slightly greater than the exciton Bohr radius. They found that the Coulomb correlations in the two-exciton complex enhances the confinement. This additional confinement is strong at zero magnetic field but an increase of magnetic field results in a reduction of the effect. The three-exciton state was found to have an energy greater than three times the single exciton energy. Therefore a three-exciton complex is confined only by the geometric confinement potential of the quantum dot. In such a complex, the exciton-exciton repulsion is strongly reduced in a magnetic field [247].
94
T. C h a k r a b o r t y
A detailed account of photoluminescence spectroscopy on multiexciton complexes in InGaAs/GaAs quantum dots in the weak-confinement regime can be found in [248]. Wojs and Hawrylak [2491 studied the coupling of an exciton and an electron, both confined in a quantum dot and subjected to magnetic and electric fields perpendicular to the plane of the dot. They found that the presence of the additional electron in the dot significantly changes the low-energy absorption spectrum of an exciton. The magnetoexciton dispersion in a quantum dot was also studied by Bockelmann [250] and the interplay of CM and relative motion for an exciton in a quantum disk was studied by Adolph et al. [251]. Heller and Bockelmann [252] performed photoluminescence experiments on single quantum dots in magnetic fields. They studied the ground state and two excited states which split into doublets in the presence of magnetic fields. They found that the ground state spin splitting is smaller than that of the excited levels.
2.8
T i l t e d - f i e l d effects
Meurer, Heitmann, and Ploog [253,254] did FIR spectroscopy of field-effect confined quantum dots with diameters ~ 100 nm in GaAs-heterojunctions in a tilted magnetic field. For zero tilt angle (0 = 0~ their results are described in Sect. 2.3.2. For the tilt angle, 0 = 18 ~ their results are shown in Fig. 64. The FIR resonances show a splitting of the dispersion caused by resonant interactions with states confined in the growth (i.e., z) direction. The results look similar to the resonant subband-Landau level coupling observed in tilted-field cyclotron resonance experiments in a two-dimensional electron system [20,255]. To understand (at least qualitatively / the experimental results described in Fig. 64, consider a model of three-dimensional quantum dots [256]. Here the electrons are confined in a three-dimensional potential Vconf -- ~m I 9 02r2 ( x 2 + y2 ) + ~1m 9 WzZ 2 2 in the presence of an external magnetic field, where 02r is the frequency of lateral confinement and Wz is the frequency of confinement in the z-direction. Assuming circular symmetry in the xy-plane, the resonance frequencies can be obtained by solving the following cubic equation in 022: 026
--
024 (022 _4_ 02r2 -~- 02z2 ) -~- w2 ( 02202r2 s i n 2 0 +
2 02c2 02z2 COS2 0 -4- (Mr4 "4- 202r2 02z)
--
0 2 r4 0 2 z2
"-
0.
(2.23) For a choice of Wz which gives a good fit to the data, Eq. (2.23) was numerically solved and the results are also shown in Fig. 64 for comparison. In Fig. 64 (a), for 02r -- 11 cm -1, Meurer et al. obtained wz - 34 c m - 1 and in Fig. 64 (b), 02r - - 2 5 c m - 1 and Wz - 100 cm -1. The model of a three-dimensional dot describes the major features of the data. In particular, it explains why the resonance frequencies at large magnetic fields are determined by the total field, rather than the perpendicular component of the magnetic field.
Q u a n t u m dots
95
F i g u r e 64 Observed magnetic field dispersion of resonance frequencies of quantum dots in a tilted magnetic field. Theoretical results (labeled "Harmonic atom") are also given for comparison.
The case of two interacting electrons in a 3D q u a n t u m dot and a tilted field was investigated by Oh et al. [257]. They noticed considerable difference in the ground state properties in the presence or absence of a tilted field. At 0 = 0 ~ no spin transitions are noticed in the ground state of a spheroidal dot, but a spin transition does appear at 0 = 55 ~ at a magnetic field between 2 and 3 tesla.
2.9
Spin
blockade
in quantum
dots
It has been recently suggested that for a correlated electron system, spin selection rules can also influence the transport properties [258,259]. In contrast to the "charging model" where the excitations are treated within a single-electron picture, spin selection rules result from fully correlated states. The spin blockade mechanism which is related to the spin polarized states, is supposed to result in negative differential conductances observed experimentally [Sect. 2.3.3]. Basically, the blockade is due to a decreased probability for states with m a x i m u m spin (S = N/2) to decay into states with lower electron number. In contrast to transitions that involve states with S < N/2, transitions in the fully spin polarized case are possible only if the total spin S is reduced. Therefore, the current is reduced at a voltage of the order of excitation energies of the S = N/2 states. Another spin blockade effect would occur in the transitions
(Eo(N), S) +~ (eo(X - ~), S'),
IS- S'l > 2,
(2,24)
T. Chakraborty
96
i.e., if the total spins of the ground states corresponding to N and ( N - 1) electrons, 1 This should affect the peak heights of the linear conductance. differ by more than 3" These spin effects are suppressed by a high magnetic which renders the ground state fully polarized [258,259]. We have already discussed in Sect. 2.5.5 that spin transitions in quantum dots are entirely due to electron correlations. Imamura, Aoki, and Maksym [260] investigated the spin blockade condition as an effect of total spin dominated by the magic angular momenta. Their system consisted of three and four electrons in parabolic quantum dots in an external magnetic field. The numerical results showed that the spin blockade condition [Eq. (2.24)] is indeed fulfilled. As an example, in their model system with confinement 1 1 3 potential ha~0 -- 6.0 meV, the ground state changes as (Y, S) - (1, 3) --~ (2, 3) --~ (3, 3) for N = 3, and (g, S) = (0, 1) --~ (2, 0) -~ (3, 1) --~ (4, 0) --~ (5, 1) --~ (6,2) for N = 4. The spin blockade is found in the region 4.96 < B < 5.18 tesla. For a double dot system (vertically Coulomb coupled), the spin blockade condition is fulfilled for a wider range of magnetic field than for a single dot.
2.10
Q u a n t u m dot molecules
In addition to the work on the electronic properties of coupled dots described above, there is also a large body of work on the transport properties of two dots in close proximity. Investigations of discrete electronic states of coupled quantum dots placed a tunneling distance apart - the quantum-dot-molecule states, began with the work of Reed and his collaborators [44]. Their starting system was a double quantum well triple barrier structure designed to have two quantum dots connected in series between quantum wire contacts. Their observed current-voltage characteristics indicated significantly sharper peak in the coupled-dot spectra compared to the single-dot spectra. Tunneling through coupled QDs was expected to be strongly influenced by the quantization of energy levels in individual dots [44,261-265]. As the tunneling between the two dots is primarily elastic, the energy states of one dot need to align with the energy states of the other dot for interdot tunneling and hence transport through the entire system to take place [44, 261,262,266]. Using a tunable double QD system with well-developed 0D states in each dot, van der Vaart et al. [263] exploited the Coulomb blockade of tunneling to control the number of electrons in the dots. They observed sharp resonances in the current when there is matching of energy of the two 0D States in two different dots. This result demonstrated that transport through a double dot is reasonably enhanced when the energies of the two 0D states match. Waugh et al. [267,268] investigated low-temperature tunneling at B = 0 through double and triple quantum dots with adjustable interdot coupling. They noticed that interdot
Quantum dots
97
A vertically-coupled double-dot device. The dashed lines define the depletion layer at the pillar surface, that confine the electrons to the interior of the pillar [64].
Figure 65
tunneling leads to a variety of phenomena not observed for single dots. One greatly discussed phenomena is that, as interdot tunnel conductance is increased, Coulomb blockade conductance peaks split into two peaks for double dots and three for triple dots. For weak tunneling, the observed peak splitting approaches zero and in the strong coupling regime where the two dots essentially merge, the splitting saturates. These results are consistent with the theoretical predictions for double dot systems [265,269]. Peak splitting in coupled QDs has been studied experimentally by Livermore et al. [270] who presented a unified picture of the evolution of the coupled dot system from weak to strong tunneling regime. In the former case, capacitive coupling is dominant, while in the other case interdot tunneling dominates. Blick et al. [271,272] investigated the charging diagrams of a double-dot system (DDS), connected in series and coupled by a tunneling barrier. Such a system was described be these authors as an artificial molecule where electrons are shared between the two sites. The charging diagram was generated by measuring the conductance through the DDS while the electrostatic potentials of two independent gates was varied. These authors
T. Chakraborty
98
40
30
~> E O
20
> I
C
>
10
0
4
8
12
16
0
4
8
12
16
B (T) F i g u r e 66 The current-step position of (a) low- and (b) high-bias I - V staircases as a function of magnetic field. The step positions are plotted with respect to the first step. The solid lines are the energy levels (with respect to the lowest energy) for a harmonic confinement potential [275].
observed a coherent tunneling mode or molecular-like state in the DDS which leads to a finite conductivity even if the gate voltages do not exactly match the positions where sequential single-electron transport through the DDS is allowed. This coherent mode manifests itself as a tunnel splitting in addition to the Coulomb interaction in the charging diagram of the DDS [271,272]. Blick et al. also investigated the Rabi oscillations between two discrete states in these artificial molecules induced by externally applied high-frequency radiation [273,274]. In Refs. [64,275], Schmidt et al. investigated single-electron t r a n s p o r t through a vertically coupled double dot system (Fig. 65) and thereby explored the single-particle regime of a strongly-coupled DDS. This system was created by imposing a submicron lateral confinement on a triple-barrier heterostructure. Their I - V curve exhibited steps similar to SET in single dots. This is in contrast to the sequential SET through two QDs in
Quantum dots
99
series that leads to sharp peaks in the I - V curve [261-263]. Schmidt et al. attributed these current steps to SET through discrete single-particle states extended over the two identical dots due to coherent interdot coupling. The DDS can then be pictured as ionized
artificial hydrogen molecule. In order to compare the position of the current steps in the I - V curve observed experimentally with the single-particle energies of the DDS, Schmidt et al. plotted the bias-voltage differences as a function of magnetic field (Fig. 66). The observed results are then compared with the computed energy levels (E~,l - E0,0) for parabolic confinement (see Sect. 2.1.1). The agreement between theoretical and experimental results is generally good except near the degeneracy points of the theoretical curves, where intricate anticrossing behavior is observed. This signifies deviations from the perfectly rotationally symmetric state assumed in the theoretical results. Austing et al. [276] investigated the addition spectra of vertically coupled double quanturn dots. They obtained the Coulomb diamond diagrams from strongly coupled (dots separated by a 25A barrier) and weakly coupled (dots separated by a 75A barrier) and extracted the shell filling just like in single dots. Not all shell fillings were observed for the DDS. The addition spectra of double quantum dots was calculated by Tamura [277], who considered a 3D electron system with parabolic confinement in the xy-plane and a squarewell potential in the z-direction. The energies were calculated via the unrestricted selfconsistent Hartree-Fock approach. It was suggested that the absence of shell-filling according to Hund's rule, observed in experiments by Austing et al. [276], is due to the effect of dot thickness. A detailed theoretical study of the electron correlation effects on the electronic states and conductance in a vertically coupled DDS has been reported by Asano [278]. For the parabolically confined system in the xy-plane and square-well potential in the growth direction, the total spin momentum of the ground states was calculated as a function of the total number of electrons and the distance between the dots by using the numerical diagonalization method. Two different regimes were considered. In the strong coupling regime (i.e., the distance between the dots is small) a correspondence was found between the ground states in the DDS and those in real diatomic molecules bound by electrons in the 2p orbital, i.e., B2, C2, N2, 02 and F2. This happens because the structures in the single-particle levels in the two systems are qualitatively similar. In the weak coupling regime (i.e., the distance between the dots is large), the effect of electron correlations on the spin structure of the ground states is quite significant. Electronic states with a small spin momentum are stable for even number of electrons while states with a large spin momentum are stable for odd number of electrons. The reason put forward by the author is that for even numbers of electrons, two electrons localized in different dots form a singlet pair due to electron correlations and interdot hopping. The ground state is then well described by a combination of such singlet pairs. For an odd
100
T. Chakraborty
number of electrons, the ground state is similar to Nagaoka's ferromagnetic state, that is the spins of all electrons align parallel in order to decrease the kinetic energy of the electrons. Interestingly, the physical picture of the ground state in the weakly coupled DDS was found to be analogous to that in the Hubbard model near half filling. The conductance of the DDS was also calculated by Asano, who found Coulomb oscillations in the DDS. For large separation of the dots, the amplitudes of several conductance peaks are suppressed due to electrons correlations (spin blockade). Finally, as in single-dot systems, PAT (see Sect. 2.4.5) has also been studied in the DDS [51][R33] [274,149,150, 279-281] coupled in series. Resonant 0D,to 0D PAT occurs when the 0D levels in the neighboring dots are separated exactly by the photon energy.
2.11
Non-circular dots
In this section, we discuss the properties of quantum dots which are somewhat different from the circular shape discussed so far. Two cases are of particular interest" elliptical dots and stadium shaped dots.
2.11.I Elliptical quantum dots Experimental work on the ellipticalquantum dots discussed above [126] inspired us to look at the energy spectrum in those systems [282]. The confinement potential of the dots studied by McEuen et al. [87][R13], [88][R14] were also anisotropic. Theoretically, anisotropy in quantum dots has been treated earlier as a perturbation [283] to the isotropic parabolic quantum dot. However, that is not expected to be correct for large anisotropy. Just like for a a circular dot, one can derive the single-electron results analytically for anisotropic dots. Let us consider a lone electron in a lateral anisotropic parabolic confinement potential in the presence of a quantizing perpendicular magnetic field. The Hamiltonian is then written 1
eA
--~ ~conf (x, Y)
(2.25)
where the confinement potential is ~conf(X, y) __ ~m~ 1 ((.dx2X2 "t- Cdyy ),
Cdx r COy
1 , O) and make the Let us choose the symmetric gauge vector potential A - ~B(-y,x following transformations
Quantum
dots
101
x
=
Y
-
X2 - ~P2 sinx, X X2 q2 c o s x - - - P l s i n x , X
qlcosx
Px
=
Pl COS X -~- m X1 q2 sin X,
Py
--
P2 cos X + X--2-1ql sin X. X
T h e s e are c o n s i s t e n t w i t h t h e c o m m u t a t i o n
X1X2
-
-
r e l a t i o n s [p~, qj] -
-ih6~j
a n d [q~, qj] -
0 if
X 2.
Accordingly,
we
~_{
rewrite the Hamiltonian
1 2 2?Tte [p2x _It- ~21X2 _it- py ..jr..~ 2 y 2 + ?TZeCOc (YPx -- X p y ) ] ,
__
2
(022x,y._~_ 1
(2.27)
2
a n d w~ is t h e u s u a l c y c l o t r o n f r e q u e n c y . It i s d i a g o n a l if _ [~1 (~2 _.]_ ~2)] 89
X
[Xl -1_
x2 x tan2 X
-~o~c [2 (a~ + a~)]~ / (al~ - a ~ )
--
L e t us define
~ = [(~ -
a~) ~ + 2 ~ e ~ ( ~ + a~)] ~
T h e H a m i l t o n i a n is t h e n f u r t h e r t r a n s f o r m e d as
= ~
1
22
22
(~P~ + ~p~ + ~q~ + &q~),
(2.28)
with t212 + 3 ~
~
=
+ ~
2 (~1 + ~ ) m l~ + a ~ - a~ 2 ( ~ + a~)
(2.29)
~i _ ~I (a~ + 3 ~ - n ~ )
T. Chakraborty
102 15
10 ;>. o v
E
0
2
''''
25
I'''
(b)
_
4
i"]",,,
6
~ I''''
1'''
'-
_
20 ,,~..._-.---.
_ ID
E 15
. ~
~10
.~
~
,
.~
.
. ', .
:--------
-
.
....
.
_
.
~-~
-
, .,,
i
O r ....
0
..
,,
I,,,,i
1
2
3 hr
4
5
c
F i g u r e 67 (a) Energy levels of an anisotropic quantum dot as a function of the magnetic field (haJc in meV) for a~x - 1.0 meV and a~y - 1.1 meV. The lines are drawn in ascending order of (nx, ny), as indicated. (b) The Chemical potential (in the CI approximation) for the energy levels of (a) [282].
T h e e n e r g y eigenvalues are t h e n easily o b t a i n e d to be [284,285] 1 1 f--'n=,nu -- (?~x + -~) ~Cdl -'t- (ny -Jr- -~) ~Cd2,
(2.30)
where (M1 - - Ctl/~l/TYt e a n d aJ2 -- a2/32/me. T h e energy Eq. (2.30) has t h e following limiting behavior" at zero m a g n e t i c field, t h e s y s t e m b e h a v e s like a pair of h a r m o n i c oscillators in t h e x a n d y directions. For a large m a g n e t i c field (~c >> a~x,ay), we get Enx = 1 (nx + 3) hCOc,i.e., L a n d a u levels form as in t h e case of isotropic p a r a b o l i c confinement. W h e n a~ - coy, i.e., t h e c o n f i n e m e n t is isotropic parabolic, nx - n + ~1 l l l - 11 a n d
Q u a n t u m dots
103
25 20
:: :~
-
L
,
.
.
.
.
~ _ _ _ _ x
.
.
.
- -
~
................... (~,) (~x=
0
2
-
-
.
~io; ~,=s.o) :I 4
6
30
~. 2O > v
10
0L, 0
"
, .... 2
,
t
I
i
I
4 he0c
Figure 68 Same as in Fig. 67, but for a~y = 5.0 meV [282].
1 ny = n + -~lll + 1l, where n and I are the principal and azimuthal q u a n t u m numbers, respectively. Also, when cox -~ coy, the energy levels are very similar to that of the isotropic case except that the (2n + Ill + 1)-fold degeneracies at B = 0 are lifted [282] as a result of breaking of the circular symmetry. A similar situation arises when the circular symmetry is broken by Coulomb coupling between two neighboring dots [71][R19]. The selection rules for the transition to higher energy levels can also be calculated from the dipole transition matrix elements [71][R19], and are as follows: polarization along the x- or y-axis, (i) An~ = 0, Any = +1, (ii) An~ = +1, An v = 0. There are just two modes as in the case of isotropic parabolic confinement [30][R7]. The only major
104
T. Chakraborty
difference here is that at B = 0 the two modes split, A E = h(cox- COy). This mode splitting has indeed been observed experimentally by Dahl et al. [126]. In Fig. 67 (a), we show the magnetic field dependence of the single-electron energy levels for a quantum dot with COx = 1.0 meV and COy = 1.1 meV. For this choice of COx and COy, the deviation from the circular dot is minimal and therefore, the energy levels are very similar to those of a circular dot except at the origin where the degeneracies are lifted. Qualitatively similar results were obtained in perturbation calculations [283]. In this figure, we also present the chemical potential calculated in the CI model discussed in Sect. 2.3.1 [Eq. (2.8)]. In those results, we include the Zeeman energy with a g-factor to be 0.44 and the effective mass of rn* = 0.067me, appropriate for GaAs. We have also used U = 0.6 meV, taken from the work by Ashoori et al. [76][R21]. The energies and chemical potentials for COx = 1 meV and COy = 5.0 and 10.0 meV are plotted in Fig. 68 and Fig. 69 respectively. Clearly, the level crossings shift to higher energies as COy is increased, and oscillations in chemical potentials are also suppressed at lower energies. For example, when COy = 5, the oscillations are suppressed for ne = 1 - 12 and for COy = 10, it happens for ne = 1 - 22. In addition, the amplitude of the oscillations decreases considerably with increasing anisotropy. On the other hand, the magnetic field threshold beyond which the oscillations in chemical potentials cease to exist increases with increasing coy. With this increase of the magnetic field threshold, the oscillations also move to higher magnetic fields like the observed experimental results of Ashoori et al. It is also to be noted that the confinement potential for the q u a n t u m dots in the experiments of McEuen et al. [88][R14] is anisotropic with COy/COx~ 4.4, which lie within the range of COx,y considered here. Madhav and Chakraborty [282] studied the two-electron states in an elliptical quant u m dot interacting via the potential of the form v(r) cv 1 / r 2, which has the advantage that most of the analysis can be performed analytically. The anisotropic system with interacting electrons were investigated via perturbation theory. For the Coulomb potential, Maksym [286] studied the eigenstates of two and three interacting electrons in an elliptical dot (2.26) by using the basis states of a circular dot [Sect. 2.5.2]. He found that the ground state in this case can undergo transitions similar to those of a circular dot but some of those in a circular dot do not survive the lowering of the symmetry. Transitions corresponding to cases where the angular momenta on either side of the transition in the circular dot differ by an odd integer do survive. However, transitions between states of the same spin whose orbital angular momenta differ by two are forbidden in the elliptical dot. Very recently, Austing et al. [287, 288] have reported their studies of ellipsoidally deformed few-electron vertical QDs. From measurements of the addition energy they found that even a small deformation of the circular dot radically alters the shell structure that they observed earlier in a circularly symmetric QD (see Sect. 2.4.6). As the deformation
Q u a n t u m dots
105
40
30 :>
E v 20
10
0 0
40
5
i
' '
I '
.... " '
10
i ' '
'-i
~
15
'
'
'
I~
'
"
____:
30 >
E
20
10 .....
,
2
,,
!hi .....
4
6 he0
F i g u r e 69
,
.....
,,
8
e
Same as Fig. 67, but for aJy - 10.0 [282].
of the circular dot is introduced, the shell structure is either disrupted or smeared out. This was a t t r i b u t e d directly to lifting of degeneracies of single-particle states present in a circular dot, as discussed above.
2.11.2 Other asymmetric potentials As we noted earlier, experiments of Tarucha et al. [48-50] on the tunneling of electrons through a q u a n t u m dot and the observation that the addition energy of an electron in the few-electron dot reveals the existence of shell structure, p r o m p t e d theoretical
T. Chakraborty
106
L_ ___.~
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
F i g u r e 70 Schematic diagram of the quantum dot stadium [294].
investigations by several groups. Ezaki et al. [54] reported exact diagonalization of a fewelectron Hamiltonian with various asymmetric potentials. They modeled the experiment of Tarucha et al. by the following form of the confinement potential,
Y) --
+
) [1 +
cos a r
(2.31)
where c~ = 0, 1 and r is the angle with respect to the specific axis in the xy-plane. For = 0, we get the elliptical dot (~x ~-~y). Setting c~ = 1 and ~x = ~y, one gets a triangular shaped confinement potential. The he-electron eigenstates were obtained by diagonalizing the ne-particle interacting Hamiltonian. The basis states were constructed from the Slater determinants composed from single-electron eigenstates of the isotropic and harmonic system. The calculated addition energy, or the energy needed to add one more electron to the q u a n t u m dot, was plotted as a function of the electron number in the dot. In a circular dot, the addition energy was found to be quite large for ne = 2 and 6. This was a t t r i b u t e d to complete shell filling. A somewhat weak peak was seen at ne = 4, which was interpreted as due to a spin polarized half-filled shell. In an elliptical dot, the addition energy does not have a clear structure (except at ne = 6) which was a t t r i b u t e d to a s y m m e t r y of this system. The a s y m m e t r y leads to splitting of the degenerate singleelectron eigenstates and mixing of many eigenstates with various angular momenta. In a triangular dot, slightly stable states due to localization of electrons at the corners were observed.
Quantum dots
2.11.3
107
Quantum dot stadium
It is well known [289] that the motion of a classical particle in a closed stadium is chaotic. Studies of quantum analog of such systems showed the eigenvalue spectrum follows the Gaussian orthogonal ensemble [290], and the most stationary states usually concentrate around narrow channels called scars which resemble the classical periodic orbits [291]. Experimental work on chaotic scattering has been reported on ballistic GaAs-heterostructures [292]. The magneto-resistance across a stadium-shaped dot has been measured. In the tunneling regime, i.e., a stadium that is classically isolated, the magneto-resistance shows periodic oscillations at high magnetic fields [293]. Theoretical results of single-electron states of a quantum-dot stadium in a magnetic field have been reported recently by Ji and Berggren [294]. They calculated the energy spectrum, two-dimensional spatial distribution of the charge and current density to look at the transition from the chaotic to regular behavior as the magnetic field is increased. Here the model is a stadium shaped wall (Fig. 70) where two semicircles of radius R are connected by two parallel, rectilinear intervals of length 2L = 2R. For spinless electrons, the single-electron Hamiltonian is 7-{ = ~
1
( - i h V + cA) 2 -Jr- ~conf,
where V~onf is the confinement potential and A - ~1B ( - y , x 0) is, as usual, the vector potential in the symmetric gauge. The electrons are confined by infinite walls. Inside the dot Vconf : 0. The Schr6dinger equation was solved numerically and for each eigenstate, the two-dimensional current density ~t
J(x, y) - - m , I m [r162
e
,
+ ~%-A I~(x y)
]2
and the probability density p(x, y) - I~(x, y)l 2 were also numerically evaluated [294]. Looking at the energy spectrum Fig. 71, these authors noticed that [294] at high magnetic fields the states were found to converge to degenerate bulk Landau levels, as seen earlier for a circular dot. At low magnetic fields, there are crossings and anticrossings of the energy levels. The anticrossings or level repulsions arise due to the nonintegrability of the problem and are a signature of quantum chaos. With increasing magnetic field, the level repulsion becomes weaker. From the charge density and the current pattern in the stadium, these authors determined how the electron motion changes from chaotic to regular behavior as the magnetic field is increased.
T. Chakraborty
108
d
.,,.
....
.-':.;
:,,~.: ............. .... .
' . .9... . . . . . . . . . : - :- . . . . - " ..... . - " :',', ."
10
. . . . ,I . . . . :o.:;;:" -" " , s ' " -...-":,',p ........ .. .z.:.. ;8-eoe:_'" .-" "ell: ..... ". ."'.'.;'/ ~ ~ ..... .,,,,'*~ oe'" 3-~ 9~ 9 * e" .e... "o ..,o o ' ~ eo ,,e ooo igiioellOOl.._ ..11 ..... . .S * .$'$~," *e** -o ..l e . 1 2 ~ II' . . . . . .-......:... . "---" "+ . . . ....... " -" ""+ ..::,, ....... ::,.s::::u '- "....... ....9.0,%8 ,.: l e ..:..:m.tim- e**.
....... ;.
........
".... ..... .
, ....
..."
:,"
9
"'.o""
,..: . . . . . . . . . .
O
,,
_~, S s
:,..; ....
,:~."
,,...:a . . . .
..:.. ....
0
::,.: .... ..~;i.':
2 B (T)
.......
...........
: . t ' l ~ +, "'--. ........
..,.:,_.~;~,
~
1
:..:,.;j
-.:, . . . . . . : | p . . .
! iiiiiii ii ili ::ii+;!!
t2
"',," - ...... . - 0'
:..:,,,:/---'--:"" ::,.:-::i;illJP- ..... .-" S'.. ::,=-'" . . . . : , j ~ .
:.:
. . . . . :.::::" . . . . . : : : : . , : ~ . , . . ~ : : - :,...,.: "-:::'... .... .,
B
"'. ........
"
...: .... ,: . . . . . .
,,,: . . . . . . . .
.... : +:'" 8 . ... ... .. . " -"" - -- 'r
.. .:,
4 2
"
"'" -
- . . . . . ,, A ".....
9. . . . . . . . . . . . . .
.................
3
4
F i g u r e T1 Energy levels of the quantum dot stadium as a function of the magnetic field. The dashed lines correspond to the Landau level energies of an ideal two-dimensional system [294].
T h e d i s t r i b u t i o n of energy-level spacings is defined as P(s) = s/{s}, where s is the level spacing and {} d e n o t e the m e a n value, and is a statistical m e a s u r e of the s p e c t r u m . P(s)ds is t h e probability of finding a s e p a r a t i o n of neighboring levels b e t w e e n s and s+ds. A t B - 0, one finds a G a u s s i a n o r t h o g o n a l ensemble d i s t r i b u t i o n P(s) - (?rs/2) e -~s2/4. At i n t e r m e d i a t e values of the m a g n e t i c field, the level spacing d i s t r i b u t i o n is G a u s s i a n u n i t a r y ensemble type, P ( s ) - (32/+r 2) s 2 e -(4/~)s2 and at higher m a g n e t i c fields, these a u t h o r s find a d i s t r i b u t i o n close to a Poisson distribution P(s) = e - s as e x p e c t e d for a regular system. T h e s t u d y of q u a n t u m dot s t a d i u m , as r e p o r t e d in [294] could be i m p r o v e d by using a m o r e realistic confinement, such as a parabolic potential. More i m p o r t a n t l y , the role of electron-electron in this s i t u a t i o n should be studied. M a n y e x p e r i m e n t a l results on q u a n t u m - d o t s t a d i u m have been r e p o r t e d in the literature which invites m o r e theoretical work on this system.
Quantum dots
109
Related topics
3.1
Q u a n t u m antidots
Antidots - a structure topologically complementary to quantum dots have been described in the literature as "an array of island voids rising out of a two-dimensional electron sea" [2]. Figure 72 shows the cases of (a) weak and (b) strong modulation of the conduction band edge in the zy-plane of a two-dimensional electron gas. In the former case, the periodically varying potential is smaller than the Fermi energy. The energy spectra of such a system in the presence of an external magnetic fields are very interesting because of the remarkable self-similarity it displays called "Hofstadter's butterfly" [295]. In the second case, the Fermi energy (shown as dotted planes) intersects the repulsive potential peaks and the electrons are excluded from these regions due to the potential barrier. The electrons are, of course, free to move in the valleys. These repulsive potentials are called the antidot potentials [296], [297][R34], [298-303], [304][R35], [305-314].
3.1.1.
Commensurability oscillations
In GaAs heterojunctions the elastic mean free path of the two-dimensional electrons is about 5-10#m. When periodic potential modulations are created, rather than the randora scatterers in a 2DEG, with a perpendicular magnetic field the transport properties are modified from those of a pure 2DEG. This happens due to the interplay of the modulated potential and the cyclotron orbit. Experiments on a 2DEG subjected to a weak one-dimensional lateral superlattice potential revealed unusual periodic low-field magnetoresistance oscillations as a function of 1/B. The periodicity was found to be in accordance with the commensurability of the cyclotron diameter at the Fermi energy:
110
T. Chakraborty
a
b
E
x
x
Figure 72 (a) Weak and (b) strong modulation of the conduction band edge in the xyplane of a two-dimensional electron gas. The dotted plane represents the position of the Fermi energy [301].
2Rc - 2 h k F / e B and the modulation period [315,316]. This has been explained with a quantum mechanical picture of Landau band formation [317,318] or with a classical picture of guiding center drift [319]. The starting point for fabricating antidots is a high-mobility 2DEG grown at the interface of GaAs/A1GaAs heterojunction. The high mobility of the 2DEG turns out to be essential to ensure that the electron mean free path is much larger than the period of the antidot potential, a. In the transport measurements of Weiss et al. [301], the antidot array was part of a conventional Hall bar geometry as sketched in the top inset of Fig. 73 b. Transport measurements reveal distinct transport anomalies that results from commensurability between the cyclotron radius Rc = hv/27rns/eB, where ns is the planar density of the 2DEG, and the period a. The Fermi wavelength ) ~ F - - V/27r/ns measures the extent of the wavefunction at B = 0 and is smaller than the period (a 2 0 0 - 400nm). Consequently, electron transport can be treated semiclassically where the electrons bounce ballistically like balls through the antidot lattice. In Fig. 73 the magnetoresistance pxx and Hall resistance pxy of patterned and unpatterned areas of the device are compared. A double peak structure in pxx for the magnetic field values where Rc ~ 0.5a and Rc ~ 1.5a reveals striking deviations from the simple Drude results, pxx = m * / e 2 n s r , where r is the Drude relaxation time, and pxy - B / e n s . The pxx peaks are accompanied by nonquantized steps in pxy as shown in Fig. 73. At higher values of magnetic field (2Rc < a), p~x drops precipitously, and pxy shows the
Quantum
dots
I 11
1200 1000I J/800I 600i' / ,J ~oo I ~-i t
a)
/1
"r--- 1.~ K
/q
a=a00nm, /'-~'i/ /" /i"~/~ i/
....//~ / ~ool _.,,,~/~//~I/I " o .r
"
." .... l..
~
,~P
~ ~, ~ , _ / 0
,
/b,T
: IUL'r-]ilOOI.Jm/"~
~,~ ,~o.mj 0 L/"
il.
'
,
1
1
/"-" / -o.~---/~ ......o.5l
......
,,
,,
2
I
3
..... l . . . . .
4
,
5
BIT) F i g u r e r a (~) Magnetoresistance and (b) Hall resistance in unpatterned (dashed line) and patterned (solid line) areas of the sample. The arrows indicate the magnetic field positions where Rc/a = 0.5 and 1.5. Inset: the sample geometry (schematic) [301].
expected quantized Hall steps. For these values of the magnetic field, the patterned and unpatterned segments of the sample display identical behavior. An interesting result is that close to B = 0, the Hall resistance is quenched (inset of Fig. 73 b). In the low field regime, the number of observed peaks and steps depend on the effective cyclotron diameter d and the period a of the antidot lattice. Figure 74 compares the magnetoresistance results of three samples, studied by Weiss et al., with periods a = 200 nm (curve 1) and a = 300 nm (curves 2 and 3). At each peak observed in Fig. 74, there is a relation between Rc and the commensurate (circular) orbit which encircle a certain number of antidots, as indicated in the inset of Fig. 74. Of course, there are some exceptions to this result, for example, the prominent low field peak in curve 2 ( R c / a ~ 1.5, derived from the peak position at B ~ 0.18 T) is not commensurate with the lattice. Some authors have suggested that this distinct peak arises from electrons on chaotic trajectories trapped on paths around four antidots [312].
T. Chakraborty
112
T-
-
"
15 K
f~
"
/
4
/ ~
"I"
" "2"
. .
~4' .(.
/'~ ~
]
"
"-"
o/i~~\-
9 ,//,
/ \,,,-,
3
"
. (~',(.'-'~1-/i.
t"\e
"
"
"
Ok, t "9
9 9
:~.
")'(~'~" ,,/~
9 9 9 e/~,4\e
c~ 0.6 0.4
0.2
.a
~/
/%
o.o 0.0
0.5
!.0 B (T)
1.5
2.0
Figure 74 Anomalies in low-magnetic field for three samples. The peaks in trace 3 can be explained by the commensurate orbits around 1,2,4,9, and 21 antidots (inset). The arrows indicate the corresponding Rc/a values 0.5, 0.8, 1.14, 1.7, and 2.53 respectively. The dashed arrow for trace 2 indicates the position of an unperturbed cyclotron orbit around four antidots (Rc/a = 1.14)[301].
The observed commensurability oscillations described above for a square array of antidots were initially explained at low fields as being a consequence of the classical cyclotron diameter fitting around groups of antidots, and thereby forming pinned electron orbits. These electrons obviously do not contribute to the transport process. To understand the physical origin of the pinning mechanism and the appearance of additional peaks [see, Fig. 74 (a)] not accounted for in the simple picture of pinned orbits, Fleischmann et al [312] solved the classical equations of motion in the presence of the potential
U(z, y) = Uo [cos(27rx)cos(27ry)] z
(3.1)
where/3 controls the steepness of the antidots and U0 is an adjustable parameter to fit the experimental situation [312]. These authors numerically traced electron motion in the antidot lattice and then used the linear response theory [320] to evaluate the conductivity. Their conclusion was that the magnetoresistance peaks are not due primarily to varying number of pinned orbits, but rather due to chaotic trajectories. Peaks in the diagonal resistance are due to chaotic trajectories of the electrons that are trapped for long times
Q u a n t u m dots
113 0.8 0.7 0.6
0.5 c~
0.4 x
Q.
0.3 0.2
0.1 0.0 0.6
A
b
0.4.
'\
/'
o~ /A ' ~ / ' "
"f~'v/z-'% 9 ~/ 9"~ L" ~" " ~ - -9~ - "")" "
,~/
,'~,
Ixl
~ \
x 0.3
t
t
t
0.2
7
3
1
0.1
T=1.5 K
-t,N"
~
" "
. . . .
~_.x
/..k ~
//~ kJ
\
0.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 .
.
.
.
.
.
.
.
.
.
-~
'
"
B(T) F i g u r e 75 (a) Magnetoresistance in a rectangular (ax = 790 nm, ay = 410 nm) and hexagonal (a = 410 nm) antidot array. In (a) L and S denote the direction of current flow with respect to the long and short period of the lattice (inset). The magnetic field positions corresponding to free cyclotron orbits around 1,2,3, and 6 antidots with Rc = 0.5, 1, 1.5 and 1.83, respectively are also shown. In the hexagonal lattice (b) the arrows indicate orbits around 1, 3, and 7 antidots at Rc/a = 0.5, 0.87 and 1.37 [301].
in the vicinity of antidots when the cyclotron orbit fits into the antidot lattice. Baskin et al. [313] calculated electron trajectories in an antidot lattice with hard walls. T h e y found an enhanced diffusion coefficient for 2Re = a (the lattice period), leading to m a x i m a in the conductivity, due to the so-called runaway trajectories t h a t bounce off neighboring antidots and channel between rows of antidots. T h e electron motion thus remains partially ordered. These authors d e m o n s t r a t e d t h a t the oscillation peaks of their calculated diffusion coefficient coincide with the peaks of m e a s u r e d resistance in the antidot lattice. In addition to the studies of commensurability peaks in square antidot arrays described
114
T. Chakraborty
thus far [296], [297][R34], [298,300-303], [304][R35], [305], transport measurements were also reported on rectangular [301,302], [304][R35], [306-309], triangular [301], [304][R35], [307,310,311], hexagonal [301,309,310], as well as disordered arrays of antidots [304][R35], [308]. These experiments suggests that the sequence of observed peaks reflects the geometry of the antidot lattice. In a rectangular lattice, pxx depends on the direction of the current flow as indicated in Fig. 75 a. Similar results in a hexagonal lattice, as shown in Fig. 75 b, show peaks when the cyclotron orbit accommodates 1, 3, and 7 antidots. Schuster et al. [306] noticed that in a rectangular array of antidots, the magnetoresistance is highly anisotropic. It reveals pronounced low-field maxima whose positions depend on the direction of current flow. For current flow along the long period direction of the antidot lattice, ballistic electron orbits around a few antidots are preferentially probed. Along the short period of the lattice, transport is dominated by trajectories around many antidots. In contrast, the Hall effect does not depend on the direction of current flow with respect to the lattice orientations. It is difficult to explain the anisotropy in terms of the simple pinned orbit theory [297][R34], [311] because occurrence of the pinned orbits does not depend on the direction of the current flow. In order to explain the anisotropic behavior, Nagao [314] performed numerical simulations on an anisotropic periodic array of hard disks (antidots). This model is the anisotropic version of Fleischmann et al.'s model for a square lattice. Considering an array of hard disks in a 2D plane, each with radius a/6 and the disks centered at points (x, y) = (2ma, na), m, n = 0, +1, + 2 , . . . , this author argued that the strong peak in pxx at B = Bc = 2m*vF/ea, where vF is the Fermi velocity, is caused by diffusive trajectories with an anisotropic character. Schuster et al. [306] argued that, as in a square lattice, electrons move on chaotic trajectories. For certain magnetic fields, there exists pinned trajectories encircling an integer number of antidots. Since the presence of pinned orbits around groups of antidots in the linear response regime does not depend on the direction of current flow, conductivity of the chaotic trajectories is responsible for anisotropic transport behavior. There are several reports on attempts to present a quantitative analysis of the commensurability peaks in antidot arrays [321-326]. Quantum mechanical calculations of magneto-transport properties [324-326] are particularly interesting because they are possibly relevant for experimental results at low temperatures where one observes oscillations superimposed on the main commensurability peaks [327]. Moreover, the classical description of antidot lattices ceases to be valid when the lattice periods approach the Fermi wavelength. Studies of antidot arrays have recently taken an interesting turn with the observation of the fractional quantum Hall effect in antidot arrays [328][R36], [329]. The original motivation behind searching for FQHE in antidot arrays was primarily to look for a
Q u a n t u m dots
115
o /~['~L% .
(o
.~o
9
.%_g.|
3
i
_
2 13
1
3
5
2
7
9
i
_
I
200fl
I
0
5
.
I I~
1
10
15
B(tesla)
F i g u r e 76 Magnetoresistance Rxx of the patterned (upper trace) and unpatterned (lower trace) sample. The antidot superlattice was nearly a square. The fractions near the top of the figure are the Landau level filling factors. Inset: schematic commensurate orbits encircling 1, 4, and 9 antidots [328][R36].
s i g n a t u r e of the presence of the " C h e r n - S i m o n s gauge field" particles [330]. In this app r o a c h one p e r f o r m s a singular gauge t r a n s f o r m a t i o n 9 t h a t represents each electron as a fermion a t t a c h e d to a & f u n c t i o n flux of size q(I)0 with q even a n d (I)0 = hc/e is the flux q u a n t u m . T h e a t t a c h e d flux can be r e p r e s e n t e d as a coupling of fermions to a gauge field - the C h e r n - S i m o n s field. In a mean-field a p p r o x i m a t i o n (no interparticle interaction a n d the a s s u m p t i o n t h a t the t r a n s f o r m e d fermions form a s t a t e of u n i f o r m density) and the absence of any i m p u r i t y p o t e n t i a l the fermions see a vanishing net m a g n e t i c field, provided the L a n d a u level filling factor is ~, - q1 a n d the sign of the a t t a c h e d flux is chosen appropriately. T h e resulting g r o u n d state is a filled Fermi sea of t r a n s f o r m e d particles with Fermi wavevector kF = (47me) 89 where ne is the areal density of the electrons. A w a y from ~ - q, 1 the particles are e x p e c t e d to execute cyclotron m o t i o n with 1 radius Re - h k F / e A B in an effective m a g n e t i c field A B - B - B ( ~ - ~). If t h a t cyclotron radius m a t c h e s with the m o d u l a t i o n period one would expect oscillations in magnetoresistance. Figure 76 shows the m a g n e t o r e s i s t a n c e Rxx of 600 n m period a n t i d o t s c o m p a r e d to 9 For details, see Refs. [330, 20].
116
T. Chakraborty
6
"
I .
.
.
.
.
.
.
.
.
.
.
I
.....
5 4 ~3 2
-
0
,
-2
-I
0 _
I
1
,
2
_
B (tesla) F i g u r e 77 Comparison of magnetoresistance Rxx near ~ - ~1 (top), near B=0 (bottom) and the B=0 trace with the magnetic field scale expanded by x/~ (middle).
Rxx from an unpatterned part of the sample. The inset shows configurations for I, 4 and 9 antidots. It is clear that while the unpatterned sample exhibits local minima near the magnetic field positions corresponding to ~ - ~3 and y- 3, 1 the resistance in the antidot trace shows overall maxima with peaks of different strength superimposed at the same fields.
Figure 77 shows the magnetoresistance Rxx around u - ~1 (top), near B = 0 (bottom) and expanded view of the electron trace (middle). The peaks around ~ - ~1 are the most sought after dimensional resonances. The middle trace is the B = 0 trace with the magnetic field scale expanded by v/2 and is to be compared with the ~ = 1 result. The peaks around B = 0 are supposed to reoccur symmetrically around ~ - 1 and their spacing is expected to differ by exactly a factor of ~/2 between electrons and the transformed fermions. T h a t is because of the different spin configurations between the two systems which results in two different Fermi wavevectors t h a t differ by a factor of v/2. In addition to magnetoresistance measurements, work on optical spectroscopy of antidots has also been reported in the literature [223,331,332]. Kern et al. [223] investigated the optical absorption spectra of antidots in modulation-doped G a I n A s / A l I n A s single q u a n t u m wells which are grown lattice matched on InP substrates (Fig. 78). Holes of typical diameter 2Re = 1 0 0 - 300 nm were punched in the two-dimensional electron
Q u a n t u m dots
117
F i g u r e 78 Electron micrograph at an angle of 45 ~ from an antidot array with period a = 300 nm and holes of diameter of 300 nm etched into a GaInAs/AlInAs single quantum well [223].
systems with a period of 300 nm. There are several resonances in the observed transmission spectra in the presence of a magnetic field. The dispersion of those resonances are presented in Fig. 79. There are, obviously, two major modes in the excitation spectrum, the high frequency (co+) and low frequency (co_) branches. The high frequency resonance co+ first decreases with increasing magnetic field and then keeps increasing with the field. At high B it approaches the cyclotron frequency coc of the two dimensional system. The low frequency branch co_ first follows coc at small B but exhibits negative dispersion at higher B. Lorke et al. [331] also observed dispersion which deviates from the cyclotron resonance. Zhao et al. [332] obtained similar results to those of Kern et al. in GaAs/A1GaAs heterostructures. Bollweg et al. [122,333] noticed that, as in the case of q u a n t u m dot arrays (Chapter 2), the co_ mode in the F I R spectra of double-layer antidot system exhibit significant oscillatory behavior (Fig. 80). For small B, co_ approaches the cyclotron frequency, as expected, but for large B, the behavior of co_ reflects skipping orbits at the outer circumference of the antidots. As is evident in Fig. 80 (b), where the low frequency antidot mode is plotted on an expanded frequency scale versus the inverse magnetic field B -1 - both the maxima and minima of t h e resonance frequency are periodic in B -1 since maxima occur at half-filled Landau levels (u odd) and minima occur at fully occupied
118
T. Chakraborty
-
"~"
200
I
I "
I
I
I'''I
I"'
l
I
"I
-
I
I
OI"
I
"-
o9,o-
:
_
o~
:
o v
150
,~
100 --
=1
~w, " "
_
"
.<-.
~.
."
9
oO
CR
50 r
i~ o--o-a. ~ . . . . . . . . . . I
1
I
!
1
I
1.
1,
1.
I
1
1
I
1
,
B (T) F i g u r e 79 E x p e r i m e n t a l l y observed dispersions of high and low-frequency modes in a sample with a n t i d o t period of 300 n m and hole d i a m e t e r of 200 n m [223].
180
.
.
.
.
.
.
.
.
9.
160 (a) 140
.
.
.
.
:
7
~ ,...~
antidotarray
26
,
.
9
,-, ,,,,
--
2 (b') ~antidotarray~
12 10
u e~
cCD 60 40
20
20 i
~
2
4
6
8 10 12 14
1~.~
'
'0:1
B (T)
0.2
0.3
B -I (T -1)
F i g u r e 80 Observed dispersion of resonant frequencies in double-layer antidot system. (b) Observed aJ_ m o d e on an e x p a n d e d frequency scale (left) versus B -1. T h e m a x i m a (filled dots) and m i n i m a (open dots) are related to the filling factor u (right scale) of the L a n d a u level. T h e m o t i o n of individual electrons in the a n t i d o t for t h e aJ_ m o d e is shown as inset [122].
Landau levels (u even) lo. As in the quantum dot case, Bollweg et al. interpreted these oscillations as manifestation of compressible and incompressible stripes at the edge of the antidots. lo In the experimental condition, spin splitting was not resolved.
Quantum dots
119
Finally, Kukushkin et al. [334] reported results of magneto-optical techniques on antidot arrays. These techniques usually provide reliable information about the DOS of the system. Radiative recombination spectrum of electrons confined in a potential well of the conduction band with photoexcited holes bound to acceptors maps the electronic states below the Fermi energy. In the antidot system, these authors find that the magnetoluminescence spectrum, in addition to providing information about the DOS, also displays interesting commensurability features, the origin of which are not entirely clear.
3.1.2. Electron correlation effects 1 In what follows we focus on the FQHE state at a-filled lowest Landau level (y - 5) described by a many-body state proposed by Laughlin [140]. The Laughlin state 11 of the electron system is identical to the state of gauge-transformed particles at y = 51 and therefore we need to consider only the system of two-dimensional electrons (and not the gauge transformed particles) in the presence of antidot arrays 12. Interestingly, the observed fractions in antidot arrays [a2S][R36] described above, did not include 13" Also, it is clear from the experiment that there is no significant difference between the patterned and unpatterned system results at 5" 2 In the case of a pure 2DEG one begins with two-dimensional electrons in a periodic rectangular geometry which is an well established method for accurate evaluation of the FQHE states [20,335]. We consider a rectangular cell containing Are number of electrons. We ignore, for simplicity, the Landau-level mixing and impose periodic boundary conditions such that the cell contains an integer number Ns of flux quanta [20, 21]. We also let the electrons be in the lowest Landau level. In the present case of antidot arrays, the rectangular cell now has, in addition, a static antidot in the middle of the cell [336]. The antidot potential, just as Coulomb interaction, is periodically repeated (with period v/2rrNst~0, for the square cell considered here, where g0 = v/hc/eB is the cyclotron radius in the lowest Landau level or the magnetic length) when the periodic boundary condition is imposed in both directions of the two-dimensional plane. The two-body terms Ujlj2jaj4 in the Hamiltonian ~-~ -- E
t J l j 2 a t 31aj2
jl ,j2
+
at32aJa aj4
E ltJlJ2J3j4at.]l jl ,j2 ,j3 ,j4
(3.2)
to be diagonalized are matrix elements of the Coulomb potential described elsewhere [20, 21,336]. The periodically repeated antidots interact with an electron at r via the potential
V(r) - E vantid~ (R -1--]~ax + l b y - r), k,1 11 For details, see Ref. [20]. 1 12 A similar L a u g h l i n - t y p e w a v e f u n c t i o n does not exist at ~ = ~.
T. Chakraborty
120
where R - (X, Y) is the position of the antidot within the cell of size a x b. Defining the Fourier transform of the antidot potential as
vantid~
= ~1 /
vantidot (r) eiq'rdr
and denoting aspect ratio a/b as A the one-body matrix elements in the Hamiltonian for the lowest Landau level can be written in the form
tjlj2 -- E eiV/2~/e~NsAXkeiV/2~A/e2~ k,l XI~ antid~ (~27r/f~NsA[k2+ A 2 ( j l -
j2 +
Nsl)2] 1/2)
X e iTr(jl +J2-Ns1)k/Nse-~[k2+A2(jl-j2+N~l)2]/2N~A. In what follows, we use a Gaussian form of potential for the scatterers:
vantid~
-- V0 e -(r-R)2/d2
(3.3)
where V0 (same units as energy, e2/ego where e is the background dielectric constant) is the potential strength, d (in units of magnetic length) is the width of the potential. In the limit d --+ 0, one gets the 5-function potential, which was considered earlier by other authors [337] within the Hartree approach and is believed to be a good approximation in the case of a steep potential of the scatterer. However, our choice of (3.3) is better in the magnetic field regime where electron correlations dominate. There are also other choices available in the literature such as the product of cosine functions [Eq. (3.1)] described in Sect. 3.1.1, but our model, where we impose periodicity of the antidot potential explicitly, should effectively be the same as that choice. The interacting Hamiltonian Eq. (3.2) for a few electrons in a periodic rectangular geometry (four electrons in the present case) is then numerically diagonalized to obtain the ground and excited state energies. Our results indicate unique spin transitions at the 51_filled lowest Landau level which is known to be fully spin polarized in the absence of antidot arrays [336]. Before we present the results for antidots, let us briefly recapitulate what we know about the state at 51 filling of the lowest Landau level, studied earlier in this model (but in the absence of the antidot potential). The ground-state energy obtained in the finite-size systems compares well with many-body calculations [20, 21]. The energy gap and the elementary excitations are also well described by our model for the pure 2DEG in the FQHE regime and are in good agreement with the many-electron results. It has long been established theoretically [338-340] and experimentally [341-344] that in the limit of low magnetic fields, several filling fractions tend to have spin-reversed states. However, the state at l-filled lowest Landau level remains fully spin-polarized, even
Q u a n t u m dots
121 I
'
. . . . . . . . . '
I
I
"
I
'
,
I
!
-0.34 (a)
-0.36 ,~
d=0.5
f
-0.38 -0.40 f
-0.42 2 1
0
1
9
I
0.0
,
1.0
I
9
L
2.0
~
I
3.0
I
,
4.0
5.0
Vo (c21~lo) I
'
I
'
I
'
I
"
I
'
I
-0.34 -0.36
(b)
d = 1.0
....
r
-0.38 -0.40
f
-0.42
I
:
2
J
:
I
.
I
;
I
~
,
.
I
:
1
,
t
1
0
....
I
.
0.0
I
.
.
2.0
1.0
3.0
1
4.0
5.0
V o (e2/do) I
"
'
I
'
I
'
'1
....
'
1
9
I
-0.34
(c) d= 1.5
-0.36
-o3s -0.40 -0.42 3
-
',
1
0 -1
..... i
0.0
,
I
1
1.0
2.0
,
I
3.0 vo
.
.1
4.0
.
|
I
5.0
(e2/~lo)
F i g u r e 81 Ground state energy (per particle) and the total spin S of 89 lowest Landau level as a function of antidot potential strength V0 and width d / ~ o - 0.5 (a), 1.0 (b) and 1.5 (c).
T. Chakraborty
122
in the limit of vanishing Zeeman energy. Our results for the ground state energy are shown in Fig. 81 for (a) d = 0.5, (b) 1.0 and (c) 1.5. For V0 = 0, we recover the earlier result of a pure 2DEG [20, 21,335] but as V0 is increased, the ground state energy increases monotonically. For a repulsive scattering center an increase in energy is, of course, expected. The interesting result here is that as V0 increases, the lowest energy state no longer remains spin polarized, but gradually transforms into a partially-spinpolarized state (S = 1) and then to a spin-unpolarized state S = 0, where S is the total spin of the four-electron system considered here. A physical explanation of this result is as follows: In the absence of electron-electron interactions but in the presence of an antidot potential, the degeneracy of the states of the non-interacting systems is lifted and the system is in the spin unpolarized state. In the absence of antidot potentials but presence of interactions, the ground state is fully spin polarized. Therefore, for strong antidot potentials (large values of V0), the ground state remains unpolarized, while for moderate to weak antidot potentials the electron-electron interaction tends to polarize the ground state. The transition also depends on the width of the Gaussian potential (Fig. 81) which works in a similar way as for V0 (i.e., the effect is dominant when d is increased). We also find that in the region of V0 where spin transitions take place, the spin of the state is not well defined because there the spin states are degenerate. Away from those regions, the spin of the ground state is well defined. Interestingly, spin transitions take place only for large values of d. A numerical study of the effect of a 5-function type impurity potential on a 2DEG was reported earlier by Rezayi and Haldane [345]. For a six-electron system in spherical geometry, they found that the Laughlin ground state is stable regardless of the impurity potential strength.
3.2
Q u a n t u m corrals
There is a very significant development taking place in the field of q u a n t u m dots thanks to the recent ability to pattern the two-dimensional electronic density of metallic surface state of electrons by manipulating single atoms on a surface [346,347]. Here electrons are confined in a rounded box (Fig. 82) and one directly measures the energy and eigenstate density. The walls of such quantum corrals are built from Fe atoms individually placed on a Cu(111) surface with a scanning tunneling microscope (STM). Electrons are confined laterally because of strong scattering that occurs between surface state electrons and the Fe adatoms. The surface state electrons are confined in the direction perpendicular to the surface because of intrinsic energetic barriers that exist in the perpendicular direction. Although the experiment was done on Cu, there is nothing unique to Cu: similar surface states exist on all noble metals. For example, some years ago xenon atoms were patterned
Q u a n t u m dots
123
Figure 82 Spatial image of the eigenstates of a quantum corral [347].
on a single crystal nickel surface to write "IBM" [348]. The quantum dots so created are different from dots created in semiconductor i n t e r f a c e s - the quantum states of corrals may be resolved spatially as well as spectroscopically. In conventional quantum dots one can, so far, only get the energy levels, albeit by an indirect method. Experiments performed with STM used single-crystal Cu samples in ultrahigh vacuum, cooled to 4K and dosed with a calibrated electron-beam Fe evaporator. Individual Fe atoms are then positioned into orderly structures on the Cu(111) surface by dragging them with the STM tip. STM measures the density of states via tunneling spectroscopy i.e., measuring the differential conductivity d I / d V of the tunnel junction, and from that, derive the density of states [349,350]. The average radius of the circle is 71.3A. The local density of states (LDOS), the probability of finding an electron with energy E at a point r, LDOS (r,E) - ~ I~k(r)l 2 ~ (E - Ek) k where ~k is an eigenstate of the surface, is shown in Fig. 82 (at the Fermi energy EF). Here the waves inside the ring are due to quantum mechanical interference of 2D electrons scattering off the walls of the ring. Analysis of the spectral and spectroscopic properties of a ring of 48 Fe adatoms reveal that this corral is well described by solutions of the Schr6dinger equation for a particle in a hard-wall enclosure. Nevertheless, the details of the confinement mechanism are not
T. Chakraborty
124
0.6
1.2
(b)
(a) 9
I--0 (hard wail model)
9 ii
I:1
1=2
1
0.4
0.2
9
~ao
I
.w=1
;~
0.0
.........
9
...................................
I
.......
0.8 e
[,T,,]
-0.2
l" ........
9
................................... ~r
i
,t
,
~'I
,"
.......
~
,",
-0.4
0.6
,.,
.
.
i.
,
,
,
,
I
.
.
.
.
l
.
.
I
.......
-50
-100
0
50
100
Distance (~)
F i g u r e 83 (a) Theoretical results for the eigenenergies of 1 = 0, 1, and 2 states of a "hard wall" round box (solid symbols). The experimental results (lines) are also shown. The dotted lines correspond to peaks at circle's center and the broken lines are for extra peaks 9A off center. (b) Eigenstates in a corral and the fit using a linear combination of [5, 0}, [4, 2} and [2, 7} eigenstate densities. The dotted curve is the theoretical result.
well u n d e r s t o o d . E i g e n s t a t e s inside a box of radius r0 can be o b t a i n e d easily. T h e radial e q u a t i o n for a free t w o - d i m e n s i o n a l electron is
h2 ( d 2 f
l df
2rn* \ d r 2 ~ r dr
12 ) r2 f - E f .
Let us now write 2rn* h2
f - r-]/2u, a n d k 2 = ~ E . T h e radial e q u a t i o n is t h e n [k
-
(l
-
which is the Bessel e q u a t i o n with solutions
~(r) -- rl/2Jl(]CT"),
0,
Q u a n t u m dots
125
where Y~(z) is t h e / - t h order Bessel function. The boundary condition, u(r0) = 0, implies gl(kro) = 0, which means that k = zl,~/ro.
Here, znz is the n t h zero crossing of
gz(z).
The energy is then given as
Enl -- h 2 k ~ / 2 m * ,
m* - 0.38me.
Figure 83 (a) depicts the calculated low-lying eigenenergies for the In, 0), In, 1) and In, 2) states in a round box having the same radius as the corral. Experimental results reveal that: [] The peaks in the observed LDOS at the center of the ring are seen to lie very close to theoretical In, 0) eigenenergies, that is, all non-zero angular m o m e n t u m states have a node at the origin, [] The peaks seen in the LDOS at 9.~ off-center lie close to the theoretical In, 1} eigenenergies, which means that these states should dominate any new spectral features just away from the origin (higher angular m o m e n t u m states have less amplitude near the origin due to the centrifugal barrier), [] Distribution of eigenstates near EF: the energies of 15, 0} and 14, 2) eigenstates fall very close to EF. The only other eigenstate lying within 25mV of EF is the 12, 7} state, i.e., one expects that the LDOS at EF inside the ring to be dominated by these states. A good fit for the experimentally measured cross section is obtained with a linear combination of J~(k5,or), J~(k4,2r) and J27(k2,Tr) [Fig. 83 (b)]. [] Observations differ from the ideal behavior of a particle in a hard-wall box in that the measured spectral lines have finite width and the energies of the lines deviate from the ideal predicted values. [] The measured linewidths are found to be narrow compared to photoemission measurements of free Cu(111) surface state electrons. They, however, correspond to a lifetime of ~ 3 x 10 - 1 4 at the Fermi energy. It is to be noted t h a t a free surface state electron at EF travels a distance equal to the ring diameter in ~ 2 x 10-14s. Electrons apparently have longer life-time confined in the corral! The quantum-corral technique is an exciting workshop to explore various phenomena in mesoscopic physics. In fact, this way of confining electrons has opened up the possibility to study quantum-confined structures with atomic precision. The particle-in-a-box wavefunctions for other geometrical patterns (squares, ellipses) or "chaotic" can be mapped. Perhaps, the size, shape, and s y m m e t r y of the confining potential can be tuned to engineer the wavefunction of the dot and study atomic physics of artificial atoms previously
T. Chakraborty
126
Figure 84 Images of self-assembled InAs islands for InAs coverage on GaAs (a) 1.6 and (b) 1.75 ML [353].
inaccessible in nature [351]. It would be interesting to investigate the wavefunctions of Aharonov-Bohm rings, or electron waveguide structures, etc. This technique has already been used recently to confine electrons in other geometries such as a stadium [347].
3.3
Self-assembled quantum dots (SAQD)
The QDs, whose properties we have discussed thus far were produced by sophisticated processing techniques including e-beam lithography and holograph patterning. As opposed to those artificial methods for creating three-dimensional confinement, dislocationfree self-organized epitaxial growth methods, which can reach much lower lateral dimensions, are lately becoming very popular. Self-assembled QDs, as they are called, confine electrons and holes on length scales down to 10 nm in all three directions. In this case, large lateral quantization results in a confinement energy which is as important as the Coulomb charging energy [15][R28], [352-358], [359][R29], [a60-a62] and other characteristic features of a zero-dimensional system have already been demonstrated in these systems. Research on SAQDs is progressing very rapidly and obviously any attempts to review most of the important developments would be futile. We therefore discuss only a few significant results. Self-assembled QDs are created during the growth of highly lattice mismatched semiconductor layer onto a substrate, leading to spontaneous formation of small islands. With an appropriate choice of surrounding materials, these islands, or dots, produce low-dimensional quantum confined structures. This happens for example in InAs/GaAs, InGaAs/GaAs, and I n P / G a I n P systems. When an InAs layer is deposited on a OaAs substrate, the growth begins two dimensionally, but after a certain critical thickness
Quantum dots
127
is reached (~ 1.5-(ML)-thick InAs), islands nucleate spontaneously, [353] and a thin "wetting layer" is left under the islands. When covered by the substrate material, these islands transform into quantum dots. The dots are remarkably uniform in size (Fig. 84); the diameter of the dots fluctuate by only about 10%. The 5-functional form of the DOS [15][R28], [16], the Coulomb blockade [363,364] and other zero-dimensional features have already been reported in the literature. Photoluminescence (PL) experiments with few isolated SAQDs have been reported by several groups [15][R28], [355,356], and quantum confinement features have been observed. Experiments with arrays of SAQDs has also been carried out and the dispersion of electronic levels as a function of magnetic field has been observed in capacitance and IR absorption spectroscopy [358]. Drexler et al. [358] employed capacitance and infrared spectroscopy to investigate self-assembled InGaAs quantum dots embedded in a fieldeffect-type GaAs/A1As heterostructure. The idea behind this particular structure was that the lowest discrete levels of quantum dots can be Charged with single electrons. As the lateral diameter of the dot is as low as d = 20 nm, the Coulomb charging energy is much smaller than the quantization energies. Combination of capacitance and IR spectroscopy enables one to gather detailed information about the quantum and charging energies of s-like and p-like states in the dots. In the presence of a high magnetic field, these authors extracted energy spacings of ~ 41 meV between the ground and first excited state. Medeiros-Ribeiro et al. [359][R29] used capacitance spectroscopy to explore the energy levels of electrons and holes in InAs self-assembled QDs embedded in GaAs. These authors derived an energy level diagram for both electron and hole states indicating large confinement energies in these systems. Zero-dimensional confinement of electrons and holes are found to be consistent with very sharp lines observed in PL experiments. In a recent interesting paper, Miller et al. [365][R30] reported results for few-electron ground states of self-assembled InAs QDs (~ 20 nm in diameter and 7 nm in height) via high-resolution capacitance spectroscopy in fields up to 23 tesla. They determined the Coulomb charging energy as a function of electron number per dot. They observed single-electron charging of the lowest energy state with two electrons and also the second lowest state with four electrons. For small dot ensembles, additional fine structures were observed in the capacitance spectra whose origin has not yet been explained satisfactorily. Fricke et al. [363] probed the few-electron ground state and excitations of SAQDs (with 1 < n~ < 6 electrons per dot) using FIR and capacitance spectroscopy. They found that, for n~ <_ 2, the dynamic behavior of the dots is practically that of a parabolically confined system, i.e., there are only two modes present and the energetic position of these modes is independent of the electron number. However, for n~ >_ 3, new transitions are possible, indicating deviations from purely parabolic confinement. In this case, their results also indicate the crucial role of electron-electron interactions. From capacitance spectroscopy measurements, these authors found that the electronic shell structure in the dot gives
128
T. Chakraborty
rise to a distinct pattern in the charging energies, different from the monotonic behavior of the Coulomb blockade usually seen in mesoscopic or metallic structures. Wang et al. [366] studied PL spectra of InA1As self-organized QDs in A1GaAs matrices in the presence of a magnetic field up to 40 tesla. From the analysis of the diamagnetic shift of the magneto-PL spectra, they estimated the total lateral electron and hole confinement energy to be 43 meV. The exciton radius was deduced to be 5 nm and the exciton binding energy in such a QD system was estimated to be 31 meV. Itskevich et al. [367] investigated the PL spectrum of self-assembled InAs QDs embedded in a GaAs matrix in a magnetic field. They found a strong anisotropy in the diamagnetic shift depending on the direction of the applied magnetic field (parallel or perpendicular to the growth direction). When the magnetic field is applied parallel to the growth direction, the spatial extent of the carrier wavefunction in the dot was estimated to be about 60 A. The electronic state of small lens-shaped SAQDs in a magnetic field can be well approximated [363] by the Fock-Darwin states of an infinite parabolic potential [60][R51. Therefore, most of the basic theoretical concepts are as described in Chapter 2, and the new physics is only in the details. Several theoretical works on the charging, and IR spectra [368] and also the magnetoexciton spectrum [369] for lens-shaped SAQDs have been reported recently. Peeters and Schweigert [370] considered a hard wall confinement to calculate the cyclotron transition energies and the results were found to compare reasonably well with the data by Drexler et al. [358]. Reports on photoluminescence spectroscopy on self-assembled QDs are now being published on a regular basis [371-373] indicating that the chapter on this topic is not yet closed. While there is not much basic physics to be learned yet, the major advantage of the SAQDs is their potential for applications, in particular quantum dot lasers even at room-temperature operations. It was long predicted theoretically [374][R31] that the 5-function density of states of QDs will lead to exceedingly low threshold current and a very weak temperature dependence, J t h - - ~0 eT/T~ i.e., a large characteristic temperature, To. Recently, injection lasers based on InGaAs and InAs QDs with promising low threshold current density and large To at low temperatures have indeed been demonstrated [375][R32]. B~c~us~ of the rapid progress in research on QD lasers [376-383] and other novel devices [384] over the last few years, it would be rather difficult to record in detail those developments and is not attempted. Present status of QD laser research is nicely documented in Ref. [24].
Quantum dots
129
Summary and outlook
The field of condensed matter physics has witnessed a revolution in recent years. It is now becoming increasingly clear that artificial systems (quantum confined) can be tailored to have unusual optical and transport properties. From quantum dot helium to quantum corrals, from quantum dot stadium to electron turnstiles, quantum-confined few-electron systems and their fascinating properties have fired the imagination of researchers in many fields of physics. In this review, we have tried to demonstrate that the close resemblance of the artificial atoms to atomic systems, alluded to in the introduction, is by no means fortuitous. The discreteness of the quantum-dot spectrum, like that of an atomic system, and other analogous properties like the spin singlet-triplet transitions, magic numbers, shell fillings, etc, have been established experimentally. Studies of collective resonances of the electrons in metal clusters, in particular, the plasmon resonance, reveal a lot of similarities in behavior with quantum dots [385,386]. Some researchers have investigated the Wigner crystal state (or, more appropriately, Wigner cluster) in a quantum dot. Calculations indicate that these clusters have ring-like or shell structure up to ne = 38 [387]. Nazarov and Khaetskii [388] proposed formation of some kind of Wigner molecules on top of a quantum dot. The similarities between QDs and atoms have been brought into a sharp focus in recent experiments [389] where coherent control of the quantum state in a single QD, well established as "wavefunction engineering" in atomic and molecular systems, was reported. Just like the confinement of electrons in a QD, it has been recently reported that photons can be confined in a 3D semiconductor microcavities [390-392]. In these structures, whose lateral size is between 5 and l#m, fully discretized optical eigenmodes are
130
T. Chakraborty
observed, much like the 3D quantization of electronic states in a QD [15][R28], [16,242], and are hence called photonic quantum dots. Pairs of interacting photonic dots can also be coupled by narrow channels to form photonic molecules [391]. With decreasing channel length, photoluminescence spectra of these coupled systems reveal splitting of the ground mode, analogous to bonding and antibonding states in a diatomic molecule such as H +. A major advantage of these systems over electronic states in quantum dot molecules is that, in this case the quantitative description of the confined level is not complicated by many-particle effects stemming from the electron-electron interaction, or by uncertainties in the confinement potential. Interaction of the photon modes in the photonic molecules is determined only by the geometry of the molecules. Photonic molecules are expected to be the building blocks for photonic crystals and other interesting structures. These devices are likely to be useful for basic physics as well as for applications, most likely in the development of efficient semiconductor lasers. In contrast to electrostatic dots where electrons are confined by an external potential, attention has also been focused recently on the properties of magnetic quantum dots where electrons are confined by a non-uniform magnetic field [393-396]. These dots can be fabricated by placing a superconducting disk close to a 2DEG subject to an external magnetic field. Alternatively, a ring of ferromagnetic material may be placed close to a 2DEC to create magnetic dots. Interesting physical problems such as the energy spectrum, persistent current as a function of electron number and magnetic field variation, effects of interactions, etc have been studied theoretically. In addition to the atomic properties of QDs, in recent transport measurements QDs were also portrayed as a lone spin (magnetic) impurity in a sea of conduction electrons (in this case, electrons in the neighboring leads) as in the Kondo effect [397-402]. Availability of local spin moments in the dot due to the presence of an odd number of electrons in the dot, and low temperatures where Kondo singlet state can form between electron spin in the leads and in the dot, can cause enhanced conductivity through the dot [401]. Our discussion of the Coulomb blockade in a QD (sect. 2.4.1) did not include the electron spin degree of freedom. However, virtual tunneling involving spin flip processes can occur [402] in the Coulomb blockade regime [Fig. 23 (a)]. The Kondo effect is experimentally observed for temperatures comparable to or lower than the Kondo temperature TK and kBTK << F, where F = FL + FR and FL (FR) is the tunneling rate through the left (right) lead. For small F, one observes the usual Coulomb blockade peaks in conductance vs Vg curves, as discussed earlier. But for large F (F of the order of single-particle level spacing in the dot), the CB peaks form pairs, with large inter-pair spacing and small intra-pair spacing. The two peaks constituting a pair have comparable widths and heights, but the widths vary significantly between the pairs. This is clear evidence that two electrons of different spin have occupied each spatial state. Goldhaber-Gordon et al. observed enhancement of linear conductance at
Quantum dots
131
low temperatures only for odd number of electrons [401]. For odd number of electrons in the dot, the spin of the unpaired electron can form a spin singlet with electrons at the Fermi level in the leads. This coupling causes a peak in the density of states at the Fermi level in the leads, #L = #R. Out of equilibrium, i.e., when a finite eVDs = #L -- # R is applied, the Kondo resonance peak in the DOS splits into two peaks which are pinned to one of the chemical potentials, #L or #R [398]. As the Kondo effect in the QD involve electrons spins, application of a magnetic field drastically affects the Kondo resonance. In fact, an applied field B splits the enhanced DOS at the Fermi level into two peaks having energies g # B B (Zeeman splitting), where g is the Land5 g-factor and #B is the Bohr magneton, above and below the Fermi level. In the out of equilibrium situation when one lead is raised or lowered by a voltage g # B B / e relative to the other lead, electrons can tunnel into the Kondo-enhanced DOS. Such a splitting was indeed observed by Goldhaber-Gordon et al. in the differential conductance at high magnetic fields. In closing, we wish to emphasize that the low-dimensional electron systems are a great laboratory for observing rich and diverse phenomena with profound implications on the basic condensed matter physics and vast potential for applications in future technology. While in this review we have discussed our understanding of the fundamental physics uncovered in these systems, the literature abounds with predictions of the potential of these systems to revolutionize computation or telecommunications [403,404]. Some obvious progress in those directions are the electron turnstile and QD lasers, the latter being most likely the first application of quantum dots [405]. For example, the former device has been suggested to provide a breakthrough in the memory chips. It is hoped that these single-electron devices will soon replace the capacitance of the state-of-the-art memory chips that currently contain a charge corresponding to ~ 200,000 electrons [10]. In this way, larger integration densities and much less heat dissipation will be achieved. We wish to close our review on that hopeful note!
Quantum dots
133
References
[1] T. Chakraborty, Comments Condens. Matter Phys., 16, 35 (1992). [2] D. Heitmann and J.P. Kotthaus, Physics Today, 46 (6), 56 (1993). [3] M.A. Reed, (Ed.), Nanostructured Systems (Academic, San Diego, 1992). [4] M.A. Reed, Scientific American, 268, 98 (1993). [5] M.A. Kastner, Physics Today, 46 (1), 24 (1993). [6] W. Hansen, J.P. Kotthaus, and U. Merkt, Semiconductors and semimetals, 32,279 (1992). [7] W.P. Kirk, and M.A. Reed, (eds.), Nanostructures and Mesoscopic Systems, [s] [9] [10] [11] [12]
(Academic, San Diego, 1992). A.D. Yoffe, Adv. Phys., 42, 173 (1993). L.P. Kouwenhoven, et al., Phys. Rev. Lett., 67, 1626 (1991). T.J. Thornton, Pep. Prog. Phys., 58, 311 (1995). C.G. Smith, Pep. Prog. Phys., 59, 235 (1996). C. Weisbuch, and B. Vinter, Quantum Semiconductor Structures (Academic, New York,
1991).
[13] T. Ando, A.B. Fowler, and F. Stern, Rev. Mod. Phys., 54, 437 (1982). [14] L.J. Challis, Contemp. Phys., 33, 111 (1992). [15] [R28] J.-Y. Marzin, et al., Phys. Rev. Lett., 73, 716 (1994). [16] M. Grundmann, et al., Phys. Rev. Lett., 74, 4043 (1995). [17] D.C. Ralph, C.T. Black, and M. Tinkham, Phys. Rev. Lett., 78, 4087 (1997). [18] D.C. Tsui, H.L. Stormer, and A.C. Gossard, Phys. Rev. Lett., 48, 1559 (1982). [19] H.L. Stormer, Physica B, 177, 401 (1992); H.L. Stormer, et al., Physica E, 3, 38 (1998). [20] T. Chakraborty, and P. Pietil~inen, The Quantum Hall Effects: Fractional and Integral (Springer-Verlag, New York, 2nd Edition, 1995). [21] T. Chakraborty, in Handbook on Semiconductors, vol. 1, ed. P.T. Landsberg (North-Holland, Amsterdam, 1992) ch. 17.
134
T. Chakraborty
[22] R. Turton, The Quantum Dot (W.H. Freeman, Oxford, 1995). [23] L. Jacak, P. Hawrylak, and A. Wojs, Quantum Dots (Springer, 1998). [24] D. Bimberg, M. Grundmann, and N.N. Ledentsov, Quantum Dot Heterostructures (John Wiley & Sons Ltd., 1998). [25] T. Ando, Y. Arakawa, K. Furuya, S. Komiyama, and H. Nakashima (Eds.), Mesoscopic Physics and Electronics (Springer, 1998). [26] L.L. Sohn, L.P. Kouwenhoven, and G. SchSn, (Eds.), Mesoscopic Electron Transport (NATO ASI Series E, vol. 345, Kluwer 1997). [27] MRS Bulletin, 23 (February 1998). [28] K. Richter, Semiclassical Theory of Mesoscopic Quantum Systems (Springer, 1999). [29] D. Pfannbook, Aspects of Coulomb Interaction in Semiconductor Nanostructures (Springer, to be published). [30] [RT] P.A. Maksym and T. Chakraborty, Phys. Rev. Lett., 65, 108 (1990). [31] D. Loss and D.P. DiVincenzo, Phys. Rev. A 57, 120 (1998). [32] S. Benjamin and N.F. Johnson, Appl. Phys. Lett., 7'0, 2321 (1997). [33] C.S. Lent and P.D. Tougaw, J. Appl. Phys., 74, 6227 (1993). [34] C.S. Lent and P.D. Tougaw and W. Porod, Appl. Phys. Lett., 62, 714 (1993). [35] C.S. Lent, P.D. Tougaw, W. Porod and G.H. Bernstein, Nanotechnology, 4, 49 (1993). [36] P.D. Tougaw and C.S. Lent, J. Appl. Phys., 75, 1818 (1994). [37] S.N. Molotkov and S.S. Nazin, JETP Lett., 63, 687 (1996). [38] H. Sato, K. Nishi, I. Ogura, S. Sugou, and Y. Sugimoto, Appl. Phys. Lett., 69, 3140 (1996). [39] P. Bakshi, D.A. Broido, and K. Kempa, Phys. Rev. B, 42, 7416 (1990). [40] D.A. Broido, K. Kempa and P. Bakshi, Phys. Rev. B, 42, 11400 (1990). [41] F.M. Peeters, Phys. Rev. B, 42, 1486 (1990). [42] T. Chakraborty, (Ed.), Proc. of the International Workshop on Novel Physics in Low-Dimensional Electron Systems, Madras, India, Physica B, 212 (1995). [43] [R1] M.A. Reed, J.N. Randall, R.J. Aggarwall, R.J. Matyi, T.M. Moore, and A.E. Wetsel, Phys. Rev. Lett., 60, 535 (1988). [44] M.A. Reed, et al., FestkSrperprobleme/Advances in Solid State Physics, 29, 267 (1989). [45] [R4] G.W. Bryant, Phys. Rev. Lett., 59, 1140 (1987). [46] M. Macucci, K. Hess and G.J. Iafrate, Phys. Rev. B 48, 17 354 (1993). [47] B. Tanatar, and D.M. Ceperley, Phys. Rev. B, 39, 5005 (1981). [48] S. Tarucha, et al., Phys. Rev. Lett., 77, 3613 (1996). [49] S. Tarucha, in Mesoscopic Physics and Electronics, Eds. T. Ando, Y. Arakawa, K. Furuya, S. Komiyama, and H. Nakashima (Springer, 1998). [50] L.P. Kouwenhoven, T.H. Oosterkamp, S. Tarucha, D.G. Austing, and T. Honda, Physica B, 249-251, 191 (1998).
Quantum dots
135
[51] [R33] S. Tarucha, et al., Physica E, 3, 112 (1998). [52] M. Macucci, K. Hess and G.J. Iafrate, Phys. Rev. B 55, R4879 (1997). [53] W.D. Heiss and R.G. Nazmitdinov, Phys. Rev. B 55, 16310 (1997). [54] T. Ezaki, N. Mori and C. Hamaguchi, Phys. Rev. B 56, 6428 (1997). [55] I.-H. Lee, V. Rao, P.M. Martin and J.P. Leburton, Phys. Rev. B 57, 9035 (1998). [56] Y. Tanaka and H. Akera, J. Phys. Soc. Ypn., 66, 15 (1997). [57] M. Fujito, A. Natori, and H. Yasunaga, Phys. Rev. B, 53, 9952 (1996). [58] [R8] A. Kumar, S.E. Laux, and F. Stern, Phys. Rev. B, 42, 5166 (1990). [59] [R2] W. Hansen, et al., Phys. Rev. Lett., 62, 2168 (1989). [60] [R5] V. Pock, Z. Physik, 47, 446 (1928). [61] [R6] C.G. Darwin, Proc. Cambridge Philos. Soc., 27, 86 (1930). [62] L. Landau, Z. Physik, 64, 629 (1930). [63] B. Su, V.J. Goldman and J.E. Cunningham, Phys. Rev. B, 46, 7644 (1992). [64] T. Schmidt, Dissertation, Max-Planck Institute, Stuttgart (1997). [65] T. Schmidt, et al., Phys. Rev. B, 51, 5570 (1995). [66] H.A. Bethe and E.E. Salpeter, Quantum Mechanics of One- and Two-electron atoms [67] [68] [69] [70] [71] [72] [73] [74] [75] [76] [77] [78] [79] [8o] [81] [82] [83] [84] [85]
(Plenum, 1977). [R9] R.B. Dingle, Proc. R. Soc. A, 211,500 (1952). [RI0] R.B. Dingle, Proc. R. Soc. A, 212, 38 (1952). F. Geerincx, F. Peeters and J.T. Devreese, J. Appl. Phys., 68, 3435 (1990). C.T. Liu, et al., Appl. Phys. Lett., 55, 168 (1989). [R19] T. Chakraborty, V. Halonen and P. Pietilb~inen, Phys. Rev. B, 43, 14 289 (1991). P. Pietils V. Halonen and T. Chakraborty, Physica B, 212, 256 (1995). [R24] V. Halonen, P. Pietils and T. Chakraborty, T., Europhys. Lett. 33, 377 (1996). R. Ugajin, Phys. Rev. B, 51, 10 714 (1995). R. Ugajin, Phys. Rev. B, 53, 6963 (1996). JR21] R.C. Ashoori, et al., Phys. Rev. Lett., 71,613 (1993). R.C. Ashoori, et al., Surf. Sci., 305, 558 (1994). R.C. Ashoori, Nature, 379, 413 (1995). M.W. Dellow, et al., Phys. Rev. Lett., 68, 1754 (1992). T.P. Smith III, et al., Phys. Rev. B, 38, 2172 (1988). T.P. Smith III, et al., Phys. Rev. Lett., 59, 2802 (1987). T.P. Smith III, Surf. Sci., 229, 239 (1990). W. Hansen, et al., Appl. Phys. Lett., 56, 168 (1990). R.H. Silsbee and R.C. Ashoori, Phys. Rev. Lett., 64, 1991 (1990). W. Hansen, T.P. Smith III, and K.Y. Lee, Phys. Rev. Lett., 64, 1992 (1990).
136
T. Chakraborty
[86] [R3] B.J. van Wees, et al., Phys. Rev. Lett., 62, 2523 (1989). [87] [R13] P.L. McEuen, et al., Phys. Rev. Lett., 66, 1926 (1991). [88] [R14] P.L. McEuen, et al., Phys. Rev. B, 45, 11419 (1992). [89] P.L. McEuen, et al., Physica B, 189, 70 (1993). [90] [R22] A.T. Johnson, et al., Phys. Rev. Lett., 69, 1592 (1992). [91] E.B. Foxman, et al., Phys. Rev. B, 47, 10020 (1993). [92] K. von Klitzing, Physica Scripta, T68, 21 (1996). [93] Y. Hirayama and T. Saku, Solid State Commun., 73, 113 (1990). [94] J. Weis, R.J. Haug, K. von Klitzing, and K. Ploog, Phys. Rev. B, 46, 12837 (1992). [95] [R23] J. Weis, R.J. Haug, K. von Klitzing, and K. Ploog, Phys. Rev. Lett., 71, 4019 (1993). [96] J. Weis, R.J. Haug, K. von Klitzing, and K. Ploog, Semicond. Sci. Technol., 9, 1890 (1994). [97] [Rll] Ch. Sikorski and U. Merkt, Phys. Rev. Lett., 62, 2164 (1989). [98] Ch. Sikorski and U. Merkt, Surf. Sci., 229, 282 (1990). [99] [R12] T. Demel, D. Heitmann, P. Grambow and K. Ploog, Phys. Rev. Lett., 64, 788 (1990). [100] K. Kern, et al., Superlatt. Microstruct., 9, 11 (1991). [101] B. Meurer, D. Heitmann and K. Ploog, Phys. Rev. Lett., 68, 1371 (1992). [102] B. Meurer, Ph. D. Thesis, MPI, Stuttgart (1992). [103] C. Schfiller, et al., Phys. Rev. Lett. 80, 2673 (1998). [104] D.J. Lockwood, et al., Phys. Rev. Lett. 77, 354 (1996). [105] G. Thurner, et al., Phys. Lett., 89A, 133 (1982). [106] [R15] P.A. Maksym and T. Chakraborty, Phys. Rev. B, 45, 1947 (1992). [107] [R16] M. Wagner, U. Merkt and A.V. Chaplik, Phys. Rev. B, 45, 1951 (1992). [108] B. Su, V.J. Goldman, and J.E. Cunningham, Surf. Sci. 305, 566 (1994). [109] P.A. Maksym, L.D. Hallam, and J. Weis, Physica B, 212, 213 (1995). [110] H. van Houten, C.W.J. Beenakker and A.A.M. Staring, in Single Charge Tunneling, eds. H. Grabert and M.H. Devoret, NATO ASI Series B (Plenum, 1991). [111] R.C. Ashoori, N.B. Zhitenev, L.N. Pfeiffer, and K.W. West, Physica E, 3, 15 (1998). [112] N.B. Zhitenev, R.C. Ashoori, L.N. Pfeiffer, and K.W. West, Phys. Rev. Lett., 79, 2308 (1997). [113] V.M. Bedanov and F.M. Peeters, Phys. Rev. B, 49, 2667 (1994). [114] F.M. Peeters, V.A. Schweigert, and V.M. Bedanov, Physica B, 212, 237 (1995). [115] F.M. Peeters, B. Partoens, V.A. Schweigert, and G. Goldoni, Physica E, 1,219 (1997). [116] D. Heitmann, Physica B, 212, 201 (1995). [117] V. Gudmundsson, A. Brataas, P. Grambow, B. Meurer, T. Kurth and D. Heitmann, Phys. Rev. B, 51, 17744 (1995). [118] I.B. Bernstein, Phys. Rev., 109, 10 (1958). [119] C.K.N. Patel and R.E. Slusher, Phys. Rev. Lett., 21, 1563 (1968).
Q u a n t u m dots
137
[120] E. Batke, D. Heitmann, J.P. Kotthaus and K. Ploog, Phys. Rev. Lett., 54, 2367 (1985). [121] E. Batke, D. Heitmann and C.W. Tu, Phys. Rev. B 34, 6951. [122] K. Bollweg, et al., Phys. Rev. Lett., 76, 2774 (1996). [123] T. Darnhofer, M. Suhrke and U. RSssler, Europhys. Lett., 35, 591 (1996). [124] [RS0] A. Lorke, J.P. Kotthaus and K. Ploog, Phys. Rev. Lett., 64, 2559 (1990). [125] S.J. Allen, Jr., H.L. StSrmer, and J.C.M. Hwang, Phys. Rev. B28, 4875 (1973). [126] C. Dahl, et al., Solid State Cornmun., 80, 673 (1991). [127] C.W.J. Beenakker and H. van Houten, Solid State Phys., 44, 1 (1991). [128] H. Fukuyama and T. Ando, (eds.), Transport Phenomena in Mesoscopic Systems, (Springer-Verlag, 1992). [129] S. Datta, Electronic Transport in Mesoscopic @stems (Cambridge University Press, 1995).
[130] L.
Geerligs, C.J.P.M. Harmans, and L.P. Kouwenhoven, (eds.), Proc. of the NATO Advanced Research Workshop on The Physics of Few-Electron Nanostructures, Physica B, 189 (1993). [131] C.W.J. Beenakker and H. van Houten, in Single Charge Tunneling, eds. H. Grabert, J.M. Martinis and M.H. Devoret, (Plenum, 1991). [132] L.P. Kouwenhoven, C.M. Marcus, P.L. McEuen, S. Tarucha, R.M. Westervelt, and N.S. Wingreen, in Mesoscopic Electron Transport, L.L. Sohn, L.P. Kouwenhoven, and G. SchSn, (Eds.), (NATO ASI Series E, vol. 345, Kluwer 1997). [133] K. Harmans, Physics World, March 1992, p. 50. [134] M. Tinkham, Am. Y. Phys., 64, 343 (1996). [135] N.C. van der Vaart, et al., Physica B, 189, 99 (1993). [136] R.J. Haug, R.H. Blick, and T. Schmidt, Physica B, 212, 207 (1995). [137] D.V. Averin and K.K. Likharev, in Quantum effects in small disordered systems, edited by, B. Altshuler, P. Lee and R. Webb (Elsevier, Amsterdam 1990). [138] J. Weis, Dissertation, Max-Planck-Institute, Stuttgart, (1994). [139] K. von Klitzing, G. Dorda, and M. Pepper, Phys. Rev. Lett., 45, 494 (1980). [140] R.B. Laughlin, Phys. Rev. Lett., 50, 1395 (1983). [141] E. Goldmann and S.R. Renn, Phys. Rev. B, 56, 13296 (1997). [142] L.P. Kouwenhoven, et al., Phys. Rev. B, 50, 2019 (1994). [143] L.P. Kouwenhoven, et al., Phys. Rev. Lett., 73, 3443 (1994). [144] T.H. Ooosterkamp, et al., Phys. Rev. Lett., 78, 1536 (1997). [145] L.P. Kouwenhoven, et al., Surf. Sci., 361/362, 591 (1996). [146] R.H. Blick, et al., Appl. Phys. Lett., 67, 3924 (1995). [147] K. Fuji, et al., Physica B, 227, 98 (1996). [148] C. Bruder and H. Schoeller, Phys. Rev. Lett., 72, 1076 (1994). [149] R.H. Blick, R.J. Haug, K. von Klitzing, and K. Eberl, Surf. Sci., 361/362, 595 (1996). [150] T. Fujisawa and S. Tarucha, Superlattices and Microstructures, 21,247 (1997).
138
T. Chakraborty
[151] T. Schmidt, et al., Appl. Phys. Lett., 68, 838 (1996). [152] B. Su, V.J. Goldman and J.E. Cunningham, Science, 255, 313 (1992). [153] M. Tewordt, et al., Phys. Rev. B, 45, 14407 (1992). [154] P.C. Main, et al., Physica B, 189, 125 (1993). [155] P. Ou~ret, N. Blanc, R. Germann, and H. Rothuizen, Phys. Rev. Lett., 68, 1896 (1992) [156] M. Stone, H.W. Wyld and R.L. Schult, Phys. Rev. B, 45, 14156 (1992). [157] JR17] P.A. Maksym, Physica B, 184, 385 (1993). [158] P.A. Maksym, Europhys. Lett., 31,405 (1995). [159] P.A. Maksym, Phys. Rev. B, 53, 10871 (1996). [160] W.Y. Ruan, et al., Phys. Rev. B, 51, 7942 (1995). [161] [R18] T. Seki, Y. Kuramoto and T. Nishino, J. Phys. Soc. Jpn., 65, 3945 (1996). [162] H. hnamura, P.A. Maksym and H. Aoki, Phys. Rev. B, 53, 12 613 (1996). [163] W.-Y. Ruan and H.-F. Cheung, Eur. Phys. 2. B, 3, 407 (1998). [164] U. Merkt, J. Huser and M. Wagner, Phys. Rev. B, 43, 7320 (1991). [165] N.F. Johnson and M.C. Payne, Phys. Rev. Lett., 67, 1157 (1991). [166] L. Quiroga, D. Ardila, and N.F. Johnson, Solid State Commun., 86, 775 (1993). [167] N.F. Johnson and L. Quiroga, Solid State Commun., 89, 661 (1994). [168] N.F. Johnson and L. Quiroga, Phys. Rev. Lett., 74, 4277 (1995). [169] P. Hawrylak and D. Pfannkuche, Phys. Rev. Lett. 70, 485 (1993). [170] P. Hawrylak, Phys. Rev. Lett., 71, 3347 (1993). [171] X.C. Xie, et al., Phys. Rev. B, 48, 8454 (1993). [172] V. Halonen, Solid State Commun., 92, 703 (1994). [173] M. Eto, 2pn. 2. Appl. Phys., 36, 3924 (1997). [174] J.H. Oaknin, et al., Phys. Rev. B, 49, 5718 (1994). [175] M. Ferconi and G. Vignale, Phys. Rev. B, 50, 14722 (1994). [176] M. E1-SMd, Phys. Stat. Sol., 193, 105 (1996). [177] A. GonzAlez, L. Quiroga and B.A. Rodriguez, Few-Body Systems, 21, 47 (1996). [178] W 9 Kohn, Phys. Rev., 123, 1242 (1961). [179] V. Gudmundsson and R.R. Gerhardts, Phys. Rev., 43, 12098 (1991). [180] D. Pfannkuche and R.R. Gerhardts, Phys. Rev. B, 44, 12132 (1991). [181] D. Pfannkuche, R.R. Gerhardts, P.A. Maksym and V. Gudmundsson, V., Physica B, 189, 6( 1993). [182] D. Pfannkuche, V. Gudmundsson and P.A. Maksym, Phys. Rev. B, 47 2244 (1993). [183] M. Taut, J. Phys. A, 27, 1045 (1994). [184] M. Dineykhan and R.G. Nazmitdinov, Phys. Rev. B, 55, 13707 (1997). [185] R. Pino, Phys. Rev. B, 58, 4644 (1998).
Quantum dots
139
[186] M. Wagner, A.V. Chaplik and U. Merkt, Phys. Rev. B, 51, 13 817 (1995). [187] K. Jauregui, W. Hs and B. Kramer, Europhys. Lett., 24, 581 (1993). [188] F. BoRon, Phys. Rev. Lett., 73, 158 (1994). [189] F. Bolton, Solid-State Electronics, 37, 1159 (1994). [190] T. Darnhofer, D.A. Broido and U. RSssler, Phys. Rev. B, 50, 15 412 (1994). [191] T. Darnhofer, U. RSssler and D.A. Broido, Phys. Rev. B, 52, R14 376 (1995). [192] F.B. Pedersen and Y.-C. Chang, Phys. Rev. B, 53, 1507 (1996). [193] F.B. Pedersen and Y.-C. Chang, Phys. Rev. B, 55, 4580 (1997). [194] E. Gornik, et al., Phys. Rev. Lett., 54, 1820 (1985). [195] J.P. Eisenstein, H.L. StSrmer, V. Narayanamurti, A.Y. Cho, A.C. Gossard and C.W. Tu, Phys. Rev. Lett., 55, 875 (1985). [196] J.P. Eisenstein, Appl. Phys. Lett., 46, 695 (1985). [197] W. Zawadazki and R. Lassnig, Surf. Sci., 142, 225 (1984). [198] R. Ugajin, Physica B, 253, 96 (1998). [199] P.A. Maksym and N.A. Bruce, Physica E, 1,211 (1997). [200] L.D. Hallam, J. Weis, and P.A. Maksym, Phys. Rev. B, 53, 1452 (1996). [201] S.-R.E. Yang, A.H. MacDonald and M.D. Johnson, Phys. Rev. Lett., 71, 3194 (1993). [202] J.H. Oaknin, et al., Phys. Rev. Lett., 74, 5120 (1995). [203] J. Kinaret, et al., Phys. Rev. B, 45, 9489 (1992). [204] J. Kinaret, et al., Phys. Rev. B, 46, 4681 (1992). [205] J. Kinaret, Physica B, 189, 142 (1993). [206] J.J. Palacios, et al., Phys. Rev. B, 50, 5760 (1994). [207] P.A. Maksym, private communications (1997). [208] Y. Tanaka and H. Akera, Phys. Rev. B, 53, 3901 (1996). [209] J.J. Palacios, L. Martin-Moreno, and C. Tejedor, Europhys. Lett., 23, 495 (1993). [210] O. Klein, et al., Phys. Rev. Lett., 74, 785 (1995). [211] K.H. Ahn, J.H. Oh and K.J. Chang, Phys. Rev. B, 52, 13757 (1995). [212] D. Pfannkuche and S.E. Ulloa, Phys. Rev. Lett., 74, 1194 (1995). [213] T. Chakraborty and V. Halonen, in Nanostructures and Mesoscopic Systems, W.P. Kirk and M.A. Reed, (eds.) (Academic, 1992), p. 347. [214] R. Ugajin, T. Suzuki, K. Nomoto and I. Hase, Y. Appl. Phys., 76, 1041 (1994). [215] R. Ugajin, Phys. Rev. B, 51, 11136 (1995). [216] K. Kempa, D.A. Broido and P. Bakshi, Phys. Rev. B, 43, 9343 (1991). [217] J.J. Palacios and P. Hawrylak, Phys. Rev. B, 51, 1769 (1995). [218] J.H. Oh, K.L. Chang, G. Ihm, and S.J. Lee, Phys. Rev. B, 53, R13 264 (1996). [219] V. Halonen, P. HyvSnen, P. Pietils and T. Chakraborty. Phys. Rev. B, 53, 6971 (1996).
T. Chakraborty
140
[220] A.S. Sachrajda, et al., Phys. Rev. B, 50, 10856 (1994). [221] C. Dahl, et M., Phys. Rev. B, 48, 15480 (1993). [222] V.A. Volkov and S.A. Mikhailov, Soy. Phys. JETP, 67, 1639 (1988). [223] [R37] K. Kern, et al., Phys. Rev. Lett., 66, 1618 (1991). [224] P. Hawrylak, Phys. Rev. B, 51, 17 708 (1995). [225] A. Lorke and R.J. Luyken, Physica B, 256-258, 424 (1998). [226] A. Lorke, R.J. Luyken, A.O. Govorov, J.P. Kotthaus, J.M. Garcia, and P.M. Petroff, condmat/9908263, and to be published (1999).
[227] J.M. Garcia, et al., Appl. Phys. Lett., 71, 2014 (1997). [228] [R25] V. Halonen, T. Chakraborty and P. Pietils Phys. Rev. B, 45, 5980 (1992). [229] G.W. Bryant, Phys. Rev. B, 37, 8763 (1988); Surf. Sci., 196, 596 (1988). [230] G.W. Bryant, Phys. Rev. B, 41, 1243 (1990). [231] Y.Z. Hu, et al., Phys. Rev. B, 42, 1713 (1990). [232] Y.Z. Hu, et al., Phys. Rev. Lett., 64, 1805 (1990). [233] E.L. Pollock and S.W. Koch, J. Chem. Phys., 94, 6776 (1991). [234] J.C. Maan, et al., Phys. Rev. B, 30, 2253 (1984). [235] D.C. Rogers, et M., Phys. Rev. B, 34, 4002 (1986). [236] W. Ossau, et al., Surf. Sci., 174, 188 (1986). [237] M. Kohl, D. Heitmann, P. Grambow and K. Ploog, Phys. Rev. Lett., 63, 2124 (1989). [238] K. Brunner, G. Abstreiter, M. Walther, G. BShm, G. Trs and G. Weimann, Phys. Rev. Lett., 69, 3216 (1992).
[239] [R26] A. Zrenner, et al., Phys. Rev. Lett., 72, 3382 (1994). [240] A. Zrenner, Surf. Sci., 361/362, 756 (1996). [241] A. Zrenner, Physica E, 2, 609 (1998). [242] M. Bayer, A. Schmidt, A. Forschel, F. Faller, T.L. Reinecke, and P.A. Knipp, Phys. Rev. Lett., 74, 3439 (1995).
[243] M. Bayer, O. Schilling, A. Forschel, T.L. Reinecke, P.A. Knipp, Ph. Pagnod-Rossiaux, and L. Goldstein, Phys. Rev. B, 53, 15810 (1996).
[s44] [R27] R. Rinaldi, P.V. Giugno, R. Cingolani, H. Lipsanen, M. Sopanen, J. Tulkki, and J. Ahopelto, Phys. Rev. Lett., 77, 342 (1996).
[245] R. Rinaldi, R. Mangino, R. Cingolani, H. Lispanen, M. Sopanen, J. Tulkki, M. Brasken, and J. Ahopelto, Phys. Rev. B, 57, 9763 (1998). [246] S. Jaziri and R. Bennaceur, Semicond. Sci. Technol., 9, 1775 (1994). [247] V.D. Kulakovskii, M. Bayer, M. Michel, A. Forchel, T. Gutbrod, and F. Faller, JETP Lett., 66, 285 (1997). [248] M. Bayer et al., Phys. Rev. B, 58, 4740 (1998). [249] A. Wojs and P. Hawrylak, Phys. Rev. B, 51, 10 880 (1995). [250] U. Bockelmann, Phys. Rev. B, 50, 17 271 (1994).
Q u a n t u m dots
141
[251] B. Adolph, S. Glutsch and F. Bechstedt, Phys. Rev. B, 48, 15 077 (1993). [252] W. Heller and U. Bockelmann, Phys. Rev. B, 55, R4871 (1997). [253] B. Meurer, D. Heitmann, and K. Ploog, Phys. Rev. B, 48, 11488 (1993). [254] B. Meurer, D. Heitmann, and K. Ploog, Surf. Sci., 305, 610 (1994). [255] Z. Schlesinger, J.C.M. Hwang, and S.J. Allen, Phys. Rev. B, 50, 2098 (1983). [256] Q.P. Li, et al., Phys. Rev. B, 43, 5151 (1991). [257] J.H. Oh et al., Superlattices and Microstructures, 16, 183 (1994). [258] D. Weinmann, et al., Europhys. Lett., 26, 467 (1994). [259] D. Weinmann, W. Hs and B. Kramer, Phys. Rev. Lett., 74, 984 (1995). [260] H. Imamura, H. Aoki, and P.A. Maksym, Phys. Rev. B, 57, R4257 (1998). [261] M. Tewordt, et al., Appl. Rev. Lett., 60, 595 (1992). [262] M. Tewordt, et al., Phys. Rev. B, 49, 8071 (1994). [263] N.C. van der Vaart, et al., Phys. Rev. Lett., 74, 4702 (1995). [264] G.W. Bryant, Phys. Rev. B, 48, 8024 (1993). [265] G. Klimeck, G. Chen, and S. Datta, Phys. Rev. B, 50, 2316 (1994). [266] D. Dixon, et al., Phys. Rev. B, 53, 12 625 (1996). [267] F.R. Waugh, et al., Phys. Rev. Lett., 75, 705 (1995). [268] F.R. Waugh, et al., Phys. Rev. B, 53, 1413 (1996). [269] J.M. Golden and B.I. Halperin, Phys. Rev. B, 53, 3893 (1996). [270] C. Livermore, et al., Science, 274, 1332 (1996). [271] R.H. Blick, et al., Phys. Rev. B, 53, 7899 (1996). [272] R.H. Blick, et al., Phys. Rev. Lett., 80, 4032 (1998). [273] R.H. Blick, et al., Phys. Rev. Lett., 81, 689 (1998). [274] C.A. Stafford and N.S. Wingreen, Phys. Rev. Lett., 76, 1916 (1996). [275] T. Schmidt, R.J. Haug, K. von Klitzing, A. FSrster and H. Liith, Phys. Rev. Lett., 78, 1544 (1997).
[276] D.G. Austing, et al., Physica B, 249-251, 206 (1998). [277] H. Tamura, Physica B, 249-251, 210 (1998). [278] Y. Asano, Phys. Rev. B, 58, 1414 (1998). [279] T. Fujisawa and S. Tarucha, Jpn. J. Appl. Phys., 36, 4000 (1997) [280] T.H. Stoof and Y.V. Nazarov, Phys. Rev. B, 53, 1050 (1996). [281] Ph. Brune, C. Bruder, and H. Schoeller, Physica E, 1,216 (1997). [282] A.V. Madhav and T. Chakraborty, Phys. Rev. B, 49, 8163 (1994). [283] R. Haupt and L. Wendler, Physica B, 184, 394 (1993). [284] B. Schuh, J. Phys. A, 18, 803 (1985). [285] O. Dippel, P. Schmelcher, and L.S. Cedarbaum, Phys. Rev. A, 49, 4415 (1994).
T. Chakraborty
142
[286] P.A. Maksym, Physica B, 249-251, 233 (1998). [287] D.C. Austing, et al., Phys. Rev. B, xx, xxxx (1999). [288] S. Sasaki, D.G. Austing, and S. Tarucha, Physica B, 256-258, 157 (1998). [289] M. Gutzwiller, Chaos in Classical and Quantum Mechanics, (Springer, 1991). [290] S.W. McDonald and A.N. Kaufman, Phys. Rev. Lett., 43, 1189 (1979). [291] E.J. Heller, Phys. Rev. Lett., 53, 1515 (1986). [292] C.M. Marcus, et al., Phys. Rev. Lett., 69, 506 (1992). [293] C.M. Marcus, et al., Surf. Sci., 305, 480 (1994). [294] Z.L. Ji and K.F. Berggren, Phys. Rev. B, 52, 1745 (1995). [295] D.R. Hofstadter, Phys. Rev. B, 14, 2239 (1976). [296] K. Ensslin and P.M. Petroff, Phys. Rev., 41, 12 307 (1990). [297] [R34] D. Weiss, et al., Phys. Rev. Lett., 66, 2790 (1991). [298] A. Lorke, J.P. Kotthaus, and K. Ploog, Phys. Rev. B, 44, 3447 (1991). [299] G. Berthold, et al., Phys. Rev. B, 45, 11350 (1992). [300] G.M. Gusev, et al., JETP Lett.. 54, 364 (1991). [301] D. Weiss, et al., Surf. Sci., 305, 408 (1994). [302] R. Schuster and K. Ensslin, FestkSrperprobleme//Advances in Solid State Physics, 34, 195 (1994).
[303] R. Schuster, et al., Phys. Rev. B, 50, 8090 (1994). [304] [R35] K. Tsukagoshi, et al., J. Phys. Soc. Jpn., 65, 811 (1996). [305] K. Tsukagoshi, et al., J. Phys. Soc. Jpn., 65, 1914 (1996). [306] R. Schuster, K. Ensslin, J.P. Kotthaus, M. Holland, and C. Stanley, Phys. Rev. B, 47, 6843 (1993).
[307] K. Tsukagoshi, et al., Jpn. J. Appl. Phys., 34, 4335 (1995). [308] K. Tsukagoshi, et al., Phys. Rev. B, 52, 8344 (1995). [309] J. Takahara, et al., Jpn. J. Appl. Phys., 34, 4325 (1995). [310] T. Yamashiro, et al., Solid State Commun., 79, 885 (1991). [311] J. Takahara, et al., Jpn. J. Appl. Phys., 30, 3250 (1991). [312] R. Fleischmann, T. Geisel, and R. Ketzmerick, Phys. Rev. Lett., 68, 1367 (1992). [313] E.M. Baskin, et al., JETP Lett., 55, 678 (1992). [314] T. Nagao, J. Phys. Soc. Jpn., 64, 4079 (1995).
[315] D. Weiss, et al., Europhys. Lett., 8, 179 (1989). [316] D. Weiss, et al., Surf. Sci., 263, 314 (1992). [317] R. Gerhardts et al., Phys. Rev. Lett., 62, 1173 (1989). [318] R.W. Winkler, J.P. Kotthaus, and K. Ploog, Phys. Rev. Lett., 62, 1177 (1989). [319] C.W.J. Beenakker, Phys. Rev. Lett., 62, 2020 (1989).
Q u a n t u m dots
143
[320] R. Kubo, J. Phys. Soc. Jpn. 12, 570 (1957). [321] S. Ishizaka, et al., Phys. Rev. B, 51, 9881 (1995). [322] S. Ishizaka and T, Ando, Phys. Rev. B, 55, 16331 (1997). [323] T. Yamauchi, Phys. Lett. A, 191, 317 (1994). [324] H. Silberbauer and U. RSssler, Phys. Rev. B, 50, 11911 (1994). [325] P. Rotter, U. RSssler, H. Silberbauer, and M. Suhrke, Physica B, 212, 231 (1995). [326] R. Neudert, P. Rotter, U. RSssler, and M. Suhrke, Phys. Rev. B, 55, 2242 (1997). [327] D. Weiss, et al., Phys. Rev. Lett., 70, 4118 (1993). [328] [R36] W. Kang, et al., Phys. Rev. Lett., 71, 3850 (1993). [329] W. Kang, et al., in High Magnetic Fields in the Physics of Semiconductors, ed. D. Heiman (World Scientific, 1995). [330] B.I. Halperin, P.A. Lee, and N. Read, Phys. Rev. B, 47, 7312 (1993). [331] A. Lorke, J.P. Kotthaus, and K. Ploog, Superlatt. Microstruct., 9, 103 (1991). [332] Y. Zhao, et al., Appl. Phys. Lett., 60, 1510 (1992). [333] K. Bollweg et al., Surf. Sci., 361/362, 766 (1996). [334] I.V. Kukushkin, et al., Phys. Rev. Lett., 79, 1722 (1997). [335] D. Yoshioka, B.I. Halperin, and P.A. Lee, Phys. Rev. Lett., 50, 1219 (1983). [336] T. Chakraborty and P. Pietil~iinen, Phys. Rev. B, 53, 4664 (1996). [337] G. Gumbs and D. Huang, Phys. Rev. B, 46, 12822 (1992); Superlatt. Microstruct., 14, 1 (1993).
[338] T. Chakraborty, Surf. Sci., 229, 16 (1990). [339] T. Chakraborty, P. Pietil~inen, and F.C. Zhang, Phys. Rev. Lett., 57, 130 (1986). [340] B.I. Halperin, Helv. Phys. Acta, 56, 75 (1983). [341] R.G. Clark, et al., Phys. Rev. Lett., 62, 1536 (1989). [342] J.P. Eisenstein, et al., Phys. Rev. Lett., 62, 1540 (1989). [343] L.W. Engel, et al., Phys. Rev. B, 45, 3418 (1992). [344] A.G. Davis, et al., Phys. Rev. B, 44, 13128 (1991). [345] E.H. Rezayi and F.D.M. Haldane, Phys. Rev. B, 32, 6924 (1988). [346] M.F. Crommie, C.P. Lutz and D.M. Eigler, Science, 262, 218 (1993). [347] M.F. Crommie, C.P. Lutz, D.M. Eigler and E.J. Heller, Physica D, 83, 98 (1995). [348] D.M. Eigler and E.K. Schweizer, Nature, 344, 524 (1990). [349] J. Tersoff and D.R. Hamanni, Phys. Rev. Lett., 50, 1998 (1983). [350] J. Tersoff and D.R. Hamanni, Phys. Rev. B, 31,805 (1985). [351] M. Reed, Science, 262, 195 (1993). [352] D. Leonard, et al., Appl. Phys. Lett., 63, 3203 (1993). [353] D. Leonard, et al., Phys. Rev. B, 50, 11.687 (1994).
144
T. Chakraborty
[354] J.M. Moison, et al., Appl. Phys. Lett., 64, 196 (1994). [355] S. Fafard, et al., Phys. Rev. B, 50, 8086 (1994). [356] R. Leon, P.M. Petroff, D. Leonard and S. Fafard, Science, 267, 1966 (1995). [357] P.M. Petroff and G. Medeiros-Ribeiro, MRS Bulletin, 21, 50 (1996). [358] H. Drexler, et al., Phys. Rev. Lett., 73, 2252 (1994). [359] [R29] G. Medeiros-Ribeiro, D. Leonard and P.M. Petroff, Appl. Phys. Lett., 66, 1767 (1995). [360] R. Heitz, et al. Appl. Phys. Lett., 68, 361 (1996). [361] R.J. Warburton, et al., Phys. Rev. Lett., 79, 5282 (1997). [362] K.H. Schmidt, et al., Phys. Rev. B, 58, 3597 (1998). [363] M. Fricke, et al., Europhys. Lett., 36, 197 (1996). [364] G. Medeiros-Ribeiro, et al., Phys. Rev. B, 55, 1568 (1997). [365] [R30] B.T. Miller, et al., Phys. Rev. B, 56, 6764 (1997). [366] P.D. Wang, et al., Phys. Rev. B, 53, 16 458 (1996). [367] I.E. Itskevich, et al., Appl. Phys. Lett., 70, 505 (1997). [368] A. Wojs and P. Hawrylak, Phys. Rev. B, 53, 10 841 (1996). [369] A. Wojs, et al., Phys. Rev. B, 5~, 5604 (1996). [370] F.M. Peeters and V.A. Schweigert, Phys. Rev. B, 53, 1468 (1996). [371] A. Kuther, et al., Phys. Rev. B, 58, R7508 (1998). [372] S. Noda, T. Abe and M. Tamura, Phys. Rev. B, 58, 7181 (1998). [373] S. Nomura, et al., Phys. Rev. B, 58, 6744 (1998). [374] [R31] Y. Arakawa and H. Sakaki, Appl. Phys. Lett., 40, 939 (1982). [375] [R32] N. Kirkstaedter, et al., Electron. Lett., 30, 1416 (1994). [376] K. Kamath, et al., Electron. Lett., 32, 1374 (1996). [377] H. Shoji, et al., Electron. Lett., 32, 2023 (1996). [378] R. Mirin, et al., Electron. Lett., 32, 1732 (1996). [379] H. Saito, et al., Appl. Phys. Lett., 69, 3140 (1996). [380] D.L. Huffaker, et al., Appl. Phys. Lett., 70, 2356 (1997). [381] K. Nishi, et al., Appl. Phys. Lett., 73, 526 (1998). [382] S. Fafard, et al., Science, 274, 1350 (1996). [383] M.K. Zundel, et al., Appl. Phys. Lett., 73, 1784 (1998). [384] S. Kim, et al., Appl. Phys. Lett., 73, 963 (1998). [385] M. Koskinen, M. Manninen and P.O. Lipas, Z. Phys. D, 31, 125 (1994). [386] V. Kresin, Phys. Pep., 220, 1 (1992). [387] F. Bolton and U. RSssler, Superlattices and Microstructures, 13, 139 (1993). [388] Y.V. Nazarov and A.V. Khaetskii, Phys. Rev. B, 49, 5077 (1994).
Quantum dots
145
[389] N.H. Bonadeo, et al., Science, 282, 1473 (1998). [390] J.M. G~rard, et al., Appl. Phys. Lett., 69, 449 (1996). [391] J.P. Reithmaier, et al., Phys. Rev. Lett., 78, 378 (1997). [392] M. Bayer, et al., Phys. Rev. Lett., 81, 2582 (1998). [393] F.M. Peeters, A. Matulis, and I.S. Ibrahim, Physica B, 227, 131 (1996). [394] J. Reijniers, F.M. Peeters, and A. Matulis, Phys. Rev. B, 59, 2817 (1999). [395] C.S. Kim and O. Olendski, Phys. Rev. B, 53, 12917 (1996). [396] G.P. Mallon and P.A. Maksym, Physica B, 256-258, 186 (1998). [397] S. Hershfeld, J.H. Davies, and J.W. Wilkins, Phys. Rev. Lett., 67, 3720 (1991). [398] Y. Meir, N.S. Wingreen, and P.A. Lee, Phys. Rev. Lett., 70, 2601 (1993). [399] A.L. Yeyati, A. Martin-Rodeo, and F. Flores, Phys. Rev. Lett., 71, 2991 (1993). [400] K. Kang, Phys. Rev. B, 58, 9641 (1998); L. Craco and K. Kang, to be published (1998). [401] D. Goldhaber-Gordon, et al., Nature, 391, 156 (1998). [402] S.M. Cronenwett, T.H. Oosterkamp, and L.P. Kouwenhoven, Science, 281, 540 (1998). [403] U. KSnig, Physica Scripta, T68, 90 (1996). [404] C. Weisbuch, Physica Scripta, T68, 102 (1996). [405] M. Grundmann, Physica E, to be published (1999).
Quantum dots
147
List of Reprinted Articles R1.
M.A. Reed, J.N. Randall, R.J. Aggarwal, R.J. Matyi, T.M. Moore, and A.E. Wetsel, O b s e r v a t i o n of d i s c r e t e electronic states in a zero-dimensional semiconductor
n a n o s t r u c t u r e , Phys. Rev. Lett. 60, pp. 535 (1988) R2.
W. Hansen, T.P. Smith, III, K.Y. Lee, J.A. Brum, C.M. Knoedler, J.M. Hong, and D.P. Kern, Zeeman b i f u r c a t i o n of quantum-dot s p e c t r a , Phys. Rev. Lett. 62, pp. 2168 (1989)
R3.
B.J. van Wees, L.P. Kouwenhoven, C.J.P.M. Harmans, J.G. Williamson, C.E. Timmering, M.E.I. Broekaart, C.T. Foxon, and J.J. Harris, Observation of zero-dimensional
states in a one-dimensional
electron interferometer, Phys. Rev. Lett. 62, pp. 2523 (1989)
R4.
G. Bryant, E l e c t r o n i c s t r u c t u r e of u l t r a s m a l l quantum-well boxes, Phys. Rev. Lett. 59, pp. 1140 (1987)
RS.
V. Fock, Bemerkung zur q u a n t e l u n g des harmonischen o s z i l l a t o r s im magnetfeld, Z. Physik 47, pp. 446 (1928)
R6.
C.G. Darwin, The diamagnetism of the f r e e e l e c t r o n , Proc. Cambridge Philos. Soc. 27, pp. 86 (1930)
RT.
P. Maksym and T. Chakraborty, Quantum dots in a magnetic f i e l d : Role of electron-electron interactions,
Phys. Rev. Lett. 65, pp. 108 (1990) RS.
A. Kumar, S.E. Laux, and F. Stern, E l e c t r o n s t a t e s in a GaAs quantum dot in a magnetic f i e l d , Phys. Rev. B 42, pp. 5166 (1990)
R9.
R.B. Dingle, Some magnetic p r o p e r t i e s of m e t a l s I. General introduction,
and properties of large systems of electrons,
Proc. Roy. Soc. A, 211,500 (1952) R10.
R.B. Dingle, Some magnetic p r o p e r t i e s of m e t a l s I I I . resonance, Proc. Roy. Soc. A, 212, 38 (1952)
Diamagnetic
T. Chakraborty
148
Rll.
Ch. Sikorski and U. Merkt, Spectroscopy of e l e c t r o n i c s t a t e s in InSb quantum dots, Phys. Rev. Lett. 62, pp. 2164 (1989)
R12.
T. Demel, D. Heitmann, P. Grambow, and K. Ploog, Nonlocal dynamic response and l e v e l c r o s s i n g s in quantum-dot s t r u c t u r e s , Phys. Rev. Lett. 64, pp. 788 (1990)
R13.
P.L. McEuen, E.B. Foxman, U. Meirav, M.A. Kastner, Y. Meir, N.S. Wingreen, and S.J. Wind, Transport s p e c t r o s c o p y of a Coulomb i s l a n d in the quantum Hall regime, Phys. Rev. Lett. 66, pp. 1926 (1991)
R14.
P.L. McEuen, E.B. Foxman, J. Kinaret, U. Meirav, M.A. Kastner, N.S. Wingreen, and S.J. Wind, S e l f - c o n s i s t e n t a d d i t i o n spectrum of a Coulomb island in the quantum Hall regime,
Phys. Rev. B, 45, pp. 11419 (1992) R15.
P. Maksym and T. Chakraborty, E f f e c t of e l e c t r o n - e l e c t r o n i n t e r a c t i o n s on the m a g n e t i z a t i o n of quantum dots, Phys. Rev. B 45, pp. 1947 (1992)
R16.
M. Wagner, U. Merkt, and A.V. Chaplik, S p i n - s i n g l e t - s p i n t r i p l e t o s c i l l a t i o n s in quantum dots, Phys. Rev. B 45, pp. 1951 (1992)
R17.
P.A. Maksym, Magic number ground s t a t e s of quantum dots in a magnetic f i e l d , Physics B 184, pp. 385 (1993)
R18.
T. Seki, Y. Kuramoto, and T. Nishino, Origin of magic angul ar momentum in a quantum dot under strong magnetic field,
J. Phys. Soc. Jpn. 65, pp. 3945 (1996) R19.
T. Chakraborty, V. Halonen, and P. Pietil/iinen, M a g n e t o - o p t i c a l transitions and level crossings in a Coulomb-coupled pair of quantum dots, Phys. Rev. B 43, pp. 14289 (1991)
R20.
A. Lorke, J.P. Kotthaus, and K. Ploog, Coupling of quantum dots on GaAs, Phys. Rev. Lett. 64, pp. 2559 (1990)
R21.
R.C. Ashoori, H.L. Stormer, J.S. Weiner, L.N. Pfeiffer, K.W. Baldwin, and K.W. West, N-Electron ground s t a t e e n e r g i e s of a quantum dot in magnetic f i e l d , Phys. Rev. Lett. 71, pp. 613 (1993)
R22.
A.T. Johnson, L.P. Kouwenhoven, W. de Jong, N.C. van der Vaart, C.J.P.M. Harmans and C.T. Foxon, Zero-dimensional s t a t e s and s i n g l e e l e c t r o n charging in quantum dots, Phys. Rev. Lett. 69,
Quantum dots
R23.
J. Weis, R.J. Haug, K. v. Klitzing, and K. Ploog, Competing channels in s i n g l e - e l e c t r o n t u n n e l i n g through a quantum dot, Phys. Rev. Lett. 71, pp. 4019 (1993)
R24.
V. Halonen, P. Pietils and T. Chakraborty, 0 p t i c a l - a b s o r p t i o n s p e c t r a of quantum dots and r i n g s with a r e p u l s i v e s c a t t e r i n g centre, Europhys. Lett. 33, pp. 377 (1996)
R25.
V. Halonen, T. Chakraborty, and P. Pietils Excitons in a p a r a b o l i c quantum dot in magnetic f i e l d s , Phys. Rev. B 45, pp. 5980 (1992)
R26.
A. Zrenner, L.V. Butov, M. Hagn, G. Abstreiter, G. BShm, and G. Weimann, Ouantum dots formed by interface fluctuations in AIAs/GaAs coupled quantum well structures, Phys. Rev. Lett. 72,
pp. 3382 (1994) R27.
R28.
R. Rinaldi, P.V. Giugno, R. Cingolani, H. Lipsanen, M. Sopanen, J. Tulkki, and, J. Ahopelto, Zeeman e f f e c t in p a r a b o l i c quantum dots, Phys. Rev. Lett. 77, pp. 342 (1996) J.-Y. Marzin, J.-M, G~rard, A. Izra~l, D. Barrier, and G. Bastard, Photoluminescence of single InAs quantum dots obtained by self-organized growth on GaAs, Phys. Rev. Lett. 73, 716 (1994)
R29.
G. Medeiros-Ribeiro, D. Leonard, and P.M. Petroff, E l e c t r o n and hole energy levels in InAs self-assembled quantum dots
Appl. Phys. Lett. 66, 1767 (1995) R30.
B.T. Miller, W. Hansen, S. Manus, R.J. Luyken, A. Lorke, J.P. Kotthaus, S. Huant, G. Medeiros-Ribeiro, and P.M. Petroff, Few-electron ground states of charge-tunable self-assembled
quantum dots, Phys. Rev. B 56, pp. 6764 (1997) R31.
Y. Arakawa, and H. Sakaki, Multidimensional quantum well l a s e r and temperature dependence of i t s t h r e s h o l d current, Appl. Phys. Lett. 40, pp. 939 (1982)
R32.
N. Kirstaedter, N.N. Ledentsov, M. Grundmann, D. Bimberg, V.M. Ustinov, S.S. Ruvimov, M.V. Maximov, P.S. Kop'ev, Zh.I. Alferov, U. Richter, P. Werner, U. GSsele, and J. Heydenreich, Low threshold, large To injection laser emission from (InGa)As quantum dots, Electron. Left., 30, pp. 1416 (1994)
149
T. Chakraborty
150
R33.
S. Tarucha, T. Honda, D.G. Austing, Y. Tokura, K. Muraki, T.H. Oosterkamp, J.W. Janssen, and L.P. Kouwenhoven, E l e c t r o n i c s t a t e s in quantum dot atoms and molecules, Physica E, 3, pp. 112 (1998)
R34.
D. Weiss, M.L. Roukes, A. Menschig, P. Grambow, K. von Klitzing, and, G. Weimann, Electron pinball and commensurate orbits in a periodic array of scatterers, Phys. Rev. Lett. 66,
pp. 2790 (1991) R35.
K. Tsukagoshi, M. Haraguchi, S. Takaoka, and K. Murase, On the mechanism of commensurability oscillations in anisotropic antidot lattices~ J. Phys. Soc. Jpn. 65, pp. 811 (1996)
R36.
W. Kang, H.L. Stormer, L.N. Pfeiffer, K.W. Baldwin, and K.W. West, How r e a l are composite fermions?, Phys. Rev. Lett. 71, pp. 3850 (1993)
R37.
K. Kern, D. Heitmann, P. Grambow, Y.H. Zhang, and K. Ploog, Collective excitations in antidots, Phys. Rev. Lett. 66, pp. 1618 (1991)
Quantum
dots
151
Reprinted Articles
153
VOLUME 60, NUMBER 6
PHYSICAL
REVIEW
LETTERS
8 FEBRUARY 1988
O b s e r v a t i o n of D i s c r e t e E l e c t r o n i c S t a t e s in a Z e r o - D i m e n s i o n a l S e m i c o n d u c t o r N a n o s t r u c t u r e M. A. Reed, J. N. Randall, R. J. Aggarwal, (a) R. J. Matyi, T. M. Moore, and A. E. Wetsel (b) Central Research Laboratories, Texas Instruments Incorporated, Dallas, Texas 75265
(Received2 October 1987) Electronic transport through a three-dimensionally confined semiconductor quantum well ("quantum dot") has been investigated. Fine structure observed in resonant tunneling through the quantum dot corresponds to the discrete density of states of a zero-dimensional system. PACS numbers: 73.20.Dx, 72.15.Rn, 72:70.+m, 73.40.Gk
Carrier confinement to reduced dimensions in a semiconductor wa~ first demonstrated in GaAs-AIGaAs quantum wells by electronic I and optical 2 spectroscopy in 1974. This achievement had led to numerous important developments in basic semiconductor physics and device technology. Structures produced by ultrathin-fllm growth are inherently two dimensional, and thus investigations have been largely confined to heterostructures where only the carrier momentum normal to the interfaces is quantized. Recent advances in microfabrication technology 3"5 have allowed the fabrication of structures with quantum confinement to one dimension ("quantum wires") 6'7 and have initiated intriguing investigations into one-dimensional physics, such as localization and electron-electron interaction, 8'9 single-electron trapping, l~ and universal conductance fluctuations, sl It is expected that the realization of semiconductor heterostructures with quantum confinement to zero dimensions ("quantum dots") will yield equally intriguing phenomeha. Attempts to observe confinement optically have been reported recently,12-16 but the spectra do not show the characteristic structure of a series of isolated peaks expected from a zero-dimensional electron-hole gas. We have therefore studied such structures by electronic transport, and in this Letter present evidence for electronic transport through a discrete spectrum of states in a nanostructure confined in all three spatial dimensions. The approach used to produce quantum-dot nanostructures suitable for electronic transport studies was to confine resonant-tunneling heterostructures laterally with a fabrication-imposed potential. 17 This approach embeds a quasibound quantum dot between two quantum-wire contacts. The initial molecular-beamepitaxial structure is a 0.5-/am n +-GaAs contact (Si doped at 2x 10 Is cm -3, graded to appf0ximately I016 cxn -3 over 200 ]t, followed by a 100-/~ undoped GaAs spacer layer), a 40-Jr A10.25Gao.v5As tunnel barrier, and a 50-A undoped InxGal-xAs quantum well. The structure was grown to be nominally symmetric about a plane through the center of the quantum well. Employing a InxGal-xAs quantum well allows one to lower the quantum well states with respect to the conduction-band edge while keeping the vertical dimensions fixed; x values
studied ranged from 0 to 0.08. Large, area (>_-2 #m square) mesas of a typical structure ( x - 0 . 0 8 ) fabricated by conventional means exhibited two resonant peaks: a ground state at 50 mV with a peak current density of 30 A/crn 2, and an excited state at 700 mV with a peak current density of 8.1 • 103 A/cm 2, both measured at 77 K. Electron-beam lithography defined an ensemble of AuGe/Ni/Au Ohmic metallization dots (single- or multiple-dot regions), nominally 1000-2500 J~ in diameter, on the top n +-GaAs contact by use of a bilayer polymethylmethacrylate (PMMA) resist and liftoff. The metal-dot Ohmic contact served as an etch mask for highly anisotropic reactive-ion etching with BCI3 as an etch gas, defining columns in the epitaxial structure. A scanning electron micrograph of a collection of these etched structures is seen in Fig. I. To make contact to the tops of the columns, a planarizing and insulating polyimide was spun on the sample and then etched back by 02 reactive-ion etching to expose the metal contacts on the tops of the columns. A gold contact pad was then
FIG. 1. A scanning electron micrograph of various size GaAs nanostructures containing quantum dots. The dark region on top of the column is the electron-beam defined Ohmic contact and etch mask. The horizontal bars are 0.5/~m.
t~) 1988 The American Physical Society
535
154
VOLUME 60, NUMBER 6
PHYSICAL
REVIEW
r n+
lal I [-, .......
GaAs
'
8 FEBRUARY 1988
LETTERS Ec,r (z)
a] :
............
/ AIGaAs barrier
ili'
InGaAs !!~!~:...I quantum well iil~,,:: AIGaAs barrier 9
b
z
.:: -~"'",, ~ I
/
I
i
|
I
....
I ........
1 b'
! iI 1 i
I
qb(r)
/ a ~
i
/
$
FIG. 2. Schematic illustration of the vertical (a-a') and lateral (b-b') potentials of a column containing a quantum dot, under zero and applied bias. O(r) is the (radial) potential, R is the physical radius of the column, r is the radial coordinate, Wis the depletion depth, (l)r is the height of the potential determined by the Fermi-level (Er) pinning, and Ec,r is the F-point conduction-band energy. evaporated over the top(s) of the column(s). The bottom conductive substrate provided electrical continuity. Figure 2 schematically illustrates the lateral (radial) potential of a column containing a quantum dot, and the spectrum of three-dimensionally confined electron states under zero and applied bias. A spectrum of discrete states will give rise to a series of resonances in transmitted current as each state drops below the conductionband edge of the injection contact. To observe lateral quantization of quantum well state(s), the physical size of the structure must be sufficiently small that quantization of the lateral momenta produces energy splittings > kT. Concurrently, the lateral dimensions of the structure must be large enough that pinchoff of the column by the depletion layers formed on the sidewalls of the GaAs column does not occur. As a result of the Fermi-level pinning of the exposed GaAs surface, the conduction band bends upward (with respect to the Fermi level), and where it intersects the Fermi level determines in real space the edge of the central conduction-path core. We can express the radial potential (l)(r) in the column [for (R - W) <_ r _< R], assumed axially symmetric, as ~ ( r ) " ~ T [ 1 -- ( R -- r ) / W ] 2,
AE - (2~r/m * ) I/2h/R,
(2)
(1)
where r is the radial coordinate, R is the physical radius of the column, W is the depletion depth, and ~ r is the height of the potential determined by the Fermi-level 536
pinning. When the lateral dimension is reduced to 2W or less, the lateral potential becomes parabolic though conduction through the central conduction-path core is pinched off. A structure that satisfies both constraints was achieved with a In0.t~Gao.92As quantum-well double-barrier structure with a physical (lithographic) lateral dimension of = 1000 /I,. Figure 3 shows the current-voltage characteristics of this (single)microstructure as a function of temperature. If we assume that the current density through the structure is approximately the same as in a large-area device, measurement of the peak resonant current implies a minimum (circular) conduction-path core of 130 ~ for this structure; thus, a lateral parabolic potential approximation is valid. This implies a depletion depth of =430/%, at the double-barrier structure, in reasonable agreement with that expected from the known doping level (at 2 x 10 Is cm - 3, W - 220 A) and with the realization that W will enlarge in the undoped double-barrier region. The splitting of the discrete electron levels in the quantum dot is then
where m* is the effective mass of the electrons in the quantum well (linearly extrapolated between that of GaAs and InAs) and R is the physical radius. With a Fermi-level pinning of 0.7 eV, the states should be split
155
VOLUME 60, NUMBER 6
PHYSICAL
REVIEW
LETTERS
8 FEBRUARY 1988
tial in the lateral dimensions) on the nanometer scale. We have performed electrical spectroscopy in the form of resonant tunneling through the spectrum of electron states, and observe resonances that correspond to the density of states of a zero-dimensional system. W e are indebted to R. T. Bate, W. R. Frensley, J. H. Luscombe, and R. H. Silsbee for helpful discussions and analysis, and to R. F',. Aldert, D. A. Schultz, P. F. Stickney, and J. R. Thomason for technical assistance. This research was supported in part by the U.S. Army Research Oitice and the U.S. Office of Naval Research.
FIG. 3. Current-voltage characteristics of a single quantum-dot nanostructure as a function of tempe~ture, showing resonant tunneling through the discrete states of the n - 2 quantum well resonance. The arrows indicate the voltage positions of the discrete states for the T-- 1.0-K curve. evenly by 26 meV. Only the excited-state resonance of this structure is observed since the current expected from the groundstate resonance is at the detection limits of the test apparatus. At high temperature, the characteristic negative differential resistance of the double-barrier structure is evident. As the temperature is lowered, two effects occur. First, the resonant peak shifts slightly higher in voltage (because of the increase of the GaAs contact xesistance with temperature) and decreases in current (because of the freezing out of excess leakage current). Secondly, there appears a series of peaks superimposed on the negative-differential-resistance peak. In the range 0.75-0.9 V the peaks are approximately equally spaced, with a splitting of ~ 50 inV. Under the assumption that most of the bias is incrementally dropped across the double-barrier structure, the splitting of the equally spaced series is 25 meV, in excellent agreement with the value determined from the physical dimension of the structure. We believe that the structure observed here corresponds to resonant tunneling through the spectra of discrete quasibound (in the z direction) states in the quantum dot which correspond to the density of states of a three-dimensional semiconductor quantum well. Another peak, presumably the ground state of the harmonic-oscillator potential, occurs " - 8 0 mV below the equally spaced serie~ The origin of this anomalously large splitting is not understood, though nonparabolicity of the lateral confining potential cannot be ruled out. ts In conclusion, we have measured electronic transport through quantum well states that have been laterally confined in all three dimensions (by heterojunction barriers in one dimension and a fabrication-imposed poten-
fa)Present address: Massachusetts Institute of Technology, Cambridge, MA 02139. (b)Present address: Harvard University, Cambridge, MA 02138. IL. L. Chang, L. Esaki, and R. Tsu, Appl. Phys. Lett. 24, 593 (1974). 2R. Dingle, A. (2. Gossard, and W. Wiegmann, Phys. Rev. Lett. 33, 827 (1974). 3A. N. Broer, W. W. Molzcu, J. J. Cuomo, and N. D. Wittels, AppL Phys. Lett. 29, 596 (1976). 4R. E, Howard, P. F. Line, W. J. Skocpol, L. D. Jackel, and H. G. Craighead, Science 221, 117 (1983). 5H. G. Craighead, J. AppL Phys. 55, 4430 (1984). 6H. Sakaki, Jpn. J. Appl. Phys. 19, L735 (1980). 7W. J. Skocpol, L. D. Jackel, R. E. Howard, E. L. Hu, and L. A. Fetter, Physica (Utrecht) UT& !18, 667 (1983). 8R. G. Wheeler, K. IC Choi, A. Go~l, R. Wisnieff, and D. E. Prober, Phys. Rev. Lett. 49, 1674 (1982). ~ 9T. J. Thornton, M. Pepper, H. Ahmed, D. Andrew~ and G. J. Davies, Phys. Rev. Lett. 56, 1198 (1986). I~ S. Raals, W. J. Skocpol, L. D. Jackel, R. E. Howard, L. A. Fetter, R. W. Epworth, and D. M. Teunant, Phys. Rev. Lett. $2, 228 (1984). IIW. J. Sckocpol, P. M. Mankiewieh, It_. E. Howard, L. D. Jackel, D. M. Tennant, and A. D. Stone, Phys. Rev. Lett. 56, 2865 (1986). 12M. A. Reed, P~ T. Bate, K. Bradshaw, W. M. Duncan, W. R. Frensley, J. W. Lee, and H.-D. Shill, J. Vac. Sei. TechnoL B 4, 358 (1986). 13K. Kash, A. Scherer, J. M. Worlock, H. G. Craighead, and M. C. Tamargo, Appl. Phys. Lett. 49, 1043 (1986). laj. Cibert, P. M. Petroff, G. J. Dolan, S. J. Pearton, A. C. Gossard, and J. H. English, AppL Phys. Lett. 49, 1275 (1986). 15H. Temkin, G. J. Dolan, M. B. Panish, and S. N. G. Chu, Appl. Phys. Lett. 50, 413 (1987). 16R. L. Kubena, R. J. Joyce, J. W. Ward, H. L. Garvin, F. P. Stratton, and R. G. Brault, AppL Phys. Lett. 50, 1589 (1987). 17j. N. Randall, M. A. Reed~ T. M. Moore, R. J. Matyi, and J. W. Lee, to be published; A. E. Wetsel, M. A. Reed, J. N. Randall, R. J. Aggarwal, R. J. Matyi, and T. M. Moore, to be published. 18M. Luban and D. L. Pursey, Phys. Rev. D 33, 431 (1986).
537
156
VOLUME 62, NUMBER 18
PHYSICAL
REVIEW
LETTERS
1 MAY 1989
Zeeman Bifurcation of Quantum-Dot Spectra W. Hansen, T. P. Smith, III, K. Y. Lee, J. A. Brum, ta) C. M. Knoedler, J. M. Hong, and D. P. Kern IBM Thomas J. Watson Research Center, Yorktown Heights, New York 10598
(Received 22 December 1988) We observe the magnetic-field-induced bifurcation of quantum levels into surface states and bulklike Landau states. The disruption of the electric field quantization by a magnetic field is most dramatic for electrons bound in two dimensions perpendicular to the magnetic field. The interplay between competing spatial and magnetic quantization mechanisms results in a pronounced and complex level splitting. The observed splitting of zero-dimensional energy levels depends critically on the size of the quantum dots, and can be explained with a calculated single-particle energy spectrum. PACS numbers: 71.45.--d,73.20.Dx, 73.40.Kp, 73.50.Jt For more than half a century confined electron systems in a magnetic field have been investigated theoretically in terms of their influence on the Landau diamagnetism of free electrons. !-3 Investigations of surface states in confined electron systems have been revived more recently in order to explain the quantized Hall effect. 4'5 Their skipping orbit nature is also demonstrated by transport measurements with ballistic point contacts. 6.7 The influence of edge states on the properties of the electron system is expected to increase with decreasing system size and even more dramatically with decreasing dimensionality. In an electron system that is free to move in only one dimension perpendicular to the magnetic field each electric subband transforms into a hybrid band when the magnetic length l a - ( h / e B ) I/2 becomes comparable to the width of the electron system. With increasing magnetic field the energy separation between adjacent hybrid bands approaches the cyclotron energy and the density of states at the bottom of each hybrid band increases, so that the bands become Landau-level-like at high magnetic fields. There is a continuous transition and a one-to-one correspondence between the electric subband structure at zero-magnetic field and Landau-level-like hybrid bands at high-magnetic field, s - ~0 In contrast a far more complex behavior is predicted for zero-dimensional (0D) systems. 1-3 At zero magnetic field the discrete energy levels are each occupied by two electrons except for degeneracies that depend on the symmetry of the confinement. With increasing magnetic field this degeneracy is lifted and hybrid levels originating from the same zero-field energy level join different Landau levels at high magnetic field. In general, there is no one-to-one correspondence between energy levels at zero magnetic field and Landau levels at high magnetic fields. The splitting of 0D energy levels at low magnetic field is similar to the normal Zeeman splitting of electronic states in atoms. However, in atomic physics the magnetic field is usually a weak perturbation of the Coulomb confinement. To observe Landau-level-like behavior of atomic electrons either the magnetic field must be 2168
several orders of magnitude larger than those experimentally realizable today tl or the atoms must be highly excited. 12 Because of the low effective mass and the high dielectric constant the hydrogenic states of shallow donors in semiconductors transform into the Landau limit at lower magnetic fields. 13 Unlike in the case of semiconductor impurities, the electrons in the microstructured heterojunctions studied here are confined by an artificially imposed potential that differs drastically from the Coulomb potential. Furthermore, the number of bound electrons can be varied systematically. The extent of the confinement potential in such devices is of the or-: der of 100 nm. 14"15 Thus at low magnetic fields all states can be considered surface states, while at magnetic fields of about 10 T (where the magnetic length is ---I0 nm), almost all states are Landau-level-like. We have observed the magnetic field rearrangement of energy levels in quantum dots for the first time. The density of states of the discrete energy levels in the quantum dots is measured with capacitance spectroscopy as a function of the gate voltage and a magnetic field applied perpendicular to the heterojunction interface. The samples are modulation-doped GaAs-AIGaAs heterojunctions. The electron system at the GaAs-A1GaAs interface is so strongly confined in the direction normal to the interface that the system is in the extreme quantum limit with respect to motion in this direction. The A1GaAs layer of the heterostructure is covered with a matrix of many (--- 105) 30-nm-thick squares of undoped GaAs with widths of 200, 300, or 400 nm. Minimal variation of the square size is achieved by lithography with a high-resolution vector-scan electron-beam writer. From process parameters such as resist resolution, beam diameter, and accuracy of positioning of the exposing electron beam we estimate the size variation of the fabricated dots to be less than 5% of the diameter. The matrix is covered with a metal gate, so that lateral confinement of the electron system is provided by the band bending beneath the Schottky barrier. The electron system resides below the GaAs squares, where the front gate has a larger separation from the heterojunction interface. Back contact is provided by a doped substrate separated
9 1989 The American Physical Society
157
VOLUME 62, NUMBER 18
PHYSICAL
REVIEW
1 M A Y 1989
LETTERS 250
z
I'=0.71<
f=10kHz
_..L_
_
200( _
0 . , 0 0 0 @ee
0 ~176176 @'~1 9 76
0
000 9 0,-,~ go
0 0 9
150 (
,o
I~ (
~) op o~~ ~ o ~~ ..o,,,,,,.,~ o
.+OO.o
.
50
J
=
go
]
oeeO
~
Ooi o 0 0 000"
oo...o| 9" ~ ~ , e " ~
o
o
~
o
o~
........
-100
9
9 9
i,,.,.o~ N~ ~ o , , . 9~
o
o
+oj - -
50
/
0 9 9 9 9
O
9
o
oO~
oo 0
-"
" Q O9 O
0
-150 B-O
J
-ee.
- ............-.~ i T
0 0 0
,, o ~ 0 0
~j,
0 0
0
9 9 0 9 0 9
-
~eeeeeeeee
-200 ~ " e e e e e -A
I
I
--0.30 -0.20 -0.10
l
0.00
, I
0.10
,,
I
0.20
0.30
-250
0.0
I 0.4
9 9 9 ooo oooo o2 9 9 9 9 9 9 9 9 ,o 9 9
~~
oo
o o o O Q O o o O O O O O 0
, I 0.8
,,
,,I 1.2
,
,I . . . . . . . . . . 1.6 2.0
B (T)
vc(v) + FIG. 1. Gate voltage derivative of the capacitance of a sample with 300-nm squares at four different magnetic fields. The orientation of the magnetic field with respect to the geometry of the confinement is indicated in the inset. The data were recorded at T - 700 m K.
FIG. 2. Gate voltage position of the maxima in the gate voltage derivative of the capacitance vs the magnetic field. The data presented in Figs. 1 to 3 were taken on the same sample, Filled circles indicate strong maxima (cf. Fig. 1). The dashed square indicates the gate voltages and magnetic field range of Fig. 3.
from the heterojunction interface by an undoped 80-nmthick GaAs layer. The gate voltage derivative dC/dVc is measured with phase lock-in techniques at modulation voltages between 5 and 10 mV and frequencies of about 10 kHz. Typical traces of the gate voltage derivative of the capacitance versus the gate voltage at four differe.nt magnetic fields are shown in Fig. I. The magnetic field is applied in the direction of strong confinement normal to the sample surface (z direction). At gate voltages below V ~ - - - 2 7 0 mV there are no electrons in the potential well at the heterojunction interface resulting in a zero derivative signal. At higher gate voltages electrons occupy the discrete energy states of the quantum dots resulting in capacitance maxima whenever the Fermi energy crosses a discrete energy level, n~ The gate voltage derivative is recorded in order to enhance the signal against the background capacitance. The period of the oscillations changes systematically with the dot size of the samples: largest in samples with 200-nm dots and smallest in samples with 400-nm dots. Also, the oscillation amplitude of samples with 400- and 300-nm squares increases significantly between T - 4 . 2 K and T - 0 . 7 K, whereas the oscillation amplitude of 200-nm squares is insensitive to temperature changes in this range. The amplitude of the oscillations varies with the gate voltage in the B = 0 T trace of Fig. 1. Such modulation of the oscillation amplitude is reproducible and clearly observ-
able in 300- and 400-nm squares. The number of recorded oscillations in samples of 200-nm dot size is too low to observe a corresponding behavior, since the applied gate voltage is limited by the onset of leakage current. The positions of the levels at low gate voltages (VG < - 0 . 1 V) change only ~lightly at low magnetic fields (B < 1 T), whereas the positions of levels at higher gate voltages exhibit a complex behavior at very low magnetic fields (B < 0.2 T). At a magnetic field of about B - 1 T a splitting of the oscillation maxima is clearly observed in samples with 300-nm squares. The field value at which this splitting occurs depends on the dot size of the samples. In samples with 400-nm squares the splitting occurs at a magnetic field of B - 0 . 6 T; in 200-nm samples at B - 1.5 T. The gate voltage positions of the oscillation maxima in dC/dVG are plotted versus the magnetic field in Fig. 2. Weak oscillations are plotted as open circles as shown in Fig. 1. The magnetic field dependence of the maxima positions at B - 1 T, where the splitting occurs, suggests a level anticrossing. Furthermore, we observe that those peaks dominating at low magnetic fields weaken as soon as satellite peaks occur at slightly higher gate voltage. With increasing magnetic field the former vanish and the latter become dominant. Figure 3 elucidates the behavior described above. The magnetic field of adjacent traces increases by 0.01 T and the traces are slightly offset for clarity. Besides illustrat2169
158
PHYSICAL
VOLUME 62, NUMBER 18 1
REVIEW
1 M A Y 1989
LETTERS 15
I
"~"'8= 1.0T
12
9
v taJ
6
=
-0.10
.
I
t
-0.05
0.00
I 0.05
0
0.10
vc Cv) FIG. 3. Gate voltage derivative of the sample capacitance in a small magnetic field range (cf. Fig. 2). Traces were recorded in 0.01-T steps.
!
0.0
!
0.5
i
i
1.0
1.5 B(T)
,i , , ,
2.0
,,~
J .....
2.5
3.0
FIG. 4. Calculated energy spectra of an electron system confined in a square of width 120 nm.
scribed by two quantum numbers (n,l):
W,,j- ~ ttcocl+(2n+ II I + l ) ( ~ h 2m2+ h 2f~2)I/2, ing in the upper half (B > 0.8 T) the onset of the strong level splitting at about B ==1 T this figure also shows that similar but less well-resolved level crossing occurs at lower magnetic fields. For instance, the lowest trace (B ==0.6 T) exhibits a pronounced asymmetry in the shape of the maximum at V G - - 0 . 0 4 V with the maximum position shifted to lower gate voltage and a shoulder evolving at the high-gate-voltage slope. The shoulder becomes a separate peak at 0.04 T higher field. At a magnetic field of about 0.68 T the asymmetry is inverted with the shoulder on the low-gate-voltage side of the maximum. The same behavior occurs at slightly higher fields in the branch starting at B==0.6 T with an oscillation maximum at Vc ==0.01 V. We have calculated the magnetic field dependence of the electron energies in a quantum dot using the decoupied approximation. Furthermore, since the electron system is in the quantum limit with respect to motion in the z direction, we can neglect the energy dispersion for motion in this direction. Because the quantum oscillations are measured in terms of gate voltage not energy, we do not anticipate perfect agreement between calculated spectra and experimental data. However, the main features of the measured data are present in the calculated spectra. An analytical calculation of the singleelectron energy spectrum was first done assuming a harmonic confinement in the x-y plane with rotational symmetry with respect to the z axis t V(x,y)=m* x f12(x2+y2)/2. The electron energies are then de2170
(1) with me the cyclotron frequency and h l the angular momentum of the electron. Model calculations previously performed for quantum wires 16 in the absence of magnetic field and extended to 0D structures in the classical limit 17'Is show that the sdf-consistent potential changes with gate voltage. It is parabolic at the threshold gate voltage but becomes square-well-like with increasing gate voltage. The results presented in Fig. 4 are calculated for electrons confined in a rectangular square-well, potential with finite (V0"-600 mcV) walls and width W - 1 2 0 nm. These values approximate the self-consistently calculated potential at a gate voltage about 200 mV above the threshold voltage. The energy spectrum is obtained by diagonalizing the Hamiltonian in magnetic field calculated with a finite basis of solutions at zero magnetic field. The spectrum at zero magnetic field represents the well-known energy levels of a particle in a box with degeneracies of different levels corresponding to the fourfold symmetry of a square. This degeneracy is lifted by the magnetic field. Each branch corresponds to a twofold-degenerate energy level since electron spin is neglected. Spin splitting of the levels is observed at far higher magnetic fields than those discussed here (B > 4 T). The evolution of each branch depends in detail on the assumptions made for the shape and symmetry of the confinement potential. 2'3 At low magnetic field (la ~. W) and high quantum numbers, a complicated level
159
VOLUME 62, NUMBER 18
PHYSICAL
REVIEW
crossing commences. This agrees invariant to a change of the scale in real space, the magnetic fields where different states cross each other scale with the dot size. As discussed+above, we observe the level crossings at higher fields with decreasing square size of the samples. At high magnetic fields different branches coalesce at the Landau energies ( v + 89)h~oe, v - 0 , 1 , 2 ..... In Fig. 4 this behavior is seen for the two lowest Landau energies at magnetic fields larger than B - 2 T. In this field regime the magnetic length is smaller than the potential width and it becomes possible to distinguish between surface states with energies between the Landau levels and bulklike states dose to the Landau energies. Experimentally, we observe an increase in the oscillation amplitudes at high fields indicating that more and more energy levels condense into bulk Landau states. However, every state is not resolved in our measurements, and the strengths of the observed oscillations vary significantly (ef. Fig. 3). In contrast, our model predicts equally strong maxima, since each branch corresponds to a twofold-degenerate energy state. Only certain maxima are expected to grow strong at high magnetic fields when many levels merge into Landau levels or, occasionally, at low fields, when s.everal levels happen to cross one another. Our calculation of the energy spectra does not inelude broadening of the energy levels by potential fluctuations originated, e.g., by impurities or surface roughness of the electrostatic confinement. The impact of fluctuations in the confinement potential on the broadening of a level depends on the probability distribution of the wave function in the quantum dot. A level with large probability amplitude in the center of the dot will be less affected by surface roughness than a state with large probability for finding the electron near the surface. We may, therefore, expect that states in which the electron wave function is predominantly in the center of the dot exhibit larger maxima in the measurement. In the harmonic-oscillator model with rotational symmetry [cf. Eq. (1)] such states have quantum numbers n - - 0 and 1 >" 0. They increase monotonically with magnetic field and are the first states to approach the Landau energies (v+ ~)hcoo v - 0 , 1 , 2 ..... In fact the dominant peaks in the measurements behave similarly except where level anticrossing occurs. A theory that explains the observations quantitatively must take into account that not only the Fermi energy, but also the confinement potential, changes with the gate voltage. Furthermore, it is predicted that electron corre-
LETTERS
I MAY 1989,
lation plays an important role.19 The exact form of the potential and the shape of the dot will determine the detailed behavior of the surface states. Our results also indicate that inclusion of surface roughness is important to explain the signal strength. A systematic study of the perimeter contributions to the Landau quantization in quantum dots may reveal a variety of detailed information about the artificially imposed confinement that has not been accessible so far. We gratefully acknowledge helpful discussions with U. Sivan, Y. Imry, D.-H. Lee, A. B. Fowler, L. Chang, and L. Esaki. Also, we appreciate the technical assistance of M. Christie and L. Alexander. This work was supported in part by the Office of Naval Research.
(a)Permanent address: Universidade de Campinas, Department de Fisica Do Estado Solido 9 Ciencia dos Materiais, i 3081, Campinas, Sa6 Paulo, Brazil. IC. G. Darwin, Pro<:. Cambridge Philos. Soc. 27, 86 (1930). 2M. Robnik, J. Phys. A 19, 3619 (1984). 3U. Sivan and Y. Imry, Phys. Rev. Lett. 61~ 1001 (1988). 4R. B. Laughlin, Phys. Rev. Lett. 50, 1395 (1983). 5B. I. Halperin, Phys. Rev. B 25, 2185 (1982). +V. S. Tsoi, Pis'ma Zh. Eksp. Teor. Fiz. 19, 114 (1974) [JETP Lctt. 19, 70 (1974)]. 7H. van Houtcn, B. J. van Wees, J. E. Mooij, C. W. J. Bcenakker, J. G. Wiiliamson, and C. T. Foxon, Europhys. LeR. 5, 721 (1988). gD. A. Poolr M. Pepper, K. F. Berggren, G. Hill, and H. W. Myron, J. Phys. C 15, L21 (1982). 9K. F. Berggren, T. J. Thornton, D. J. Newson, and M. Pepper, Phys. Roy. Lett. 57, 1769 (! 986). I~ P. Smith, III, J. A. Brum, J. M. Hong, C. M. Knocdler, H. Arnot, and L. Esaki, Phys. Rev. Let~ 61,585 (1988). IIH. Herold, H. Ruder, and G. Wunner, J. Phys. B 14, I I40 (1981). 12j. Main, G. Wiebusch, A. Holle, and K. H. Wclgr Phys. Roy. Lett. 57, 2789 (1986). InC. J. Armistead, P. C. Makado, S. P. Najda, and R. A. Stradling, J. Phys. C 19, 6023 (1986). 14T. P. Smith, lIl, K. Y. Lee, C. M. Knoedler, J. M. Hong, and D. P. Kern, Phys. Roy. B 38, 2172 (1988). 15M. A. Reed, J. N. Randall, R. J. Aggarwal, R. J. Matyi, T. M. Moore, and A. E. Wetsel, Phys. Rev. Lett. 60, 535 (1988). 16T. P. Smith, III, H. Arnot, J. M. Hong, C. M. Knoedler, S. E. Laux, and H. Schmid, Phys. Rev. Lett. 59, 2802 (1987). tTS. E. Laux and F. Stern, Appl. Phys. Lett. 49, 91 (1986). 18A. Kumar, S. E. Laux, and F. Stern (unpublished). 19G. W. Bryant, Phys. Rev. Lett. 59, 1140 (1987).
2171
160
VOLUME62, NUMBER 21
PHYSICAL
REVIEW
LETTERS
22MAY 1989
O b s e r v a t i o n o f Z e r o - D i m e n s i o n a l S t a t e s in a O n e - D i m e n s i o n a l E l e c t r o n I n t e r f e r o m e t e r B. J. van Wees, L. P. Kouwenhoven, and C. J. P. M. Harmans Department of Applied Physics, Delft Unit,ersity of Technology, P.O. Box 5046, 2600 GA Delft, The Netherlands J. G. Williamson, C. E. Timmering, and M. E. I. Broekaart Philips Research Laboratories, 5600 JA Eindhoven, The Netherlands C. T. Foxon and J. J. Harris Philips Research Laboratories, Redhill, Surrey RHI 5HA. United Kingdom (Received 9 March 1989) We have studied the electron transport in a one-dimensional electron interferometer. It consists of a disk-shaped two-dimensional electron gas, to which quantum point contacts are attached. Discrete zero-dimensional states are formed due to constructive interference of electron waves traveling along the circumference of the disk in one-dimensional magnetic edge channels. The conductance shows pronounced Aharonov-Bohm-type oscillations, with maxima occurring whenever the energy of a zerodimensional state coincides with the Fermi energy. Good agreement with theory is found, taking energy averaging into account. PACS numbers: 72.15.Gd,72.20.My, 73.20.Dx Advancing technology has made it possible to study electron transport in systems with reduced dimensionality. The - ultimate limit is reached when the electron motion is restricted in all directions, and discrete, zerodimensional states are formed. =-3 An elementary way to study these zero-dimensional (0D) states is to construct a one-dimensional (ID) electron interferometer. It is technologically feasible to fabricate such an interferometer by defining a I D wire by lateral confinement of the electrons, together with controllable barriers at both ends of the wire. 4 We have taken an alternative approach and have employed the one-dimensional nature of the electron transport along the boundary of a two-dimensional electron gas (2DEG) in high magnetic fields. This transport takes place through magnetic edge channels, which consist of the current-carrying states of each Landau level, which are located at the boundaries of the 2DEG. By defining a disk in a 2DEG, we have made an interferometer in which 0D states are formed by the constructive interference of electron waves traveling along the circumference of the disk in one-dimensional edge channels. Quantum point contacts (QPC's), which are attached to the disk, function as barriers with controllable transmission. This device has the additional advantage that the interference can be tuned by the magnetic flux which penetrates the disk. A system similar to ours, consisting of a 2D quantum dot to which narrow leads are attached, was studied theoretically by Sivan, Imry, and Hartzstein. 5 Its conductance showed Aharonov-Bohm (AB) type oscillations which were attributed to interference of electrons traveling in edge channels. Jain 6 has predicted similar ABtype oscillations in the high-field conductance of narrow rings, van Loosdrecht et al. 7 have observed AB-type oscillations in the conductance of a single quantum point
contact. They were explained by the partial reflection of edge channels near both the entrance and exit of the QPC. Compared to Ref. 7, our device provides a clearcut distinction between the barriers where the reflections take place (these are formed by Q P C ' s ) a n d the 1D conductor which carries the current in between the barriers (the latter is formed by a magnetic edge channel inside the disk). A schematic layout of our device is given in Fig. 1(a). Current (I1,I2) and voltage (VI,V2)contacts are at-
A
(b)
B
r'l 2/] 1,1,3
~1,1T~2 1
(
><->~
~
)-<-->C~T8
~2
FIG. I. (a) Layout of the device. (b) Current flow in high magnetic fields. In this example the first edge channel is fully transmitted through the device and the second edge channel forms a one-dimensional interferometer. The interference can be tuned by the flux ~.
1989 The American Physical Society
2523
161
VOLUME 62, NUMBER 21
PHYSICAL
REVIEW
tached to the 2DEG of a high-mobility GaAs/Alo.33Gao.rvAs heterostructure. Its electron density is 2.3x10tS/m 2 and the elastic mean free path is 9 /~m. Two gate pairs A and B are defined by electron beam lithography and liftoff techniques. Application of a negative voltage (--0.2 V) to both gate pairs depletes the electron gas underneath the gates. The narrow channels in between the gate pairs are already pinched off at this gate voltage. A disk of 1.5 pm diam is created, which is connected to the wide 2DEG regions by two 300-nmwide QPC's. A further reduction of the gate voltage widens the depletion regions around the gates. At the QPC's these depletion regions overlap, and a saddleshaped potential is created, the height of which may be controlled by the gate voltage. Application of a negative voltage to only one gate pair (and zero voltage to the other) also makes it possible to measure the conductanees of the individual QPC's, and compare them with the conductance of the complete device. In high magnetic fields the location of the wave functions of the electrons at the Fermi energy Er, which constitute the edge channels, is determined by their guiding-center energy EG: 7.8
E~--Et;--.(n-- ~ )htoc .-T-~ glanB. e2 [ Go---h- Nq-
(1)
GA.s -- (e Vh ) (N + TA.a ) ,
(2)
where N indicates the number of fully transmitted spinsplit edge channels and TA and TB the partial transmission of the upper edge channel through QPC's A and B. Equation (2) illustrates that in high magnetic fields no scattering between different edge channels takes place in the QPC's. 9 The conductance Go of the complete device is the sum of the (quantized) conductance of the N fully transmitted edge channels and the conductance of a 1D I interferometer: s.6.1~
TATs
We have measured the conductance of the complete device as well as both individual QPC's as a function of magnetic field. The measurements were performed in a dilution refrigerator, the mixing chamber being at ~ 6 mK. A lock-in technique was used with a current of 0.5 nA. The gate voltage was fixed at --0,35 V. Increasing the magnetic field has two effects. First, the number of edge channels transmitted through the QPC's is gradually reduced. The conductance of the individual QPC's shows quantized plateaus in those B intervals in which the edge channels are either fully transmitted or completely reflected 9 (TA, T a - 0 ) . In the intervals between the plateaus the upper edge channel is only partially transmitted (TA,TB~0). As a second effect, the magnetic field changes the phase 0. Equation (3) predicts that Go is quantized when both GA and Gn are quantized, and predicts regular oscillations when GA and Gs are not quantized, with maxima occurring whenever the
22 MAY 1989
Electrons with different Landau-level index n and spin orientation flow along different equipotential lines V(x,y), which are determined by the condition - e • V(x,y) -Ec. Figure 1(b) illustrates the electron flow for the case of two occupied (spin-split) Landau levels in the bulk 2DEG. In this example the potential in the QPC's is such that the first edge channel is fully transmitted, and the second edge channel, which follows a different equipotential line, is only partially transmitted through the QPC's. As can be seen in Fig. 1(b), a one-dimensional interferometer is formed by this upper edge channel. In high magnetic fields the eonductances GA and Ga of QPC's A and B can be written as 9
I.F(I _ T,4)(I _ TB)._2[(I _T,~)(I _ TB)]I/2cos(O )
In this expression 0 is the phase acquired by a wave in one revolution around the disk. The relation between this phase and the area A enclosed by the upper edge channel is given by O--2zr(BA)/r in which r is the flux quantum. If TA and Ts are zero, discrete zerodimensional states are formed at those energies for which 0 equals an integer multiple of 2~. At these energies the edge channel encloses an integer number of flux quanta. 12
2524
LETTERS
(3)
energy of a 0D state coincides with the Fermi energy [this implies c o s 0 - 1 in Eq. (3)]. Figures 2(a) and 2(b) show the measured eonductances of the individual QPC's, illustrating the transition from the 3e 2/h to the 2e 2/h plateau. In contrast to Ref. 7, no oscillations are observed in the individual QPC's. Irregular structure is present instead, with a typical scale of ~. 0.03 T. This corresponds ,to one flux quantum in an area of 350• nm 2, which is the approximate area of the QPC's. The structure in GA and Ga can therefore be attributed to random interference effects in the QPC's themselves. Figure 2(c) shows the conductance of the complete device. Large oscillations are observed, with a period B0 which slowly varies from 2.5 mT at B - 2 . 5 T to 2.8 mT at B - 2 . 7 T. As can be seen in Fig. 3(a), which shows them on an expanded scale, these oscillations are extremely regular. The amplitude of the oscillations, as well as their period, does not change significantly when the magnetic field is reversed (and the current and voltage leads are interchanged). 13,14 Because of the different location of different edge channels [Eq. (1)], they will enclose different areas and their oscillations will therefore have different periods (see below). The observation of a single, well defined period therefore shows that the conductance of only a
162
VOLUME62, NUMBER 21 '
....
PHYSICAL
i . . . . . . . . .
i . . . . . . . . .
3
REVIEW
i . . . . . . . . .
. . . . . . . . . . . .
22MAY 1989
LETTERS 3
.
.
.
.
.
.
.
(Ai 1
2.5 .---.
~
r-
...-. f-
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
~
(
B
)
2
cuo~
r~
.
2.5 .
.
2.55 .
.
2.6
i
U.I Z
~
< 2.
. . . . . . . . . . . . . . . . . . . . . . . . . .
F--
o Z o ~
0 0
0 ~
o
.....
..... 1 2.65
2.7
I"2~rV~AAAA "
2.75
(C) l
2 2.4
2.5
2.6
2.7
MAGNETICFIELD (T) FIG. 2. (a) Conductance GA of QPC A. (b) Conductance Ga of QPC B. (c) Measured conductance Go of the complete device. Oscillations occur in the region where both GA and Gs arc not quantized. The maxima occur when the energy of a 0D state coincides with the Fermi energy, due to resonant transmission. (d) Conductance Go calculated from GA and Ga with a fixed period Bo-3.0 mT and T - 2 0 mK. single edge channel is modulated. The fact that the conductance in the oscillating region does not drop below 2e2/h, nor exceed 3e2/h, also shows that a truly onedimensional interferometer has been realized. The oscillations disappear when the temperature is raised above 200 mK. They also vanish when the voltage across the device is raised above 40 pV. From this we estimate the energy separation between consecutive 0D states to be about 40/aeV. ~5 Figure 2(d) shows the conductance calculated with Eq. (3). The values for TA and Ta have been determined from the measured GA and Ga [Eq. (2)]. A fixed period (3.0 mT) was chosen for the calculations. The finite temperature has been taken into account by including energy averaging: The conductance Go(T) at finite temperatures is given by Go(T)--fGo(E)[dfCE,T)/ dE]dE, in which f(E,T) is the Fermi distribution function, and Go (E) is the energy-dependent conductance at zero temperature. The latter can be obtained from Eq. (3) by noting that by changing 0 by 2~r one obtains the next 0D state, and this corresponds with an energy change of 40/~eV. Figures 2(c) and 2(d) show a good agreement between the amplitude of the oscillations as well as the amount of modulation, when a temperature of 20 mK is chosen for the calculation. The fact that this temperature is higher than the actual temperature of the device ( ~ 6 mK) can
5.05
5. I
MAGNETICFIELD {T}
5.15
FIG. 3. (a) Measured conductance Go, showing transmission resonances of the third edge channel. (b) Resonant conductance through zero-dimensional states. In this region the 9conductance of the third edge channel is almost zero, except when the energy of a 0D statecoincideswith the Fermi energy. The width of the peaks corresponds with an effectivetemperature of about 30 inK. (c) Measured conductance Go, showing transmission resonances due to 0D states belonging to the second edge channel.
be accounted for by the additional energy averaging due to the finite voltage ( ~ 6 g V) across the device. This ,t' results in an effective temperature T, ~. 20 mK. It was not possible to make a detailed comparison between the structure in GA and Ga and the structure in Go. This is probably due to the fact that the application of a voltage to gate pair B results in a slight change in GA and vice versa. In the region where TA and Ta are low, the conductance exhibits very narrow peaks when the energy of a 0 D state coincides with EF, as a result of resonant transmission. This is shown experimentally in Fig. 3(b). Narrow peaks with regular spacing occur in the conductance. Their height is modulated by the structure in GA and Ga. As a result of the finiteeffectivetemperature Te these peaks are broadened and acquire a half-width of approximately 4kTe. From the ratio between half-width and peak spacing (the latter corresponds with an energy of 40 ~eV) we obtain Te ~ 30 m K This is in reasonable agreement with the expected effective temperature of 20 m K resulting from the finitetemperature and finitevoltage across the device. Zero-dimensional states belonging to other edge channels have also been observed. Figure 3(c) shows the os2525
163
VOLUME 62, NUMBER 21
PHYSICAL
REVIEW
cillations from the second edge channel. Their period ( B 0 - 5 . 3 mT at B--5.1 T) is different from the oscillations from the third edge channel discussed above. Also oscillations from the fourth ( B o - 2 . 1 mT at B - - 1 . 8 5 T ) and fifth edge channels ( B o - 1 . 4 m T at B - 1.25 T) have been observed. The observed oscillations as a function of magnetic field are different from the Aharonov-Bohm oscillations observed in small metal 16 or semiconductor 17 rings. In our device the edge channels which carry the current are only formed when the magnetic field is applied. A variation of the magnetic field changes the location of these edge channels [Eq. (1)]. The change in enclosed flux 6 ~ resulting from the change in field A l / c a n now be written as A + --A (BTrr 2) -- 7rr 2AB + B2~rrAr
-[ttr2+ B2ttr dEt; dB ]
"
(4)
The change in edge channel radius is given by Ar -AEo/eE, in which E is the radial electric field at the location of the edge channel. Evaluation of (4) with r - 7 5 0 rim, B - - 2 . 5 T, and the rough estimate ts E - 3 x 104 V/m shows that the second term (which is negative) can be of the same order as the first one. Therefore, the observed period B 0 - - r is not simply related to the area enclosed by the edge channel, but depends on the magnetic field and the form of the electrostatie potential in which the electrons are confined, t9 In summary we have reported a realization of a onedimensional electron interferometer, in which the discrete electronic states show up in a very pronounced way. We thank L. W. Molenkamp. A. A. M. Staring, and C. W. J. Beenakker for valuable discussions, S. Phelps at the Philips Mask Centre and the Delft Centre for Submicron Technology for their contribution in the fabrication of the devices, and the Stichting your Fundamenteel Onderzoek der Materie ( F O M ) for financial support.
Ij. Cibert, P. M. Petroff, G. J. Dolan, S. J. Pearton, A. C. Gossard, and J. H. English, Appl. Phys. Lett. 49, 1275 (1986). 2M. A. Reed, J. N. Randall, R. J. Aggarwall, R. J. Matyi, T. M. Moore, and A. E. Wetsel, Phys. Rev. Lett. 60, 535 (1988).
2526
LETTERS
22 MAY 1989
3T. P. Smith, III, K. Y. Lee, C. M. Knoedler, J. M. Hung, and D. P. Kern, Phys. Rev. B 38, 2172 (1988). 4C. G. Smith, M. Pepper, H. Ahmed, J. E. F. Frost, D. G. Hasko, D. C. Peacock, D. A. Ritchie, and G. A. C. Jones, J. Phys. C 21, I..893 (1988). 5U. Sivan, Y. Imry, and C. Hartzstein, Phys. Rev. B 39, 1242 (1989); U. Sivan and Y. lmry, Phys. Rev. Lett. 61, 1001 (1988). 6.I.K. Jain, Phys. Rev. Lett. 60, 2074 (1988). ?P. H. M. van Loosdrecht, C. W. J. Beenakker, H. van Houten, J. G. Williamson, B. J. van Wees, J. E. Mooij, C. T. Foxon, and J. J. Harris, Phys. Rev. B 38, 10162 (1988). 8The Quantum Hall Effect, edited by R. E. Prange and S. M. Girvin (Springer-Verlag, New York, 1987). 9B. J. van Wees, E. M. M. Willems, C. J. P. M. Harmans, C. W. J. Beenakker, H. van Houten, J, G. Williamson, C. T. Foxon, and J. J, Harris, Phys. Rev. Lett. 62, 1181 (1989). i~ B/ittiker, IBM J. Res. Dev. 32, 63 (1988). l ij. K. Jain and S. A. Kivelson, Phys. Rev. Lett. 60, 1542 (1988). 12B. I. Halperin, Phys. Rev. B 25, 2185 (1982). 13Due to a slow drift of the device parameters the fine structure in consecutive traces did not fully reproduce. This prevented a detailed test of the symmetry of the magnetoconductance. However, the amplitude of the oscillations as well as their period were approximately the same for both field orientations. t4van Loosdrecht et al. (Ref. 7) have observed a large asymmetry in the amplitude of the oscillations in forward and reverse fields. Also, two sets of oscillations with slightly different periods were observed, which were attributed to spin splitting. In our device the spin splitting is fully resolved, as a result of which we only expect and observe a single set of oscillations. 15The Fermi energy is ~ 9 meV, which means that a large number of 0D states are occupied. 16R. A. Webb, S. Washburn, C. P. Umbach, and R. B. Laibowitz, Phys. Rev. Lett. 54, 2696 (1985). 17G. Timp, A. M. Chang, J. E. Cunningham, T. Y. Chang, P, Mankiewich, R. Behringer, and R. E. Howard, Phys. Rev. Lett. 58, 2814 (1987); C. J. B. Ford, T. J. Thornton, R. Newbury, M. Pepper, H. Ahmed, C. T. Foxon, J. J. Harris, and C. Roberts, J. Phys. C 21, L325 (1988). JSAt the 2DEG boundary the electrostatic potential changes by an amount Erie ( ~ 9 mV) in a depletion region which is about 300 nm wide. This gives a typical field strength E ~--3 x 104 V/m. 191n a fixed magnetic field (2.5 T) oscillations in Go are observed with a period ~ i mV, when the voltage on both gate pairs is swept. From the dependence of this period on the gate voltage we obtained an estimate for the edge channel radius: r ~.~ 350 nm.
164
VOLUME 59, NUMBER 10
PHYSICAL
REVIEW
LETTERS
7 SEPTEMBER 1987
Electronic Structure of Uitrasmali Quantum-Well Boxes Garnett W. Bryant
McDonnell Douglas Research Laboratories, St. Louis, Missouri 63166 (Received 11 May 1987) The electronic structure of interacting, few-electron systems confined in quasi-zero-dimensional, ultrasmall, quantum-well boxes has been calculated by use of the multielectron effective-mass Schr'6dinger equation. The configuration-interaction method is used to include electron correlation. Correlation effectS are dominant in large boxes; the electrons form a Wigner lattice. In smaller boxes subband spacing becomes dominant and the carriers become frozen in the lowest subbands. The calculations determine how and on what size scale this transition occurs. PACS numbers: 73.20.Dx, 71.45.Lr
Individual atoms are the microscopic limit for very small, confined-electron systems, in which the motion in all three spatial dimensions is quantized. Bulk systems bounded by surfaces are the macroscopic limit for very large', confined-electron systems. Recently, systems in the intermediate, mesoscopic regime, where the crossover from atomic to bulk behavior occurs, have begun to receive attention. Studies t-5 of semiconductor microcrystallites, with dimensions L from one to several tens of nanometers, extend the investigation of carder-confinement effects away from the atomic limit. With the recent advances in the art of microfabrication, ultrasmall (0.02 / z m ~ L < 0 . 5 /tm) quasi-zero-dimensional quantum-well boxes can be made ~9 which exhibit carrier confinement, extending the investigations away from the macroscopic limit and into the submicrometer size regime. The properties of ultrasmall structures are governed by the physics of the mesoscopic regime. Because ultrasmall structures give promise of novel, device applications, 10.t I there is strong motivation to develop quickly an understanding of mesoscopic physics. The carriers in a bulk (L > 1.0 ~um) structure form a many-electron system of weakly interacting particles which can be modeled by the single-particle effectivemass equation. In ultrasmall structures (L < 0 . I /zm) the effective-mass approach still provides a good description of the motion through the lattice. However, the carriers cannot be assumed to be a weakly correlated many-particle system. Consider a quantum box constructed with use of the confinement at an interface to define one of:'the confined dimensions. For a typical inversion-layer charge density of I0 il cm -2, a twodimensional uniform gas in a square, 0.1-/zm-wide box would contain ten carriers; in a box 0.01 /zm wide, less than one carrier. Carriers in ultrasmall boxes must be treated as interacting few-particle systems. In this Letter I calculate the electronic structure of interacting, few-particle systems confined in ultrasmall quantum-well boxes to determine how and on what size scale the carders in ultrasmall structures become correlated. The few-particle (N__< 6) systems have been studied i2 by solving the multiparticle effective-mass SchrSdinger 1140
9
equation for two-dimensional, interacting particles confined in a box modeled as a strictly two-dimensional quantum well. No effects of inversion-layer width are included. For simplicity the well is rectangular and has infinite barriers. Because the barriers are infinite, a basis set of wave functions which are separable in the two directions that define the well can be used. The singleparticle one-dimensional eigenstates (sines and cosines) are used as the basis functions. The particle interaction is the Coulomb interaction screened by the background dielectric constant. The correlations are included by use of a configurationinteraction approach. The multiparticle wave function is expanded in terms of Slater determinants constructed from the single-particle noninteracting eigenstates. The kinetic-energy and interaction matrix elements are found by used of the Slater-determinant basis and the Hamiltonian is diagonalized to find the eigenstates. The evaluation of the Coulomb matrix elements is straightforward. 13 In the infinite-barrier model, all kinetic-energy matrix elements scale as 1/L 2 and all interaction matrix elements scale as IlL when the dimension L of the box is changed without changing the box shape. This scaling determines the nature of the electron system. For small L, the Coulomb interactions are insignificant compared to the single-particle level spacings; the electrons are independent, uncorrelated, particles. As L increases, the interactions become significant and the multielectron states become correlated. The multielectron states evolve continuously, as L increases, from the exact, independent-particle states of the small-L limit. In the infinite-barrier model, the results are independent of the electron mass m, and dielectric constant e if all'energies are scaled by the effective Rydberg, Re--eZme/2a~E 2, and the lengths are scaled by the effective Bohr, ae -aoe/me. To illustrate the important size scales, I present results for GaAs wells, me--0.067mo and LE--13.1. The lengths are scaled by a0 and energies by Re. The evolution of the energy levels of an interacting, confined system that occurs when the box size changes is shown in Figs. 1 and 2 for two simple systems: two in-
1987 The American Physical Society
165
VOLUME 59, NUMBER 10
PHYSICAL
REVIEW
7 SE~rE~BER 1987
LETTERS
1.0 ,-,-
oo0 e e l
PxP, S ,
-.
~J i O0~i
zoO
el 90
"
.,, : e
.:~,~
o0
000
--"
el 0~--
ooO
e0 r oo
~
ol oO 0.1 --
1 "<J
el eeO 1
~
"J.,e o o
- - ]
'
"~l
oO
~ ~m~+-+
,, e
02;
::g ~ - ::~)
160
,,~
..
~
o0
~
oo0 +80 ee4~
0 -- eeO 0 -- e
O (,)
~ Co)
(c)
(d)
(e)
~ eO
FIG. 1. Energy levels for two electrons in a long, narrow ( L y - l Os GaAs quantum-well box. The spacings are measured relative to the ground-state energy E: and scaled by R,. (a) For noninteracting electrons Ly--20ae, E t --1.905R,, and /~--10sxR,. For interacting electrons I.~/ae, Es/R,, and Rj/P~ are (respectively) (b) 20, 1.907, 10s; (c) 200, 1.923, 10~; (d) 2000, 1.987, 10; and (e) 2• 2.314, 0.1. The parity P (e -- even, o - odd) and total spin S of each state are indicated.
teracting carriers in (a) long, narrow (L------Ly= 10Lx) quantum boxes (Fig. I) and (b) square quantum boxes (Fig. 2). The energies are scaled by a factor Ra which is different for each L to account for the 1/L 2 scaling of the kinetic energies. 14 If the Coulomb interactions were unimportant, then the scaled energy levels would be independent of L. The increases in the scaled ground-state energy and the changes in the level spacings that occur as L increases show how important the electron-electron interaction becomes in large boxes~ The internal motion of electrons in long, narrow structures is quasi one dimensional because the carriers are in the lowest subband of the narrow (x) direction. Mixing of higher x subbands is insignificant in the size regime covered in Fig. I. Ten y subbands were used to account for correlations along the long direction. Energies calculated with 10 y subbands differ from energies calculated with 8 y subbands by less than 0.1%. The internal motion in the square box is two dimensional. Six x and six y subbands were used to obtain energies accurate to O. 1% in square boxes. The effective Coulomb interaction increases as the dimensionality is lowered, s5 The Coulomb contribution to the ground-state energy E t is larger, when measured on a common energy scale, in a
(,)
Co) (c)
~
"
(e)
FIG. 2. Energy levels for two electrons in a square GaAs quantum-well box. f.~/ao, Er/P~,, and R J R , are (respectively) (a) for noninteracting electrons, 20, 0.377, 104; and for interactin8 electrons, (b) 20, 0.387, 104; (c)200, 0.456, 10~, (d) 2000,0.996, 1; (e) 2x104, 3.925, 0.01; and (19 6x104, 4.373, 0.0022. The x and y parity and total spin of each state are indicated. States with odd-x, even-), parity are degenerate with even-x, odd-), parity states and are not shown. Additional degeneracie~ not due to parity or spin are shown in parentheses. 9 long, narrow box than in a square box with the Same long f l y ) dimension. For Ly < 1 nm, the scaled energy l~vels shift slightly from the noninteracting levels, and exchange splitting occurs. Typically, states with total spin S - - 1 have lower energy (Hund, s t-dle) than ,5"-'0 states with the same parity. When Ly .~ 10 rim, the scaled energy levels shift substantially from the levels of the noninteracting system a n d have begun to cross in long, narrow boxes. Significant Coulomb contributions to E : begin to occur on this length scale. When Ly ~-100 nm, substantial reordering of the levels occurs in both square boxes and long, narrow boxes. In the large-L limit, where interactions dominate the kinetic energy, the system should become strongly correlated, with the electrons located to minimize the repulsive interaction as in a Wigner lattice ( W L ) f o r unconfined systems. The signature of the W L states in a confined system is the degeneracy of the levels. For example, in a long, narrow box there is one way to put two particles with the same spin on the box axis to minimize the direct Coulomb interaction. For two particles with opposite spin, there are two configurations. In a square box the particles would sit on opposite ends of the same diagonal in the WL limit. The degeneracy would be double the degeneracy for a long, narrow box since there 1141
166
VOLUME59,
NUMBER I0
P, H Y S I C A L
REVIEW
are two equivalent diagonals. In long, narrow boxes, the evolution of the states into levels with the degeneracies of the WL limit ~6 occurs when L ~>0.1 pm and the splitting of levels that are degenerate in the WL limit is no longer apparent. The W L limit is reached at larger L for square boxes because the effective Coulomb interaction is weaker. Ceperly 17 found that a two-dimensional electron gas becomes a WL when r, > 33ae. In GaAs the WL would occur for r, > 0.34 pro. This is consistent with the length scale on which the confined square system approaches the W L limit. The evolution of the energy levels is more complicated for systems with more than two particles. When N > 2 the level mixing is more complex, degeneracies in the WL limit are higher, and the transition to the WL limit occurs at larger L, requiring more accurate calculations to reach the W L limit. I calculated the cases of three and four particles in long, narrow boxes with accuracy adequate for tracing the evolution to the W L limit. For square boxes, the level degeneracies of the W L limit are not obvious when the number of particles is incommensurate with the symmetry of the structure; for example, when a square box contains three particles. The present results suggest that three spin-parallel electrons in a square box have four degenerate states in the WL limit. This is the degeneracy expected from the symmetry of the box. However, I have not yet been able to calculate with adequate accuracy the states for two parallel- and one opposite=spin particle to confirm that the degeneracy in the W L limit is twelve, as needed to be consistent with the results for parallel-spin particles. The spatial correlations of two particles in square boxes is shown in Fig. 3. For boxes in which the confining potential dominates, the carrier density a ( r ) ap= proaches the independent-carrier density. As L in= creases, the carriers move apart along the diagonals with a f t ) peaking farther from the center, and the density on the diagonals increases relative to the density off the diagonals. In addition, as L increases, the particle positions become more strongly correlated. As L increases, the probability, o(r,r0), for finding one particle at r if the second is on a diagonal at r0 increases for r at the opposite end of the same diagonal and decreases for other r.
The multiparticle wave functions evolve, as L in= creases, from the single-configuration Slater-determinant states of the noninteracting system by a mixing in of other configurations with the same parity and spin. For L ~<0.01/am, the only important configuration (probability ~ 0 . 9 5 ) in the interacting ground state is the noninteracting ground-state configuration. For L >~0.I /am, the excited noninteracting configurations are mixed in with comparable or greater probability than the noninteracting ground state. The important excited configurations of an infinite-barrier confined system are different from those of an atomic system. In an atom the excited single-particle levels get closer together (like 1142
7 SEPTEMBER I987
LETTERS [
'
'1
x0
'
i
.....
i''
y
~'/
!~
~
1.0t
0
~
I~
(e)
I
I
I
I
-
I
I
F -.0.5
-0.25
0 0,,9..5 0.5 y,,L FIG. 3. Ground-state single-particle density ~(r) and the two-particle conditional density o~r,ro) for two particles in a square box: (a) for noninteracting particles; and for interacting particles, (b) L--2OOao, (c) L--2000ao, and (d) L~ --20000a~ The paths in the insets define the contours on which the densities are determined. The dot in the inset and the arrow on the axis indicate the position re. The densities'are scaled for each L so that the densities of the noninteracfing states are independent of L.
l/n 2) as the excitation level n increases, and the most important excited configurations in an atomic correlated state are the single-particle excitations from the highest filled level. In contrast, the excited single-particle levels of the infinite-barrier box get farther apart (like n 2) as the excitation level increases, and the most important excited states in a correlated state are excitations to the lowest empty level. To conserve parity, such excitations must be two-particle excitations from the highest filled level or single-particle excitations from deeper in the core. For three- and four-particle systems, the most probable excited configurations are those with empty cores and those with unpaired spins in the core. In small (L
167
VOLUME 59, NUMBER. I0
PHYSICAL
REVIEW
of electrons. For example, in the smallest square box (L ~ 1.'nm) considered in Fig. 2, the energy cost to pair two electrons is 0 . 0 1 / ~ - - 5 . 3 eV, more than any realistic finite barrier in a GaAs structure. In conclusion, particle-particle correlations occur on the mesoscopic length scale in quantum microstruetures. The correlations are a few-particle, rather than a manyparticle, effect. As a consequence, these structures display a rich variety of electronic properties--unpaired electrons, weakly correlated states, and confined Wigner lattice states---that make microstruetures intriguing systerns to study. This work was performed under the McDonnell Douglas Independent Research and Development program.
IC. J. Sandroff, D. M. Hwang, and W. M. Chung, Phys. Rev. B 33, 5953 (1986). 2.i. Warnock and D. D. Awsehalom, AppL Phys. Lett. 48, 425 (1986). 3L. Brus, IEEE J. Quantum Eleetrom 22, 1909 (1986), and references therein. 4H. M. Sehmidt and H. Weller, Chem. Phys. Leth 129, 615 (1986). Sy. Kayanuma, Solid State Commun. 59, 405 (1986). 6M. A. Reed, R. T. Bate, K. Bradshaw, W. M. Duncan, W. R. Frensley, J. W. Lee, and H. D. Shih, J. Vae. Sei. Teeh-
LETTERS
7 SEPTEMBF.~ 1987
noL B 4, 358 (1986). 7K. Kash, A. Seherer, J. M. Worloek, H. G. Craighoad, and M. C. Tamargo, AppL Phys. Lett. 49, 1043 (1986). 8.i. Cibert, P. M. Petroff, G. J. Dolan, S. J. Pearton, A. C. Gossard, and J. H. English, Appl. Phys. Lett. 49, 1275 (1986). 9H. Tcmkin, G. J. Dolan, M. B. Panish, and S. N. G. Chu, AppL Phys. Lett. SO, 413 (1987). I~ lnoshita, S. Ohnishi, and A. Oshiyama, Phys. Rcv. Lctt. 57, 2560 (1986). I tM. Asada, Y. Miyamoto, and Y. Sucmatsu, IEEE J. Quantum Electron. 22, 1915 (1986). taA preliminary report on these calculations was given as part of G. W. Bryant, D. B. Murray, and A. H. MacDonald, in "Superlattices and Microstructures," Proceedings of the Second International Conference on Superlattice~s, Mierostructures and Microdevices, GSteborg, Sweden, 1986 (to be pubfished). 13G. W/Bryant, Phys. Rev. B 31, 7812 (1985). 14R, scales as l/L2 except for ease CO in Fig. 2. In that ease, P~ was chosen to be factor of 2 larger than would be obtained for I/L~ sealing, so that the sealed energy levels would fit on the figure. In Fig. 2(19 the energy-level spacings would be twice as large if l i l y 2 sealip~ was used. 15G. W. Bryant, Phys. Rev. B 29, 6632 (1984). 16To see that the correct degeneracies are obtained in the WL limit, one must remember that the levels are independent of s:, so that each S--1 level has three degenerate levels (sx-- -t-_1,0), and that the odd-x, even-y parity states are not shown in Fig. 2. 17I). Ceperley, Phys. Rev. B 18, 3126 (1978).
1143
168
446
Bemerkung
zur Quantelung des harmonischen im Magnetfeld.
Oszillators
Von u F o c k * , zurzeit in GSttingen. (Eingegangen am 12. Januar 1928.) Die exakten Werte der Energieniveaus des isotropen ebenen harmonischen OsziUators in einem zu seiner Ebene senkrechten homogenen Magnetfeld sind
E -
( . ~ - . ~ ) h ~ + (.~ + .~ + ~) h V;o~ + ~ (n 1, n2 - - O, 1, 2 . . . ) , eH m c
wo ~o die Eigenfrequenz des ungestSrten Oszillators. und ~'1 : 4 =
die Larmor-
frequenz bezeichnet. Der isotrope ebene harmonische Oszillator in einem zu seiner Ebene senkrechten homogenen Magnetfeld liefert das einfachste Beispie! tier Anwendung der St(irungsrechnung auf ein entarfetes mechanisches System. Es kiinnte daher die exakte Liisung dieses einfachen Problems yon einigem Interesse sein. Der Zweck dieser Notiz ist die Ableitung einer so]chen L(isung. Die H a m i l t o n s c h e Funktion fiir das betrachtete Problem lautet --
1
,j) +
1
,-
eH
+~
e2 H ~
( , j ~ - x~.) + S~c~ (~ + ~). f-Oo
tiler bezeichnet m die Masse, e die Ladung, 2
--- % die Eigenfrequenz
des Oszillators, H die magnetische Feldst~rke. frequenz
Fiihrt man die Larmor-
co, ?'1 "--- 9 " ~ '
601 =
eH 2 W'~ C
ein, so l~L~t sich die H a m i l t o n s c h e Funktion schreiben 1
H
__
1
- - 2,~ (l~ + p~) + ~ (y l~ - - x ~,) + ~ m (COo~ + ~o~) (x ~ + ~:,~).
* International Education :Board Fellow.
169
V.Fock, Bemerkung zur {~uantelung des harmonischen 0szillators usw. 447 Die entsprechende S c h r S d i n g e r s c h c Amplitudengleichung ist
(:,
z/0_~4~im
+
s,~',n[
0~
1
]
l Y .... E - - g m f l o g + r o ~ ) ( x 2+?p-) ~ , - - 0 .
Wir ftthren als L~ngeneinheit die Gri~13e
hm
(cog + ~D-'t~
ein und bezeiehnen co~
2~
1/~0~+ ~ = ~'
E
h t,~o~ + ~,
Benutzt man die dimensionslosen Polarkoordinaten x =
y -'- b r s i n cp,
brcos(p,
so wird die Amplitudengleichung 02 r ~ ~
rl O r
~- r 2 O q~ ~- 2 i
~
-+-
W--
,:'2) ~p --- O.
Diese Gleichung liiilt sich leicht durch Separation der Variablen 15sen. Setzt man
so erhalt man ftir 2~ (r) die Gleichung
(
d22 1 d/~ ~_ 2IV: dr 2 ~ r dr
n,r2
)
r2 /~--0,
d. h. genau die Gleichung, die man bekommt,, wenn man den isotropen ebenen 0szillator ohne M a g n e t f e l d in Polarkoordinaten behandelt. Dutch die Substitution r2--Q [indet man daraus d~/~
ld/~ Q dQ
(
1
B~ 2Q
Diese Gleichung is_t bereits yon S c h r S d i n g e r * Eigenwerte sind bier ~V~ - -
2k-l-n-l-
1
n2 ) 4 untersucht worden.
Die
( k - - O, 1, ...)
* E. Schr6dinger, Quaatisierung als Eigenwertproblem, 3. Mitteilung. Ann. d. Phys. 80, 484, 1926.
170
9448
V.Fock, Bemerkung zur Quantelung des harmonischen 0szillators usw.
und die Eigenfunktionen mit L~
n+k (@)
m
dn
d o n f-"n+k(@)
[Ln+k(Q) L a g u e r r e s c h e s Polynom]. Bereehnet man daraus die Energie E und fiihrt man wieder die Fre.quenzen v o- und v~ ein, so erh~lt man
L' - - +_ ~ h v, + (2 k + ~ + 1) h Vvo' + v~' oder
E-
(,~-
~ ) h ~ + (~, + ~, + ~)hl/~2
§
wo n~ und ~ zwei nicht-llegative ganze Zahlen sin& Dem International Education Board b i n ich fiir die ErmSglichung meines GSttinger Aufenthal~s zu herzlichem Dank verpflichtet.
171
86
Mr Darwin, The dia,mgnetism of the free electron
The Diamagnetism of the Free .Electron. By Mr C. G. DARWIN, Christ's College, Professor of Natural Philosophy, University of Edinburgh. [Received 7 :November, read 8 December 1930.] 1. In a recent paper Landau* has shown that, when electrons are moving freely in a magnetic field, they exhibit, in addition to the paramagnetic effect of their spin, a diamagnetic effect due to their motionS. This result is rather unexpected, since it is quite contrary to the classical case. There it might appear as though the circles described by the electrons must produce a magnetic moment, but the error was long ago pointed out by Bohr. The motion of the electrons must be confined to some region by means of a boundary wall, and the electrons near the wall describe a succession of circular arcs, repeatedly bouncing on the wall, and slowly creeping round it in the direction opposite to that of the uninterrupted circles; when the moment of these electrons is taken mto account, it exactly cancels out that due to the free circles. In Landau's work it is of course necessary to consider the boundary, but he shows how allowance is to be made for it by an appropriate process. The complete justification is rather subtle, and so it may be worth considering a special case, admitting of exact solution,i which takes the boundary into account, and so makes it possible to follow more closely the analogy between the classical and quantum problems. With regard to the general case with a boundary wall of any type, we shall only observe that the-different results arise, because in the wave problem ~ must vanish at the bounding "potential wall," and so will be small near it; this upsets the balance of the electric current near the wall, and yields the magnetic moment. 2. We shall replace the sharp boundary by a weak field of force proportional to the distance from a central axis. Thus the electron is mo~lng under the influence of a uniform magnetic field H along z, and of a fbrce represented by potential energy 89 (x~+y~), where B will ultimately be allowed to tend to zero. The motion along the z direction is of no interest and will be omitted. The Hamiltonian of the system is then +89 . . . . . . . . . (2.1). Landau, Zeits. f..Phys., 64, p. 629 (1930). f Throughout the present work it is supposed that the electron density is so small that the Exclusion Principle need not be applied.
172
87
M r D a r w i n , T h e d i a n u ~ n e t i s m of t h e f r e e electron
We form the SehriSdinger equation in polar coordinates,writing h for the quantum divided by 2rr, and co for eH/2mc, the Larmor precession. Then
. . . . . . . . . .
(9.-2).
Substi.tuting ~ = f (r) eae we have d~f , l d f d-# -r r dr
l~ rz f-
m ~oo~b~ 2m h~ r * / = - - ~ ( W -
lho~) f
...(2"3),
where we have written 1+
b=
.........
The parameter b is greater than unity, but will ultimately be made to tend to unity, as B tends to zero. There is no need to describe the solution of (2"3) in detail. The form of the function f is evidently independent of b, and s'o is s'uggested by considering the case b = 1, where the system degenerates so that it can be solved by x and y. In this form the solutions will depend on Hermite polynomials, and can be conveniently written as
e~'/2 d--~.~ e-~" eY'/2 d-~ e-'J', but when these solutions axe reeombined into the types appropriate for polar coordinates, the resulting functions turn out to be much simpler than Hermite polynomials. The solution is best expressed by the substitution z = r2moJb/h . . . . . . . . . . . . . . . . . . . . . . . . . (2"5), and we have f(r)
with
= z 89
e89 \d-~]
W = h~ [l + b (2n + ] / [ + 1)] ............... (2"7).
In this, n is any positive integer or zero, while 1 m a y have any integral value positive, negative or zero. We can see that there will be important differences according as it ispositive or negative. 3. It is worth examining how the quantum numbers are related to the corresponding classical orbits. The ttamiltonian is ~. =~
2+
+ copo + 89nzco~b2,2 ...... (3"1).
173
88
M r D a r w i n , The diamagnetism of the free electron
The angular momentum gives pc= P, and the e n e r g y equation is then . . . . . . . . . (3"2),
= -~z~r ~+ m(W-~P)-b~co~r2
whence we find I t follows that W >1~ P + b~[ P I. Again ~ = r + _P/mr ~, from which we derive tan (0 - cot) =
W - coP - ~/{(W- w P ) ~ - b~co22~} tan bo~t. bcoP
To bring out the analogies with the wave Solution we shall substitute 39 = hl, and (2"7). Then 1 may have any value, and n any value greater than - 8 9 The orbit is given by r ~m~b/h=
2n + 1
+IZl
+ cos
2b t
+
+
I Zl) ......... (3-3),
tan (t9- cot)= 2 n + l +
l l l - ~ / { ( 2l n + l + ! l t )
~
12}tanbcot ......... (3"4).
Consider first the orbits which have r constant. These require 171
n =-
89 and hence tan (t9- cot)=-~ tan bo~t. Thus if 1 is positive
we have t9 = (b + 1)cot, or nearly 2o~t, and the orbit is a circle described in the ordinary ~ a y at twice the Larmor speed. For negative 1 we have 8 = - (b - 1 ) cot, which means an orbit creeping very slowly in the opposite direction. In the limit with b = 1, it re.presents a stationary electron, at distance ~/(h l l I/mo~) from the omgm. :Now consider the general case, when n is greater than - 8 9 first with b = 1 so that the orbits are circles. Evideritly if 1 = 0 we have an orbit passing through the origin and so we see that positive 1 corresponds to an orbit surrounding the origfn eccentrically, while negative 1 implies a circle not surrounding the origin. W h e n we have . . b > 1, the orbit with . P ositive l will~sti]l swee P round the origin at a rate nearly given by angular velocity 2o~, but for negative 1 the orbit will be approximately a cycloidal motion consisting of a small circle described positively at rate 2oJ superposed on a large circle round the origin at slow negative rate ( b - 1 ) c o . I t is thus evident that we have got the features of Bohr's a r g u m e n t about the creeping of the electron round the boundary wall. Furthermore "
174
89
M r Darwin, The diamagnetism of the free electron
we can see that nearly all the important orbits are those with negative l, since those with positive 1 mostly have large energy; and so we are led to anticipate that the whole process, both of Landau, and of the vanishing in the classical case, will depend on these negative orbits. 4. Associated w i t h e a c h state n, l there is a magnetic moment, which could b~ worked out in detail by calculating the current function. I t is quite unnecessary to do this, since we can obtain it at once by differentiating the energy. Thus Wa ~z =
OH
=-~B
+ Ill + I}
l+
b
. (4.1),
...........
where/~B is the Bohr magneton eh/2mc. In forming this it is of course necessary to remember that b involves ~ and so must be db 1 differentiated; from (2"4)it is seen that ~ ~ = ~ - b . I t is then easy to set down a series for the average value of/~. But it is quicker to write down at once the partition function S = ~,~,z e-wz''kr . . . . . . . . . . . . . . . . . . . . . (4"2), for then
~ = kT ~ h~
If we write
log S.
~BH
k--T= kT - a
we have
S - - x,
x,
e-[t+b(2n+ltt+l)],~
n=O l = - - ~
e~ = [e(b+l) <<-
1] [1 - e "(b'l);']
.............
(4"3).
This is exact, but w e now approximate by putting b = 1 + Bi2mo~ ~ with B small. Then logS=a-log
I
e
2+2~
:
-1
B -log~a,
which tends to a - log (e~ - 1) + log a ................... (4"4), if we substitute for co in terms of a, and reject constant terms, including one in log B. We thus obtain _
a [a - log (e ~ - 1) + log a]
=/~B [ - c o t h a + 1 ] . . . . . . . . . . . . . . . . . . . . . . . . ( 4 " 5 ) ,
175
Mr .Darwin, T h e diamagnetism of the free electron
9O
which is the Langevin formula, but with the opposite sign. This reduces to Landau's result when a is small, and would seem to be correct in all cases where t h e density is not so great or the temperature so low that the Exclusion Principle must be applied. .,
5. In conclusion it must be remembered that this is to be added to the paramagnetic effect of the spin, which yields pB tanh a. I t is quite easy to obtain both effects together b.y the use of Dirac's four wave-equations. The four split into two pairs (in the standard form @x goes with ~4, ~ with ~3) and if it is assumed that the temperature is not so enormous that kT (and with it the energy) becomes of the order of mc~, it is possible to eliminate ~1 and @2 respectively, and obtain second order equations in @3 and @4. These only differ from (2"3) in that the energies come out as
1)+ b
+ lzl + 1)3
and ho~ [ ( / + 1) + b (2n + Ill + 1)] respectively. There are now twice as many terms in S as before, and it is easy to see that (4"3) is to be multiplied by a factor r + e~. This gives to ~ a t e r m / ~ tanh a and so the total mean resultant moment of the spinning electron is ~
sinh 2a '
where a = ~ H/kT. At high temperatures a free electron gas has susceptibility 2 p,~ 3 k/' per electron.
176
VOLUME 65, NUMBER 1
PHYSICAL
REVIEW
LETTERS
2 JULY 1990
Q u a n t u m Dots in a M a g n e t i c Field: Role o f E l e c t r o n - E l e c t r o n Interactions P. A. Maksym (a) and Tapash Chakraborty Max-Planck-lnstitut fiir Festki~rperforschung, Heisenbergstrasse 1, D-7000 Stuttgart 80, Federal Republic of Germany (Received 26 JanuAry 1990) The eigenstates of electrons interacting in quantum dots in a magnetic field are studied, The interaction has important effects on the magnetic-field dependence of the energy spectrum. However, when the confinement potential is quadratic, the optical excitation energies of the many-body system are exactly the same as those of a single electron. This makes the interaction effects difficult to observe directly but they could be seen by measuring the thermodynamic properties of the electrons. This is illustrated with the calculations of the electronic heat capacity. PACS numbers: 73.20.Dx,71.45.Gm, 72.!5.Rn Rapid advances in semiconductor technology have led to the fabrication of zero-dimensional structures called quantum dots. Essentially, they are little islands of twodimensional electrons which are laterally confined by an artificial potential. Alternatively, they can be thought of as artificial atoms where the confining potential replaces the potential of the nucleus. Typical dot sizes are about 100 nm and each dot typically contains between 2 and 200 electrons, i-4 While several current experiments 1'3"4 have focused attention on various properties of quantum dots in a magnetic field, their theoretical understanding is still in its infancy. In particular, the effect of the electron-electron inter~:tion has not yet been investigated although Bryant 5 has emphasized its importance in the case of zero field. The purpose of this Letter is to present a detailed study of quantum dots in which both the magnetic field and the electron-electron interaction are fully taken into account. First, numerical calculations of the electron states are used to show that the electron-electron interaction is highly important and leads to unusual magnetic-field dependence of the ground state and its excitations. Next it is shown that, when the confining potential is quadratic, far-infrared (FIR) spectroscopy is insensitive to interaction effects because the center of mass (cm) and relative motions then separate in the same way as for the free electrons. This explains recent experimental results which show that the effect of the interaction is apparently very small. ~ Finally, results for the heat capacity are presented to suggest that the effect of the interaction could be probed by measuring the thermodynamic properties of the electrons. The first step in calculating the electron states is to find the eigenstates of a single electron. It is believed that the dots currently studied experimentally confine the electrons by a potential that is quadratic to a good approximation when the electron number ne is small. ~,6 Therefore the calculations reported here have been done for the case of ideally two-dimensional electrons, in a circular dot, confined by a radial potential of the form ~-m*m~r 2 with a magnetic field B perpendicular to the plane of the dot and m* is the electron effective mass. The corresponding single-electron states were first de108
rived by Fock7 and Darwin s and later studied in detail by Dingle. 9 Apart from a normalization constant they have the form v,(r) - r Ill exp( - ilO)Lltl(r2/2a 2)exp( - r2/4a 2), where L/tl is a Laguerre polynomial, aa--(h/m*)(o~ 2 +4~o~) -t/2, and OJc--eB/m*. The single-electron energies depend on the two quantum numbers n and l,
Ent--(2n+ 1 + Ill)h(~ o,2+o,~) t/2- ~ lhoJc, but in the limit, when oJo--* 0, they reduce to E , t - I n + ~ + (111 -l)/2]h~o~ and depend only on the quantum number N - n + ( I I I - l)/2. Physically, N is the L a n - dau-level index a n d - ' - I is the angular momentum quantum number. Without the confining potential the energies of the positive I states would be independent of I but in its presence they increase with 1. This is the key difference between the behavior of free and confined electrons. It is reponsible for much of the new physics reported here. To calculate the states of the interacting electrons it is supposed that B is strong enough to keep them spin polarized. This assumption allows the interplay between the effects of confinement and interaction to be studied without including extra complications. The role of spin is briefly discussed later in this Letter. In the spinpolarized case the constant Zeeman term can be ignored so the Hamiltonian is 2m* ,-~, ( P i + e A " 2 + 2m*~~ 1
+~
e2 4xeeo
1
~ Ir,-rjl
e (1)
"
where eeo is the dielectric constant. The interaction between electrons in different dots is neglected but this should be a good approximation because the dot spacings are typically larger than dot sizes. The neutralizing positive background present in real systems is also ignored. For an infinite system it cancels the divergence caused by the Coulomb repulsion but for a single dot all the matrix elements of the Coulomb interaction are finite. In the case of a periodic array of dots with a large spacing of the background cancellation merely leads to a constant
O 1990 The American Physical Society
177
65,
VOLUME
NUMBER
PHYSICAL
1
REVIEW
shift of the energy levels for the single dot. The eigenstates of the system are eigenstates of the total angular momentum, which is conserved by the electron-electron interaction. They can be classified by a quantum number J, which is the sum of the single-electron l values. The states are calculated by numerically diagonalizing the Hamiltonian. Landau-level mixing is taken into account but is found not to influence the physics. Its effect is estimated quantitatively from trial computations where an extra Landau level is included. For the parameters considered here the low-lying energy levels converge to 7% (ne "-4) or 2% ( n e - 3 ) when only two Landau levels are taken into account. The convergence of the ground state is better and its optical excitation energies agree to within 10% (ne - 4 ) or 2% ( n , - 3 ) with the expected cm excitations (see below). For higher electron numbers or lower B the effect of Landau-level mixing would be more important. The energy levels of the three and four electron systems are shown in Fig. 1. They have been calculated using parameters appropriate to GaAs and a value of 4 meV for h~oo. The energies are given relative to what would be the lowest Landau level, that is, the constant of h(88 u2 per electron is not included. Each frame of the figure shows total energies plotted against J at magnetic fields representative of low- and high-field behavior. Clearly there are always two sets of broadened levels separated by a gap. In the limit of zero confining potential these would be the lowest two Landau levels. The general trend is that the energies increase with J because the single-electron energies increase with L This is most clearly seen at B - 2 T; at high field the increase is much weaker. Calculations for five and six electrons show the same physics but they are not so well converged. The main difference between high- and low-field behavior is the angular momentum of the ground state. At B - 2 T this occurs at the lowest available J, that is, the
" 70[ne=4
[ ine=3
B=8T
tU 40[
7o[ne=4 >_=6o~ -
'
7
,
B= 2T .
B=IOT_=__
2 4
8 101214 J
-
.....
. . . . . .
_ _,,iii
6
9" \
20
E LU
2
_,
I~e--;
B:2-T
./
e/o,,';~ J
~o/,-~
/"
totot
"~176
7\ 0~
/"
/~176 ~
7"'\
__~_i
40 f
/'"
singte
7.-,. / etectron " " " i " " 7 ........ /./" ~ "
_
=~
-
./-"
interoction
o/
20 0
he=3 B = I O T no L a n d a u Level -mixing
30
_-:--~;_-~_~
3 0 ' ~ ' '----'.-C-'
smallest angular momentum compatible with placing all the electrons in N - - 0 states. For example, for three electrons the ground state is at J - 3 which is formed from the 1 values 0 + 1 +2. If the electrons did not interact, the ground state would have the lowest available J provided B is so high that only N - 0 is relevant. Because of the interaction, however, the ground-state J increases with B. This effect is caused by the interplay of the single-electron energies and the interaction energy. It can be understood in terms of a simple calculation in which only the N - 0 states are taken into account. In this case the single-electron contribution to the energy is simply h[( 88a,c2+ a,~) ! / 2 - 89a,c]J (relative to the lowest Landau level), but the interaction contribution has to be obtained by numerical diagonalization. Figure 2 shows these two contributions together with their sum. The single-electron contribution increases linearly with J because electrons in high angular momentum states see a higher confining potential. In contrast, the interaction term decreases because electrons with higher angular momenta move in orbitals of larger radii, thereby reducing their Coulomb energy. The net result is that the total energy as a function of J has a minimum. At low field this occurs at the lowest available J because the single-electron ene[gy increases steeply with J. At high field the increase is much weaker so the minimum occurs at a higher J value. Since each J value has its own set of energy levels, changing the field has a dramatic effect on , the excitation spectrum as well as the ground state. The ground state does not take all possible values of J 9 because the interaction energy does not change continuously with J. Instead it contains steplike structures which occur at particular J values (see arrows in Fig. 2) and are possibly analogous to the energy cusps in the fractional quantum Hall effect, no Because the total en-
.....
-:;~=iii=iiiiii
Eso
2 JULY 1990
LETTERS
0
0 1 2 3 4 5 6 7 8
1
o
1
lO
I
1
20
J
FIG. I. Energy levels as a function of J for three and four electrons in a GaAs quantum dot. Each frame corresponds to a different magnetic field as indicated.
FIG. 2. Contributions to the total energy as a function of J. The points give energy values and the lines are to guide the eye. Arrows indicate the steps in the interaction energy. 109
178
VOLUME
65,
NUMBER
1
PHYSICAL
REVIEW
ergy is the sum of the single-electron and interaction terms, the steps lead to minima in the total energy as a function of J and the ground state always occurs at one of these. For example, in Fig. 2 the global minimum occurs at J - 6 but there is another minimum close to it at J - 9 . If the field was increased to 10.4 T, this would become the global minimum; further increasing the field would cause the global minimum to occur at J - 1 2 and so on. Using the well-known formula V-he(he-l)/2J (which is obeyed by the Laughlin states to), one can convert the sequence of favored J values to effective filling factors. We find that the three electron states go through the sequence of v values 1, ~, }, ~I . . . . and the four electron states go through 1, ] , {, ~ . . . . . It is remarkable that even denominators occur in addition to the usual odd ones. ll We emphasize that the multiple minima in the ground-state energy only occurs in the presence of confinement: With no confinement the steps in the interaction term persist but the single-electron term vanishes, consequently there are no minima at finite J. The rich structure of the energy spectra shown in Fig. 1 is in complete contrast with current experimental re=ults on FIR absorption. These have just two features whose energies seem to correspond to single-electron excitations. To explain this apparent contradiction it is necessary to consider the perturbation due to the electromagnetic radiation. Typical wavelengths are 50 #m while typical dot sizes are 100 nm so the dipole approximation holds to a high degree of accuracy. In other words, the electrons see a perturbing vector potential which is independent of position within the dot, or equivalently, a position-independent electric field E0 x exp(-ia~t). The perturbing Hamiltonian therefore has the form 7t'-~.,]'-teEo'rjexp(-ia~t). This depends only on the sum of the electron coordinates; hence, it can be expressed in terms of the cm coordinate R-Y~iri/ne and the total charge Q-nee, that is, H'-QEo. Rexp(-ia~t). To understand the effect of this it is convenient to rewrite the Hamiltonian (1) in terms of the cm and 3(he - 1 ) coordinates relative to it. These are most conveniently chosen to be r i ' - r i - R , where i ranges from 1 to ne - 1. The coordinate of the neth electron is then given by r,,-R ~7"..~I r,.' and the Hamiltonian becomes 1
ff --~-~'(P+QA)2+
~ M a ~ R :'+ f i r e , ,
where P-~7"--i Pi, A is the vector potential of the cm, and M--nero*. The last term ff~m is a function of only the relative coordinates and contains all the effects of the interaction. The Hamiltonian clearly separates in the same way as established long ago t2 for the case of no confining potential. This is a very special property of quadratic confinement--if the potential was different, the cm motion would couple to the relative motion. It 110
2 JULY 1990
LETTERS
follows that FIR radiation excites the cm but does not affect the relative motion. Further, the cm Hamiltonian has the same form as the Hamiltonian of a single confined electron. In addition, it has exactly the same energy eigenvalues because a~c in the expression for E,t depends only on the charge-to-mass ratio Q/M-e/m *. Consequently, FIR absorption experiments see only features at the single-electron energies. 13 How then is it possible to observe the effects of the electron-electron interaction? There seem to be two possibilities. One is to change the shape of the dot to force coupling of the cm and relative motions. It is possible that this is responsible for some level-crossing effects observed in recent experiments. 4 The second possibility is that the interaction should affect thermodynamic properties, for example, the electronic heat capacity (7,, (which is observable t4). To test this idea the magnetic-field dependence of C,, has been calculated. As usual, this is found from the temperature derivative of the mean energy. For simplicity Landau-level mixing is neglected, but this does not leave out any essential physics. Figure 3 gives C~, excluding the Zeeman contribution (which is a small, slowly varying background). For interacting electrons s (solid lines) is clearly very different from that of noninteracting electrons (dotted lines). In particular, when the electrons interact C,, oscillates as a function of B and has minima that are associated with crossovers from one ground-state J value to another. (The dashed lines in Fig. 3 indicate the ground-state J.) The oscillations in C,. are a many-body effect, unlike the low-field oscillations in (7,. for a two-dimensional electron gas. t5 Their origin is best understood by considering the curves for T - i K. At this very low temperature the dominant contribution to C,. comes from two competing ground states. This causes the doublet structure around the crossovers and can be understood in terms of the B dependence of the gap between the corresponding ground
0"3[ne=4
0.2
T=3K
~]
~ 0.2
/ne=3
T=3K
," . . . .
O'2[ne=4,
......"0.2
T=IK
"~ ~0.I
~:,_~
,]
J=,8 ~-.....
J-lO~
J =t2
lne=3
_
6
J=,2
iu. r I
~
J =3 8 i0 12 14 16 18 20 0 2
B(T)
J=9
J=6 ' - . . . . . .
9
:6, 0 2 4'
T=IK
I"^ /
4
r .....
"
r 6
8 10 12 14 16 18 20
B(T)
FIG. 3. Heat capacity (7,. as a function of magnetic field for three and four electrons in a GaAs quantum dot. Each frame corresponds to a different temperature as indicated.
179
VOLUME65, NUMBER 1
PHYSICAL
REVIEW
stateS. Far away from a crossover the gap is large so C,, is small. Similarly, it is small exactly at a crossover because the gap i~ then zero. However, on either side of a crossover the gap is nonzero but not too large. Consequently, C,. is nonzero because neither the probability of a thermal excitation nor the heat absorbed in one are vanishingly small. This picture of an oscillatory heat capacity holds when other spin polarizations are taken into account. Indeed, for B > I0 T the ground states are expected to be fully polarized but at lower fields there is more structure in C,, when the ground-state spin depends on B. In conclusion, the interaction of electrons in quantum dots leads to rich structure in their energy spectrum. However, FIR spectroscopy cannot probe it when the confining potential is quadratic because the optical excitations are then excitations of the cm and have exactly the same energies as single-electron excitations. The structure could be probed by deliberately engineering the dots so that the cm and relative motions are coupled or by measuring the thermodynamic properties of the electrons. We would like to thank Rolf Gerhardts, Detlef HeRmann, and Klaus yon Klitzing for helpful discussions. One of us (P.A.M.) thanks Peter Fulde for his kind hospitality durin~ a visit to the Max-Planck-lnstitute, Stuttgart, and is grateful for the support of the Nufiield Foundation. The other (T.C.) thanks D. E. Khmel'nitzkii for pointing out Ref. 7. (a)Permanent address: Department of Physics, University of
LETTERS
2 JULY 1990
Leicester, Leicester LEI 7RH, United Kingdom. ICh. Sikorski and U. Merkt, Phys. Rev. Lett. 62, 2164 (1989); U. Merkt, Ch. Sikorski, and J. Alsmeier, in Spectroscopy of Semiconductor Microstructures, edited by G. Fasol, A. Fasolino, and P. Lugli, NATO Advanced Study Institutes, Ser. B, Voi. 206 (Plenum, New York, 1989); Ch. Sikorski and U. Merkt, Surf. Sci. (to be published); A. V. Chaplik, Pis'ma Zh. Eksp. Teor. Fiz. 50, 38 (1989) [JETP Lett. 50, 44 (1989)]. 2M. A. Reed, J. N. Randall, R. J. Aggarwal, R. J. Matyi, T. M. Moore, and A. E. Wetsel, Phys. Rev. Lett. 60, 535 (1988). 3W. Hansen, T. P. Smith, 111, K. Y. Lee, J. A. Brum, C. M. Knoedler, J. M. Hong, and D. P. Kern, Phys. Rev. Lett. 62, 2168 (1989); T. P. Smith, III, K. Y. Lee, C. M. Knoedler, J. M. Hong, and D. P. Kern, Phys. Rev. B 38, 2172 (1988). 4"1".Demel, D. Heitmann, P. Grambow, and K. Pioog, Phys. Rev. Lett. 64, 788 (1990). 5G. W. Bryant, Phys. Rev. Lett. 59, 1140 (1987). 6S. E. Laux, D. J. Frank, and F. Stern, Surf. Sci. 196, 101 (1988). 7V. Fock, Z. Phys. 47, 446 (1928). sC. G. Darwin, Proc. Cambridge Philos. Soc. 27, 86 (1930). 9R. B. Dingle, Proc. Roy. Soc. London A 211, 500 (1952). I~ Chakraborty and P. Pietil~inen, The Fractional Quantum Hall Effect (Springer-Verlag, Heidelberg, 1988). ! liThe fractional states at filling factors T and ~ in quantum dots are discussed in T. P. Smith, III, Surf. Sci. (to be published). 12Yu. A. Firsov and V. L. Gurevitch, Zh. Eksp. Teor. Fiz. 41, 512 (1961) [Soy. Phys. JETP 14, 367 (1962)l. i3L. Brey, N. F. Johnson, and B. I. Halperin, Phys. Rev. B 40, 10647 (1989), obtained the same result in a different way. 14E. Gornik, R. Lassnig, G. Strasscr, H. L. St6rmer, A. C. Gossard, and W. Wiegma~n, Phys. Rev. Lett. 54, 1820 (1985). 15W. Zawadazki and R. Lassnig, Surf. Sci. 142, 225 (1984).
111
180
PHYSICAL REVIEW B
15 SEPTEMBER 1990-I
VOLUME 42, NUMBER 8 E l e c t r o n s t a t e s in a G a A s q u a n t u m d o t in a m a g n e t i c field
Arvind K u m a r I B M Research Division, Thomas J. Watson Research Center, P.O. Box 218, Yorktown Heights, New York 10598 and Department o f Electrical Engineering and Computer Science, Massachusetts Institute o f Technology, Cambridge, Massachusetts 02139* Steven E. Laux and Frank Stern I B M Research Division, Thomas J. Watson Research Center, P.O. Box 218, Yorktown Heights, New York 10598 (Received 12 March 1990; revised manuscript received 6 June 1990) Self-consistent numerical solutions of the Poisson and Sehr&linger equations have been obtained for electron states in a GaAs/AlxGam_xAs heterostructure with confinement in all three spatial dimensions. The equations are solved in the Hartree approximation, omitting exchange and correlation effects. Potential profiles, energy levels, and the charge in the quantum dot are obtained as functions of the applied gate voltage and magnetic field. First, the zero-magnetic-field ease is considered, and the quantum-dot charge is allowed to vary continuously as the gate voltage is swept. Then, in connection whh the phenomenon of Coulomb blockade, the number of electrons in the quantum dot is constrained to integer values. Finally, the calculation is extended to examine the evolution of levels in a magnetic field applied perpendicular to the heterojunction. Our results indicate that the confining potential has nearly circular symmetry despite the square geometry of the gate, that the energy levels are quite insensitive to the charge in the quantum dot at a fixed gate voltage, and that the evolution of levels with increasing magnetic field is similar to that found for a parabolic potential.
I. INTRODUCTION Modern fabrication techniques have made possible confinement of a two-dimensional layer of electrons into wires, grids, or dots where quantum-mechanical effects are strongly manifested. Of particular current interest are quasi-zero-dimensional systems, which have been made by selective etching of a GaAs cap on a GaAsA l x G a l _ x A S heterostructure, 1 by depositing a cross-grid gate structure on a GaAs heterostructure 2 or on Si, a by using crossed holographically defined gratings, 4 and by using an array of small Latex particles as an etch mask, s to cite some recent examples. Such quantum-dot structures offer a dispersionless system with an electronenergy spectrum that can be modulated either by varying gate bias voltage or by applying an external magnetic field. Smith et al. l have reported oscillatory structure in capacitance versus gate voltage in zero magnetic field and have attributed it to the discrete energy states of a quantum dot. Recently, Hansen et al. ~ have reported observing Zeeman splitting of quantum-dot capacitance features, as expected when a magnetic field is applied perpendicular to the heterojunction. There axe many papers that treat the energy-level structure of related systems, including the paper by Darwin that treats a twodimensional harmonic-oscillator potential in the presence of a normal magnetic field, 7 Robnik's paper on a disk in a magnetic field, s and the recent calculations by Bruin and co-authors on a model quantum dot. 6'9 Sivan and Imry !~ have described the evolution of states in a quantum dot versus magnetic field in relation to magnetization and 42
persistent currents, which are not considered here. In this paper we present numerical self-consistent results in the Hartree approximation for potential profiles, energy levels, envelope wave functions, and charge distributions for quantum dots like those studied by Hansen et aL 6 Self-consistent numerical treatments of electron states in quasi-one-dimensional systems in the absence of a magnetic field have been carried out for a narrow channel in silicon by Laux and Stern II and for a split-gate G a A s / A l x G a l _ ~ A s heterostructure by Laux et al. L~ Numerical methods for such systems, which have quantities that vary in two spatial dimensions, have also been used by Kojima et al. 13 and by Kerkhoven et al. 14 However, the analogous calculation for a totally confined system requires a coupled solution of Poisson's equation and SchrSdinger's equation in three spatial dimensions, increasing the computation requirements significantly. Application of a magnetic field, which leads to complex wave functions and a Hermitian rather than a real symmetric eigenvalue problem, also adds to the computational burden. In Sec. II we discuss our formulation of this problem. In See. III we discuss results of such a calculation on a G a A s / A l x O a 1 _ x A s quantum-dot structure in zero applied magnetic field. In particular, we discuss the effect of varying the charge in the quantum dot on the energy-level structure and the quasi-Fermi-level, and its relation to the Coulomb blockade. 15 In See. IV we extend our self-consistent calculation to include the effect of an applied magnetic field perpendiculax to the heterojunetion on the potential, charge density, and electron states of the quantum dot. We find good 5166
9
The American Physical Society
181
42
5167
ELECTRON STATES IN A GaAs QUANTUM DOT IN A . . .
qualitative agreement with the results of earlier calculations for model potentials. II. FORMULATION OF THE PROBLEM The structure we consider is a model of a single quantum dot from the array of dots used in the experiments of Hansen et al.,6 described above. It is based on a heretostructure with an n-type G a A s substrate layer with a net ionized donor concentration of 10 ts cm -3, an 80-rim layer of undoped G a A s (a background acceptor concentration of 1014cm -3 is assumed throughout a n d diffusion of donors from the substrate is ignored), a 20-nm layer of undoped Alo.4Gao.6As, a 20-nm layer of the same m a t e d al with a donor concentration of 1.5 • I0 cm -3, and a 30-nm G a A s cap layer. The cap is etched away, except in the central 300 n m X 3 0 0 n m portion of a 500 n m • n m area, and the structure repeats on a square lattice. Finally, a metal gate is deposited over the entire top surface. A negative voltage on the gate depletes the charges in the G a A s channel, except u n d e r the remaining G a g s cap, and this three-dimensionally confined " p u d d l e " of electrons is the q u a n t u m dot being studied. The n-type G a A s substrate allows a low-impedance capacitative contact to the dot.." Figure I shows the conduction-band edge in the structure versus vertical distance along a line through the center of the dot, and in the inset is sketched
TM
the semiconductor region included in the m o d e l For the GaAs, we use an electron effective mass of 0.07m 0 and a dielectric constant of 13; for the AlxGat_xAS, we use 0.1 l m 0 and 11.8, respectively, corresponding to a n AlAs mole fraction x - - 0 . 4 . The conduction-band offset is taken to be 0.3 eV. The binding energy of the deep donor in the Alo.4Gao.6As is taken to be 0.15 eV, and the effective Schottky-barrier heights of the gate electrode to the G a A s and the Al0.4Ga0.6As are taken to be 0.7 and 0.95 eV, respectively. Although the present calculations d e a l with the structure of Hansen et al.,6 the methods t o b e described in this paper can be used for a wide class of structures in which three-dimensional confinement of electrons is achieved by a combination of band offsets and electrostatic means. We solve the SchrSdinger and P0isson equations selfconsistently. Image effects 16 in the Schr'6dinger equation are ignored and we use the Hartree approximation, ignoring exchange and correlation effects. Bryant t~ showed that many-electron interactions can have significant quantitative and qualitative influence on the energy spect r u m of a q u a n t u m dot with a small number of electrons. Similar effects are expected for the structures studied here, but have not been included in our e.alculation. T h e electrostatic potential 4, is governed by the Poisson equation
(1)
V . [ e(x,y,z)V4( x,y,z) ] = - p ( x , y , z ) , 1.8
L4 i
.-
1.2
>e
1.0
,-
o.a:
.
x
i
v
o.e-
T
i i
!
<
0.4 0.2
where e is the permittivity (in the present case it depends only on the z coordinate), with boundary conditions determined by voltages applied at the contacts. A t boundaries where there are no contacts, the normal derivative of the potential is taken to be zero. The total charge density p in Eq. (1) includes the charge in q u a n t u m states, calculated as described below, as well as the contribution from ionized impurities in the A l x G a l _ x A s , and of any electrons outside the Schr6dinger domain. In particular, any electrons in the cap layer are treated classically. In a magnetic field (Bx,By,B z) the three-dimensional Schr&iinger equation for the electron envelope function (in the effective-mass approximation) becomes
o.o -0.2
0.0
0.05
0.10 z
0.15
0.20
+[U(x,y,z)-E.]~.=o
,
(2)
(~)
FIG. I. Conduction-band edge along a vertical line through the center of the OaAs-AJxGat-xAs structure considered here, for a gate voltage of -- 1.03 V. The layers of the structure, from left to right, are 30 nm of n +-GaAs, 80 nm of undoped GaAs, 20 nm of undoped Alo.4Ga~e,~, 20 nm of Alo.4Gao.6As with No = I.SX 10ts cm -3, and a 30 nm GaAs cap. The repeating unit is 500 nm square, and the GaAs cap layer is etched away, except under a central 300-nm square mesa. A metallic gate is then deposited over the structure. The Schottky barrier assocb ated with the gate suppresses induced charge in the GaAs, except under the central portion of the mesa. All calculated results are for 4.2 K and the zero of energy is taken at the Fermi level in the substrate.
where mj is the electron effective mass in the j t h direc= tion and the electron charge is --e. We choose the symm e t r i c gauge
A~ =(a,z -B.y)/2, and
cyclic
permutations.
(3) In
the
present
case
mx--my----rn z and Bx--By --0. The electron charge density in the q u a n t u m dot is
Pinv(X,Y,z)=--2e ~_. ~*a(x,Y,z)~a(x,Y,z) n
x f CCE~--B~ ~/kB r~ ~ ,
(4)
182
5168
A.RVIND KUMAR, STEVEN E. LAUX, AND FRANK STERN
where the sum is over all states n, the factor 2 is for spin degeneracy (spin splitting is ignored in this calculation), EqF is the quasi-Fermi-energy, and f is the Fermi-Dirac occupation function at temperature T. If the Fermi energy in the quantum dot is equal to the Fermi energy in the n-type substrate, then the calculated charge in the dot will be a continuous function of the gate voltage. The charge per quantum dot will, in general, be a nonintegral multiple of the electron charge, and will represent the average for a large ensemble of dots. Physically, however, the charge in an isolated dot should be an integral multiple of the electron charge. If we constrain the charge in the dot to be art integral multiple of the electron charge, then we apply Fermi-Dirac statistics to determine the quasi-Fermi-level that gives the prescribed charge from the calculated energy levels. Among many simplifying assumptions in our calculation is the neglect of the interface image potential and of many-electron contributions to the potential. Then the potential energy is U------'e~-I-AEc, where the second term is the position-dependent conduction-band offset relative to the bottom of the conduction band in the GaAs. Level broadening has not been included expficitly, but some broadening, small compared to typical level spacings, is~simulated because we carry out the calculations at T = 4 . 2 K. Both the Poisson and Schr~dinger equations are cast into discrete form on a nonuniformly graded, tensorproduct (finite-difference) mesh, with no interior meshline terminations, IS and the resultant matrix equations are solved numerically. The Schr~linger mesh includes only the region of significant dot charge; elsewhere electrons are treated semiclassically. Electrostatic potential, envelope functions, and charge-density values are defined at "mesh nodes, whereas material properties such as dielectric constant, effective mass, and effective bandedge shift A E c are piecewise constant in the individual rectangular parallelepiped elements defined by the mesh. Equations for the potential and envelope function at each node are obtained by integrating Eqs. (I) and (2) over the box defined by the six planes bisecting the lines connecting the node to its nearest neighbors (for nodes on the boundary, only the volume inside the boundary is included). For the Poisson equation, this results in a real symmetric matrix problem L 0 - - --flp, where F, is the operator V-eV integrated over the boxes, 45 and P are vectors of the nodal potentials and charge densities, and I~ is a diagonal matrix of the nodal box volumes. For the Schr6dinger equation, one similarly obtains ~J[~ =E~f1~n, where H is l~l:,e Hamiltonian integrated over the boxes and ~'n is the complex vector of the envelope function for state n at each node. This equation is readily transformed into a standard matrix eigenvalue problem by premultiplying both sides by ~ - I / ~ and substituting I=[1-1/2111/2 to give (l'~-ll2H[]~-ll2)(l'~ll2~n)~-~En(l'lll2~n), or simply H y n -- E, Yn, where H ~- I1- t / 2 ~ - i/2 is still an Hermitian matrix and Yn----~1/2~n. In zero magnetic field, the Hermitian matrix reverts to a real symmetric matrix. The Poisson equation in discrete form is nonlinear, since the charge density depends on the potential. The
42
solution to this nonlinear problem constitutes the search for self-consistency between the charge and the potential. The solution to the SchrSdinger equation enters as part of the evaluatlon of the total charge density in the device, for a given potential. We linearize the Poisson equation via Newton's method. The vector ~ whieh is the zero of the function F ( ~ ) ' - L ~ + lip(#5) is sought by iterating
F'( ~t)~ t'-" --F( c~t) ,
(5a)
r + l=c~t+ t ~ l ,
(Sb)
until convergence is obtained. Here, I is the iteration index and the scalar damping factor t is selected according to a modified Bank-Rose damping scheme as discussed in Ref. 19. The evaluation of the Jacobian matrix F ' is a possible stumbling block because the dependence on #$ of the charge density in the channel given in Eq. (4) is nonlocal, which would destroy the seven-diagonal structure of the Jacobian, rendering the matrix solution significantly more diflieult. Instead, as discussed in Ref. 19, a rather crude approximation to the depen.dence of the channel charge on local potential is made for purposes of calculating IF' only, in order to circumvent this difficulty. While this precludes a second-order convergence rate of the Newton iteration, converged solutions can still be ol;tained in an acceptable number of iterations. The linear matrix equations in (5a) above are solved via a conjugate-gradient metho& Such methods require a preconditioner to accelerate convergence; we have selected a polynomial preconditioner, 2~ as it has proven robust and highly vectorizable. The discrete Schr6dinger equation is solved by one of two methods. Far away from self-consistency, between charge and potential, a Lanczos method is employed. 21 This method forms an approximate tridiagonalization T of the matrix H. No reorthogonalization is used in this process. Then, the eigenvalues of T are found in a specified energy interval (from the minimum of the quartturn dot potential to 5 - 1 0 meV above the Fermi energy) by a bisection search together with Sturm sequencing. 21 Care must be taken in discarding potentially "spurious'" eigenvalues of T, that is, eigenvalues of T which are not good approximations I~o true eigenvalues of H. 21 Finally, inverse iteration is used to find the associated eigenvectors. Gaussian elimination is used to solve the tridiagonal matrix equations involved in inverse iteration. Neac self-consistency between the charge and the p o tential, a simple Rayleigh quotient-iteration algorithm ~ is used to solve the eigensystem. This algorithm requires an initial guess for the eigenfunctions, and can be summarized as follows. Let cr(y)--(yHHy)/(yHy) be the usual Rayleigh quotient (superscript H denotes Hermitian conjugate) and let y 0 be an initial guess for the nth eigenfunction. Then solve [H--~r( yt)I ]x/+ l = y t ,
(6a)
y l + t = x l + ! / Iix 1+I] I ,
(6b)
. .
n--I
y:+l._~+a_
~
(yHyt+l)y t (this step only if n >0) ;
i-0
(6c)
183
42 if
5169
ELECTRON STATES IN A GaAs QUANTUM DOT IN A . . .
IlHyl+'-cr(y'+')y~+~ll
<e,
then done .
III. ZERO MAGNETIC FIELD
(6d)
The solution becomes y , = y/+ 1 and E. = cr(yt + t). Note that the step (6c) above is not a part of the standard Rayleigh quotient iteration. This step serves to remove components related to previously determined eigenfunctions Yi, i = 0 , 1 , 2 , . . . , n -- 1, from the vector yt + t which is evolving into the eigenfunetion Y,. In practice, this orthogonalization step has an important benefit: In solving the equations for a series of gate voltages or m a g netic fields, the time-consuming Lanezos method can be omitted from the iteration for self-consistency between charge and potential, provided the new solution is not too distant from the previous solution. The orthogonalization ensures that energy levels which are "close" at some initial solution do not converge to the same level at a later step. This procedure gives correct eigenstates using significantly less computation time, but may eventually miss some intermediate eigenvalues if extended over too wide a range of gate voRage or magnetic field without an intervening Lanczos solution. The Hermitian matrix system in (6a) above is solved with the polynomial-preconditioned conjugate-gradient method. 2~ Although the matrix H - - ~ r I in (6a) is not positive definite, this method of solution has always been robust for the class of problems we have encountered. The boundary condition used in our numerical metl~od is that the normal derivative of ~'n vanish on the SehrSdinger mesh boundary. In the lateral (x and y) directions this condition occurs sufficiently far from the region of induced charge that it has no appreciable effect on the results. In the direction normal to the inversion layer, we truncate the SchrSdinger mesh 36 nm below the GaAs/AlxGal_xAs interface to avoid the quasicontinuum of eigenstates arising from the heavily n-type doped substrate. This may lead to significant errors in the values of energy levels and thresholds. In particular, some of the qualitative results for the present structure may not apply to a dot with stronger vertical confinement, as could be obtained if a p-type substrate were used. The convergence criterion for seE-consistency is that the nodal potential energies of successive iterations differ by no more than 0.01 meV anywhere on the Poisson mesh. The necessarily limited mesh size (51 X51 X35 for the Poisson mesh and 43 X43 • 18 for the SchrSdinger mesh, in the x, y, and z directions, respectively) and other approximations made in the calculation will lead to errors that are larger than this convergence criterion. A n IBM 3090 computer with vector processor was used for these calculations. A single Newton's loop, in which the Poisson and SchrSdinger equations are each solved once, required approximately 15 min of computation time for B = 0 and 45 min for B=/=0 if the Lanczos recursion was used. If the Rayleigh quotient algorithm was used in place of the Lanczos method, the solution of the SchrSdinger equation (nearly all the computation time) ranged from 5 to 50 times faster, depending on the quality of the initial guess. A typical bias point required 4 - 2 0 Newton's loops to converge.
Potential contours in a plane 8 nm below the GaAsA l x G a l _ x A S interface, near the maximum of the electron charge distribution, are shown in Fig. 2. Note that the potential contours are nearly circular, especially at the lower energies, although the defining gate geometry is a square. T h a t follows from the attenuation of higher Fourier components of the potential in regions some distanee from the gate, as found previously for fluctuations in the width of a gate opening. ~ Also, the effective size of the q u a n t u m dot, given by the contour at the Fermi level, is considerably smaller than the size of the defining structure in the gate. In Fig. 2 and throughout this paper we make cuts in representative planes or along representative fines to display functions Of three spatial coordinates. T h e figures are intended t o indicate the main features of the calculated results, but should not be considereal to be complete. T h e raggedness of some of the later curves is a consequence of the necessarily coarse mesh used in the discretization. Figure 3 shows the number o f electrons in the q u a n t u m dot, the lowest-energy levels, and the quasi-FermMevel as functions of the voltage on the gate at 4.2 K for zero magnetic field., The notion of "'quasi-Fermi-level" does not arise in calculations for quasi-one-dimensional wires, for which the charge can be considered to vary continuously, provided a suitable means of equilibrating with an adjacent gate or contact exists. F o r the very small structures considered here, where a dot may contain only a
0.35
500nm D;T
'
I
'
EF 0.30
~0.25
0.15 0.15
I 0.20
0.25 x (/./.m)
0.30
035
FIG. 2. Lateral potential contours in the plane 8 nm below the GaAs/AI~Gaj_xAs interface, near the peak of the vertical charge density, for a gate voltage of --1.03 V. The innermost contour is 15 meV below the Fermi level, which is indicated by the heavy line, and the remaining contours are at 10-meV intervals from -- 10 to + 50 meV. Note the nearly circular symmetry despite the square geometry of the cap. The effective quantum dot size, with a diameter of about 100 nm, is considerably smaller than the 300-nm square mesa in the OaAs cap layer.
184
5170
ARVIND KUMAR, STEVEN Eo LAUX, AND F R A N K STERN 40
9
't
F~'4"44~,
:
. .
-~ .'
.i .
. j;
20
:~
....
.
10
--.....
~1 ......
J
"
J. . . . . . .
~""~.~QUASI-FERMi
E ~-
3o "-o
i
20
,
~: ~ ~9 '9 : : i : 9 : :
/
1 9
: i
/
4.2K
~
~
!
" i .-'~ : : : ! i ! : ~ ~ : : i ! i i !
!
~
~ . : ~
-1.10
e
L . .I
~
-1.05
GATE VOLTAGE
f
~:
!~ 4
- ~ is
1
10 ~
! 9- 3 0
0
ENERGY
! ~BOTTOM
OF WELL
0
-1.00
(V )
FIG. 3. Energy levels (solid lines) and Fermi level (dotteddashed line) relative to the bottom of the potential well, and number of electrons in the quantum dot (dashed line), vs gate voltage. The second level is doubly degenerate, and the next three levels lie very close to each other. In addition, each level has a twofold spin degeneracy. These energies reflect the combined effect ~f vertical and lateral confinement. The energy of the lowest state with a node in the z direction is indicated by the plus signs. Only integer electron occupations, indicated by t h e vertical dotted lines, correspond to physically realizable states of an isolated quantum dot.
few electrons, discontinuities can arise because transfer of just one electron can have a significant effect on the energies in the problem. This gives rise to the Coulomb blockade, as found in m a n y experiments. 1~ Because o f this effect, the curves in Fig. 3 have no physical signficance for an isolated quantum dot at points where the number o f electrons in the dot is different from an integer. Some o f the energy levels we calculate are degenerate (apart from the spin degeneracy, which applies to all levels in our calculation) and others are nearly so. F o r example, the second and third levels are exactly degenerate at zero magnetic field because of the square symmetry of t h e structure we consider. The fourth and fifth levels would be degenerate at B "-0 if our system had circular symmetry. T h e small splitting results from the weak remnant of the square symmetry of the cap. Finally, the sixth level, w h i c h is close to the fourth and fifth, would be exactly degenerate with them if the system had circular symmetry a n d had the perfectly parabolic potential treated by D a r w i n . 7 Similar considerations apply for higher-lying levels, except that they are increasingly influenced by the deviations from a circularly symmetric potential. Figure 4 shows the quasi-Fermi-energy and the bottom of the potential relative to the Fermi energy in the G a A s substrate when the n u m b e r of electrons in the q u a n t u m dot is six, seven, o r eight. A t a given gate voltage, several different charge states o f the dot are possible, although t h e state with the quasi-Fermi-level closest to the F e r m i
--4O
--5O --1.05
--1.04
--1.03
GATE VOLTAGE
-1.02
-1.01
(V }
FIG. 4. Quasi-Fermi-level and energy of the bottom of the well, vs gate voltage for six, seven, and eight electrons in the quantum dot. The energy difference between the quasi-Fermilevel and the Fermi level gives a driving force for electrons to move between the dot and the substrate. The circles correspond to gate voltages for which the dot is in equilibrium with the substrate for an integer electron occupation.
level in the substrate contact is the one most likely to be observed. The buildup o f potential difference before a charge transfer occurs is a signal of the Coulomb blockade. 15 One measure of capacitance of our structure is obtained by using the lower curve in Fig. 3 to calculate a gate-to-dot capacitance C g = d Q Q D / d V g . That capacitance varies from about 1 X 10 -17 F at small values of dot charge to about 3 X 10 -17 F when there are about 12 electrons per dot. M o r e directly relevant to the experiment of Hansen et al. 6 is an effective substrate-to-dot capacitance C s, which we obtain by dividing the electron charge by the vertical separation between successive quasiFermi-level curves in Fig. 4, to obtain a value of about 6 X 10-t7 F for a dot occupation between seven and eight electrons. Both of these effective capacitances will increase with increasing dot charge. The dynamical behavior of this system depends on charge-transfer rates between the q u a n t u m dot and adjacent electrodes, a problem which is outside the scope of the present static calculation, i5 N o t e that the barrier between the dot and the substrate is very small, as indicated in Fig. 1 for a line through the center of the dot. This barrier would have been larger h a d we used a larger value than I0 t4 cm -3 for the net acceptor doping in the nominally undoped GaAs. Figure 5 shows a few of the lowest-energy levels versus gate voltage when the number of electrons in the quan-
185
4.2
ELECTRON STATES IN A GaAs QUANTUM DOT IN A . . . ->-
4O -
3 0 0 nm DOT
~
~
"
v
- -
36 0
34
E
32
(.9 ee"
.f...--
~4~ ~?.J~ '"
.t"
Er F (7)
,...,--~
E~=
............
ELECTRONS/
7
---8 ~
30
!
--1.06
-1.08
-
-"
..~"
,..~.t"
L_
m
t/"
~ . / -
9
_
_ EqF (8)
~
I
PeR D O T I -
-1.04
~
J
/
--1.02
GATE VOLTAGE
--1.00
(V )
FIG. 5. The six lowest-energy levels (note that the second and third levels are degenerate), and the quasi-Fermi-energies, for six, seven, and eight electrons per quantum dot, vs gate voltage. Each level has a twofold spin degeneracy. For the upper three levels ~nly the values for seven electrons per dot are shown; the results for six and eight electrons per dot almost coincide. These results are for 4.2 K and B =0.
--
1.6
,'"
,
,
,
5171
tum dot is fixed at six, seven, or eight with zero magnetic field. The energy levels depend remarkably little on the charge state, but are quite sensitive to gate voltage. Some details of potential and charge density are given in Figs. 6 - 8 , both as functions of charge in the quantum dot at B = 0 and as functions of magnetic field (as discussed in the next section) for fixed charge in the dot. Figure 6 shows the charge density along a vertical line through the center of the dot. The charge density peaks about 8 nm below the GaAs/AlxGat_xAS interface, but is t r u n c a t e d - - a s described above---before the rise of charge density in the substrate begins. A lateral cut through the charge density near the peak in Fig. 6 is shown in Fig. 7. Finally, Fig. 8 shows the variation of the conduction-band edge in the X-direction, in the same plane as in Fig. 7. The effective size of the dot is about 100 rim, considerably smaller t h a n the 300-nm square mesa in the G a A s cap layer. T h e potential somewhat resembles the truncated parabola found previously for ni-p-i dopin~ superlattiees 24 and for wires in Si (Ref. 11) and G a A s , 2 but with more structure, which can be attributed to the small n u m b e r of discrete states that contribute to the charge in the cases shown. IV. NONZERO NIAGNETIC IrIELDS W h e n a magnetic field is applied normal to the surface, the SchrSdinger equation, Eq. (2), becomes complex, and
-
?
o:
3~gOT
,.2
.-.
A8
1.6 8
-o
1.2
0.8
~ 0.4 u uLi
0.4
0.0 .--
1.6
~'~
'
,
' ,~
,
..
-~...........
0.0
7 ELECTRONS/DOT ) A
1.6
1.2 0.8
~
2
(hi
(b)
0.8
0.4 0.4
'~
0.0
I
0.07
0.08
0.09
O. 10
-
O. 11
-, O. 12
z (~m)
'~
0.0 0.20
0.25
x
FIG. 6. Total charge density in the vertical direction along a line through the quantum dot center for (a) six, seven, and eight electrons per dot, with B ---0, and (b) B =0, 1, 2, 3, 4, and 5 T, with seven electrons per dot. The z coordinate and the gate ~olta~e a~e the same as in Fig. I.
0.30
(wn)
FIG. 7. Lateral cut of total charge density in a plane 8 nm below the GaAs/AI~GaI_.As interface. The cut is taken through the center of the quantum dot. Other quantities as in Fig. 6.
186
5172
A R V I N D K U M A R , STEVEN E. LAUX, A N D F R A N K S T E R N
42
TABLE I. Calculated quantities for states of the quantum dot at 4.2 K for B = 5 T and a gate voltage of --1.03 V, with seven electrons in the dot. The energy is relative to the bottom of the potential well in the dot, (l=) is the expectation value of the z component of the "'canonical" angular momentum r X p, ( R ) is the expectation value of the two-dimensional radial distance from a vertical axis through the center of the dot, fiR is its standard deviation, and ( L : ) is the expectation value of the z component of the total angula r momentum r X m_v. The 0' state is the lowest state with a node ,in.khe z direction.
10
State
Energy (meV)
(1=)/ti
(R) _ .(rim)
0 1 2
35.4 35.5 35.7
0.07 -- 1.05 -- 1.98
14.3 21.2 25.9
3 4 5 6 7 8 9 O"
36.2 37.0 38.1 39.5 4 I. I 42.9 43.6 43.7
-- 2.91 --3.83 -- 4.75 -- 5.62 -- 6.46 --7.32 0.98
--
- - - ~
>-'
-o
0 " ~
300 n r n ~ T ' ~ - ~ r ' ~ ' ~
-,J 0
--
4.2
-
'
u'
....f
1
.................
(a)
~R (rim)
29.4 32.3 34.8 37. I 39.1 41.0 21.0 ~ . : 15.9__. . . . . .
I
(L,)/f/
7.5 7.5 7.3
1.06 0.87 0.76
7.1 6.9 6.8 6.6 6.6 6.5 7.4 : , 9.4
0.57 0.32 0.03 --0.24 --0.49 --0.78 2.87 1.30
.....
its discretized f o r m leads to a n H e r m i t i a n m a t r i x . In o u r case, this m a t r i x h a s a b o u t 30 000 r o w s a n d c o l u m n s , a n d a c o r r e s p o n d i n g n u m b e r o f eigenstates, but we t y p i c a l l y look for o n l y t h e ---20 eigenstates w i t h t h e lowest e n e r g y . N e v e r t h e l e s s the c a l c u l a t i o n , as d e s c r i b e d above, is v e r y t i m e c o n s u m i n g . W e s h o w in Fig. 9 t h e e n e r g y levels f o r t h e case o f seven e l e c t r o n s p e r q u a n t u m d o t , w i t h a g a t e
~I0
-15
42
-20 -25
6 I
'
.
.
.
.
.
.
'
'
I
40
I
10 ,.-..
>
5
{
(1)
E
o
>s n--10
36
LU
--15 --20 O. 15
38
-...,.
34 0.20
0.25 x
0.30
0.35
(tan)
32 0
FIG. 8. Potentials along the same line as in Fig. 7. The Fermi energy is at zero. The quasi-Fermi-energies for six and eight electrons per quantum dot are indicated in (a). The quasiFermi-energy is within I meV of the Fermi energy for seven electrons per dot for the range of magnetic fields shown, and has been omitted. Also omitted in (b) are the curves for B = 1 and 2 T, which lie very close to the curve for B =0.
1
2
3
4
5
B (T} FIG. 9. Energy levels vs magnetic field for a quantum dot with seven electrons and a gate voltage of -- 1.03 V. The labels give approximate values of the z component of the canonical angular momentum r X p in units of/L
187
42
ELECTRON STATES IN A GaAs QUANTUM DOT IN A . . .
voltage of --1.03 V. As shown in Fig. 2, the potential has nearly circular symmetry, and therefore angular m o m e n t u m is approximately a good q u a n t u m number. The curves are labeled with an integer to represent the approximate z component of angular m o m e n t u m (in units of ~i), but the calculated expectation values for the points shown differ from an integer by up to 10%, and by less than 0.1 for the zero-angular-momentum states. T h e s e labels should therefore be considered to have only qualitative significance. A t B - - 0 , where the envelope eigenfunctions are real, the angular m o m e n t u m is zero for all
....,.
?
6
o
....
|"
i'
I
I
i
i
i
i
I
!
I
']
i
" 'i
5 4
8Eo
.-
I
8 9
.....
I
B=5 T
e 0
8 2 E 1 0 --0.05
O LATERAL POSITION
0.05 ( 9.rn )
FIG. I0. Probability densities (absolute squares of the norrealized envelope wave functions) for the four lowest eigenstates, in a plane 8 nm below the OaA.s/Al, Gat_=As interface. The gate voltage is --1.03 V and there are seven electrons in a quantum dot. Results are shown for (a} B ==0.05 T and (b} B---5 T. The labels give the approximate value of the z component of the canonical angular momentum in units of/f. The states labeled 1,--1 for B =0.05 T have probability densities which are almost the same (they correspond to opposite angular-momentum combinations of the x- and y-like degenerate solutions for B---0). The probability densities for the states shown in both (a) and (b) are approximately circularly symmetric, except for the state labeled --2 in (a); for which cuts along the x direction (dashed line) and along the diagonal x --y klotted line) are shown.
5173
the states. The calculated curves are in good qualitative agreement with the results found by Darwin 7 for states in a two-dimensional harmonic-oscillator potential in a magnetic field. The curves are shown to cross, as would apply for states with different angular m o m e n t u m in a circularly symmetric potential, although we expect that small antierossing gaps would appear if the calculation were c a r d e d out with greater resolution. The difference between the crossing behavior in a circularly symmetric case and the anticrossing for positive-parity states in a rectangular box is nicely illustrated in Figs. 1 and 2 of the paper by Robnik. s The angular m o m e n t u m referred to in the preceding paragraph is what Van Vleek 25 has called the canonical angular momentum. It is the expectation value of l = r X p , where p is the operator - - i h V . T h e "'true" angular m o m e n t u m , L = r X m v , has:an additional term ~'26 ( e / 2 ) r X ( B X r ) , analogous to the additional term in the Hamiltonian in the presence of a magnetic field. The angular-momentum q u a n t u m n u m b e r associated with the z component of the canonical angular m o m e n t u m is the integer I that appears in the angular factor exp(il~} in the wave function in a circularly symmetric potential. Table I gives some additional information for the lowest states four B ----5 T. W e show the expectation value of the energy relative to the bottom of the well, of the z component of the canonical angular m o m e n t u m , of R = ( x 2 + y Z ) 1/2, with lateral position measured relative to a vertical axis through the center of the q u a n t u m dot, of 8R -----(( R 2) _ ( R )2)t/z, and of the z component of the "'true" angular m o m e n t u m , ( L z ) = ( / z ) + ( e B / 2 ) ( R 2 ) " The last state in the table is the lowest state with a node in the z direction. The expectation value of the true angular m o m e n t u m for a one-electron problem is related to the magnetic mo -~ m e n t / z by 25 /zz = -- d E / d B "-------- ( e / 2 m ) ( L z ), where we assume the magnetic field to be in the z direction, as in the example treated in this paper. Our numerical results deviate somewhat from this relation, a difference which we attribute to the inclusion of the Hartree terms for the electron-electron interaction in the potential energy. The energy levels in Fig. 9 are all associated with states that have no nodes in the z direction. States with such nodes, which would correspond to the first excited subband in a two-dimensional electron gas in an unpatterned G a A s heterojunction, appear at energies above 42 meV. As already shown in Figs. 6 - 8 , the character of the solution changes with increasing magnetic field. The radial wings of the charge density contract, with a corresponding increase in charge density near the center of the q u a n t u m dot and a change in the shape of the bottom of the potential well. The shape of the charge density of the four lowest states in a dot with seven electrons is shown in Fig. I0 for magnetic fields of 0.05 and 5 T. Even at 5 T, for which the magnetic length, ( r nm, is considerably smaller than the effective dot radius, about 50 nm, a distinction between bulklike and edgelike states is not obvious from the charge densities or angular momenta of the occupied states. Note that spin splittings, which we have ignored, will become significant at the upper end of the magnetic field range that we use.
188
5174
ARVIND KUMAR, STEVEN E. LAUX, AND FRANK STERN V. DISCUSSION
42
value corresponding to an integer electron occupation, the difference between the quasi-Fermi-level in the dot and in the adjacent substrate electrode increases, related to the Coulomb blockade. The gate voltage at which the charge changes discretely is not considered here. Finally, we gave some pictures of energy levels and wave functions, with approximate values of angular momentum, for a range of values of gate voltage, charge in the dot, and magnetic field." At least one of the authors began this work expecting to find a clear qualitative distinction between bulklike and edgelike states. Our computed envelope wave functions do not show any abrupt qualitative differences, which can be considered to be a consequence of the rather soft potential at the walls of the q u a n t u m dot. Note added in proof. Since completion of this work we have become aware of two related publications. The eigenfunctions and eigenvalues o f the two-dimensional harmonic oscillator in a magnetic field were obtained by Fock 33 three years before the paper by Darwin. 7 Maksym and Chakraborty 34 have treated the energy levels of quantum dots with three and four electrons moving in a two-dimensional harmonic oscillator potential with an applied magnetic field, including effects of electronelectron interacltion.
As noted earlier, many approximations have been made in these calculations. In particular, the substrate structure of the sample we have modeled required trunca9tion of the SchrSdinger mesh on a plane where the wave functions had not yet decayed to zero. There must be another, for the present not well understood, approximation in our description of the sample, because the calculated voltage threshold is about --1 V, while the measured threshold is about - - 0 . 2 V. 6 The measurements are made in the dark, and the calculations use a deepdonor binding energy consistent with that condition. The large discrepancy between calculated and measured threshold voltages may be due to changes in the properties of the top layers and of the interfaces caused by the processing steps used in defining the lateral sample geometry. The neglect of many-body interactions is also significant. W e expect, however, that many of the qualitative results for the internal structure of the quantum dot remain valid. We found that the energy-level structure can be considered to be a perturbation of the states of a parabolic potential in a magnetic field, with angular m o m e n t u m a rough g u i d e to the properties of the states. We also found, in contrast to our original expectations, that the energy levels measured from the bottom of the potential well are quite insensitive to the number of electrons in the quantum dot, for a fixed gate voltage. A weak dependence of level separations on electron population was obtained theoretically by Chaplik. 27 A number of authors have found theoretically that optical transitions for a parabolic potential in superlattices, z8 q u a n t u m wells, 29 wires, 3O and dots 31 reflect the underlying structure of the bare harmonic-oscillator potential and are unaffected by electron-electron interactions. Experiments on q u a n t u m wires 32 and q u a n t u m dots 2'4 are consistent with this result. We have shown how the quasi-Fermi-level in the quantum dot depends on gate voltage for different charge states of the dot. As the gate voltage changes from a
We are indebted to Ralph Willoughby and Jane Cullure for access to the Lanczos eigenvalue program used here, to Mark Amidon, Len Borucki, Orest Bula, Steve Furkay, and Fred Pileggi for graphics and database support in earlier stages of our work, to Trey Smith for information about the samples and for discussions of the experiments and their interpretation, to Jose Brum for discussions of his calculations, a n d to Boris Alt'shuler, Pradip Bakshi, Gottfried DShler, Alan Fowler, Wolfgang Hansen, Joe Imry, Khalid Ismail, Rolf Landauer, Uri Sivan, and Phil Stiles for helpful discussions and comments.
*Present address. 1T. P. Smith IH, K. Y. Lee, C. M. Knoedler, J. M. Hong, and D. P. Kern, Phys. Rev. B 38, 2172 (1988). 2(2. T. Lith K. Nakamura, D. C. Tsui, K. Ismail, D. A. Antoniadis, and H. I. Smith, Appl. Phys. Lett. 55, 168 (1989). 3j. Alsmeier, E. Batke, and J. P. Kotthaus, Phys. Rev. B 41, 1699 (1990). 4U. Sikorski and Ch. Merkt, Phys. Rev. Left. 62, 2164 (1989). SH. Fang, R. Zeller, and P. J. Stiles,AppL Phys. Lett. 55, 1433 (1989). 6W. Hansen, T. P. Smith III, K. Y. Lee, J. A. Bruin, C. Knoedler, D. Kern, and J. M. Hong, Phys. Rev. Lett, 62, 2168 (1989). 7C. G. Darwin, Proc. Cambridge Philos. Soc. 27, 86 (1931). SM. Robnik, J. Phys. A 19, 3619 (1986). 9j. A. Brum and G. Bastard, in Science and Engineering of Oneand Zero-Dimensional Semiconductors, edited by S. P. Beaumont and C. M. Sotomayor-Torres (Plenum, New York, in press).
I~ Sivan and Y. Imry, Phys. Rev. Lett. 61, 1001 (1988). llS. E. Laux and F. Stern, Appi. Phys. Lett. 49, 9I (1986). 12S. E. Laux, D. J. Frank, and F. Stern, Surf. Sci. 196, 101 (1988). 13K. Kojima, K. Mitsunaga, and K. Kyuma, Appl. Phys. Lett. 55, 882 (1989). 14T. Kerkhoven, A. T. Galick, J. H. Arends, U. Ravaioli, and Y. Asad, J. Appl. Phys. (to be published). lSSee, for example, R. Wilkins, E. Ben-Jacob, and R. C. Jaklevic, Phys. Rev. Lett. 63, 801 (1989), and references therein to earlier work. 16The image potential associated with the different dielectric constants of GaAs and AlxGa~_xAs has been shown to have a very small effect on energy levels in a heterojunction IF. Stern and S. Das Sarma, Phys. Rev. B 30, 840 (1984)] and is expected to have an insignificant effect on the results of the present calculation. 17G. Bryant, Phys. Rev. Lett. 59, 1140 (1987). l SSee, for example, G. E. Forsythe and W. R. Wasow, Finite
ACKNOWLEDGMENTS
189
42
5175
ELECTRON STATES IN A OaAs QUANTUM DOT IN A . . .
Difference Methods for Partial Differential Equations (Wiley, New York, 1960). 19S. E. Laux, in Proceedings of the Fifth International Conference on the Numerical Analysis of Semiconductor Devices and Integrated Circuits (NASECODE I0, edited by J. J. H. Miller (Boole, Dun Laoghaire, Ireland, 1987), pp. 270-275. 2~ G. Johnson, C. A Mieehelli, and G. Paul, SIAM J. Numer. Anal. 20, 362 (1983). 2lj. K. Cullum and R. A. Willoughby, Lanczos Algorithms for Large Symmetric Eigenvalue Computations (Birkh~user, Boston, 1985), Vols. I (Theory) and II (Programs). HB. N. Parlett, The Symmetric Eigenoalue Problem (PrenticeHall, Englewood Cliffs, NJ, 1980). 23A. Kumar, S. E. Laux, and F. Stem, Appl. Phys. Lett. 54, 1270 (1989). 24p. Ruden and G. H. DShler, Phys. Rev. B 27, 3538 (1983}. 25j. H. Van Vleck, The Theory of Electric and Magnetic Susceptibilities (Oxford University Press, London, 1932), Secs. 7 and
36. 26H. L. Zhao, Y. Zhu, and S. Fcng, Phys. Key. B 40, 8107 (1989). 27A. V. Chaplik, Pis'ma Zh. Eksp. Teor. Fiz. 50, 38 (1989) [JETP Lctt. 50, 44 (1989)]. 2sp. Rudcn and G. H. D6hlcr, Phys. Rev. B 2,7,3547 (1983). 29L. Brcy, N. F. Johnson, and B. L Halperin, Phys. Rev. B 40, 10647 (1989). ~V'. B.'Shikin, T. Dcmel, and D. Heitmann, Zh. Eksp. Teor. Fiz. 96, 1406 (1989) [Soy. Phys.mJETP 69, 797 (1989)]. 31K. Kempa, D. A. Broido, and P. Bakshi, Bull. Am. Phys. Soc. 35, 768 (19190);P. Bakshi, D. A. Broido, and K. Kempa, Phys. Rev. B (to be published). 32W. Hansen, M. Horst, J. P. Kotthaus, U. Merkt, Ch. Sikorski, and K. Ploog, Phys. Rev. Lett.S8, 2586 (1987). 33V. Fock, Z. Phys. 47, 446 (1928). 34p. A. Maksym and I". Chakraborty, Phys. Rcv. Lett. 65, 108 (1990).
~-
190
Some magnetic properties of metals I. General introduction, and properties of large systems of electrons BY R. B. :DINGLE
.Royal Society Mend .Laboratory, University of Cambridge (Commun~
by Sir .Lawrence Bragg, F.R.S. Received 2 3 J u l y 1951m .Revised 15 November 1951)
A general introduction surveying the problems to be examined in a series of papers is followed by a detailed treatment of the magnetic behaviour of a large system of electrons. The SchrSdinger equation is solved on the assumption that the system is unbounded, and the modifications caused by the finite size of the system are then determined for the limiting case in which the system is much larger t h a n the electronic orbits. An expression is then obtained for the density of states, and the free energy of the system found assuming t h a t k T < E0, where EQ is the degeneracy parameter. The magnetic susceptibility, thermodynamic potential and specific heat are discussed for the two c a s e s N constant andEG constant. Explicit formulae are given for the temperature-dependence of the field-independent term in the susceptibility. I n the final section the corrections due to electron spin are introduced. G E N E R A L INTRODUCTION
In classical theory the magnetic susceptibihty of any dynamical system is identically zero. Special cases of this remarkable theorem had already been found--particularly by Bohr (x9I x)--before Miss van Leeuwen (i919) demonstrated its general validity. The absence of magnetic behaviour is due to the fact that on classical theory electric charges may occupy any position and pursue any path consistent with Maxwell's equations. Under such lax conditions the average electric current at any point vanishes (see van Vleck 1932, p. 100). The essential new element introduced by quantum theory is t h a t of preference for certain orbits. In these conditions the current at each point no longer averages out, and magnetic behaviour becomes possible. To a first approximation a metal may be treated as ff it consisted of electrons, with a certain effective mass m, enclosed within a box. In pictorial terms, the electrons will revolve around the magnetic field, their radii of curvature r being related to their velocity v by the expression
eH/c = mv_L/r, where v• is the component of v in the plane perpendicular to the magnetic field H. It is a well-kno~m consequence of Fermi-Dirac statistics that the electron gas in a metal is degenerate at all but the highest temperatures. In other words, the magnetic behaviour depends only on electrons having an almost standard velocity v, the Fermi velocity. The ratio of the maximum radius of the orbits, rmax., given by eH/c = mv/rma• , to the dimension of tile system in the plane perpendicular to the magnetic field, R say, therefore appears at once as an important parameter of [ 500 ]
191
Soma magnetic properties of metals. I
501
the theory. If R ~ rma~_ practically all the electrons can complete their orbits without colliding with the surface of the system, whilst if/~ ~ rma~ hardly any of them can do so. Inserting typical values for m and v, it is clear that there are three cases which must be considered: (a) large systems: R ~ rmax., HR~ 5 gauss cm, (b) small systems: R~rma~, H R ~ 5 g a u s s c m , 9(c) intermediate systems: RNrmax., HR~ 5gausscn~.
Large systems The magnetic susceptibility of large systems consists of a steady diamagnetism together with some terms v a r y ~ g periodically with the field. The steady diamagnetic term, which remains of significant magnitude a~ all temperatures, was first cMculated by Landau (I93o), and later studied also by Darwin (x93o), Teller (i 93 x) and van Vleck (I 932). The periodic susceptibility is a purely low-temperature phenomenon first observed experimentally by de Haas & van/~Aphen (I93oa, b) and named after them. I t was accounted for theoretically by Peierls (x933), and the theory considered in greater detail by Blackman (x 938), Landau (I 939), Akhieser (i939) , Rumor (1948, 1949), and very recently by Sondheimer & Wilson (1951). At attempt has been made to generalize the theory in the followhag directions:* (1) Because the phase of the oscillations at high fields plays an important role in comparing theory and experiment, the complete power series in the field has been calculated. The complete series in rising powers of lcT[Eo has also been obtained, because the effective vMue of the degeneracy parameter E0 may be quite small for the electrons contributing to the de Haas-van Alphen effect. (2) The influence of the electron spin has been taken into account by applying the general formula (valid for any system whatever)
5'sptn(E0)=~ F
0+2m0c ]+-F
O-2moc/I,
where F is the flee energy ignoring spin, ~pta that taking it into account, and m 0 the actual mass of the electron. The periodic terms in the susceptibility acquire a new factor cos (zrpm/mo), where p is the harmonic considered and m the effective mass of the electrons in their motion through the lattice. Since m is usually much less than m 0 for the de Haas-van Alphen effect, the lower harmonics are therefore hardly affected by spin. (3) Collision broadening has been shown to diminish the amplitude of the periodic susceptibility by a factor e -~/~*, where ~ is the mean collision time, and v = eH/27rmc is the orbital frequency of the electrons in their orbits around the magaetic field. (4) The finite size of the system has been found to cause an apparent broadening of the energy levels relevant to the de Haas-van Alphen effect, but the consequences of this turn out to be rather insignificant. This problem will be treated as part of the theory of intermediate systems. 9 Some of the results, and their relation to recent experinaental work, were discussed in a previous note (Dingle & Shoenberg 195o).
192
502
R . B . Dingle
(5) The lattice field of the metal has been found to produce both a change in the spacing of the energy levels--which may be exactly allowed for by introducing an equivalent mass-tensor, and gives rise to a change in frequency of the periodic terms--and a broadening, which diminishes their amplitude but leaves the frequency unaltered. Of these five points, (1) and (2) will be discussed in the present part, (3) in part II, and (4) and (5) will be published later. In addition to these generalizations of the theory of the magnetic susceptibility of large systems, a calculation has been made of the absorption of electromagnetic radiation incident on a system of electrons moving in a uniform constant magnetic field. The absorption is found to increase to about twice the normal skin-effect value at a resonance frequency v--eH]2~mc, but no pronounced peak will be observed unless rv >~1. This work will appear in part HI.
S m a / / s y s ~ (1oartI V) Comparatively little work has been done on the theory of the magnetic behaviour of small systems. Huud (I938) has pointed out that near absolute zero the susceptibility should contain terms periodic in the magnetic field. Welker (i938) has attempted to calculate the steady diamagnetism for a cylinder, but obtained the answer zero to the degree of approximation to which he worked. I t will be shown that this was due to his replacement of summations by integrations. In fact, whatever the system, all integrations over quantum numbers give quantities independent of the magnetic field, and magnetic behaviour arises only from the differences between the sum and the corresponding integral. The slightest error in calculation is liable to result in extremely large values for the susceptibility, since the various contributions arising from integration may then no longer cancel; we believe that errors of this type are inherent in the work of Papapetrou (I937a) and Osborne (I95i). In a later paper Papapetrou (I939) showed that the susceptibility of small systems depends on their geometrical shape. A quantitative calculation has been made of both the steady diamagnetism and the periodic terms in the susceptibility of an infinite cylinder with its axis parallel to the field, and a sphere. I t is found that the volume susceptibility is always greater than the Landau value true for high fields, but that it never appr6aches the value necessary for perfect diamagnetism except in the rather impracticable limit of a radius greater than a hundred miles and a field less than i0 -7 gauss (Dingle I95I). The influence of the interaction energy due to the electron spin has been calculated by means of the general formula given above. As far as the terms independent of field are concerned, the usual paramagnetic susceptibility due to the spin is simply added to the diamagnetic susceptibility. The effect of spin on the periodic terms is to modulate them with a frequency about (d/_R)~ times their own, where d is the mean spacing of the electrons, and/~ the radius of the cylinder or sphere.
Intermediate systems The wave-functions and energy levels for electrons in systems of intermediate size have been determined by means of the well-kno~m Wentzel-Brillouin-Kramers method. It is found that there are two entirely different types of wave-function
193
Some magnetic properties of metals. I
503
possible, one type which gives riseto the large susceptibility of small systems, and another which gives the Landau diamagnetism of large systems. The actual susceptibility of intermediate systems may be found by Calculating the relative number o f occupied states of the two types, and therefore falls continuously from the high value found for small systems to the Landau value valid for large systems. This work will be published later. T H E MAGNETIC BEHAVIOUR OF LARGE SYSTEMS OF ELECTRONS
(1) The Schr6dinger equation and its formal solution Let us consider a system of free electrons contained within a cylinder of radius/~." A magnetic field H is supposed applied parallel to the axis of the cylinder. Now the classical Hamilton]an of a free electron in a constant magnetic field with vector potential A is ~f= ~ p. (t.1) A magnetic field H directed along the z-axis may conveniently be represented by the vector potential (- 89 89 0), so that I
~f = ~
I
eH
I
eH
p'+ ~
1
= ---P~+
2m
2~
e~H~
(xp~- yp=) + ~
(x, + y,)
e~H~ ~"
T~ + - - - - - -
(1.2)
8mc~'
where p~ is the angular momentum of the electron about the z-axis, and r its radial distance from the axis. The quantum Hamilton]an is obtained on substituting T = - / ? / . g r a d . The wave-equation 9/Z~F = E~F, where E is the energy, then reads in cylindrical polars
The term ~tF/az ~is the only one depending on z, showing that the contributions to the wave-function and to the energy arising from the translational motion along the cylinder are unchanged by the introduction of a magnetic field. This term may therefore be dropped from the equation if we confine our attention to the quantized energy due to motion in the plane perpendicular to the field. Assuming that the angular dependence of the wave-function is given by tF = e~Ur H(r), where 1 is an integer (positive or zero), since tF must repeat itself for a rotation through 2~r, we obtain the equation r~r where
+ 2a-~-~#r
9 m (.E •
~= ~
eHti.~
II = 0,
eH
~ -- 2~c"
(1.4) (1-5)
The substitution x = r 2 reduces (1-4) to
8~II 18II i(_~,~ 2cz lxO ~x2 + x ~ x + ~
+ - x- - -
II = 0.
(1.6)
194
R . B . Dingle
504
Removing the s e c o n d t e m by the substitution II = x-ig, ~g+l (
2=
i -Y2~ x
/9-1)
x~
g=0.
(1-7)
g=0,
(a.s)
Or in dimensionless form 32g
~+~
1(
2~
-l+~y
l~-l)
y,
where y = 7x. (This is essentially Whittaker's form of the confluent hypergeometric equation; in the notation of Whittaker & Watson I927, chap. 16, the solutions are W~=,~z(y) and W _ ~ z ( - y ) . ) The further substitution g : yt~Z)e-4Yh transforms (1.8) into ~2h ~h
y~+q+
where so that by (1.5)
a - y ) N + ~ h = 0,
(z.9)
~ = T(2n + l + 1),
(1.10)
Ez = ~e~H (2n _+ l + l + 1).
(1.11)
Equation (1.9) is one form of the confluent hypergeometric equation. In the notation of Jahnke & Erode (i945, p. 275), the required solution* is M ( - n , l+ 1, y). It is given by the series n n(n-1) y2_ (l.12) M ( - n , l + 1,y) = 1 - / - u (l+ 1) (/+2).2! "'" and has the asymptotic expansion (obtained from that given by Jalmke & Erode with the aid of the relation (-z)! (z)! = 7rz]sin (Trz))
M( - n,l+ 1,y) ~ ~ - (- - yl ') " (1 n(n +l) ~-...} (n+l)! y l'n, eUsin(Tr(n+ l)} ( (n+ l ) ( n + l + l) } + " 7ry,~+z+z 1+ Y + ....
(1.13)
:For unbounded space, the wave-function must vanish as y-->c~. Equation (1-13) shows that this can only be the case if n is zero or a positive integer, so that the energy given by (1.11) is quantized in a simple way. In this particular case, the confluent hypergeometric function reduces to a polynomial of degree n; in fact
M ( - n , l + 1,y) = ( - 1)Zn!l!L~+z(y ) E(n+~)!]~ , where
and
L~(y) =
Lk(y) = O' ~
Lk(y),
(yk e-~')
(1.14) (1.15)
(1.16)
is the Laguerre polynomial of degree/c. * Since h is the solution of a second-order differential equation, there is another solution, but it turns out to be inadmissible since it is not finite at the origin; eL ~rebb & Airey 1918 and Stoneley i934.
195
S o m e m a g n e t i c p r o p e r t i e s o f metals.
I
505
Any practical system is, however, of finite radius, and we must inquire how this alters the energy levels and wave-functions. I t follows from the general properties of second-order differential equations t h a t if t h e expression in brackets in (1-7) or (1.8) is positive, the wave-function will be oscillatory; if it is negative, the wavefunction will fall offin amplitude roughly exponentially. Taken together with (1-10), this means t h a t a given wave-function ~F~z is of significant magnitude only if
eHr ~
l + 2n + 1 + {41n + 2l + 4n ~+ 4n + 2}t > ~
> l + 2n + 1 - {4ln + 21 + 4n 2 + 4n § 2}89
(1.17) I t will shortly be shown t h a t l m ~ ~ nma~ for a large system. Thus for practically all occupied states the corresponding wave-functions "~n,t are localized according to the relation
eHr ~]2~c ~--l.
(1.1s)
The spread of the wave-functions (Ar) is given b y
(eH/2~) ar~ -~ 4{l(n + 89 so t h a t
Ar[r ."- 2{(n + 89
~ 1.
(1-19)
(1-20)
The effect of the b o u n d a r y of the system will t h e n be accurately simulated by merely excluding from consideration all wave-functions (of the t y p e for unbounded space) whose m a x i m a would lie outside the system. I n other words, we m a y use the wave-functions and energy levels valid for u n b o u n d e d space provided we restrict the q u a n t u m n u m b e r l so t h a t (by (1-18))
l < lmax. = eHR~/2hc = eHA/hc,
(1-21)
where A is the cross-sectional area of the system. We have now to prove that/max_ >~nmax.. B y ( 1-11) nmax. N
Eomc/e~H ,
(1.22)
where E o is the Fermi energy, i.e. the degeneracy parameter. Hence lmax. >~nmax. if
eHR/c>~ (2mE0)89 = mVFerml,
(1"23)
which is just the initial postulate t h a t the electronic orbits are of much smaller radius than the system. By a precisely similar a r g u m e n t we m a y show that it is 9justifiable to neglect the energy levels with the upper sign in (1-11). The estimation of the actual error in the calculated susceptibility resulting from these various approximations will be given in a later p a p e r on systems of intermediate size. Collecting our results, we have t h a t for R>~ rma~.
e?~H E~_ = --~(~+ 89
(1.24)
a n d there are eHA/hc degenerate states for each value of the energy.* * According to the old quantum theory, energy is quantized in units of ?~o),where ~ois the angular velocity v• Since eH/c = rav• this leads to the energy spectrum E• = neIiH/mc. The distance of the centre of the orbit is also quantized, the corresponding quantum number l having a maximum value lmax.,"mvj.l~/~, because no electron can be at a greater distance from the centre than the radius. It must be stressed that both in the old quantum theory and in the new, each single value of l corresponds to a whole group of electron orbits.
196
506
R.B.
Dingle
This method of derivation has been outlined because some of those in the literado not seem to be quite complete. Landau (I930), i n his original work on the subject, solved the SchrSdinger equation not in the cylindrical polar co-ordinates used here, but in cartesians. His solution has the drawback that the variables have not been completely separated, and the wave-function does not consist of a product of separate functions of the co-ordinates. I t is therefore not easy to introduce the b o u n d a r y condition ~F = 0 in order to calculate the error caused by ignoring the existence of the bounding surface. Van Vleck (193z), in his derivation, appealed to the o1(1 quantum theory. The calculations of the magnetic susceptibility presented by Darwin (I93O), Mort & Jones (i936), and Sondhehner & Wilson (x95i), do not deal with quite the who!aproblem, since it is taken for granted t h a t the susceptibility is purely a volume effect. If this assumption is admitted, the actual finite physical system may be replaced by a fictitious one which is mathematically more tractable. For instance, Darwin considered a model in which there is acting on the electron a force of restitution directed towards the origin and described by an energy proportional to r 2. Such an energy leaves the form of the wave-function unchanged, and the result obtained for the susceptibility becomes identical with that found by Landau if the limit is taken of the force tending to zero. Mort & Jones assume a one-to-one correspondence between the levels with and without a magnetic field, whilst Sondheimer & Wilson's calculation is tantamount to finding the susceptibility per unit volume of some infinite system. Although mathematically elegant, these demonstrations do not seem to be quite self-sufficient. For instance, they yield no rigorous internal evidence of their own limits of validity. In particular, they do not show whether or not the contributions from the surface are negligible for most systems and fields used in practice. (Comparison with our calculations shows that in fact they reqaire eHR2/2~c>~1 and eHR/c~ (2mE0)89, where ~ is the least dimension in a plane perpendicular to H.)
, ture
(2) The density of states The contribution to the energy arising from translational motion along the direction of the magnetic field is En = E - / ~ . (2.1) By Heisenberg's uncertainty principle these states can be distinguished only if Az Ap~ 1>h. Taking into account the two possible orientations of the electron spin, there are therefore 4.Cf I Pz 1/h = 4..~(2mEu)89 distinguishable states with momenta (and energies) numerically less than p~ (and E~,), .Cf being the length of the system in the z-direction. Hence the number of states per unit volume with total energy less than E is given by
Z(E,H)
eH 4(2m)89 (
-hc
h
eliH
Z E - - m - t-(n+ 89
}'
.
(2.2)
:For later purposes it is more convenient to evaluate I Z dE rather than Z itself. :Now
f
ZdE
= ~(flH)a
Z {c- (n + 89 n
(_0.3)
197
S o m e m a g n e t i c p r o p e r t i e s o f metals. where
fl = eh/mc,
a = (2m)l/3~r~h a,
I
507
e = E[flH.
(2-4)
The summation in (2-4) is to be taken over all the positive integers n, starting from zero, for which { e - ( n + 89 is real and positive. I t m a y be shown (see appendix) from the Poisson summation formula* t h a t
E f ( n + 89 = nffiffiO
E :p=
( -- I)P
f ( n ) e 2n'~n dn.
(2-5)
--co
Thus Z {e-- (n + 89 = n-O
E
"
( - 1)P
fo'
p----~
( e - x)' e 2n'p= dx =
Z
"
( - 1)~ e*"~
p=--r
"f:
y' e -2*'~u dy. (2-6)
Now
t
io
(~=0).1
Putting y = t/27rp,
f~y
(2.8)
-t e-2,,,,,,, dy = (2rrp)-tf:"~'t -~ e -u dt = (2ip)-i r {(2rripe)i},
where r is the error function r
= 27r-~
for r is (Jahnke & Erode i945, p. 24)
f
2
e -4' dt. Now the asymptotic expression 0
---r-2e-Z" ~ ( - I)~ (29-- 1)}
r
(2-9)
so that by (2-6) and (2.7) zd_E
=~el+
Z (--i)~ :p------ aO
r.3~/e
eI
• ks.,f
3
Z
(--l)~(2v--1)!
3e '(9n'-{"O']
(2.10)
2.ip ~16.~i~---~._~0p.+~(sm~).(~-1)!- i g ~ ~ j'
where 2] means summation excluding the t e r n 19 = 0.
Now
~
Z
(-l)pp-"=
_g(v)(l_21_. ) = _ rrB89 ~
=--~
.-I- 1 )
(veven)
(2-1I)
]2!
where the ~'s are Riemann zeta functions and the B's Bernoulli numbers. Thus finally
ZdE
= 5~el ~/e 16
3
( _ 1 ......... /~ )I(~-i)B~(~+3)(2~+2 I) ( 2 9 - I)' 1,3.... (8e)" (v-- I)! (9 + 3)!
3
,3 ( - 1)~ oo~ (2~p~- t.)
8n2~/2~-I~
~l
.
(2.12)
* The introduction of this method of evaluating the required re,ruination is due to Landau ('939)-
198
508
R . B . Dingle (3) The evaluation of the free energy
The free energy of a system of N electrons is F = N E o - k T Z In {1 + e(~*-E0/kT},
(3-1)
i
where E~ are the available energy levels and E o the degeneracy parameter determined by the condition N = ~] 1/{1 +eCE~-~d/kr}, i.e. by ~$']~]~o = 0. The number of states per unit volume with energies between E and E + dE is dZ(E, H), so that 1~-- N E o = - k T V
In {1 + e ( ~ , - z ~ v - r } d Z = -
V
e0g-mO 1k2 + 1
v f | ( f Z dZ) e~-'~kr dZ = - k-T
- ~o
(e(E-~o)/kr + 1)s
(3-2)
on successive integrations by parts. The energy range has been taken from - 0 3 to + 03 because the electrons are moving in a field of force, and it mustnot beassumed without proof that there are no bound states; actually, all terms involving energies less than +e?iH/2mc automatically drop out. The cosine term of t Z d E may be integrated exactly. Putting
(E-Eo)lkT = 7, ~ = 2~rpkTiPH we have
e ~(l+u) d7
and
e=
E/ZH = (~,kT + Eo)/ZH,
t l~dt = (i3)Y( - i(?)I =
(i+e~) 2
(I+0
2
"
"
sin(m~)
sinh (~z~)'
(3-3)
the last steps following from formulae given, for instance, in Jahnke & Emde (I94 5, pp. 20 and 11 respectively). The remaining terms of tZ dE m a y be integrated by noting that the integrand Q/
in (3.2) is appreciable only in the energy region immediately surrounding E0. Putting again (E-Eo)/kT = 7 and expanding a function f(E), consisting of the remaining terms in I Z dE, in a Taylor series, we have 3 co
f(E) = ~., (kT)~'7gf~)(Eo)l(/z)!.
(3.4)
!~ = 0
The integral
I~=
f?
yze~ d7
~(e~+l)~=
f~_
7~d7
~(e~+e-i~)~
vanishes if # is odd. For even values 89 = E (- 1)'-'s s=l
7"e-'yd7 = (#)! 7: (- I)'-~ s=l
8/~
= (,u.)! ( 1 - 2'-~) U/~) = (2~'-~- 1).~Bi~,
(3-5)
199
Some magnetic properties of metals. I
509
where the B's are Bernoulli numbers, B o being taken as - 1 since I o = 1. Thus
I
1 ~ f(E)exp{(E-Eo)/kT}dE 2 Z (kT)~fO)(Eo)(2,'-l- 1)~gBjg. k-T -.o iexp{(E-Eo)/kT}+i]~ = ~=o.2..... ( t t ) ! "
(3.6)
This is essentially the same as a formula given by Sommerfeld (x9z8). Finally, we obtain* the following result for the free energy: F
- NE o aV
•
3kT(flH)! ~ ( - 1)v cos A 9
=" 442
[~ m _ ,~
v: 1 f
(fill )~
|
3
Z"
(;tkT)~'
2 ~,=o, Z2..... ------(2~-~~)B~ (,a)l
sinh ~
(- ~)~-~,(~H),+ _a Bt4"+a)(2"+' --
(~oo)l' 1) (2~- 1)"] (3-7)
where
~= (4)
2n~TkT/flH
and
A=
2r~TEo/flH-88
The evaluation of X assuming that N is constant
The susceptibility is given by
(4.1) g
~
H ~H
V-H , , o o , E. +
u
s.
the second term vanishing since 8F/~E o = O. If N is constant,
--
~
= 2 42 ( f i l l ) t , . 1 sinh ~ - ~
[~/p.Eo tanh ~ + 47rEoTtU
1 s (rrkT)~'(2~,_1_ 1) B~, + i~,=0.~.... (/~)!
E~/ x
3
~
Eo +.~o,=~, ~....
( -1)t~"-l) (flH)"+x (2"+~- l ) (2v-1) , Bt6,+a)] (8Eo)" ( v " i)! (V+ 2)!
"
"
(4.2)
In this expression, of which the dominant terms were given by Landau (i939) , a = (2m)t/3n~ 3, fl = e?i/mc and the B's are Bernoulli numbers, the first few being It is of interest to consider the origin of the terms in (4.2). At absolute zero the electron gas is completely degenerate, and the occupied energy levels are those possessing an energy less than E o. The number of occupied energy levels such as E• therefore decreases with H in a series of jerks of frequency Eo/~tt, and the magnetic susceptibility contains terms periodic in H. At finite temperatures, the active electrons do not all have the same energy Eo. The width of the energy spread is about kT. As the magnetic field is increased the energy levels will be expelled past this blurred energy surface, but now they can be distinguished from each other only if their separation ~H is greater than kT. (In classical statistics there is no energy maximum at all, and therefore no periodic susceptibility.) Each energy level E~ is accompanied by a taft (stretching up to E0) due to the unquantized energy of motion along the direction of H, and there is a steady diamagnetism in addition to the periodic terms. Since the tails stretch from energy * Dr 8ondheimer has informed me that his work leads to the same expansion. Vol. 21~.
A.
33
200
510
R.B.
Dingle
levels placed at all d i ~ c e s up to ~o from the energy maximum, their amplitude is not significantly altered by the spread (kT) of the energy surface provided J~o>>k T . iV, determined from (3-7) by the condition 8F]0E o - 0, is found to be N 3. ~ ( - 1)" sin ,~ a'-'V = ~ kT(flH)t Z ~ + ~-1
(fl//)*
•
~/I'
a
sinh~
2
~ Z
~-o.~,...
~
OrkT)~' 0 [ ' i~' (2~-~- 1 ) ~ t , (/~)!
(~//),+~(2,+*-
~)']
...................
...
(8~0)" ( ~ - ~)! (~+ 3)!
,J " (4.3)
If N is constant, as assumed above, Eo must first be found from (4.3) as a function of temperature and field. This is easily achieved by a process of iteration, i.e. by inserting the value of ~o for zero temperature and field in all the smaller terms, and then repeating the process until sufficient accuracy has been obtained. If E~0 - (2~]aV)! is the thermodynamic potential per electron for T --- 0 and H = 0, an approximation to E0 at a finite temperature and field is
1 (.kT,SIt)" /
+
-
42
3s4
(E~)'
4 p ....8mh V
(4.4)
Thus if h r is constant, E o falls as the temperature rises, and increases with the field. In addition to these steady changes, there is an oscillation in E o approximately proportional to sin ( 2 ~ ] , 8 H - 88 Since the dominant periodic term in the susceptibility is approximately proportional to sin (27rEo[~H- 88 the susceptibility-field curve will not be perfectly sinus0idal, but will be distort~
E, +
=
r -d-~ =
E.,
(4.5)
since OF/~E o = 0. The specific heat at constant volume and field is thus obtained from (3-7); it is CV, H aV
T(~S/ST)H aV
_ 3~/(2)k~(flH)t ~ (-1)VcoSXEs 1 ~ ~ ] 4 pt inh ~ tanh ~ - 2 sinh ~ tanh ~~ - 2 sir~ 3 p~=l + 2
x
~ Bt~,(2~'-l~,= 2, 4 .... (/~ - 2) !
1)
+ # - 1 dT
bee
2 j ~/Eo(flH) 2 3 X ( - 1)t(~-x)(flH)~+3(2~+'-l)(2v-1)!Bi(~ E~ 16 8 ,~/Eo ~=1~,s .... (8Eo) ~(v - 1)! (v + 3 )! (4.6)
whcre E o is given by (4.4) ff N is constant.
201
S o m e magnetic properties of metals. I
511
(5) The evaluation of X assuming that E o is constant If practically all the available electrons contribute to the magnetic effects, E 0, the thermodynamic potential per electron, will be a parameter determined by the condition that N is constant, and the theory of the last section will be the valid one. However, the fact that the periodic terms are observable only if flH > k T leads to a number of interesting consequences. I f we insert numerical values, we find that for observability H ~ lcT/~ ~ 1(PT gauss (5-1) 9ff the electrons are free. The fundamental period of the oscillations AH is given by
aJtlIt
= PHIEo N 10-'H
(5.2)
for free electrons ia most metals. These two conditions would be difficult to satisfy, (5.1) necessitating very high fields and very low temperatures, and (5-2) requiring a magnetic field of quite exceptional homogeneity. Actually the situation is even worse, because any broadening of the energy levels (such as that caused by impurities) decreases the amplitude of the periodic terms still further. The net result is that the observed de H a a s - v a n A1phen effect is normally not due to free electrons at all, but to any electrons (even though their number may be very small) which happen to have both an unusually small effective mass (giving a large value for p) and an unusually small value of E o. These conditions are satisfied by electrons lying near the bottom of a BriUouin zone which is separated by a small e n e r g y gap AE from the zone below. For such electrons/? is increased by a factor* (1 + 4E/AE), and E 0 is much smaller than for free electrons because it must now be measured from a zero corresponding to the lowest energy for which the small effective mass is possible, i.e. from the energy gap. Thus measurements on the de Haas-van Alphen effect inevitably yield information only on t h e finer details of the structure of the Brillouin zones. The steady diamagnetism, 0nthe other hand, is contributed roughly equally by all the available electrons, and therefore enables us to study the broad outlines of the zone structure. If only such a small fraction of the available electrons (or, of course, positive holes) contribute to the diamagnetic susceptibility, the thermodynamic potential per electron (which must be the same for all electrons in the system, although its apparent value E 0 may vary due to measurement from different points of the zone structure), will be practically unchanged by the presence of a magnetic field, and therefore E o must be taken to b e a constant,% and the number of electrons contributing to the susceptibility found from the relation ~F/aE o = O. In this case N is still given by (4-3), since this depends only on the relation N = ~] 1 / (1 + eCE~-Ea/kr), i
which is true whether or not N is constant. The appropriate formula for X, on the other hand, requires investigation. * l~ott & Jones (I936, PP. 65, 84 and 210). t First suggested by Blaekman (z938). 33"2
202
512
R.B.
Dingle
W h e n N is considered as a variable, a n e x t r a t e r m E od ~ m u s t be introduced into t h e equation of conservation of energy ( L a n d a u & Lifshitz z938 , w43), so t h a t
dU = TdS-TdV-
xVHdtt + Eod~ ,
(5.3)
where U is the total energy, the sum of t h e internal energy a n d the potential energy in the field of force. Introducing the Helmholtz free energy b y the relation F = U - TS, we have d E = - S d T - T d V - x V H d H + E0dN, (5-4) d(F-NE0) = -SdT-pdV-xVttdtt-Nd-Eo.
(5-5)
E q u a t i o n (5.5) shows t h a t ff E 0 is to be considered as constant, the susceptibility a n d entropy are no longer given b y the c u s t o m a r y formula~ X VH = - ( a F / 0 / / ) r , H and S = - ( a F / a T ) v , H - - w h i c h indeed are shown b y (5.4) to be normally valid only ff AT is c o n s t a n t - - b u t by X -- - - - z
VII ~
( F - - NEo)
S = -- ~-~ ( F - - 2VEo) '
v,r
,
(5.6)
.
(5-7)
V,H
E q u a t i o n s (5-6) and (5.7) lead to formally precisely the previous relations (4-2), (4.5) and (4-6), although now, of course, no such correction a s (4-4) m u s t ' b e m a d e to E o, The differences between the magnetic properties calculated according to the two assumptions hr constant and E o constant are thus significant only at relatively large fields.* (6) The non-periodic susceptibility at higher temperatures If we introduce (4.4) into (4.2), a n d retain only the terms independent of field, we obtain the asymptotic series Xnon-perlodic --
8 VkT
o
""
'
where E ~ = k T o and fl = e ~ / m c is twice the Bohr magneton. The firsttwo terms of this expression were given by Stoner (1935). W h e n T > T o the expansion (6.I) breaks down. W h e n E o is negative, a series m a y be obtained by expanding in powers of g = eEo/kr; thus
exp ((E - Eo)/IcT } + 1 -- ,=IZ ( - 1)'-~ ~"
Z(E) e-sE/kr dE.
(6.2)
The parameter ~ may then be eliminated in the same way as E 0. After some tedious reiteration, we obtain the series Xn~176
=
12VkT
1-0"2660
+0"0761
-0"0044
+...
,
(6"3)
* Some of the results for the m a g n e t i c susceptibility for E0 constant quoted in a preliminary note (Dingle & Shoenberg I95o)were erroneous, since the usual relation xVH = - ( a F / a H ) T , r h a d been assumed. I am grateful to Professor L. Onsager for pointing out t h a t equation (5-6) m u s t be used when E o is constant.
203
Same magnetic properties of ~ ~ .
I
513
which is convergent for (To]T)! 4 1-018. In the case of the equality, the quantity in square brackets is equal to 0-788. The first two terms of (6-3)have been given by Stoner (I935) and by Mott (I936). The gap between the regions of temperature covered by (6.1) and (6.3) must be filled by numerical integratiqn. Mott (z 936) has shown in this way that the susceptibility-temperature curve is smooth in this region. (7) The influence of the electron spin So far, we have taken no account of the interaction between the electron spin and the applied magnetic field, i.e. we have neglected the spin paramagnetism. A l ~ e s e r (i939) has given a treatment which is valid for perfectly free electrons. In his work it is assumed that the effective mass of an electron moving through the metal lattice is exactly the same as that giving the intrinsic magnetic moment of the electron spin---the actual mass. The influence on the de Haas-van Alphen effect may be foreseen. The spacing of the energy levels is still fill, but they have now all been shifted in energy by 89 the interaction energy between the spin and the magnetic field. The fundamental periodic term and all odd harmonics are therefore shifted in phase by 180~ (i.e. changed in sign), whilst the even harmonics are unaltered. This result is, however, rather academic, since we have already noted that the observed periodic term is contributed only by electrons of unusually small effective mass, so that the mass assumed for the spin and that for motion through the lattice are in practice entirely different.* The shift inenergy due to the electron spin is now only a small fraction of the energy spacing, and the de Haas-van Alphen effect therefore hardly altered. The accounts of spin paramagnetism usually given in. the literature are inaccurate, because of the implicit assumption made that paramagnetism and diamagnetism are algebraically additive. In these demonstrations, the spin paramagnetism is found by dividing the energy levels into two groups corresponding to the two possible orientations of the spin, and then integrating over the energies of each group, finally subtracting the results. Integration over energy levels, however, leads to the same results as classical theory, and the erroneous conclusion of absence os diamagnetism results. Stoner (z935) and Akhieser (I939) have avoided this difficulty by starting afresh with the energy levels as modified by the spin. Sondheimer & Wilson (I95I) re-evaluate the partition function and determine from its'singularities in the complex plane the free energy on FermLDirac statistics. Throughout our work we shall employ a very simple artifice which reduces to a trivial mechanical procedure the task of taking into account the electron spin for any system whatever. The interaction energy due to the spin is equal to + 89 where flo = e~i/moc, mo being the actual mass os the electron. If the number of states with energy less than E is Z(E, H) when no account is taken of spin interaction, the number of states taking spin into account is Zsptn(E, H) = 89
+ 89 H, H) + Z(E - 89
* This was pointed out to m e b y D r Shoenberg.
H)}.
(7.1)
204
514
R . B . Dingle
:Now, (3-I) shows that in evaluating %he free energy the terms + 89 be transferred to E o, so that we have
~.(Eo)
=
~{F(Zo + 89
may'simply
+ ~(~o- 89
(7.~)
Similar relations also hold for all t h e r m o d y n a m i c functions not involving differentiation with respect to the magnetic field--for instance, to the n u m b e r of electrons, the entropy, the internal energy a n d the specific heat. Thus (3.7) becomes F - NEo 3 =---V-- = ~
kT(flH)'
G0 ~: ( - I)~'cos A cos (TrlV,8o/,8)
9' sinh
~_t
-2 #- Z| (=kT)~.u~r ,7,.-...,.--I) ,o.-x (~oo)" O, 2 . . . .
3
9
(- l)~:'-x)(~H)~+s (2~+s- 1) (2v- I) !Bir
%
16~=x,sz"....
8,(v- 1)! (v+ 3)!
• {(~o + 89
(7.z)
+ (Zo- 89
The sole effect of the electron spin o n the periodic terms in the susceptibility and specific heat is therefore to introduce a factor* cos (~1ofl0]fl) = cos (Tr2m#no). This result holds irrespective of whether N or E o is constant. For perfectly free electrons m = m 0, and the cosine factor cancels the ( - 1)~; this is Ak~eser's (i939) result. In practice, however, m is usually m u c h less t h a n m 0 for the observed periodic t e r m s , and the cosine factor is nearly unity, i.e. the influence of electron spin is negligible. For t h e non-periodic terms we obtain -
\C~Vl
10240
O~
"'"
X ) = Zo~(~ - b~) + EfftH~(30/?e/~~ 15/?~ - 7/?4) + ... 8 3840 (3/~o - fl 2) (E~ = (Eo~ +
(~) g constant
8
Hg ( 5,B2ff~o- 5 ~ - , 8 ' ) + 320
(7.5)
(7-6) "'"
where a = (2m)0/3n2h3,/? = el~/mc, ]3o = el~/moC and E ~ = k T o = (N/a V) t. Equation (7-6) agrees with Stoner's result (I935) in the special case fl =/?o. The temperature variation of the field-independent terms in these total susceptibilities is exactly the same as t h a t given in the last section for the diamagnetism alone. Taking into account the electron spin a n d assuming constant N, the specific heat is found to be
CV'H=
2T oV
1-
-
48kZT~
-....
* The same conclusion has been reached by Sondheimer & ~Vilson (1951).
(7.7)
205
Some magnetic properties of metals. I
515 CD
APPenDIX. THE ~.VXLUXTIO~ OF S V ~ , ~ O X S
OF THE FOm~ Z $(n + 89 nffi0
Poisson's summation formula enables one to sum any function (subject to certain restrictions) provided one can sum its Fourier transform. Quantitatively (cf. Titchmarsh I937, p. 60) Z| f(s) + 89
=
Z~
j/0| (s) e 2'r~ds.
(A 1)
:For summation of functions of haft-integers, it is much more convenient to transform this relation as follows:
Z| f(n+~)= 89189
n==O
Z
-
r=-r
Z
|
r ~ --~O
fo~ n+ 89 e2"ln'dn = 89189
f(n+89
•-t
Z
rffi-~o
-
(n+ 89
(A2)
B u t if a Fourier analysis of a function is made over an Luterval L, such that f(x)=
a(r)e -9"trx,
Z
a ( r ) = L -x
then
e2"~'xf(x)dx.
I"'--- a0
Taking L = 89 we see that the first two terms i n (A 2 ) a r e equal and opposite. Changing the variable in the remaining term, we obtain
Zco f(n+89
- -
~==0
. Zao
(-- 1)r f n ~
r----~o
f(n) e2"tnrdn.
(A3)
~0
The usual theory of the non-periodic term in the susceptibility of free electrons involves the use of a form of Euler's summation formula. In order to facilitate comparison between such theories and our use of (A 3), we give here the relation between the formulae. By (A 3)
Z f(n+89 =
n=O
f:=of(n)
dn+
,o ( - 1)r Z ~ f(v-I)(0) ( - 1)v, X (27rir)~
r = - - oo
(A 4)
r=l
on expanding the integrals for r 4:0 by successive integrations by parts. Now
{
0
x (-1), rffi-~ r~ --- - 2 ~,,( v ) ( 1 - - 2 i - ~ ) = -- (--~) ~ t.B89
(As) (veven),J
whence CO
.=o
~
f(n+{) = j,~ f(n)dn =o
-
O0
Z
( - 1)t.B~.(1-2~-~)fr
. = 2 , 4 ....
(v)!
= f:f(n)dn+~-~f'(O)-v~f'(O)+...,
(A6)
which is the form taken by Euler's summation formula when applied to functions of haft-integers. The first two terms of this expansion are those usually employed in the discussion of the non-periodic term in the diamagnetism.
206
516
R . B . Dingle
This work would never have been accomplished without constant stimuli and encouragement from Dr D. Shoenberg. I am also grateful to Dr E. H. Sondheimer and Mr A. H. Wilson, F.R.S., for keeping me informed of the progress of their recent work, and for showing me their paper before publication. l~EPERENCES Akhieser, A. I939 G.R. Acad. Sci. U~R,9.S., 23, 874. Blackman, M. x938 Prec. Ray. ~ac. A, 166, 1. Bohr, N. x9xx Dissertation, Copenhagen. Darwin, C.G. x93o Prec. Gamb. Phil. ~oc. 27, 86. Dingle, R . B . x95x Phys. Rew. 82, 966. Dingle, R. B. & Shoenberg, D. x95o 2qature, ~ . , 166, 652. de Haas, W. J. & van Alphen, P . M . x93oa Gonmtun. Phys. f ~ . Univ. ~ , 212a. de H a ~ , W. J. & van Alphen, P . M . x93ob Prec. Acad. Sci. Anat., 33, 1106. Hund, F. x938 Ann. Phys., f/pz., 32, 102. Jahnke, E. & Erode, F. x945 To~/~ offu~ct;~;ts, 4th ed. New York: Dover. Landau, L, x93o Z. Phys. 64, 629. Landau, L. x939 See appendix to Shoenberg, D., Prec. Roy. Soc. A, 170, 341. Landau, L. & Lifshitz, E. x938 St~/~t/r~ physics. Oxford University Press. Mott, N . F . x936 Prec. Camb. Phil. Soc. 32, 108. Mort, N. F. & Jones, H. x936 Properties of ~ and o77~s. Oxford University Press. Osborne, M. F . M . x95x Phys. 7~ev. 81, 147. Papapetrou, A. x937 a Z. Phys. 106, 9. Papapetrou, A. x937 b Z. Phys. 107, 387. Papapetrou, A. x939 Z. Phys. 112, 587. Peierls, R. x933 Z. Phys. 81, 186; Z. Phys. 80, 763. Ruiner, Y . B . x948 J . E ~ . Th~rr. Phys. U.S.S.R., 18, 1081. Rumer, Y . B . x949 J . Exp. Theor. Phys. U.S.S.R. 19, 757. Shoenberg, D. x939 Prec. 7~ay. Soc. A, 170, 341. Sommerfeld, A. ,928 Z. Phys. 47, 1. Sondheimer, E. H. & Wilson, A . H . x95x Prec. Roy. Soc. A, 210, 173. Stone]ey, R. x934 7~fon. 1Vet. R. Astr. Soc. Geophys. Suppl. 3, 226. Stoner, E . C . x935 Prec. Roy. Soc A, 152, 672. Teller, E. x93x Z. Phys. 67, 311. Titchmarsh, E. C. I937 Introduction to the theory of Fourier integrals. Oxford University Press. van Leeuwen, J . H . x9x 9 Dissertation, Leiden. Summary in J. Phys., Paris, , 92,, 2, 361. van Vleck, J. H. x932 Theory of electric and magnetic susceptibilities. Oxford University Press. Webb, H. A. & Airey, J. R. ,9,8 Phil. ~a!7. 36, 129. Welker, H. 1938 S.B. bayer Akad. Wiss., 14, 115; Summary in Phys. Z. 39, 920. ~rhittaker, E. T. & 3Aratson, G. N. x927 Modern analysis, 4th ed. Cambridge University Press.
207
Some magnetic properties of metals I I I . D i a m a g n e t i c resonance BY R. B. Dn~oT.v.
Royal Society Mond Laboratory, University of Gambridge (Gommunicate~ by Sir Lawrence Bragg, E.R.S.---Received 23 July 1 9 5 1 Revised 15 ~ovember 1951) It is shown Chat electromagnetic radiation incident on a large system of electrons moving in a constant m a g n e t i c field H in a metal is strongly absorbed near a frequency v=e2~t/2nmc, where m is the effective mass. The resonance absorption is found to be of r same order of magnitude as the absorption due to the akin effect. (1) Introduction According to wave-mechanics, free electrons give rise to diamagnetism as a result of the quantization of their orbits in a plane perpendicular to the magnetic field. The absorption of electromagnetic radiation incident on such a system might be expected to show a marked peak when its frequency coincides with t h a t of the revolution of the electrons in their orbits (v = eH/2~mc, where m is the effective mass of an electron in its motion through the lattice), or perhaps with a simple multiple of t h a t frequency. Such a phenomenon would be roughly analogous to nuclear, ferromagnetic and paramagnetic resonance, and may be called 'diamagnetic resonance', since it would be due to transitions between the quantized states giving rise to the Landau diamagnetism.~ The extent of the absorption is determined by the transition probabilities between the different states. Provided the wave-length of the incident radiation is much greater than the spread of the electronic wave-functions, these transition probabilities are proportional to the squares of the 'mixed dipole moments' of the unperturbed wave-functions. In the first part of the present paper this diamagnetic absorption is calculated in the following stages: (a) normalization of the wavefunctions of the u n p e r h ~ b e d system, (b) calculation of the mixed dipole moments, (c) calculation of the absorption of energy, and (d) comparison with the absorption due to the skin effect. In the second part of the paper, the influence of quadrupole moments and other electronic states are considered. (2) The wave-functions and their normalization As shown in part I (Dingle 1952a), in cylindrical co-ordinates the wave-functions and energy levels of the unperturbed system are,:~ for a large system (HR>~ 5 gauss cm),
~F = e-ur yiZe-tu L~+z(y),
e~H
E• - - ~ - (n + 89
(2.1) (2.2)
Paramagnetic resonance due to reversal of electron spins is also possible, but could be distinguished from diamagnetic resonance by its different dependence on the angle between the constant magnetic field and the incident radiation, and by its different frequency v=eH[2~moc, m o being the actua/mass of an electron. ~: There are also some sparsely occupied states for which E• = (ehH/mc)(n + 1+ 89 Their influence will be considered in w9. [3S]
208
Some magnetic properties of m e t a l .
III
39
where y = eHr~/2?ic and
i~+z(y) = ( - 1):((n+!)!}2M(-n,l+ 1,y),
(2-3)
n!l!
with
l
=
eHA/lw,.
(2.4)
Normalization of the wave-functions (2-1) requires that 1 =
where
* dr = N~ z =
f:
a , . - o j #-o
~F~F*r dr dq5 =
yle-v {L~a(y)}" dy =
e-Hdy-odd-o
f: O,,~a(y) e-v
XF~* dy dr a N, z, (2"5) '
L,+~(y) dy.
(2-6)
Here G,,+t(y)= yrL~.u(y ) is a polynomial of degree (n+/), with leading terms (ay '~+l+ byn+Z-t+ ...), say. On l successive integrations by parts
f: L,,+z(y)
N,,~ = ( - 1)z
[e-U(ay n+z+ by "+z-x +...)] dy
= f:e-uL,,+z(y){ay"+Z+by'~+z-Z-al(n+l)~+z-~+...}dy.
(2.7)
d Inca (yn~ae-u), e-u Ln~a(y) = /~dy]
But
(2.8)
so that after (n + l) further successive integrations by parts, (2.7) becomes
(-1)n+Z f:a(n+l)!y"+~e-Udy = (-1)n+:a{(n+l)!} ~.
(2.9)
Now a was defined as the coefficient of the leading term in ylL~a(y ). 'But L,+z(y ) = ev / d '~-+z(yn+~e-u) = ( - 1).+,(yn+Z_ (n + l)' yr,+z,1 + .. .},
kay/
(2.10)
So that
~/L~(y) - ~/
L.~(y)-
~/(-1)-+z (n + Z)! y- _ (n + l) (n +l)!y--1 + (n)! ( n - 1)!
(2-11) .
.
.
.
The coefficient of the leading term is thus
a = ( - 1)n+t(n+l)!/(n)!.
(2.12)
Substituting this value in (2-9), we obtain
, f o ~ ~ " (L~n+z(Y))"dY =
=
((n +l)!)a/(n)!.
(2.13)
Wave-functions with differing values of n are orthogonal, since they correspond to different energy levels. The same is not obviously true for differing values of l, for (2-2) shows that the energy is independent of l. If, however, i is replaced by - i in (2.1)--a change which will not affect the presence or absence of orthogonality--
209
40
R.B. Dingle
(2-2) is replaced by F~• -- (e~H/mc) (n + l + 89 which does depend on l; wave-functions with differing values of this quantum number are therefore orthogonM.
(3) 8elee,ti,on rules and moments for dipole radiation The probability of transitions between the states (n, l) and (n', l') is proportional to the square of the interaction energy. For the interaction between the electric field ~ of the perturbing radiation and the electric dipole moments of the electrons in their orbit~s around the magnetic field H directed along the z-axis, the transition probability (n, l)~,-~--(n',l') is proportional to
'r*~, F~re:~l~/ r d r d ~ /
9
(3.1)
By (2.1) this is se~n to be proportional to
e~ F~ _ /
-o
e~r-4)r
t..l,t
\u
/
1
(3-2)
The integral over the angle r in (3.2) vanishes unless l - l ' = _+1 for radiation with electric vector perpendicular to the z-axis, and l - l ' = 0 for radiation with electric vector along the z-axis. Since the wave-functions are orthogonal, (3-1) shows that there are no transitions between different states due to the component of the perturbing electric vector directed along the z-axis, i.e. parallel to the constant magnetic field. By (3.2), the mixed electric dipole moment~ for transitions (n,l),-~-(n',l-1) is proportional to
D'(n,l; n ' ,1 - 1) = [ ~fle-uL~n+z(y ) L,rz-x 3o
(3-3)
The simplest method evaluating this integral is to make use of the following recurrence relation between confluent hypergeometric functions (cf. Jahnke & Erode I945, p. 275):
aM(cz + 1, "/+ 1, y) = (~z- )i) M(~z, )I + 1, y) + yM(a, y, y).
(3-4)
Substituting from (2-3), this gives
z,~,+,_l = " l--1
l l t Ln" +t- I - Ln.+z/(n + l) .
(3.5)
Since wave-functions with different values of n are orthogonal, (3.3) then leads to
D'(n,l; n + 1 , / - 1) = N,.l,
D'(n,l; n , / - 1 ) = Nml/(n+l ).
(3.6)
It must be stressed that such moments are not related to the moments of individual orbits, since each wave-function with a single value of l corresponds to a whole group of orbits.
210
S o m e m a g n e t i c properties of metals. I I I
41
Now, y = eHlrr2/hc, so that the corresponding dipole moments are [ehc(n + 1)}t D(n, l; n + 1, l - ]) = \rrH/ (N,. l, 2~n+l,l_l) 89= t ..... ~ H
and'
D(n, l; n, l - 1) = ~-~-~] (n + l) (Nn,z,Nn,~_l)~ = (
7rH
(3.7)
"
(3-8)
All other electric dipole moments vanish. For the vast majority of occupied states l>> n (part I, w1), so that the second moment is generally by far the larger. The most likely transitions therefore are those between states with the same quantum number n--i.e, with the same quantized energy. Such transitions do not lead to absorption of energy. The magnetic dipole moment between two states is in the direction of the magnetic field and of magnitude e
J t ( n , l; n', l') oc ~--~ f t F ,.zP~ tF*, ~,r dr de. Since
p~
~F*n,z =
--i?~
~r ~F*,z, =
]~l'W.*,t, ,
(3.9) (3.10)
only the diagonal elements of (3-9) survive. The magnetic dipole moment therefore does not lead to transitions between different states. (4) Energy absorbed due to dipole interactions We have seen in w3 that there are only two electric dipole moments which do not vanish. One of these, D(n, l; n, l - 1), leads to transitions between states with the same quantized energy, and thus does not contribute towards the absorption. The other, D(n, l; n + 1, l - 1), leads to transitions between states differing in energy by hv = e~H/mc = fill,
(4.1)
say, so that the absorption frequency is just the classical frequency of a quasi-free electron in its orbit around the imposed constant magnetic field. If the electric vector of the radiation incident on the system is perpendicular to H, the transition probability between the two states (n, l) and (n + 1, l - 1) is 2zr 9., 47mc P(n; n + 1) = ~ D in; n + 1)I = - ~ f f ( n + l ) I ,
(4.2)
where I is the total energy in the incident radiation per unit volume per unit frequency range. The transition probability in the reverse direction is identica], but there is a net absorption because in thermal equilibrium there are more occupied states with quantum number n than with (n + 1). Since there are eH 2(2m)~ hc h El~-~ 4E n
211
42
Dingle
R.B.
states per unit volume with unquantized energy contributions between /~a and E~ + dE,, the net absorption of energy in unit volume per unit time is
x exp{E +,SH(n+ 89
=
-
+ 1-exp{.g+,SH(n-l-~)'.go}/IcT;i
m . - 0 : 0 exp {~ + f i B ( . + 89 Eo}lkT + 1 = - - W - '
where N is the number of electrons per unit volume. The resonance absorption lines will not be sharp, owing to natural and collision broaden~g. Let us suppose the width of the resonance line is Av as compared to the band width Af of the incident radiation. Normally A f ~ Av, and the absorption will be reduced by a factor Af/Av, so t h a t the total fraction of energy absorbed per
W
unit time is
4rrNe~
-ff = : ~ v
(4.4)
per unit volume, where U = I Afis the total energy density of the incident radiation. (5) The line breadth The contribution of dipole transitions to the natural breadth of an absorption line is equal to the Einstein coefficient of spontaneous emission: 64u 4 A~na t -- ~ Z v D 9,
(5" 1 )
where the sum is taken over all possible transitions. Substituting for D from (3.7), we have
32u9" e~ A2nat"
= - - - 3- ( n$7"t~ + l )2
(5.2)
(e~/mc~ is the so-called 'electron radius'). The natural width of the energy levels therefore increases roughly proportionally to the energy. AAnat. may be estimated by using the relation so that
nmax.~ Eo mc/eliH,
(5.3)
A2nat. ~ lOOeEo/~H~ lO-3/H cm.
(5.4)
The line breadth due to collisions will be given by AVco~ = AEcon./h~ I/v,
(5-5)
where r is the mean collision time. For usual field values, Aveon.>~Avnat., and (4.4) then gives at resonance W) 4rrNe~r (5.6) res.
m
(6) Comparison with the absorption due to the 8kin effect The real problem is to determine whether or not the absorption given by (5.6) could be detected in a metal amidst the unavoidable energy absorption due to the skin effect. If the frequency of the radiation is such that the current is related to
212
Some magnetic properties of metals. I I I
43
the electric field at the same point (which requires that the classical skin depth far exceeds the electronic mean-free-path), the energy absorbed per second in unit volume is W = 0-F~, where 0- is the conductivity and • the electric field. The total energy density is F ' / 4 , , so that for the skin effect (W) skin
- 4 . 0 " - 4"yezT-~ ~ '
(6"1)
which is identical with (5-6). This result means that at resonance the energy absorption due to diamagnetic resonance is equal to that due to the skin effect. In other words, the ~ factor of the system drops at resonance to 50 % of its normal value. Within the limits of validity of the assumptions made, this result is independent of both the constant applied magnetic field and the collision time. I t does not follow, however, t h a t the resonance will be perceived whatever the values of r and H, because unless there is a pronounced peak in the absorption curve the effect could not be ascribed to diamagnetic resonance rather than to anomalies in the magneto-resistance. Such a pronounced peak in the absorption will be seen only if the frequency spread due to collision broadening is much less than the resonance frequency itself, i.e. only if 1 >>Avcon./v For free electrons
1/TV
(6"2)t
-- )t/fir.
]t = mc2/eH = 1700/Hcm
(6-3)
and (6-2) reduces to the condition H >~5-67 • 10-S/r gauss.
:
(6.4)
For most metals at room temperature V~ 10-14 s, so that diamagnetic resonance will only be observed ff 2,~ 3 x 10-4 era; this requires that H >~5 x 10e gauss, an impossibly high field. At extremely low temperatures, where only the residual resistance remains, r might be increased to N 10 -xl s, requiring that 2,~ 0-3 cm and H >>5 x 10a gauss, a reasonable field but a very small wave-length. Since micro-waves of wave-length less than a centimetre are at present difficult to manipulate, it appears that diamagnetic resonance could in general be conveniently studied only if the collision time could be increased to a value greater than about 10-1~s, corresponding to a residual resistance of less than about 10-11 ohm cm. Alternatively, it might prove possible to use residual rays of wave-length ~ 10-~ cm (e.g. from the alkali halides), together with a field ~ 105 gauss, in which case we require only that r>~ 10-1~ S. I t is difficult to generalize the theory to the case of frequencies at which the skin effect is anomalous, because the variation of the electric field then becomes a very complicated function of the depth (cf. Reuter & Sondheimer I949, appendix III). A rough estimate of the effect m a y be made by noting that whilst the absorption due to diamagnetic resonance is proportional at each point to F*$', t h a t due to the skin effect is proportional to F ' J , where J is the current density. Since F falls off This condition a u t o m a t i c a l l y ensures also t h a t the amplitude of the r a d i a t i o n canno~ v a r y a p p r e c i a b l y in a distance c o m p a r a b l e to :the orbital radius measured along the direction
of H.
213
44
R . B . Dingle 0
with depth in a manner which is very roughly exponential, J , b e r g given by Maxwell's equations as d~F co~ 9 4row -~-~+~ ~ -~J, (6.5) varies with depth in a roughly similar manner. F / J is therefore approximately independent of z, and an idea of its average value may be obtained from that prevailing at the surface, i.e. from the surface imtmdance calculated by Reuter & Sondheimer. The relative absorption due to diamagnetic resonance is therefore increazed at high frequencies by just about the same ratio as the surface impedance is increased over and above its classical value. THE INleLUENCE O:F HIGHER MOMENTS AN~ OTHER STATES
(7) Eel.~cti~n ~
and moments for quadruTole radiation
The mi~ed electric quadrupole moment between the states (n, l) and (n', l') is, in cartesian co=ordinates u, v, z, (7.1)
ef ~,~F*.z.{U ~, v ~, z ~, uv, uz, vz} d V.
Since the wave-functions are independent of z, the z-component of the moment vanishes unless n' -- n and l' ffi l, so that there are no transitions ff the electric vector of the perturbing radiation is in the same direction as the s ~ y magnetic field causing the diamagnetism, just as for dipole interactions. The components of the quadrupole tensor which involve uz and vz are just z times the corresponding ones for the dipole vector. Taking u - r cos r and v - r sin ~, uSoc rS(e~ir + e-2S~ + 2),] /
(7.2)
V2(22 r2(e2t, -{-e-21r -- 2), f
Since the angular dependence of ~F~z~F*u.is etr ~'-~, the selection males of 1 for these components of the quadrupole radiation are l - l ' = + 2, 0. If l' = l - 2 , the mixed quadrupole moment is proportional to
fo
"
fO
*
L~,+z_~(y) dy
fo
= Q'(n,l; n',l-2).
(7.3)
A double application of (3.5) yields the relation z-2 L..+z_~
2
z
z
]
L..+z-2- n, + l - 1Ln'+z-1 + (n' + l - 1)(n' +/) L~,+z ,z
(7.4)
so that (7.3) leads to Q'(n,1;n+2,1-2)-Nnz,
'
Q'(n, 1 ; n + l l - 2 ) '
2Nn'z
=n+l'
Q'(n,l; n , l - 2 ) =
Nn.z (n+l)(n+l-1)"
(7-5)
214
S o m e magnetic proTerties of metals. I I I
45
Normalizing these results, the corresponding electric quadrupole moments are found to be he Q(n,l; n + 2 , / - 2 ) = ~-~{(n+ 1) (n+2)}t, (7.6) 2he Q(n,l; n + 1 , / - 2 ) = ~-~{(n+ 1) (n+/)}t, Q(n,l; n , l - 2 )
he = -~--~{(n+l)(n+l-1)}L
(7.7) (7-8)
Again the maximum moment is that for transitions between states with the same energy. If l' = l, the mixed quadrupolo moment is proportional to
(7.9)
This integral may be evaluated by maMng use of the following recurrence relation (cf. Jahnke & Erode I945, p. 275)a M ( ~ + 1,y,y ) -- ( y + 2 a - T ) M ( o : , y , y ) + ( y - a ) M ( a - l , v , y ) .
(7.10)
Substituting from (2.3), this becomes yL~,~q = (2n' + l + 1) L~,~ -- (n' + l)" L ~~' + ' - I - (kn' . n'+ +l +! i)-/ L'n,+z+l,
(7.11)
so that (7-9) leads to Q'(n,l; n,l) = ( 2 n + / + l) N,,,~, Q'(n,l; n + 1,/) - (n+Z+ 1)2Nn, z,
Q'(n,z; n - 1,/) = (Z-7"~,7) N-,r" (7.12) Normalizing these results, the corresponding electric quadrupole moments are found to be (7.13) Q(n, l; n, l) = hc (2n + l + 1), hc
Q(n,l; n + 1,/) = ~-~((n + 1 ) ( n + l + 1))t.
(7.14)
The value of Q(n, l; n - 1, l) derived from Q'(n, l; n - 1, l) is, of course, the same as (7-14) with n replaced by ( n - 1). Q(n, l; n, l) is just the permanent electric quadrupole moment of the state (n,/), i.e. it is er ~. There is no such corresponding permanent electric dipole moment, since D(n,l; n, l) = O. (8) Energy absorbed due to quadrupole interactions Of the five different quadrupole moments calculated in the last section, those given by (7.8) and (7.13) do not interest us, since they do not lead to transitions between different states. Of the remainder, (7-7) and (7.14) are much more
215
46
Dingle
R.B.
important than (7-6), because for practically all occupied states l>>n. The resonance frequency is therefore again given by (4-1), and we may take h,.c~ Q*(n,n+ 1) ~n.z//~ (n + 1)/.
(8-1)
If the electric vector lies in the plane perpendicular to the field H, the interaction energy of the quadmapole is 21r]Atimes that of the dipole, where Xis the wave-length ofthe incident radiation. C o m p a ~ with the re~lt (4.3), wesee that the absorption of energy due to such quadrupole interactions is, per unit volume and time, 16rr~Ne helI ~
S~-ce
mHA~
~ = ~==
(8-2)
.
ZdZ = 2he'
the average absorption due to quadrupoles would appear to be m
'
(8-4)
where/~ is the linear dimension in the plane pertmndicular to the imposed constant magnetic field. The corresponding absorption Would be about (R/A) 9 times that given by (4.4). Owing to quad_rupole transitionS, the actual line breadth given by (5-2) is also increased by a similar term, (R]A) ~ times the dipole term. As shown by (8-2), these quadrupole effects are largely due to wave-functions with large values of l. These will be much perturbed by collisions--we have seen (part I_I, (5.6)) that the transition probability is in fact roughly proportional to l. In practice, therefore, the quadrupole contributions to the absorption will probably be less than these calculated values. (9)
The influence o/ states for which g~. = (e~B/mc,) (n + l + 89
The dipole moment .D(n,l; n , l - I) given by (3.8)is so much larger than that of (3-7) that it might at first sight be expected to compensate for the smallness of the fraction of occupied states for which such transitions would lead to an energy change. A similar argument to that of w4 shows that the net absorption of energy in unit volume per unit time would be 2(2m)lei~H4rrecI 2,o Zm=x. N ( n + l + 1) f / E-89 hA mc ~H ,~=o z=o
{ 1 • exp(E+flH(n+l+ 89 2(2m)~ ehH 4rrecI hA mc hH
1 1-exp{E+flH(n+l+~)
=o 'z=ofo ~exp +Zn n:o
+ + 89 Eo}/kT
} (9-1) Eo}/kTq- 1
1
exp {E +/3g(n + 89 Eo}/kT + 1 "
(9.2)
216
Some magnetic properties of metals. I I I
47
The ratio of the first term of (9-2) to (4.3) is just the same as the ratio of the total numbers of electrons in each of the types of states; as we have already seen (part I, w1), this ratio is negligible. The ratio of the second term of (9-2) to (4.3) is of the order of nmax./Zmax_, which is again negligible. These states therefore give rise to a quite insignificant additional absorption. I am grateful to D r D. Shoenberg for drawing m y attention to the possibility of the effect discussed in this paper.
REFERENCES Dingle, R.B. I952a Prec. Roy. See. A, 211,500 (part I). Dingle, R . B . I952b Prec. Roy. See. A, 211,517 (part II). Jahnke, E. & Erode, F. I945 Tables of functions, 4th ed. New York: Dover. Reuter, G. E. H. & Sondheimer, E . H . x949 Prec. Roy. Soc. A, 195, 336.
217
VOLUME 62, NUMBER 18
PHYSICAL
REVIEW
LETTERS
1 MAY 1989
S p e c t r o s c o p y o f E l e c t r o n i c S t a t e s in I n S b Q u a n t u m D o t s Ch. Sikorski and U. Merkt institut fiir Angewandte Physik, Unioersit~t Hamburg, D-2000 Hamburg 36, West Germany (Received 22 December 1988) We have realized arrays of quantum dots on InSb and observe intraband transitions between their discrete (zero-dimensignal) electronic states with far-infrared magnetospectroscopy. In our devices, the number of electrons can be adjusted by a gate voltage and less than five electrons per dot are detectable. PACS numbers: 73.20.Dx,72.15.Rn, 78.30.Fs Progress of nanofabrication technology now renders it possible to laterally confine electrons on semiconductors to quantum wires and quantum dots. s Both systems take advantage of quasi-two-dimensional (2D) electron gases present in heterostruetures, quantum wells, or metaloxide-semiconductor devices. 2 Quantum wires are obtained by confinement in one of the two directions in the plane of the 2D gas. Consequently, the electrons can move freely only along the remaining direction and thus constitute a 1D gas of a quantum wire. Confinement in both directions results in a 0D gas of a quantum dot. Zero-dimensional electronic behavior recently has been demonstrated unambiguously by resonant tunneling 3 through laterally constricted lnGaAs quantum wells and by capacitance oscillations 4 of microstructured GaAs/GaAIAs heterojunctions. Here, we report the direct observation of resonance transitions between discrete states of quantum dots on lnSb. The particular advantage of this narrow-gap semiconductor is its small effective electron mass m* -0.014m,, at the conductionband edge. It gives rise to comparatively high quantization energies -~ 10 meV for electronically active widths of typically 100 nm which can be achieved laterally with present semiconductor technology. 5.6 The application of a magnetic field to quantum dots offers interesting possibilities to study few-electron systems. Since the cyclotron energy h toc readily can be made much larger than the binding energy of the confining electric potential, we can examine the transition from electrically bound states to Landau-type magnetic levels in this system. In real atoms, observation of transitions between magnetic-type levels, e.g., quasiLandau resonances, only is feasible when the electrons are excited to high Rydberg states 7 or when the atoms are exposed to megatesla fields 8 present near pulsars. More closely, our system is related to the one of shallow donors in semiconductors. 9 In contrast to donor atoms, however, we can adjust not only the size of our dots but also their electron number. Arrays of "-'10 8 dots are prepared on p-type lnSb (I 1 I) surfaces of typical areas 3x3 mm2. Samples covered with photoresist are exposed twice in a holographic setup which employs an argon laser (~.-458 nm) whose expanded beam is split into two partial 2164
beams. The partial beams interfere near the sample and thus create a periodic intensity pattern of grating constant a - 250 nm in the photoresist. After the first exposure, the sample is rotated by an angle of 90* and exposure is repeated. Subsequent to development the samples are etched in an oxygen plasma. This removes resist residues between the dots and reduces the dot heights to values below 50 rim. We then evaporate a NiCr film which acts as a Schottky depletion gate s at the NiCr/ I n S b interface between the dots; i.e., there we pin the Fermi energy EF within the lnSb band gap. A monitor sample metallized with Au is shown in Fig. 1 together with a schematic sketch of the band structure across the
FIG. !. Scanning electron micrograph of resist dots, with a 125 nm marker, together with a schematical sketch of the band structure across the dots right at the lnSb surface. The bright disks give an idea of the geometrical dot size. This monitor sample is shadowed with gold for contrast enhancement.
9 1989 The American Physical Society
218
VOLUME62, NUMBER 18
PHYSICAL
REVIEW
dots. After deposition of = 4 0 0 nm SiO2, we evaporate a second NiCr film as a gate contact. When a gate voltage Vs is applied between this contact and the InSb substrate, the number of mobile inversion electrons under the resist dots can be controlled by the field effect due to the finite resistivity (R ~--1 M f~) of the InSb substrate. The device becomes completely depleted of mobile electrons at the threshold voltage V, - --98 V. Spectroscopy is carried out with an optically pumped far-infrared laser at liquid-helium temperature. The light impinges perpendicularly onto the sample and, hence, is polarized parallel to the surface. The relative change of transmittance t - - [ T ( V , ) - T ( V I ) ] / T ( V , ) is recorded versus the strength B of a magnetic field applied perpendicular to the surface. Spectra for various laser energies hca and gate voltages A Vs - V t - V t are shown in Fig. 2 for linearly polarized light. However, the spectra are almost independent of the polarization direction in the plane as is expected by virtue of sample preparation. Spectra for the energy h c a - 10.4 meV resemble cyclotron resonances of a homogeneous 2D inversion layer but the resonance magnetic fields are already shifted considerably (AB-'0.4 T) to lower field strengths. I~ This directly reflects the additional spatial quantization in the I
-
I
6Vg( V ~ / ~
I
I~.
LETTERS
1 MAY 1989
confining lateral potential. For energy h c a - 7 . 6 meV, we no longer observe a distinct resonance maximum at finite fields but a monotonic decrease of the relative transmittance when the magnetic field is increased. We will show below that this is indeed expected when the characteristic quantization energy of the lateral potential approximately coincides with the laser energy. For the energy h c a - 3 . 2 meV, we again observe distinct but weak resonances at B--1.5 T. As we show next, these resonances are characteristic of a system which is confined in both lateral dimensions. To obtain a simple description of electrons in quantum dots, we consider the harmonic-oscillator potential ~-m* xca~(x2+y 2) with eigenfrequency cao in a magnetic field directed along the z direction, tl This parabolic model is expected to be a good approximation for lowelectron numbers, t2 The single-electron eigenenergies of the lateral motion,
E.. - 1 2 , + I,.I + l)n [(~d2)~+.=~J '12+(hcacl2)m,
(1) depend on the radial n - 0 , 1 . . . . and azimuthal m - 0 , __ I . . . . quantum number. At low-electron numbers, only the lowest 2D subband is occupied, t0 Figure 3 depicts the lowest energies versus magnetic field strength and the allowed dipole transitions which have resonance frequencies
hta=10.~,meV
co__. -[(cad2)2+ca~] I/2+_ ~ c l 2 ,
I 0
I
eL3 ~
~Vg(V)
I
I
i
,
'~I /o,/
(a) !, hu= 7.6meV
(2)
,.
X
~ 0.2
-'
p-InSb 11111 T=~K
0.1 0 0.3--
> I 6Vg(V)
O.S
f ' ~
1.0
BIT)
w.
2.0
2.S
-
i~ ~
~ 10
FIG. 2. Far-infrared spectra for three laser frequencies co and three gate voltages AVt. (a) co+ resonances at B'-I.0 T for a laser frequency above the quantization frequency oJo, (b) traces for oJ--oJo, and (c) ~ - resonances for B--'I.5 T for ~ < ~oo.
I "~"
(b) I ~d= 3.2 meV
o.2 0
-- 30~11
0 0
~
I 1.0
.j(O.O)
-
l B(T)
~ % : 7.5 meV m* = 0.01/,m, .I . . . . . . 3.0 /,.0
FIG. 3. Calculated level diagram of the oscillator potential a magnetic field B IIz. Levels are indicated by their quantum numbers (n,m) with ~ - - m . For some initial states, transitions r allowed for two circular light polarizations, respectively, are marked by arrows.
~m*~(x2+y 2) in
2165
219
VOLUME62, NUMBER 18
PHYSICAL
REVIEW
and which are excited with circular light polarizations _ , respectively. At B - - 0 we have oscillator levels (n' + 1 ) h too, with the abbreviation n ' - 2n + I m I. In high magnetic fields (cac>> ca0), all levels with quantum numbers n - 0 , m __<0 converge and form a highly degenerate ground state of energy ~-hcac. Levels n - 1, m __<0 converge to a common excited state ~ ha}. States with angular momentum m > 0 have much higher energies and 9do not contribute to the signals since they are no longer occupied. In this high magnetic field limit, transitions ~o+ become cyclotron resonances between Landau levels hcac(n + af ) and the electron gas exhibits 2D behavior. Simultaneously, the oscillator strength of transitions oJvanishes. Therefore, these resonances are not observed in strong magnetic fields. To be more specific, they are characteristic of a confined system with a radius comparable to or less than the cyclotron radius I - ( h / e B ) ]/2. To model our line shapes, we first calculate classical conductivities a +_(ca)
(3)
en~ 1.4- [(ca(;t/ca) -- ca "t- cac]2r 2
for circular light polarizations from the classical equation of motion with a phenomenological relaxation time r and a m o b i l i t y / ~ - e r / m * . We then take into account the quantum-mechanical oscillator strengths, l i The relation between conductivity and relative change of transmittance
l
t~
2cr(ca)/Yo
(4)
a 2 1 + , f 6 + crJYo is adopted from its 2D counterpart t3 with InSb dielectric constant e = 17, wave admittance Yo"(377 1~) -1, and effective sheet conductivity cro- (18 f~) - l of the two NiCr films. The conductivity c r " ( c a + c r + + c a - c r - ) / (ca+ + c a - ) for linearly polarized light incorporates the oscillator strengths and proves to be nearly independent of level occupation. Equation (4) qualitatively describes the observed line shapes. In particular, it explains cyclotronlike ca+ resonances at frequencies ca > coo, monotonic intensity decreases proportional to (1+/~2B2) -I at ca-"cao, and weak intensities of ca- resonances at ca < cao.
Most important, it allows us to determine the average number no of electrons in a dot. This is most readily
I MAY 1989
LETTERS
done with spectra taken at frequencies cameo0 where we have the relation t ( B - - 0 ) ~ 2enop/r 2 for conductivities cro:>>Y~ Mobilities/~--(BI/2) - ! ~ 20000 cm2V - I s - i are obtained from the fields B i/2 where the transmittances have dropped to half of their maximum values at B--0. Electron numbers no, quantum numbers n ' - 2 n +[ml of the highest populated B--O level, and electronic dot radii r r - [ 2 t t ( n ' + l)]m*~oo] I/2 at the Fermi energies E r ~ 15, 22.5, and 30 meV are summarized in Table I for the gate voltages of Fig. 2. Almost the same electron numbers no are obtained when cyclotronlike resonances for a much higher energy hco-26.6 meV are fitted with theoretical line shapes Io of 2D cyclotron resonance. Difference between numbers obtained for frequencies ~==e)e (lateral electric oscillator) and e):~ e)o (Landau oscillator) arc given as experimental uncertainties in Table I. For reasons =we do not yet understand in
detail, the number of electrons saturates at. voltages above AVg - 2 0 V. Experimental resonance positions for gate voltage A V s - 8 V are given in Fig. 4 together with theoretical curves calculated from Eq. (2). At the highest energy (hco~--hoJc>>hcoo) there is a shift AB--0.8 T between the experimental and theoretical results. This shift is almost quantitatively explained by the influence of band nonparabolicity t4 which at the lower energies is less important. For lower energies, Eq. (2) provides a qualitative description and we can estimate the quantization energy hco0-7.5_.+ 1 meV. This value agrees with the one which we already deduced from the shape of the h c a - 7 . 6 meV spectra in Fig. 2. Within experimental error, the quantization energy does not depend on electron number in the range n 0 - 3 to 20. This provides strong evidence that collective depolarization modes which might be expected to become important at higher electron numbers 15'=6 are strongly suppressed in our devices. In fact, macroscopic electric fields are effectively screened by the NiCr Schottky gate since it is evaporated in very close vicinity to the electron ............. I. . . . . . . . . . I ..... p- ]n~3 (111) 3 0 - T= Z,K ~v,= 8v fheory /
I'
/ /
-
. -
TABLE I. Average electron number no per dot, quantum number n' of the highest populated ( B - 0 ) oscillator level, and electronically active dot radius re. Values are given for three gate voltages AVs -- Vs -- I/I. AVt (V)
n0
,'
rF (nm)
3 8
3--+! 9+_ 1 20 4" 2
1 2
54 66 76
!8
2166
3
0
1.0
B IT }
30
/,.0
FIG. 4. Resonance positions for gate voltage AFs=8 V. The solid lines are calculated from Eq. (2) for parameters hwo-?.5 meV and m* -O.014m,.
220
VOLUME 62, NUMBER 18
PHYSICAL
REVIEW
systems. 6 For the same reason, we do not expect electromagnetic coupling between dots. To conclude, we directly detect intraband transitions between discrete states of quantum dots on lnSb. The number of electrons no per dot can be controlled and switched by a gate voltage. We determine quantization energies of about 7 meV for zero magnetic fields and deduce electron numbers n o - 3 to 20. This means that we approach an ultimate limit set to the miniaturization of electronically active semiconductor devices, namely one electron per dot. We thank J. P. Kotthaus for valuable discussions and acknowledge financial support of the Deutsche Forschungsgemeinschaft and the Stiftung Volkswagenwerk.
t Physics and Technology of Submicron Structures, edited by H. Heinrich, G. Bauer, and F. Kuchar, Springer Series in Solid-State Sciences Vol. 83 (Springer-Verlag, Berlin, 1988). 2T. Ando, A. B. Fowler, and F. Stern, Rev. Mod. Phys. 54, 437 (1982). 3M. A. Reed, J. N. Randall, R. J. Aggarwal, R. J. Matyi, T. M. Moore, and A. E. Wetsel, Phys. Rev. Lett. 60, 535 (1988). 4T. P. Smith, Ill, K. Y. Lee, C. M. Knoedler, J. M. Hong,
LETTERS
1 MAY 1989
and D. P. Kern, Phys. Rev. B 38, 2172 (1988). 5W. Hansen, M. Horst, J. P. Kotthaus, U. Merkt, Ch. Sikorski, and K. Ploog, Phys. Rev. Lctt. 58, 2586 (1987). 6j. Alsmcicr, Ch. Sikorski, and U. Merkt, Phys. Rev. B 37, 4314 (1988). 7j. Neukammer, H. Rinnebcrg, K. Vietzke, A. K6nig, H. Hieronymus, M. Kohl, H. J. Grabka, and G. Wunncr, Phys. Rev. Lctt. 59, 2947 (I 987). 8H. Ruder, H. Herold, W. R~ner, and G. Wunner, Physica (Amsterdam) 127B, 11 (1984). 9y. Yafet, R. W. Keyes, and E. N. Adams, J. Phys. Chem. Solids 1, ! 37 (I 956). I~ Merkt, M. Horst, T. Evelbauer, and J. P. Kotthaus, Phys. Rev. B 34, 7234 (1986). t tR. B. Dingle, Proc. Roy. SOc. London, Scr. A 211, 500 (1952); 212, 38 (1952). This treatment of the Landau quantization in the symmetric gauge allows us to include the lateral oscillator potential in an easy way. 12S. E. Laux, D. J. Frank, and F. Stern, Surf. Sci. 196, i01 (1988). 13S. J. Allen, Jr., D. C. ~'sui, and F. DeRosa, Phys. Rev. Lett. 35, 1359 (1975). 14U. Merkt and S. Oetling, Phys. Rev. B 35, 2460 (1987). 15S. J. Allen, Jr., H. L. St6rmer, and J. C. M. Hwang, Phys. Rev. B 28, 4875 (1983). 16W. Que and G. Kirezenow, Phys. Rev. B 38, 3614 (1988).
2167
221
VOLUME64, NUMBER 7
PHYSICAL
REVIEW
LETTERS
12 FEBRUARY 1990
N o n l o c a l D y n a m i c R e s p o n s e a n d Level C r o s s i n g s in Q u a n t u m - D o t S t r u c t u r e s T. Demel, D. Heitmann, P. Grambow, and IC Ploog
Max-Planck-lnstitut f~r Festk~rperfor$chung. HeLsenber&$wasse1. D-7000 Stuttgart 80. Federal Republic of Germany 0t.eceived 21 August 1989) Very small quantum-dot structures containing 210 to 25 electrons per dot have been prepared starting from modulation-doped AIGaAs/GaAs hetereetructures. The far-infrared response c~ashts of a set of resonances which split, in a magnetic field B, into brsachet with negative and positive B dispersion. The inte~ of these ~ n c e s , in c l ~ analogy edge magnetopinsmons, leads to an anticrmsing of the disper~hms. This resonant coupling is induced by nonlocal interaction which becomr important at small dimensions. PA(~ n~mbcrg 73.20.D~,72. I&ga, 73.20.Mf Currently there is an increasing interest in the investigation of ultrasmall, laterally microstructurcd, originally two-dimensional electronic systems (2DES). Because of the reduced dimensionality, quantum confineJnent and novel transport phenomena are observecL t-6 One ultimate goal is the realization of artificial atoms in quantum-dot structures. We have prepared quantumdot structures containing 210 to 25 electrons per dot in AIGaAs/GaAs heterostructures by deep-mesa-etching techniques, v T h e dynamic far-infrared (FIR) response shows a set of resonance absorption peaks at frefluencies moi which split with increasing magnetic field B into two branches, a~:+(B) and oJl-(B), with positive and negative B dispersion, respectively (i--1,2 .... ). The dispersion is similar to that observed for excitations in InSb quantum-dot structures 6 and for edge magnetoplasmons in wider (radius R--1.5 ~m) GaAs structures s and finite-sized 2DES on liquid He surfaces (Refs. 9 and 10; for a recent review, see, e.g., Ref. I 1). However, in our systems we can for the first time observe higher-order branches ( i - - 2 ) in ultrasmall dimensions ( R ~ 10a~, a~ the effective Bohr radius). For these small structures we find, very surprisingly, a resonant anticrossing of the m t + and the to2- branches. We show that this anticrossing arises from nonlecal interaction. We find that the coupling is similar in strength, but very different in the a~ vs B dependence, as compared to resonant nonlocal interaction in infinite 2DES. t2 The nonlocai interaction is governed by the parameter a~q (q--fiR is the wave vector, see below). Thus the smallness of our structures makes it possible to observe the interaction which is, as such, an inherent property of quantum-dot structures. For a system with a very small number of electrons, which behaves more single-particle-like, we expect that the observed anticrossing corresponds to transitions of anticrossing single-particle energy states in a magnetic field, i.e., the bifurcation that has been observed indirectly in magnetocapacitance measurements on quantum-dot structures. 5 The anticrossing that we observe is a demonstration of electron-electron interaction in the artificial atom in the 788
sense that it is P~nsitive to a fine structure in the excitation spectrum beyond non-self-consistent "empty-atom" models, e.g~, Ref. 13. The +samples were prepared by starting from modulation-doped AIGaAs/GaAs heterostructures. An array of photoresist dots (with a period of 1000 nm both in the x and in +y directions) was prepared by a holographic double exposure. Using an anisotropic plasmaetching process rectangular 200-nm-dcep grooves were etched all the way through the 10-nm-thick GaAs cap layer, the 53-nm-thick Si-doped AIGaAs layer, and the 23-nm-thick undoped AIGaAs spacer layer into the active GaAs, leaving quadratic dots with rounded corners and geometrical dimensions of about 600 by 600 nm (see inset of Fig. 1). With this technique of "deep mesa etching" it was possible to realize in linear stripe systems IDES with typical energy separation for the I D subbands of about 2 meV. TM It was f o ~ d that the actual width of the electron channels was smaller than the geometrical width, indicating a lateral edge depletion of 100 to 120 nm. For the dot structures here, with increased etched surface area, this depletion is even more pronounced. Actually, we have prepared samples which had, in the dark, no mobile electrons. Via the persistent photoeffect we could then increase the number of electrons in steps up to 210 per dot. Since lateral transport I is inherently not possible in dot structures, we used the strength of the FIR absorption to determine the number of electrons per dot, N. In particular we used the Drude-type model for bound electrons in Ref. 15, formula (6). The potential that confines the electrons and thus determines the radius of the 2D disk, depends on the remote ionized donors and, in a self-consistent way, on IV. We have estimated the radius from the observed resonance frequency and formula (I) which will be explained below. The FIR experiments were performed in a superconducting magnet cryostat, which was connected via a waveguide system to a Fourier-transform spectrometer. The transmission T(B) of unpolarized FIR radiation through the sample was measured at fixed magnetic
O 1990 The American Physical Society
222
VOLUME fi4, NUMBER 7
PHYSICAL
REVIEW
12 FEBRUARY 1990
LETTERS
(a)
2OO 1oo
~"1-
I
u l + "~
150 A ,,4
,oo
N ' - 210
R - 1(10rim 5O
~;'
N = Zl.O
it m llNksm
leO
0
r""-
200
i
i
l
i
i
i
i
i
J
i
i
|
i
i
i
(b)
150
| m4
0
50
I00
V.Lvenumben
150
200
(era-t)
FIG. 1. Relative transmission of unpolarized FIR radiation for a quantum-dot structure with radius R--160 nm and H--210 electrons per dot. In a magnetic field B > 0 the resonance q)lits into two branches, mt+ and mr-, with positive and negative B ~ respectively. For B > 4T an additional r~_sonance, tea+, is observed. At ran-40 a n - ' ( t ) there is a prommnced anticrmsiag intersection. Inset: The dot structare shown schematically. The actual dots have rounded corners. ES dcnotcs the confined electron system. fields, B, oriented normally to the surface of the sample. The spectra were normalized to a spectrum T(B0) with a fiat response. The resolution of the spectrometer was set to 0.5 cm - t . The temperature was 2.2 IC The measured sample area was 3 x 2 mm2;, thus we measure 107 dots. Experimental spectra for a sample with a dot radius R - - 1 6 0 nm and H - - 2 1 0 electrons per dot are shown in Fig. 1. For B - - 0 one resonance is observed at m0--32 c m - t . With increasing B the resonance splits into two resonances; one, m t - , decreases in frequency, the other oh+, increases. For B > 4 T a second resonance, m2+, can be resolved which also increases with B. The most interesting observation is the obvious resonant coupling which occurs at a frequency of about 40 c m - l. The experimental resonance positions for this sample are depicted in Fig. 2(a). Figure 2(b) shows the resonance positions from a sample containing only 25 electrons per dot. The dip in T(B)/T(Bo) is only 0.4% which is the experimental limit for reliably determining position and signal strengths for the current spacings of the dots. Here we cannot clearly resolve the m l - branch because of the limit.ed sensitivity of our spectrometer at small frequencies. Similar dispersions as in Fig. 2(a) were measured on a series of samples with slightly different values of R, N, and corresponding m0. The interesting observation is the significant resonant anticrossing at m ~ 1.4m~
i
,00
N--25
R m 100rim 50
0
2
4
6
8
I0
12
14
16
I/agneUe Field B (T) FIG. 2. Experimental B dispersion of resonant absorption in quantum-dot structures with (a) R -- 160 nm and N - 210 and (b) R--100 nm and N--25. The full lines are fits with the theoretical dispersion [Eq. (2)]. Both structures show an anticrosaing of the m2- with the mt+ mode, which is caused by nonlocal interaction.
which was found for all our samples. In the following we will discuss that this anticrossing is caused by noalocal interaction. For our discussion we first adopt a description for a classical 2DES of finite size and discuss effects of q u a n t u m confinement later. The F I R resonances observed here, are, except for the resonant splitting, very similar to earlier observations on larger, finite-sized 2DES in G a P s (dots with R - - 1 . 5 pro) s and electrons-on-liquid-He systems (R ~ 1 cm) 9"10. These resonances can be explained either in an edge-magnetoplasmon model 9-11 or, at least at small B, equivalently as a depolarization resonance, s A simple way to describe the F I R response of a finite-sized 2DES is to start f r o m linear edge magnetoplasmons (e.g., see Refs. 11, 16, and 17) which have the dispersion OJ~p --0.81m2(q), where m~(q)--Nse2q/2m*eoew is the 2D plasmon frequency. The circumference of the disk quantizes the q vectors in values q - - i / R ( / - - 1 , 2 .... ). Thus that at B - - 0 is a ~ --0.I 8N:e 2i/2m * eoe~R .
(1)
In a magnetic field one calculates a set of double 789
223
VOLUME 64, NUMBER 7
PHYSICALREVIEW
branches (e.g., see Refs. g and 10), OJI'l" " [ a i l ~ t @at2 (~c/2 ) 2] I/2 "4"ai3oJr
,
(2)
with al: --at2 ==ai3==1 (co is the cyclotron frequency). These models already agree very well with more sophisticated theories which determine more accurately the coefficients air ( k - - 1 , 2 , 3 ) of e ~ and ~t. The latter and the spacings of higher modes depend slightly on the exact modeling of the 3D density profile for the electron distribution. We have fitted in Fig. 2(a) the dispersion with formula (2), using a2t as a fitting parameter and all other air -- I, and find, taking account of the nonparabolicity of m* in Cut,As, a very good agreement with the experimental dispersion. In particular we find me2 ~ 1.5a~0:, which is very close to the simple-model value of ~o2--,~met. From this fit it becomes clear that the splitting is caused by an anticrossing of the ~02- and wt+ modes. The interesting question is which interaction causes this splitting and what d~termines its strength. The fact that this splitting is not observed on larger GaAs systems with a very similar shape, s in particular, not on the liquid-He systems of Ref. 9, where beautifully sharp intersecting edge magnetoplasmons were found with, however, no interaction at all, leads to the conclusion that the smallness of our structures is the important parameter. (For large R and thus small q, o~+ and mr-, represent, respectively, left- and right-circular-polarized eigenmodes, which are as such decoupled.) This automatically draws one's attention to the nonloca! interaction. Nonlocal effects are well known for the homogeneous 2DES. They arise from the inherent finite compressibility of the Fermi gas and lead to corrections q2v~ (vF--Fermi velocity) for the squared plasmon frequency co~(q). 12,:7 These effects are very small under usual experimental conditions. However, they can be clearly observed in a resonant-magneticfield experiment, where the nonlocal interaction leads to an anticrossing with 2coo.:2 That is, the "'local" magnetoplasmon dispersion, ~02mp--oJ~0+~0e, 2 2 splits into two branches. One branch starts at B - - 0 with the local dispersion at copo and with increasing B approaches asymptotically 2we. The other branch starts at o~--0 asymptotically with 2coc and with incre~ing B approaches the local dispersion. Thus, the frequency dependence of the nonlocal interaction in 2D is very different from the dot systems shown in Fig. 2. There is so far no theoretical treatment of resonant nonlocal interaction in 0D system. Only nonresonant nonlocal effects on the linear 1D edge-magnetoplasmon dispersion have been considered until now. 17 Nevertheless it is interesting to compare the 2D nonlocal interaction strength with the splitting that is observed here, in particular, since the splitting occurs at small B, and thus one is not too far away from a 2D-plasmon-type behavior. The strength of the nonlocal resonant coupling in 2D, measured in terms of the frequency splitting, t2 is 790
LETTERS
12 FEBRUARY 1990
Aco/oJ0--2.6~" (q in units of n m - t ) . If we use for the interaction here the same model and for the eJ2- branch q - 2 / R , we find for the sample of Fig. 2(b) with R - 100 nm Am/me--0.37 which agrees surprisingly well with the experimental value of 0.33. This very close agreement might be to a certain degree accidental and should be compared with a so far not available rigorous theory. In particular, we expect that the nonlocal interaction also depends on the exact three-dimensional density profile of the electron system. However, this close agreement demonstrates that the experimentally observed splitting is of the expected order for nonlor interaction. Moreover, independently of the absolute value we expect within our simple analogy Am/me ~ (l/R)!/2. If we scale the experimental splitting of 0.33 for the sample in Fig. 2(b) ( R t - - 1 0 0 rim) with ( R : / R z ) !/2 we find 0.26, which agrees very well with the experimental splitting of 0.25 for the sample in Fig. 2(a) (R2"-160 nm). (For all our samples with R ranging from 100 to 180 nm we found that the experimmtal splitting agrees within a factor of 0.85 to 1.0 with AodoJo--2.6,~'.) Thus also the dimensionality dependence confirms that the interaction is caused by nonlocality. The question arises whether deviations from a circular shape might influence the splitting. We do not believe that this is a significant effect: (i) Such effects are independent of the dimensions and should thus also be present in the experiments o f Ref. 8 where samples of a very similar shape show no splitting. (ii) For our samples one would expect that a splitting due to the geometrical shape effect should be especially pronounced at large N and R when the electronic system extends closer to the geometrical edge. This is in contrast to the experimental observation. The smallness of the structures, needed to make nonlocal effects essential, leads inherently to the regime of quantum confinement. Concerning our structures we note that linear stripes with the same dimensions, prepared with the same techniques as used here, show a quantum-confined I D energy spectrum with a typical subband separation of about h ~ 0 - - 2 meV. 7 Therefore we believe that, in particular, in our sample with only 25 electrons per dot, the electrons occupy discrete quantized energy levels. In particular, in a two-dimensional harmonic-oscillator potential, :3 V ( x , y ) - ~ m* ft~(x 2 + y 2), only five discrete energy levels are occupied. It is well known (e.g., see Refs. 14 and 18) that the level spacing h fl0 is not directly observed in a FIR experiment, but rather the observed resonance frequency oJr is shifted, cot2 - - f t ~ + r 2, where cop characterizes the collective depolarization effect which increases with increasing N. For the sample with N - - 2 1 0 the resonance frequency h m , - 4 meV is strongly governed by the depolarization effect. Thus here a classical, plasmonlike resonance behavior is the adequate description of the FIR response. However, for the sample with N - - 2 5 electrons per dot we have estimated h rio--2 meV; thus OJp is about equal
224
VOLUME 64, NUMBER7
PHYSICAL
REVIEW
to or smaller than the one-part/cle energy separation/~0. Thus here the FIR response should reflect strongly the single-particle aspects of the dot structure. The collective effects make it di/~cult to determine exactly the level spacing of a 0D system from FIR spectroscopy. In particular, from the experimental B dispersion only, one cannot distinguish between a classical collective plasmon type of response [Ref. 8 or formula (2) here] and 0D-level transitions, e.g., in a harmonicoscillator confinement. 6'~3 With a theoretical modeling of the splitting that we observe here, one can perhaps gain insight into the energy s t n ~ u r e s of the dots. As was pointed our for linear systems ti the edge magnetoplasmons correspond in a quantum-confined system to transitions between the discrete energy state~ As such, in the limit of quantum confinement, and for the N - 2 5 sample with ~ep < / ~ we approach this limit, the observed splitting corresponding to transitions between anticrouing one-particle energy levels in a magnetic field. In conclusion, in the FIR response of quantum-dot structures containing 25 to 210 electrons per dot a resonant antiorossing of edge-magnetoplasmon-type excitations is observed. The interaction arises from the nonlocality which becomes important for the very small dimensions. The interaction is, compared to 2DES, similar in strength, but very different in its ~ vs B dependence.
IIC-F. Berggren, T. J. Thornton, D. J. Newson, and M. Pepper, Phys. Rev. Lett. $7, 1769 (1986).
LETTERS
12 FEBRUARY 1990
2B. J. van Wees, H. van Houten, C. W. J. Beenakker, J. O. WiIliamson, L. P. Kouwr D. van der Marel, and C. T. Foxon, Phys. Rev. Left. 60, 848 (1988). 3M. A. Reed, J. N. Randall, R. J. Aggarwal, R. J. Matyi, T. M. Moore, and A. E. Wetscl, Phys. Rev. L~t. 60, 535 (1988). 4T. P. Smith, llI, K. Y. Lee, C. M. Kno~lr J. M. Hon~ and D. P. Kern, Phys. Rev. B 38, 2172 (1988). sW. Han=en, T. P. Smith, ill, K. Y. Lee, J. A. Brum, C. M. Knocdler, J. M. Heng, and D. P. Kern, Phys. Rev. Left. 62, 2168 (1989). 6Ch. Sikorski and U. Merkt, Phys. Rev. LetL 62, 2164 (1989). 7T. Demel, D. Heitmann, P. Orambow, and K. Ploog, Appk Phys. Lett. $3, 2176 (1988). SS. J. Allen, Jr., H. L. StSrmer, and J. C. Hwan& Phys. Rev. B 28, 4875 (1983). 9D. C. Glattli, F.. Y. Andrei, G. Deville, J. Poitrenaud, and F. L B. William& Phys. P,.ev. Lett. 54, 1710 (1985). I~ B. Mast, A. J. Dahm, and A. L. Fetter, Phys. Rev. Lett. 54, 1706 (1985). "v. B. Sandomln~ V. & Von~oV, O;:R. ~ and S. A. Mikhailov,Electrochim.Acta 34, 3 (1989). 12F- Batkc, D. Heltmann, J. P. Kotthaus,and K. Ploog,Phys. Rev. Lett. 54, 2367 (1985). 13(2.G. Darwin, Prec. Cambridge Philos. Soc. 27, 86 (1930). 14T. Demel, D. Heitmann, P. Grambow, and K. Ploog, Phys. Rev. B 38, 12732 (1988). 15B. A. Wilson, S. J. Allen, Jr., and D. C. Tsui, PhyL Rev. B 24, .5887 (1981). t6J.-W. Wu, P. l.'lawrylaic, and J. J. Quinn, Phys. Re,,'. Lett. 55, 879 (1985). i~A. L. Fetter, Phys. Rev. B 3Z 7676 (1985). isw. Que and G. Kirczenow, Phys. Rev. B 38, 3614 (1988).
225
VOLUME 66, NUMBER 14
PHYSICAL
REVIEW
8 APRIL 1991
LETTERS
T r a n s p o r t S p e c t r o s c o p y of a Coulomb Island in t h e Q u a n t u m H a l l Regime P. L. McEuen, E. B. Foxman, U, Meirav, (a) M. A. Kastner, Yigal Meir, and Ned S. Wingreen Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 S. J. Wind IBM Thomas J. Watson Research Center, Yorktown Heights, New York 10598 (Received 30 January 1991) Transport measurements of a Coulomb island, a semiconductor dot small enough that Coulomb interactions dominate transport, are presented. At moderate magnetic fields ( B - 2 - 4 T) the amplitude and position of the Coulomb-regulated conductance peaks show distinct periodic structure as a function of B. This structure is shown to result from the B dependence of the quantized single-particle energy states on the island. Analysis of successive peaks is used to map out the single-particle level spectrum of the island as a function of B. PACS numbers: 73.'20.Dx,72.20.My, 73.40.Gk Transport through nanometer-seale electron gases such as small metal particles i or lithographically patterned semiconductor dots 2 is currently a subject of great experimental and theoretical interest. In these structures the quantization of the charge and energy of the electron gas has important implications for transport. Charge quantization is important since it means that adding an extra electron to the dot can require a finite charging energy. Transport is suppressed if this charging energy exceeds ks T, creating a "Coulomb island" - - a small electron gas electrically isolated from the leads by Coulomb interactions. This suppression is lifted whenever the charge fluctuations required for transport do not change the total energy of the system, and a peak in the conductance results. A semiclassical stochastic model of these devices, called the Coulomb-blockade model, ! has been remarkably successful in explaining experiments on small metal structures. Although also capable of explaining some aspects of experiments on semiconductor dots, 3,4 this model is inappropriate at low temperatures since it ignores the quantization of the dot's energy spectrum. The discrete spectrum of dots, which has been explored by various spectroscopic techniques, 5-7 causes such transport effects as Aharonov-Bohm-type oscillations 7 and resonant tunneling. 6,7 While charge and energy quantization effects are, taken separately, well understood, the regime in which both are important is only beginning to be explored. Recent theoretical work s-zz has predicted that the properties of the Coulomb-blockade conductance peaks are affected by the single-particle electronic eigenstates of the dot. In this Letter, we present an experimental study of a semiconductor dot in the quantum Hall regime, where the properties of the single-particle states are well known, t2'13 We find that the conductance peaks reflect the properties of the quantized energy levels of the island in surprising and dramatic ways. We further show that these measurements allow spectroscopy of all energy levels of the island--including levels through which negligi1926
ble current flows. The geometry of the device used here is shown schematically in Fig. 1 (a); a detailed description may be found in Refs. 3 and 14. Briefly, it is an inverted GaAs/ AIGaAs heterostructure in which electrostatic gates are used to confine and adjust the density of a twodimensional electron gas. A negative bias applied to a lithographically patterned split upper gate defines the island z5 while a positive bias applied to a lower gate ad-
12.9 12.8 "~ 12.7 12.6 12.5 7.4
7.6
7.8
%1% FIG. I. (a) Schematic top view of the device, showing the path of the edge states associated with the lowest two Landau levels (LLs). The upper gate (shaded) defines a dot whose lithographic dimensions arc 500 nm by 700 nm. (b) Energy i m,ca~r 2 levels of a dot with a parabolic confining potential ~as a function of cu,.-eB/m* in a parameter range where two LLs are present (Ref. 12). The heavy line represents the energy of the single-particle state that is 78th lowest in energy.
1991 The American Physical Society
226
VOLUME
66,
NUMBER
PHYSICAL
14
REVIEW
LETTERS
8 APRIL 1991 Theory
Experiment
131.0
,
0.03
1
,]
o.1 K
0.1 o
o~ 10-t
130.0 =
,
(a) Ir
0.02
0~ .,.a o
0.26 K 0.17 g
"0
0.01
o "~o 1 0 - 2
-
129.5 Ix.
128.5
~" 10-3
2.'0
L0
uagnetic Field (T)
L0
FIG. 2. Height and position of a conductance peak as a function of magnetic field at base temperature. The temperature of the electron gas is approximately 0.1 K (Ref. 3). Inset: Conductance vs Vt for the device at B - 3 T. Full scale is 0.03e 2lb.
(d)
(b}
O ...=, .,.a . -
128.0
0.4
0.2
0 In.,
t~.~
z.~ B-Field (T)
justs the electron density. The conductance G versus gate voltage Ve applied to the lower gate is shown in the inset of Fig. 2. As reported previously,3 the conductance consists of a periodic series of sharp peaks. We have studied, in detail, the dependence of the amplitude and position of these conductance peaks on magnetic field B. At low fields (B < 1 T), the amplitude shows strong random fluctuations with B which give way to more systematic behavior at higher magnetic fields. Figure 2 shows the position and amplitude of a particular conductance peak for B - 1 . 5 - 4 . 5 T. At roughly periodic values of B, the peak amplitude drops by as much as an order of magnitude. Commensurate with these dips, oscillations are observed in the position of the peak. This structure washes out rapidly with increasing temperature and is almost entirely destroyed by T---0.3 K [Figs. 3(a) and 3(b)], although the peaks in the conductance versus gate voltage remain well defined. W e now discuss the origin of this behavior. The basic periodicity of the series of conductance peaks shown in the inset of Fig. 2 can be understood within the standard Coulomb-blockade model. !.4 A valley corresponds to a gate voltage where an integer number of electrons minimizes the electrostatic energy of the dot. Changing the occupancy of the dot requires a finite charging energy, and transport is suppressed. A conductance peak, on the other hand, corresponds to a gate voltage where a halfinteger charge ( N - ~- )e on the dot would minimize the electrostatic energy. Since the actual charge on the dot is restricted to integer values of e, it fluctuates between ( N - 1 )e and Ne with no cost in charging energy, and transport can occur at T - 0 . The spacing of the peaks is determined by the gate-voltage change required to change the occupancy of the dot by one electron.
!
I
129.0
2,0
~ E(z)/AE (t)
FIG. 3. Temperature dependence of (a) the peak amplitude in e 2/h and (b) the peak position in mV of a peak over a narrow B range containing one dip. Also shown arc the predictions of the three-level model described in the text for (c) the peak amplitude in units of (eVh)r")lae (') and (d) the peak position in meV, both for AE o)-0.05 meV. All but the lowest peak-position curves have been offset for clarity.
If ks T is less than the single-particle level spacing, the discreteness of the energy levels of the dot must be considered, since in this case the charge fluctuation between ( N - - 1 )e and Ne involves emptying and filling the Nth single-particle state in the dot. The energy Ere of this single-particle state directly affects the position of the peak. For example, if E~v increases, the peak occurs at a higher VI since, roughly speaking, the state is more difficult to fill. Elementary arguments show that, for constant Coulomb energy U, the position of the Nth conductance peak at T - 0 can be written as ~'4't~
Vz(N) - ( l / a e ) [ ( N - ~ )U + (EN - p ) l +const , where p is the chemical potential in the leads and a is a dimensionless constant relating changes in gate voltage to changes in the electrical potential of the dot. The constant a can be determined from the temperature dependence of the width of a conductance peak 3 and is found to be 0.4 for this device. 16 The position of the Nth conductance peak is thus determined by a Coulomb term proportional to (N -- ~- )U and by a single-particle term proportional to Ely --/a. In our device, the dominant term is the Coulomb term, producing conductance peaks roughly periodic in Vg. The Coulomb energy does not vary with magnetic field, however, so the variation of the position of the peak 1927
227
VOLUME 66, NUMBER 14
PHYSICAL
REVIEW
shown in Fig. 2 results from variations in Et~. To understand why Ejv exhibits a periodic modulation, consider Fig. 1 (b), a plot of the single-particle energylevel spectrum of a dot in a high magnetic field. In the parameter range shown, the spectrum consists of two Landau levels (LLs), which, in turn, are composed of discrete nondegenerate states because of the confinement potential n2"~3 (spin is suppressed for clarity). States in the first LL fall in energy with increasing B while those in the second LL rise. The thick line in Fig. 1(b) shows the behavior of Ere, the state occupied by the Nth electron on the dot, as a function of B. This electron alternately occupies a state in the first LL and a state in the second LL as the magnetic field is increased. Consequently, the position of the Nth peak oscillates, as is evident in the data of Fig. 2. These oscillations will be clear if there are two Landau levels occupied. With many more than two, the simple oscillations give way to complicated fluctuations, while with less than two there are no oscillations because all. the electrons are in the lowest LL. l0 In Fig. 2, the oscillations become clear around B - 2 T, and then die out around B - 4 T. We thus attribute these field values to filling factors of v--4 and v - 2 , respectively. In addition, the oscillations change character above a B value (2.5 T) that roughly corresponds to v - 3 . In this regime, the second LL is likely spin polarized due to the enhancement of the g factor. ~7 We note that each oscillation in Fig. 2 represents the transfer of one electron from the second LL to the first LL. We are thus watching the magnetic depopulation of the second LL, one electron at a time. We now consider the behavior of the amplitude of a conductance peak. The behavior evident in Figs. 2 and 3 follows if (a) only two LLs are occupied, and (b) the states of the second LL do not couple to the leads. TM These assumptions are schematically illustrated in Fig. l (a). At a particular B, if the Nth single-particle state is in the first LL [the outer-edge state shown in Fig. 1(a)], it couples well to the leads and transport can occur by resonant tunneling through this state. If the Nth state is in the second LL [the inner-edge state in Fig. l(a)], however, the peak amplitude is suppressed since the coupling to this state is minimal. A dip in amplitude is thus expected whenever the Nth state is in the second LL, i.e., when the position of the peak is moving up in energy. 19 This is indeed what is observed in Fig. 2. The dip in conductance disappears when kBT becomes comparable to the single-particle level spacing in the first LL, since transport can then occur by thermal activation to the nearest energy state in the first LL. The arguments above can be made quantitative using the theory of Meir, Wingreen, and Lee. ~ This theory gives an explicit expression for the conductance in terms of the interaction energy U, the single-particle energies El, and the single-particle elastic-tunneling widths Fi. The main features of the experimental data can be ac1928
LETTERS
8 APRIL 1991
counted for by a simple three-level model: two states (representing states in the first LL) with energy separation AE ci) and equal elastic widths F (~) and a single state (representing a state in the second LL) with an energy E (z) that increases with B and has negligible coupling to the leads (F (2) --0). In Fig. 3, the temperature dependence of the height and position of a particular conductance peak are compared with the theoretical predictions for the three-level system. The agreement between theory and experiment is excellent, 2~ considering that the only free parameter determining the shape of the theoretical curves is the energy-level spacing in the first LL, AE 0). This spacing is found to be 0.05 meV for this peak at this magnetic-field value. Further, the elastic width can be obtained from the height of the conductance peak and is found to be F 0 ) - 0 . 0 0 0 6 meV, assuming symmetric barriers. The theory also predicts a significant broadening of the width of the conductance peak at the dip in amplitude, which is also observed (not shown). Having understood the behavior of a single conductance peak, we now turn our attention to the behavior of successive peaks, as sho~n in Fig. 4(a). As indicated by the arrows, a single-particle level within a given LL moves continuously through successive conductance peaks, allowing it to be tracked over a wide range of magnetic field, n9 The Coulomb portion U/ae of the peak separation in Fig. 4(a) is approximately constant and
FIG. 4. (a) Peak position vs B for a series of consecutive conductance peaks. The arrow follows a particular state in the first LL as it moves through successive peaks. (b) Singleparticle energy-level spectrum inferred from (a) as described in the text. The zero of the energy scale is arbitrary.
228
VOLUME 66, NUMBER 14
PHYSICAL
REVIEW
can be removed by subtracting a constant gate-voltage spacing (chosen to be 0.685 mV) between successive peaks. Furthermore, the resulting peak positions in Vz can be converted to energies using the factor a - 0 . 4 determined from the temperature dependence of the width of a conductance peak. 3 Doing this, we obtain the results shown in Fig. 4(b). This plot represents the single-particle energy-level spectrum o f the island as a function of B. The level spectrum of Fig. 4(b) is qualitatively very similar to the theoretical spectrum shown in Fig. 1(b). The curves moving to higher (lower) energy with B are states in the second (first) LL. 19 The parts of the curves inferred from different conductance peaks match up well, indicating that the assumption of a constant Coulomb term is a reasonably good one. There are deviations, however, such as the discontinuity in the inferred singleparticle states starting near 0.2 meV. Other regions of magnetic field show even more unusual behavior. These deviations indicate the importance of interaction effects beyond the scope of the constant-Coulomb-energy model and will be explored in future experiments. The quantitative aspects of Fig. 4(b) are also in excel' lent agreement with expectations. For example, the increase in energy of the states in the second LL relative to those in the first LL with increasing B is approximately 3.6 meV/T. This value compares favorably with theoretical predictions for parabolic confinement 12 (--2h~oc/B - 3 . 2 meV/T) or hard-wall confinement 13 (--3h~oc/B - 4 . 8 meV/T) when the second LL is nearly depopulated. The single-particle energy-level spacings Within a LL can be found directly from Fig. 4(b); they are AE O) --0.05 meV and AEt2)--0.1 meV. The level splitting AE O) inferred in this way agrees with the value of AE tiJ --0.05 meV obtained earlier from the temperature dependence of a peak dip. The level splitting in the second LL is about twice that in the first LL, again suggesting that the second LL is spin resolved at this field value (and hence has half as many states per unit energy). The periodic spacing of the states in the first LL is somewhat unexpected, si.nce spin splitting would in general group the states into twos. We note, however, that the anticipated bare spin splitting g l a a H - 0 . 0 6 meV at 3 T, and so the observation of a single energy spacing may simply be, because the spin splitting is approximately half the spin-resolved level spacing. In conclusion, we have shown that the B dependence of the conductance peaks of a Coulomb island in the quantum Hall regime are determined by the B dependence of the single-particle energy levels. The amplitude of the Nth peak reflects the coupling of the Nth singleparticle state to the leads. The position of the Nth conductance peak reflects the energy of the Nth state. The Coulomb part of the energy spacing between peaks can be subtracted to obtain the single-particle energy spectrum. These measurements show the importance of the
LETTERS
8 APRIL 1991
single-particle energy states to transport in Coulomb islands and also demonstrate a powerful new tool for probing the quantized energy levels of these structures. We wish to thank P. A. Lee, K. K. Likharev, and R. G. Wheeler for useful discussions. In addition, we thank N. R. Bclk for help with the instrumentation. This work was supported by NSF Grant. No. ECS-8813250 and by the U.S. Joint Services Electronics Program under Contract No. DAAL03-89-C-001. One of us (Y.M.) acknowledges the support of a Weizmann Fellowship.
(a)Present address: Department of Nuclear Physics, Weizmann Institute of Science, Rehovot 76100, Israel. IFor a review of Coulomb-blockade phenomena in metals, see D. V. Averin and K. IC Likharev, in Mesoscopic Phenomena in Solids, edited by B. L. Altshuler, P. A. Lee, and R. A. Webb (Elsevier, Amsterdam, 1991). 2For a review of quantum transport in semiconductor nanostructures, see C. W. J. Beena~er and H. van Houten, Solid State Phys. 44, ! (1991). 3U. Meirav, M. A. Kastner, and S. J. Wind, Phys. Rev. Lett. 65, 771 (1990). 4L. I. Glazman and R. I. Shekhter, J. Phys. Condens. Matter 1, 5811 (1989); H. van Houten and C. W. J. Beenakker, Phys. Rev. Lett. 63, 1893 (1989). ST. P. Smith, III, et al., Phys. Rev. B 38, 2172 (1988); W. Hansen et al., Phys. Rev. Lett. 62, 2168 (1989); Ch. Sikorski and U. Merkt, Phys. Rev. Lett. 62, 2164 (1989). 6M. A. Reed et al., Phys. Rev. Lett. 60, 535 (1988); C. G. Smith et al., J. Phys. C 21, L893 (1988). ?B. J. van Wees et al., Phys. Rev. Lett. 62, 2523 (1989). SL. I. Glazman and K. A. Matveev, Pis'ma Zh. Eksp. Teor. Phys. 51, 425 (1990) LIETP Lett. 51, 484 (1990)]. 9D. V. Averin and Yu. V. Nazarov, Phys. Rev. Lett. 65, 2446 (1990). I~ W. J. Beenaklcer, H. van Houten, and A. A. M. Staring (to be published); C. W. J. Beenakker (to be published). i ly. Meir, N. S. Wingreen, and P. A. Lee (to be published). 12V. Fock, Z. Phys. 47, 446 (1928); C. G. Darwin, Proc. Cambridge Philos. Sot:. 27, 86 (1930); R. B. Dingle, Proc. Roy. So<:. London A 216, 118 (1953);-2"19, 463 (1953). 13U. Sivan and Y. Imry, Phys. Rev. Lett. 61, 1001 (1988). 14U. Meirav, M. Heiblum, and Frank Stern, Appl. Phys. Lett. 52, 1268 (1988). 15In all of the experiments reported here, the voltage on the upper gate is held fixed at -0.3 V. 16The inferred value of a is smaller than expected from the known upper-gate and lower-gate capacitances (.see Ref. 3). The origin of this discrepancy is not known. 17See, e.g., T. Ando, A. B. Fowler, and F. Stern, Rev. Mod. Phys. 54, 437 (1982). 18B. J. van Wees et al., Phys. Rev. Lett. 62, 1181 (1989). 19At finite temperatures the energy of a state in the second LL is only indirectly related to the position of the peak since conduction is mostly by thermal activation to states in the first LL [see Fig. 3(d)]. 2~ remaining minor discrepancies with experiment in Fig. 3 can be accounted for within the theory by including additional single-particle states and small variations in the elastic widths. 1929
229
PHYSICAL REVIEW B
VOLUME 45, NUMBER 19
15 MAY 19924
Self-consistent addition spectrum of a Coulomb island in the quantum Hall regime P. L. McEuen,* E. B. Foxman, Jari Kinaret, U. Meirav, t and M. A. Kastner Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139
Ned S. Wingreen NEC Research Institute, 4 Independence Way, Princeton, New Jersey 08540 S. J. Wind IBM Thomas J. Watson Research Center, Yorktown Heights, New York 10598 (Received I! March 1992) Coulomb interactions are shown to influence the addition spectrum of a small electron gas in the quantum Hall regime in ways that cannot be described by a classical charging energy. The interaction energy between electrons is observed to depend upon Landau-level index, and the evolution of the addition spectrum with magnetic field is found to depend strongly on Coulomb interactions. A selfconsistent model of the island is introduced that can account for these results. The energy-level spectrum of a two-dimensional island of electrons in a high magnetic field is a subject of considerable recent interest, s- == For noninteracting electrons residing in a circularly symmetric external confinement potential Vcxt(r), the behavior is well known. In the limit where V=xt(r) varies slowly on the scale of the magnetic length I s - ( h / e B ) =/2 the kinetic and spin energies of the electrons are quantized into Landau levels (LL's), and the single-particle energies are given by
E ( n , m , S = ) ~ [ ( n + ~ )hcoc+gpsBSz]+V=xt(r,n) ,
trons on the island and EN is the Nth quantized singleparticle energy state of the island. U is assumed to be a constant, independent of both magnetic field and particle number, and E# is calculated for a noninteracting system [cf. Fig. I(a)]. Despite its simplicity, the CI model has been quite suc-
(I)
_
where oJc--eB/m* is the cyclotron frequency, n "=0, I, 2. . . . is the orbital Landau-level index, S: - _+ ~- (denoted [ and | ) is the spin LL index, and rm - ( 2 m h / e B ) i / 2 is the radius of the drifting cyclotron orbit that encloses m flux quanta. The behavior of these single-particle energies, calculated for V c x t ( r ) - ( ~ )m*o~r 2, is shown in Fig. 1(a). A rich spectrum of level crossings is observed with increasing B as the LL degeneracy (given by l / 2 x / ~ ) increases, and the number of occupied LL's (given by v - 2 x l ~ n , where ns is the sheet electron density) decreases.
Coulomb interactions are expected to strongly alter this picture, however, as studies of both microscopic 3-? and phenomenological 1"2 models have shown. In the microscopic models, Coulomb repulsion causes electrons to spread out and partially occupy higher rm orbitals, 4 and correlations lead to the formation of fractional quantum Hall states. 6 These microscopic models have been solved, however, only for small numbers of electrons 3-6 or for oversimplified forms for the interaction, 6"? making comparison to experiment difficult: Because of this, a phenomenologicai model, which w e call the constantinteraction (CI) model, I'e has typically been employed to interpret experiments. =.s.9 In this CI model, the electrochemical potential, i.e., the energy required to add the Nth electron to the island, is given by (Ref. l)
~,0v)- 0v- } )u+E.,
(2)
where U is the Coulomb interaction energy between elec45
FIG. I. (a) Dots: Calculated noninteracting level spectrum of an island vs 8 (in units of oJJa~). Thick solid line: Energyof the 39th electron. The filling factor v of the island is as indicated. (b) Position in back-gate voltage Vt of a conductance peak as a function of B (in T) at T ~ 30 inK. The measured filling factors v of the 2DEG are also shown. Inset: Conductance vs Vg at B 12.5 T. (c) Electrochemical potential of the 38th and 39th electron vs B (in T) calculated using the self-consistent (SC) model described in the text. 11 419
9 1992 The American Physical Society
230
P. L McEUEN et al.
II 420
eessful in describing a variety of transport experiments by our group s and others, i In this Rapid Communication, however, we present new experimental results that are inconsistent with the CI model. These experiments lead us to reinterpret our previous results s and to develop a different self-consistent (SC) model of the island. This SC model, whose predictions are in good agreement with experiment, gives an interesting picture of a small electron gas in the quantum Hall regime. We begin with some experimental observations. The device, which has been discussed in detail previously, s'l~ consists of a two-dimensional electron gas (2DEG) in an inverted GaAs/AIxGal-xAs heterostructure with electrostatic gates above and below it. A negative bias applied to the upper depletion gate defines an island of 2DEG (lithographic dimensions, 450x900 nm 5) containing fewer than 100 electrons. The island is probed by weakly coupling it to two large 2DEG regions and measuring the conductance as a function of the voltage Vz applied to the lower (n +-type GaAs) gate...Tbe positions in Vz of the observed conductance peaks [Fig. I (b) inset] are a direct measurement of the energies for adding additional electrons to the island: mV z ( N ) - p e ( N ) / e a , where a [ ~ 0 . 4 (Ref. 8)] is a constant. Figure i (b) shows the position of a particular conductance peak as a function of magnetic field. Also shown are the filling factors v of the 2DEG adjacent to the island, as determined by Shubnikov-de Haas measurements, j .
.
E,~(N)-
.
.
.
.
.
.
.
.
.
.
.
.
.
.
~[(n+ + )h
where p,=(r) is the electron density in the n, Sz spinpolarized Landau level, p(r) is the total electron density, V~t(r) is the bare (unscreened) confining potential created by. the upper gates, and Vcc(r,r') is the electronelectron interaction. The electron density in each LL is limited by the Landau-level degeneracy: (4a)
We further assume that the charge in each Landau let,el
is quantized, i.e.,
f dSro.,(r)-N.,. ~,~,N.,-N. n
be smaller. We are thus led to a different interpretation" than the one given in Ref. 8; namely, that 1,9 T corresponds to v - 2 , and the region above 1.9 T to v ~ 2. The structure in the peak position above 1.9 T thus represents crossings between states in the two lowest spin-split LL's. This interpretation presents a dilemma, however. The experimental data does not resemble the predictions of the constant-interaction model. The CI modelpredicts very infrequent level crossings in the v_.< 2 regime [Fig. I (a)] since, in GaAs, the spin-splitting glzaB issmall. In the experiment, however, frequent level crossings are observed [Fig. I (b)]. Motivated by this discrepancy, we have developed an alternative self, consistent (SC) model of the island, which we now discuss in detail. In this model, the total energy of the island, Etot(N), is given by
.
{z
pn.,(r) <--- l/2xl~.
In Ref. 8, we presented similar results from a different device. Based on the predictions of the CI model, the structure in Vt (N) in the region above !.9 T was attributed to level crossings between states in the lowest two orbital Landau levels. In that interpretation, B - 1 . 9 T in Fig. l(b) would correspond to a filling factor of v - 4 in the island. Comparison with the filling factors in the 2DEG, however, show that this interpretation is unreasonable. It would require the filling factor (and density) of the island to be larger that that of the 2DEG, while other experiments 1,9 (and common sense) indicate that it should
(4b)
$
and N,., is an integer. This is expected to be a valid approximation if the coupling between states in different Landau levels is small. According to simulations of our device by Kumar, ll Yext is roughly parabolic with oscillator frequencies given by hoJr - 3 . 5 meV, and ha~x-0.8 meV. For computational simplicity, we will assume radial symmetry: Vext(r)-(~-)m*oJ~r 2 and use a single parameter, h~oo-[(h~ox)(hoJr)] I/2-1.6 meV, to characterize the potential. Electrons added to this bare confinement potential interact with each other and screen the potential. This interaction V,,(r,r') is cut off at short distances by the finite z-extent 6z of the 2DEG wave function (--10 nm) and is screened at long distances by the image charges associated with the nearby metallic n + region. We use the following form for the interaction to ac-
"I"y
dSrp(r)[ Vcxt(r) +
~"y d~r 'p(r')Vec(r,r')
],
(3)
i
count for these effects:
V~ (r,r') -- e 2/~(lr -- r'l 5+ az 5) m
- e Ve(Ir - r'l 5+ 4d 5) an.
(5)
where d is the distance from the 2DEG to the gate (100 nm) and e - 1 3 . 6 is the dielectric constant of GaAs. Finally, we will use the bare g facto[ of GaAs g - - 0 . 4 4 to describe the spin splitting, although the exchange enhancement of the g factor 15 can be included in a straightforward manner. As we will see, the size of the spin splitting does not qualitatively affect the predictions. We now discuss the results of the SC model, obtained by numerically minimizing (3) subject to the constraints (4), and by using the definition of the electrochemical potential:
pe (N ) =--Etot(N) - Etot(N -- 1).
(6)
Figure 2(c) shows the calculated p , ( N ) for N - - 3 8 and 39 as a function of magnetic field. The overall shape, as well as the scale, of the structure in/1, is quite similar to that in the experimental data of Fig. l(b). In addition, the separation between successive/z,, curves in the SC model [--0.6 meV between / z , ( N - 3 8 ) and p c ( N - 3 9 ) ] is in reasonable agreement with the experimentally observed peak spacing (aAVz---0.48 mV). The density of the island in the model is ---30% less than in the experiment, but uncertainties in the number of electrons and in the pa-
231
SELF-CONSISTENT ADDITION SPECTRUM OF A COULOMB . . . 1~
B=1.15z
(a)
3
2 ~
,
.... ~.... ib,o
'-':
:~ o.s 0 l~
/
i
0 . . . . 0
40
80
"
....
incomprcssiblr
0
120 160
radius (nm)
B = 2T
FIG. 2. Charge density and electrostatic potential in the self-consistent model for 39 electrons and (a) B --I. !5 T and (b) 2.0 T. The dashed line is the classical electrostatic solution. Partially filled (cornprcssiblc) LL's screen the confining potential, while full (incompressible) LL's do not. (c),(d) Schematic top views of the islam showing tim compressible regions of the 01 LL (region !), the 0l LL (region ll), and the ! • LL (region Ill).
rameter mo can easily account for this difference. Most importantly, there are frequent level crossings in the v __<2 regime; the SC model can thus account for the level crossings observed in the experiment [Fig. I (b)]. To understand the physics underlying this behavior, we first note that in the SC model the shape of the charge
distribution closely approximates the classical electrostatic solutiolt 13This can be seen in Figs. 2(a) and 2(b), where the SC charge density p(r) (solid lines) is given at two very different magnetic fields and compared to the classical result (dashed line). While electrons can lower their kinetic and spin energies by shifting to lower LL's, these quantum effects are only a small perturbation. We can now immediately understand the origin of frequent level crossings in the v__<2 regime. An electron transfers from the upper to the lower LL roughly whenever a flux quantum hie is added to the area of the island and an additional electron can be accommodated in the lower LL. The frequency of these level crossings is governed by classical electrostatics and LL degeneracies, and not by the spin-splitting energy between LL's. To understand the oscillations in greater detail, we first note that the readjustment of charge to minimize kinetic and spin energy leaves the island in a configuration, as shown in Figs. 2(c) and 2(d), where compressible regions (partially filled LL's) are separated by incompressible regions (full LL's). The compressible regions can be thought of as metallic regions that screen the external potential, ,4 leaving the self-consistent electrostatic potential flat [see Figs. 2(a) and 2(b)]. The incompressible regions, on the other hand, can be thought of as insulators that do not screen the external potential. Now consider adding an additional electron to the island, e.g., in the v_< 2 regime shown in Fig. 2(d). The electron may be added to either compressible region I or If, and the behavior of/ze(N) with B depends upon where it is added. To scc this, note
11 421
that when the magnetic field is increased slightly, the degeneracy of the LL's increases. The charge density in the lower LL then increases in the center of the island and thus decreases near the edges, thereby increasing the electrostatic potential in region If and decreasing it in region i. The rising (falling) portions of the pc(N) vs B curves in Fig. I(c) thus correspond to the Nth electron being added to region II (region I). As the magnetic field is further increased the electrostatic potential disparity between regions If and I becomes large and discrete charge e will move from the upper to the lower LL, as shown in Fig. 2(d). These electron transfers show up as crossovers between the rising and falling parts of the p,,(.N) traces. The oscillations of/z, with B are thusdue to the periodic buildup and release of electrostatic frustration in the island that occurs because the charge must distribute itself among the LL's in multiples of e. Another important result from the SC model is that the magnitude of the interaction between electrons depends upon the shape of the charge distributions and upon the width of the incompressible region separating them. This can be seen by constructing . level spectra', $ 1 59 in different magnetic-field regimes, as is done in Fig. 3. These spectra are made by subtracting a constant between successive pc curves in the model, or between peak position curves in the experiment. Physically, this constant represents the interaction energy U between an electron in one metallic region and an electron in another. In particular, it is tbe U between electrons whose electrochemical potentials are crossing with increasing B. Figures 3(a) and 3(b) are level spectra in the 3 >_. v >_ 2 regime. The amount subtracted (Uo.,-0.45 meV in the model, AVso.I--1.175 r a v i n the data) is the Coulomb interaction between an electron in the n - 0 LL and an electron in the n - I LL. A
0.5
,-. . . . .
Cc>
--
030
.~, 0.4 >" E 0.3
0.25 0.20
:~0.2
0.10
0.I
0.05
"-"
0.15
0
1.5 .
,
l.? ,
1.9 .....
,
2.1. . . . . .2.3 2,6 . . . . .
~d~.
2.7
......
2.8
1.oo
.... ::> r 0.75 E "~=0.50
0
2.9 !
......
0.8
o.6 0.4
0 25
0.2
0
1.0
1.2
1.4
B (T)
1.6 1.7
1.8
1.9
2.0
B (W)
FIG. 3. (a) Experimental spectrum and (b) SC model spectrum in the 3>-- v>- 2 regime. These spectra arc constructed by subtracting a constant between successive peak position traces. (c) Experimental spectrum and (d) SC mode! spectrum in the v ~ 2 regime, constructed by subtracting slightly larger constants than in (a) and (b).
232
11422
P. L McEUEN et al.
in the n -- ! LL. A different, larger constant must be subtracted to construct a level spectrum in the v < 2 regime. This is done in Figs. 3(c) and 3(d), where the amounts subtracted are Uoi,0|==0.55 meV in the model, and AV~I,01 ==1.35 mV in the experiment. The Coulomb interaction U thus depends upon LL index. In addition, U depends on the magnetic field, since the level spectra in general line up only over a limited range of B. 7 These variations i n U again illustrate the limitations of the constant-interaction model and the necessity of a selfconsistent approach. 9 In conclusion, we have studied the addition spectrum of a small electron gas in the quantum Hall regime. We find that the magnitude of the Coulomb interaction between electrons is a function of the Landau-level index and mag-
"Present address: Department of Physics, University of California at Berkeley, Berkeley, CA 94720. tPresent address: Department of Nuclear Physics, Weizmann Institute of Science, Rehovot 76 I00, Israel. t For a review, see C. W. J. Beenakker and H. van Houten, i n Single Charge Tunneling, edited by H. Grabert, J. M. Martinis, and M. H. Devoret (Plenum, New York, 1991). 2C. W. J. Beenakker, H. van Houten, and A. A. M. Staring, Phys. Rev. B 44, ! 657 (i 99 ! ). 3Garnett W. Bryant, Phys. Rev. Lett. 59, !i40 (1987). 4p. A. Maksym and Tapash Chakraborty, Phys. Rev. Lett. 65, 108 (1990). 5Arvind Kumar, Steven E. Laux, and Frank Stern, Phys. Rev. B 42, 5 i 66 (! 990). 6Jari M. Kinaret, Yigal Meir, Ned S. Wingreen, Patrick Lee, and Xiao-Gang Wen, Phys. Rev. B (to be published). 7N. F. Johnson and M. C. Payne, Phys. Rev. Lett. 67, !157
netic field, and that Coulomb interactions strongly influence the evolution of the addition spectrum with B. The experimental results are in good agreement with a self-consistent model of the island. We thank Yigal Meir, Patrick Lee, and Xiao-Gang Wen for useful discussions. We also thank Arvind Kumar for his expert simulations, and Nathan Belk and Paul Belk for their help with the instrumentation. The work at MIT was supported b y the N S F under Grant No. ECS8813250 and by the U.S. Joint Services Electronics Program under Contract No. DAAL03-89-C-001. One of us (J.K.) acknowledges the support of the Academy of Finland.
(1991); N. F. Johnson and M. C. Payne (unpublished). Sp. L. McEuen et al., Phys. Rev. Lett.66, 1926 (1991). 9A. A. M. Staring, H. van Houten, C. W. J. Ikenakkcr, and C. T. Foxon (unpublished). t~ Meirav, M. A. Kastner, and S. J. Wind, Phys. Rev. Lett. 65, 771 (1990). l lArvind Kumar, in Proceedings of Electronic Properties of Two-Dimensional Solids-9 [Surf. Sci. (to be published)], 12See, e.g., T. Ando, A. B. Fowler, and F. Stern, Rev. Mod. Phys. 54, 437 (1982). 13V. Shikin, S. Nazin, D. Heitmann, and T. Demcl, Phys. Rev. B 43, II 903 (1991). 14A. M. Chang, Solid State Commun. 74, 271 (1990). tSNote that these are addition spectra, not excitation spectra. Nonlinear measurements must be used to explore the excitation spectrum.
233
PHYSICAL REVIEW B
VOLUME 45, NUMBER 4
15 JANUARY 1992-11
Effect of electron-electron interactions on the magnetization of quantum dots P. A. Maksym
Department of Physics, Unit,ersity of Leicester, Leicester LEI 7RH, United Kingdom Tapash Chakraborty
Institute for Microstructural Sciences, National Research Council of Canada, Montreal Road, M-50, Ottawa, Canada KIA OR6 (Received 12 August 1991) The low-temperature magnetization of parabolic quantum dots is calculated and is shown to be a sensitive probe of interaction effects. The interaction causes the ground state to occur at certain magic values of the total angular momentum, the strength of the magnetic field determining which of them is selected. Increasing the magnetic field causes the ground-state angular momentum to jump from one magic value to another and this causes the discontinuities in the magnetization. The effects of spin lead to extra discontinuities at low magnetic field. The magic angular momenta for the spin-polarized case are derived by a simple physical argument.
The electron-electron interaction in quantum dots subjected to a magnetic field leads to interesting effects which are highly elusive. For example, Maksym and Chakraborty I (MC) have shown that the ground state of electrons in a magnetic field occurs only at certain magic values of the total angular momentum, and that transitions from one magic value to another should occur as the magnetic field is increased. This cannot be probed by infrared spectroscopic techniques because far-infrared radiation couples to the center-of-mass motion and hence is insensitive to the interaction when the confinement is parabolic.~-3 Nevertheless, there are probes that are sensitive to the interaction and MC showed that the heat capacity is one of them. In the present work the magnetization is found to be another, which can in principle be measured. (St6rmer and co-workers 4 have measured the magnetization of a two-dimensional electron gas.) It is shown that the field dependence of the magnetization is oscillatory with discontinuities that occur when the ground-state angular momentum changes. In addition, the effect of spin is considered, and it is shown that this leads to rich behavior in the low-field regime, where both the spin and angular momentum of the ground state vary discontinuously with magnetic field. Finally, the key physics is explained in terms of a simple model that includes only the states in the zeroth Landau level, and the rule for determining the magic values of angular momentum in the spin-polarized case is given. The starting point for calculating the magnetization is the calculation of the energy eigenvalues of the electrons interacting in a parabolic dot. This is done by numerically diagonalizing the Hamiltonian
~t = ]E r n .l .s
+
Z
~'~n II i,n212,n~l~m414Cnt d o's t
,
nil..n 4 It...I 4 s,s'
where ~,t are single-electron energies and ..4 is the 45
Coulomb matrix element
..4 ~. f drldr2r162 X V ( r l - r2)r162
).
The single-electron energies are obtained from 5"6 ~,/ -(2n+l+ ltl)ht~- ~lhto,., where t~( 88 '/2, to,,--eB/m*, and hto0 is the confinement energy. The single-electron wave function (ignoring the normalization constant) is written as 3 ~,a-rlflexp(-ilo)Ltnti(r2/ 2a 2 ) e x p ( - r2/4a2), where the effective magnetic length a is given by a 2 - h / ( 2 m * fl). The quantum number - / i s the angular momentum and the quantum number n is related to the Landau quantum number N - n + ( l l 1 - 1 ) / 2 (referred to as Fock-Darwin-level index in Ref. 3). In the absence of confinement this becomes the usual Landaulevel index and the single-election energies become (N + 89)htoc, but in the presence of confinement, ~,t is a function of both N and 1. The relevance of these quantum numbers is that they are used to select the basis states for the numerical diagonalization. The basis includes all contributions of single-electron states that are compatible with the desired total angular momentum, subject to the constraint that the sum of the single-electron N values does not exceed the specified maximum. The exact eigenstates have the property that they can be factorized into a product of a function of the center of mass and a function of relative motion, t'2 and this method of truncating the basis ensures that the same holds for the numerically generated eigenstates. Once the many-body eigenvalues and eigenstates are available, the magnetization can be calculated in one of two ways. The first is to evaluate matrix elements of the magnetization operator ~ = ( - e / 2 m * ) ~ ? l r i x (p/ +eAi), where n,. is the number of electrons. The second is to differentiate the eigenvalues with respect to the magnetic field B. While these two procedures would give the same result if the basis was infinite, the results obtained from a truncated basis differ and the results given by the differentiation procedure are superior. The reason is that 1947
9 1992 The American Physical Society
234
P. A. MAKSYM AND TAPASH CHAKRABORTY
1948
the magnetization operator couples states whose n quantum numbers differ by • I, as well as states with the same n. Therefore accurate evaluation of magnetization requires a larger basis than accurate calculation of the energy eigenvalues. The differentiation procedure is superior because calculating the magnetization by differentiation of the eigenvalues obtained in the truncated basis is equivalent to including the omitted states by first-order perturbation theory and then calculating matrix elements of the magnetization operator. This can be proved by considering d E / d B explicitly:
d__EEdB<--dB"l~tlw~+<w~
Wo)+(wol~l--~/,dw~% (1)
where ~o is a state formed from the truncated basis and E is its energy. The derivatives of ~o can be estimated by first-order perturbation theory: d~o dB
~o(B'I- 6B ) - ~o(B ) 6B
-~ Z~j Lila~/aBl~ + Z Ej -- Eo
j
~,.
<.la~/aBIo> E.
o,,o
-
Eo
(2)
where the ~'a come from the diagonalization within the truncated basis and the Cj are the remaining basis states. Substituting (2) into (I) shows that differentiation of the energies automatically generates a first-order contribution
k
0 (,0
\
-zoi ~
~ 1 2 ,
from the omitted basis states; hence differentiation is the superior caiculational method. In practice the calculations are done at. finite temperature; that is, the numerically generated eigenvalues are used to compute the free energy, which is then differentiated to get the magnetiaation. All the Zeeman terms for each spin are included in the calculation of the free energy. The results are shown in Figs. 1 and 2. The top panel of each figure gives the magnetization as a function of B, calculated both with and without interaction for three electrons (Fig. !) and four electrons (Fig. 2). The remaining panels show the ground-state total-angular-momentum quantum number J and the ground-state spin S. All results are for GaAs quantum dots with ha)o=4 meV. The calculations were done with the maximum value of N taken to be !; that is, one electron was allowed to have N > 0 and the other electrons had N--O. This truncation is surprisingly accurate, even at low magnetic fields. The absolute value of the magnetization is insensitive to the upper value of the N sum, and the only effect of increasing it is that the position of the discontinuities changes. This is illustrated in the inset in Fig. I, where the results of allowing the upper limit of the N sum to rise to 2 are shown. The curves coincide on either side of the discontinuity but the position of the discontinuity is shifted by about 0.15 T. This level of accuracy is typical for fields greater than about 2.5 T; at lower fields the positions of the discontinuities are estimated to be accurate to __+0.4 T. Physically, the discontinuities correspond to changes of the ground state J, or both J and S, as can be seen by comparing the three panels of the figure. When the system is spin polarized, the sequence of magic J values is the same as found in Ref. 1. Qualitatively the discontinuities resemble the structure discussed by Sivan and Imry 7 but their ori-
5 A (10
3; -2 t
-3 10
0
J
E
-2
8
--J
y
6 4
r-'
2 0
~
V--
-4 16
, , / ~
12 -'3
8
o
1
t
5
10
B (T) FIG. i. Magnetization .:~[(meV/Tesla) at T=0. I K of a parabolic quantum dot containing three electrons ( N = I ) , the ground-state angular momentum J, and the ground-state spin S. The dash-dotted line corresponds to the noninteracting case. A comparison of the results for the N = 2 case (dotted line) is liven in the inset.
4 0 2 03 1 0 0
1
J.
5
10
15
B rr) FIG. 2. The same as Fig. !, but for the four-electron system.
235
EFFECT OF ELECTRON-ELECTRON INTERACTIONS ON T H E . . . gin is quite different; the latter are an edge-state effect that occurs when several Landau levels are occupied. In contrast, the discontinuities found here occur even when a Landau level is partly full, and are clearly a consequence of the interaction. The magnetization for noninteracting electrons has no discontinuities because the lowest-two single-electron levels are unaffected by level crossings as the field is increased [curves of dnt(B) for the parameters used here are given in Ref. 8]. Hence systems of up to four noninteracting electrons in the lowest spin state stay in the same angular-momentum state throughout the field range, so the magnetization curve is smooth. All the discontinuities in this case are a consequence of the interaction. For five or more noninteracting electrons the magnetization would be affected by negative I levels crossing positive 1 levels; however, the position of these crossings would be drastically affected by the interaction. In addition, there are relatively few of them when the electron number is small (for five electrons in the lowest spin state there is only one) and they tend to occur at low field. In contrast, the discontinuities due to the interaction occur at a regular sequence of J values throughout the field range. Because of the small magnetization per dot, experimental observation of the discontinuities would require measurements on an array of dots, and so would be affected by statistical fluctuations in electron number. Nevertheless, the discontinuities should be observable when the fluctuations are small because the discontinuities ~for different numbers of electrons occur at different magnetic fields. There are two aspects to the physics underlying the results shown in Figs. I and 2. The first is the question of how the interaction affects the magnetization, and the second is the reason why the ground-state angular momentum changes with magnetic field. Both aspects can easily be understood by considering the simplified case of a spin-polarized system with electrons restricted to occupying N - 0 states. In this case the expression for the total energy simplifies considerably because the interaction energy can be diagonalized independently of the confinement energy. In addition, the confinement energy in this case is only a function of J, so the total energy of each state takes the form
1949
high-field limits. When B - 0 , the confinement term is heJ/2m*, and as B increases, it smoothly decreases and approaches -hene/2m* as B - - o o . In contrast, the interaction term approaches 0 both when B - - 0 and when B - - oo. For the parameters used here this term contributes at the I% level when B < 2.5 T, and -,-15% when B-" 10 T; it is most significant when 2.5 < B < l0 T. The field dependence of the magnetization at fixed J is essentially determined by the first term, and if J was independent of B, the magnetization of the interacting system would be qualitatively similar to that of the noninteracting system. The major effect of the interaction is that the ground state J changes with magnetic field. Every time this happens the magnetization curve shifts to a different 9track, and this causes the discontinuities shown in Figs. i and 2. The jumps in the ground state J occur because at the magic J values there are basis states in which electrons are kept very. effectively. The ground state always occurs at one of these J values and the competition between interaction and confinement determines I the optimum J. Physically, the preference for certain J values can be understood in a number of equivalent ways; the simplest is to consider the diagonal elements of the Hamiltonian. These have,he form ~.,~tt'nlnr, where nt is the numberoperator and o4#,"o4otororot-o~otorotor. The quantity ~tr is the difference between the Hartree energy and the exchange energy of a pair of electrons with angular momenta I and r. It is plotted in Fig. 3 as a function of !' for the case when 1-'5, and this illustrates its behavior for typical values of I and 1'. When 1 "1', ~ t r - O then increases as I I - l'l is increased, and then it decreases again. Thus it is energetically favorable to have t l - l ' l either large or small for all pairs of electrons; however, large values can only occur when the total angular momentum is large. The optimal way of making I 1 - rl small is to put all the electrons onadjacent orbitals, but this can only be done when J satisfies J ~'ne (ne -- ! )/2 + kne, where k -'0, !, 2 . . . . . For three electrons this leads to the magic angular momenta 3, 6, 9, 12. . . . . while for four electrons it gives 6, 10, 14, 18. . . . . These are exactly the values found in numerical calculations for the spin-polarized case. These calculations also confirm that basis states in which all
! e2 e -(n~ + j )h ~ - .~Jh co,.+ 4'=~eoa~(J) + g* ~.BS:,
0.35
where the first two terms are the confinement energy, the third is the interaction energy, the fourth is the Zeeman energy, k(J) is a dimensionless eigenvalue that depends only on J, and g* is the effective g factor. For GaAs, g* is small, so the Zeeman term only affects the magnetization at the 1% level, and the physics is determined by the first three terms. Differentiating them yields two contributions to magnetization: .4(.,,
-
h...__~e (J + ne ) o9c
2m*
~
-J
I 1
e 2
e
wc
8n-~:e0 (hm*)l/2 'i21"1)3/2z ( J ) " These two terms behave very differently in the low- and
,
0.300=250~'0-
p.
..o.- 4--,i,...
.e..+.
/ .. .+
~, -
I<':' o.~0.100.05
9
.
..
-
..
0.00 0
5
I
I
10
15
20
FIG. 3. The 1' dependence of ~st' (units: e2/4Jreeoa). The points give values of ..4 and the dashed line is to guide the eye.
236
1950
P. A. MAKSYM AND TAPASH CHAKRABORTY
electrons occupy adjacent orbitais occur with high probability; for example, 53.5% for three electrons at J - ' 9 and 48.8% for four electrons at J - 14. in this physical picture the magic angular momenta are favored because the exchange term efficiently reduces the energy of basis states in which all electrons sit on adjacent orbitals. An alternative way of looking at the situation is to consider the motion of a pair of electrons about its center of mass. In this picture small relative angular momenta occur with very small probability if the electrons occupy adjacent orbitais. This leads to a reduction in energy because the large-angular-momentum matrix elements of the Coulomb interaction are the smallest. The remaining item to consider is spin. From Figs. 1 and 2 it is clear that spin effects are important at fields B < 10 T, and that the system is spin polarized at higher fields. It is perhaps surprising that the occurrence of full spin polarization in the range 10__
values lead to discontinuities because the lowest-energy state at a given spin is not always the absolute ground ! state. In the case of three electrons at S = 7, the magic J values for states formed from the zeroth Landau level are 2, 5, 8, II . . . . . but only 2 and 5 lead to discontinuities because S - ~ states have lower energies than the remaining S - ~- states. In addition, J "- I occurs at low magnetic fields, where the contribution of N "- I states is important. The four-electron case is more complicated. Although magic J sequences exist, not every member of them occurs at all strengths of the confining potential. For example, the magic J values for states formed from the zeroth Landau level at S "-1 are 5, 9, 13, 17 . . . . . and some of these occur in Fig. 2. However, J " 9 does not occur if the confinement is very weak and for moderate values of confinement J ' - 13 shifts to J - ! 2. Because of this sensitivity to the confinement, it is difficult to give a rule that determines the magic J values for an arbitrary number of electrons at an arbitrary spin. In contrast to the spin-polarized ease, it seems that the magic J values have to be determined numerically. In summary, the low-temperature magnetization of quantum dots provides a sensitive probe of the effect of the electron-electron interaction on the ground state. The key physics can be understood in terms of a simple model in which only the zeroth-Landau-level states are taken into account.
Ip. A. Maksym and Tapash Chakraborty, Phys. Rev. Lett. 65, 108 (1990). 2L. Brey, N. F. Johnson, and B. i. Halperin, Phys. Rev. B 40, 10647 (1990); see also, F. M. Peeters, ibid. 42, 1486 (1990); V. Shikin, S. Nazin, D. Heitmann, and T. Demel, ibid. 43, 11903 (1991). 3Tapash Chakraborty, V. Halonen, and P. Pietil/iinen, Phys. Rev. B 43, 14 289 (i 99 ! ). 4T. Haavasoja, H. L. St6rmer, D. J. Bishop, V. Narayanamurti, A. C. Gossard, and W. Wiegmann, Surf. Sci. 142, 294
(1984); J. P. Eisenstein, H. L. St6rmer, V. Narayanamurti, A. Y. Cho, A. C. Gossard, and C. W. Tu, Phys. Rev. Lett. 55, 875 (1985). 5V. Fock, Z. Phys. 47, 446 (1928). 6C. G. Darwin, Proc. Cambridge Philos. Soo. 27, 86 (1930). 7U. Sivan and Y. lmry, Phys. Rev. Lett. 61, 1001 (1988). 8Tapash Chakraborty and P. Maksym, in Proceedings of the CAP-NSERC Workshop in Theoretical Physics: Excitations in Superlattice and Multi-Quantum Wells, London, Canada (unpublished).
One of us (P.A.M.) is grateful for the support of the United Kingdom Science and Engineering Research Council. The other (T.C.) would like to thank U. Merkt for helpful discussions.
NRCC 32801
237
PHYSICAL REVIEW B
15 JANUARY 1992-II
VOLUME 45, NUMBER 4
Spin-singlet-spin-triplet oscillations in quantum dots M. Wagner Hitachi Cambridge Laboratory, Hitachi Europe Limited, Cambridge CB3 0HE, United Kingdom U. Merkt and A. V. Chaplik* lnstitut fiir Angewandte Physik, Unieersitiit Hamburg, Jungiusstrasse ! I, 2000 Hamburg36, Germany (Received 12 August 199 i ) Two interacting electrons confined to a disk on a semiconductor surface are considered in a perpendicular magnetic field. As it is appropriate for experimental realizations, we use a two-dimensional harmonic-oscillator well to confine the electrons in the plane of the disk. We predict oscillations between spin-singlet and spin-triplet ground states as a function of the magnetic field strength. Phase diagrams describing this peculiar manifestation of the electron-electron interaction in a quantum dot are calculated for GaAs and experiments to verify them are proposed.
Nanostructure technologies allow the lateral confinement of two-dimensional electron gases in heterojunctions or metal-oxide-3emiconduetor structures to widths comparable to the effective Bohr radius a* of the host semiconductor. I In this case we have electron systems with discrete energy spectra that are commonly called zero-dimensional systems or quantum dots. 2-9 Since their widths in the x-y plane are much larger than their extent in the z direction, which is the growth direction of the underlying semiconductor structure, quantum dots may be regarded as artificial atoms with disklike shapes. Electron numbers as low as one or two per dot have already been realized. 6.9 So far, quantum dots have been investigated experimentally by capacitance-voltage spectroscopy 3 and transport measurements, 4'5 as well as by far-infrared spectroscopy.6-9 Capacitance-voltage and transport measurements are not favorable for the study of isolated dots since they require coupling to external contacts. Particularly for small dots their interpretation is additionally hampered by Coulomb blockade. Despite these difficulties, much information on the single-electron energy spectra could be deduced from transport data.5 In many cases the value of far-infrared spectroscopy 6-9 is limited as a consequence of the approximately harmonic shapes of the confining potentials and the associated validity of the generalized Kohn theorem.~~ This theorem states that, for strictly harmonic potentials, dipole radiation can only probe the center-of-mass motion of all electrons but is inadequate to see any effect due to the electron-electron interaction. Here, we predict spin oscillations of the ground state of two electrons in a harmonic quantum dot as a function of the magnetic field strength, which are a peculiar consequence of the electron-electron interaction and the Pauli exclusion principle. Hence, they are a direct manifestation of the two-electron states in the quantum dot. The oscillations should be accessible to a different type of experiment, namely spin susceptibility12 and magnetization measurements 13 that previously have been successfully applied to study electronic properties of two-dimensional electron gases in GaAs/Gai-xAlxAs and related hetero45
structures.
In experimentally realized dots, the motion in the z direction is always frozen out into the lowest electric subband Ei-o. Since the corresponding extent of the wave function is much less than the one in the x-y plane, we can treat the dots in the two-dimensional limit of thin disks. For most dots, a harmonic oscillator is a very good approximation to describe the lateral confinement of the electrons. 2-6'14 Hence we consider two electrons of effective masses m* in the z - 0 plane in the harmonic potential 89m * r 2+y 2) of characteristic frequency Cooor oscillator length lo-(h[m*Coo) I/2. The perpendicular magnetic field (BIIz) is in the symmetric gauge described by the vector potential A " ~ - ( - y , x , 0 ) B . A dielectric constant e accounts for the host semiconductor. Ignoring the Zeeman spin splitting for the present, the Hamiltonian can be separated into center-of-mass and relative-motion terms as H - [ P + Q A ( R ) 1 2 + ~. MoJo~2+ [ p + q A ( r ) ] 2 2M + 89
2// 2h
e2 1 4~ree0 r
(1)
by introducing the center-of-mass coordinates R - ( r l +r2)/2, P ' - p t +p2, the total mass M = 2 m *, and charge Q - 2 e > 0, as well as the relative coordinates r - r l - r 2 , p - - ( p l - p 2 ) / 2 , the reduced mass/1-'-m*/2, and charge q -e/2. This separability and the cylindrical symmetry of the problem allow us to write the two-particle wave function in plane polar coordinates r - ( r , r in the form ~(R)~(r)exp(imr The spatial part of the total wave function is symmetric or antisymmetric with respect to particle permutation (r r for even, respectively odd, azimuthal quantum numbers m. Since the Pauli exclusion principle requires the total wave function to be antisymmetric, we therefore have spin singlet ( S = 0 ) and triplet (S - I ) states for even and odd m. 15 The energy eigenvalues of the Hamiltonian in Eq. (1) are the sum of the center-of-mass energy and the energy of the relative ! 951
9 1992 The American Physical Society
238
1952
M. WAGNER, U. MERKT, AND A. V. CHAPLIK
motion. The former is given by io !/2
+h-yM (.O c
EN.M--h(2N+,M.+I) [ a ~ + [ - ~ -
(2) with cyclotron frequency oo,.-eB/m*, radial ( N - - 0 , 1 , 2 . . . . ), and azimuthal ( M - - 0 , __+I, __. 2 . . . . ) quantum numbers. The energy of the relative motion has corresponding quantum numbers n and m and includes the electron-electron interaction. The spin of the two electrons leads to an additional Zeeman energy
Es - g * #aBS: - g * m* .h t~ S: me 2
(3)
described by an effective Land~ factor g*. Thus triplet states split into three distinct levels, while singlet states remain unchanged. The exact eigenvalues E,,.,, of the relative motion including the Zeeman energy are calculated numerically. Is We now investigate the ground state of the two-electron system as a function of dot size and magnetic field strength. Since the center-of-mass quantum numbers N,M and the quantum number m are conserved by the Coulomb interaction, the ground state has the quantum numbers N " 0 , M - 0 , n '10, m _< 0, and only m is to be
i
i
i
i
lola*=3
(a) - I0
independent electrons
2
-]
I
"-~ m--
0
t
t r
t
l~
(b)
I
=3
interacting electrons
aa
~
1
=0 tl m=-I IS=01
0
S=l t
0
1
I
m=-2
I~ m=-3
~ II 1
S=0
I S=!
I
t
1
oJ,, / coo
1
4
t
5
FIG. !. Eigenenergies in units of the effective Rydberg constant R* vs the ratio to,./too for a dot size lo/a* ==3 and Land~ factor g* z0. The family of states N 50, M 50, n "=0, m _< 0 is shown (a) without and (b) including Coulomb interaction between the two electrons. As the ratio to,./t~ increases, the Coulomb interaction leads to a sequence of different ground states m "=0, - I, - 2. . . . . and concomitant changes of the total spin S =~0,1,0. . . . .
45
determined. The important feature of the ground state to be discussed here is that its angular momentum hm does depend on the Coulomb interaction. In Fig. I we have plotted the energy of the states N - 0 , M - 0 , n --0, m __<0 for vanishing Land6 factor g * - ' 0 and dot size Io/a* "-3 as a function of the ratio toe~tOoof cyclotron and oscillator frequency. In Fig. I(a) we neglect the Coulomb interaction and the m " 0 state is always the ground state. If, however, we include the Coulomb interaction in Fig. i (b), the state m - 0 remains as a ground state only for low magnetic fields. As the magnetic field increases, this state rims in energy while the states m - - 1 , - 2 , - 3 .... drop, thus leading to a sequence of different ground states m-0,I , - 2 , - 3 . . . . as the magnetic field is swept. Since the total spin of the two electrons is S - ' [ I - ( - I ) ' ] / 2 , this entails an alternating sequence of singlet and triplet states. The reason for these changes of symmetry of the ground state is found in the competition of the various energies contributing to the energy of the relative motion. On the one hand, a higher angular momentum h m means higher rotational energy, but on the other hand, the average distance between the two electrons is then increased and hence the Coulomb energy gets smaller. With the relative strength of the Coulomb interaction varying as [l+(toc/2too)Z]-t/41o/a *, the optimum number m thus depends on the dot size and on the magnetic field. By properly designing the dot size lo/a* and adjusting the relative strength of the magnetic field toe/too, it should therefore be possible to investigate different ground states. A suitable tool to visualize this is a phase diagram in the lo/a* - to,./too plane. The transition m --~ m - l between a singlet and a triplet ground state is given by the condition Eo.,, - Eo.,,-! (m <_0). These are the only possible transitions as long as the Zeeman spin splitting Is ignored. For a negative Land~ factor g* < 0, the spin-splitting energy in magnetic fields will lower the energy-of the spin S= - + I component of the triplet states while leaving the singlet states unchanged. Thus the phases of the triplet states will increase at the cost of the singlet phases, and eventually, for high magnetic fields, the singlet ground states are totally suppressed. Singlet-triplet transitions only exist for Eo.,, < Eo.,,,+! (m even), and for Eo,,,-i < E o .... -2 (m odd). In particular, the relation Eo.... --'Eo....- i " E o . , , , - 2 (m odd) defines triple points where singlet phases cease to exist. Beyond this point we are left with phase transitions between triplet states described by the condition Eo.,, ==Eo.m-2 (m odd). In Fig. 2 we show phase diagrams based on an exact numerical diagonalization of the Coulomb interaction. In Fig. 2(a) the Land~ factor is set to zero. Then we have only singlet-triplet phase transitions m--* m - I,m - 2 . . . . starting with the m - 0 singlet phase at zero magnetic field, in Fig. 2(b) we assume the Land~ factor g*--0.44 and effective mass m*-O.O67me of bulk GaAs conduction-band electrons. For higher magnetic fields the singlet ground states now completely vanish and only triplet-triplet phase transitions m (odd)--* m - 2 , m - 4 . . . . are left. For a typical experimental dot size of lo/a*-3 the last singlet phases present are at to,./wo < 0.69 and 2.2 < w,.Itoo < 2.9.
239
45
SPIN-SINGLE'T-SPIN-TRIPLET OSCILLATIONS IN QUANTUM DOTS ,0
T
0
.....
(a) 8
i
'1
.~
g* =0
1. . . . . . . . . . . . . .
l~ a* = 3
o \k,
"2 4 ,r = - _
i
1953
"'"
2 0 8
6
I ....
2
0
0
0
2
4
o9,.Io9o 8
10
2
FIG. 2. Phase diagram for singlet and triplet ground states calculated (a) in the absence and (b) in the presence of Zeeman spin splitting with a Land~ factor g * = - 0 . 4 4 appropriate for GaAs. in the absence of spin splitting the singlet phase m - 0 (S--0) in zero and low magnetic fields is followed by m " - ! (S=I), m--2 ( S - ' 0 ) . . . . phases as the magnetic field strength is increased, in the presence of spin splitting the singlet phases are strongly suppressed, and only the hatched m -'0, -- 2, - 4 singlet phases are left.
Essential features of these phase diagrams can already be understood, if we calculate the Coulomb energy Eco,h,,,b in first-order perturbation theory, which is valid for [l+(m,./2a~o)2]-1141o/a*<
(n - 0 ) Ecou~,~b
"
l~
!+
o9'"
a
21/4
~
(2lml- I)!!
[ 2 1 / 2
(2lml)!! (4)
For dot sizes lo/a *> !, strong magnetic fields (m,. >>ogo), and Ig*lm*/m~<
2o90 3/2 [ ~ + 2 o9'
oge
I/2 . ( - 1 )"'g* m
[~
(5)
me
and for triplet-triplet phase transitions m - * i n - 2 odd) 10
a*
1
~
2(21mJ +4)!!
[ 2o90
(21ml-l)!!(41ml+5)t--d~,.
3/2
(m
(6)
By using Eqs. (5) and (6), the exact triple point at
2
-,! .....
4
-"
,1 ..... ~
B (T)
~
8
........
10
FIG. 3. Magnetic moment P,,~s in units of the Bohr magneton pn of a single quantum dot with two electrons vs magnetic field strength for GaAs parameters. The curves for various temperatures have been successively displaced for clarity. Oscillations caused by changes of the azimuthal angular momentum and the tota!~ spin become visible for temperatures less than about i K.
m,.[mo'4.48, Io[a* " 1.27 in Fig. 2(b) is approximated to ~ be m,./mo=4.76, io/a* "1.16. Hence, the perturbational results can be used advantageously to estimate phase diagrams for other effective masses m * or Land~ factors g*. Based on the exact eigenenergies, the magnetic moment p,,~g(B) of a single dot with two electrons is plotted in Fig. 3 in units of the Bohr magneton pa'=eh/2me for various temperatures. At T = 4 . 2 K we find a significant but smooth diamagnetic behavior wtfile for temperatures less than ! K strong oscillations become visible. These contributions with amplitudes of more than 10ps exceed the paramagnetic spin contribution - g * p B of two independent electrons by more than an order of magnitude and show most obviously the two-particle nature of the ground state at intermediate magnetic field strengths. The fact that these oscillations are Coulomb induced is easily understood for zero temperature where we have Pmag(B,T=O) = -OEground/OB. By comparing Figs. ! (a) and I(b) we see that discontinuities in this derivative occur at the phase boundaries and are therefore a clearcut consequence of the Coulomb interaction. For instance, the sharp drop of the magnetic moment at B =0.51 T is caused by the first singlet-triplet transition m =0---* m = - I . In high magnetic fields (m,. >> o90) the oscillations vanish and we are left with the saturation moment I~./PB = - 2mdm* - g * = - 2 9 . 4 , valid for two independent electrons in GaAs. To conclude, we predict spin singlet-triplet and triplettriplet transitions of the ground state of two interacting electrons in quantum dots in a perpendicular magnetic field. In principle, our prediction can be verified by spin
240
1954
M. WAGNER, U. MERKT, AND A. V. CHAPLIK
susceptibility or magnetization measurements at low temperatures ( T < I K). We are aware of the intensity problems resulting from the low number of electrons, even if arrays of 10 9 dots/cm 2 are used. Perhaps it is particularly challenging that the analogous singlet-triplet transition from para- ( S - 0 ) to ortho- ( S - I ) helium, predicted at
about B - - 4 x 10 5 T in the vicinity of white dwarfs and pulsars, i6 also remains to be observed.
*Permanent address: Institute of Semiconductor Physics, U.S.S.R. Academy of Science, Novosibirsk, U.S.S.R. IG. W. Bryant, Phys. Rev. Lett. 59, ! 140 (1987). 2M. A. Reed, J. N. Randall, R. J. Aggarwal, R. J. Matyi, T. M. Moore, and A. E. Wetsel, Phys. Rev. Lett. 60, 535 (1988). 3W. Hansen, T. P. Smith 111, K. Y. Lee, J. A. Bruin, C. M. Knoedler, J. M. Hong, and D. P. Kern, Phys. Rev. Lett. 62, 2168 (1989); 64, 1991 (1990); W. Hansen, T. P. Smith I!1, K. Y. Lee, J. M. Hong, and C. M. Knoedler, Appl. Phys. Lett. 56, 168 (1990). 41.. p. Kouwenhoven, F. W. J. Hekking, B. J. van Wees, C. J. P. M. Harmans, C. E. Timmering, and C. T. Foxon, Phys. Rev. I=tt. 65, 361 (I 990). 5p. L. McEuen, E. B. Foxnlan, U. Meirav, M. A. Kastner, Y. Meir, N. S. Wingreen, and S. J. Wind, Phys. Rev. Lett. 66, 1926 (1991). 6Ch. Sikorski and U. Merkt, Phys. Rev. Lett. 62, 2164 (1989); 64, 3100 (1990); Surf. Sci. 229, 282 (1990). 7A. Lorke, J. P. Kotthaus, and K. Ploog, Phys. Rev. Lett. 64,
2559 (1990). ST. Demel, D. Heitmann, P. Grambow, and K. Ploog, Phys. Rev. Lett. 64, 788 (1990). 9B. Meurer, D. Heitmann, and K. Ploog (unpublished). lop. A. Maksym and T. Chakraborty, Phys. Rev. Lett. 65, 108 (1990). I !p. Bakshi, D. A. Broido, and K. Kempa, Phys. Rev. B 42, 74 i 6 (1990). 12M. Dobers, Surf. Sei. 229, 126 (1990). 13j. p. Eisenstein, H. L. Stormer, V. Narayanamurti, A. Y. Cho, A. C. Gossard, and C. W. Tu, Phys. Rev. Lett. 55, 875 (1985); J. P. Eisenstein, Appl. Phys. Lett. 46, 695 (1985). 14A. Kumar, S.E. Laux, and F. Stern, Phys. Rev. B 42, 5166 (1990). 15U. Merkt, J. Huser, and M. Wagner, Phys. Rev. B 43, 7320 (1991). 16G. Thurner, H. Herold, H. Ruder, G. Schlicht, and G. Wunner, Phys. Lett. 89A, 133 (1982).
We thank J. Huser for the supply of numerical routines and acknowledge financial support from the Deutsche Forsch ungsgemeinsehaft.
241
PHYSICA
Physica B 184 (1993) 385-393 North-Holland
Magic number ground states of quantum dots in a magnetic field P.A. M a k s y m Department of Physics and Astronomy, University of Leicester, UK The electron-electron interaction in quantum dots leads to interesting effects which are highly elusive. For example, the magnetization of dots cont'aqningvery small numbers of electrons is predicted to oscillate with magnetic field. The reason is that the ground state prefers to be at certain magic values of the total angular momentum which are field-dependent and the oscillations mirror jumps from one magic angular momentum value to another. This behaviour is a direct consequence of the Pauli principle which enables the electrons to reduce their energy optimally only at the magic angular momenta. An expression for the magic angular momenta in the spin-polarized case is given and electron probability distributions are computed to illustrate the physical difference between magic and nonmagic states. In the limit of large angular momentum the ground state appears to be the molecular analogue of a Wigner crystal.
1. Introduction
The interaction between 2D electrons confined in q u a n t u m dots in a magnetic field has a -dramatic effect on their energy spectrum. Indeed, numerical studies of very small interacting systems [1] have shown that physically observable ground states only occur at certain magic values of the total angular m o m e n t u m , - h J . Usually several different magic values compete and which one is selected depends on the strength of the magnetic field. This is because the cyclotron radius decreases with field and this has a drastic effect on the equilibrium between the effects of repulsion and confinement. Several authors have presented calculations of this effect [1-5] and have shown that it leads to oscillations in the thermodynamic properties of the dots. A detailed explanation of the underlying physics and the origin of the magic numbers is still lacking although Maksym and Chakraborty [2] have presented a physical argument which correctly predicts the magic J values of very small spin-polarized systems. The purpose of the presCorrespondence to: P.A. Maksym, Department of Physics and Astronomy, University of Leicester, University Road, Leicester LE1 7RH, UK.
ent work is to explore this underlying physics which turns out to be remarkably simple. First a novel approach is used to visualise the states, then it is shown that the occurrence of energy minima at the magic J values is a direct cons e q u e n c e of the Pauli principle and finally the large-J limit is studied in quantitative detail. In this limiting case the ground state is analogous to one of the Lagrangian solutions of the classical n-body problem and its excitations correspond to small oscillations about the classical Lagrangian orbits [6]. This state could-also be termed a 'Wigner molecule' by analogy with the Wigner crystal.
2. Ground states
The electrons are taken to be confined by a circularly symmetric quadratic potential. Therefore their quantum states in the absence of the interaction are the D a r w i n - F o c k states as described in refs. [1,2,5] for example. Except for a normalisation constant they have the form
q~,,=c
,
(r2) (_r2)
t..,, ~ - y
-0921-4526/93/$06.00 (~ 1993- Elsevier Science Publishers B.V. All rights reserved
exp - ~ f
,
(1)
242
P.A. Maksym / Magic number ground states of quantum dots
386
where L~ 1 is a Laguerre polynomial. The energies are
E.,-' ha(2n +
Itl +
scale and this roughly determines the ground state angular momentum: because of relation (2) large angular momentum implies a large confinement energy and small angular momentum implies a large repulsive energy. The competition between these effects leads to a broad minimum in Eo(J ) as shown in fig. 1, and the position of this minimum is field-dependent because A depends on B according to eq. (3). The Eo(J ) curve has some downward cusps around the broad minimum and these occur at the magic J values which are all multiples of 3 in the threeelectron spin-polarized case. Changing the field changes the J values of the ground state and hence affects the excitation spectrum. This causes oscillations in the thermodynamic properties such as magnetization and heat capacity [1-3,5]. For example, oscillations in magnetization are shown in fig. 2. Some of these are due to changes in the total spin S although this work is mostly concerned with spin-polarized systems. The origin of the magic numbers can be under-
1 1) - ~ hw c
where ,O 2 = w 02 + w2/4, h~o0 is the confinement energy and ~0r is the cyclotron frequency. In the limit of zero confinement these states approach the usual Landau states and it is convenient to use the term Landau levels to denote those levels whose energies differ by about h~0r in the large field limit. Physically the Darwin-Fock states are localised on rings whose radius R is given by R 2"~" 2 A 2 ( 2 n +
Izl +
1)
(2)
where the length parameter A, given by A2 =
h 2m*a
(3)
'
is a most important parameter. It sets the length 80-
:2>
O9
n = 3, hw ~ - 4 m e V
78
0.36 0.30
.... "
0.26 0.20 76
I<{'~
0.16 0.I0
~9 or3
@
006
%
9
o
@ 0
C.~
@
0.00
74
9
72
9
@
9 0
0
0
0
0
I
I
I
I,
I
I
5
i0
15
20
25
30
Angular momentum Fig. 1. G r o u n d state energy of 3 interacting electrons at B = 20 T against angular momentum quantum number J. The inset shows the angular m o m e n t u m dependence of the difference between the direct and exchange energies of a pair of electrons as discussed in the text.
243
387
P.A. Maksym / Magic number ground states of quantum dots
j = n ( n - 1) 2 + kn,
where n is the number of electrons and k is an integer. These are precisely the magic numbers for spin-polarized systems of 5 or less electrons. The (1, 1) particle-hole symmetric states found by Kingaret et al. [7] have the same angular momenta. At first sight it may seem paradoxical that the energy of the system is minimized when the electrons occupy adjacent states but some reflection shows that it is not. If two electrons in an antisymmetric state are at the same radial coordinate and are adjacent in angular momentum space their wave function contains the angle dependent factor approximately equal to
5
:~
-2
Ir
Bm
~
_-......
-3 10 8 6
f
,U
4
,/--
2 0 CO
i
1
0
IU 5
(4)
/
exp(-i(/+ 89 10
+ q:,~))sin~, e
~2 \
.....}
15
B (T) Fig. 2. Magnetic field dependence of magnetization for 3 interacting electrons. The results are calculated with 2 Landau levels and the inset shows the effect of increasing the number of Landau levels to 3.
stood most simply as a consequence of the exchange interaction [2]. The exact states can always be expanded as anti-symmetrized products of the single-electron states defined by eq. (1). Each product corresponds to a configuration of electrons and the configurations that occur with the highest weight are generally those with the lowest energy. A configuration's energy is the sum of the energies of all pairs of electrons and the energy of a pair with angular momenta I and l' is the difference A of a direct (or Hartree) term and an exchange term. This difference is shown in the inset to fig. 1 for the case when one electron has 1 = 5 and both have n = 0 . It is clearly smallest when the electrons are adjacent in angular momentum space, i.e. on adjacent rings in real space. The energy of a configuration is particularly low when all the electrons are adjacent but this can only happen when the total angular momentum satisfies
o f which the modulus is largest when (qh - q~2)= +_,rr, that is when the electrons are opposite to each other and this of course reduces their Coulomb energy. To visualise the distinction between magic and, nonmagic states it is convenient to use the conditional probability P(r, to) of finding an electron at r given that there is one at r 0, that is the ground state expectation value of
Z $(r-- ri)~(r o -- rj) . i#j
This is shown in fig. 3. The frames show P for magic ( J = 9 ) and nonmagic ( J = 8 ) ground states of 3 electrons. The dot indicates the position of the fixed electron and r 0 is chosen to be the radius of a classical triangular Lagrangian orbit. Each calculation is done with basis states formed from one and two Landau levels but the qualitative appearance of P is independent of the number of Landau levels which shows that the essential physics can be understood in terms of zeroth Landau level states. Clearly, P has wellseparated peaks when J = 9 but its form is much broader when J = 8. This reflects the tendency for the electrons to stay apart when J = 9.
244
388
P.A. Maksym / Magic number ground states of quantum dots n = 3, J = 8, N,,,~ = 0
,,.
n = 3, J = 8, N,~,~ = 1
n =3, J=9, N,,,~=O
Fig. 3. Probability distributions P(r, ro) for magic J = 9 and nonmagic J = 8 states of 3 interacting electrons in a quantum dot.
3. Permutational and rotational symmetry W h i l e the physical a r g u m e n t for the e x i s t e n c e of m a g i c n u m b e r s is c o m p e l l i n g it d o e s n o t p r o vide a full e x p l a n a t i o n of t h e effect. It t u r n s o u t
t h a t this c an be d o n e solely o n the basis of s y m m e t r y c o n s i d e r a t i o n s , so it is n e c e s s a r y to c o n s i d e r the s y m m e t r y of t h e i n t e r a c t i o n p o t e n tial. Since the c o n f i n e m e n t is p a r a b o l i c the centr e of m a s s ( C M ) m o t i o n s e p a r a t e s o u t e v e n in
245
P.A. Maksym / Magic number ground states of quantum dots
the presence of the confinement [1]. However, the CM wave function is symmetric under all permutations so its form is irrelevant to the permutational symmetry of the full wave function. The Coulomb potential is only a function of coordinates that describe the relative motion (RM). More precisely, it depends on n - 1 distances and n - 2 angular coordinates. The remaining angular coordinate X specifies the orientation of the electrons in the relative coordinate system. There is a great deal of freedom in choosing the relative coordinates. One convenient choice is to specify the instantaneous positions of the electrons as displacements from their classical equilibrium positions (i.e. the positions given by the Lagrangian solutions of the classical n-body problem). Further, it is also convenient to use normal coordinates and a moving frame of reference as employed in molecular physics [8,9]. The RM hamiltonian then becomes
HaM
89
=
_
L*)2+ ~
h2/z + V +
8
ha~r
--~
1 2~3
LRM
k=,
'
p~
(5)
where LR M _ h
19
i 19X'
Lv
= E ~kPk, k
~k = ~-,(Qij • Qik)" fcQ y q
and
,to tx -- ( I 0 + m * ~i] ai
" QijQ j)2 9
Q j a r e normal coordinates, Pj are the associated momenta, Qij are elements of the matrix that transforms cartesian to normal coordinates, a i are the equilibrium positions, 10 is the equilibrium moment of inertia and V is the total potential (including quadratic terms from the magnetic vector potential). In addition, LRM is the RM
389
angular momentum, L v is angular momentum associated with 'vibrational' motion and these two momenta are coupled because Coriolis forces appear in the moving reference frame. Although this hamiltonian is intended for the study of vibrational excitations it is exact provided no approximations are made to the potential. Since V is independent of X the RM wavefunction can be written in the form exp(--iJRMX)~RM where the effective hamiltonian for ~RM is obtained by substituting in eq. (5). The effective hamiltonian thus acquires a centrifugal potential term and the normal modes can be found by diagonalizing the matrix of second derivative of the total effective potential (i.e. the sum of Coulomb, confinement and centrifugal potentials). Antisymmetric states of the original RM hamiltonian can be constructed from states of Herf and the advantage of this otherwise complicated formulation is that this problem can be solved by using the methods of molecular physics [10,11]. The general idea is that some of the cyclic permutations are equivalent to rotations, so an antisymmetric state can only be constructed from a state of Heff if it has the correct rotational symmetry. An antisymmetric state must be even under these special permutations if n is odd and odd if n is even. Both exp(--iJRMX) and ~RM change phase under a rotation so the only possible combinations are such that the phase change is +_1, depending on whether n is odd or even. Now if the ground state of Heft has no nodes it will be symmetric under rotations; therefore the angular momentum factor is required to change phase by + 1. However, this is only possible if JRM is magic. In this case the RM ground state can be constructed from the ground state of Heff. At nonmagic J values the antisymmetric ground state must be constructed from an excited state of Heft~ This leads to the possibility of a downward cusp in E(JRM ) but numerical calculations are required to determine if one actually occurs. There is an alternative coordinate system in which the equivalence of permutations and rotations can be exhibited explicitly and this offers further insight into the nature of the magic states. Consider the orthogonal coordinate transformation defined by
246
P.A. Maksym / Magic number ground states of quantum dots
390
Zk = ~
1~
jk
to zj
(6)
./=1 ..
where z i = xj - iyj and to = exp(2-rri/n). Z, is the CM coordinate and the remaining Zj are RM coordinates. A cyclic permutation of zj clearly changes Z k by a factor tok so it is equivalent to a rotation. The RM hamiltonian for noninteracting electrons separates when it is expressed in terms of the Z k because the coordinate transformation is orthogonal. The eigenstates of this hamiltonian are therefore products containing n - 1 of the functions defined in eq. (1) and the eigenstates of the interacting system can be expanded in antisymmetrized combinations of these products. The l values in each product are restricted n--1 by E j= t lj = JRM and the equivalence of permutational and rotational symmetry leads to the fur--1 9 ther restriction that the sum E nj=l llj is equal to n p if n is odd and np + 89n if n is even, where p is an integer. After some algebra it can be shown that an important consequence of these restrictions is that basis states in which all the I values except one are zero and the remaining I value is magic have the property that their squared modulus is largest when all the electrons sit at the corners of a regular n-sided polygon whose radius is equal to the radius of a classical circular orbit in the confining potential. This state only occurs at the magic J values and has precisely the angular distribution of electrons that is required to minimise their Coulomb energy. Exactly that distribution is evident in fig. 3.
ground states at the indicated field. It is clear that the peaks are very sharp in this limit and their positions correspond to the corners of a regular polygon. This suggests that the electron state in this limit is a 'Wigner molecule' in which the equilibrium positions of the electrons are given by a classical Lagrangian orbit and whose excitations are small oscillations about this steady-state orbit. This idea can be tested quantitatively by calculating the excitation energies. Detailed examination of Her f shows that when the potential is expanded about the classical equilibrium positions all the terms of 3rd order or higher in the displacements are of order 1/ VTRM or smaller. If these terms and the Coriolis coupling are discarded Heft becomes the hamiltonian for a vibrating rotator; however, this is poor approximation because the Coriolis coupling is of the order hg2 which is large. This effect can be taken into account exactly by a canonical transformation similar to the one employed in the classical theory of small oscillations about a steady state [12] and the resulting vibrational energies agree very well with the results of exact calculations. This is shown in fig. 5. The two parts of the figure show approximate (solid line) and exact (points) excitation energies of 2 and 3 , interacting electrons. The curve for 3 electrons actually shows 3 times the vibrational energy since a minimum of 3 vibrational quanta have to be excited to preserve antisymmetry in this case. The fluctuations in the exact~3-electron excitation energy are correlated with changes in the size of the basis set. The RM ground state energy in the large-J appro~iination is given by
4. The large angular momentum limit Eo(J.M) = Em,n(J.M)+
As discussed the introduction, ground states with large angular momentum are expected to occur in the limit of large magnetic fields because the confinement is then less important. Probability distributions for 2 . 3 , 4 and 5 electrons in this limit, calculated from only n = 0 basis states, are shown in fig. 4. The value of r 0 is chosen as for fig. 3 and the states are all magic states with k = 9. The figure illustrates the typical behaviour of very small numbers of electrons in the large-J limit but the states shown are not absolute
Ez.(J.~)+ Ex(J.M)
(7)
where Emi n is the minimum energy of the effective potential, Ezp is the zero point energy and E x is an excitation energy determined by the lowest vibrational state compatible with antisymmetry. According to the rules given in the previous section E x is nonzero only when the ground state is magic. Figure 6 shows the RM ground state energy of the 3-electron system. There are clearly downward cusps a t magic JRM values and the
247
P.A. Maksym / Magic number ground states of quantum dots
391
n=3, J=30
n = 2 , J = 19
n = 4, J = 42
n = 5, J = 55 6O
~0'
4~
3O"
2O+
tn
?o
10
;0
'
l~
;~
,'o
?o
,'~
,',
I~
'
'0
9Fig. 4. Probability distributions P(r, ro) for 2, 3, 4 and 5 electrons in the limit of large angular momentum. a b s o l u t e g l o u n d state is at JRm = 12, in a g r e e m e n t with the exact results. T h e d i f f e r e n c e bet w e e n g r o u n d state e n e r g i e s at JRm = 3p a n d JRm = 3p + 1 is h i g h e r t h a n the smallest C M e x c i t a t i o n e n e r g y . T h e r e f o r e the g r o u n d state of
t h e full h a m i l t o n i a n at J - 3p + 1 is an excited s t a t e of the C M with JRM = 3p and Jcm -- 1. T h e l o w e s t R M state a p p e a r s as the first excited state of the full h a m i l t o n i a n a n d this again agrees with t h e exact results. It is r e m a r k a b l e that an ap-
248
P.A. Maksym / Magic number ground states of quantum dots
392 3.5-
9
~:
2.5'
r~
9
= 3, intra g.g. mode
1.5'
<1
0.5'
I
10
I
15
'
I
20
I
25
''
'
I
30
Angular momentum 0
0
-0.25
-0.5
-0.75"
n = 2, inter L.L. mode I
r~ <1
- I .25'
-1,5,
-1.75-
i
Fig. 5. C o m p a r i s o n of exact a n d a p p r o x i m a t e excitation energies for 2 a n d 3 e l e c t r o n s at B = 20 T.
proximation expected to work in the large-J limit gives this level of agreement at fairly small J values.
S. Effects of spin The results described so far are for the case of
a spin-polarized system but numerical studies indicate that there are magic states at other values of the total spin (fig. 2). These are most easily identified from numerical studies for hto 0 = 0. Numerical studies also show that their presence or absence is dependent on the strength of the confinement. It is likely that the physics is similar to that of the spin-polarized case.
249
393
P.A. Maksym I Magic number ground states of quantum dots
80-
n = 3,. h w o
= 4 m eV
78'
Q)
76.
~ 0
74.
O
9
9
9
O 0
000 9 O:
9 I 5
9 I 10
''
00
00 I . . . . 15
I 20
......
I .......... 25
I 30
Angular m o m e n t u m Fig. 6. Ground state energy' of RM motion for 3 electrons at B = 20T calculated in the approximation of large angular momentum.
6. Conclusion
References
Magic number ground states of very small numbers of interacting electrons in quantum dots have been described and their effect on the thermodynamic properties of the dots has been explained. The underlying physics has been shown to be a direct consequence of the Pauli principle. The extent to which similar results hold for larger numbers of electrons is a fascinating open question.
[1] P.A. Maksym and T. Chakraborty, Phys. Rev. Lett 65 (1990) 108. [2] P.A. Maksym and T. Chakraborty, Phys. Rev. B 45 (1992) 1947. [3] M. Wagner, U. Merkt and A.V. Chaplik, Phys. Rev. B 45 (1992) 1951. [4] D. Pfannkuche, V. Gudmundsson and P.A. Maksym, submitted for publication in Phys. Rev. B. [5] T. Chakraborty, Comments on Condensed Matter Physics 16 (1992) 35. [6] C.L. Siegel and J.K. Moser, Lectures on Celestial Mechanics (Springer-Verlag, Berlin, 1971). [7] J.M. Kingaret, Y. Meir, N.S. Wingreen, P. Lee and X. Wen, Phys. Rev. B 45 (1992) 9489. [8] J.K.G. Watson, Mol. Phys. 15 (1968) 479. [9] J.D. Louck, J. Mol. Spectrosc. 61 (1976) 107. [10] E.B. Wilson Jr, J. Chem. Phys. 3 (1935) 276. [11] P.R. Bunker, Molecular Symmetry and Spectroscopy (Academic Press, New York, 1979). [12] E.T. Whittaker, Analytical Dynamics of Particles and Rigid Bodies (CUP, Cambridge, 1952).
Acknowledgements Part of this work is the result of a very fruitful collaboration with Dr. T. Chakraborty. Support from the Computational Science Initiative of the UK Science and Engineering Research Council is gratefully acknowledged.
250
Journal of the Physical Society of Japan Vol. 65, No. 12, December, 1996, pp. 3945-3951
Origin of Magic Angular Momentum in a Quantum Dot under Strong Magnetic Field Taku SEKI*, Yoshio KURAMOTO** and Tomotoshi NISHINO
Department of Physics, Tohoku University, Sendal 980-77 (Received April 16, 1996) This paper investigates origin of the extra stability associated with particular values (magic numbers) of the total angular momentum of electrons in a quantum dot under strong magnetic field. The ground-state energy, distribution functions of density and angular momentum, and pair correlation function are calculated in the strong field limit by numerical diagonalization of the system containing up to seven electrons. It is shown that the composite fermion picture explains the small magic numbers well, while a simple geometrical picture does better as the magic number increases. Combination of these two pictures leads to identification of all the magic numbers. Relation of the magic-number states to the Wigner crystal and the fractional quantum Hall state is discussed. KEYWORDS: quantum dot, fractional quantum Hall state, composite fermion, exact diagonalization, magic number, Wigner crystal
w
Introduction
The two-dimensional electron system with a small number N of electrons is realized as the q u a n t u m dot formed at a semiconductor interface. 1) Correspondingly there is a growing theoretical interest in the system. It has been found 2"x~ that if a system is placed in a strong magnetic field, extra stability arises for a special set of angular momentum M. These values of/~f are called magic numbers. According to numerical diagonalization, the magic number states occur with an interval A M --- N for systems with N < 5, and that these states have a polygonal pattern with N apexes in the configuration of electrons. 5, 7) The magic number states prevail the ground-state phase diagram in the plane of magnetic field vs the confining potential for N = 5 and 6. s) On the other hand, it has been shown 6' 9) that many of the magic number states can be explained in terms of the composite fermion (CF) picture 11) for the fractional quantum Hall (FQH) states. The overlap is close to unity 9' 12) between a trial wave function based on the CF theory and the exact ground state wave function in the presence of the Coulomb interaction. However, some magic number states do not fit into the interpretation of the CF theory. The purpose of this paper is to clarify the origin of the magic numbers on the basis of exactly derived results for the energy, and one- and two-body distribution functions. In addition to the previously known series with A M = N, we find another series with A M = N - 1 of the magic number states for N >_ 6. We provide simple geometrical interpretation on the origin of new series of magic numbers. Combining with previous interpretations, we can account for all the magic numbers. * Present address: Information Systems Division, Hitachi Ltd., 890 Kashimada, J(awasaki 211. ** E-mail:
[email protected]
The paper is organized as follows: In w we define the model which involves truncation to the lowest Landau level. Section 3 presents numerical results on the ground state energy as a function of the total angular momentum. The magic nmfibers are identified for systems with 6 and 7 electrons. The results on one- and two-body distribution functions are shown in w where the characteristic features of the magic-number states are made explicit. We provide in w a geometrical interpretation of the magic number states, and compare the resultant magic numbers with those derived by the CF picture. 9) In w we combine the~geometrical and CF pictures, and argue that the ranges of validity for each interpretation overlap considerably: This reveals significance of magic-number states as showing crossover from an incipient fractional quantum Hall state to an incipient Wigner crystal. w
M o d e l and Choice of Basis
We consider a model for a quantum dot in two dimensions with the Hamiltonian H = H0 + V1 + V2.
(2.1)
Here I Io, 171, and V2 are given by I H ~ = ~-:m* i= 1 1 N V1 -- ~ m'co 2 ~ Izi 12, i=1
V~ +
Ai
e 2 X-~ 1 V2 = - - ~ s i<j Izi - zjl'
(2.2)
(2.3)
with w being the fi'equency of the harmonic oscillator, and z = x + iy. Other notations are the standard ones. The vector potential is chosen to be A i = (yiB/2,-xiB/2). ~u assume that the magnetic field B is so strong that all electrons are in the lowest Landau level (LLL), with spins being completely polarized.
3945
251
3946
One-electron state is specified by its orbital angular momentum only. We ignore the spin degrees of freedom and mixing between the LLL and higher Landau levels. Hence the Zeeman term has been dropped in our model. The normalized single electron wave function in the LLL is given by 1 zmexp([z[2) era(z) = ~/2rg22mm! -,~
o~
oo
m=O
m=O
,
/o
m*w2~ 2
(2.8)
The length is scaled by g, and energy is scaled by e 2/(eog) from now on. Since the confining potential and the Coulomb interaction have the rotational symmetry, the total angular momentum M can be used to label the many body eigenstates. We introduce the distribution function of angulax momentum by
n m = (~o[a~am[Oo),
(2.9)
where IqSo > is the normalized ground state. Then the total angular momentum is given by
M = ~
mnm.
(2.10)
m--0
Next we introduce a field operator r the LLL:
r
truncated within
= Zr
(2.~)
m
Then the charge density n(z) is given by
n(z) = (qb0lr162
,
(2.12)
and the total number N by its integral over the whole space. Finally the pair (two-body) correlation function is defined as
n(z, z') - (~o[r (z)r (z')r
2 m* w2g2(rn + 1),
(2.7)
r ( j - k + 1) r ( k + d + 1)
(2.13)
T w o S e r i e s of Magic N u m b e r s By using the Lanczos method to diagonalize the
r(d+
1) 2 . . . .
(
j-k+l;d+l;-
ml = j - k , m2 = k, m3 = k + d, m4 = j - k - d with d > 0. We discard in the following the constant kinetic term (mlHolm) - 89 The confinement parameter 7 is defined by
2~2/(~0e ).
(2.6)
and the Coulomb matrix element is computed numerically from
(
Here aF1 is Kummer's hypergeometric function, and
=
1 ~t~,
(mlHoim) --
2-'~ dxx~dFa
(2.5)
where am is the annihilation operator of an electron with m. The matrix elements are given by
~ o t . ~/(j - k)!k!(k + d)!(j -- k' " d)! . . . . . x
,a~2amsam4,
rnl
(mlV11m) --
( e2 ) (mamulV2lm3m4) -"
+-~1 ~(mam21V2irn3m4)a~
(2.4)
where m is the angular momentum and g the magnetic length defined by g = (hc/eB) 89 The Hamiltonian in the second quantization is written as
w
(Vol. 65,
Taku SEKI, Yoshio KURAMOTO and Tomotoshi NISHINO
aFa k + d + l ; d + l ; -
.
Hamiltonian numerically, we calculate the lowest energy and corresponding eigenvectors for each M and N. For given N and M, the one body angular momentum takes a value from 0 to M - N ( N 1)/2. The upper limit less than M comes from the Pauli exclusion principle. In constructing the Hamiltonian matrix, we take all the necessary basis for m without truncation. We note t h e following relation: V1JqSo) = 7 ( M + N)[45o},
(3.1)
which follows from eq. (2.7). Thus the wave function [4~0) is actually independent of 7, and the numerical result for 7 = 0 is sufficient to derive the energy for other cases of 7Figure I(a) shows the results for the ground-state energy vs M in the case of N = 6. A number of downward cusps appear clearly. Upon closer inspection one finds that a part of the cusps with M = 15, 21, 27, 33, 39, 45 are represented by the formula
1
M -2N(N-1)+Nk,
(k = 0,1, .--)
(3.2)
with N - 6. This set of states with the interval A M = N are called the series SN in this paper. The other set of cusps appear for states with M = 25, 30, 35, 40, 45, 50 with another interval A M = N - 1 which is called the series SN. The_pm'ticular state with M = 45 belongs to both SN and SN. According to Fig. l(a), 1/4~ = 20 in the series obN is not actually a magic number. W i t h 7 = 0, each magic-number state and the adjacent state with one more angular nmmentum often have the same energy. This degeneracy comes from the degrees of freedom associated with the center of mass motion as pointed out in ref. 4. The angular momentum of the absolute ground state is determined by competition between the Coulomb repulsion and the strength of the confinement. As 9' increases, the optimum angular momentum shifts to smaller val-
,
252
I996)
3947
Origin of Magic Angular Momentum in a Quantum Dot can interpret the magic number M w
Distribution Functions
4.1
Angular momentum
36.
In order to clarify the electronic property associated with the magic-number states, we calculate the distribution function nm of the angular momentum. In this section, we report mainly on results for N = 6. Figure 2 shows representative results. As M increases, two types of distribution appear by turns in the magic-number states. One is the distribution with double peaks at the origin as well as at a finite angular momentum. We call the double-peaked distribution the type-D hereafter. This is the case with M -- 35 and 45. The other type is the distribution with a single peak which we call the type-S, as for M - 33 and 39. An important observation is that the magic number states with the type-D distribution are all in the series SN, while those with the type-S distribution are all in the series SN. These features are common to cases other than N - 6. It is instructive to interpret the result in terms of the Laughlin wave function ff'L(zl, "-', Zg) for finite N. It is given for general N by
i<J
(4.1)
The odd integer p is related to the filling v = lip of the LLL in the limit of N -+ r In the case of finite N, the maximum one-body angular momentum m.m~ is given by mm~x -- p N ( N - 1 ) / 2 . Withp=3andN=6. in particular, we obtain mm~, = 15 and the total angular momentum M is given by M = 3 N ( N - 1)/2 = 45. \Ve note that the numerically obtMned results for M = 45 in Fig. 2(b) has the ma.ximum 15,.of the significantly occupied angular momentum in good correspondence to the Laughlin wave function with p = 3.
4.2
Charge density
The larger angular momentum corresponds to the wave function more extended from the origin. More explicitly we have the relation Fig. 1. The ground-state energy vs the total angular momentum in the case of (a) N = 6 and (b) N - 7. The parameter ~/ represents the strength of the harmonic confinement as explained in the text.
ues. In the case of-y = 0.04 for example, the state with M = 45 takes the absolute minimum of the energy. In Fig. l(b) we show the ground-state energy in the case of N = 7. We find clear downward cusps at the series S~v. This includes M = 21, 33, 39, 45. 51, 57. 63 and 69. However. the number M = 27 belonging to SN is not a magic number. On the other hand, the cusps corresponding to the series SN are not clear except for M = 28. This is in strong contrast with the case of N _< 6. We note that a magic number state with M = 36 is seen in Fig. l(b). This state belongs to neither the series SN nor SN. As we discuss later, the CF picture
(mlr21m) = 2t2(m + 1).
(4.2)
We can therefore expect that n(z) of the type-D distribution should have double peaks both near the origin and near the edge of the dot, and that of the type-S distribution should have only a single peak near the edge. We have actually calculated the charge densities for these states with N = 6 and 7. Since the result n(z) depends only on r = Izl, the density is written as n(r) in the following. Figure 3 shows some exemplary results. It is found that the state with the type-S distribution (M - 33) indeed has a single peak near the edge of the quantum dot, and those with the type-D distribution (M = 35, 45) have double peaks. This feature is common to systems with different electron numbers. Since the Laughlin-type state with M = p N ( N - 1)/2 belongs simultaneously to series SN and SN, it is natural to expect that the charge distribution is also a superposition of the type-D and the
253
3948
(Vol. 65,
Taku SEKt, Yoshio KURAMOTOand Tomotoshi NISHINO
[
0.8 -I .91=33 9
N=6 7 -0.04
~ ]
~-'
''
I
'
I
'
'
'
i''
'".
'
-
N=6
0.1
(a)
0.6
0.05 : : .9 .. 99 o.
Z oo.4
.e!
-
:
0
!
0
2
4
6
8
r
2 i rj rj 9
Fig. 3. T h e c h a r g e d e n s i t y n(r) v s t h e d i s t a n c e r f r o m t h e c e n t e r o f t h e d o t in t h e c a s e o f M -- 33, 35 a n d 45.
-t
0.2
-7
It is seen from eq. (4.2) that the average density of electrons in the q u a n t u m dot decreases as the total angular m o m e n t u m M increases. In w we use this fact for understanding of all the magic numbers.
~ [=35
.4.3 Pair correlation function !
0
,
,
i
,
I
,.,
,
i
|
!
,
,
,
I
t.
N=6 ,~ --004
0.6
|
10 20 30 ANGULAR MOMENTUM
-
~0.4 Z
z
0 0
20 40 ANGULAR MOMENTUM
F i g . 2. O n e - b o d y a n g u l a r m o m e n t u m d i s t r i b u t i o n f u n c t i o n for (a) M -- 33 a n d 35 a n d (b) M = 39 a n d 45. T h e d i s t r i b u t i o n w i t h d o u b l e p e a k s as for M -- 35 a n d 45 is c a l l e d t h e t y p e - D , w h i l e t h e o n e w i t h a s i n g l e p e a k a s for M -- 33 a n d 39 is c a l l e d the type-S.
type-S distributions. As a result, the distribution should have double peaks. The result in Fig. 3 shows that this is indeed the case with M -- 45 and N -- 6.
In order to see the correlated motion of electrons more closely we calculate the pair correlation function n(z, z t) exactly. This quantity gives us a quantum analogue of a snapshot picture of electrons. More precisely, it gives the distribution of electrons on condition that one of the electrons is nailed down at z'. The pair correlation function for N = 5 or less has been calculated by Maksy~n, 7) who found a pattern corresponding to the polygon w i t h N apexes for the series SN. For N = 6 or more, however, we are not aware of exact results reported so far. Moreover nothing is known about the pair distribution for the series SN. We define rp as, the peak position of the charge density, which in the case of type-D distribution is to be taken at the position of the outer peak.. Then we calculate n(z, z') numerically setting ]z'] ~ rp. We have checked that resultant patterns are qualitatively the same as [z' 1 decreases from rp to rp/v/~2. Here we show the results in the case of IZ'] = rp/X/2 = ro since this choice shows the contrast between the case of M = 45 and other magic numbers most clearly. From Fig. 4 with N = 6, it is clear that electrons are arranged like a hexagon in the magic number state (M = 33) belonging to the series SN, and like a pentagon in the magic number state (M = 35) of the series SN. Vfe recall that a Laughlin-type state with v = l i p belongs simultaneously to the series SN and SN. Then n(z, z') should be a superposition of pentagon- and hexagon-like patterns. To demonstrate this, the pah" correlation function for p = 3 (M - 45) is calculated and is shown in Fig. 5. It clearly confirms the expectation. Similar calculation is also performed for N -- 7. Figure 6 shows the result for M - 39 which belongs to the series SN. It shows the hexagon-type pattern as expected. On the contrary, we have confirmed that non-magic-number states such as M -- 35 (N -- 7) and M = 34 ( N -- 6) do not show a regular polygonal pattern.
254
1996)
Origin of Magic Angular Momentum in a Quantum Dot
3949
Fig. 6. Contour m a p of the pair correlation function n(z, z ~) for M = 39, N = 7 w i t h z ~ = 2.442i. T h e interval of contours is 1.343 x 10 - 3 .
w
Interpretation
5.1
Geometrical interpretation
of
Magic Numbers
As we have seen in the previous section, all the magic number states with N = 6 belong either to the series Se or $6- In this section, we discuss the relation between the total angular m o m e n t u m and the s y m m e t r y of the many-electron wave function. The argument extends the previous one which explains t h e series SN. TM Let us consider the wave function ~(Zl, z2, . - - , ZN) in the case where t h e c o o r d i n a t e s correspond to apexes of a regular polygon: zn = R e x p ( i n r with n = 1, 2, . . . , N and r = 27r/N. By applying exp(iJQr where M is the total angular m o m e n t u m operator, we obtain z2, . . . , Zg) = ~.(ZN, Zl, ' ' ' , ZN-1)
exp(i/~rr
= (-1)N-~(zl,
z 2 , . . . , zN).
The second equality follows from the antisymmetry of the fennion wave function. Then we get [exp(iMr
+
( - 1 ) N ] ~ ( z l , z2, " " , ZN)
=
0.
(5.1)
In order for the regular polygonal p a t t e r n with N apexes to be realized, 4i(zl, z2, - . . , Zg) must be nonzero. This gives a selection rule on M as follows:
M=N(j+I),
(j = 0, 1, -..)
(5.2)
if N is even and
M --- N j , if N is odd.
(j = 0, 1 , - . . )
(5.3)
The Pauli principle in addition requires
M >_ N ( N - 1)/2. This selection rule leads to the series SN. In order to examine the case of the polygonal p a t t e r n with N - 1 apexes for N electrons, we p u t Zg - O, zn = R e x p (inr with n -- 1, 2, - . . , N - t and r - 2 7 r / ( N 1). Then we obtain
255
3950
Taku SEKI, Yoshio KURAMOTOand Tomotoshi NISHINO
exp [" 21rM • ~(z~, z~, . . . , zN_~, 0) = 0.
(sA)
This leads to another selection rule: M-(N-1)j,
(j - 0,1, .-.)
(5.5)
if N is even, and
(Vol. 65,
The other extreme case is the one where each of the six lowest Landau levels has only one electron, which we write as (1, 1, 1, 1, 1, 1), resulting in M0 = -15. If one excludes an occupation where a higher LL has more electrons than a lower one, the number of compact states is counted as 11 for N = 6. However, both occupations (3, 1, 1, 1) and (2, 2, 2) give M - 27, and both (4, 1, 1) and (3, 3) give M = 33. The set of the magic numbers with q = 1 are then M = 15, 21, 25, 27, 30, 33, 35, 39, 45.
if N is odd. This selection rule leads to the series SN. These rules nicely explain the occurrence of the magic number states in the case of N <_ 6. However for larger N, the configuration with N-apex pattern costs more energy than another apex pattern with one or more electrons in the interior. This is the reason why the magic number series $7 is hardly seen in Fig. l(b). The exceptional appearance of $7 is the magic number state M - 28 with N - 7. The stability of the state, however, is better explained in terms of the CF picture as discussed below. Now we turn attention to the v - 1/3 Laughlin-type state which belongs to the series SN and SN simultaneously. Since this state has superposition of the polygonal pattern with N apexes and the one with N - 1 apexes, an extra stability is expected due to their resonant energy.
5.2
Composite fermion picture
In ref. 9 an interpretation of magic numbers in terms of the CF picture is presented. In this picture the wave function is constructed in the following form:
~c~(z~, . . . ,
zN) = ~I(z~ - z ~ ) ~ p ~ o ( z ~ , . . . , ZN),
i<j
(5.7)
where ~P0(zl, ---, ZN) is a wave function of free electrons, and q is a natural number. The Jastrow-type factor in eq. (5.7) is interpreted as binding a magnetic flux with strength 2q to each electron, hence the name of the CF. 11) Although ~V0(zl, . . . , zN) is not restricted to the LLL, the projection operator P picks out only such component that belongs to the LLL. Thus the total angular momentum M of ~PCF(Zl, "", ZN) is the sum of the part M0 associated with g~0(zl, " ' , Zg) and that coming from the Jastrow-type factor. Namely we obtain
M = qN(N - 1) + M0.
(5.8)
According to ref. 9 the magic number corresponds to such g'0(zl, ---, zy) that has compact occupation of each Landau level from the lowest possible angular momentuna. In the case of N < 5 the CF picture explains all the magic numbers. 9) The CF magic numbers for N = 5 are the ahnost the same as the series $5 with additional ones M = 18 and 22 from the series $5. For N = 4 all magic numbers in $4 are also given by the CF picture, while the CF picture gives M = 12 as the only magic number in the series $4. The series SN with the electron number N _ 5 have been studied by other authors 4, 5, 7) as well. For the case of N = 6 and q = 1, the state with M = 45 has six electrons in the LLL. This state has M0 = 15.
All these states belong to either $6 or $6- With q = 2 the lowest magic number is M - 45 and the second lowest is 51. A nice feature of the CF picture is to reject M - 20 as a magic number in accordance with the numerical result in Fig. l(a). On the other hand, the numerical result shows clear cusps at M - 40 and 50 in contradiction to the CF picture. Similar analysis is carried out for N = 7. The number of compact states is 15 for each q and the magic numbers with q = 1 are given by M = 21, 28, 33, 35, 36, 39, 41, 42, 43, 45, 48, 49, 51, 56, 63. By comparing with Fig. 2(b) we see that many of them correspond to cusps in the ground state energy. However, the numerical result shows clear cusps also at M = 57 and 69 both of which do not correspond to compact states. As N increases, the magic numbers predicted in the CF picture appear more densely than given in the series SN and SN. Some of them appear clearly in the numerical result, while others have ahnost no cusp. Unfortunately the relative stability among magic number states is not given in the CF picture. We remark that among the magic numbers which ~ e not explained in the CFpicture, M=40 forN=6andM=57forN=7 are both next to smaller magic numbers, 39 and 56. w
Discussion and Conclusion
We have seen t h a t there are apparently conflicting interpretations of magic number states: the geometrical one which emphasizes the real space configuration, and the CF one which emphasizes the compact occupation of the one-body angular momentum. The two pictures m'e in fact not conflicting to each other. Namely over the wide range of M, many of the magic numbers are common to both pictures. By combining the geometrical and CF pictures properly we have succeeded in accounting for all the magic numbers. In the case of small M, the CF picture works better in rejecting/l~r - 20 with N = 6, or M = 27 with N - 7. However, in the case of large M, the geometrical interpretation works better in explaining M - 40 and 50 with N - 6, and M = 57 and 69 with N - 7. In order to demonstrate that the two pictures have a considerable overlap in their ranges of validity, we take an example of a magic number M - 39 with N = 6. We point out that the wave function for M - 39 constructed by the composite fermion picture has been shown to be an excellent approximation to the exact one?) although
256
1996)
Origin of Magic Angular Momentum in a Quantum Dot
M = 39 is adjacent to another magic number M = 40 which is explained only by the geometrical picture. It should be noted that the smaller M corresponds to higher density of electrons in the macroscopic limit. Thus the crossover in the effectiveness of the geometrical and the CF interpretations seems to reflect the transition from a FQH liquid at high density to the Wigner solid at low density in the macroscopic limit. The success of both interpretations in the wide range of M is intriguing. This suggests a smooth crossover for finite N from the incipient F Q H liquid to the incipient Wigner solid with decreasing density. In this connection we remind the early work 14) which argues that particular densities for the stable FQH states also stabilize the Wigner crystal with large zero-point motion and the ring exchange. It is thus not surprising t h a t some of the electronic states in the q u a n t u m dot connect both to the FQH state and the Wigner crystal. Experimentally, a particular advantage of the q u a n t u m dot is the wide range of controllable electron numbers N. Thus one may observe how the electronic states change from the atomic type to the macroscopic type as N is increased. This feasibility is in strong contrast to genuine atoms. In summary, we have investigated the q u a n t u m dot system in a strong magnetic field by numerical diagonalization of the Hamiltonian. We have extended the geometrical interpretation and have shown that combi-
3951
nation of it with the composite fermion picture accounts for all the magic numbers in the total angular momentum. Acknowledgment T h e authors are grateful to Y. Kato, T. Nihonyanagi, S. Tokizaki and H. Yokoyama for fruitful discussions. The numerical calculations were performed by SX3-44R at the computing center of Tohoku University. This research was supported by a Grant-in-Aid for Scientific Research on Priority Area from the Ministry of Education, Science, Sports and Culture. 1) 2) 3) 4) 5) 6) 7) 8) 9) 10)
11) 12) 13) 14)
R. C. Ashoori: Phys. Rev. Lett. 71 (1993) 613. R. B. Laughlin: Phys. Rev. B 27 (1983) 3383. S. M. Girvin and T. Jach: Phys. Rev. B 28 (1983) 4506. S. A. Trugman and S. Kivelson: Phys. Rev. B 31 (1985) 5280. P. A. Maksym and T. Chakraborty: Phys. Rev. Lett. 65 (1990) 108. B. Rejaei and C. W. J. Beenakker: Phys. Rev. B 46 (1992) 15566. P. A. Maksym: Physica B 184 (1993) 385. S. R. E. Yang and A. H. MacDonald: Phys. Rev. Lett. 70 (1993) 4110. J. K. Jain and T. Kawamura: Europhys. Lett. 29 (1995) 321. W. Lai et al.: Solid State Commun. 52 (1994) 339. J. K. Jain: Adv. Phys. 41 (1992) 105. G. Dev and J. K..lain: Phys. Rev. B 45 (1992) 1233. W. Y. Ruan et al.: Phys. Rev. B 51 (1995) 7942. S. Kivelson et al.: Phys. Rev. Lett. 56 (1986) 873.
257
PHYSICAL REVIEW B
VOLUME 43, NUMBER 17
15 JUNE 1991-I
Magneto-optical transitions and level crossings in a Coulomb-coupled pair of quantum dots Tapash Chakraborty* Max.Planck-lnstitut fiir Festki~rperforschung, Heisenbergstrasse 1, D-7000 Stuttgart 80, Federal Republic of Germany V. Halonen and P. PietiHiinen Department of Theoretical Physics, University of Oulu, Linnanmaa, SF-90570 Oulu 57, Finland (Received 9 October 1990) Optical transitions for a pair of Coulomb-coupled quantum dots in a magnetic field are investigated. It is shown that for a parabolic confinement potential, the electric-dipole transition energies reveal interesting anticrossing behavior which is due to the many-electron interaction. The results for the ground-state single-particle density in these systems also reflect the correlated behavior of the electrons. Qualitatively similar results are obtained for electron and hole dots (type-ll quantum dots).
Recent experimental realization of quasi-zero-dimensional electron systems (quantum d o t s ) a t the semiconductor interface ~-8 has heightened interest on the physics of a few electron systems subjected to quantum confinement in three dimensions. They are often referred to as artificial atoms, each having a discrete energy spectrum where the electrons are confined by an artificially created potential which is very different from the Coulomb potential. The experimental information about the electronic properties of quantum dots is primarily from capacitance spectroscopy, ~-3 resonant tunneling spectroscopy, 5 and far-infrared (FIR) magnetospectroscopy. 6-s An oscillatory structure in the capacitance observed by Smith et al. ~,2 has been attributed to the discrete energy levels of a quantum dot. In the presence of a perpendicular magnetic field, Zeeman bifurcation of the energy levels of a quantum dot has also been observed. 1.2 The splitting is believed to occur due to the interplay between competing spatial and magnetic quantization. Another interesting result in capacitance spectroscopy is the observation by Hansen et al. ~ of fractionally quantized states, similar to the fractionally quantized Hall effect in two-dimensional electron systems, 9 in quantum dots at gate voltages and magnetic fields corresponding to the filling factors ~ and FIR spectroscopic measurements on quantum-dot structures in InSb by Sikorski and Merkt 6 showed that the measured resonant frequency is independent of electron number within the experimental error. These resonances are, in fact, related to single-particle transition energies in a bare confinement potential. Demel et al. 7 recently created small quantum-dot structures in AlxGal-xAs/GaAs heterostructures. The FIR resonance absorption they observed shows similar dispersion with a magnetic field as seen for the excitations earlier in Ref. 6. However, a resonant anticrossing in the energy levels was resolved in the experiments of Ref. 7. The appearance of anticrossing is thought to be primarily due to a nonlocal interaction in a single dot which is important in low dimensions. In the absence of a magnetic field, Bryant l0 has shown that the electron-electron interaction has a significant 43
influence on the energy spectrum of a quantum dot. In a magnetic field, the nature of the many-electron states is a challenging problem. The effect of the many-electron interaction on the energy spectrum of a dot in a magnetic field is indeed quite intricate, t l One consequence is that different ground states are expected to occur at different magnetic fields. From self-consistent numerical solutions of the Poisson and SchF6dinger equations for a quantum dot in the Hartree approximation, Kumar, Laux, and Stern 12 showed that the confining potential has nearly circular symmetry and therefore angular momentum is approximately a good quantum number. Also with increasing magnetic field, the evolution of energy levels was found to be similar to that for a parabolic potential. ~2-,14 It is shown in Ref. I1, however, that for the parabolic confinement of the electrons, F I R spectroscopy is sensitive only to the center-of-mass (c.m.) motion of the electrons which has exactly the single-electron excitation energy of a bare confinement potential. This)s in line with the experimental observation of Ref. 6. Brey, Johnson, and Halperin t5 obtained similar results for an electron gas in a perfect parabolic quantum well. "It is possible, of course, to retain the parabolic confinement but break the circular symmetry by working with a pair of quantum dots which are coupled only via the Coulomb interaction (tunneling of electrons between the two dots is not allowed). It is then possible to study in our scheme the effect of the many-electron interaction on the optical excitations. In this paper, we present evidence that in such a system the energy levels do show the effect of many-electron interactions by exhibiting anticrossing behavior in the optically induced transition energies first observed experimentally in Ref. 7. The Hamiltonian for our system is chosen to be ~ / = ~ o + ~ec, where
7[0 ~ a ~~l , 2 ~i 2-~[Pai-(e/c)A(r~i)]2 + ~ m* [12(r,i - R a ) 2 14 289
9 1991 The American Physical Society
258
43
TAPASH CHAKRABORTY, V. HALONEN, AND P. PIETILAINEN
14290
and e"~'2 X
Z
1
e 2
where x - ( b / 2 1 ~ ) t / 2 r , b=(l+4fl2/co~) I/2, cor=-eB/ m * c, !0 ~ (hc[eB) |/2, and
1
_r.j------~+--/~j
Here, the sums over i and j run over the number of electrons in a single dot, m* is the effective mass, fl is the strength of the confining potential, e is the background dielectric constant, and Ra is the position of the center of the ath dot. No attempt is made to include any singleparticle interaction between the dots, i.e., the electrons feel the other dot only through the Coulomb interaction fro:. For a parabolic dot in a perpendicular magnetic field, the single-particle wave function is of the form iJ b
n!
I/2
2trig (n+lll)!
~~
x ~'~ C(n,l,a)e-it~
(l)
a "0
C(n,l,a) -(-
....(n + It/)!
1 )a
(n
a)!(lll+a)!a!
"
In Eq. (I), n =0,1 . . . . is the radial quantum number and 1 - 0 , __+1. . . . is the azimuthal quantum number. In our numerical work to be presented below, we have considered only the lowest two Fock-Darwin levels (FDL) [ N - n + (l/l-1)/2 is the level index] for convenience. As noted by Dingle, t4 the most likely transitions are those between states with the same quantum number n. In the quantum-dot systems the dipole approximation is very accurate. ~j In this approximation, the optical transition matrix elements s't4"16 are evaluated from
I i/2
(~on,t,ire +-iol~o.d2)==-(nll l lr +_ln212) -- ~t, +_t.h [ 2
lo
nl
n i!n2!
(,,
I/2
+ It, l)t(,,2 + It21 ~.
n2
x ~_. C(n,,l,,al) a! ~ 0
~_. C(n2,12,a2)[a~+a2+~-i(il~l+ll2l+l)]!.
(2)
a 2 fmO
|
Since we exclude light-induced transitions between the dots, initial and final states must be in the same dot. The electric-dipole transition probability from the ground state to an excited state is calculated from t6
~;0-~ I<*01~1.;>12 = ~ I<,I, olr+l*;>l 2+ ~-I(*olr-I*~>l 2 , (3) where q'o and % denote the ground state and the ith excited state and r +_ is defined in the second quantized notation (ignoring spin and dot label) r _+ =
~-.
(nllllr +_In212)an,l,a.d2. *
(4)
nl,li,n~.12
where a t and a are the usual creation and annihilation operators. The dipole transition energies for a pair of quantum dots with three and four electrons per dot and the electron-dot-hole-dot pair are presented in Fig. 1 as a function of the magnetic field B (in tesla). The excitation energies are obtained by numerically diagonal|zing the Hamilton|an where the electrons are assumed to be spin polarized. The numerical method for diagonalization of the Hamilton|an is described in Refs. 9 and 11. It is to be noted that the spin polarization of the ground state depends on the material parameters (effective mass, dielectric constant, and the confining potential strength) one uses. For the parameters used here in the case of GaAs, two- and four-electron dots have the spin-polarized ground state. For the parameters corresponding to dots in InSb, the ground state is usually spin unpolarized. One should note, of course, that in our present study of optical transition in quantum dots, spin polarization does not play an essential role. The basis states are constructed in two steps: First, we consider a single dot with three or four electrons, where
the ground state occurs at a total angular momentum L0. When L >__Lo, we consider electrons to be in the lowest FDL, but when L < Lo, it is necessary to put, in each basis state, one electron in the second FDL. The next step is to take a Cartesian product of the two single-dot basis. We would like to emphasize that the present results explicitly include electron correlations in the many-electron system. Furthermore, within the dipole approximation, the excitation energies presented here are the exact results for the systems considered. The numerical resuits for the transition energies are presented such that the diameters of the circles are proportional to the calculated intensities of the transitions. The results for the transition energies of the electrondot (and hole-dot) pair show several interesting features. At B - 0 , the degeneracy present in the single-particle result is lifted due to the lack of circular symmetry. The lower mode of the transition energies for the electron-dot pair [Figs. l(a) and l(b)] is always close to the singleparticle mode. On the other hand, the upper mode is seen to behave quite differently from the single-particle result and exhibits interesting ant|crossing behavior. This is clearly a consequence of the Coulomb interaction between the two dots. Due to this interaction, the c.m. excitations with different angular momentum for individual dots couple to each other or with excitations due to relative motion of the electrons and is responsible for the discontinuities in the upper mode of the dipole transition energy in Fig. 1. For example, in Fig. l(a), the discontinuity at low magnetic field is due to coupling between the L = 2 (c.m.) excitation energy level (L is total momentum for a single dot; angular momentum is no longer a good quantum number for the dot pair), and the lowest excitation energy level with L --5 (relative) of an individual dot. 17 The other splitting is caused by the coupling of the excitations
259
MAGNETO-OPTICAL
TRANSITIONS
with L - - 2 (c.m.) and L--5 (c.m.) of an individual dot. Considering the fact that the anticrossing around B---0.5 T is due to coupling of the upper mode with a higherenergy mode, we find that at B - 0 the higher-energy mode has energy lower than twice the energy of the lowest mode, contrary to what one expects from a single-electron
9 0008
^ooO~ j ..,
ooooO~
E (meV}
I
i
0.0
0.5
1 9
(b)
,
I,
1.0 B(T)
,~ I ................ 1.5 2.0
d=120 nm
5 ~
AND
LEVEL
. .
.
14291
theory. This lowering is indeed observed in experiments of Ref. 7 and can be understood as a many-electron effect, ts The lower mode in Fig. 1 remains a singleparticle type because it is caused by the transition to the L - - 4 (c.m.) level which has no other level to coupleto. One other interesting result is that with increasing elect r o n number, the dots can be set further apart in order to observe similar structures in the upper mode of the transition energies. If this trend persists, for dots containing a considerably large number of electrons, as in Ref. 7, the separation ne.~! not be as small as in the present work. The resulls for a quantum-dot pair where one of the dots is made up of holes is presented in Fig. 1(c). Such a system has not yet been obtained experimentally, but preparation of similar systems in one dimension (type-lI quantum wires) has been reported recently. 19 The results in this case are qualitatively similar to that of the electron-dot pair, except that t h e modes are shifted much higher in energy from the single-particle excitation energies and a l~-intense second level of upper and lower modes is visible. The antierossing behavior of the upper mode is also present in this ease. Transition energies for various values of the dot separation are shown in Fig. 2. For very large separation of the
oo~176176
ooooOO~
I ~
CRO~
6 .......................
ooO o ~ 1 7 6- .
oooo o~
E(meV)
5
....
9. . . . . ,
3 o ~
2
!
4 E (meV) 3
I
I
0.0
0.5
I
l
1.0
1
1.5
~~
2.0
-
5
i
C)
d:]O0
r'~ r"~0
:1
'
~ 1 70 6
. . . . . oO
0.0
05
"
1.0 I '
B(T)
1.5 ....... 22.O
oooOO o o ~ 1 7 6 1 7 6 1 7 6
E(meV)
43
~
I(b)
d.120n m
5~ . ~176176176176
2
" " 99
.........
I
0.5
LOB (T) 1.5
- ~ . _ ~ o o & ~ , . , , , ~
E (meV)
,
0.0
oooo~OO
2.0
FIG. I. Electric-dipole transition energies and intensities of a (a) three-electron per dot pair (d ==I00 nm), (b) four-electron per dot pair (d-120 nm), and (r electron-dot-hole-dot pair (three particles per dot, m* :=mh*, and d - I00 nm). Here, d is the distance between the dots. Confinement potential energy is fixed at h ~ " 2 . 5 meV. Solid lines are one-particle transition energies (Rcf. 7). Diameters of the circles are proportional to the calculated intensity of the transition.
3
~
o
0.0
^_
0.5
"
1.0 1.5 2.0 B(T) FIG. 2. Electric-dipole transition energies and intensities of a three-electron per dot pair for two different separations of the dots. The other parameters are the same as in Fig. 1.
260
14292
TAPASH CHAKRABORTY, V. HALONEN, AND P. PIETIL~,INEN 0.8
0.8 \
p(?)
0.6
,o-31o.,
i
-~ d--lO0nm
rim
0.6
\
p(F) (nml2J
Inm)2], 0.2
0.0 -100 -50
tt
0.2
0 50 100 distance (nm)
150 200
0.0 100
50
0 50 100 distance (nm)
150 200
FIG. 3. One-particle density of the ground state in the case of an (a) electron-dot pair and (b) electron-dot-hole-dot pair. The dots contain three electrons or holes. Distance between the dots is d - I00 nm and the confinement potential energy is h 11-2.5 meV. Magnetic field is i T. The solid line represents density along a line that goes through the centers of the dots (x axis) and the dotted line represents density along a line that goes through the center of one dot and is perpendicular to the x axis.
dots, the interdot Coulomb interaction is vanishingly small and in this case the results quite naturally approach the single-particle results (solid lines); ll the upper mode is almost featureless, but the degeneracy at B - 0 is still lifted. This shows once again that the anticrossing is a consequence of the many-electron interaction. With decreasing separation of the two dots, the interdot interaction increases and more structures appear in the upper mode of the transition energy. In Fig. 3 we present the one-particle density p ( r ) for the (a) electron-dot pair and (b) electron-dot-hole-dot pair. In the former case, the electron-electron interaction, albeit weak due to the attractive confining potential, is responsible for the dot centers to move slightly apart. The density of a single dot (dotted line) is, of course, symmetric. The interesting situation appears for the electron-dot-hole-dot case where the attractive interaction causes the electron densities of the dots to overlap strongly with p ( r ) peaking slightly away from the center of each
dot. In conclusion, we have studied a system of quantum-dot pairs where the circular symmetry is broken and, as a result, the radiation couples to the internal motion of the electrons. In that case, the effect of the many-el~ztron interaction is manifested by the anticrossing behavior in the transition energies. We have also studied the optical transitions in type-ll quantum dots. The results for singleparticle density exhibit some structures due to correlations. The actual systems consist of arrays of quantum dots and therefore, as a next step, one could consider more dots surrounding a single dot. In this way, circular symmetry could be partially restored and the calculations could be made feasible. Details will be published elsewhere.
*Present address: Institute for Microstructural Sciences, National Research Council of Canada, Ottawa, Canada K IA 0R6. 1For a brief review, see T. P. Smith !il, Surf. Sci. 229, 239 (1990). 2W. Hansen, T. P. Smith III, K. Y. Lee, J. A. Brum, C. M. Knoedler, J. M. Hong, and D. P. Kern, Phys. Rev. Lett. 62, 2168 (1989); T. P. Smith II!, K. Y. Lee, C. M. Knoedler, J. M. Hong, and D. P. Kern, Phys. Rev. B 38, 2172 (1988). 3W. Hansen, T. P. Smith II1, K. Y. Lee, J. M. Hong, and C. M. Knoedler, Appl. Phys. Lett. 56, 168 (! 990). 4S. J. Allen, H. L. St6rmer, and J. C. M. Hwang, Phys. Rev. B 28, 4875 (1983). 5M. A. Reed, J. N. Randall, R. J. Aggarwal, R. J. Matyi, T. M. Moore, and A. E. Wetsel, Phys. Rev. Lett. 60, 535 (1988). 6Ch. Sikorski and U. Merkt, Phys. Rev. Lett. 62, 2164 (1989); Surf. Sci. 229, 282 (1990). 7T. Demel, D. Heitmann, P. Grambow, and K. Ploog, Phys. Rev. Lett. 64, 788 (! 990). 8D. C. Tsui et al., Appi. Phys. Lett. 55, 168 (1989). 9Tapash Chakraborty and P. Pietii~iinen, The Fractional Quantum Hall Effect, Springer Series in Solid State Sciences Vol.
85 (Springer-Verlag, New York, 1988). W~ W. Bryant, Phys. Rev. Lett. 59, 1140 (1987). lip. A. Maksym and Tapash Chakraborty, Phys. Rev. Lett. 65, 108 (1990). i2A. Kumar, S. E. Laux, and F. Stern, Phys. Rev. B 42, 5166 (1990). 13V. Fock, Z. Phys. 47, 446 (1928); C. G. Darwin, Proc. Cambridge Philos. Soc. 27, 86 (1930). 14R. B. Dingle, Proc. R. Soc. London, Ser. A 212, 38 (1952). ~-~L. Brey, N. F. Johnson, and B. I. Halperin, Phys. Rev. B 40, 10647 (1989); J. Dempsey, N. F. Johnson, L. Brey, and B. I. Halperin, ibid. 42, Ii 708 (1990); P. Bakshi, D. A. Broido, and K. Kempa, ibid. 42, 7416 (1990). 161-t. A. Bethe and E. E. Salpeter, Quantum Mechanics o f Oneand Two-Electron Atoms (Springer-Verlag, Berlin, 1957), p. 248. 17As the Hamiltonian for the single dot in a parabolic confinement separates into c.m. and the relative terms (see Ref. ! I ), the energy levels can be classified accordingly as the c.m. and relati~,e energies. 18D. Heitmann (private communication). 19A. Pinczuk et al., Phys. Rev. Lett. 63, 1641 (1989).
One of us (T.C.) would like to thank Detlef Heitmann for helpful discussions and Klaus yon Klitzing for his comments on the manuscript.
261
VOLUME 64, NUMBER21
PHYSICAL
REVIEW
LETTERS
21 MAY 1990
Coupling o f Q u a n t u m D o t s on G a A s A. Lorke and J. P. Kotthaus Institut fiir Angewandte Physik, Universitdt Hamburg, Jungiusstrasse 11, D-2000 Hamburg 36, West Germany and Sektion Physik, Universitdt Miinchen, Geschwister.Scholl-Platz l, D-8000 Miinchen 22, West Germany K. Ploog Max-Planck-lnstitut fiir Festkiirperforschung, Heisenbergstrasse I, D-7000 Stuttgart 80, West Germany (Received30 October 1989) With far-infrared spectroscopy, coupling between electron quantum dots becomes visible in the electronic excitation spectrum. We employ gated GaAs-AIGaAs quantum wells that enable field-effect tuning of the coupling between adjacent dots. For noninteracting quantum dots in a magnetic field we observe the characteristic edge- and bulk-mode spectrum. The coupling of dots is reflected by a branching of the bulk mode into a cyclotron-resonance-likeand a magnetoplasmonlike mode and a splitting of the edge mode. The latter is caused by formation of new edge orbits embracing two adjacent dots. PACS numbers: 73.20.Mf,73.40.Kp,78.65.Fa
A laterally periodic modulation of the confining potential of electrons in a quasi-two-dimensional electron gas formed at a semiconductor interface has proven to be a powerful tool for the creation and investigation of lowdimensional electron systems. Direct insight into the energy spectrum of such systems is promoted by the application of magnetic fields such that characteristic magnetic and electric lengths are o f comparable size. As predicted by Hofstadter I this should eventually result in an energy spectrum of intriguing richness and beauty. Several attempts to realize such systems have already been made: In the limit of weak one-dimensional modulation Weiss et al. 2 and Winlder, Kotthaus, and Ploog 3 observed a novel type of oscillation in the magnetoresistance of GaAs heterostructures. The other limit of a very strongly modulated potential results in separated, virtually noninteracting electron systems, i.e., quantum wires or quantum dots. Quantum-dot systems have recently been realized on different semiconductors employing various confinement schemes. 4-1~ In this Letter we report on the electrostatic generation of quantum dots such that the coupling strength between dots can be tuned by the applied gate voltage. We present farinfrared investigations that show new features in the excitation spectrum in the regime of gate voltages where isolated quantum dots transform into an electron mesh of connected dots. We demonstrate that the onset of these features reflects the competition between electrostatic confinement and magnetic-field-induced delocalization. The experimental observations can be explained by both a billiard-type trajectory model and a tunneling model of 9coupled dots. The samples are grown by molecular-beam epitaxy. On a semi-insulating substrate the following are deposited: a 200-nm short-period superlattice (SPS) consisting of 2.5-nm GaAs and 2.5-nm AlAs layers, a 2.5-nm layer of ~-doped GaAs, a 20-nm SPS spacer, a 50-nm GaAs
quantum well, a 20-nm SPS spacer, 2.5-nm Si 6-doped GaPs, and a 50-nm SPS cap layer. A 4• 5-mm2 large piece of the wafer is cut and wedged to avoid interference phenomena in the far-infrared measurements. Small pads of an ln-Ag alloy are evaporated at the edges of the sample and diffused for 2 min at 420"C to serve as Ohmic contacts to the two-dimensional electron gas. 5 • 107 photoresist dots of 200 nm diam forming a crossgrating of period a - 4 5 0 nm are defined using holographic lithography. An evaporated layer of 10-nm NiCr serves as a modulated transparent gate that is used to deplete the electrons around the dots. 3 Illuminating the sample at liquid-helium temperatures creates a bypass in the doped GaPs below the well which serves as a back contact when depleting the electrons in the well. This is essential, since without a back contact the problem arises that as the regions around the dots become depleted electrical contact to large areas of the sample is lost and the electron density can no longer be tuned. It To characterize the electron'ic properties of such a quantum-dot array we study the far-infrared excitation spectrum with a Fourier-transform spectrometer. The sample is cooled to 2 K in the center of a superconducting solenoid with the direction of the magnetic field parallel to the surface normal, Figure I shows transmission spectra of the sample at various magnetic fields and a gate bias Vs at which the electron system consists of isolated dots containing about N o - 5 0 electrons each. At magnetic field B--0 a dimensional resonance is visible as a minimum in the normalized transmission at 33 c m - i . With increasing magnetic field this resonance splits into a bulklike mode that approaches the cyclotron resonance in high magnetic fields_and an edge mode decreasing in frequency with increasing field. Such a behavior is characteristic for electron systems confined in all three spatial dimensions. We can also deduce that isolated dots have formed at gate bias V g - - 3 . 1 V from
9 1990 The American Physical Society
2559
262
VOLUME 64, NUMBER 21
PHYSICAL
REVIEW
LETTERS 100
21MAY 1990 ,
"
t00 80
-
'
i
"
,
iv
VO=-3.
"'1
|
_1
=,
60
I00
~.. 4o _.>
v-- 100
E
20 (a)
~- t00
g
-
-
---'!
t20 I00
m
9
BO
9 ~
m -
99 0
I
I
I
I
J
20
40
60
80
t0o
wave number
~
6O
4O
(cm- I )
20
FIG. 1. Transmission spectra of isolated quantum dots normalized by the transmission at thi~shoid voltage V, at which the dots are fully depleted. The dimensional resonance that is visible at magnetic field B--0 as a minimum in transmission at 33 cm -! splits into a high-frequency bulk and a low-frequency edge mode with increasing B. Here about fifty electrons per dot occupy approximately seven quantum levels at B - 0 .
capacitance-voltage measurements. The average number of electrons per dot can be extracted from fits to the cyclotron resonance at high magnetic fields and magnetocapacitance studies. 4,6 The resonance positions for traces such as in Fig. 1 are summarized in Fig. 2(a) for the same gate voltage. The solid lines are calculated with the following equation that can be derived from electrodynamics, t2 classical mechanics; 13 and quantum mechanics Z4and holds for mesoscopic electron disks 12 as well as for quantum dots: +_ -- ((o~ + m2c14 ) 2/2 + r
(I)
Here r162 is the cyclotron frequency and r is the resonance frequency at B - 0 . The upper mode is not very well described using w o - 3 3 cm - ! and for B > 1 T rather follows Eq. (1) with r cm - l . Such a behavior of the bulk mode has also been observed for electron disks on liquid He (Ref. 15) and can be attributed to the fact that the confining potential of dots is not perfectly parabolic as assumed in deriving Eq. (1) and rather softens towards the edges. Measurements on silicon metal-oxide-semiconductor field effects where the shape of the confining potential can electrostatically be tuned support such a picture, io The single-particle energy-level spacing h cosp can be calculated for a parabolic potential from h tOsp- 2h 2(n -I- 1) / m * r 2 ,
(2)
with m* being the effective mass, rt, the radius of the 2560
0
0
t
2
3
mgnetic field
4
UI
FIG. 2. Measured resonance positions as a function of the magnetic field for (a) isolated and (b) connected quantum dots. The solid lines reflect the dispersion of Eq. (i). Coupled dots show additional modes. Inset: Trajectories responsible for the upper- (A) and the lower- (B) frequency edge modes. The dashed line gives the dispersion expected for a type-B edge mode.
electron orbit at the Fermi energy, and n the quantum number of the highest oecupiexl level (n--~6 for N o - 5 0 ) . If we assume r e - 100 nm agreeing with the geometric size of the dots, Eq. (2) results in a level spacing of 1.6 meV. This is comparable tO h~sp derived for quantum wires of similar confinement and electron density ~6"!7 and accounts for about 40% of the observed resonance energy h too. For parabolic confinement h tosp reflects the level spacing in the screened potential whereas h too is understood as the characteristic energy of the bare potential. Is Alternatively, hto0 can be expressed as a combination of single-particle and collective contributions htoo as ~o~-tO~p+tO~. We thus conclude that singleparticle and collective effects are of comparable size for the dots studied here. The large tunability of our device enables us to study the transition from isolated to strongly coupled quantum dots by simply changing the applied gate voltage. Capacitance measurements show that at V g - - 2 . 7 V the dots are still electrically connected, although there are already voids in the two-dimensional electron gas such that an electron mesh has formed. In the following we wish to concentrate on the gate-voltage regime where the transformation takes place from an electron mesh to a system of isolated dots. Figure 3 displays spectra taken
263
VOLUME64, NUMBER 21
PHYSICAL
REVIEW
I00
Ioo --
~J
"
-2
.9
~> soo I-.8
al
a-
t00 .7 v)
g9
4T g8 0
20
40
wave number
60
80
t00
(cm-tl
FIG. 3. Transmission spectra at B-2.4 T for different gate voltages. Two edge modes appear (arrows) as the bias is increased such that the dots couple to form an electron mesh. A shoulder appearing on the high-energy side of the bulk mode is identified a s a magnetoplasmon.
at B--2.4 T for various gate voltages in this regime. At V s -- - - 2 . 7 V the edge mode is split into two that merge as the bias is changed to --3.1 V, i.e., as the dots become decoupled. The bulk mode ~o+ centered at about 60 can -~ rapidly loses strength with decreasing bias. For the coupled dots ( V s >_ - 2 . 9 V) and B = 2 . 4 T an additional resonance becomes apparent as a shoulder on the high-energy side of ~o+. This resonance shifts away from the co+ mode as the magnetic field is increased. Figure 2(b) summarizes the resonance positions at a gate potential Vz - - 2.9 V where we have well defined but already strongly coupled electron dots. With increasing magnetic field we observe additional branches close to the m+ and r branches that appear at about 2 and 1 T, respectively. From the electron number per unit area N o / a 2 we can estimate a local electron density N s - - ~ 2 N o / a 2 ~ l . 6 x l 0 : t cm -z. At B = 2 T this corresponds to a classical cyclotron diameter at the Fermi energy of 2Re--2(2JrNs):/2(h/eB)"'_70 rim. This is comparable to the width of the constriction that connects adjacent dots in the situation realized experimentally. As long as 2Re is larger than this constriction the system is expected to behave similarly to an array of isolated dots as is observed experimentally in low fields. As 2Re becomes smaller then the constriction electrons on bulklike orbits can communicate between adjacent dots. Then the periodic configuration of the electron mesh should make it possible to excite "tw~176 magnetoplasmons. 19.20 This is, in fact, what we observe in the experiment. To unambiguously identify the highestlying mode oJ++ in Fig. 2(b) the large tunability of the
LETTERS
21MAY
1990
sample is of great advantage. Since the entire range between a two-dimensional electron gas (Vt - 0 V) and an electron mesh ( V z - - 2.9 V) is accessible in the experiment, it is possible to study how the cyclotron resonance and the magnetoplasmon develop with decreasing bias. We observe that in magnetic fields where the r + mode has established it develops Out of the magnetoplasmon with decreasing bias whereas the m+ mode develops out of the cyclotron resonance. In magnetic fields > 5 T where 2Re is very small compared to the width of the constriction we observe that the highest mode follows a dispersion relation m{ + - m 2 + ai~, with mp the plasmon frequency similar to the two-dimensional magnetoplasmon dispersion, t9 Figure 3 shows that the edge mode persists even for relatively high bias. This indicates that the edge charge can still move on the perimeter of the dots though the dots are already connected. In addition, we observe a second edge mode at lower frequencies. This additional mode is lower in frequency than the fundamental mmode that merges with the r mode at B - 0 . Hence it should not be mistaken for higher harmonics of the mmode that have been observed for electrons on liquid He (Ref. 15) and in etched GaAs quantum dots.9 In a classical picture of edge modes in sufficiently high magnetic fields, the charge moves along boundaries and the mode frequency is determined by Eq. (1) taking a~m l/p, with p the dot perimeter. We therefore identify the new edge mode as a charge moving along a boundary of about twice the length of the perimeter of an isolated dot. This is supported by the magnetic-field dependence calculated for such a mode [dashed line in Fig. 2(b)]. Thus a charge moving along a peanut-shaped orbit enclosing two dots as indicated in the inset of Fig. 2(b) explains the second observed edge mode. Such an orbit needs only tWO transmission events through a constriction and therefore appears the most likely involving more than one dot. The observation that the development of a second edge mode takes place-at much lower magnetic fields than the splitting of the bulk mode is consistent with the picture of a charge moving along the edge on "'skipping orbits." On a skipping orbit electrons can pass the constriction between adjacent dots even if 2Re is larger than the width of the constriction. The branching of the bulk mode and the appearance of an additional edge mode can be equally explained by coherent transmission through the narrow constriction connecting adjacent dots and magnetic breakdown of the barrier imposed by the constriction. Then the spectrum can be understood in a ~imilar fashion as that of a twodimensional electron system under the influence of a one-dimensional periodic potential. 2: In such a picture the development of the splitting with increasing bias as shown in Fig. 3 becomes clear as a lifting of the degeneracy of the electronic levels in adjacent dots as the coupling becomes stronger. Thus the far-infrared spectra directly reflect the strong coupling of adjacent dots in an 2561
264
VOLUME64, NUMBER21
PHYSICAL
REVIEW
electron mesh. In conclusion, we have prepared quantum dots with variable interaction strength and investigated the farinfrared transmission spectra at low temperatures. For isolated quantum dots in a magnetic field an edge mode (co-) and a bulk mode (co+) are observed as expected. For strongly interacting dots new spectral features become apparent: An additional edge mode at frequencies lower than co- develops and the bulk mode branches into a cyclotron-resonance-like and a magnetoplasmonlike mode. These features display the transition from isolated dots to an electron mesh quite similar to the transition from atoms to molecules or from atoms to a solid state. The former is manifested in the appearance of an edge mode enclosing two adjacent dots; the latter in the formation of a collective excitation, the magnetoplasmon. We wish to thank J. Alsmeier and W. Hansen for valuable discussions and acknowledge financial support by the European Strategic Program for Research in Information Technology Basic Research ACtion.
ID. R. Hofstadter, Phys. Rev. B 14, 2239 (1976). 2D. Weiss, K. v. Klitzing, K. Ploog, and G. Weimann, Europhys. Lett. 8, 179 (1989). 3R. W. Winkler, J. P. Kotthaus, and K. Ploog, Phys. Rev. Lett. 62, 1177 (1989). 4Ch. Sikorski and U. Merkt, Phys. Rev. Lett. 62, 2164 (1989). ST. P. Smith, III, K. Y. Lee, C. M. Knoedler, J. M. Hong,
2562
LETTERS
21 M A Y 1990
and D. P. Kern, Phys. Rev. B 38, 2172 (1988). 6W. Hansen, T. P. Smith, III, K. Y. Lee, J. A. Brum, C. M. Knoedler, J. M. Hong, and D. P. Kern, Phys. Rev. Lett. 62, 2168 (1989). 7C. T. Liu, K. Nakamura, D. C. Tsui, K. Ismail, D. A. Antoniades, and H. I. Smith, AppL Phys. LetL 55, 168 (1989). SM. A. Reed, J. N. Randall, R. J. Aggarwal, R. J. Matyi, T. M. Moore, and A. E. Wetsel, Phys. Rev. Lett. 60, 535 (1988). 9T. Demel, D. Heitmann, P. Grambow, and K. Ploog, Phys. Rev. Lett. 64, 788 (1990). 10j. Alsmeier, E Batke, and J. P. Kotthaus, Phys. Rev. B 41, 1699 (1990). l lThis might have been the reason why Liu et al. (Ref. 7) were not able to observe the oJ- mode. 12S. J. Allen, Jr., H. L. St~rmer, and J. C. M. Hwang, Phys. Rev. B 28, 4875 (1983). 13B. A. Wilson, S. J. Allen, Jr., and D. C. Tsui, Phys. Rev. B 24, 5887 (1981). 14C. G. Darwin, Proc. Cambridge Philos. Soc. 27, 86 (1930). 15D. C. Glattli, E. Y. Andrei, G. Deville, J. Poitrenaud, and F. I. B. Williams, Phys. Rev. Lett. $4, 1710 (1985). 16F. Brinkop, W. Hansen, J. P. Kotthaus, and IC Ploog, Phys. Rev. B 37, 6547 (1988). 17T. Demel, D. Heitmann, P. Grambow, and K. Ploog, Phys. Rev. B 38, 12732 (1988). 18L. Brey, N. F. Johnson, and B. I. Halpcrin, Phys. Rev. B 40, 10647 (1989). l~l'. N. Theis, J. P. Kotthaus, and P. J. Stiles, Solid State Commun. 24, 273 (1977). 2Oy. Zhu, D. Huang, and S. Feng, Phys. Rev. B 40, 3169 (1989). 2IT. G. Matheson and R. J. Higgins, Phys. Rev. B 25, 2633 (1982).
265
VOLUME 71, NUMBER 4
PHYSICAL
REVIEW
LETTERS
26 JULY. i 993
N-Electron Ground S t a t e E n e r g i e s of a Quantum Dot in M a g n e t i c Field R. C. Ashoori,* H. L. Stormer, J. S. Weiner, L. N. Pfeiffer, K. W. Baldwin, and K. W. West AT& T Bell Laboratories, Murray Hill, New Jersey 07974 (Received 8 December 1992) Using single-electron capacitance spectroscopy, we map the magnetic field dependence of the ground state energies of a single quantum dot containing from 0 to 50 electrons. The experimental spectra reproduce many features of a noninteracting electron model with an added fixed charging energy. However, in detailed observations deviations are apparent: Exchange induces a two-electron singlet-triplet transition, self-consistency of the confinement potential causes the dot to assume a quasi-twodimensional character, and features develop which are suggestive of the fractional quantum Hall effect. PACS numbers: 73.20.Dx,71.50.+L 72.20.My,73AO.Gk
For a given atomic species, isolated atoms have identical spectra; this property has facilitated the study of atomic physics, as signal levels in experiments can be enhanced by simply creating samples containing many atoms. In contrast, the study of artificially structured atoms in semiconductors, or "'quantum dots," has been impeded by nonuniformity in systems of many dots. Recently, two techniques have been developed which allow spectroscopic study of the ground state (g.s.) energies in indioidual quantum dots with a resolution limited only by the temperature of the electronic system [1,2]. A key question to be answered by spectroscopic studies on quantum dots is the role of the electron,electron interaction in modifying the dot's electronic level structure. Bryant [3] has addressed this question for quantum dots containing just two electrons. He finds a continuous evolution of the level structure, from single-particle-like states in the limit of a very small dot, to a level structure dominated by the electron-electron interaction in larger dots. Since the confinement potential in semiconductor quantum dots can be controlled at will, a large range of this continuum which is not accessible in atomic physics can be examined. In a previous paper Ill, we have introduced singleelectron capacitance spectroscopy (SECS). The method allows the direct measurement of the energies of quantum levels of an individual small structure (dot) as a function of magnetic field (B). When the Fermi energy of an electrode becomes resonant with a quantum level of a nearby dot, single electrons can tunnel back and forth between the electrode and the dot through a tunnel barrier [see Fig. l(a)]. The resulting charge induced by this motion on the opposite electrode of a "tunnel capacitor" is detected by an on-chip, highly sensitive transistor. Using this technique we were able to detect spatially distinct localized states in a small tunnel capacitor. In this Letter, we now use SECS to measure the g.s. energies of a single quantum dot containing N electrons in which charge nucleates in only one central location. The basic configuration of our GaAs samples has been described previously [1], although the semiconductor structure has been slightly modified for the present exper-
iments. A schematic of the sample is shown in Fig. 1(a). The layer sequence is as follows: a 3000 ~ n +Moped (4xl017 cm -3) GaAs bottom electrode; a 600 ,~ undoped GaAs spacer laye~ a 125 A. undoped Alo.3Gao.TAs/GaAs superlattice tunnel barrier; a 175 ]~ quantum well (vertically confines the quantum dot); a 500 A Al0.3Gao.TAs blocking barrier; and a 300 A GaAs cap layer. The blocking barrier contains a Si delta doped layer 200 A from the well edge. The wide 600 A spacer layer and the superlattice tunnel barrier [4] were implemented to prevent Si atoms from migrating into the well [1]. Lateral confinement is provided by first patterning a 3500 A. diam circular metallic disk on top of the sample surface and using this as an etch mask to etch down to the AlGaAs blocking barrier surface. The 3500 ,~ diam top electrode is contacted for measurement by overlaying it
FIG. 1. (a) Schematic of sample. (b) Capacitance data vs gate bias for the quantum dot sample in zero magnetic field. The top and bottom traces show the signal resulting from electron tunneling in phase and electron tunneling in 90~ lagging phase with the 210 kHz excitation voltage, respectively. 613
266
VOLUME 71, NUMBER 4
PHYSICAL
REVIEW
with a !.5 gm diam metal disk. All measurements are taken at 0.35 K. Figure 1(b) displays capacitance versus gate bias data for the quantum dot sample. The top trace is the signal observed in phase with the excitation voltage. A first peak appears at - 3 7 3 mV and arises as the lowest electronic state of the dot becomes resonant with the Fermi energy of the n + electrode. With increasing positive gate bias subsequent electrons tunnel onto the quantum dot. Unlike our previous results in a larger dot, the peaks are spaced rather uniformly, with their separation decreasing slightly with increasing electron number. The constancy of the peak heights attests to the quantization of charge that is being moved onto the dot. Beyond the 25th peak, the peak heights in the top trace of Fig. l(b) drop due to a decrease in the tunneling rate. This interpretation is confirmed by measuring the signal at the dot in 90 ~ lagging phase, shown in the bottom trace of Fig. l(b), where peaks occur only for N > 25. This behavior is unambiguous evidence that the tunneling rate of electrons is becoming smaller than the 210 kHz excitation frequency. A slow tunneling rate causes an electron to "wait" a length of time before it tunnels in response to the excitation voltage, and its motion thus lags the excitation. Measurements on several wafers suggest that the decreasing rate is attributable to the thick 600 A spacer layer, which itself acts as a long and low ( < 20 mcV) tunnel barrier. The regime of a few electrons in a dot has been probed by relatively few experiments [5]. We now use SECS in the B field to study this domain with unprecedented resolution. Figure 2 is a color scale image of the dot capacitance as a function of gate bias and the B field applied perpendicular to the plane of the dot. The white, red, and black regions correspond to the highest, intermediate, and lowest capacitance, respectively. The gate bias scale is converted to an energy scale [1] by division by a lever arm of 2.0 +_ 0.1 for this structure. Figure 2 represents the B-field evolution of the first 35 N-electron g.s. energies of the quantum dot. The field dependence of the lowest energy state in Fig. 2(a) is smooth and is well described by the first electron in a cylindrically symmetric parabolic potential [6] ~-m'toUr 2 with hto0=5.4 meV. The high field asymptote of this curve follows the dashed line in Fig. 2(a) with slope htoc/2. From the classical turning points of the lowest bound state we deduce a dot diameter of 408 A. In contrast to the first electron, the evolution of the ground state energy of two electrons shows a pronounced "bump" and a change of slope at about 1.5 T (see dot on second electron). We interpret this feature as a singlettriplet crossing. Considering noninteracting states, the first two electrons in the dot fall into a twofold (spin) degenerate ground state for B - 0 . At higher field, the energy difference between the ground orbital state and the first excited state shrinks, and the Zeeman effect causes a level crossing at 25 T for h too = 5.4 meV. 614
LETTERS
26 JULY 1993
Electron-electron interactions significantly reduce r the B field for this singlet-triplet crossing. Wagner, Merkt, and Chaplik [7] have calculated its position for parabolic quantum dots. For htoo-5.4 meV, the crossing is expected at 3.6 T, about a factor of 2 higher than seen in Fig. 2(a). The discrepancy may arise from the assumption of a strictly parabolic potential in the calculation. Such a singlet-triplet crossing has not been observed in atomic physics experiments due to the exceedingly high B field required (4x l05 T for He). The weak binding of electrons in our quantum dot along with the small electronic mass shifts it to attainable fields. The singlet-triplet crossing should exist et,en in the absence of a Zeeman splitting, arising solely from the electron-electron interaction [7]. The angular momentum quantum number m of the two electrons in the ground state increases with B, being equal to zero only at low field [7,8]. The energy difference between single-particle states of progressively larger angular momenta decreases with increasing B; in higher fields, it becomes advantageous for the system to place electrons in states of successively higher angular momenta (larger orbit radii) in order to decrease the Coulomb repulsion between electrons. To maintain exchange antisymmetry of the two-electron wave function, the system undergoes singlet-triplet (triplet-singlet) crossings as m switches from even (odd) to odd (even) numbers. The Zeeman energy moves the first singlet-triplet crossing to yet lower fields. Moreover, at higher fields the Zeeman effect may force the system to remain in a spin triplet, allowing only transitions between odd m states. For our GaAs dot, the nature of transitions beyond the initial singlet-triplet crossing depends sensitively on the value of h too for the dot as well as on the precise shape of the bare confining potential. These transitions cause smaller changes of slope in the two-electron g.s. energy, and we do not attempt to label them here. The data of Fig. 2(a) display several unexpected features. The bump seen in the g.s. energy of the twoelectron system seems to progress through ~'ihe fewelectron system (white dots). Its position shifts monotonically to higher fields with increasing N, producing a clear "ripple" through the data set. It seems likely that these features are also spin related. Finally, selected traces of Fig. 2(a) show a distinct intensity loss with increasing B resulting from an unexplained decreased tunneling rate. Figure 2(b) shows the ground state energies of the dot for N - 6 - 3 5 on an expanded field scale. In order to interpret the general features of this data set, we turn first to Fig. 3(a). This graph reproduces the highly intertwined single-particle states of a cylindrically symmetric parabolic potential with h to0 = 1.12 meV in a B field. N electrons in this system fill the N lowest energy states. The g.s. energy of the Nth electron should thus oscillate as levels cross as indicated in bold red for the fourteenth electron g.s. The oscillations cease at about 2 T. The density of electrons at the center of the dot is larger than
267
VOLUME 71, NUMBER 4
PHYSICAL
REVIEW
FIG. 2. Color scale plots of the sample capacitance as a function of both magnetic field and gate bias. The vertical bars in both (a) and (b) represent an energy of 5 meV. The dashed line shows hazel2. Numerals along the traces indicate the electron number N. The magnetic field and energy scales are different in both (a) and (b). The symbols are discussed in the text.
at the dot's edges. Taking the Landau level index v for the dot to be given by the Landau level occupancy at the dot center, the position of the last crossing in Fig. 3(a) can thus be identified with v - 2 , with two electrons at dot center per flux quantum passing through the dot. In order to incorporate the electron-electron interaction to lowest order into this picture, we follow the constant interaction (CI) model [2,9,10]. It consists of singleparticle states each separated by a charging energy, similar to what is shown in Fig. 3(d) and observed in Fig. 2(b). In Fig. 2(b), the development of the v - 2 positions are clearly visible (white triangles). Beyond N - 1 0 , the v - 2 positions for each successive electron agree well with the CI model using a constant hco0-'l.I meV. Curiously, the tunneling rates are attenuated around v - ' 2 at large N. At v ' - 2 , the electrons in the dot center are in a quantum Hall state, and we speculate that tunneling suppression arises from the incompressibility of this state. Figure 3(b), taken at 125 kHz, zooms in on the v---2 region for N " 2 7 - 3 2 . The oscillations expected from the CI model are clearly visible. To follow the traces more carefully, we fitted each capacitance peak of the original data set and plot their central positions in Fig. 3(c) for N - 2 1 - 3 3 . The traces have been moved together in the vertical direction for clarity. For comparison, Fig. 3(d) shows the results of the CI model for the same N values in a parabolic dot with hco0-1.12 meV using an arbitrary charging energy of 0.6 meV to separate the traces. Although the qualitative agreement between experi-
LETTERS
26 JULY 1993
FIG. 3. (a) Theoretical Darwin-Fock states for a parabolic quantum dot with halo- !. ! 2 meV. The bold red curve displays the magnetic field evolution of the fourteenth electron. (b) Color scale capacitance data for N--27-32. (c) Measured Nelectron ground state energies as a function of magnetic field
for N - 2 1 - 3 2 , extracted from the data set which includes (b). (d) Evolution of N-electron ground states for N - 2 1 - 3 2 , calculated from the single-particle model. The vertical bar in (b) represents 5 meV and applies to all four windows. ment and the simple model is satisfying, there exist some remarkable differences. The CI model of Fig. 3(d) presents a pattern of oscillation-s with nearly uniform period and amplitude in each of the traces. The experi-
FIG. 4. Sample capacitance as a function of gate bias and magnetic field for N'-39-46. The vertical bar represents an energy of 5 meV. The white symbols are discussed in the text. 615
268
VOLUME 71, NUMBER 4
PHYSICAL
REVIEW
mental traces of Fig. 3(c), on the other hand, show such oscillations only near v - ' 2 and drop considerably for higher fields. In the CI model, we expect [Fig. 3(d)] that the g.s. energies for successive N-electron states will alternately oscillate in phase (due to spin degeneracy) and 180~ out of phase. In the experiment [Fig. 3(c)] different pairs of successive traces display 180 ~ phase shifts depending on the B-field value [red and blue bars in Fig. 3(c)]. While we presently have no explanation for the existence of oscillations only close to the v " 2 region and their relative phases, we believe that the energy drop beyond v - ' 2 is related to the nonparabolicity of the selfconsistent potential. Hartree calculations [! I] show that the bottom of the dot's confinement potential is "'flattened" considerably by the presence of electrons, and in the interior can be considered as a small two-dimensional (2D) system. In a 2D system there exist well-known sudden drops in the chemical potential as Landau levels depopulate in the B field. As N is increased, the dot approaches a 2D system, giving rise to the enhanced chemical potential drop at v - 2 seen in our data. The identification of the v ' - 2 position allows us to determine the size of the dot, calculate its charging energy, and compare it with the observed gate bias spacing between successive electrons [9]. Since for large N the potential around the dot center is approximately constant, we can define a capacitance C of the dot to the electrodes. Ignoring the comparably small quantum level spacings, successive electron additions occur when the electrostatic potential in the dot changes by e/C. In a dot with a flatbottom potential, the area of the dot A is related to the Landau level filling fraction v by A - N ( h / e B v ) . For the 30th electron, v ' - 2 occurs at about 2.2 T, which translates into a dot diameter of 1900 A,. Assuming parabolic confinement with h t o o " l . l meV rather than fiatbottom confinement decreases the dot area by only 2%. A simple parallel plate capacitor model neglecting fringing fields suggests peaks spaced 4.2 mV apart in gate bias, only ---25% larger than the measured spacing. As we move to yet higher N, approaching the 2D limit, additional features become apparent in our spectra. Figure 4, taken at 125 kHz, displays the chemical potentials of the dot containing 39-46 electrons. Similar to Fig. 3(b), we observe the steep drop in chemical potential at B fields just beyond v - 2 (white triangles). The same behavior is now apparent at v - 4 (white dots). We attribute the accentuation of these features to the increasingly 2D character of the system at high filling. Pursuing further the transition between a quantum dot and a finite-sized 2D electron system, we now examine the region v < 2 at B above 4 T. We observe a sequence of "bumps" shifting only slightly to higher B with higher N. These features are inexplicable in terms of any CI model which all predict that successive traces oscillate 180 ~ out of phase [2,10]. We hypothesize that the bumps 616
LETTERS
26 JULY 1993
seen in Fig. 4 are of many-particle origin reminiscent of the fractional quantum Hall effect (FQHE). In the FQHE the chemical potential of the system undergoes maxima between FQHE steps and minima at the steps [12]. The features seen in Fig. 4 are 0.2-0.5 meV in height, not unlike the characteristic energy range of the FQHE at such B fields. Moreover, the decrease in tunneling rates (intensity) between the bumps (see arrows in Fig. 4) may reflect the energy gaps in the FQHE. These features grow monotonically in prominence as more electrons are added to the dot, suggesting a two-dimensional origin. The size and distribution of the electron density within the dot vary with the B field, and it is thus difficult to assign a precise value of v at dot center for fields beyond v - 2 . While the nonuniform electron density in the dot complicates the problem, one still expects FQHE minima [I 3] when the central portion of the dot is at a v value appropriate for the FQHE. We thank S. J. Pearton for help preparing the quantum dots and L. 1. Giazman, P. Hawrylak, P. A. Lee, A. H. MacDonald, B. I. Shklovskii, and N. Wingreen for helpful discussions.
*Present address: Department of Physics, Massachusetts Institute of Technology, Cambridge, MA 02 i 39. [I] R. C. Ashoori, H. L. Stormer, J. S. Weiner, L. N. Pfeiffer, K. W. Baldwin, and K. W. West, Phys. Rr Lett. 68, 3088 (1992). [2] P. L. McEuen, E. B. Foxman, U. Meirav, M. A. Kastner, Yigal Meir, Ned S. Wingreen, and S. J. Wind, Phys. Rev. Lett. 66, 1926 (1991). " [3] Garnctt W. Bryant, Phys. Rev. Lett. 59, 1140 ([987). [4] U. Meirav, M. Heiblum, and l~rank Stern, Appl. Phys. Lett. 52, 1268 (I 988). [5] B. Meurer, D. Heitmann, and~K. PIoog, Phys. Rev. Lett. 68, 1371 (1992); Bo Su, V. J. Goldman, and J. E. Cunningham, Science 255, 313 (1992). [6] C. G. Darwin, Proc. Cambridge Philos. Soc. 27, 86 (1930). [7] M. Wagner, U. Merkt, and A. V. Chaplik, Phys. Rev. B 45, 1951 (1992). [8] P. A. Maksym and Tapash Chakraborty, Phys. Rev. Lett. 65, 108 (1990). [9] R. H. Silsbee and R. C. Ashoori, Phys. Rev. Lett. 64, 1991 (1990). [10] P. L. McEuen, E. B. Foxman, Jari Kinaret, U. Meirav, M. A. Kastner, Ned S. Wingreen, and S. J. Wind, Phys. Rev. B 45, ! ! 419 (1992). [I !] Arvind Kumar, Steven E. Laux, and Frank Stern, Phys. Rev. B 42, 5 i 66 (1990). [i 2] T. Chakraborty and P. Pietil/iinen, The Fractional Quantum Hall Effect (Springer-Verlag, Berlin, 1988). [13] Jari M. Kinaret, Yigal Meir, Ned S. Wingreen, Patrick A. Lee, and Xiao-Gang Wen, Phys. Rev. B 45, 9489 (1992); A. H. MacDonald and M. D. Johnson, Phys. Rev. Lett. 70, 3107 (1993).
269
VOLUME 69, NUMBER i0
PHYSICAL
REVIEW
LETTERS
7 SEPTEMBER !992
Z e r o - D i m e n s i o n a l States and Single Electron Charging in Q u a n t u m Dots A. T. Johnson, h) L. P. Kouwenhoven, W. de Jong, N. C. van der Vaart, and C. J. P. M. Harmans Faculty of Applied Physics, Delft University of Technology, P.O. Box 5046, 2600GA Delft, The Netherlands C. T. Foxon (b) Philips Research Laboratories, Redhill, Surrey RHI 5HA, United Kingdom (Received 19 May 1992)
We observe new transport effects in lateral quantum dots Where zero-dimensional (0D) states and single electron charging coexist. In linear transport we see coherent resonant tunneling, described by a Landauer formula despite the many-body charging interaction. In the nonlinear regime, Coulomb oscillations of a qunatum dot with about 25 electrons show structure due to 0D excited states as the bias voltage increases, and the current-voltage characteristic has a double-staircase shape. PACS numbers: 73.20.Dx,72.20.My,73.40.Gk Transport through semiconductor quantum dots shows striking effects due to the electron wave nature and its finite charge. The first leads to the formation of zerodimensional (0D) states with discrete energies in a system confined in all three directions [1,2], and the possibility of coherent resonant tunneling [2], as with photons in a Fabry-P6rot cavity. The latter induces Coulomb effects, which cause a strong shift in the dot energy upon addition of a single electron [3]. Experiments are beginning to be done on quantum dots where these effects coexist. McEuen et al. [4] used transport measurements to determine the magnetic field dependence of N-electron ground-state energies in such a system, and related it to the calculated energies of single-particle levels in the absence of charging. Similar issues have been addressed in double barrier resonant tunneling structures [5], and using capacitive [6] and optical [7] techniques. Here we report new results from dots where 0D states and charging coexist. We observe coherent resonant tunneling in the linear regime (low bias voltage), surprisingly well described by a Landauer formula (until now applied only to noninteracting systems) despite the electronic Coulomb repulsion. We also see the signature of combined 0D states and charging in nonlinear transport. The current-voltage characteristic has a double-staircase shape, and we can measure the tunnel spectroscopy of the excitations of a quantum dot containing about 25 electrons. The two quantum dots of this work are defined by metal gates on top of a GaAs/AIGaAs heterostructure with a two-dimensional electron gas (2DEG) 100 nm below the surface. The ungated 2DEG has mobility . 230 m2/Vs, I e2
Gdot "- ~
and electron density 1.9x 1015 m -2 at 4.2 K. Applying - 3 0 0 mV to the gates depletes the 2DEG under them, making a quantum dot coupled to large reservoirs via barriers at the center of two quantum point contacts (QPCs). Making the voltage on the QPC gates more negative (positive)decreases (increases) the barrier transmissions. The inset of Fig. 1 gives the geometry of dot 1, with pattern size 0.8/~m x I pm. The "finger" gate F forms one side of the dot, while QPC gates 1 and 2, and center gate C form the other side. With depletion, we expect this dot to be circular with a diameter of about 0.6 /zm. The inset of Fig. 3 shows the layout for sample 2, with QPC gate pairs 1 and 2, and center gate pair C. The central region is 0.2/zmx0.6/~m. For this sample, we apply a more negative voltage to the center gates (typ, ically - 9 0 0 mV), enhancing the depletion region around them. The dot is again circular, but now with diameter 0. I/zm. Measurements were done in a dilution refrigerator at its base temperature below 20 mK. We took data on sample 1 in a magnetic field of 7 T, when transport through the dot is via the lowest-energy edge channel, lying along the dot ~cireumference, and is essentially one dimensional [2,8]. If a quantum particle of proper energy moves between two barriers without loss of phase memory, coherent resonant tunneling occurs through a 0D state formed by constructive interference of multiply reflected partial waves. As in an optical FabryP6rot cavity, the transmission probability can approach 1, even if each barrier alone is highly reflecting. At zero temperature in one dimension and in the absence of charging effects, the conductance of this interferometer is given by a Landauer formula [2]"
TIT2
h l+(l-Tl)(l-T2)-2[(l-Ti)(1-T2)]i/2cos~p
"
( 1)
Tl and T2 are the barrier transmissions, and ~ is the phase acquired by a wave in one round trip between the barriers. Finite temperature leads to energy averaging by the derivative of the Fermi function, reducing the peak transmission. Biittiker [9] has described the transition from coherent to incoherent (sequential) tunneling using the Landauer-Biittiker formalism. Figure 1(a) shows the conductance G of dot I as a function of voltage Vi on QPC gate 1, when V2 is set so QPC2 has transmission T 2 - 1 , and gates F and C are formed. Transport at B - ' 7 T is adiabatic over distances much larger than 1592
9 1992 The American Physical Society
270
VOLUME 69, NUMBER I0
0.6
PHYSICAL
REVIEW
.i~
(a)
04
~
0.2
~
0
J
i
'
~
1
0.2
z o
o.1
0
-850 -800 GATE VOLTAGE [mV]
-750
FIG. 1. (a) Conductance of QPCt vs gate voltage v! for dot i. I,'2is set so 7'2- I. (b) Coulomb oscillations of dot I as l,'t is swept. T2 is about 0.02. The maximum peak height calculated with Eq. (1) at a temperature of 50 mK is shown by the heavy line. The dashed line is the classical prediction. The magnetic field is 7 T. Inset: Gate geometry for dot i. the dot size, SO G measures Tt: G-Tte2/h. When I/2 is more negative, so QPC2 is a tunnel barrier (T2~0.02), sweeping Vt gives the periodic conductance peaks of Fig. I(b). These are the Coulomb oscillations [3] of the charging regime, caused by sweeping a gate voltage that is capacitively coupled to the dot. In contrast to the usual experiments, here sweeping VI simultaneously changes Ti, the transmission of QPCt. The peak height of the oscillations shows a dramatic modulation, correlated with Ti [Fig. l(a)], but in a nonclassical manner. Near V t - - 770 and - 8 5 0 mV, for example, the peak con-
ductance is strongly suppressed, even though T! is at a maximum of 0.6. The classical, one-dimensional sequential tunneling prediction for the conductance maxima is shown by the dashed line in Fig. l(b): Gd--(e2/ h)TtT2/(Tt+T2--TIT2). The actual conductance peaks exceed this prediction by as much as a factor of 15. In contrast, the peak conductance ( c o s t - 1 ) predicted by the quantum formula (1) agrees well with the data, when thermal averaging of about 50 mK is included [Fig. 1(b), heavy line], in line with the temperature and bias voltage (5 pV) ofthe experiment. In Eq. (1), Tl and T2 must match to have total transmission well above the sequential value. Since T2 ~ 0.02 in Fig. l(b), increasing the transmission of QPCt above 0.02 reduces the total transmission predicted by Eq. (1), just as in the data. This is the first demonstration that coherent transport described by an independent-electron Landauer formula occurs despite the many-body charging interaction. Although initially surprising, this result is in agreement with the idea that transport in the linear regime occurs when the electrochemical potential of the dot is equal to
LETTERS
7 SEPTEMBER 1992
that of the reservoirs [10]. Transport of the Nth electron is an energy-conserving process, where phase coherence is maintained even though the other N - 1 electrons experience a Coulomb energy change. Muir and Wingreen [1 I] have recently developed a Landauer-type formula for interacting systems. Along with coherent resonant tunneling in the linear regime, combined 0D states and charging lead to novel nonlinear transport effects, clearly shown in experiments on the smaller dot 2. If a set of 0D states of energy E t,E2 . . . . coexists with charging, the dot's electrochemical potential changes discontinuously as the number of electrons increases: pdOV+I)--pd(N)--Ec+SE [10]. Here E c - e 2 / C is the electrostatic energy cost of charging the quantum dot by one electron (C is the total capacitance from the dot to ground), and 8 E - E n + l - - E n is the energy between 0D states, also the minimum energy needed to excite the N-electron ground state. With diameter d - 0 . 1 pm, dot 2 contains about N - 2 5 electrons at the bulk density. We estimate the charging energy E c - e 2/C ~- 9 2/4~60d-3.5 meV, where ~ - - 13 for GaAs, and excitation energy 8E ~ E F / N - 3 0 0 geV. Both energies far exceed ks T at 20 mK. Although we refer here to separate charging and 0D state energies of an independent (uncorrelated) electron system, the concepts can be generalized: A correlated system also has a change in electrochemical potential upon adding one electron and a minimum excitation energy at a fixed number of electrons, the analogs of Ec+ 6E and BE, respectively. Calculations exist for nonlinear transport [12], but the dot potential energy landscape of Fig. 2(a)gives us a qualitative understanding. At zero temperature, states of the left (fight) reservoir are fully occupied UP to /~L (~ar) and empty at higher energies. Solid lines in the dot show ~ad(N) and /~d(N+l) characterizing the N- and ( N + l)-electron ground states, )vhile dashed lines are discrete 0D excited states of the dot. Suppose 0 < ~az --PR < gd(N+ l ) - - ~ d ( N ) , SO at most one charge level lies between g r a n d ~aR. Wheh'the transport condition ~az > ~ad(N)> ~ r is satisfied, current flows as electrons tunnel one by one from left to fight via states in the dot with energy between ~aL and #s. If, on the other hand, I~d(N + 1) > gL,I~R > ~d(N), no current flows due to the Coulomb blockade. Changing the center gate voltage Vr shifts the conduction-band bottom and with it all charge levels /~d(N), producing Coulomb oscillations in the current as the transport condition is alternately satisfied and not satisfied. In the metallic limit, when the broadening of the 0D states is much larger than the splitting 6E, the dot excitation spectrum is continuous. As the bias voltage V " ( g L - / ~ r ) / e increases, the Coulomb oscillations broaden and grow in amplitude, but remain featureless. This is not true when discrete GD states exist. At small bias voltage eV<<Ec,~E, the number of states in the allowed energy range between/~L and ~ar changes from 0 to 1 to 0 as Vr is varied, giving a smooth oscillation. 1593
271
VOLUME 69, NUMBER I0
PHYSICAL
(') ~
vcll~
,,
REVIEW
LETTERS
7 SEPTEMBER 1992
2
g 2"I ()b"
'
...... '
'
[-.,
z~
"Ii
~Zl rj
o
-0.75 -0.7 CENTER GATE VOLTAGE IV] .
FIG. 2. (a) Potential energy landscape (left) and Coulomb o6cillation with 0D shoulders (right) for a quantum dot with bias voltage eV--t~t. --I~m"1.8aE. Solid lines in the dot are the electrochemical potentials ,u#(/V) and/td(N+ 1). Dashed lines show excitations with splitting 6E. The number of states available for transport, noted by the peak, changes as 0-2-1-2-0 as Vc varies. (b) Evolution of 0D shoulders with increasing bias voltage in dot 2. The curves are offset for clarity. From the bottom, the bias voltages are 100, 400, and 700 vV. The magtactic field is 4 T. When eV is of order 6E, however, this model predicts the appearance of "'0D shoulders" in the oscillations. Suppose ~s(N) is just less than VR, and we scan Vc so that v s ( N ) and the 0D state energies increase. The current is first zero (Coulomb blockade). When ; t s ( N ) " # R current flows via the p + 1 0D states with energy between ,at and/~R, where p < eV/~E < p + l (assuming the same energy 8E separates all 0D states). As Vc varies further, one 0D state first becomes unavailable for transport as its energy exceeds /~L (leaving p current-carrying states); next a second becomes available as its energy rises above ~aR ( p + l current-carrying states again). More allowed states give a larger transition rate and more current. After p such cycles,/as(N) exceeds/~t, and the current drops to zero. So when V > 8E/e, the Coulomb oscillations acquire structure, 0D shoulders, due to the discrete spectrum of 0D excitations. This is shown in Fig. 2(a) for eV-- 1.8 6E (p -" 1 ). The number of states available for transport changes as 0-2-1-2-0. The appearance of 0D shoulders with increasing bias voltage in the Coulomb oscillations of dot 2 is shown in Fig. 2(b). The magnetic field is 4 T. Starting with the bottom curve, the bias voltage is I00, 400, and 700/~V, and current flows via a maximum of one, two, and three 0D states ( p - 0 , l, and 2), respectively. Above the shoulders we show the number of states contributing to transport. We can determine the typical 0D splitting ~E by noting that since two or three 0D states appear in 700 /~V, then 270/~V < 6E < 350/~V. This measured 6E is in good agreement with that given above based on the 1594
-2
0 2 V O L T A G E [mV] FIG. 3. Zero-field I-V curves at various center gate voltages for dot 2, showing the double,staircase structure. From the bottom, the center gate voltage is -920, -910, -907, and -905 inV. The curves are offsetfor clarity;all traces have 1 - 0 when V-0. Inset: Sample 2 gate geometry. Transport is from left to right through QPCs 1 and 2.
dot size, and confirms that the dot contains about N - - E r / 6 E ~ 25 electrons. The dot 0D excitation spectrum also causes structure in I-V characteristics, as predicted in Ref. [12]. Suppose in the energy landscape of Fig. 2(a) that the bias voltage V is swept so ItL increases from/~L " ; t R with/~R fixed., At small bias, the Coulomb blockade suppresses current until /~s >/zd(N) >/zR. As/ZL increases from this point, the current grows in small steps as the window between /~L and /IR expands to include additional 0D states one by one, each contributing to transport. Eventually an extra charge level/aa(N+ 1) is included between /1/. and/zR. There is a larger current jump at this point, since transport can now occur two elect/bhs at a time. The I-V characteristic has a double-step structure, with small 0D excitation steps and larger steps of the Coulomb staircase familiar from the metallic regime. This double-step structure is clearly visible in the I-V characteristics of Fig. 3, taken at zero magnetic field. The curves are for different values of the center gate pair voltage ranging from P c ' - - 9 2 0 m V (large Coulomb blockade, bottom curve) to V c - - 9 0 5 mV (zero Coulomb blockade, top curve). The typical spacing between smaller 0D state steps is 300/aV, in good agreement with the above estimates of 6E and N. Regions of negative differential resistance (NDR), not predicted by usual theories, appear at both positive and, more clearly, negative bias voltage. These can be caused by 0D states that for some reason (e.g., dopant-induced irregularities in the dot confining potential) are more weakly coupled to the reservoirs than the other levels [13]. If an electron tunnels into this state, further transport is blocked by the
272
PHYSICAL
VOLUME 69, NUMBER I0 i
i
w
1
]
w
,
,
i
1
REVIEW
1
>
N
-3
-2
-1
0
1
2
3
VOLTAGE [mV] FIG. 4. dl/dV vs V for sample 2 for equally spaced magnetic fields from 0 (top) to 2.56 T (bottom). Peaks in dl/dV correspond to discrete excitations of the quantum dot. The curves are offset for clarity.
charging energy until the state empties. Increasing V making this extra 0D state available for transport, reduces the current, causing a NDR. Current steps in the I - V characteristic caused by 0D states cause peaks in the differential conductance dl/dV. Tunnel spectroscopy of the 0D levels is possible, although complicated by the capacitances between the electrodes and the quantum dot, which shift the energy levels as the bias voltage V is scanned. Figure 4 shows traces of d l / d V vs V for magnetic fields from 0 (top curve) to 2.5 T (bottom curve). At B "-0 the center gate voltage is tuned to a conductance maximum. Since the largest blockade in I - V characteristics was 3.5 mV, at zero field peaks in d l / d V at IV[ < 3.5 mV correspond to excitations of the dot at fixed electron number. These measurements let us track the field dependence of the discrete excitation energies of the quantum dot in the charging regime. This field evolution at times resembles results of theories of confined, noninteracting electrons [14], for example, the behavior of the peak near V - - 1 mV at B - 0 (top curve in Fig. 4). As B increases, this peak moves towards the peak at V - 0 , then near B = 1.5 T it reverses direction and moves back to more negative bias, similar to what is expected for noninteracting electrons in the intermediate-field re-
LETTERS
7 SEPTEMBER 1992
gime. In general, however, the excitation energies do not evolve as predicted by the usual theories. A full discussion will appear in a later publication. In summary, small lateral quantum dots show the combined effect of 0D states and single electron charging in linear and nonlinear transport. Coherent resonant tunneling occurs, described by a Landauer formula despite the charging interaction. Coulomb oscillations and I-V characteristics show extra structure due to the 0D excitation spectrum. We thank J. E. Mooij, C. W. J. Beenakker, and J. J. Palacios for useful discussions; D. J. Maas, W. Kooi, and A. van der Enden for sample fabrication; and the Delft Institute for Microclectronics and Submicron Technology for the use of their facilities. This research was supported by FOM and ESPRIT (NANSDEV, Project No. 3133). Note a d d e d . - - W h i l e revising this manuscript for publication, we received a prcpdnt from E. B. Foxman et al., with data from a sample much like our dot 2. Their results and interpretation are similar to those presented here.
(a)Present address: Division 814.03, NIST, 325 Broadway, Boulder, CO 80303. (b)Prcsent address: Department of Physics, University of Nottingham, Nottingham NG72RD, United Kingdom. [l] M. A. Reed et al., Phys. Rev. Lett. 60, 535 (1988). [2] B. J. van Wees et al., Phys. Roy. Lett. 62, 2523 (1989). 9 [3] D. V. Averin and K. K. Likharev, in Quantum Effects in Small Disordered Systems, edited by B. Artshuler, P. A. Lee, and R. A. Webb (Elsevier, Amsterdam, 1990). [4] P. L. McEucn et al., Phys. Rev. Lett. 66, 1926 (1991). [5] Bo Su, V. J. Goldman, and J. E. Cunningham, Science 255, 313 (1992); P. Gu~ret e~al., Phys. Rcv. Lett. 68, 1896 (1992). [6] R. C. Ashoori et al., Phys. Rev. Lett. 68, 3088 (1992). [7] B. Meurer, D. Hcitmann, amt-K. Ploog, Phys. Rcv. Lctt. 68, 1371 (1992). [8] B. I. Halpcrin, Phys. Rev. B 25, 2185 (1981). [9] M. Biittiker, IBM J. Res. Dev. 32, 63 (1988). [ 10] L. P. Kouwcnhovcn et al., Z. Phys. B 85, 367 (1991 ). [i 1] Yigal Meir and Ned S. Wingreen, Phys. Rev. Lctt. 68, 2512 (1992). I! 2] D. V. Averin, A. N. Korotkov, and K. K. Likharev, Phys. Rev. B 44, 6199 (1991). [13] Ned S. Wingreen (private communication). [14] V. Fock, Z. Phys. 47, 446 (1928); C. G. Darwin, Proc. Cambridge Philos. So<:.27, 86 (I930).
1595
273
VOLUME 71, NUMBER 24 Competing
PHYSICAL
Channels
REVIEW
in Single-Electron
13 DECEMBER 1993
LETTERS
Tunneling
through
a Quantum
Dot
J. Wets, R. J. Haug, K. v. Klitzing, and K. Ploog*
Max-Planek-lnstitutf~r FestkSrperforschung, Heisenbergstrasse1, 70569Stuttgart, FederalRepublic of Germany (Received 3 August 1993) Coulomb blockade effects are investigated in lateral transport through a quantum dot defined in a two-dimensional electron gas. Tunneling through excited states of the quantum dot is o b served for w~rious tunneling barriers. It is shown that transport occurring via transitions between ground states with different numbers of electrons can be suppressed by the occupation of excited states. Measurements in a magnetic field parallel to the current give evidence for tunneling processes involving states with different spin. PACS numbers: 71.50.+t, 71.70.Ej, 72.20.My,.73.40.Gk The Coulomb interaction determines the behavior of electro n transport through mesoscopic electronic systems weakly coupled with two electron reservoirs. Transport is inhibited if the energy necessary to add an additional electron to the mesoscopic island exceeds the electrochemical potential of the reservoirs and the thermal energy kBT. This is known as the Coulomb blockade of tunneling [1]. For quantum dots realized in semiconductor nanostructures the discreteness of the energy spectrum also has to be taken into account. By increasing the voltage of a gate electrode, capacitively coupled to a quantum dot, the levels of the quantum dot are shifted in energy relative to the levels of the electron reservoirs, allowing the number of electrons enclosed in the quantum dot to increase one by one. Applying a magnetic field changes the energy necessary to add an electron to the quantum dot. Different techniques were used to measure this energy as a function of a magnetic field orientated perpendicularly to the plane of disklike quantum dots realized in A1GaAs/GaAs heterostructures [2,3]. Here we use magnetic fields orientated in the plane of the disk to resolve Zeeman spin splitting. This is combined with measurements at finite bias voltage between the reservoirs, where excited states of the confined electron system become accessible in transport, providing new tunneling channels [4-6]. Therefore, for the new magnetic field orientation we will present results of spectroscopy of the ground states and excited states of our system. To define the quantum dot, metallic split gates, as shown schematically in the inset of Fig. 1, were deposited on the top of a Hall bar etched in a GaAs/Alo.33Ga0.67As heterostructure with a 2DEG (electron density 3.4 x 10 Is m -2, mobility 60 m2/Vs at a temperature of 4.2 K). The diameter of the area between the tips of the gates is about 350 nm. In addition to these top gates, a metallic electrode (back gate) on the reverse side of the undoped substrate was used to change the electrostatic potential of the quantum dot. The distance between the 2DEG and the top gates was 86 nm, the distance between the 2DEG and the back gate was 0.5 mm. The sample was mounted in a 3He/4He dilution refrigerator with a base temperature of 22 mK. The two-terminal conductance through the quantum dot was measured by using an ac lock-in technique at a frequency of 13 Hz and an effec-
tive ac source-drain voltage of 5 #V. In addition to the ac source-drainvoltage, a dc voltage VDS in the range of mV could be applied. For the measurements, the top gates were kept at fixed voltages (around - 0 . 7 V). The tunneling barriers could be tuned by slight changes in the voltage applied to the different top-gate fingers. In Fig. 1, a typical curve of conductance versus backgate voltage is shown. Only a few such well separated conductance resonances are observable in our system. For more negative back-gate voltages the conductance is completely suppressed, whereas for more positive backgate voltages the widths of the conductance resonances are broadened and a finite conductance is measured between adjacent peaks. In Fig. 2(a) the differential conductance dI/dVDs is shown as a function of the back-gate voltage VB for the different bias voltages VDS (between - 3 mV and 3 mV in 0.1 mV steps). In the linear grey-scale plot, white regions correspond to dI/dVDs below -0.1 #USand black ones to dI/dVDs above 2 #uS. For clarity the main structures visible in Fig. 2(a) are sketched in Fig. 2(5). At vanishing VDS the conductance resonances are observed. By increasing the absolute value of Vz)s, the range in back-gate voltage VB where transport through the quantum dot occurs is broadened linearly with IVDsl. These regions of transport enclose almost rhombically shaped regions between them, where transport through the quantum dot
J"''
3
9o
0
1
'
'
Z
I.l..
I
I
-10
I
I
'
I
I
-5
'
;
'
1
I
I
I
'
'
'
'
I
I
I
I
I
0
I
'
'
'
'
I
I
I
I
I
5
10
Gate Voltage (V)
FIG. 1. Conductance versus back-gate voltage. Inset: Scheme of the center of split-gate structure used to define the quantum dot.
0031-9007/93/71 (24)/40 ! 9 (4)$06.00 9 1993 The American Physical Society IIII
m
4019
I
I
274
VOLUME 71, NUMBER 24
PHYSICAL
REVIEW
~IG. 2. (a) Differential conductance dI/dVDs given in linear grey scale (white <_ -0.1/~S, black > 2 p~S) as a function of back-gate voltage Va for different bias voltages VDS. (b) At the top: regions of SET are hatched, regions where the number of electrons can change by two at a time are cross-hatched. Lower part: The main structures visible in (a) are sketched. Dashed fines show regime of negative differential conductance; dotted lines show suppressed conductance.
is blocked [Coulomb blockade regime (CB)]. Here, the number of electrons in the quantum dot is fixed to an integer (e.g., to N), as the energy ~N+I(VB,VDS ) necessary to add the next electron [the (N + 1)th] to the quantum dot lies above the electrochemical potentials of the emitter (#E) and of the collector (#c), and the energy #N(VDs, VB) lies below #E and # c [7]. With N electrons enclosed in the quantum dot, the electrostatic potential of the quantum dot is continuously shifted up by sweeping the back-gate voltage lib to positive values. Transport through the quantum dot from occupied states in the emitter to unoccupied states in the collector can begin when #N+I(VB, VDS) falls below #E- As soon aS ~tN+I(VB, VDS ) falls below # c by further increase in VB, the number of electrons gets fixed to N + 1. Thus the boundaries between transport and blockade regimes in the lib vs VDS plane are defined by the condition DE = lZN+I(VB, VDS) and lZN+I(VB, VDS ) -- PC, respectively, where #E -- #C = eVDs. As is visible in Fig. 2(a) these resonance positions shift linearly in lib when changing VDS, indicating that the confinement potential of the quantum dot is only weakly influenced by the small changes of VB or VDS. Thus, the shifts of the electrostatic potential of the quantum dot can be modeled by a capacitance circuit. From the two slopes dVB/dVDs characterizing the boundaries between transport and blockade, the scaling factor a between the change A r of electrostatic potential and the change AVB of back-gate voltage VB is obtained [a = A r = (4.5 :t= 0.2) x 10 -4] [4]. This implies the difference #N+t - - # g in our dot to be around 1.3 meV. From the dot size the electron number in the dot is estimated to be about 50. Because of the charging energy, the number of electrons in the quantum dot can change only one at a time 4020
LETTERS
13 DECEMBER 1993
in the transport regime neighboring the regime of blockade. This is called the regime of single-electron tunneling (SET) [hatched regions in the upper part of Fig. 2(b)]. At vanishing VDS during transport, the quantum dot changes between the ground states of two electron sys, terns (e.g., between a N and a N + 1 electron system). At finite bias voltage, excited states for both electron systems also become accessible, providing new tunneling channels through the quantum dot. There are two possibilities: a new channel opens at the emitter side, or a new channel opens at the collector side. In the first case, an excited state of the N + 1 electron system becomes accessible for putting the (N + 1)th electron from the emitter to the N electron system in the quantum dot. In the second case, the quantum dot is left in an excited state of the N electron system, as the N + lth electron leaves the quantum dot to an unoccupied state in the collector. Both kinds of new channels are visible in Fig. 2 as additional structure within one S E T regime and can be distinguished as the resonance position for the opening of a new channel from the emitter (to the collector) shifts to negative (positive) Va with increasing [VDsl [4]. The distance of VB positions between differentialconductance peaks within one S E T regime, shifting parallel in VB when increasing IVDsl,reveals the energy difference of states with the same number of electrons. This simple interpretation is true only if a fast and complete relaxation to the ground state of the confined electron system in the quantum dot occurs before the next tunneling process through one of the barriers starts. In this case the two systems (N or N + 1 electrons) can be distinguished, allowing the spectroscopy of the N + 1 electron system and of the N electron system separately within one S E T regime. But, it has been shown that relaxation in zero-dimensional systems can be strongly suppressed [8]. Therefore, transitions between excited states of the N electron system and excited states of the N + 1 electron system occur, making the interpretation more complex. The time in which relaxation can occur is roughly estimated by the mean time At between successive electrons passing through the quantum dot. From the mean height A I of current steps within the S E T regime around lib = -12 V in Fig. 2, we obtain At -- e/AI = 0.5 ns. For the measurements shown in Fig. 2(a), the tunneling barriers were tuned to obtain (at vanishing bias voltage VDS) a m a x i m u m amplitude for the conductance peak observable around VB = --5 V in Fig. 2(a), i.e., to have resonant tunneling through symmetric barriers. In Fig. 3, measurements for asymmetric tunneling barriers are shown. Because of the capacitive coupling of the top gates to the quantum dot, the conductance peaks observed in the back-gate voltage are shifted. For asymmetric tunneling barriers the transport process through the quantum dot is expected to be governed by the tunneling process through the "thick" barrier (thick means "less coupling of the quantum dot to the neighboring lead"). For such a barrier the capacitance between the quantum
275
VOLUME 71, NUMBER 24
PHYSICAL
REVIEW
FIG. 3. "(a) Same as for Fig. 2(a), but for asyrametric tunneling barriers. (b) The main structures visible in (a) are sketched. Dotted lines show suppressed conductance. dot and the lead is decreased, as indicated in Fig. 3 by a change in the slopes characterizing the boundaries between transport and blockade regimes. In Fig. 3, for negative bias voltages VDS, electrons are injected into the quantum dot through the thick barrier, whereas for positive bias voltages, they are injected through the thin barrier. In contrast to the more symmetric case, where the opening of new channels at both barriers has the same importance (causing the gridlike structure within the SET regime in Fig. 2), here the conductance through the quantum dot is dominated by the thick barrier. Even more pronounced are the effects observed beyond the SET regime, where the number of electrons in the quantum dot could change, e.g., between N - 1, N, and g + 1 IN + ( 1 / N ) / ( N - 1) regime, see for identification Fig. 2(b) where similar regions are crosshatched]. In Fig. 3 at negative VDS, the structures indicating new transport channels in the N + 1IN regime (SET) stop at the boundary of the g + ( l / N ) / ( N - 1) regime, since in this case the thick barrier is at the emitter side and the quantum dot is preferably filled with N - 1 electrons. Thus, transitions from the N - 1 to the N electron system dominate in the transport process. On the contrary, if the thick barrier is on the collector side, the quantum dot is filled up to N + 1 electrons. Transitions from N + 1 to N electrons dominate the transport. The most striking features observable in Figs. 2 and 3 (dotted lines) indicate the interplay of transport channels opened on the emitter and the collector side leading to a suppression of conductance. For instance, the structure visible in Fig. 2(a) around VB = --12 V at negative VDs may be interpreted by the following: At small negative VDS, the transport through the quantum dot occurs via transitions between the ground states of the N - 1 and the N electron system. Going to negative VDS and negative VB by following the boundary between the transport and the blockade region ( N - 1 electrons enclosed in the dot), this transport channel becomes suppressed (dotted line) by a channel opened on the collector side (solid line
LETTERS
13 DECEMBER 1993
which shifts to positive I/8 with negative VDS). Thus, the occupation of an excited state of the N - 1 electron system blocks transport. Penetrating into the SET regime, an excited state of the N electron system becomes accessible (solid line which shifts to negative VB with negative VDs), again increasing the conductance. Thus, the c o n d u ~ c e through the quantum dot is not always increased but can also be decreased by accessing excited states. This is a property of the SET process: Using one channel blocks the transport through the other channels as the number of electrons in the quantum dot can be changed only one at a time. Electrons from the emitter reservoir compete in entering the quantum dot by different channels and then electrons in the quantum dot compete in leaving the quantum dot by different channels ....... to the collector reservoir. The change of the conductance depends on the traversing time of the electrons through the dot via the different channels, and is also influenced by relaxation processes. Within the SET regime, negative differential conductance (NDC) appears [white regions in Fig. 2(a), dashed lines in Fig. 2(5)] as also observed by Johnson et al. [5]. In our measurements the NDC peaks are shifting parallel to the boundaries of transport and blockade regimes, which again implies that the conductance through the quantum dot is decreased by accessing excited states as explained above. Pfaff et al. [9] modeled NDC by taking into account the different spin degeneracy and the spin selection rule for transition between states of the N and N + 1 electron system. To obtain more information about the different tunneling channels, measurements in a magnetic field B orientated parallel to the plane of the 2DEG and parallel to the current were performed. This orientation diminishes orbital effects, allowing us to follow a conductance peak up to high magnetic fields where Zeeman splitting of the energies of states with different spin quantum numbers is resolvable [at 15 T gDaB is 0.38 meV for the g factor of bulk GaAs (Ig[ = 0.44)]. In Fig. 4(a) the differential conductance measured at VDS = --0.7 mV is plotted in grey scale (white dI/dVDs <_ -0.01 #S, black dI/dVDs _> 1 #S) as a function of back-gate voltage Vs for the magnetic field values between - 1 5 and 15 T in 0.5 T steps. The peak maxima visible in Fig. 4(a) are plotted, in Fig. 4(b), indicating the resonance condition for opening a new channel. The tunneling barriers have been tuned at B -- 0 T roughly to the situation of Fig. 2. The upper and lower boundaries of each SET regime show the same shift in Va position with magnetic field as predicted by the conditions I~(VDs, V ~ , B ) = #c and p,(VDs, V~,B) -- ftE, where # F , - #C -- eVDs is magnetic field independent. At finite bias voltage the magnetic field dependence of a state which is a ground state in some range of the magnetic field can be followed into the range where this state has become an excited state. In Fig. 4(b) for the conductance resonance around VB - - 4 V, four arrows indicate magnetic field posi4021
276
VOLUME 71, NUMBER 24
PHYSICAL
REVIEW
FIG. 4. (a) Differential conductance dl/dVDs given in a linear grey scale (white < -0.01/zS, blac~ > 1/zS) as a function of back-gate voltage I/8 for different magnetic fields B. The bias voltage VDS was --0.7 mV (see Fig. 2). (b) Positions of peak maxima visible in (a). The lines are fitted curves (as discussed in the text). tions, where such transitions occur. In corresponding measurements at vanishing VDS, where only transitions between the ground states are allowed, the shift of the VB position of this conductance peak shows bends accompanied by an amplitude modulation at these magnetic field values. Several different magnetic field dispersions are observable but only two different dispersions can be followed over a magnetic field range large enough to enable a fit to be made to the data ( a B 2 +/3B with same a and different/3). These two dispersions are found in all SET regimes [solid lines in Fig. 4(b)]. The difference in the shifts can be fitted linearly. This linear dependence suggests that the tunneling channels are due to states with different spin. Relating the difference in shift to the Zeeman splitting we obtain a g factor of Ig[ = 0.31 • 0.04. A splitting of the differential conductance peaks is not observed. Malcher et al. [10] calculated the spin splitting of levels in a 2DEG at B - 0 T due to the nonparabolicity of the bulk band structure of GaAs and spin-orbit coupling. This is in the range of a few tenths of a mV which is comparable to typical distances of energy levels observable here in our quantum dot. For the overall shift of the conductance resonances in VB, the change in the chemical potential of the 2DEG has to be taken into account, which was discussed elsewhere [11]. The magnetic field dependence (Fig. 4) can now be compared with the results of Fig. 2, where suppression of the conductance is seen in several regions (dotted lines in Fig. 2). For instance, the feature observable around VB = --12 V in Fig. 2 is interpreted by the following: The transition between the ground states of the N - 1 and N electron system (channel 1) is suppressed because an excited state of the N - 1 electron system has become accessible, blocking transport. The conductance is increased again when allowing the transition to an excited state of the N electron system (channel 2). The differ4022
LETTERS
13 DECEMBER 1993
ence between the transition energies of the two channels increases linearly with magnetic field (Fig. 4). This suggests that states with different spin are responsible for the suppression of the conducCumce through the dot. In summary, transport measurements through a single quantum dot in the single-electron-tunneling regime allow spectroscopy of ground and excited states. New tunneling channels opening at finite drain-source voltage are classified by their shift of position in the back-gate voltage when changing the bias voltage. Transport through new tunneling channels increases or decreases the total conductance through the quantum dot, depending on the interplay between the different channels. Different conductance channels show different dependence on a magnetic field parallel to the current, which we correlate to states with different spin. We gratefully acknowledge stimulating discussions with D. Pfannkuche, R. Blick, H. Pothier, D. Weinmann, W. H/itLsler, J. J. Palacios, C. Tejedor, and P. Maksym. We thank M. Pdek, A. Gollhardt, and F. Schartner for their expert help with the sample preparation. Part of the work has been supported by the Bundesministerium ffir Forschung und Technologie.
* Present address: Paul-Drude Institut fiir FestkSrperelektronik, Hausvogteiplatz 5-7, 10117 Berlin, Germany. [1] Special Issue on Single Charge Tunneling, in Z. Phys. B 85, No. 3 (1991); Single Charge Tunneling, edited by H. Grabert and M.H. Devoret, NATO ASI Ser. B, Vol. 294 (Plenum, New York, 1992). [2] P.L. McEuen, E.B. Foxman, U. Meirav, M.A. Kastner, Y. Meir, N.S. Wingreen, and S.J. Wind, Phys. Rev. Lett. 66, 1926 (1991). [3] R.C. Ashoori, H.L. StSrmer, J.S. Weiner, L.N. Pfeiffer, S.J. Pearton, K.W. Baldwin, and K.W. West, Phys. Rev. Lett. 71,613 (1993). [4] J. Weis, R.J. Haug, K. v. Klitzing, and K. Ploog, Phys. Rev. B 46, 12837 (1992). [5] A.T. Johnson, L.P. Kouwenhoven, W. de Jong, N.C. van der Vaart, C.J.P.M. Harmans, and C.T. Foxon, Phys. Rev. Lett. 69, 1592 (1992). [6] E.B. Foxman, P.L. McEuen, U. Meirav, N.S. Wingreen, Y. Meir, P.A. Belk, N.R. Belk, M.A. Kastner, and S.J. Wind, Phys. Rev. B 47, 10020 (1993). [7] L.P. Kouwenhoven, N.C. van der Vaart, A.T. Johnson, W. Kool, C.J.P.M. Harmans, J.G. WiUiamson, A.A.M. Staring, and C.T. Foxon, Z. Phys. B 85, 367 (1991). [8] U. Bockelmann and G. Bastard, Phys. Rev. B 42, 8947 (1990); U. Bockelmann and T. Egeler, Phys. Rev. B 46, 15574 (1992). [9] W. Pfaff, D. Weinmann, W. Hgusler, B. Kramer, and U. Weiss (to be published). [10] F. Malcher, G. Lommer, and U. RSssler, Superlattices Microstruct. 2, 267 (1986). [11] J. Weis, R.J. Haug, K. v. Klitzing, and K. Ploog, Proceedings of the 10th International Conference on Electronic Properties of Two-Dimensional Systems, Newport, 1993 [Surf. Sci. (to be published)].
277
EUROPHYSICS LETTERS
10 February 1996
Europhys. Left., 33 (5), pp. 377-382 (1996)
Optical-absorption spectra of quantl!m dots and rings with a repulsive scattering centre V. HALONEN1, P. PIETILAINEN 1 and T. CHAKRABORTY2 1 Theoretical Physics, University of Oulu - Fin-90570 Oulu, Finland 2 Institute of Mathematical Sciences - Taramani, Madras 600113, India
(received 19 September 1995; accepted in final form 20 December 1995) PACS. 73.20Dx- Electron states in low-dimensional structures (including quantum wells, superlattices, layer structures, and intercalation compounds). PACS. 71.45Gm- Exchange, correlation, dielectric and magnetic functions, plasmons. PACS. 73.20Mf- Collective excitations (including plasmons and other charge-density excitations). A b s t r a c t . - We have studied electron correlation effects in quantum dots and rings which
include a repulsive scattering centre and are subjected to a perpendicular magnetic field. The results for the dipole-allowed absorption spectrum show good qualitative agreement with the observed magnetoplasmon dispersion in similar systems. This work provides a unified description of the electron correlations in quantum dots and quantum rings in a magnetic field. We also demonstrate that optical absorption is a direct route to explore the effects of impurity and interactions in a quantum ring. Quasi--zero-dimensional electron systems, or quantum dots, in a magnetic field have been under intense investigations in recent years [1]-[6]. These systems exhibit phenomena reminiscent of atoms (and are therefore commonly called artificial atoms) and yet their size, shape, etc. can be controlled in the experiments. Theoretical results on the electronic properties of these quantum-confined few-electron systems [4], [5] have been generally in good agreement with the experimental results [6]. The electronic and optical properties of these systems are essential elements in developing the mesoscopic devices in the future [7]. Ever since the first theoretical work on interacting electrons in quantum dots subjected to a magnetic field was reported [4], a large number of papers on variations of such systems have been published in the literature [3], [5]. Almost all of these theoretical studies involve impurity-free quantum-confined few-electron systems. Here we report on the results of our work on dipole-allowed absorptions of a quantum dot and a quantum ring with a repulsive scatterer at the centre. Experimental work on the magnetoplasma resonances in a two-dimensional electron system confined in a ring geometry has been reported recently [8], and transport properties of the quantum dots with an impurity which can be controlled independently are also under active investigations [9]. Interestingly, such systems are related to another system called antidot array - - a two-dimensional electron system with a periodic array of scatterers, whose transport [10], [11] and optical properties [12] are also of much current interest. Finally, in a mesoscopic ring, the most common problem studied so far, both experimentally [13] as well as theoretically [14], [15] has been the persistent current. We have recently developed a model for a quantum ring [15] where in addition to the persistent current, other electronic properties can also be studied very accurately. The energy spectrum calculated earlier by us for such a system with and without an impurity can be used to explore the dipole-allowed absorption spectrum. (~ Les Editions de Physique
278
378
EUROPHYSICS LETTERS
Our present work, therefore, unifies our understanding of electron correlations in t w o very interesting systems: a quantum dot with a repulsive scatterer and a quantum ring. We also demonstrate here that in optical-absorption studies one makes a direct probe of impurity and correlation effects in such systems. We model the quantum dots and rings like in the earlier works [4], [5], [15]. We consider electrons of effective mass m* moving in the (x, y)-plane confined by a parabolic potential and subjected to a perpendicular magnetic field. The single-electron Hamiltonian is written as
2m---7 p -
A
+ -~m w2(r - r0) 2 ,
(1)
where r0 is the radius of the ring (r0 --- 0 for the dot). We use the symmetric gauge and the vector potential is A = 8 9 Bx, 0). The impurity is modelled by a Gaussian potential V imp ( r ) -- V0 exp[-
(r - R) 2/d2],
(2)
where V0 is the potential strength, d is proportional to the width of the impurity potential (the full width at half-maximum is .~ 1.67d), and R is the position of the impurity. In the present work the position of the impurity is located such that IRI - r0. We apply the exact diagonalization method by constructing the basis using the singleparticle wave functions of the Hamiltonian (1). These wave functions are of the form Cnt = / ~ l ( r ) exp[i/0],
n = 0, 1, 2 , . . . , l - 0, +1, + 2 , . . . ,
(3)
where n and l are the radial and orbital angular-momentum quantum numbers, respectively. For parabolic quantum dots (r0 = 0) the radial part can be expressed explicitly as
Pint (r) - C exp[-r2/(2a2)lr Itl L~'(r2/a2),
(4)
where C is the normalization constant, a = V/h/ (m*Yl), Y2 - X/w 2 + w2/4, and nk(x) is the associated Laguerre polynomial. In our quantum ring model (r0 ~ 0) the radial part P~z has to be determined numerically. Intensities of the optical absorption are calculated within the electric-dipole approximation. If the single-particle matrix elements are defined as dx~, = (A'lr exp[iO]lA ) -- 27r(~l+l,l,
r2R,x,(r)Rx(r) dr,
where A represents the quantum number pair {n, l}, the dipole operators can be written as
{ 1 x = Y =
E
,k),'
+
Z
(5)
~,
The probability of absorption from the ground state I0) to an excited state If) will then be proportional to the quantity A = I(flrl0)l 2 = I(flXl0)l 2 + I(flYI0)l 2 .
(6)
In the figures for t h e a b s o r p t i o n spectra presented below, the areas of the filled circles are proportional to A.
279
V. HALONEN e t a~.: OPTICAL-ABSORPTION SPECTRA OF QUANTUM DOTS ETC.
379
Q u a n t u m dots. - In the numerical calculations that follow, we have used material parameters appropriate for GaAs, i.e. E = 13 and m* -- 0.067me. We have included spin in our quantum dot calculations but ignored the Zeeman energy. The confinement potential strength is chosen to be hw0 = 4 meV and the parameters for the repulsive Gaussian potential at the dot centre are V0 - 32 meV and d -- 5 nm. With these parameters the electrons are confined in a wide ring. Both the effective radius and the width of this ring are about 20 nm for a single electron. Figure 1 shows electromagnetic absorption energies and intensities of the system with one, two and three electrons as a function of the magnetic field. The one-electron results reveal four distinct modes. The strongest of the upper two modes can be interpreted as a bulk magnetoplasmon mode according to its asymptotic behaviour, i.e. its energy approaches hWc as the magnetic field is increased. The origin of the discontinuities near 5 and 8 teslas can be traced back to the fact that the potential forming the ring, in our case, is highly asymmetric. We have a steep Gaussian potential near the centre of the dot and the outer edge is formed by a soft parabolic potential. For a symmetric potential we expect that the bulk magnetoplasmon mode is a smooth function of the magnetic field. If we ignore the above-mentioned discontinuities, the two upper modes of the one-electron spectrum behave clearly the same way as seen in the experimental results of Dahl et o2. [8]. However, the two lower modes behave quite differently (in the one-electron case) when compared with those experimental results. The lower modes, i.e. edge magnetoplasmon modes, reveal a periodic structure similar to the results of a parabolic ring [15] (see below). That is, however, true only for the one-electron system. When the number of electrons in the system is increased, the periodic structure of the edge modes (the two lowest modes) starts to disappear. This is, of course, due to the electronelectron interaction. The Coulomb interaction is very important in wide rings. It should be emphasized that because the spin degree of freedom is also included in these calculations, the difference between the one- and two-electron results is entirely due to the Coulomb force. The model quantum rings we have studied so far [15] are extremely narrow and the interaction does not play an important role. The lowest mode (which is also the strongest) behaves (even only for three electrons) much the same way as does the lowest mode in the experiment [8] (where the system consists of the order of one million electrons). It is quite safe to speculate that the second lowest mode would be the same as the second lowest mode in that experiment. It is interesting to note that this mode is also similar to the observed magnetoplasma resonance in antidot arrays [12]. In the high-field regime, the upper mode observed in antidot systems is also qualitatively reproduced in the quantum dot case. Q u a n t u m rings. - We have demonstrated earlier [15] that our model in the appropriate limit, correctly reproduces the behaviour of an ideal one-dimensional ring [13] and that of a two-dimensional electron gas. The energy spectrum in the case of non-interacting and interacting electrons, magnetization and the susceptibility have been studied earlier in this model. The two-body Coulomb matrix elements were evaluated numerically, with the result t h a t in the loweSt Landau level and for an impurity-free system, the Coulomb interaction simply shifts the non-interacting energy spectrum to higher energies [15]. There is no discernible effect of interaction on the magnetization which was explained as due to conservation of angular momentum in the system. The impurity potential is found to lift the degeneracies in the energy spectru m and the persistent current is then reduced from the impurity-free value. The effect of the Coulomb interaction on the persistent current even in the case of impurity interactions is, however, still insignificant [15]. In our ring model the length is measured in units of r0 (radius of the ring). The energy and impurity strength V0 are expressed in units of h 2 / 2 m * r A , where A - ~rr2 is the area of the
280
380
EUROPHYSICS
'n ...... | ne--'~l
n
u
....
.-
"'oeO0
.
'
00 ~ 1 7 6 1~7 6
16 12
u 9
LETTERS
O
"~ ~ 1 7 6
0.0 !
1.0
2.0
!
!
3.0
1.5 4 1.0 ~
~'~ ~"
16 -
,~
12 -
he--2 gO ~
~
9 O~176176
I
~
oo
0.5
1
o *~176176 9 * 9
<1
." o o
8
9
O0 O O O 0
~
-'" 1.5 -
4-~t~LO~-aO
a "o
oo
(b)_ 9 o.
. . . . . . . . . .
. .....
~ ....
~ ......
1.0 0_8 ~
n e ----3 0000000000000000000000000000000
0.5
4 L.dindL~oOo . 9
0
2
9
OOo0
9
4
6
~---
8
I
10
9 . . . . . . . .
,
0.0
B (T)
I
,,
1.0
I
2.0
,
m. . . .
,
I
3.0
~/~0
Fig. 1.
Fig. 2.
Fig. 1. - Absorption energies and intensities of a quantum dot including a repulsive scatterer with one, two and three electrons as a function of the magnetic field. The areas of the filled circles are proportional to the calculated absorption intensity. Fig. 2. - Dipole-allowed absorption energies of a single electron in a quantum ring vs. ~h/fho for c~ - 20 and a) V0 -- 1.0, d - 0.2; b) V0 -- 4.0, d -- 0.5. The areas of the filled circles are proportional to the calculated absorption intensity.
ring [15]. In these units the confinement p o t e n t i a l is 1
*
2"
U ( r ) = -~m w o ( r -
to) 2 = 4 c ~ 2 ( x - 1) 2,
where c~ = w o m * A / h , x - r / r o . T h e p a r a m e t e r c~ is related to t h e w i d t h of t h e ring. For c~ - 20, the single-electron energy s p e c t r u m closely resembles t h a t of an ideal o n e - d i m e n s i o n a l ring, while for c~ - 5, it has the characteristics of a t w o - d i m e n s i o n a l electron gas [15]. In our present work we have used r0 - 10 n m a n d c~ - 20. Given these p a r a m e t e r s , electrons are confined in a narrow ring whose radius is a b o u t four t i m e s its effective width. T h e o t h e r p a r a m e t e r s are the s a m e as in the q u a n t u m d o t case, namely, c - 13 and m* - 0.067me. In a p u r e one-electron ring the dipole-allowed a b s o r p t i o n f r o m t h e g r o u n d s t a t e can h a p p e n w i t h equal p r o b a b i l i t y to the first two excited s t a t e s and all o t h e r t r a n s i t i o n s are forbidden. A n i m p u r i t y in the ring will mix t h e a n g u l a r - m o m e n t u m eigenstates of the pure s y s t e m into new s t a t e s between which dipole t r a n s i t i o n s are allowed. In the case of an i m p u r i t y of m e d i u m
281
V. HALONEN
e
_0.0
al.:
1.0
U
lo5,1,
2.0
I ......
I
,~I.0 ~~
.
o I
"o
I'
e " ".oo%.;0o, .',0,, g e"o,...o-
OI.
" @ ~
9
eo
9
9
.
l.
/
.
~
9
gO
-
9
1.0
:,,O~
"e'oa.~176
. ' o 0 8 "- .
omw--
r~ 1.oL;-"-'w'~2"" T~176 ~
I II
't
I
I 1.5 - @@@__ .~..
0@@@@__
"...~..@@@O@@..... @@-
0 0 0 @~ "@0000000
1.0-
1.0
0.5 _ ~ o ~ o ~ o ~ ~ o ~ % .
0.5
gOo @
~0 OOOO@"
e@@OO@OO@''~
1"-Ooo OO~
I 2.0
I
3.0
OOO
,.,
.,
(b)
Fig. 3.
.:oO
(a)
1o
! 1.0
-_
./Ooo.
0.5
(a)
n 0.0
3;0
.......... 2 . 0 ........
.
0.5-
[
.....
l'Sr'oo-.
"--
%;..- -...;... % 9
0;0
3.0 .......
9 I
381
O P T I C A L - A B S O R P T I O N S P E C T R A OF QUANTUM DOTS ETC.
(b)., t
0.0 Fig. 4.
I
I
1.0
2.0
I
3.0
~/~o
Fig. 3. - Dipol ~allowed absorption energies for four non-interacting electrons in a quantum ring vs. ~/~o and c~ = 20, where only the impurity potential is included. The other parameters V0 and d are the same as in fig. 2. Fig. 4 . - Same as in fig. 3, but with Coulomb and impurity potential included.
strength, as st ~own in fig. 2 a), an appreciable part of the transition probability still goes to the first two excit ed states while in the case of a strong impurity, absorptions taking the electron to the lowest ~xcited state become more favourable (fig. 2 b)). One very important result here is that, in a s 'stem with broken rotational symmetry the transition probability depends strongly on the polari; ~tion of the incident light. T h a t is, if instead of unpolarized light considered in this work we vrere to consider the case of light polarized for example along the diameter passing through the ir apurity (which is the case in [16]), the absorption would prefer the second excited state. Anothe r prominent feature observed in fig. 2, i.e. the periodic behaviour of absorption energies as fur Lctions of the applied field, follows closely the behaviour of the persistent current. The blocking of this current caused by a strong impurity is reflected as the flat behaviour of absorption fre quencies as a function of the magnetic field. Finally, as we pointed out earlier, the oscillator, behaviour in fig. 2 a) is indeed qualitatively similar to that seen in the quantum dot with the repulsive scatterer at the centre (fig. 1). In order to study electron correlations, we consider rings with four spinless electrons. The main difference to the pure single-electron ring is that the dipole transitions to the first excited state are forbidden (IALI > 1). Just as in the one-electron ring discussed above, the introduction of an impurity will permit transitions to the previously forbidden states. In general, the effect of an impurity and the behaviour of the absorption spectrum as a function of the external magnetic field can be qualitatively explained by the single-particle properties9 For example, when we compare fig. 3 a) and b) we notice that the lifting of the degeneracy in the
282
EUROPHYSICS LETTERS
382
energy spectra of non-interacting electrons is reflected by a smoother behaviour as a function of the applied field. The sole effect of the Coulomb interaction on the energy spectrum is to shift it upwards and to increase the gap between the ground state and the excited states [15]. Consequently, as shown in fig. 4, the Coulomb interaction moves the absorption to higher frequencies. The intensities clearly show the effect of the electron-electron interaction: In the non-interacting system (fig. 3) the intensity of each absorption mode does not depend on the magnetic field at all, whereas in the interacting system (fig. 4) there is a strong variation of intensity as a function of the field. In closing, we demonstrate here t h a t the optical-absorption spectra in a quantum ring not only reflects the behaviour of the persistent current, but it also reveals the subtle effects of the interaction and broken symmetry caused by an impurity. Quite clearly, the magnetoplasma excitations in the q u a n t u m dots and rings with a repulsive scatterer in the middle provide an ideal ground for a detailed study of the impurity and correlation effects in low-dimensional electron systems. REFERENCES
[1] [2] [3] [4] [5]
[6] [7] [El [9] [10] [11] [12] [13] [14]
[15] [16]
For a review see, CHAKRABORTY T., Comments Condens. Matter Phys., 16 (1992) 35. FUKUYAMA H. and ANDO T. (Editors), Transport Phenomena in Mesoscopic Systems, (SpringerVerlag, Heidelberg) 1992. CHAKRABORTY T. (Editor), Proceedings of the International Workshop on Novel Physics in Low-Dimensional Electron Systems, Madras, India, January 9-14, 1995, Physica B, 212 (1995). MAKSYM P. A. and CHAKRABORTY T., Phys. Rev. Lett., 65 (1990) 108; Phys. Rev. B, 45 (1992) 1947. CHAKRABORTY T., HALONEN V. and PIETIL)~INEN P., Phys. Rev. B, 43 (1991) 14289; MERKT U., HUSER J. and WAGNER M., Phys. Rev. B, 43 (1991) 7320; 45 (1992) 1950; JOHNSON N. F. and PAYNE M. C., Phys. Rev. Lett., 67 (1991) 1157; HALONEN V., CHAKRABORTY T. and PIETIL~,INEN P., Phys. Rev. B, 45 (1992) 5980; MAKSYM P. A., Physica B, 184 (1993) 385; BOLTON F., Phys. Rev. Lett., 73 (1994) 158; OAKNIN J. H. et al., Phys. Rev. B, 49 (1994) 5718; MADHAV A. V. and CHAKRABORTY T., Phys. Rev. B, 49 (1994) 8163; HALONEN V., Solid State Commun., 92 (1994) 703; UGAJIN R., Phys. Rev. B, 51 (1995) 714. ASHOORI R. C. et al., Phys. Rev. Lett., 71 (1993) 613; ZRENNER A. et al., Phys. Rev. Lett., 72 (1994) 3382; SIKORSKI CH. and MERKT U., Phys. Rev. Lett., 62 (1989) 2164. THORNTON T. J. , Rep. Prog. Phys., 58, (1995) 311; WEISBUCH C. and VINTER B., Quantum Semiconductor Structures (Academic, New York, N.Y.) 1991; REED M. A. (Editors) Nanostructured Systems (Academic, San Diego) 1992. DAHL C. et al., Phys. Rev. B, 48 (1993) 15480. SACHRAJDA A. S. et al., Phys. Rev. B, 50 (1994) 10856. WEISS D. et al., Europhys. Lett., 8 (1979) 179; Surf. Sci., 305 (1994) 408; KANG W. et al., Phys. Rev. Lett., 71 (1993) 3850; CHAKRABORTY T. and PIETIL~.INEN P., Phys. Rev. B, February 15, (1996). ENSSLIN K. and PETROFF P. M., Phys. Rev. B, 41 (1990) 307. KERN K. et al., Phys. Rev. Lett., 66 (1991) 1618; ZHAO Y. et al., Appl. Phys. Lett., 60 (1992) 1510. MAILLY D., CHAPELIER C. and BENOIT A., Phys. Rev. Lett., 70 (1993) 2020. IMRY Y., in Quantum Coherence in Mesoscopic Systems, edited by B. KRAMER, (Plenum, New York, N.Y.) 1991, p. 221; in [2]; LEGGETT A. J., in Granular Nanoelectronics, edited by D. K. FERRY, J. R. BERKER and C. JACOBONI, NATO ASI Set. B, Vol. 251 (Plenum, New York, N.Y.) 1992, p. 297. PIETIL)kINEN P. and CHAKRABORTY T., Solid State Commun., 87 (1993) 809; CHAKRABORTY T. and PIETILAINEN P., Phys. t~ev. B, 50 (1994) 8460; in [2]; Phys. Rev. B, 52 (1995) 1932. PIETIL)i,INEN P., HALONEN V. and CHAKRABORTY T., in [3].
283
15 MARCH 19924
VOLUME 45, NUMBER 11
PHYSICAL REVIEW B
Excitons in a parabolic quantum dot in magnetic fields V. Halonen Department of Theoretical Physics, University of Oulu, Linnanmaa, SF-90570 Oulu 57, Finland Tapash Chakraborty Institute for Microstructural Sciences, National Research Council, Montreal Road, M-50, Ottawa, Canada KIA OR6 P. Pietil~iinen Department of Theoretical Physics, University of Oulu, Linnanmaa, SF-90570 Oulu 57, Finland (Received 30 August 1991; revised manuscript received 6 November 1991) The properties of an exciton in a parabolic quantum dot in an external magnetic field are studied theoretically using an effective-mass Hamiltonian. The results for the energy and the optical absorption of the ground state and the low-lying excited states are presented. The Hamiltonian is written in terms of the center of mass and relative coordinates, and it is shown that, due to the coupling between the center of mass and relative motion, optical-absorption energies reveal an interesting antierossing behavior. It is also shown that the ground-state properties are approximately determined by that part of the total Hamiltonian that depends only on the relative coordinates.
I. INTRODUCTION A system of electrons and holes moving in two dimensions with their transverse motion quantized in the lowest level and subjected to a strong perpendicular magnetic field is known to exhibit many interesting properties. 1-5 It should_be mentioned that in the single-component case of electrons (or holes) in a similar situation with the lowest Landau level partially filled, a remarkable manyelectron phenomenon known as the fractional quantum Hall effect was discovered some years ago. 6 It is therefore quite natural to investigate what a two-component (electron and hole) system has in store. In the ideal twodimensional case where the electron and hole wave functions are considered to be identical, Lerner and Lozovik l (and later, Rice, Paquet, and Ueda 2) found that the exact ground state is a Bose condensate of noninteracting magnetic excitons. Another interesting result found by Rice, Paquet, and Ueda was that there is no plasma oscillation in this s y s t e m - - a consequence of the confinement to the lowest Landau level. The collective excitation is simply given by the single-exciton dispersion relation which is a result of the ideal Bose character of the ground state. In this paper, we have added another dimension to our present understanding of the excitons in a magnetic field 9discussed above by placing an exciton in a zerodimensional parabolic quantum dot structure. 7 These systems are of much current interest in order to develop an understanding of the mesoscopic physics in reduced dimensionality. Recent experimental work on quantum dots in a magnetic field 8 has demonstrated the interplay between the competing spatial and magnetic quantization and other subtle features due to electron correlations. Theoretical studies 9'1~ have revealed the interesting role of electron correlations in these quantum confined systems. Earlier work by Bryant on excitons 11 and biexcitons 12 in quantum boxes (in the absence of a magnetic field) demonstrated the competing effects of quantum 45
confinement and Coulomb-induced electron-hole correlations. Excitons and biexeitons have also been studied recently in semiconductor mierocrystallites by Koch et al. 13-t5 It should be pointed out that the measurement of the exciton binding energy in the presence of a magnetic field has been reported in quantum wells 16 and quantum wires. 17 In Sec. II, we briefly describe the formalism and numerical techniques used to calculate the ground-state properties of an exciton in a quantum dot subjected to an external magnetic field. For simplicity, we have considered only the parabolic confinement of the electrons and holes. In some of the calculations, we also left the hole unconfined in the two-dimensional plane. Some of the computational steps are discussed briefly in this section. The results for the ground-state and low-lying excitation energies, electron-hole separation, and normalized intensity of the optical absorption are presented and discussed in Sec. III. A brief discussion and conclusion are given in Sec. IV. II. THEORY Our model Hamiltonian for a two-dimensional hydrogenic exciton in a parabolic confinement potential and in a static external magnetic field is ff~ --- J'[e "~- ~ [ h "~-Jl[ e-h
(1)
,
where the electron, hole, and electron-hole terms are 2
~e
--
~
I
e
--ihVe----Aec
1
e
J'[ h -- ~ m h
--i~iV h -k- --C A h
e2 Y-le-h -5980
' 22 , -.}-TmetOere
]2
d- TmhO~h 1 2 r2 ,
(2)
1
e Ir e - r hI 9 1992 The American Physical Society
284
45
EXCITONS IN A PARABOLIC QUANTUM DOT IN MAGNETIC FIELDS
Here e is the background dielectric constant. We calculate the eigenfunctions and eigenvalues of the system using the method of numerical diagonalization of the Hamiltonian. In this method the Hamiltonian of the system is divided into two parts, Yt=Y/o+Y/', where Yfo is the Hamiltonian for the noninteracting system. The term Y/' then includes all the interactions between the particles. The eigenfunetions of Y / a r e expanded in terms of the eigenfunctions of Y/0- The original problem of finding eigenfunctions and eigenvalues of Y / i s now transformed to a problem of diagonalizing a matrix whose components are (q0i [Y/lq0s ), where the q~i's are the eigenfunctions of Yf0. In the actual numerical calculations the number of basis functions qg~ must be finite. Usually the basis functions are chosen such that they are the lowestenergy states of the Hamiltonian Y/w One possible approach within the diagonalization scheme is to expand the wave functions of the system in terms of the eigenfunctions of the noninteracting
5981
electron-hole pair, i.e., we select Y~C0=Yat~+y~r Y/' =Y'/~-h. The eigenfunctions of Y/e and Yat~ are 1/2
aa
ha!
r
(na+llal)l
~/n"t"=
and
X exp[ -- ila O a -- ( a ar a )2/2 ]
)l* L nIt~l[(aara)~ ], a
X(aara
where
a
denotes
the-
a~=[(~o~: + ~ o ~ / 4 ) ~ / 2 m a / h ]
(3)
electron
or
hole,
~/2, L I~ are the associated
Laguerre polynomials, a~c is the cyclotron frequency, and I and n are the angular and radial quantum numbers, respectively. The advantage of this method is that the interaction matrix elements between the noninteracting electron-hole pair states can be expressed in a closed form:
I
i i i " ( nelenhl~ [..~(e.~iInilln],l~ ) e2 6. aeati + I~,l/+l/
1/2
(I/~l+ne ')! r
(It~l+~eJ) '- (It/l+n/)t
x ~, ]~ 2~ Y_,[a+B+ 89189
l+lt/,I-k)]!
a=0//=0 y=08=0
(Igl+ni)!(It~l+n/)!
(-l).+~ X - a~
i (ll~l+a)!(n,-a)!(ll~l+B)!(n/-B)!
- l)V+6
(Ith I + ng )!( It/l+ n/)!
r!a! ~lthl+r)ttng-r)t
~' p=o
[a+lZ+ 89
[a+B+ 89
+p)!
V+8+ 1/2(II~I+ II/I-k)
2
[~, +a+ 89
I + I/h/I +k)]!
[~,+a+ 89
s=0
+s)!
(_l)P+s (ae/ah)2S+kr'(k +p +s + 89 X- , pls!
where k --II~-l~l. Unfortunately this approach leads to a poor convergence of the eigenvalues as a function of the number of basis states. A better way is to introduce the center of mass (c.m.) and relative coordinates R=(1/M)(mere+rnhrh), r = r e - - r h , where we have adopted the usual notations: M = m e -1- m h , l ~ = m e m h / M , and y - - ( m h - - m e ) / M . We also choose the symmetric gauge vector potentials for the electrons and holes as A e = 8 9 and A h = - - 8 9 The Hamiltonian (1) can then be written in the form Y~=Y~c.m +Y~rel + ~ x ,
(5)
where the c.m., relative, and the cross term of the Hamiltonian are
fit[c"m"-
[l+(ae/ah)2] k+p+s+l/2
2M Vc2"m"+ 89M h2
itle
(4)
[_:
( m e coe2 +mha,~, ) R 2
.,.
~rel-- -- 2"-~-~Vr21 "at-2-~C ~t 1:1"r X Vrel (6) e2B 2
1
q--~/.t 4/.t2c2 q- -M(mhC~ i~ie _
Y'/x = - ~ c t j ' r •
2 q_ met02 ) r E
e2 1 6 r '
.... q-~(C02- C02 ) R . r .
In Eq. (6), Y/c .... which depends only on the c.m. coordinate, is the Hamiltonian of a well-known twodimensional harmonic oscillator with energy spectrum
285
45
V. HALONEN, TAPASH CHAKRABORTY, AND P. PIETIL.'AINEN
5982
(7)
Ec.m. =(2nc.m. + Ilc.m. I+l )ICk0c.m. and
!
,oo..~ = ~1( meto ~ +mhto ~ )
!
(8)
with n c.m. >0. Also, ~ is a Hamiltonian of a twodimensional charged particle in a magnetic field and in parabolic and Coulomb potentials. It can be further separated into radial and angular parts. The radial part of the Schr'ddinger equation [~[
R"-F1R"t-r --
e21
(Str))=~,c*c,RJ i.j
, ,~ ( o ) g . i tJ (0)
n rer tel
which gives the probability of finding the electron and hole at the same position. IlL RESULTS
eB Ere I = E -- 7' "~--7Irel 9 z/~c
(9)
(10)
Finally, all interactions between c.m. and relative motions are included in the cross term Y~x. Eigenfunctions and eigenvalues of the system can now be calculated accurately using the eigenfunctions of Y'/c.m.+Y/rel as basis functions for the numerical diagonalization of the total Hamiltonian Y/. Because the contribution of interaction Y/x was found to be relatively small only a small number of basis states was needed in the actual calculations. One final note about Y/tel: here the harmonic term is i toc2 + ( 1 / M ) ( mhto 2e + me~02 )]r E, where -~[ ( 1/M)(mha~ e2 -!-meto 2 ) is different from O)c.2 m. = ( 1/M)(meto 2 +rnhCO2h). One cannot therefore simply replace coc in the two-dimensional calculations of Ref. 5 by (COc 2 +toc2.m.)1/2 to obtain our results described below. In addition to the ground-state and low-lying excitation energies we also present the numerical results for the electron-hole separation and the intensity of the optical absorption. The electron-hole separation can be calculated as an expectation value of the relative coordinate:
lrl n Yrel I yrel )
i,j X(~ni
.... ",;.m. '
121
can be solved numerically using a standard method called shooting to a fitting point, is Briefly, the method is to guess the value of E and to integrate the differential equation from zero to some point rf (called the fitting point) and from infinity (i.e., from a point which is far enough from the origin) to the same point rf. For the correct E we require that both the solution R and its first derivative are continuous at the fitting point rf. The eigenfunctions are labeled by the angular momentum/tel and by the principal quantum number n rel. The contribution of the relative motion Hamiltonian Y~relto the total energy is
i I rel i rel
~ tel
x'~,~.,,t~'s.~.,..,.'~..,.'~g
E-t'----6 r --'-~-
2+me<02h) 1r 2 R = 0 4~2C 2e2B2 -t---~.~.~(mhto Mfr'2
(r)-'~,c*cy(n
cal integration. In order to compare with the results obtained in Ref. 12 for a confined exciton (in the absence of a magnetic field) we have also calculated the quantity ( ( r 2 ) )1/2. The normalized intensity of optical absorption is calculated as an expectation value,
J.m. ~1/ . . . . lJ.m.
'
where the c i's are the expansion coefficient of the ground state. Because the eigenfunctions In irel l irel ) of the Hamiltonian Y/rel can be obtained only numerically, the matrix i i elements (n rel Irel [r[ n jrellrel J ) are calculated using numeri-
In this section we present the numerical results for the parameters appropriate to GaAs, i.e., dielectric constant e = 13.1, electron effective mass me=0.067m, and hole effective mass mh =0.090m for light holes and m h =0.377m for heavy holes, t2 We should point out that by a light (heavy) hole we mean a hole that has light (heavy) in-plane mass. In Ref. 12, the opposite convention was employed. Let us first compare the results of the two approaches described in the preceding section. The ground-state energy of a heavy-hole exciton as a function of confinement potential energy (toe = t Oh) in the absence of the magnetic field is shown in Fig. 1. As the interaction between c.m. and relative motions Y/x in this case is equal to zero, the c.m. and relative motion separation approach is exact within the numerical accuracy of the shooting method that was used to calculate the eigenvalues of ~rel" It is seen from Fig. 1 that at low confinement energies when the electron-hole pair is strongly correlated due to the Coulomb force, the noninteracting electron-hole pair state basis approach needs a very large number of basis states to converge. When the confinement energy is increased the noninteracting
75
o s"
s""""
50
• LU
25
,...s ,"' ' S " ' " ~"S ' " ~ -
j
........... ,-SS
0 -25 I
-50 0
10
I
I ,
20 30 hco (meV)
40
5O
FIG. 1. The ground-state energy of a heavy-hole (mh=O.377m) exciton as a function of a single-particle confinement potential energy (ha~e=~Oh). The solid curve is calculated using c.m. and relative motion separation and the dot-dashed curve is calculated using the noninteracting electron-hole pair state basis (500 basis states). The energy of a noninteracting electron-hole pair (dashed curve) and the Coulomb-interaction energy (dotted curve) are also shown.
286
5983
EXCITONS IN A PARABOLIC QUANTUM DOT IN MAGNETIC FIELDS
45
20 .___(_b.) 15
ILl 5
-
0 -'-------T"0
5
I
l
10 B('r)
15
20
FIG,. 2. The ground-state energy of a heavy-hole (mh =0.377m) exciton as a function of the magnetic field B (T).
Confinement potential energy for both electron and hole is (a) 15.0 meV and (b) 25.0 meV. Solid curves are calculated using c.m. and relative motion separation and dashed curves are calculated using the noninteracting electron-hole pair state basis (500 basis states) as explained in the text.
electron-hole pair states give a better description of the system, although the magnitude of the Coulomb interaction Y/e-h is also increased. Another comparison between our numerical approaches is given in Fig. 2, where we have plotted the ground-state energy of a heavy-hole exciton as a function of the external magnetic field. It is quite apparent that both approaches give the same magnetic-field dependence. This indicates that the correlations do not change as a function of the magnetic field. In the numerical results presented in Figs. 1 and 2 we have used the same confinement potential energy for both electron and hole. One other possibility is to leave the hole unconfined within the two-dimensional plane. In that case the hole is moving freely in the plane and it feels only the electrostatic potential of the electron and the external magnetic field. In Fig. 3 we have plotted the
ground-state energies of heavy- and light-hole excitons as a function of the confinement potential energy of the electron when the hole is not confined within the plane. Comparing Figs. 1 and 3 it is evident that the groundstate energy as a function of the confinement energy is not increasing as rapidly as in the case of equal confinement potential energy for both particles. The main contribution for this comes from the behavior of ~/lfc.m. and ~[rel" In Fig. 3 we have also plotted the ground-state energy calculated without the cross term ~ x - Interaction between c.m. and relative motions is attractive for both light- and heavy-hole excitons, thereby lowering the ground-state energy. We have calculated the ground-state energy, the electron-hole separation, and the relative intensity of the optical absorption as a function of the magnetic field with various confinement potential energies (Figs. 4-6). In all these calculations we have left the hole unconfined within the plane. In Fig. 4 the ground-state energies with and without the cross term between c.m. and relative motions are shown. As a comparison, the result for a purely twodimensional exciton is also shown. For a twodimensional exciton (~uoe = irtcoh = 0 ) our numerical calcu' lations reproduce the results of Shinada and T a n a k a : The effect of the cross term ~t(x to the ground-state energy is relatively small. Therefore, as a first approximation, we can explain the ground-state properties of the exciton using the Hamiltonian YZ(c.m.+]Z(re I. At low confinement potential energies the ground-state energy is increasing approximately quadratically as a function of the magnetic field. This is due to the B 2 coefficient in the harmonic term in YZ(reI. But because the harmonic term is proportional to the sum of the squares of the magnetic field and of the confinement potential energy, the effect of the magnetic field is decreased when the confinement potential energy is increased. A n o t h e r interesting feature seen from Fig. 4 is that the effect of the cross term Yz(x to the ground-state energy is nearly independent of the magnet-
40 3O ,_.
20
E iii
kkl
-10
~O
-15
-10
-20
~
c
0
5
-20 0
10
20 h~ e
30
40
50
(meV)
FIG. 3. The ground-state energy of a heavy-hole (mh =0.377m) and a light-hole (mh =0.09m) exciton as a function of confinement potential energy of the electron with (solid curve) and without (dashed curve) the cross term between c.m. and relative motion. The hole is unconfined in the twodimensional plane.
FIG.
4.
l
The
t 10
~
..... i 15 B (T)
ground-state
20
energy
25
of
a
30
heavy-hole
(mh =0.377m) exciton as a function of the magnetic field (T). Confinement potential energy for the electron is (a) hcoe=5.0 meV, (b) fuo~= 15.0 meV, and (c) fuoe =25.0 meV. The hole is unconfined within the two-dimensional plane. The lowest curve is the result for a two-dimensional exciton. Energies without the cross term between c.m. and relative motion are also plotted (dashed curves).
287
V. HALONEN, TAPASH CHAKRABORTY, AND P. P I E T I I J d ~ 4
5984
7 "-". . . . . . --.(a) E r
"9 ~rc:z .~
"',,.
(b) "" 9%,,
.....
6
Ic
(b)
"-.
~ ...... -'-'22""":-...
5 - (c)
4 0
c i
, I
........,t
I
5
10
15
20
,
t 25
,
r
30
B(T) FIG. 5. The ground-state electron-hole separation of a heavy-hole (mh =0.377m)exciton as a function of the magnetic
field (T). Confinement potential energy for the electron is (a) r meV, (b) hto,=15.0 meV, and (c) ht0e=25.0 meV. The hole is unconfined within the two-dimensional plane. Electron-hole separation is calculated as ( r ) (solid curves). We have also plotted (( r 2) )1/2 (dashed curves). ic field. In Fig. 5 the numerical results for the ground-state electron-hole separation of an exciton as a function of the magnet[c field are shown. The separation is, as expected, largest at low confinement and at low magnetic fields and approaches zero when the confinement potential energy and magnetic field are increased significantly. The decrease of the magnetic-field dependence is biggest near 10 T. The influence of the external magnetic field on the electron-hole separation is surprisingly small. For example, when the confinement potential energy for the electron is 5 meV the size of the exciton is reduced only by about 30% a s t h e magnetic field is increased from 0 to 30 ~,
O
T. For higher values of the confinement energy the effect of the magnetic field is even smaller, The behavior of the ground-state optical-absorption intensity as a function of the magnetic field and confinement potential energy is presented in Fig. 6. The results are in line with the results of the ground-state energy and the electron-hole separation calculations. The increase of confinement suppresses the effect of the magnetic field and vice versa. The intensity of optical absorption increases with increasing magnetic field. In Fig. 7 we have presented the results for the lowest energy levels of a light-hole (m h =O.09m)exciton as a function of the magnetic field without (a) and with (b) the cross term Yarx in the Hamiltonian. We have plotted only those levels that have the total angular momentum L = 0 . Although the effect of the cross term -'q~cxis much larger for the excited states than for the ground state, we can identify at least some of the lowest levels using the noninteracting energy levels [Fig. 7(a)]. Because of the confinement, the relative angular momentum of the exciton can have different angular momentum values to give the total angular momentum L = 0 . In Fig. 7(a) the energy levels for the relative angular momentum l ~ = 0 (solid curves) and l r e l = + l (dashed curves) are shown. For each energy level of the relative motion there is a spectrum of the c.m. levels separated from each other by the amount of the confinement potential energy. To illustrate this feature more clearly we have labeled each level by its (exact) quantum numbers: nc.m., lc.m., rtrel, and lrel" When the cross term Y/x is included in the Hamiltonian, many anticrossings are revealed in the energy spectrum [Fig. 7(b)]. In addition to these anticrossings, it can be
~~oxo,o)
(
80
'
60
,,,o~'
,
Lo.~o
LLI "~ 40 ~
26
45
2
0
,,
"~ 24
o
~
~.
uJ 4~ 20 I
2D exciton
16 0
60
!
I
I
I
I
5
10
15
20
25
30
BIT) FIG. 6. Normalized intensity of the optical absorption calculated for the ground state of a heavy-hole (mh --0.377m) exciton as a function of the magnetic field (T). Confinement potential energy for the electron is (a) ha~e = 5.0 meV, (b) ~toe= 15.0 meV, =rid (c) ha~e= 25.0 meV. The hole is unconfined within the two9 mensional plane. The lowest curve is the result for a twodimensional exciton. The normalized intensity is measured in units of a--2, where a =Eh2/l.te 2 is the effective Bohr radius.
~ ,
,
(b)
0 0
20
30
B (T)
FIG. 7. The ground-state and low-lying excitation energies of a light-hole (mh =0.09m) exciton as a function of the magnetic field (T) without (a) and with (b) the cross term between c.m. and relative motion. The angular momentum is lc.m.+l,et=0. Confinement potential energy is 15 meV for both particles. In (a) the energy levels are labeled by their quantum numbers, which are shown in parentheses as (n ..... l .... )( n rel,Irei)-
288
45
EXCITONS IN A PAI~.BOLIC QUANTUM DOT IN MAOI~BTIC FIELDS . . . . ...
9
80-
t eo o ooOOe
9 O o . o * * 9O o
~O-
oo
*e'~ ~ oo~
OOOooe.o A
o o~
9
g*~
~ oooeeeoOO~176176
. e . .
ILl
4o- OIOOO0+oooooOOlOOOOlO
@IoO0+O0 9
3985
culations. In reality the energy levels of light- and heavy-hole excitons are coupled leading to an even more complicated structure of the magneto-optical energy spectrum than we have in our calculations. Nevertheless, our results predict that it should be possible to evaluate from magneto-optical measurements the binding energy of the exciton and the strength of the confinement potential.
20 00000000000000000000000000000
0 0
I
I
5
~
IV. DISCUSSION AND CONCLUSIONS
I
I
I
9
~
~
3O
Bm FIG. 8, Optical-absorption energies and intensities of a light-hole (m,--0.09m) exciton as a function of the magnetic field (T). Confinement potential energy is 15 meV for both particles. Diameters of the filled points are proportional to the calculated intensity of the absorption. seen that when the magnetic field is increased some of the energy levels begin to form the first and second Landau levels. We have also calculated the intensity of the optical absorption for all of the energy levels shown in Fig. 7(b). The results of these calculations are plotted in Fig. 8, where a rich anticrossing structure of the optically active energy levels is still present. Because we have used an effective-mass approximation, the effects of the valenceband mixing have not been taken into account in our cal-
tl. V. Lerner and Yu. E. Lozovik, Zh. Eksp. Teor. Fiz. 80, 1448 (1981) [Soy. Phys. JETP 53, 763 (1981)]. 2T. M. Rice, D. Paquet, and K. Ueda, Phys. Rev. B 32, 5208 (1985); HeN. Phys. Acta 58, 410 (1985). 3C. Kallin and B. I. Halperin, Phys. Rev. B 30, 5655 (1984). 4L. P. Gorkov and I. E. Dzyaloshinskii, Zh. Eksp. Teor. Fiz. 53, 717 (1967) [Soy. Phys. JETP 26, 449 (1968)]. 5M. Shinada and S. Sugano, J. Phys. Soc. Jpn. 21, 1936 (1966); O. Akimoto and H. Hasegawa, ibid. 22, 181 (1967); M. Shinada and K. Tanaka, ibid. 29, 1258 (1970). 6D. C. Tsui, H. L. St6rmer, and A. C. Gossard, Phys. Rev. Lett. 48, 1559 (1982); Tapash Chakraborty and P. Pietil/iinen, The Fractional Quantum Hall Effect (Springer-Verlag, New York, 1988). 7A preliminary report on these calculations was given by Tapash Chakraborty and V. Halonen, in Proceedings of the International Symposium on Nanostructures and Mesoscopic Systems, Santa Fe, New Mexico (Academic, New York, in press). ST. Demel, D. Heitmann, P. Grambow, and K. Ploog, Phys. Rev. Lett. 64, 788 (1990); W. Hansen, T. P. Smith III, K. Y. Lee, J. M. Hong, and C. M. Knoedler, Appl. Phys. Lett. 56, 168 (1990); Ch. Sikorski and U. Merkt, Phys. Rev. Lett. 62, 2164 (1989). 9p. A. Maksym and Tapash Chakraborty, Phys. Rev. Lett. 65,
We have presented results for a single exciton in a parabolic quantum dot in the presence of a magnetic field. We have discused in detail different numerical approaches employed in obtaining the results.+ The numerical results for the ground-state and low-lying excitation energies are presented. We have also presented the results for the electron-hole separation and the intensity for optical absorption for an exciton where the electron is confined in a parabolic potential and the hole is left unconfined within the two-dimensional plane. Magnetooptical measurements in quasi-zero-dimensional exciton systems have just begun. 19 It is expected that the theoretical results presented here might provide useful insights on the experimental investigations of excitons in a quantum dot in magnetic fields. ACKNOWLEDGMENTS One of us (T.C.) would like to thank Dr. K. Kash (Belicore) for several helpful discussions.
108 (1990); Tapash Chakraborty, V. Halonen, and P. Pietil~inen, Phys. Rev. B 43, 14 289 (1991). I~ W. Bryant, Phys. Rev. Lett. 59, 1140 (1987). ttG. W. Bryant, Phys. Rev. B 37, 8763 (1988); Surf. Sci. 196, 596 (1988). 12G. W. Bryant, Phys. Rev. B 41, 1243 (1990). 13y. Z. Hu, M. Lindberg, and S. W. Koch, Phys. Rev. B 42, 1713 (1990). 14y. Z. Hu, S. W. Koch, M. Lindberg, N. Peyghambarian, E. L. Pollock, and F. A. Abraham, Phys. Rev. Lett. 64, 1805 (1990). ]SE. L. Pollock and S. W. Koch, J. Chem. Phys. 94, 6776 (1991); E. L. Pollock and K. J. Runge (unpublished). n6j. C. Maan, G. Belle, A. Fasolino, M. Altarelli, and K. Ploog, Phys. Rev. B 30, 2253 (1984); D. C. Rogers, J. Singleton, R. J. Nicholas, C. T. Foxon, and K. Woodbridge, ibid. 34, 4002 (1986); W. Ossau, B. J/ikel, E. Bangert, G. Landwehr, and G. Weimann, Surf. Sci. 174, 188 (1986). 17M. Kohl, D. Heitmann, P. Grambow, and K. Ploog, Phys. Rev. Lett. 63, 2124 (1989). xSW. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes in C: The Art of Scientific Computing (Cambridge University Press, Cambridge, 1988), p. 606. t9K. Kash, J. Lumin. 46, 69 (1990).
289
VOLUME 72, NUMBER 21
PHYSICAL
REVIEW
LETTERS
23MAY 1994
Quantum Dots Formed by Interface Fluctuations in AIAs/GaAs Coupled Quantum Well Structures A. Zrenner, L. V. Butov,* M. Hagn, G. Abstreiter, G. Bfhm, and G. Weimann Wailer Sekottky lnstitut. Teehnisehe Unitr Miinchen, D-85748 Garching. Germany (Received 21 December 1993) We report about optical experiments on electric field tunable AIAs/GaAs coupled quantum well structures in the regime of the electric field induced F-X transition. Using the energetically tunable X-point state in the AlAs layer as an internal energy spectrometer and charge reservoir we are able to map out the electronic states in the neighboring GaAs quantum well in great detail, in spatially resolved and bias voltage dependent photoluminescence experiments we find sets of extremely narrow emission lines below the fundamental band gap energy of the GaAs quantum well. The new emission lines are shown to originate from natural quantum dots which are formed by well width fluctuations of the GaAs quantum well. P A C S numbers: 6 8 . 5 5 . - a , 71.50.+t, 73.20.Dx, 73.61.Ey
The investigation of the microscopic structure of the heterointerfaces was the subject of many scientific contributions in the past. In most of those molecular beam epitaxy (MBE) grown GaAs/AIGaAs quantum wells (QW's) have been investigated: some are listed below [ ! - I !]. Those contributions can be divided into a group of investigations on optical properties [ i - 9 ] and into a second group on atomic scale structural investigations [I0,1 I]. Previously there was some discrepancy between the results of optical and structural investigations. Whereas by optical techniques the existence of huge monatomicaily smooth islands with a size of up to several /~m has been-claimed in high quality growth interrupted QW's [6], structural methods revealed substantial roughness and alloy fluctuations on an atomic length scale [10,1 I]. In more recent work, however, data from optical investigations also was found to be inconsistent with the existence of huge monatomically smooth islands [9]. Currently interface roughness with an amplitude of at least I monolayer (M L) in growth direction is believed to appear on a broad range of length scales reaching from atomic scale to wafer scale, in a narrow QW well width fluctuations of several monolayers will result in sizable lateral potential variations. On this basis it is justified to describe a narrow two-dimensional QW sample as a disordered array of quantum dots with arbitrary dimensions. In theoretical work well width fluctuations have been shown to strongly affect the properties of QW's in terms of eigenvalues and charge localization [12]. in the present Letter we demonstrate the existence of zero-dimensional states in narrow QW's. For our investigations we use an electric field tunable AIAs/GaAs coupled QW structure [I 3] to overcome the intrinsic lifetime limitations of single QW's and to benefit from resonant carrier injection into GaAs quantum dot levels. Our coupled quantum well structures are configured as heterotype n +-i-n + diodes. The active pair of AIAs/GaAs layers is contained between two 400 ,~ wide intrinsic (i) Alo.48Gao.s2As layers. The layer sequence was grown by M BE without growth interruptions. A band diagram of the active part of the structure is shown for negative bias voltage VB as an inset in Fig. I. The nominal width of 30 A 3382 9
has been chosen for the GaAs QW to get enough confinement energy to compensate the internal offset between the GaAs F conduction band (CB) edge and the AlAs X CB edge. Three different samples with AlAs layer widths of 30, 40, and 50 A have been grown and investigated. Within the tuning range of Ve we obtain the electric field induced F-X transition [14]. A detailed study of the electric field induced F-X transition in AIAs/GaAs coupled QW's can be found.in Ref. [13]. Important for this work is the huge enhancement of the exciton lifetime in the indirect regime, which can reach 500 nsec as compared to less than 500 psec in the direct
V-CB---~ "i B'*
1780l CB__
~1,40 ........ : . . . . . . . . . .
I~ Depth" ,~,
GaAs/AIAs 30A/50~,
"~ .....
T=4.gK kL=632.8nm PL=5OttW
u,
-1.5 -1.0 -o.$ 0 E X-HHo
0,5 1.0 1.5
vs(v)
dL=lO0ttm 56
dL:
m
,
Energy (meV)
FIG. I. (a) PL response of a GaAs/AIAs 30 A/50 A coupled QW structure in the indirect regime for Va ' = - 0 . 2 V. The diameter of the optically probed area dL is 100 ~m. A schematic band diagram of the structure and the observed PL energies as a function of VB are shown in the inset. (b) Same as (a) only with higher spatial resolution (dL ==2 ,urn). New narrow emission lines (labeled from n - I to 7) appear in the region of t ~ indirect PL.
0031-9007/94/72 (2 ! )/3382(4)$06.00 1994 The American Physical Society
290
VOLUME 72, NUMBER 21
PHYSICAL
REVIEW
regime [15], and also the linear Stark shift which is caused by the electric field acting on the spatially separated electron-hole system. The photoluminescence (PL) energy as a function of Va for a GaAs/AIAs 30 ,~/50 A, structure (solid squares) is shown in the second inset of Fig. I. The solid lines represent theoretical results for excitonic transitions (no r - x coupling), assuming the indicated well width (GaAs/AIAs 28.8 J~/50/~; see Ref. [13] for details). For the present work only the transitions between the individual ground states E~-HHo and Eff-HHo are relevant. The F-X transition occurs at Va-0.25 V. From numerical model calculations we find that a width variation of the GaAs QW by 2 monolayers (between l0 and 12 ML) will result in a 43 meV shift of the direct transition (E~-HHo). The major part of this shift (36 irieV) appears in E~'. Since Eox can be tuned by more than 70 meV with respect to E~', the X-point level E x can be used as an internal energy spectrometer to map out the local energy minima of the GaAs QW in an energy range which is relevant for well width fluctuations. Experimental evidence for the existence of zerodimensional states in the GaAs QW is provided by different optical and magneto-optical experiments. In Figs. l(a) and I(b) we show PL results in the indirect regime ( V s - - 0 . 2 V) for two different sampling areas. For the spectrum shown in Fig. I (a) the laser was focused to a diameter dL of-100 pro. The direct (E~-HHo) and the indirect transition (EX-HHo) is observed. The corresponding linewidth in terms of the full width at half maximum (FWHM) is I I meV for the direct and 4.5 meV for the indirect line [16]. Whereas the direct line has a Gaussian shape as expected for the inhomogeneously broadened line of a QW grown without interruptions, a structured tail is detected on the low energy side of the indirect line. The structure in this tail appears tremendously enhanced as the laser is focused down to dr. "-2 pm and moved to an appropriate position in the x - y plane of the mesa diode [see Fig. l(b)]. New extremely narrow emission lines (labeled from n - I to 7) emerge from the background of the indirect line. The F W H M of those new emissions is about 0.5 meV in the observed spectrum, limited by the choice of slits in the spectrometer (the natural linewidth is about 0.2 meV). Both the narrow linewidth and the level sequence are reminiscent of an emission spectrum from a fully quantized system such as a single quantum dot [I 7,18]. The low energy cutoff of the narrow emission lines is about 30 meV below the position of the direct PL and 38 meV below the direct absorption edge, which corresponds to an almost 2 M L variation in well width. It can be further shown that a 1000/~ wide Gaussian-shaped potential well with a depth of about 20 meV has an almost similar level spectrum as the one shown in Fig. I(b). The observed level spectrum changes as we choose a different position in the x-y plane of the mesa diode, in all of our samples narrow emission lines qan be observed.
23 MAY 1994
LETTERS
Results as a function of Va from a GaAs/AIAs 30 J~/40 ~!, structure are shown in Fig. 2. Compared to Fig. I(b) the observed level spectrum is more complicated. The total number of observed lines decreases with decreasing Vs. The linear Stark shift of the indirect line is still evident as a global redshift of the background from which the narrow emission lines emerge. There is, however, only negligible Stark shift on the position of the narrow emission lines. From the absence of Stark shift and the strength of the emission lines we conclude that the origin of the new lines cannot be from a real- and k-space indirect recombination process. We rather think that we observe direct transitions from zero-dimensional states in local potential minima of the GaAs QW. in general we expect to find more than one local potential minimum in an optically probed area of d L - ' 2 pro. For small r - x separations carrier injection into a big variety of shallow and deep local potential minima is possible and a complex superposition of sets of narrow emission lines results (Vs-0.0 V). For large separations (Va--0.6 V) only sufficiently deep potential minima can be populated and the spectra simplify. Influence on the strength of the emission lines is expected from the relaxation and injection processes. As sketched in the inset of Fig. 2, electron-hole pairs are photoexcited selectively in the GaAs QW. The electrons transfer subsequently in a fast U-X relaxation process [I 9] into the electric field tunable, long-lived Eox state of the AlAs QW. in this sense the X-point level E0x is a tunable charge reservoir which can be used to inject electrons resonantly back from the X point into local energy minima of the direct GaAs QW at the U point. A detailed understanding for this injection
I
/
I
..... .
'i
-.iv.
t"NW-L~~Z ~ .~
.
,.,..,
PL=0"lmW
k
v~(V)o.oo -0.06
-0.24 -0.30
~_.__...._..._jA ~
-o.4s
,
, -0.60 1715 1720 1725 1730 1735 1740 Energy(meV) FIG. 2. PL response in the indirect regime as a function of Va. The diameter of the optically probed area dL is 2 pro. The relevant relaxation and recombination processes after optical excitation in the GaAs QW (HHo--*E~) are shown in the inset. 3383
I
I
II
291
VOLUME 72, NUMBER 21
PHYSICAL
REVIEW
process is still missing. As in the initial F-X relaxation the huge momentum transfer for the X-F process probably comes from resonant F-X mixing and eventually also from zone-edge phonon emission [20]. Some evidence for the participation of phonons can be seen in Fig. 2. The three energetically lowest emission lines around 1720 meV do not dominate the radiative emissions at low Vs. Instead the envelope of their intensities vs Vn seems to follow the position of the indirect line like a phonon replica, displaced by 10-15 meV. Combining the principle of resonant charge injection with spatially resolved measurements we are able to map out the in-plane potential fluctuations. As indicated in the inset of Fig. 3 we have performed a series of spatially resolved PL measurements over an area of 45 pro x36 pm. Within this area we have recorded PL spectra with a pitch of 3 pm in the x and y directions and a laser spot diameter of about 2 pm. For each position we have plotted a PL spectrum as indicated in the inset. The displayed energy range contains only the indirect line and the new emission lines. For each given Vn the direct line appears undistorted throughout the scanned area. Part of the scanned area is masked by the gold metallization on
U 1710
1735
LETTERS
23 MAY 1994
the mesa which is shown grey in the inset. Figures 4(a) (Va-0.1 V) and 4(b) (1"8--0.1 V) are two reprcsen, tative examples from a larger series of measurements between 1/8-0.2 and - 0 . 2 V. The position and strength of the narrow emission lines in Fig. 4(a) (Ve--0.1 V) is a strong function of the in-plane coordinates. Narrow emission lines with different amplitudes can be detected over almost the entire scanned area. There exist also larger clusters with different amounts of structure in the PL spectrum. All features are strictly reproducible (even after warm-up to room temperature) and can be regarded as a fingerprint of a particular region, which is defined by a specific x-y dependence of the potential in the GaAs Q w . The data are consistent with our earlier proposed picture of the GaAs QW as a disordered array of arbitrarily sized quantum dots. The spatial resolution in this experiment is twofold: First, there is the geometrical resolution given by the step size of 3 pm. in addition, there are the level spacings between the narrow emission lines which contain information about the lateral extent of the potential minima on a mesoscopic scale. Although a detailed analysis of the roughness spectrum is difficult and not the subject of this work it is still obvious that the well width fluctuations appear on a broad distribution of length scales. The amplitude of the well width fluctuations can be explored by varying the injection energy via Vs. Results for V s - - 0 . 1 V are shown in Fig. 3(b). As compared to Vs-O.I V, the injection energy, which is
Energy(meV) 1732 1311 A
.A JN .,h A - ~
h A~.~ A
~ )'k ^ A. A ...,~,2 ^ ~.,,~k
x
A.A
~,,A~/k A
A A
~N. A
A A ~ A / ~ A AA
.~[AA
AA
A A A~.~ A
A. A A , ~
_~J~A
/k
AA
.~ A A N
"
A
~
A~A AA~r
A
A
IA
. , 4 ~ L / ~ I ~ . ~ L~', I ^
IA
A
L.~'~
AIA
A
A
AIA
IA
~-
A
J~
A
1~'~
A
A
AIA
.
.
.
K
(b) VB='0"iV
A~ .
.
.
.
A | !
[A
A
. . . .
~
X
Ve=.0.tV
--
T =4.21(
1728
1732
I /k. LA
(a) ,,
,!
!
~
4
;
~
'
,
A I A
A
~-~
L
A
[x
.
FIG. 3. Topology of the PL response for two different Ve. As indicated in the inset an area of 45 pmx36 pm has been scanned with a pitch of 3 pm in the x and y directions. A PL spectrum is plotted for each position. (a) For Va-O. I V huge variations of the PL response are observed, indicating substantial potential fluctuations below the injection energy. (b) At Va--0.1 V only a few very deep potential minima fall below the reduced injection energy. 3384
30A/40~
=_ _ .... _ _ = - - -
1724
AIA
1730 ~
~ .
GaAs/A|As --:
21 1726
.~ _.~ ~ J . , , _ A
AI~IA
.....
- =. .-. . . . -. .- .-.-. . . . . . . . . . .=.. - =
A K A ~ A A A A ~ A
A A ANA.~A
~._
~ " 84
....
1726
172,t
17220
2
;
I'0
1'2
14
B ('13 FIG. 4. Position of the narrow emission lines as a function of B, (a) and Ba. (b). Different positions in the x-y plane are probed for Bn and B.. Whereas for Bu the diamagnetic level shift is too small to be observed, complicated shifts, splittings, and anticrossings are observed for Ba..
292
VOLUME 72, NUMBER 21
PHYSICAL
REVIEW
given b y the energetic position of the indirect line, is lowered by about 6 meV. At this reduced injection energy allowed energy states in the GaAs QW are harder to lind. Over most of the scanned area only the undistorted indirect PL line is observed. Narrow emission lines emerge only at some selected locations. Those are the regions where the local width of the GaAs QW has maxima. in additional PL excitation measurements we found corresponding variations in the local onset of the direct gap absorption in the GaAs QW. With the detection energy on the emission line of a ground state quantum dot level we find narrow absorption lines from excited dot levels as precursors of a redshifted (up to 10 meV) twodimensional absorption edge of the GaAs QW. This observation denotes that those electron-hole pairs which feed the narrow emission lines are photogenerated in regions with locally reduced band gap, namely, in the vicinity of the natural quantum dots. Since our dots are by principle very asymmetric with strong confinement in the z direction (30 /~) and weak confinement in the x - y plane, their response on parallel (Bn) or perpendicular magnetic field (B• has to be totally different. Experimentally we performed our magneto-optical measurements in a superconducting magnet using an optical fiber. Since the optically probed area was about 10/~m in diameter, emission lines from several local potential minima are contained in the spectra. The positions of the observed emission lines are plotted as a function of Bu and B• in Figs. 4(a) and 4(b). The probed positions on the mesa are different for BIt and B• and different sets of lines appear in both spectra. Experimentally we find negligible influence of Bu on the position of the narrow lines [see Fig. 4(a)]. According to z~E "-e2(zi2)B~/2m*, we calculate for E~" and H H o at Bit " 8 T a total diamagnetic shift of only 0.2 meV, in reasonable agreement with our experimental findings (e is the electron charge, (z/2) the expectation value of z;2, i the subband index, and m* the effective mass). For B• complicated level shifts, splittings and anticrossings are observed [see Fig. 4(b)]. The complexity of the data is partly due to the fact that contributions from several potential minima are contained, if we concentrate, however, on the 3 to 4 emission lines on the low energy end of the spectrum, which originate as we think from a single potential minimum, we find qualitatively very similar behavior as theoretically predicted by Halonen, Chakraborty, and Pietilainen [21] for excitons in quantum dots in magnetic fields. For the present work we only conclude from our magneto-optical analysis that the observed quantum dots do indeed have the proposed geometry, namely, weak confinement in the x - y plane and strong confinement in the z direction. With our experiments we have shown that the observed narrow emission lines do originate from natural quantum dots formed by well width fluctuations in the GaAs QW. The new emission lines appear up to 40 meV below the
LETTERS
23 MAY 1994
direct absorption edge, which relates to a maximum local enhancement of the well width of about 2 MLs compared to the average width. Similar or even larger variations are expected to appear in artificially made arrays of dots and wires. Potential fluctuations as reported in this work will destroy the desired overlap between the strongly peaked contributions of the density of states in those structures and will lead to unavoidable inhomogeneous broadening effects. in summary, we have observed natural quantum dots in electric field tunable AIAs/GaAs coupled quantum well structures. The dots appear in disordered arrays and originate from width fluctuations of the GaAs QW. Using the tunable X-point state in the AlAs layer as an internal energy spectrometer and charge reservoir we have been able to populate those quantum dots by resonant charge injection from the X point. In spatially resolved and bias voltage dependent PL experiments we have observed the emissions from the quantum dots as sets of extremely narrow emission lines below the fundamental band gap energy of the GaAs QW. This work was supported in part by the BMFT (Photonik project No. 01BV219) and by the DFG (SFB 348). L.V.B. thanks the FVS Foundation for financial support. *Permanent address: Institute of Solid State Physics, Russian Academy of Science, 142432 Chernogolovka, Russia. [I] C. Weisbuch et aL, Solid State Commun. 38, 709 (1981). [2] L. Goldstein et aL, Jpn. J. Appl. Phys. 22, 1489 (1983). [3] D. C. Reynolds et aL, Appl. Phys. Lett. 46, 51 (1985). [4] R. C. Miller et aL, Appl. Phys. Lett. 49, 1245 (1986). [51 F. Voillot et aL, Appl. Phys. Lett. 48, 1009 (1986). [61 D. Bimberg et aL, J. Vac. Sci. Technol. B 5, ! 191 (1987). [7] P. M. Petroff et al., J. Yac. Sci. Technoi. B S, 1204 (1987). 18] M. Kohl et al., Phys. Rev. B 39, 7736 (1989). [9] C. A. Warwick et al., Appl. Phys. Lett. 56, 2666 (1990). [101 A. Ourmazd et al., Phys. Rev. Lett. 62, 933 (1989). [11] H. W. M. Salemink and O. Albrektsen, Phys. Rev. B 47, 16044 (1993). [12] A. Catellani and P. Ballone, Phys. Rev. B 45, 14197 (1992). [13] A. Zrenner, in Festk6rperprobleme: Ad~,ances in Solid State Physics, edited by U. R~issler (Vieweg, Braunschweig/Wiesbaden, 1992), Vol. 32, p. 61. [14] M. H. Meynadier et aL, Phys. Rev. Lett. 60, 1338 (1988). [15] J. Feldman et aL, Phys. Rev. Lett. 59, 2337 (1987). [16] The differences in linewidth are partly caused by the differences in the effective electron mass (see Refs. [13] and [201). [171 K. Brunner et aL, Phys. Rev. Lett. 69, 3216 (1992). [I 81 B. Adolph, S. Glutsch, and F. Bechstedt, Phys. Rev. B 48, ! 5 077 (I 993). [19] J. Feldmann et aL, Phys. Rev. Lett. 62, 1892 (1989). [201 A. Zrenner et aL, Surf. Sci. 263, 496 (1992). [211 V. Halonen, Tapash Chakraborty, and P. Pietil/iinen, Phys. Rev. B 45, 5980 (I 992). 3385
293
VOLUME 77, NUMBER 2
PHYSICAL
REVIEW
LETTERS
8 JULY 1996
Zeeman Effect in Parabolic Quantum Dots R. Rinaldi, P. V. Giugno, and R. Cingolani UniM INFM-Dipartimento Scienza dei Materiali, Universitd di Lecce, Lecce, Italy H. Lipsanen, M. Sopanen, and J. Tulkki Optoelectronic Laboratory, Helsinki University of Technology, 02150 Espoo, Finland J. Ahopelto VTT Electronics, Otakaari 7B, 02150 Espoo, Finland (Received 2 January 1996) An unprecedentedly well resolved Zeeman effect has been observed when confined carders moving along a closed mesoscopic path experience an external magnetic field orthogonal to the orbit plane. Large Zeeman splitting of excited higher angular momentum states is observed in the magnetoluminescence spectrum of quantum dots induced by self-organized InP islands on InGaAs/GaAs. The measured effect is quantitatively reproduced by calculations including the vertical quantum well confinement and strain induced, nearly parabolic, lateral confinement, together with the magnetic interaction. [S0031-9007(96)00469-3] PACS numbers: 71.70.Ej, 73.20.Dx, 78.55.Cr In the limit of strong confinement, the band structure of a semiconductor quantum dot (QD) forms a sequence of deltalike states corresponding to the quantized levels of a "macroatom." The study of the electronic and physical properties of these dots is of great relevance for all branches of fundamental physics. The interaction of an external magnetic field with the quantized levels of the dot has been investigated in field-effect quantum dots [15] and in dots obtained by electron-beam lithography and shallow etching [6]. In this Letter we report evidence of the Zeeman effect occurring in the atomiclike states of parabolic InGaAs/GaAs quantum dots. Magnetoluminescence experiments clearly show the splitting of the interband transitions corresponding to quantum dot states with n + I ml - 5, where n is the principal quantum number and m is the angular momentum quantum number. The splitting occurs when the magnetic field is parallel to the symmetry axis of the structure. This is attributed to the breaking of the degeneracy of states with different values of the angular momentum quantum number induced by the external magnetic field (Zeeman splitting) [7-9]. State-of-the-art quantum dot samples were fabricated by self-organized growth of InP islands on the topmost barrier of an ln0.]Gao.9As/GaAs quantum well (QW) [10]. The propagation of the strain field induced by the InP islands through the GaAs barrier layer results in a parabolic potential in the QW plane (see inset in Fig. 1). The samples consisted of a 6.5 nm In0.1Ga0.9As QW and a 5 nm GaAs cap layer, covered by InP islands of 20 nm height and 80 nm diameter. The resulting depth of the lateral confining potential was about 70 meV for electrons and 25 meV for heavy holes [11 ]. In Fig. 1 we show the cw photoluminescence spectra as a function of the excitation power density. The spectra
FIG. 1. Photoluminescence spectra of InGaAs _quantum dots excited with an argon laser. For Icxc = 75 W/cm 2 we estimate, on the basis of peak intensities, that there are approximately 6 - 8 electrons and 6 - 8 holes in a quantum dot. Inset: schematic diagram of the strain-induced confining potential of the dots.
342
9 1996 The American Physical Society
0031-9007/96/77(2)/342(4)$10.00
show up to five intersubband trar~sifions involving quantized electron and hole states in the dots. At the lowest excitation intensity (514 nm line of an argon laser, Icxc = 5 W cm -2) the spectrum exhibits three well resolved bands due to radiative transitions involving the first three quantized states. The luminescence line at 1.439 eV is the lowest excitonic line of the lnGaAs quantum well. This signal arises from the sample area between adjacent
1r in
Ec"~'"
i
21:
2n 31: w/ 9
t
17,oo'\ Q W
~
..... i ,, I 1.30 1.35 1.40 1.45 ENERGY (eV)
294
VOLUME
77,
NUMBER
PHYSICAL
2
REVIEW
1'
islands. With increasing excitation intensity, we observe the filling of higher-energy quantized states, up to the fifth level. Under all pumping conditions, the separation of the recombination lines is rather constant (about 17 meV). The energy position of the observed transitions is consistent with the calculated confinement energies, according to Ref. [12]. Clearly, these dots evidence strong threedimensional confinement properties. In Fig. 2(a) we report the cw magnetoluminescence spectra obtained under Iexc = 39 W cm -2 excitation intensity and with the magnetic field parallel to the growth axis of the structure (z). The zero field spectrum exhibits four well resolved lines due to intersubband recombination of carriers from the four lowest QD energy levels. With increasing magnetic field the recombination lines corresponding to excited states broaden and split into two or more lines of different amplitude and width. This is clearly observable in Fig. 2(a), where the split transitions arising from the single lines at 0 T are connected by dashed lines. Note that changing the magnetic field orientation by 90" (B parallel to the quantum well layer), the excited transitions in the spectra exhibit neither splitting nor appreciable shift with increasing field, as shown in Fig. 2(b). In Figs. 3(a) and 3(b) we present the fan plots obtained from-the magnetoluminescence spectra under different band filling conditions (i.e., different excitation intensity). All lines corresponding to excited states exhibit clear splitting. The data can only be explained starting from the quantum mechanical treatment of magnetic interaction affecting a particle in a potential of axial symmetry [13], i.e., a system which can be described in terms of cylindrical coordinates. As shown by us recently [12], the confinement energies of our quantum dot system can be reproduced by the Luttinger-Kohn model. The effective potential for electrons and heavy holes includes the QW confinement potential (vertical confinement) and the strain induced deformation potential (lateral confinement). Because of the axial symmetry, the envelope wave functions can be written as Rnm(r,z)e ira4', where the principal quantum number n specifies the number of radial modes and m the z component of angular momentum Lz = mti. The states are labeled n E, n II +-, n A-* for m - 0, • 1, • • 3..... where states with the same I m I are degenerate at zero field. Therefore our ground state is I E, the first excited state 1 H - , and the second excited state 2~ and 1A --- (nearly degenerate). Because of the decoupling of heavy and light
{ :[( +
)( r , z ) 7 z
+
O--S. . . .
,tEe,h) I \ ,,(e,h) : I V Q W LZ) + V s t r a i n L Z ' r ) I ( emh ) ( .., Z ) I]in,
!
'
i
'....
i
-'i
' /
""
-(a) A
'1
Im39Wom"2
. _
& ~
"
6T
z bl
z_
.32
1.35
1.38 1.41 ENERGY (eV)
;?,;wo.,
i
i
1.32
1.35
i
i
1.38
1.4.1
1.44
1.47
,
.47
ENERGY (eV)
FIG. 2. Magnetoluminescence spectra in different configurations. (a) B II growth axis and lexr = 39 W cm -2 and L,(b) B l growth axis and lexc ----75 W cm -2. The dashed lines in (a) connect split transitions arising from single bands at 0 T. hole states the Schr6dinger equation for electrons (e) and holes (h) in extemal magnetic field reads
o) o( r o) m
8 JULY 1996
LETTERS
r
tr,
z)
T;r
= P .,-.(.e. .' h. .). d t ( e 'mh ) (" r ' Z) '
1
H(e,h)]
H(ze,h)
m2
m ~ e,h) r 2
+
+
(1)
343
295
VOLUME
77, N U M B E R ~ i ,
(a)
PHYSICAL
2 s ;'|'",'i
, i,
i,
i
, I,
REVIEW
tively [14]. V~w~)(z) is the rectangular band edge confinement due to the quantum well, and V~'hi)n(Z,r ) i s the approximately harmonic strain potential calculated by the theory of elasticity [12]. For B [[ z the Zeeman effect is determined by H(ze~) = --(e/2mr(e~)mo)BLz, where Lz is the angular momentum. The total Zeeman effect, linear in B, is thus given by
~r , l ' ,
1.44 ill I11 1.42
.= E
N e Z 0
SO
8
1.40
p..
(Toolo)
//'I--'
eh ( m ~ + m~h ) Bm. AEz = ~mo
(/1 0 n v < bJ I1.
1.38
.~.IAJ....
7t11.1
e2
lr, . . , , . . , . . , , . . , . . . . - ; - - - , 1.34
, i I s.
l,
0 1
2
3
4
i, 5
I,
6
|,
7
8
|
9
,-
9
MAGNETIC FIELD (TESt.A)
Co)
i',
1.44
i
9 i 9 i"'J'~ ',
OW @
@
@
i',
J',
i-
9
o
9
@
@
m--4
,,, , , . . / - - . y - / . .
z
o
/ 9
-1 |
. m-3
~
,,p--/.9
" ...-0
.-" . -' ""
9
rn--1
1.40
p-
2
C/)
o n
/,
Jz,~.lr/ / _
>
i ,i"i ~'1
9 me,2
m
1.38
< tLl rt
~
-",,Ira__,-"" - V
V
I1'111 1
1.36
0 1 2 3 4 5 6 7 8 9 MAGNETIC
FIELD (TESLA)
FIG. 3. (a) Fan plots extracted from the magnetoluminescence data with B II growth axis with l~,c --- 5 w cm -2 (symbols). Transitions with the same n value are indicated by the same symbol. Curves are calculated according to Eqs. (2) and (3). The dashed lines indicate unsplit (I m l ~- 0) transitions. Continuous lines indicate split ( I m l # 0) transitions associated with nil-*, hA-*.... states. Note that symbols associated with I m I - - 0 transitions change slope at high fields due to anticrossing of different states. Inset: magnetic field dependence of the Zeeman splitting of the 111 transition. (b) The same as (a) but for l=c --- 75 Were -2. The O's indicate the shift of the QW transition on the highest energy side of the spectrum. where isotropic electron mass is assumed for the electron (m e --- mz). The radial and vertical heavy hole masses are given in terms of Luttinger parameters by mrn = 1/('yn + 72)----0.143m0 andm h --- 1/(Tl -- T2),respec344
((r2)"
(r2)h ~B2
A Eo = 8moc2~ m----f" r + ~] e2 .~ ~ c 2 (r2)B 2.
--9 m-0
i 9 i,
(2)
The diamagnetic shift determined by the Ho term in the Harniltonian is given by
*.. - ~ 1 7 6
~ ~
8 JULY 1996
LETTERS
(3)
The diamagnetic shift is quadratic in the field up to a few teslas before obtaining a linear asymptotic limit at very high field strengths [14]. If Coulomb interaction is taken into account, i.e., electrons and holes form excitons, the total diamagnetic shift of the exciton will be smaller than that forecast by Eq. (3), due to the additional confinement of the exciton wave function in the dot. This effect has been neglected in the present calculations [15]. Furthermore, the atomic part of the "atomic" Zeeman splitting should be taken into account (this is often called spin splitting, although it also involves the interaction of the orbital magnetic momentum with the magnetic field). This effect is related to the electron magnetic moment of the atomic parts of the Bloch wave functions. For electrons we have Ee = ge(eti/2mo)msB, where ms = + 1 / 2 and ge ranges between -0.598 and -0.948 [14]. For holes (at lowest order) we have Eh = gl*l(etl/2mo)myB, where my - +_3/2 and g~ is the atomic g factor. However, even with the higher order corrections, [14], this elfeet is of the order of 1 meV, comparable to the quantum well case. This cannot be resolved in the luminescence spectra, and can safely be neglected here. The results of the theoretical analysis are shown by the lines in Fig. 3, where we plot the transition energies calculated by means of Eqs. (2) and (3) for states"of different m value, split by the Zeeman effect. The selection rules for these transitions are An = 0 and Am - 0 [16]. Figure 3(a) clearly shows the lifting of the degeneracy of the 111--- levels. The experimentally observed splitting of this level as a function of the magnetic field is shown in the inset. The dependence is clearly linear, as expected, for the Zeeman splitting of an axially symmetric quantum dot. With increasing excitation power, it is possible to observe the splitting of several filled states (up to the fifth luminescence peak consisting of 3~, 2A, and 1F transitions). It is worth noting that the ~ states (I m I = 0) do not split and exhibit only a weak diamagnetic shift in
296
VOLUME 77, NUMBER 2
PHYSICAL
REVIEW
the field [see dashed lines in Figs. 3(a) and 3(b)]. The fine structure of the fan plots becomes more and more complicated with increasing the quantum number n. The measured diamagnetic shift of the ground level (2 meV at 8 T) is smaller than that obtained by Eq. (3) (4 meV at 8 T for free carriers), due to the neglect of excitonic effects. If we add the experimental diamagnetic shift to the Zeeman term [rather than Eq. (3)], we find an even better agreement between theory and experiment. This indicates that excitonic effects could play some role in our experiment. A quantitative analysis of these effects will be presented in a forthcoming study [17]. In the opposite configuration (B perpendicular to the growth axis) [Fig. 2(b)], the average magnetic energy due to the angular part of the wave function is zero because the angle between the z component of the angular momentum and the magnetic field direction is 90*. In this configuration the magnetic field direction is no longer a symmetry axis for the Hamiltonian, and the magnetic field induced shift of the photoluminescence bands is very small because the orbit of carriers crosses the rectangular well potential boundary along the z direction. Very recently, Bayer et aL [6] have reported the magnetoluminescence spectra of QD's fabricated by shallow etching. However, their interpretation was based on a Zeeman splitting given by ( 2 t i e / I z ) B (for m = _'L-1transitions) which is by far larger than our experimental data and our Eq. (2). In conclusion, we have presented the first well resolved quantitative measurement and theoretical interpretation of a fundamentally new phenomenon, the breaking of the degeneracy of quantum dots states with different angular momentum quantum numbers induced by an axial magnetic field. This Zeeman effect is peculiar to quantum confined nanostructures. We gratefully acknowledge D. Cannoletta, A. Melcame, and M. Corrado for expert technical help.
[1] B. Meurer et al., Phys. Rev. Lett. 68, 1371 (1992). [2] D. Heitmann and J.K. Kotthaus, Phys. Today 46, 56 (1993); D. Heitmarm, Physica (Amsterdam) 212B, 201 (1995).
LETTERS
8 JULY 1996
[3] R.C. Ashoori et al., Phys. Rev. Lett. 68, 3088 (1992). [4] Ch. Sikorski and U. Merkt, Phys. Rev. Lett. 62, 2164 (1989). [5] W. Hansen et aL, Phys. Rev. Lett. 62, 2168 (1989). [6] M. Bayer et aL, Phys. Rev. Lett. 74, 3439 (1995). [7] G.W. Bryant, Phys. Rev. Lett. 59, 1140 (1987). [8] U. Merkt et aL, Phys. Rev. B 43, 7320 (1991). [9] P.A. Maksym and T. Chakraborty, Phys. Rev. Lett. 65, 108 (1990). [10] M. Sopanen, H. Lipsanen, and J. Ahopelto, Appl. Phys. Lett. 66, 2364 (1995). [1 l] H. Lipsanen, M. Sopanen, and J. Ahopelto, Phys. Rev. B 51, 13868 (1995). [12] J. Tulkki and A. Heinim/tki, Phys. Rev. B $2, 8239 (1995). [13] L.I. Shift,Quantum Mecham'cs (McGraw-Hill, New York, 1949). [14] Th. Wimbauer et al.,Phys. Rev. B 50, 8889 (1994). [15] Electron-electron interaction is neglected in our lowest order theoretical calculation. However, in our system, there is evidence that the experimentally observed dipoleallowed transitions can be analyzed by the one-particle picture. In fact,our present and previous (Ref. [I I]) experiments show that the luminescence energies are insensitive to the state filling. Similar observations have been reported in Refs. [2,4,9] by far infrared measurements in field effect quantum dots. Further, our calculations have shown that when several dot states are filled, the calculated intensity of the emission lines is very close to the measured intensity distribution under intense photoexcitation. This suggests that the transition energies of the dipole-allowed lines are insensitive to correlation. A possible explanation of this surprising result is that a properly modified generalized Kohn's theorem can be applied to our dots. Note that this theorem, which has been experimentally found to be valid for field effect quantum dots, cannot be applied as such to our dots. [ 16] J. Pankove, Optical Processes in Semiconductors (Dover, New York, 1971). [17] A coupled heavy hole-light hole band calculation has been carried out for hole states in axially symmetric quantum dot in external field. The results show that the band coupling is significant and accounting for it improves the agreement with experiment [M. Brasken, ]. Tulkki, and M. Lindberg (unpublished)].
345
297
VOLUME73, N U M B E R 5
PHYSICAL
REVIEW
LETTERS
1 AUGUST1994
P h o t o l u m i n e s c e n c e o f Single I n A s Q u a n t u m D o t s O b t a i n e d b y S e l f - O r g a n i z e d G r o w t h o n G a A s J.-Y. Marzin, J.-M. G&ard, A. Izra#.l, and D. Barrier France Telecom, Centre National d'Etudes des Tdldcommunications-PAB, Laboratoire de Bagneux, BPI07, 1:92225 Basneux, France
G. Bastard Laboratoire de Physique de la Mati~re Condens~e, F.cole Normale Supdrieure, 24 rue Lhomond, F75005 Paris, France
(Received 11 March 1994) We present photoluminescence data on InAs quantum dots grown by molecular beam epitaxy on CmAs. Through the reduction of the number of emitting dots in small mesa structures, we evidence narrow litw.s in the spectra, each associated with a single InAs dot. Beyond the statistical analysis allowed by this technique, our results indicate short capture and relaxation times into the dots. This approach opens the mute towards the detailed optical study of high quality easily fabricated single semiconductor quantum dots. PACS numbers:71.50.+t, 73.20.Dx, 78.55.Cr Low dimensionality semiconductor structures constitute very attractive objects both for their fundamental properties and their potential application.s in micro- and optoelectronics. While quantum well structures are already widely used in optoelectronic devices, quantum wires and quantumdots appear to be much more difficult to fabricate for this purpose [ 1]. Though the observations of low temperature photoluminescence (PL) on a single quantum dot [2] or of an isolated short wire [3] were recently reported, most results concern so far collections of objects with large fluctuations of sizes, which both restrains drastically the interest for applications and mask the influence of their intrinsic properties. In the case of quantum dots, besides the technological difficulties of their fabrication, fundamental limitations as the slowed down relaxation between confined states [4-8] were theoretically predicted. They are, however, difficult to study on an inhomogeneous collection of dots, while the fabrication and experiments on a single dot still constitute a challenge. In Ref. [2], the multiple lines detected in the PL spectrum of a single dot fabricated by local intermixing were interpreted in terms of emission from confined excited states. However, due to the fabrication process, the interpretation of the details of the PL speca'a for various dot sizes appears to be very difficult. In this Letter, we demonstrate the validity of an alternative approach to observe high quality single InAs/GaAs quantum dots, similar in spirit to the methods used to study the optical properties of single molecules [9]. Because of the large calculated energy distance between confined levels, such single dots constitute ideal objects for studying relaxation phenomena. It was indeed shown several years ago that the growth of a highly lattice mismatched semiconductor layer onto a substrate could lead to the spontaneous formation of semiconductor clusters with sizes in the quantum range. Such a situation was in particular observed in the InAs/GaAs [10-16] (7% mismatch) and In_GaA.s/GaAs systems [17,18]: When an InAs layer is deposited on 716
a GaAs substrate, the growth is first obtained with a bidimensional (2D) mode and, beyond a limit of the order of 1.7 InAs [11,16] monolayer (ML), InAs islands are nucleated on the surface. :If the growth is interrupted, these islands evolve and a quasiequilibrium is reached after typically 10 s [19]. This quasiequilibrium distribution could be studied by atomic force microscopy (AFM) [20]: The nucleated islands are small InAs square based pyramids (2.8 nm high, base dimension around 24 nm with a relative fluctuation of 15% for an InAs amount of 1.8 ML), laying on one 2D InAs layer. The typical center to center interisland distance is of the order of 55 rim. The angles of the pyramid merely depend on its size and correspond on the average to (410) limiting planes [20]. If the growth proceeds with GaAs deposition, one obtains InAs clusters which are free of nonradiat/ve defects, as shown in previous studies [I I ]. These objects constitute local potential wells for electrons and holes and exhibit an efficient photoluminescence. From the point of view of their size and crystalline quality, these InAs clusters are ideal quantum dots mad their size fluctuations are very small, as compared to the state of the art of artificial dots. We clearly evidence in the following the influence of the statistical distribution of their sizes on th'e PL, and observe for the first time the emission of single such quantum dots. The samples were grown by molecular beam epitaxy at 520 "(2. Sample A (B) consists of a (I 00) GaAs substrate onto which a 500 nm GaAs buffer layer was first deposited, followed by a 2.2 ML (1.7 ML) deposition of InA_s, and a I00 nm GaAs cap layer. Both samples contain InAs clusters and are further processed using e--beam lithography to remove the InAs clusters except in mesas. Those square mesas with a side of 5000, 2000, 1000, 500, 200, and 100 nm are insolated in polymethylmethacrylate using a JEOL 5DIIU machine, transferred to a Ni mask using a standard lift-off procedure and etched using SiCl4 reactive ion etching. The distance between mesas is 15/zm.
0031-9007/94/7 3 (5)/716(4)$06.00 9 1994 The American Physical Society
298
V O L U M E 73, N U M B E R 5
PHYSICAL
REVIEW
The cw ph9toluminescence spectra were obtained using a Ti sapphire tuneable laser, pumped by a ha .+ ion laser. The laser beam is first cleaned by a prism monochromator and focused through a microscope objective onto the sample situated in a circulation He cryostat. The spot diameter is 2 / z m and the power density around 1 kW/cm 2. The PL signal is collected through the same microscope objective, dispersed by a double I m focal length monochromator and detected by a photon counting system based upon an avalanche Si photodiode. Figure 1 shows the 10 K PL spectrum of a 5000 nm mesa of sample A, with an excitation energy of 1.5 eV. This emission is typical of InAs clusters [ 11 ]. Its limited full width at half maximum (50 meV for both samples) resuits from a careful optimization of the growth conditions. It is attributed to an inhomogeneous distribution of cluster sizes and the emission line can be fitted by a Gaussian curve (standard deviation o" -- 25 meV). In order to calculate the dots' energy levels, we assume that GaAs overgrowth affects only the top lnAs island monolayer through indium segregation effects, consistently with what was evidenced on 2D InAs/GaAs structures [21]. However, the size of the InAs islands is smgller than what is observed by AFM, due to the quenching of their evolution by the GaAs deposit. We nevertheless assume that they already have a pryamidal shape with the same base angle at the early stage of their formation, and this is our main approximation. Figure 2 shows the low temperature calculated energies of the lowest energy transitions for the InAs clusters, in a simple effective mass treatment, of which a detailed presentation is beyond the scope of the present Letter. Basically, we approximate the pyramid by a cone of axis Z, with the same base surface and height. We include the effect of the indium segregation at the InAs/GaAs interface [21] during the GaAs overgrowth by spreading the top InAs ML with an exp(-Z/L) profile (L -- 1.1 rim) and assuming for the In(Ga)As layer the same strain state as for the 2D case. This latter approximation is justified because the clusters have a very fiat shape. Finally, the nonseparable Hamiltonian is diagonalized numerically for electrons, heavy holes, and light holes. The calculated transitions can be shown to be, for a given extension of ! !
.e,
!
1.2
,I . . . . t . , , ~ g . , . , , 1.25 1.3 1.35 1.4 Energy in eV
FIG. I. I0 K PL spectrum of a 5000 n m mesa in sample A.
1 AUOUST 1994
LETTERS 'n"
i
i ,1 i
n
i
'i.....
m
~
I
_1
I i
ii
i I
1.4 1.2 1.0 I
50
1. I
1 ~
lOO
I
I
~5o
Radius r (A) FIG. 2. Low temperature calculated fundamental and first excited transitions as a function of the radius r of the InAs cluster whose schematic structure is shown in the inset, for d - 0.33 nm (e-hh: m : e-lh: - - ) , d -~ 0.66 nm (e-hh: ---; e-lh: .--) and a cone base angle of 12.4". the potential on the Z axis and a given InAs volume, quite insensitive to the detailed shape of the cluster. The calculation whose results are displayed in Fig. 2 was done assuming that the clusters are lying onto a 1 or 2 ML uniform InAs layer. Note that, as shown schematically in the inset of Fig. 2, the radius indicated on the abscissa takes into account this underlying layer, and that we refer to the geometry of the islands prior to the GaAs overgrowth. At low r values, the fundamental transition energy tends to the band gap energy of an InAs quantum well of thickness d. At larger r values (r > 8 nm), the first states are strongly localized in the central part of the clusters so that this transition energy does not depend on d. This will allow us to determine the dots' radii from their emission energies. The calculated energies can be compared with experimental data for quasiequilibrium distributions of dots where the islands' radii (and shape) are known from AFM. Such dots grown with a 1.8 ML InAs deposit and a 20 s growth interrupt before GaAs overgrowth emit at 1,07 eV at 10 K. From Ref. [20], r --- 13.5 nm for these islands so that the calculated first and second transition energies are, respectively, 1.06 and 1.211 eV. Besides validating our calculation, this result clearly indicates that the PL peak corresponds to the fundamental transition. The comparison of the experimental emission energies o~" sample"A 03) with the c~culated values yields r = 9.5 nm (8 rim) and h -- 2.1 nm (1.8 nm). For these parameters, there is only one electron, one light hole, and one heavy hole level bound in the dot for d = 2 ML. For d -- 1 ML, a second electron (and heavy hole) state is marginally bound in the dot. The standard deviation in r is 0.5 nrn. This figure, significantly smaller than that deduced by A F ~ in the equilibrium case, is impressive when compared to state of the art artificial dots. Figure 3(a) shows the typical PL spectra of a 500 nm mesa obtained on sample A. Whereas the spectrum displayed in Fig. 1 could be nicely fitted by a Gaussian, we resolve in this spectrum a forest of narrow lines. 717
299
VOLUME 73, NUMBER 5 ~
I
PHYSICAL '
'
'
'
I "
REVIEW
'~"~'"I
~ 5oo
o
1.25
_i._
1.3 Energy in eV
' " l "
"l~""l
1.35 -', ' ' '
0 1.2g 1.2/}5 1.29 1.295 Energy m eV FIG. 3. (a) 10 K PL spectrum of a 500 nm mesa in sample A. (b) is a blow up of a part of the spectrum displayed in (a). Figure 3(b) shows a blow up of this spectrum in a narrower energy range. The narrowest lines have a full width at half maximum smaller than the 0.1 meV resolution of our experimental setup. About 90 peaks can be counted in this particular spectrum. We checked carefully that the peaks positions are reproducible, independent of the laser energy, and change from one mesa to the other. We also studied on some of the more intense transitions the evolution of their energies with temperature between 5 and 90 K: They decrease with temperature, with variations intermediate between the very similar lnAs and GaAs band gap dependences. Finally and contrarily to what is reported in Ref. [2], we cannot attribute several peaks to different optical transitions of a given cluster. All of them are indeed observed in a 80 meV energy range, smaller than the calculated energy separation between the first two transitions shown in Fig. 2. We therefore attribute each of these peaks to the emission of a specific InAs cluster. The evolution (not shown) between the spectra of 5000 nm m'6sa displayed in Fig. 1, in which the reproducible stmctureg are already due to the limited number of emitting dots, to the spectrum of Fig. 3 is consistent with the reduction of the average number of transitions per unit energy as the total number of clusters in the studied mesa decreases. In order to corroborate our interpretation, wr have analyzed in detail the statistics of the peaks of the spectrum displayed in Fig. 3. Though there are large fluctuations of their intensities, the density of peaks per unit energy (regardless of their intensity), estimated by 718
I AUGUST 1994
LETTERS
a floating average over I0 meV, ~ n a b l y follows the Gaussian curve deduced from the spectrum of Fig. 1, as can be seen in Fig. 4. Finally, on most 200 am mesas of sample B, less than 10 peaks were observed as shown in Fig. 5. Figures 5(a)-5(c) are spectra obtained on different such mesas, whereas Fig. 5(d)was obtained by adding the spectra obtained on 20 different 200 nm mesas. This latter spectrum reproduces satisfactorily the typical density of peaks observed for larger mesas. All these results are consistent with the assignment of the observed narrow wansitions to PL line.s each associated with a single cluster. These spectra also allow us to get a deeper insight into the cluster formation. As we observe about IOO peaks for a 500 nm mesa, we can deduce a typical surface occupied per cluster around S ,= 50 x 50 rim2. This figure is very close to what is observed for "equilibrium" islands by AFM [20] so that the density of islands is likely to be fixed at the earliest times of their formation. This result suggests the following picture for the islands' formation: When evolving during a growth interrupt towards quasiequilibdum, the primary (quickly formed) islands grow in size at constant number at the expense of the 2D InAs layer. This is corroborated by the small variation (55 to 61 n m ) o f the intefisland distance for a large increase of the InAs deposited amount (1.8" to 3.6 ML) reported in Ref. [20]. For sample A, where the InAs was deposited in 1 s and immediately overgrown by GaAs, the evolution of the islands is efficiently quenched when they are still in an early stage. Knowing the size of the clusters (PL peak energy) and their density (number of peaks in a given mesa) which both do not depend on the average thickness d of the underlying 2D InAs layer, we can extract this latter parameter by writing the conservation of the total amount of InAs. It would correspond to 2.1 ML for sample A. Unfortunately, this determination of the average value of d does not give us the spatial distribution of the underlying layer. AFM shows that the thickness of this film is inhomogeneous and corresponds to roughly 1 ML in the vicinity of "equilibrium" islands. The existence of this underlying InAs film explains the observation of '~3,.l l ' ' '
~
v
'j v-l 1.25 1.3 Energy in e V
1.35
HG. 4. Number of peaks per unit energy observed in the spectrum of Fig. 3 (full line) compared to the estimate from the Gaussian fit of the spectrum of Fig. 1 (broken line).
300
VOLUME 73, NUMBER 5 ii,
1.3
PHYSICAL ,
REVIEW
, , i , ,
1.35
Energy in eV FIG. 5. (a), (b), and (c): 10 K PL spectra of three different 200 nm mesas of sample B. (d) sum of 20 spectra recorded on different such mesas.
LETTERS
1 Auous'r 1994
semiconductor quantum dots. Among those, it will be important to check experimentally the microscopic structure of the dots and to detect by PL excitation the transitions between excited levels, if any. Finally, time resolved experiments on larger clusters with several bound electron and hole levels should allow us to deepen our understanding of energy relaxation in quantum dots. The authors gratefully acknowledge J.M. Moison, R. Raj, B. Jusserand, F. Laruelle, and M. Voos for their fruitful comments and L. Ferlazzo for the etching of the mesa structures. Part of this work was partly supported by NANOPT EEC ESPRIT Basic Research Action.
[1] For a t~view of quantum wires and dots fabrication techniques and optical studies see IC KaY, J. Lumin 46,
69 (1990). intense PL signals with an excitation energy of 1.5 eV (used for all our experiments), below the GaAs band gap. This energy is above the fundamental transition energies calculated for 1 and 2 ML thick InAs quantum well in GaAs: Most carriers are created in a quasi-2D InAs layer and are further captured into the clusters. By varying the laser energy from 1.45 to 1.52 eV, we observed that the onset of PL of the individual lines observed in the spectrum of sample A is around 1.46 eV, closer to the gap of a 1 ML InAs quantum well. While the fundamental levels are not sensitive to the local environment of this InAs film, the first excited levels are, so that a more detailed microscopic study of the dot is necessary to know whether there are several bound states in those dots or not. In any case ( d - - 1 or 2 ML), there is a calculated distanca of at least 94 meV (54 mcV) between the lowest electron (heavy hole) level and the first excited level for the average cluster of sample A. These figures are much larger than the longitudinal optical (LO) phonon energy in GaAs (36 meV). At the present time,, the existing theoretical models [4-8] fail to explain the high PL efficiencies and short PL rise times [19] (in unprocessed samples) which we observe. To summarize, we have observed the low temperature PL of single InAs clusters embedded in GaAs in samples where small mesa structures were designed by nanolithography. The statistics of the energies of these emissions in such small mesas are consistent with the spectra observed on unprocessed samples or large mesas. From the comparison with our calculations, we deduce a very homogeneous distribution of cluster sizes, as compared to state of the art artificial fabrication techniques, and get new insight into the formation of InAs clusters. These objects constitute very attractive test systems for the electronic properties of quantum dots because of their relatively easy fabrication, of their intrinsic regularity of size and of the large spacing between electronic levels as compared to room temperature thermal energy and to the LO phonon energy. Beyond these first observations, numerous additional experiments are still to be performed on these single
[2] If.. Brenner, U. Bockelmann, (3. Abstreiter, M. Walther, G. Bohm, G. Tranlde., and G. Weimann, Phys. Rev. Lett. 69, 3216 (1992). [3] L. Birotheau, A. Izra~l, J.Y. Marzin, R. Azoulay, V. Thierry-Mieg, and F.R. Ladan, Appl. Phys. Lett. 61, 3023 (1992). [4] U. Bockelmann and G. Bastard, phys. Rev. B 42, 8947 (1990). [5] H. Benisty, C.M. Sottomayor-Torres, and C. Weisbuch, Phys. Rev. B 44, 10945 (1991). [6] T. Inoshitaand H. Sakaki, Phys. Rev. B 46, 7260 (1992). [7] U. Bockelrnann and T. Egeler, Phys. Rev. B 46, 15 574 (1992). [8] U. Bockelmann, Phys. Rev. B 48, 17 637 (1993). [9] Th. Basch~, W.E. Moerner, M. Orrit,and H. Talon, Phys. Rev. Left.69, 1516 (1992). [I0] W.J. Schaffer, M.D. Lind, S.P. Kowalczyk, and R.W. Grant, J. Vac. Sci.Technol. B I, 688 (1983). [II] L. Goldstein, F. Glas, J.Y. Marzin, M.N. Charasse, and G. Le Roux, Appl. Phys. Left.47, 1099 (1985). [12] F.J. Grunthaner, M.Y. Yen, R. Fernandez, T.C. Lee, A. Madhukar, and B.F. Lewis, Appl. Phys. Left.46, 983 (1985). [13] F. Houzay, C. Guille, J.M, Moison, P. Henoc, and F. l~_rthe, J. Cryst. Growth 81, 67 (1987). [14] F. Glas, C. Guille, P. Henoc, and F. Houzay, Int. Phys. Conf. Ser. 87, 71 (1987). [15] O. Brandt, L. Tapfer, K. Ploog, R. Bierwolf, M. Hohenstein, F. phillip, H. Lage, and A. Hebede, Phys. Rev. B 44, 8043 (1991). [16] J.M Gtrard, Appl. Phys. Lett. 61, 2096 (1992). [17] C.W. Snyder, B.G. On', D. Kessler, and L.M. Sander, Phys. Rev. I.att. 66, 3032 (1991). [18] D. Leonard, M. Krishnamurthy, C.M. Reaves, S.P. Denbaars, and P.M. Petroff, Appl. Phys. Latt. 63, 3203 (1993). [19] J.M. Gtrard, in "Confined Electrons and Photons: New Physics and Applications," edited by C. Weisbuch and E. Burstein, NATO ASI Series (Plenum, New York, to be published). [20] J.M. Moison, F. Houzay, F. Barthe, L. Leprince, E. Andrt, and O. Vatel, Appl. Phys. Latt. 64, 196 (1994). [21] J.M. Moison, C. Guille, F. Houzay, F. Barthe, and M. Van Rompay, Phys. Rev. B 40, 6149 (1989). 719
301
Electron and hole energy levels in InAs self-assembled quantum dots G. Medeiros-Ribeiro, a) D. Leonard, and R M. Petroff
Materials Department, University of California, Santa Barbara, California 93117 (Received 16 September 1994; accepted for publication 3 February 1995) Capacitance spectroscopy is used to determine the allowed energy levels for electrons and holes in InAs self-assembled quantum dots embedded in GaAs. Using this technique, the relative energy of the electron and hole states is measured with respect to their respective energy band minima in the GaAs. This allows the construction of an energy level diagram for these quantum dots which correlates well with previously observed photoluminescence data. By tuning the device geometry, a fine structure in the electron ground state is revealed and attributed to Coulomb charging effects. 9 1995 American Institute of Physics. Due to their unique properties, zero-dimensional semiconductor systems have been extensively investigated in the ,past years. Initially, these systems were produced by sophisticated processing techniques, including e-beam lithography and holographic patterning, l These artificial systems opened new perspectives for devices as well as basic physics research, s=~For example, single artificial quantum dots2 exhibit Coulomb charging effects, and in this case the Coulomb energy overwhelms the quantum confinement energies. As opposed to artificial methods for creating 3D confinement, self-organized epitaxial growth methods can reach much lower lateral dimensions. In this case, the lateral quantization is larger, and the confuting energy is as important as the Coulomb charging energy. Self-assembled quantum dots (SADs) can be created by crystal growth in the Stranski-Krastanow 3-6 (SK) growth mode yielding islanding. These islands, or dots, produce low dimensional confining structures when an appropriate choice of surrounding materials is made. Photoluminescence experiments with very few isolated SADs have been performed, TM and the effects of quantum confinement have been observed. Experiments with arrays of such objects were also carried out, and dispersion of the electronic levels as a function of magnetic field could be observed in capacitance as well as far-infrared absorption spectroscopy.9 The existence of hole states however has not been demonstrated as clearly. Nevertheless, sharp linesTM observed in photoluminescence (PL) suggest that radiative transitions may occur between electron and hole confined states in the SADs. The present experiment is aimed at clarifying this issue by probing both electron and hole states separately. The existence of confined levels for both holes and electrons is demonstrated using capacitance spectroscopy in p- and n-type doped structures, respectively. The studied system consisted of InAs SADs grown on GaAs. The detailed growth procedure of the dot layer is described elsewhere. 3'9't~ As in Ref. 10, we did not rotate the substrate during the InAs deposition. We have therefore a variation in the InAs coverage throughout the 2 in. wafer, which produces a variation in the SAD areal density ranging from zero to ---l0 t~ cm -2. Transmission electron microscopy confirmed this density variation for the samples studied here. Samples were taken from high dot density regions of ")Electronic mail: medeiros~engrhub.ucsb.edu Appl. Phys. Lett. 66 (14), 3 April 1 9 9 5
the wafer, which correspond to the regions closer to the indium cell. The InAs coverage is not exactly the same in each of the samples, differing by small fractions of a monolayer. The quantum dots are embedded in an undoped C~_As matrix, with a 25 or 15 nm thick tunneling layer separating, respectively, an n- or p-type back contact from the dot layer (see Fig. I)o The thickness of this layer is chosen to provide a transparency adequate for the complete charging of the dots. Since the tunneling rate for holes is smaller than for electrons, we used a thinner tunneling layer in the p-type case. For both sm,'cmres we used a 15 nm u n d ~ GaAs spacer following the dot layer, and a GaAs/AIAs short period superlattice (SPS) with a 2 nm period on top of the spacer. The structure was capped with 5 nm GaAs to prevent oxidation and obtain reproducible Schottky barriers. Figure 1 represents schematically the structure and the band diagram. The 1.5 ML thick wetting layer, ~~ characteristic of the SK growth mode, is also shown schematically. We also investigated n-type samples with different total thickness. Using conventional photolithography, circular gates ( 1 5 0 / a n diam) were fabricated consisting of a 15 nm thick Cr layer as the Schottky interface, a 10 nm thick Ni layer and a 200 nm thick Au layer for bonding the devices. The back contact was a Ni/AuGe alloy annealed at 400 ~ for the n-case and Cr/ ZnAu alloy annealed at 450 ~ for the p-case, in b o ~ cases, the annealing time was 120 s, and the current voltage characteristics observed for both cases indicated an Ohmic behavior in the voltage range used. The capacitance measurements using these structures were carried out at 4.2 K. The
FmV~§ v~ "1
.t,j~ t
t
r'%~,,~-,,..-./... I ! i~om]
1-- -
ii~~I
i._~_.i I
-
-~ ,
FIG. 1. Samplesstructureand banddiagram.The tunnelingbarrier is 15 and 25 nm thick, for the p- and n-typesample, respectively.The wetting layeris approximately 1.5 monolayersthick for the InAs/GaAssystem(see Ref. 10).
0003..6951195166(14)1176713156.00
9 1995 American Institute of Physics
17157
302
40
.........
,
' '
'
9
p-type muq~e ~ " 24
I -
-0.5
'
i
.
. . . .
,..
!
' .
|
.
.
'
!
n - ~ e mm)ple I
i
0
0.5
.
0
1
HG. 2. ~ t a n c e - v o R a g e (C-V) characte~t~ for p (left-handside) and n (right-handside) type samples. The formationof the wetting layer twodimensionalelectron/holegas is shown togetherwith the estimatedflatband voltage. The 2DEG is not completelyformed in the n-type sample,due to leakage in the gate. frequency f u s e d was kept less than RC (R and C are, respectively, the tunneling resistance and the capacitance of the tunneling layer), in order to assure that the system was in equilibrium and the dots could be charged and uncharged within one period of the measurement frequency. W e used a 5210 E G & G dual phase lock-in amplifier,and the ac bias is added to the dc bias using an R C network. The ac bias used was 5 m V rms in most cases. W e restrained the measuremcnts to an applied bias which would allow detection of a purely capacitive signal. By changing the dc bias, one can move the Fermi level inside the GaAs band gap and through all quantized levels in the quantum dots. With low frequencies and no dc current, thermal equilibrium is established. The mcasuremcn~ were reproducible and did not show any hysteresiseffectwith changes in the applied bias, which confirms this assertion. Using fundamental thermodynamics and electrostatics, the differential capacitance at 0 K is proportional to the density of states, as can be verified by the equation
dQ dn dlz c = -z~ = q -j-~ -j-~ = o , ( e F ) ,
(I)
where dQ is the infinitesimal charge induced by a change in voltage dV, q is the electronic charge, n is the number of particles (carders), /z is the chemical potential, and D s = dn/d/~ is the density of states at the Fermi energy E F. Therefore, the capacitance-voltage characteristics reflect the density of states (DOS). The SAD energy spectrum includes an energy q2/2C increase for each added electron to a dot of capacitance C. The DOS described by Eq. (1) does not include these effects. The measured capacitance is a sum of a geometric capacitance Cgeo, the DOS capacitance CDOS and a capacitance related to 2D carrier gases C2DCO formed in both wetting layer and in the GaAs/SPS interface. Cgeo is the background observed in all capacitance voltage (C-V) curves (Figs. 2 and 3), and it is inversely proportional to the sample thickness. C2DCGexists only when the carrier gas is formed, and it Appi. Phys. Lett., Vol. 66, No. 14, 3 April 1995
.
.
0.2
.
0.4
0.6
0.8
Vo~ge (Voits)
voaage (vons)
1768
.
FIG. 3. C-V characteristics for samplesA, B, and C. Differentlever arms
increase the resolution of the experiment and we can better resolve the shoulder in the low energy side present in all samples as we decrease tb/tt~. C-V characteristicsfromeach sampleare plotted in differentscales, and for ~unpleA the wholecurve is displaced by 0.15 V for clarity. is responsible for the capacitance increase and plateau observed at - 0 . 5 and - 0 . 8 V for the p-case and at 0.7 V for the n-case (Fig. 2). The two plateaus observed in the p-case are the wetting layer hole gas and the GaAs/SPS interface hole gas. In the n-type sample, the conductivity sharply increased after that flatband condition was established and, therefore, only the plateau due to the wetting layer electron gas was observed. In the p-type C - V characteristics,the structure H 0 in Fig. 2 observed at -0.45 V is the ground state for holes in the dots, as no other peaks appeared at greater voltages. In the low dot density side of the wafer, the hole signature was too small to be measured, and therefore, we believe that H0 is a hole level in the SADs. We could not infer from the capacitance measurements in this particular system if we were probing a heavy or light hole state or mixture of both. For the n-type sample, the first peaks which appear at approximately 0.23 and 0.53 V are labeled respectively as E0 and E 1, the ground and first excited state. The magnetic field dispersion of these levels and the infrared absorption characteristics 9 indicate further that they are indeed the first and second electron state in the dots rather than two different size distributions of the SADs. A negative differential capacitance is consistent with a peaked DOS for E0 expected for quantum dots. The same effect cannot be observed for E1 because of the increasing importance of Cgeo and C2DE~' at this voltage range. The existence of both hole and electron states in S A D s indicates that optical transitions inside the S A D occur between confined electron and confined hole states.This is consistentwith the sharp linesobserved in P L spectra of few dots.TM Neglecting band bending in the contact layer,the change in the Fermi energy inside the structure is linear with changes in the bias and obeys the following relation
tb Ae/q=~AV,
(2)
where AE is the change in the Fermi energy position when subject to a bias change of A V. tb and ttot are, respectively, the thickness of the tunneling layer and the total thickness of Medeiros-Ribeiro, Leonard, and Petroff
303
TABLE L Predictedand measured valuesof the relative increasein capacitance with respect ot the zero bias capacitance.C,v=is the capacitanceof the sample when the wettinglayer is loaded withcarriers; it corresponds to the first plateau in Fig. 2. Cmco is the capacitance of the sample when the two-dimensionalcarrier gas at the GaAs spacer/SPS interface forms; it corresponds to the second plateau in the p-type sample.Co is the capacitanceof the structures at zero bias. Sample
C,~1 theor
C,,t exp
Cm~-xj theor
Cmca exp
p n
1.29 1.43
1.26 1.38
1.70 1.93
1.54 ...
the structure, from the doped layer to t h e surface ot" the sample. The other parameters of the structure are depicted in ,Fig. 1. The ratio tb/t ~ is also referred to as the lever arm, which sets a reduction factor from applied bias into energy levels. The smaller the level arm is, the smaller the influence of the external bias on the dots levels. The intrinsic resolution of the capacitance spectroscopy in this system is thus a function of both the lever arm and the ac bias amplitude. Samples A, B, and C were grown with different total thickness to achieve lever arms 0.4, 0.3, and 0.25, respectively. Figure 3 depicts the capacitance characteristics of these samples, showing the expected increase in the spectral resolution of the experiment. The higher energy resolution provided by sample C shows a splitting of the E0 electronic level-in the dots. The gap between the split peaks is ---20 meV. The Coulomb charging energy e2/2C, where C is the self-capacitance of the dot, is --18 meV assuming a disk shaped SAD with a 10 nm radius. This suggests Coulomb charging may be the origin of the ground state splitting observed in the dots. Future experiments are aimed at confirming this hypothesis and extending the analysis for the excited state. It should be pointed out that in both p- and n-type cases, these structures are broadened by statistical size fluctuations. Nevertheless, in view of the very large number of sampled dots (---106), the fine structure observed demonstrates a remarkable uniformity in the SAD sizes. Equation (2) is valid when there is no dc current flowing and when the structure has no net charge. When the dots are loaded there will be some band bending, but we neglect this effect and use Eq. (2) to translate the voltage changes into energy changes. As seen in Table I, for the samples shown in Fig. 2 the calculated increases in the capacitance for the loading of the wetting layer and the interface gas are in good agreement with the value predicted by this equation, which supports the assumption that band bending effects are not pronounced. In order to set a correct energy origin, we use the Schottky barrier values determined by the flatband condition. This condition is determined when a two-dimensional electron/hole gas is formed at the GaAs spacer/SPS interface. We evaluate the flatband condition by looking at the change in the slope of the C - V curves immediately after the formation of the carrier gas in the wetting layer. The change in the slope of the plateau assigned to the wetting layer occurs at ----0.66 V for the p-type, whereas for the n-type this change occurs at 0.91 V. We take these values as the Schottky barrier heights. The Schottky barriers (alPB, n and dp~,p) and the Appl. Phys. Lett., Vol. 66, No. 14, 3 Apdl 1995
FIG. 4. Energy levels in self-assembling quantum dots. The energy scale is set with respect to the GaAs conduction band edge.
GaAs gap (Eg=dPn,,,+dPa, p) determined in this way are overestimated values, since we no longer have a flatband condition when the 2D carrier gas forms at the spacer/SPS interface. We obtain a value of 1.57 eV for the GaAs gap, 50 meV larger than the expected value at 4.2 K. The energy difference E0-Ho extracted from this experiment is 1.3 eV, quite close to the 1.27 eV value measured from p~hotoluminescence in similar structures, s A more accurate comparison would require C - V and PL measurements from exactly the same sample area. As demonstrated by Eq. (1), the capacitance is directly related to the density of states. By means of Eq. (2), and using the values for Schottky barrier height, an absolute energy level diagram can be constructed as shown in Fig. 4, where both electron and hole states for the InAs/GaAs quantum dot system can be seen. Such a diagram directly shows the large confinement energies in InAs SAD embedded in GaAs. The 0D confinement for electrons and holes provided by SAD is thus consistent with very sharp lines observed in PL experiments. Good uniformity of SAD systems has been inferred from capacitance spectroscopy. Improvements in uniformity however may allow a more detailed study of Coulomb charging effects in SAD. The authors would like to acknowledge financial support from QUEST and AFOSR. One of us (G.M.R.) would like to thank CNPq (Brazilian Agency) for financial support. IT. Chakraborty, Comm. Cond. Mat. Phys. 16, 35 (1992). 2R. C. Ashoori, H. L. Stfrmer, 1. S. Weiner,L. N. Pfeiffer, K. W. Baldwin, and K. W. West, Phys. Rev. Lett. 71, 613 (1993). 3D. Leonard, M. Krishnamurthy,C. M. Reaves, S. P. Denbaars, and P. M. Petroff, Appl. Phys. Lett. 63, 3203 (1993). '~J. M. Moison, F. Honzay, E Barthe, L. Leprince,E. Andre, and O. Vatel, Appl. Phys. Lett. 64, 196 (1994). 5S. Ohkouchiand N. lkoma, Jpn. J. Appl. Phys. 33, L471 (I 994). 6R. N&zel, J. Temmyo, H. Kamada, T. Furuta, and T. Tamamura,Appl. Phys. Lett. 65, 457 (1994). 7S. Fafard, R. Leon, D. Leonard, J. L. Merz, and P. M. Petroff, Phys. Rev. B 50, 8086 (1994). sj. y. Marzin, J. M. Ggrard, A. lzra~l, D. Barrier, and G. Bastard, Phys. Rev. Lett. 73, 716 (1994). 9H. Drexler, D. Leonard, W. Hansen, J. P. Kotthaus, and P. M. Petroff, Phys. Rev. Lett. 73, 2252 (1994). roD. Leonard, K. Pond, and P. M. Petroff, Phys. Rev. B 50, 11687 (1994). Medeiros-Ribeiro, Leonard, and Petroff
1769
304
PHYSICAL REVIEW B
15 SEPTEMBER 1997-I
VOLUME 56, NUMBER 11
Few-electron ground states of charge-tunable self-assembled quantum dots B. T. Miller, W. Hansen,* S. Manus, R. J. Luyken, A. Lorke, and J. P. Kotthaus Sektion Physik, LMU Miinchen, Geschwister-SchoU-Platz 1, 80539 Miinchen, Germany S. Huant LMCI-CNRS, 38042 Grenoble, France
G. Medeiros-Ribeiro t and P. M. Petroff Materials Department and QUEST, University of California, Santa Barbara, California 93106 (Received 4 October 1996; revised manuscript received 20 March 1997) The few-electron ground states of self-assembled InAs quantum dots are investigated using high-resolution capacitance spectroscopy in magnetic fields up to 23 T. The level structure reveals distinct shells which are labeled as s-, p-, and d-like according to their symmetry. Our measurements enable us to resolve the singleelectron charging not only of the lowest (s) state with two electrons but also of the second lowest (p) state with four electrons as pronounced maxima in the capacitance spectra. Furthermore, two peaks at higher energy can be attributed to charging of the d shell with the first two electrons. We discuss the energy spectrum in terms of spatial quantization energy, Coulomb blockade, and many-particle effects. At around B= 15 T we observe a magnetic-field-induced intermixing of the p and d shell. Additional fine structure in the capacitance spectra is observed and discussed both in terms of nearest-neighbor Coulomb interactions and monolayer fluctuations of the dot size. [S0163-1829(97)01835-3]
I. INTRODUCTION Semiconductor quantum dots can be considered as artificial atoms. In fact, as in natural atoms, these small electronic systems have a discrete spectrum of energy levels. The confining potential of quantum dots, however, is different from the Coulomb potential of atoms. It arises from the interplay of band offsets and charges that surround the confined electrons. Using field effect devices it is possible to fabricate quantum dots with a voltage-tunable number of electrons. The few-electron ground states of such dots have been studied experimentally by several techniques such as singleelectron capacitance spectroscopy I and single-electron tunneling.~ Most studies have been performed on lithographically defined quantum dots with a lateral confinement length of about 100 nm? '4 For these systems, the energy for adding a single electron is usually dominated by Coulomb charging effects. For smaller dots, which can be directly grown by self-assembly,5'6 the spatial quantization energy becomes more important. Furthermore, the artificial atoms discussed here are not exactly identical. Therefore the energy spectra of dot arrays may show the interplay between intrinsic effects of individual dots and properties of the dot ensemble. Previous studies of capacitance spectroscopy on largescale arrays of self-assembled InAs quantum dots have demonstrated the strong spatial q uantization due to the small dot diameter of about 20 nm.' Recently the single-electron charging of the s shell was reported, s'9 The goal of the studies presented here was to resolve all discrete many-electron ground states and to study the magnetic-field dependence of the p shell. A high-resolution capacitance technique t~ enables us to examine small dot ensembles. The number of dots is thereby drastically reduced as compared to prior experi0163-1829/97/56(11)/6764(6)/$10.00
56
ments on self-assembled dots which averaged over more than 106 dots. 7-9 The results presented here are obtained on ensembles where the number of dots ranges from one thousand to several tens of thousand. Due to the reduced inhomogeneous broadening we observe the single-electron charging of the p and d shell in the capacitance spectra of such self-assembled quantum dots. We are able to determine the Coulomb blockade as a function of electron number per dot. A magnetic field B applied perpendicular to the plane of our oblate dots lifts the orbital degeneracy of the p shell. From the dispersion of the splitting we derive a transport effective mass which is distinctly different from the one obtained from spectroscopic studies. At B = 15 T we observe a magneticfield-induced crossover from a p- to a d-like ground state. All this information gives us a complete picture of the evolution of the ground-state energies of self-assembled dots with electron occupation and magnetic field allowing a quantitative comparison with theory. In addition to the quantized energy states of noninteracting dots the capacitance spe&ra exhibit a highly reproducible fine structure that reflects ensemble properties. Interdot interactions and monolayer fluctuations of the dot heights as possible causes for this additional structure are discussed. H. EXPERIMENTAL DETAILS The samples are grown by molecular-beam epitaxy, generating lnAs dots in the Stranski-Krastanow growth mode. 5-9'11-14 The dots are embedded into a suitably designed MIS (metal-isolator-semiconductor)-type GaAs/A1As heterostructure, as described in Refs. 7 and 8. Figure 1 shows the essential layer sequence and a sketch of the conductionband edge. The layer sequence starting from the substrate is as follows: buffer consisting of an A1As/GaAs superlattice 6764
9 1997 The American Physical Society
305
676,5
FEW-ELECTRON GROUND STATES OF CHARGE-...
56
inAs quantum dots
(a)
,,
A (pm 2)
,~-"
/ •
> o
..
~)
l. . . .
-1
ZE
hot FIG. 1. (a) Layer sequence of our devices. The InAs dots are distributed within the plane sandwiched between two Gabs layers. (b) Sketch of the conduction-band edge Ec with respect to the Fermi level EF along the growth direction for gate voltages at which no electrons are in the InAs dots. The indicated distances define the lever arm according to ttot/tb (in our case equal 7) which converts voltage into energy differences. (period 4 nm); GaAs back contact layer, Si doped to 4• Is cm-3; GaAs tunnel barrier; the self-assembled quantum dots within the plane of an InAs wetting layer; undoped GaAs layer; blocking barrier consisting of an AlAs/ GaAs superlattice (period 4 nm); GaAs cap layer. The dots are distributed within the plane of the wetting layer with a density in the range of 101~ cm-2. s From atomic force micrographs of similarly grown samples we estimate the InAs dots to be approximately 20 nm in diameter and 7 nm in height. 5 They are remarkably uniform in size with their diameters and thicknesses fluctuating by only about 10%. 5,6 Ohmic contacts to the back contact are made with alloyed AuGe. On the crystal surface metal electrodes are defined by electron-beam lithography and thermal evaporation. The area A of this front gate determines the number of dots in the ensembles under investigation. Samples with A = 14, 89, and 656/zm 2 were studied. The number of electrons per dot can be tuned with the bias applied between the gate and the back contact. Measuring the capacitance-voltage (CV) characteristics of our devices allows us to study the electronic ground states of the dots. An increased capacitance signal with respect to the background reflects the gate voltage at which single electrons are injected into the dots. In a simple perturbative model of the N-electron ground-state energies the difference in gate voltage between two successive peaks is separated into terms for the electron-electron interaction and, whenever the two peaks are attributed to the filling of energetically
I
,
I
. . . . . . . .
-0.5 0 gate voltage (V)
FIG. 2. Differential capacitance as a function of gate voltage recorded on samples with gate areas A = 14, 89, and 656/zm 2 (from top to bottom) at B=0. The capacitance scale is given for the sample with A = 656/an 2 which has a total capacitance of 0.5 pF at V z = - 1 . 2 V. All other traces have been scaled by the indicated factors (within about 30% accuracy) and offset for clarity. Assuming a uniform density of 100 dots per/zrn: the different gate areas of the samples contain dot numbers from 1400 to 70 000. The amplitude and frequency of the excitation voltage are d V = 4 mV and f=170 kHz, respectively. For the samples with A= 14 and 89 /.an2 the single-electron charging of the p shell can cle~-ly be resolved (denoted by the arrows for A = 89/an2). different single-electron states, for the spatial quantization energy. 4,15 The small capacitance of the self-assembled dot arrays is measured by a high-resolution capacitance bridge? ~ The technique is similar to the one applied by Ashoori et a/. 1'16 The signal at the balance point is detected with a phasesensitive amplifier via an on-chip impedance transformer. Voltage differences at the amplifier are proportional to capacitance values via a conversion factor containing the shunt capacitance of the balance point. The CV characteristics of the sample with the largest area A = 656/zm 2 are measured using a more direct technique which yields current signals proportional to the absolute capacitance. The advantage and principal reason for the high resolution of the bridge measurements is the drastically reduced shunt capacitance and therefore noise level. In the measurements of the small-scale samples the noise level was reduced to about 20 nV at the balance point for a time constant of a few seconds enabling us to resolve capacitance changes as small as 4 aF. All measurements discussed below were performed at liquid-helium temperature ( T = 4 . 2 K) with an excitation amplitude of 4 mV and a frequency of 170 kHz. III. RESULTS AND DISCUSSION A. Quantized energy states of noninteracting dots Figure 2 shows the CV traces of three samples with different gate areas recorded at B = 0. The traces have been scaled by different multiplication factors and are offset for
306
6766
56
B.T. MILLER et al.
clarity. 17 The samples are prepared from the same wafer, diced out of an area of about 20 mm 2. Assuming a uniform density of 100 dots per/xm 2, the number of dots underneath the gates ranges from 1400 to 70 000. At low gate voltage, Vg<-1.2 V, the capacitive signal is determined by the parallel-plate capacitor formed by the back contact and front gate. The increase of the signal at large positive gate voltage, V g > 0.3 V, reflects the charging of a two-dimensional electron gas in the InAs wetting layer. The well-resolved double structure at around V s = - 0 . 8 V arises from the charging of the dots with the first two electrons. We attribute these maxima to the s shell. We expect a fourfold degeneracy of the p shell, and thus four peaks at a higher gate voltage. In previous studies, however, the individual charging peaks of the p shell were not resolved due to inhomogeneous broadening in the dot ensemble investigated with mm 2 large gate areas. 7-9 Figure 2 shows that in our largest sample with A =656/~m 2 the broadening is still too strong for the four peaks of the p shell to be resolved. One broad shoulder around V s = - 0 . 2 5 V is observed. By decreasing the characteristic gate length below about 10 /an, inhomogeneous broadening due to long-range dot size variations is sufficiently reduced so that for the samples with A = 89 and 14 /zm2 the charging of the p shell with four individual electrons is observed. The classification of the capacitance maxima according to the angular momenta of the shells is further confirmed by the magnetic-field dependence of the corresponding charging peaks. Figure 3(a) shows the CV traces of the sample with A = 89/~m 2 for magnetic fields between 0 and 23 T, oriented perpendicular to the sample surface. The curves have been offset for clarity. The double structure of the s shell is only little affected by the magnetic field whereas the four maxima of the p shell exhibit a magnetic-field-dependent splitting, with two peaks decreasing with magnetic field and two increasing for B < 15 T. At small positive gate voltage two more peaks can be identified exhibiting a negative magneticfield dependence for B < 15 T. We associate them to the charging of the d shell with the first two electrons. At around 15 T the magnetic-field dependence of the four highest observed maxima changes. Figure 3(b) displays the gate voltage positions of the dominant features against magnetic field as extracted from Fig. 3(a). To get a rough estimate for the corresponding energy scale we can divide the gate voltage differences by the lever arm (see Fig. 1). Screening of the gate potential by charges in the dots as well as by image charges is neglected in this approximation. The level structure is found to be atomlike with energy states comparable to shells. The many-electron ground states of self-assembled quantum dots are resolved as individual charging peaks for up to eight electrons in the dots. Although deviations from a parabolic confining potential in our self-assembled dots have been observed, 9 we compare for the sake of simplicity the magnetic-field dependence of the data with the Fock model, i.e., a single-particle model which assumes a two-dimensional parabolic lateral confinement. '9 For B = 0 the energy-level diagram consists of equidistant energy states E n , l = (2n + [l I + 1 )hto 0 , where hto 0 is the quantization energy and 1 is the angular quantum number, 1= O, +_ 1, + _ 2 .... (for all states observed in our experiments the quantum number n is zero). The degeneracy of
(a)
B=0 "2
8 o o
>; 11 :" -1
I
-0.5
B = 23 T I
0
0.5
gate voltage (V)
(b)
~ g ..0.4
0
4
8 12 16 20 magnetic field (B)
24
FIG. 3. (a) Differential capacitance of the sample with A = 89/zrn 2 at different magnetic fields applied in the growth direction. The traces are offset for clarity. From top to bottom the magnetic field is increased in steps of 1 T from B = 0 to 23 T. (b) Magnetic-field dependence of the individual charging peaks'extracted from (a). The size of the symbols shows the accuracy for the determination of the peak positions which is about _+5 inV. a state with energy ( m + 1)6to 0 is 2 ( m + 1)with m = 2 n + [l[. As mentioned above we approximate the action of the electron-electron interaction by adding a corresponding term to the Fock states which lifts the degeneracy of the states. According to the quantum number l we have used the term s shell ( l = 0 ) for the double peak at large negative gate voltages, p shell (l = _+ 1 ) for the four maxima at small negative gate voltages and d shell ( / = - 2)for the two maxima at small positive gate voltage. Due to the charging of the wetting layer at V g > 0 . 3 V, we are only able to observe two of the expected six d levels at high magnetic fields. As can be deduced from Figs. 3(a) and 3(b), the Coulomb charging energy depends on the number of electrons in the dots. At B = 0 we measure A V g12 = 132 mV for the two
307
56
FEW-ELEC'IRON GROUND STATES OF CHARGE-...
maxima in the s shell and AV34=61 mV and AVe56= 6 7 mV for the first two and last two maxima in the p shell, respectively. The two peaks associated to the d shell can only be observed for B > 7 T because of the strong increase of the capacitance in the corresponding gate voltage range due to the wetting layer. For B = 7 T we get A V~7s=65 mV. The dependence of the Coulomb charging energy on the number of occupied states can be explained by the effective dot size which changes for different energy states according to the corresponding wave functions. 2"3 Our data show that the Coulomb charging energy of the present self-assembled quantum dots is drastically different for the s and p shell, whereas the difference between the p and d shell is negli~gible. The observed dependence of the Coulomb blockade on the number of electrons in the dots agrees well with a recent many-particle theory by Wojs and Hawrylak in which the few-electron ground states of similar self-assembled quantum dots is calculated. 2~ For the gate voltage difference between the states with two and three electrons in the p shell at B = 0 we measure the large value of A V4.s = 128 mV which is nearly twice as 34 s56 ' 9 large as AVe and A V~ . The observatmn of the enhanced value for A V~5 is corroborated by the strong curvature of the corresponding p levels for B < 2 T, which can be observed in Fig. 3(b). Anisotropy of the dots may partly explain this obse~ation. 2t An additional quantization term due to the anisotropy of the dots will increase A V45 with respect to AV s34 and A V e56 . Far-infrared (FIR) experiments on very similar dots show a splitting of the r + and ca_ modes a t B = 0 which is explained by the anisotropy of the dots. 9 This splitting, however, amounts to about 2 meV corresponding to a gate voltage difference of only about 15 mV. Wojs and Hawrylak predicted that even for isotropic dots A V~g5 should be considerably larger than A V~4 and A V56 due to an exchange-interaction term.zo The filling of the dots with electrons should obey Hund's rules similar to the situation in atomic spectra. For a single lithographically defined quantum dot such a behavior has recently been observed in studies by single-electron tunneling. 3 Similarly, we attribute the large value of A V~g5 in our system predominantly to this exchange interaction. With the gate voltage difference A V et2= 132 mV we can determine the s-shell charging energy of the isolated dot to be 21.5 meV where the lever arm as well as screening by the gates has been taken into account.9~22 With this energy we estimate the characteristic length of the ground state l o and therefore the quantization energy h coo to lo=5.3 nm and h r 0= 44 meV. 23 These values nicely demonstrate that in our system the quantization energy is about a factor or two larger than the Coulomb charging energies. The values for the energies are in very good agreement with the ones obtained by capacitance and infrared transmission spectroscopy on largescale dot arrays. TM We will now analyze in detail the magnetic-field dependence of the capacitance spectra. According to the theory of Fock the energy E0,t depends on magnetic field as
E0~-(l/l+ 1)h~[(Wc/2)2+w~+lhoJc/2,
(1)
where h ~ c = h e B / m * is the cyclotron energy. The slight increase in gate voltage of the structure of the s shell
6767
(1=0) with magnetic field reflects the diamagnetic shift conrained in the first term of Eoj. The splitting of the two branches of the p shell (1 = _+ I ) is expected to be linear in magnetic field according to Eq. (1) with a slope yielding an effective mass of the electron system. For 2 T < B < 13 T we observe a line&,"dependence and can thus extract an effective mass of m* =(0.057• e . The change of the magnetic-field dependence of the highest four observed maxima at a_ro~d B = 15 T can only be explained by a change of the qua.rRum numbers for the corresponding ground states. In fact, calculations of Wojs and Hawrylak confirm this statement, z~ They predict that a magnetic-fieldinduced intermixing of the p and d shell should occur at around B = 15 T in remarkably good agreement with our observation. We can extract a gate voltage difference a t the crossover point of A V~67= 115 mV. It should be mentioned that the Fock energy states--with hr meV and m* =0.057m:--describe the magnetic-field dependence of the experimentally determined capacitance maxima well within the accuracy of the measurements when the different singleelectron states are offset by phenomenological energies corresponding to the Coulomb charging energies and the zerofield splitting of the two branches of the p shell (plot not shown here). The effective mass of m * = ( O . O 5 7 • e as derived from the orbital splitting of the individual charging peaks of the p shell in capacitance measurements is distinctly different from the one obtained from spectroscopic studies. FIR experiments yield a value of m * = ( O . O 8 2 + - O . O O 8 ) m e for a sample from the same wafer. The observation indicates that the Fock model is insufficient for a quantitative description of the experimental results. Considerable nonparabolic terms in the confinement potential might explain the discrepancy between the masses. On the other hand, in a parabolic potential where FIR experiments probe the bare effective m a s s - unaffected by electron-electron interactions--24 in transport measurements electron-electron interactions contribute to the mass, resulting in a dressed mass. 25 For two-dimensional electron gases this effect is well known. Different values for the effective mass were observed for transport and spectroscopic experiments with the dressed mass being slio~htly larger than the bare one.25 However, we observe the contrary for quantum dots where the transport mass is about 20% smaller than the FIR one. It is also important to note that both masses are considerably higher than the conduct~bnband edge mass of InAs, m * = O . O 2 3 m e , and are closer to the one of GaAs, m * = O . O 6 7 m e . This can be explained by the penetration of the dot wave function into the GaAs (Ref. 26) and additional effects of strain and nonparabolicity in k space.9,27 All of the discussion above reflects that we can quantitatively explain the dominant features in the measured capacitance spectra by the shell structure of the electron states in our artificial atoms assuming the dots to be equal and noninteracting. A more complex model involving the. entire dot ensemble, however, is needed to explain small additional structure which can be observed in the CV spectra of the small area samples. B. Ensemble properties The gray scale plots in Figs. 4(a) and 4(b) depict the capacitance as a function of gate voltage and magnetic field
308
6768
56
B. T. MILLER et al.
One possible explanation for the systematically shifted spectra are monolayer fluctuations of the dot heights, Fluctuations where the height for different dots in the ensemble is assumed to vary uniformly by one or more monolayers will essentially result in a discrete offset of the threshold gate voltage. The lateral quantization energy will only be slightly affected. Numerical calculations of the ground-state energies of dots with different geometries support this argument. 2s An energy difference of approximately 6 meV is obtained for a monolayer change of the height assuming both lens-shaped dots and tnmcated pyramidal dots. Another possible cause for the observed fine structure could be the nearest-neighbor interdot Coulomb interaction. Assuming clustering of dots an additional Coulomb energy will be necessary to charge a dot whose nearest-neighbor dot has already been charged. For an estimate of such an interdot interaction energy the nearest-neighbor distance of our dots has to be knowrL Recent results for similar self-assembled quantum dots indicate that this distance is typically twice the dot diameter and often much smaller than the average spachag between two dots as estimated from the dot density, tt~9~ With such small nearest-neighbor distances interdot Coulomb interactions again yield energies comparable to the observed energy difference between the replica.
FIG. 4. Gray-scale plots of the capacitance of the same sample as in Fig. 3(a) for different thermal cycles. Data are recorded up to 13 and 14.5 T, respectively. The background of the raw data has been subtracted to obtain a better contrast. It can clearly be seen that the amplitude of the additional structure changes for different measurements, the energetic spacing between the replica, however, does noL for the same sample as in Fig. 3(a). Between the measurements of Figs. 3(a), 4(a), and 4(b) the sample was thermally cycled. White areas in Fig. 4 correspond to maxima, black areas to minima in the capacitance. Here the background of the raw data is subtracted to obtain a better contrast. The additional maxima within the s, p, and d shells essentially show the same magnetic-field dependence as the dominant ones which, as shown above, correspond to the quantized energy states of noninteracting dots. A thorough investigation of the correlation between these peaks reveals that they can be grouped in sets. Each of these sets is essentially a replica of the spectrum of the main maxima shifted by roughly __.40 mV [corresponding to approximately +_(5-6) meV] or multiples thereof with respect to the main spectrum. Systematic measurements show that warming up and cooling down the sample changes the amplitude of this fine structure. The energetic spacing of the additional structure, however, is highly reproducible, even when different forward and reverse gate voltages are applied during the cooling procedure or when the cool down time is varied. Therefore the replica cannot be explained by a random background potential caused by frozen charges of the intentional or unintentional doping. One may, however, invoke fluctuations of the background charge in our samples to account for the change in the amplitude of the additional peaks with thermal cycling. The energetic fluctuations due to the background charge must be much smaller than the observed spacing of the replica.
IV. CONCLUSION In summary, we employ high-resolution capacitance spec .... troscopy to study the few-electron ground states of selfassembled quantum dots. We resolve Coulomb charging peaks in the p and d shell of the capacitance spectrum. We are therefore able to determine the Coulomb charging energy as a function of electron number per dot. A splitting of the states with two and three electrons in the p shell indicates the importance of the exchange energy similar to Hund's rule for atomic spectra. Furthermore, the orbital splitting of the p shell yields a transport effective mass of m*=(0.057 -0.007)me. This mass is significantly lower than the effective mass deduced from FIR spectroscopic studies. This might be explained by the nonparabolicity of the confinement potential or by many-particle effects that are known to result in different masses in high- and low-frequency try.asport studies, Experiments in high magnetic field up to 23 T show a magnetic-field-induced ground-state transition. The p shell and the d shell intermix at B = 15 T. Below 15 T the fifth and sixth electrons are filled into p-like states which become d like for higher fields. The observed behavior is in very good agreement with model calculations. Additional fine structure is observed in the capacitance spectra of small dot ensembles and discussed in terms of ensemble properties such as interdot interactions and monolayer fluctuations of the dot heights.
ACKNOWLEDGMENTS We would like to thank S. E. Ulloa, A. O. Govorov, Schmerek, and R. J. Warburton for continuous support well as M. Grundmann, S. J. Allen, F. Simmel, and A. Efros for stimulating discussions. We would also like
D. as L. to
309
56
FEW-ELECTRON GROUND STATES OF CHARGE-...
6769
thank Dr. Ponse (Siemens AG, Munich) for supplying us with transistors. We gratefully acknowledge financial support by the DFG, BMBF, and the High Magnetic Field Laboratory, Grenoble. The work in Santa Barbara was funded by
QUEST, a NSF Science and Technology Center. The collaboration between the LMU and QUEST is supported by an EC-US grant and by the Max Planck Society and the Alexander yon Humboldt Foundation.
*Permaneaat address: lnstitut flu" Angewandte Physik, Universit~t Hamburg, Jungiusstr. 11, 20335 Hamburg, Germany. tPermanent address: Hewlett-Packard Labs, 3500 Deer Creek Rd., Palo Alto, CA 94304. I R. C. A_shoori, H. L. Stormer, J. S. Weiner, L. N. Pfeiffer, S. J. Pearton, K. W. Baldwin, and K. W, West, Phys. Rev. Lett. 71, 613 (1993). 92T. Schmidt, M. Tewordt, R. H. Blick, IC J. Haug, D. Pfannkuche, K. v. Klitzing, A. Foerster, and H. Iaieth, Phys. Rev. B 51, 5570 (1995). 3S. Tamcha, D. G. Austing, T. Honda, R. J. van der Hage, and L. P. Kouwenhovem, Phys. Rev. Lett. 77, 3613 (1996). 4M. Kastner, Phys. Today 46(1), 24 (1993), and references therein. 5D. Leonard, M. Krishnamurthy, C. M. Reaves, S. P. Denbaars, and P. M. Petroff, Appl. Phys. Lett. 63, 3203 (1993). 6j. M. Moisofl, F. Houzay, F. Barthe, L. Leprince, E. Andre, and
17It can be seen for the sample with A =89/zm 2 that the gate voltage range for charging the dots with six electrons is smaller compared to the other samples. A similar small gate voltage range is observed for another large-scale sample prepared from a part of the wafer very close to A = 89/an 2 (not shown here). This reflects a slightly larger dot diameter as compared to the other samples and is caused by the intentional gradient of the In flux across the wafer (Refs. 8 and 18). A larger average dot diameter reduces the separation between the energy levels and therefore compre~sses the features in the capacitance spectrum. Is D. Leonard, K. Pond, and P. M. Petroff, Phys. Rev. B 50, 11 687
o. Vatel, Appl. Phys. Lett 64, 196 (1994). 7H. Drexler, D. Leonard, W. Hansen, J. P. Kotthaus, and P. M. Petroff, Phys. Rev. Lett. 73, 2252 (1994). SG. _Medeiros-Ribeiro, D. Leonard, and P. M. Petroff, Appl. Phys.
Lett. 66, 1767 (1995). 9M. Fricke, A. Lorke, J. P. Kotthaus, G. Medeiros-Ribeiro, and P. M. Petroff, Europhys. Lett. 36, 197 (1996). l~ Schmerek, S. Marius, A. O. Govorov, W. Hansen, J. P. Kotthaus, and M. Holland, Phys. Rev. B 54, 13 816 (1996). t t L. Goldstein, F. Glas, J. Y. Marzin, M. N. Charasse, and G. Le Roux, Appl. Phys. Lett. 47, 1099 (1985). 12Q. Xie, A. Madhukar, P. Chen, and N. P. Kobayashi, Phys. Rev. Lett. 75, 2542 (1995). 13M. Grundmann et al., Phys. Rev. Lett. 74, 4043 (1995). 14j..y. l~arzin, J.-M. Gerard, A. Izrael, D. Barrier, and G. Bastard, Phys. Rev. Lett. 73, 716 (1994). lSC. W. J. Beenakker, Phys. Rev. B 44, 1646 (1991). 16R. C. Ashoori, H. L. Stormer, J. S. Weiner, L. N. Pfeiffer, S. J. Pearton, K. W. Baldwin, and K. W. West, Phys. Rev. Lett. 68, 3088 ( 1992).
0994). 19V. Fock, Z. Phys. 4"/,446 (1928). 2~ Wojs and P. Hawrylak, Phys. Rev. B 53, 10 841 (1996). 21C. DaM, F. Brinkop, A. Wixforth, J. P. Kotthaus, M. Sundaram, and J. H. English, Solid State Commun. 80, 673 (1991); S. K. Yip, Phys. Rev. B 43, 1707 (1991). 22A. Lorke, M. Fricke, B. T. Miller, H. I-laslinger, J. P. Kotthaus, G. Medeiros-Ribeiro, and P. M. Petroff, in Proceedings of the 23rd International Symposium on Compound Semiconductors, St. Petersburg, 1996, Inst. Phys. Conf. Ser. 155 (Institute of Physics and Physical Society, London., 1997), Chap. 11, pp. 803-808. 23U. Merkt, J. Huser, and M. Wagner, Phys. Rev. B 43, 7320 (i991). 24Q. p. Li, K. Karrai, S. K. Yip, S. Das Sarma, and H. D. Drew, Phys. Rev. B 43, 5154 (1991). 25j. p. Kotthaus, Surf. Sci. 73, 472 (1978), and references therein. 26F. M. Peeters and V. Schweigert, Phys. Rev. B 53, 1468 (1996). 27C. Gauer et aL, Semicond. Sci. Technol. 9, 1580 (1994), and references therein. 28M. Grundmann (private communication). 29F. Heinrichsdorff, A. Krost, M. Grundmann, D. Bimberg, A. Kosogov, and P. Wemer, Appl. Phys. Lett. 68, 3284 (1996). 3~ P. Kobayashi, T. R. Ramachandran, P. Chen, and A. Madhukar, Appl. Phys. Lett. 68, 3299 (1996).
310
M u l t i d i m e n s i o n a l q u a n t u m well laser a n d t e m p e r a t u r e
dependence
of its
threshold current Y. Arakawa and H. Sakaki
Institute of lndustrial Sciencr Universityof Toky~ Minato-ku, Tokyo 106, Japan (Received 19 January 1982; accepted for publication 23 March 1982) A new type of semiconductor laser is studied, in which injected carriers in the active region are quantum mechanically confined in two or three dimensions (2D or 3D). Effects of such confinements on the lasing characteristics are analyzed. Most important, the threshold current of such laser is predicted to be far less temperature sensitive than that of conventional lasers, reflecting the reduced dimensionality of electronic state. In the case of 3D-QW laser, the temperature dependence is virtually eliminated. An experiment on 2D quantum well lasers is performed by placing a conventional laser in a strong magnetic field (30 T} and has demonstrated the predicted increase of To value from 144 to 313 "C. PACS numbers: 42.55.Px, 73.60.Fw, 78.45. + h, 78.20.Ls The two-dimensional (2D) nature of electron motion in the quantum well (QW) structure introduces several unique features to semiconductor lasers. For instance, the threshold current J,~ of QW lasers is found less temperature sensitive than that of conventional double heferostructure (DH) lasers. ''z Such improved behavior of J',~ is ascribed to the change in the state density Pc(6) of electrons, which is brought forth by the decreased dimensionality of the freeelectron motion from 3D to 2D. Consequently, further improvements are expected if one modifies the form ofp,(6). In this letter, we propose and analyze a new type laser "'the multidimensional (2D or 3D) quantum well (MD-QW) laser'Las an extension of the conventional QW laser, which we call l D-QW laser, hereafter. The most remarkable feature to be shown is that Jth of MD-QW lasers is much less temperature sensitive than that of the ID-QW laser. We show, further, that a conventional DH laser placed in a strong magnetic field behaves as a 2D-QW laser and the observed temperature sensitivi'ty indeed decreases in accordance with our theoretical prediction. Figure l(a) shows an illustration of the active layer in conventional DH lasers, in which the z axis is taken normal to the active layer. 1D-QW lasers are realized by reducing the thickness Lz of the active layer to the order of the de Broglie wavelength 2 c ofcarriers, as shown in Fig. l(b). MDQW lasers are defined as lasers, in which not only the thickness Lz but also the length Ly, and/or the width Lx are reduced down to the order of Ac, as shown in Figs. l(c) and l(d). Although the fabrication of such structures at present is still technically difficult even with the most advanced device technology, 2D-QW or 3D-QW structures can be effectively achieved if we place conventional DH lasers or 1D-QW structures in a strong magnetic field, in which the electron motion is confined in two dimensions, as will be discussed later. To achieve the efficient population inversion and also the efficient optical confinement, a number of mutually isolated quantum wells should be stacked in practice., so that the group of QW occupies the volume identical with the active layer of the conventional DH laser. As the dimension of QW increases from ID to 2D or 3D, the degree of freedom in the free-electron motion de-
939
Appl. Phys.Lett. 40(11)01 June 1982
creases, leading to a change inp,(e). For the (3-0-dimensional electron gas in the/-dimensional QW, p~e) is expressed as follows:
(2m,/~) 31~ p~t~ ----(2~r2----------~
rtIr
p,:~e) = _~. (.r
V6,
(1)
~ re- ~.)],
(2)
(m c/2~) ~lV(~'LvLz)
p~)(e)= ~ [ e - r
-
~n)]'~ '
1
P())(6) = __~/.k(LzLyI./x) ,5[e - 6x(k ) -- ey(l ) -- ez(n) ],
(3) (4)
where mc is the electron effective mass, 6 is the energy measured from the conduction-band edge Ec, h is Planck's constant, H (6) is a unit step function with H (6>0) = 1 and H(6 <(3) --0, and 8(6) is the delta function. 6z(n), 6y(/), and ex(k ) denote the quantized energy levels with the quantum numbers n, l, and k, respectively, over which summation should be carried out. In case the potential barrier is sufficiently high, the quantum levels are given by 6z(n) ---- ('~2"[r2/2mc )(n/Lz) 2, 6y~l) = (?i2~'2/2m)(1/I.,y) 2, and e'x(k ) = (p~r2/2mc )(k /Lx)2. Note thatp(#')(6),p~)(6), andp~)(6) are very different from the parabolic state densityp~)(6). Similar behaviors are also expected for the state density p~(6)
".
idl/I'x (a) (b)
FIG. 1. Illustration of various active layers for the conventional laser (a), and the multidimensional QW lasers. (b), (c), and (d) correspond to 1D-, 2D-, and 3D-QW structures.
0003-6951/82/'110939-03501.00
9 1982AmericanInstituteof Physics
. 939
311
of holes. The analysis of d~ and its temperature dependence of the conventional laser has been done by Lasher and Stern 3 (referred to as LS, hereafter) and by Adams." The LS theory is recently extended to the analysis of a ID-QW laser, s and the steplike nature ofp~'~e) is shown to be mainly responsible for the reduced temperature dependence of Jt~. In the following, we extend the L$ theory further to evaluate the behavior of 2D-QW lasers as well as 3D-QW lasers. If we use "no k-selection rule", ~ the gain coefficient ~ E ) o f the i-dimensional QW laser for the photon energy E can be formally . expressed as
X [ f,(~')--f,,(r -- E l i d e '
(5)
in which n, is the refractive index, c is the fight velocity, Eg is the energy gap, and B ~ is a constant representing the probability of dipole transitions. Consequently, ~4(E ) can be calculated, once the distribution functions~ (el and f , (el of electrons and holes are fixed. To determiner, (el we follow LS and assume the active layer to be heavily doped with acceptots such that the hole concentrationp0 is constant. To determinef~(e) we adjust the quasi-Fermi level in such a way that the maximum gain ~ ( E , ~ ) satisfies the threshold condition, i.e., g~(E,~) is equal to the total optical loss in the laser cavity which we assume to be independent of the temperature. The rate R ~ of the total spontaneous emission is then calculated by the energy integral of the spontaneous emission rate r~ (E) which can be uniquely determined whenp~, p,,f~, and f , are given [see Eq. (6a)ofRef. 3]. By using R ~ at just below the threshold, Jt~ can then be expressed as
J ~ = qdR~,/~l,
(6)
where q is the electronic charge, dis the active layer thickness, and 7/is the quantum efficiency. The threshold current Jth thuS formulated can be calculated only numerically in general, but analytical expressions can be obtained for 1DQW and 3D-QW lasers as given by j ( i l ~_.
,h
qd
mc poB(')kTln( 1 + Q ), ~h2Lz
91 9di2Lz P~
\Po ~li2Lz]'
(7)
(at high temperature)
jo,
qd(at3'Vlo) ~ + -r - p p ~ o,,
'~ = ---v
(8)
9where V = LxLyLz, A ('~== (Ir~c~tP/n2,Eg2)B~o, Q = I [ ( v ' c ~ + ( D - I)'~(I + C)]/(1 + C - CD)}~, D = exp [ a(')(~rr2Lz}2/A ~ ], C = 1/[ exp(po.mrr~Lz/m,kT) - 1 ], and m, is the hole effective mass. Equations (7) and (8) indicate that Jth of a 1DQW laser is proportional to Tln(T/const) near room temperature, whereas Jth ofa 3D-QW laser is independent of T. We have also calculated numerically Jth for GaAs/AIGaAs 2D-QW lasers as well as for conventional DH lasers. We 940
Appl. Phys. Lett., Vol. 40, No. 11,1 June 1982
_ Jth(TL - j~,-expi~}
~ 1.5 ~= "o
$ 1.0 I
.=.
c
l
~
(al % =104.
~/ I,,)
,
I
-40
to! 10 (al
i
"C "C "C "C
3; = 2 8 5 T. =4.81 "1; = o o
I
9 I
-20
0
,
I
20
l
I
4.0
, A
60
temperature ( "C ) FIG. 2. Numerical example of threshold current J ~ calculated by extending the model of Lasher and Stern for conventional lasers (a) and quantum well lasers for (b) ID-, (c) 2D-, and (d) 3D-QW structures. Jth is normalized byJ,u =t0"C,
assume here that electrons populate only in the ground subband, which is valid when Lx, Ly, and Lz are chosen sufficiently small. The results for T near room temperature are summarized in Fig. 2, and clearly show that the temperature dependence of Jth depends drastically on the dimension of quantum well. If we express the result in terms of Jth = .Toexp(T/To), To for 1D, 2D, and 3D are equal to 285 "(2,481 "C, and infinity (oo), respectiv'8Iy, and exczcds by for Jr, (104 "C) of conventional DH lasers. To understand the reason for such dramatic increase of T~ we like to remind here that the temperature dependence o f Jth of the conventional GaAs-Ga~dAs laser is ascribed to the thermal spreading of the injected carriers over a wider energy range of states, which leads to decreases of the maximum gain g(E~x ) at a given injection level. Consequently, in 1D-QW lasers, where p~)(E ) and p~)(E ) are steplike, the effect of such thermal spreading is expected to be smaller? In case of 2D-QW lasers, one expects further suppression of the temperature effect becausep~)(E ) has a peaked structure and is a decreasing function of E. In 3D-QW lasers, the thermal spreading of carriers should vanish because the state density is delta-functionlike. Hence, the temperature dependence of Jth will totally disappear, as long as the electron popula~on in higher subbands remains negligibly small. We consider next the possibility of demonstrating this unique feature of the MD-QW laser experimentally. In view of the technical difficulties of fabricating such structures, we have investigated the conventional GaAs-GaA1As DH laser placed in a strong magnetic field. As we have used a channelled substrate planar (CSP) laser with the nondoped active layer having relatively high carrier mobility, and applied the high (pulsed} magnetic field Bz up to 30 T perpendicularly (llz) to the active layer,/~B ( = ~0cT)is much greater than 1. Hence carriers in the active layer are expected to complete their cyclotron orbit. 6 The motion of such electrons is known Y. Arakawa and H. Sakaki
312
1.5 Jth(T)
II JC
I-r
8 ~ .
u
El
1.0
O
O
II
! II
(,i O
B = 24 T
( T. = 313 "C ) F (bl
II
B
=
0
T
( T. = i ~ ' C }
0"5
J
i
-40
, j
I
-20
,,
I
9
i
0 20 temlx~ature { "C )
,
i
.
40
HG. 3. Temperaturedependenceof thresholdcurrent 3'0. with and without magnetic fieldB (24 T)../'0. is normalized by J'o, at 0 "C, which is 52 mA at B - - 0 and 54 mA at B - - 24 T.
313 "C, which is in fair agreement with our prediction. A nonlinear behavior of In J,h with respect to Tis possibly due to the contribution from higher Landau levels; we leave the detailed description of the experiment and its interpretation as the subject of a separate paper and simply note here that the mean time between collision is much larger than 1/~oc (since co, ~-> 1), and that r c term is greater than k T when B exceeds 20 T. Hence, the two-dimensional confinement of carriers by the field Bz is effective, and introduces the peaked structure in the state density although there are some contributions from higher subbands. In summary, we have proposed a new type multidimensional Q W laser and have shown theoretically that the dramatic increase of T is expected. An experimental proof of such prediction has been successfully done for a 2D-QW laser by placing a conventional laser in a strong magnetic field. We wish to express our sincere gratitudes to Professor N. Miura and Dr. G. Kido for allowing the use of pulse magnets, Dr. M. Nakamura and Dr. K. Aiki of I-Iitachi Ltd. for supplying the laser diodes, Professor J. Hamasaki and Professor Y. Fujii for their support and encouragement, and Mr. M. Nishioka for his excellent technical assistance. The work is supported by the Ministry of Education, Science, and Culture.
to be quantized in the two transversal directions (x andy) and forms a series of discrete Landau levels [ E = '&o=(n + 1/2) + ( ~ / 2 , n < ) c z ;
<.<,== ~lql B
Ira, ].
Hence, the conventional laser placed in a strong magnetic field can be regarded approximately as a 2D-QW laser, which is indeed demonstrated quite recently by the independent work of Bluyssen. 7 We, therefore, have measured the temperature dependence of./th with and without the magnetic field B and the results are shown in Fig. 3. While To is found to be 144 "C for B -- 0, T o for B -- 24 T is shown to increase especially at lower temperature. When averaged (in the sense of least-mean-square fitting) over the temperature range (230--. 300 K) To for B = 24 T is determined to be
'R. Chin, N. Holonyak,Jr., and B. A. Vojak,Appl. Phys. Lett. 36,19 (1980). iN. Holonyak,Jr., R. M. Kolbas,11.D. Dupuis, and P. D. Dapkus, IEEE J. Quantum Electron. QE-16, 170 (1980). 3G. Lasher and F. Stem, Phys. Rev. 133, A553 (1964). "M. J. Adams, Solid-StateElectron. 23, 585 (1980). SK. Hess, B. A. Vojak, N. Holonyak,Jr., R. Chin, and P. D. Dapkus, SolidState Electron. 23, 383 (1980). eR. B. Dingle, Proc. R. Soc. London A 211, 517 (1952). 7H. J. A. Bluyssenand L J. van Ruien, IEEE J. Quantum Electron. QE-17, sso{1981). W. Arakawa, H. Sakaki, M. Nishioka, G. Kido, and N. Miura (unpublished).
I I
313
ways for tn situ quantum dot fabrication. In this work the authors present photoluminesoenee (PL), electroluminescenee (EL) and lasing e h a r a e t e ~ of (In,Ga)As-GaAs quantum dots formed on a GaAs (I00)surface.
Experiment:The structures are grown by elemental source moleculax beam r (MBE) on G a A s (100) substrates. Two laser structures referred to as A and B in the following, with different average deposited thickness t,. of Ino.sGa0_d,s, are compared in this work. They are grown on Si-doped subsh'-ate at 6000C and represent double heterostructure lasers with p- and n-type 1,51,tm th'r.k Ale~Gae.~As cladding layers. The GaAs waveguide layer is 0.2rim (A) and 0.1ttm (B). Short superlattices are introduced to provide better morphology. The Ino,sGao~As deposition is performed at 460"C (A) and at 490~
(13)in a submonol-
ayer growth mode [6] using 0.1rim growth cycles. Formation of dots started after the deposition of a 0.9nm thick In0.sGao.dts layer and leads to the transformation of a 'streaky' reflection highenergy electron diffraction (RHEED) pattern to a 'spotty' pattern. For s ~ B, InozGao_sAs deposition is stopped at the initial stage of layer transformation to dots (t,. - Inm). For structure A, deposition is continued up to t,, = l.Snm. A 10rim thick CmAs contact layer is grown on the top of both structures. Samples for TEM studies arc prepared in a sirmqar manner at 460"C without a cladding layer and superlattice buffer layer. Planview TEM studies are performed using a JEOL JEM 1000 (1 MV) high voltage electron microscope.
Fig. 1 Schematic laser structure and shallow mesa stripe geometry
The fabricated shallow mesa stripe laser geometry is schematically shown in Fig. 1. The laser cavity length is 10001am and the stripe width 10 and 201am for structures A and B, respectively. Photoluminescenec (PL) is excited using the 632.8nm line of an He,-Ne laser and d e t ~ l using a cooledgermanium photodetector.
Low threshold, large To injection laser emission from (lnGa)As q u a n t u m d o t s N. K i r s t a e d t e r , N . N . Ledentsov, M. G r u n d m a n n , D, Bimberg, V . M . Ustinov, S.S. R u v i m o v , M . V . M a x i m o v , P.S. K o p ' e v , Zh.I. Alferov, U. Richter, P. W e r n e r , U. G6sele a n d J. H e y d e n r e i c h
Indexing terms: Semh:onductor junction lasers, Semiconductor quantum dots
Low threshold, large To injection laser emission via zerodimensional states in (InGa)As quantum dots is demonstrated. The dots are formed due to a morphological transformation of a pseudomorphie Ino.sGae.sAs layer. Laser diodes are fabricated with a shallow mesa stripe geometry. Introduction: Quantum dot lasers are expected to exhibit properties much superior to quantum well based devices, such as reduced temperature sensitivity [1]. The self-organisation of nanoscale 3-D clusters due to the strain-induced transformation of the pseudomorphic layers [2-5] has emerged as one of the most promising
1416
Fig. 2 Pldn-view TEM images of Ino.sGao.sAs quantum dots in GaAs
matrix
Average thickness of In0.sGa0.r,As deposited : a lnm b 2.2nm Experimental results: At the initial stage of dot transformation (t,,
= Into, Fig. 2a) a small scale granular structure with details up to 10nm in size as well as their agglomerations of 20-50nm lateral
ELECTRONICS LETTERS
18th August 1994
VoL 30
No. 17
314
III
I
I!
II
I
I
[I
I
I
I II
I
extent are present. For t,,, = 1.6nm coherent and elongated islands of size lO-30nm are formed. A further increase of Ir~sGa~sAs deposited up to t,, = 2.2nm (Fig. 2b) results in the formation of well developed dots, with --70% of them having a size between 12 and 17nm. Thus for laser structure A the strain induced island formation results in larger, better developed and more uniform islands than for structure B.
9
;-0
:~ >, .c_
sincjlemoded ~ 32mA:d~7 B
1-314
5o 100 150 temperotur e, K
7 K r -1-316
energy.eV
.... ~ .-" . . . .
--
9-~
PL ...- ....... 9..... .-_
In laser structure B, the characteristic temperature is 7", = 330 K between 77 and 140K and the threshold current density is equal to 120A/cm 2 at 77K. In the range 150 - 300K the temperature dependence on the threshold current is much stronger, described by a much lower 7", of 120K. We attribute this dramatic decrease of 7", above 150 K to the thermal excitation of carriers in the GaAs barrier region, which increases the injected carrier density necessary to maintain a given threshold gain.
f ' _ Y
;- O l - ~ t ~ I, I ,--.-
.
t 300
1OO 200 ternperott.~'e,K -. lnmQW F_.g~OAS)
"-. , PL
1.3
range 50 - 120 K. This is much higher than the theoretical prediction of 285K for quantum well lasers [I]. However the threshold current is observed to increase dramatically for temperatures above 130 K. We attribute this effect to the non-optimised growth conditions which result in a x5 lower PL intensity as compared to structure B and do not permit room temperature laser operation.
I'
1-4
T
1-5
energy.eV ~g. 3 Electrolmninescence spectra below and above threshold ....
Conclusion: We have demonstrated lasing from (lnGa)As quantum dots in a (Ga,AI)As injection laser structure at room temperature with a threshold current density of 950A/cm 2. A To value of 350K at low temperatures between 50 and 120K and a threshold current density of 120A/cm" are found.
PL spectra at low excitation density (1W/cm"); arrows denote energetic position of GaAs bandgap and Into thick uniform Ine.sGae~,s/GaAs quantum well
PL and EL spectra at 77K for both laser structures are shown in Fig. 3. The calculated E I - H H I optical transition energy assuming a uniform Into thick lntsGa,.sAs layer is 1.42eV and more than lOOmeV higher than all observed luminescence features [6--8]. Similar results [5] are obtained from a comparison of the PL spectra of l~.sGaQ.~As dots (t~, = 1.2nm) and an lrt,.sGa,.sAs SQW of 4nm thickness grown on GaAs (100). The quantum dot induced PL peak energy for laser structures A and B at low excitation density is nearly independent of the average thickness of ln0.~Gao..~As because relatively large dots or agglomerations of smaller dots are already formed at the initial stage of layer transformation in agreement with the TEM data. Non-equilibrium carriers are also more effectively trapped by larger dots. Further Ino.sGao.~s deposition results in an increase of the number per unit area of large and uniform dots rather than in an increase in their size, as can be seen in Fig. 2. In laser structure A (well developed larger dots), the EL intensity maximum is at the low energy side of the PL spectrum. This shift is typical for quantum well lasers and indicates pronounced self-absorption in the waveguide geometry. Fig. 3 shows that an increase in injection current density results in a small blue shift of the EL maximum but lasing remains on the low energy side of the PL spectrum. This is direct proof that lasing occurs through the zero-dimensional states in uniformly developed quantum dots. In laser structure B (initial stage of dot formation) the EL maximum coincides with the maximum of the PL spectrum indicating only minor self-absorption in the waveguide geometry. The EL maximum shifts towards higher energies with increasing injection current density (this shift is also found for the PL maximum) and the lasing occurs on the high energy side of the PL spectrum. This effect is not observed for structure A and is explained by gain saturation for larger dots in structure B because there axe fewer per unit area. The difference in energy between the laser emission and the E I - H H I transition of a uniform Into-thick ~ . ~ t s layer still exceeds 100meV. Therefore we conclude that the smaller dots, characteristic of the initial stage of layer transformation, are responsible for lasing in this case. The high resolution gain spectrum in Fig. 3 shows single longitudinal mode oscillation with a sidemode suppression ratio of 6dB and more than 50% of the lashag intensity in the central mode. This is not typical for broad area quantum well lasers and reflects the very narrow gain function expected in quantum dot lasers. The temperature dependence of the threshold current density for laser structure A is shown in the inset of Fig. 3. The characteristic temperature To is measured to be 350K in the temperature
ELECTRONICS LETTERS 18thAugust 1994 Vol. 30
Acknowledgments: Part of this work is supported by the Volkswagen Stiftung. Germany and the D F G in the. framework of SFB 296.
O IEE 1994 Electronics Letters Online No: 19940939
29 June 1994
N. Kirstaedter, N. N. Ledentsov, M. Grundmann and D. Bimberg
(lnstitut ffir FestkOrperphysik. Technische Hardenbergstr. 36. D-10623 Berlin, Germany)
Universitdt
Berlin,
V. M. Ustinov, S. S. Ruvimov, M. V. Maximov, P. S. Kop'ev and Zh. I. Alferov (A. F. loffe Physical-Technical Institute of Academy of Science, 194021 Politekhnicheskaya 26, St. Petersburg, Russia) P. Wemer, U. GOsele, J. Heydenreich and U. Richter (Max.Planck-
Institut far Mikrostrukturphysik und Labor far Elektronenmikroskopie in Naturwissenschaft und Medizin, Weinberg 2, D-06120 Halle(S), Germany) References 1 ARAKAW^,Y., and SAKAKI.H.: 'Multidimensional quantum well laser and temperature dependence of its threshold current', Appl. Phys. Lett., 1982, 40, pp. 939-941 2 MO, Y.M., SAVAGE.D.E., SWAR'I2ENTRUBF.R,B.S., and LAGALLY,M.G." 'Kinetic pathway in Stranski-Krastanov growth of C-e on Si(001)', Phys. Rev. Lett.o 1990, 65, pp. 1020-1023 3 GRUNDMAHN,M., CHRISTEN,J., LEDENTSOV,N.N., BOHRER,J., mMU~O. D,, XUVIMOV,SS., W~RN~ V., x~ocrBx, u., C,~EL~ U., ~ENREICH. s US'ITNOV,V.M., EGOROV,A. YU., KOP'EV.P.S., and ALFEROV,ZH.I.: 'Ultranarrow luminescence fines from single quantum dots', to be published in Phys. Rev. Lett. 4 TEILSOFF,J., and TROMP,R.M.: 'Shape transition in growth~ of strained islands: Spontaneous formation of quantum wires', Phys. Rev. Lett., 1993, 70, pp. 2782-2785 5 LEONARD,D., KRISHNAMURTWf,M., REAVF..S,C.M., DENBARS,S.P., and Pe'rROFF. P.M.: 'Direct formation of quantum,sized dots from uniform coherent islands of InGaAs on CraAs surfaces', AppL Phys. Lett., 1993, 63, pp. 3203-3205 6 wANG,P.D., LEDENTSOV.N.N., SOTOMAYORTORRES,C.M., KOP'EV,P.S., and USTI~OV.V.M.: 'Optical characterization of submonolayer and monolayer InAs structures grown in a GaAs mattrix on (100) and high-index surfaces', AppL Phys. Lett., 1994, 64, pp. 1526-1528 7 vn~N, R.K., and 8,~% ^.c.: "Strained-layer superlattices: Physics, semiconductors and sernimetals' (Academic Press, Inv. Boston San Diego New York London Sydney Tokyo Toronto 1990, 32), pp. 17-53 8 HELLWEGE,K.-H.: 'Numerical data and functional relationships in science and technology', /n: 'Landolt-B6rnstein new series groups III-V' (Springer-Verlag Berlin Heidelberg New York 1982, 22a), pp. 82-144
No. 17
1417
315
PHYSICA!] ELSEVIER
Physica E 3 (1998) 112-120
I
Itl
It
I
I"111
I I IIIII ~jlll
II
I
I
Electronic states in quantum dot atoms and molecules S. Tarucha a,,, T. Honda a, D.G. Austing ", Y. Tokura a, K. Muraki a, T.H. Oosterkamp b, J.W. Janssen b, L.P. Kouwenhoven b a NTT Basic Research Laboratories, 3-1, Morhtosato Wakamo,a, Atsugi-shi, Kanagawa 243-4)124, Japan bDepartment of Applied Physics and DIMES, Dell? Univetwit.I, of TechnohJq.l'. PO Box 5046, 2600 GA Delft, The NetherlamL~"
Abstract We study electronic states in disk-shaped semiconductor artificial atoms and molecules containing just a few electrons. The few-electron ground states in the artificial atom show atomic-like properties such as a shell structure and obey Hund's rule. A magnetic field induces transitions in the ground states, which are identified as crossings between single particle states, singlet-triplet transitions and spin polarization. These properties are discussed in conjunction with exact calculation in which the effect of finite thickness of the disk is taken into account. An artificial molecule is made from vertically coupling two disk-shaped dots. When the two dots are quantum mechanically strongly coupled, the few-electron ground states are de-localized throughout the system and the electronic properties resemble those of a single artificial atom. 9 1998 Elsevier Science B.V. All rights reserved. PACS: 73.20.Dx; 72.20.My; 73.40.Gk Keywords: Artificial atom; Artificial molecule; Quantum dot; Excitation spectroscopy; Coulomb oscillations
I. Introduction Semiconductor quantum dots are often referred to as artificial atoms since their electronic properties, for example the ionization energy and discrete excitation spectrum, resemble those of real atoms [ 1,2]. We have recently fabricated a circular disk-shaped quantum dot, and observed atomic-like properties by measuring Coulomb oscillations [3]. Electrons bound to a nuclear potential experience sufficiently strong quantum mechanical confinement and mutual Coulomb in* Corresponding author. Tel.: +81 462 40 3445; fax: +81 462 40 4723" e-mail:
[email protected].
teractions that they are well arranged in ordered states. This leads to the ordering o f atoms in the periodic table. The ionization energy has large principal maxima for atomic numbers 2, 10, 18 . . . . when certain orbitals are completely filled with electrons. In addition, for the filling of electrons in similar orbitals Hund's ~ l e favours parallel spins until the set of orbitals is halffilled. This also gives rise to secondary maxima in the ionization energy. The disk-shaped quantum dot we can fabricate is formed in a laterally gated double barrier structure, and contains a tunable number of electrons starting from zero. Associated with the rotational symmetry of the lateral confinement, we observe a two-dimensional ( 2 D ) " s h e l l structure" from the addi-
1386-9477/98/$ - see front matter 9 1998 Elsevier Science B.V. All rights reserved. PII" $1386-9477(98)00225-2
316
s. Tarucha et al. / Phrsica E 3 (1998) 112-120
tion energies, analogous to the three-dimensional (3 D ) shell structure for atomic ionization energies. In addition, spin effects such as a pairing o f Coulomb oscillation peaks due to spin degeneracy, and modifications of the pairing in line with Hund's rule are all observed. In real atoms, electrons are so strongly trapped that their quantum mechanical properties are not accessible by means of conventional spectroscopic techniques. In contrast, the electrons in our quantum dot are bound in a relatively large region of the order o f 100 nm. This allows us to use readily accessible magnetic fields not only to identify the quantum mechanical states, but also to induce transitions in the ground states which are expected but never tested in real atoms on earth [4]. In this paper, we first discuss the addition energy spectrum of the ground states at zero magnetic field, and magnetic field induced transitions in the ground states for a different number o f electrons, N, in a diskshaped dot. We employ an exact diagonalization technique incorporating many body interactions, and the effect of a finite thickness of the dot disk, to understand the magnetic field induced transitions of ground states. The effect o f finite thickness weakens the Coulomb interactions relative to the quantum mechanical confinement. In our previous calculation [4] we neglected this effect, and assumed that the Coulomb interactions were weaker than those reproduced in the experiment. We show here that a good agreement with experiment is now obtained with a realistic interaction model. For the next set after artificial atoms, we outline how vertically coupled disk-shaped dots can be employed to study the filling of electrons in artificial molecules. We show that in a quantum mechanically strongly coupled double dot system the electronic states are delocalized.
Fig. !. (a) Schematic diagram of the device containing a diskshaped dot. (b) Schematic energy (horizontal axis) diagram along the vertical axis of the pillar. Hatched regions are occupied electron states in source and drain contacts. For the case shown, two electrons are permanently trapped in the quantum dot. The third electron can choose to tunnel through the N = 3 ground state (solid line) or through one of the two excited states which lie in the transport window. (c) Schematic of tunneling current vs. gate voltage for a sufficiently small ~d (dashed line) that only the ground state contributes to the current, and for a larger ~d (solid line) that allows both the ground and excited states to contribute to the current. For the case of the large ~, a small jump in the current stripe identifies where an excited state enters the transport window.
!13
2. Device fabrication and experimental set up Fig. l a shows a schematic diagram o f the device which consists o f an n-doped GaAs substrate, with undoped layers o f 7.5 nm A10.22Ga0.7sAs, 12 nm In0.05Ga0.9sAs, and 9.0nm A10.22Ga0.TsAs, and a 500 nm n-doped GaAs top layer. A sub-micronmeter pillar (geometrical diameter, D) is fabricated using
317
114
s. Tarucha et al. / Phvsica E 3 (1998) 112-120
electron beam lithography and etching techniques [5]. Source and drain electrical wires are connected to the top and substrate contacts. A third wire is attached to the metal that is wrapped around the pillar. This electrode is the side gate. The energy landscape along the vertical axis is shown in Fig. lb. The ln0.05Ga0.~sAs layer has a disk shape. By making the gate voltage, 1~, more negative we can electrically squeeze the effective diameter of this disk from a few hundred nanometers down to zero. Application of a bias voltage, ~ , between the source and drain opens a "transport window" between the Fenrti energies of the source and drain for detecting both the ground and excited states in the dot (Fig. l b). Ground and excited states lying within the transport window can contribute to the current. If the gate voltage is made more positive, then the levels in Fig. I b shift down in energy. When ~d is smaller than the energy difference between the ground and lowest excited state, only the ground state contributes to the current, 1, because the electron tunneling into the excited state is blocked by the charging of the ground state. This is the usual case for the measurement of Coulomb oscillations, so we see a series of current "peaks" as a function of gate voltage corresponding to the one-by-one change of electrons in the ground states of the dot (Fig. lc). When l~ is large enough, however, both ground and excited states can be within the transport window, and contribute to the current. Electron tunneling into the excited states can occur following the electron escape from a ground state. 1 vs. F~ therefore becomes a series of current "stripes". Small jumps inside the stripe measure when excited states enter the transport window. Each current stripe falls off when the ground state leaves the transport window, so it has a width in energy given by e(d. We employ this technique to study the excitation spectrum. Our samples are measured while mounted in a dilution refrigerator. Due to pick-up of noise the effective electron temperature is about ! 00 inK.
3. Electronic states in quantum dot atoms 3.1. Grottnd state . w e c t r o s c o p y
Fig. 2a shows I vs. ~ for single quantum dot (D = 0.5 l.tm) measured at a small ~d of 0.15 mV so that only the ground states contribute to the current.
Fig. 2. (a) Change of electrochemical potential vs, electron number tbr a single dot (D = 0.5 }.tinop.,cncircles) and also for two strongly coupled dots (D=O.561am solid circles; .~e Section 4 in the text). The inset shows the Coulomb oscillations measured for gd=0.15mV at B=OT for the single dot. (b) Change of the electrochemical potential calculated tbr the single dot shown in Fig. 2a. Three different h~,~0 values of 3. 4 and 5 meV are taken as parameter. The experimental data for the single dot in Fig. 2a is also shown by the crosses for comparison.
The current oscillations arise from the one-by-one change of electrons trapped in the dot. The absolute values of N can be identitied in each zero-current region (Coulomb blockade region) between the peaks, starting from N = O, because for ~ < - 1.6 V the dot is empty. When N becomes smaller than 20, the oscillation period depends strongly on N. In contrast, Coulomb oscillations observed for a large dot containing more than 100 electrons look very periodic (data not shown; see Ref. [6]), as expected from classical Coulomb blockade theory. The current peak to the left o f a Coulomb blockade region with N trapped electrons thus measures the N-electron ground state energy (or electrochemical potential, I t ( N ) , of the Nelectron dot). For example, the first, second and third peaks from the left measure the one, two and three electron ground state energies, respectively. The peak
318
S. Tarucha et al. / Physica E 3 (1998) 112-120
spacing labeled by " N " therefore corresponds to the energy difference ll(N + 1 ) - F ( N ) between the N and N + 1 electron ground states. This energy difference, which can also be determined from measurement of the widths of the so-called "'Coulomb diamonds" [2], is plotted as a function of N in Fig. 2a. In correspondence to the spacings between the Coulomb oscillations, the energy difference is unusually large for N = 2, 6 and 12, and is also relatively large for N = 4, 9 and 16. The values of 2, 6 and ! 2 arise from the complete filling of the first, second and third shells, respectively, while those of 4, 9 and 16 are due, respectively, to the half filling of the second, third and fourth shells with parallel spins in accordance with Hund's rule [3]. We compare the data in Fig. 2a with an exact calculation for N = I to 7 shown in Fig. 2b. This calculation incorporates the effect of the finite thickness of the disk. We note that this effect weakens the Coulomb interactions in the dot, and thus makes the shell structure more visible. The thin disk thickness freezes the electrons in the lowest state in the vertical direction. We therefore only have to consider the confinement in the plane of the disk for which we take a parabolic potential V(r) = ~i m * % r 2 , where m* = 0.06m0 is the effective mass of electrons in the lnGaAs disk, ~o0 is the characteristic frequency of the lateral confinement and r is the distance measured from the center of the disk. Details of the calculation technique are given in Ref. [7]. The strength of the Coulomb interactions can be represented by a parameter Q = e 2/~:1o, where /0 ( = V/h/m'to0) is the spatial extension of the lowest state's wave function for parabolic confinement. As to0 becomes large, or as the quantum mechanical confinement becomes strong, the strength of Coulomb interactions relative to that of quantum confinement ( = Q / h t o o ) becomes progressively small ( , t~ ) [8]. Consequently, as h.too varies from 3 to 5 meV in the calculation, the N - - 2 and 6 peaks linked to the complete filling of shells become significantly large as compared to the N - 4 peak. If we inspect carefully the peaks relative to the background, we find that the agreement with the experimental data is good for/~co0 > 4 meV when N ~<2, and becomes better for a smaller ht')0 value as N increases. Note that in the calculation the N = 2 peak is almost missing for h~,)0--3 meV and the N = 6 peak is too high for h~'90 = 5 meV to reproduce the experiment. These arguments on hog0 are also supported by the experiment
115
Fig. 3. !( It, B) for N = 0 to 5 measured with small V...~= O.! mV such that only ground states contribute to the current. Different types of ground state transitions are indicated by different labels. The arrows in the squart.~ indicate the spin configuration. The lowest square corresponds to a single particle state with angular momentum / = O. For squares to the right ! increases to i, 2, 3, etc. For N =4 and 5, near B=OT, also the I = -1 square is shown to the left of the ! = 0 .square.
on the B-field dependence of the Coulomb oscillation peaks (see Ref. [3] and Section 3.3 ). Decrease of ho)0 with increasing N can be explained by the effect of Coulomb screening from the leads and gates which is not incorporated in the calculation [9]. We note that the background relative to the peaks is significantly smaller in the calculation than that in the experiment. This can also be due to the screening effect. Fig. 3 shows the magnetic field dependence of the first five current peaks for ~ =0.1 mV. The single dot device is similar to, but not the same as, that used for the experiment of Fig. 2a. The B-field dependence of the peak positions in gate voltage reflects the evolution of the ground state energies. Besides an overall smooth B-field dependence, we see several kinks, which we indicate by different labels. As we discuss in the following section, these kinks are assigned to transitions in the ground states, so for the regions between the kinks, we can identify the quantum numbers, including the spin configurations.
319
116
X Tarucha et al. / Physica E 3 (1998) 112-120
Fig. 4. I(I~,B) for N =0 to 2 measured with ~ = 5mY up to 16T. /<0.1 pA in the dark blue regions and / > 10pA in the dark red regions. Both the ground state and the first few excited states can contribute to the current. Current stripes between the Coulomb blockade regions (black) for N = 0 and ! electrons, and for N = i and 2 electrons are the first and second current stripes, respectively. The states in the first stripe are indexed by the quantum numbers ( n. I).
Fig. 5. Exact calculations of the ground states (thick lines) and excited states (thin lines) for N = I and 2. The N = ! states are indexed by single-particle eigenenergies En. t. The N = 2 many-body states are indexed by the total spin and total angular momentum (boxed number). The lowest thick line of each stripe is the ground state energy. The upper thin dashed line of each stripe is the ground state energy shifted upwards by 5 meV. The solid and dotted lines indicate S = 0 and I configurations, respectively. The label A is discussed in the text.
3.2. E x c i t a t i o n spectroscopy
state (lower edge of the stripe). In the second stripe of Fig. 4, however, we see the first excited state cross with the ground state at B = 4.15 T, i.e. the first excited state for B < 4.15 T (seen as the current change from blue to red inside the second stripe) becomes the ground state for B > 4 . 1 5 T. Located exactly at this magnetic field is the kink labeled by T in Fig. 3, so it is assigned to a crossing between the ground state and the first excited state, in a similar fashion, we are able to identify a crossing between the ground state and the first excited state corresponding to each kink
To investigate what kind o f many-body states are responsible for the kinks observed in Fig. 3, we measure I vs. Vgfor a large V~dof 5 mV. The data are shown in Fig. 4 for N --- 1 and 2. For this particular voltage, the two stripes just touch at B = 0 T. A pronounced current change, as indicated by the colour c h a n g e from dark blue to red (i.e. from < I p A to > 10pA), enters the upper edge of the first stripe at B -- 0.2 T. This change identifies the position of the first excited state for the N -- 1 dot (we discuss the index Eo, i below). Note that at higher B values two higher excited states also enter from the upper edge of the stripe at 5.7 and 9.5 T, respectively. The energy separation between the ground state and the first excited state can be read directly from therelative position inside the stripe. So, the excitation energy (--hOgo) is slightly larger than 5 meV at B = 0 T and decreases for increasing B. Note that even over this wide magnetic field range of 16 T, the first excited state never crosses with the ground
in Fig. 3 [4].
3:3. E x a c t cak'ulation o f m a n y - b o d y states For a few electrons the energy spectrum can be calculated exactly [ 10]. Fig. 5 shows exact calculations of the electrochemical potential vs. B for the N = 1 and 2 ground states (thick lines) and excited states (thin lines). W e use hco0 of 5.5 meV and make the same assumptions as those taken above in Section 3.1. For
320
S. Tarucha et aL/ Physica E 3 (1998) 112-120
each stripe the lower thick line is the ground state, and the upper thin dashed line is the lower thick line shifted upwards by 5 meV. This indicates the transport window for Vsd= 5 mV. For the first electron the exact calculation gives single particle states. The eigenenergies with radial quantum number n = 0, 1,2 .... and angular momentum quantum number l = 0, + !, :1:2.... are given by [11]
32~-
117
'
'
'
::'1'-".r
9 ." " _ . J-.~. ...'" -/_'.-.:. I~ "-~..:..:IS"/ .,,s"
",,. :'~ k.[~l
:/'.:"'.."'" S~ S
28
~ g e~ ~ c| o "~
24 i
28
l
....
i
.....
!
i
.....
1 ......
!
S=2
t
|
.................... i!!::::::
N= 3
where the cyclotron frequency COc= eB/m*. (We neglect the much smaller Zeeman energy.) The states lying within this transport window can be compared with the observed current changes seen in the first stripe of Fig. 4. The agreement is very good for both the ground state and the first excited state over the whole B-field range. For the N = 2 case, many excited states are lying within the transport window. We index the ground and excited states by the total spin, S, and the total angular momentum, M (boxed number). The solid and dotted lines indicate the S = 0 (spin singlet) and S = 1 (spin triplet) states, respectively. Whilst the _ N = i ground state E0.0 never crosses with the first excited state, we see a crossing (labeled by 9 between the ground state with (S, M ) = (0, 0) and the excited state with ( S , M ) = (1,0), which is referred to as singlet-triplet transition [12]. For h~o0 = 5.5 meV this singlet-triplet transition is expected at B--4.0 T, which is in good agreement with the experimental value in Figs. 3 and 4. Note that the calculated excited states with (S, M ) = ( 1, - 1) and (0, 2) for N = 2 can also be seen in the second stripe of Fig. 4 (i.e. the lines between blue and red current regions which show a maximum near -,-2 T). The ( 1, - 1) is located ~3 meV above the ground state of (0, 0) at B = 0 T. This position is well predicted by the calculation of Fig. 5. The excitation energy is significantly smaller than the single-particle excitation energy of Eo. t ( = hco0) due to the exchange effect of the parallel spin of the electrons for the (S, M ) = ( 1, - 1) state. Now we discuss the energy spectrum for N -- 3 and 4. Exact calculations of the ground and excited states are shown in Fig. 6. The three electron ground state has two transitions labeled by e . On increasing B, the total spin and the total angular momentum of the manybody states change from (S,M) = (1/2, 1) to (1/2,2) at 4.3 T, and then to (3/2,3) at 4.8 T. The transitions to
i :"
N=~
} 24 ~
22 20
~ I
................. $=1.5
1
Magnetic field (T) Fig. 6. Exact c a l c u l a t i o n s o f g r o u n d states (thick l i n e s ) a n d e x c i t e d states ( t h i n l i n e s ) lbr N = 3 a n d 4. M a n y - b o d y states are i n d e x e d b y the total s p i n a n d a n g u l a r m o m e n t u m ( b o x e d n u m b e r s ) . For N = 3, the solid a n d d o t t e d lines i n d i c a t e S = 0.5 a n d !.5, r e s p e c t i v e l y . For N = 4, the solid, d o t t e d and d o t - d a s h e d lines indicate S = 0, ! and 2 respectively. The II, 9
labels are d i s c u s s e d in the text.
larger angular momentum states reduce the Coulomb interactions. In addition, the total spin increases to gain exchange energy. A double transition in the ground state energy is indeed observed in the form of two kinks in the third peak trace of Fig. 3. In most regions in Fig. 3 there is one main configuration for the occupation of single-particle states. For N = 3, however, in the region between the two 9 labels there are two important configurations, which both have the same total spin, and total angular momentum. The four electron ground state has five transitions: (S,M) = (1,0) to (0, 2) at 0.43 T, then to (1, 3) at 4.0T, to (1, 4) at 4.9 T, to (1, 5) at 5.2 T, and finally to (2, 6) at 5.4 T, respectively. The first transition (labeled by I I ) is associated with breakdown of Hund's rule [3] and the other transitions can be understood in the same way as discussed above for the three electron ground state. These transitions are indeed observed as kinks in Fig. 3. The transitions between the two 9 labels are not so evident in the experimental data. This is prob-
321
!18
S. Tarucha et al. / Physica E 3 (1998) 112-120
ably because different states in this region lie close in energy, and the ground state does not show a critical change in electrochemical potential at the transition points (see Fig. 6). We note that the first few excited states in the calculation of Fig. 6 are also observed in the measurement of the excitation spectra [4]. For B larger than the right most 9 for N = 3 and 4 there is again a distinct ground state in which the electrons are fully spin-polarized and occupy sequential momentum states.
4. Electronic states in quantum dot molecules A quantum dot molecule can be realized in the same vertical device configuration as for a quantum dot atom except that the double barrier structure is replaced by a triple barrier structure [13]. The outer barriers have the same thickness of 7.0 nm. Quantum mechanical coupling between the two dots form symmetric and anti-symmetric states. By changing the thickness of the central barrier, b, from 7.5 to 2.5 nm we are able to increase the energy splitting between symmetric and anti-symmetric states, As AS, from about 0.09 to 3.4 meV. Quantum mechanically, we consider the dots separated by a 7.5 nm barrier to be "weakly" coupled, and the dots separated by a 2.5 nm barrier to be "strongly" coupled. As a rough guide, for the case of two electrons trapped in the system (N = 2 ), the lateral confinement energy, ~xo0, = 4 meV, a typical average "classical" charging energy, Ecla~.~ical= 3 meV, and an electrostatic coupling energy, Eetectrostati c "-- 0.7 rneV respectively are comparable to As AS for b - 2.4, 2.8, and 4.8 nm. The hog0 value is slightly smaller than that for a single dot probably because the lateral electrostatic confinement is weaker as the system is larger. We note that quantum mechanical coupling is not the only coupling mechanism in artificial molecules. In the regime where (h~o0 > ) Eelectrostatic >~ AS AS, it is electrostatic coupling between the dots which becomes important [14]. Competition between the two mechanisms as b is varied is expected to have a profound effect on the transport properties of the two dot system. Here we focus on the strongly coupled double dot system. Fig. 7 shows a grey scale plot of d//dVsd in the Vsd-Vg plane for a D = 0 . 5 6 t a m "strongly" coupled double dot device (b--2.5 nm). Black (positive val-
Fig. 7. Grey scale plot of d / / d l ~ in the I~-I~ plane lbr a D = 0.56 ~tm quantum mechanicall)" "strongly" coupled double dot device. Coulomb diamonds similar to those for a single dot are formed from N = 1 to 22 close to zero bias. The half width i n e ~ of a diamond shaped region is a direct measure of the change of electrochemical potential when one more electron is added to the double dot system.
ues of d//d ~d ) and white (negative values of dl/d ~d ) lines criss-crossing the plot and running parallel to the sides of the diamonds identify bound and excited states- details of which will be published elsewhere. Well formed Coulomb diamonds (grey regions where I = 0 p A ) close to zero bias from N - - 1 to 22 are evident. The symmetry of the diamonds with respect to the bias direction confirms that the states responsible are indeed delocalized over both dots. Notice that the N = 2, and N = 6 diamonds are unusually large compared to the adjoining diamonds. As for the single dot the half width of the Nth diamond is a direct measure of lt(N + 1 ) - p ( N ) . The ll(N + 1 ) - l l ( N ) values obtained from Fig. 7 are shown by the black circles in Fig. 2b. For this double dot device we see the same magic numbers 2, 6, and 12 as for the single dot device although, intriguingly, and for reasons which are not yet understood, 4, 9, and 16 are apparently absent. The 3 meV value of Eclassical is also in
322
S. Tarucha et aL / Phvsica E 3 (1998) 112-120
line with the experimental data for N = 1,2, and 3 in Fig. 2b. Note that for N > 15, /~(N + 1 ) - p(N) is approximately half that of the single dot. This is reasonable because the double structure dot occupies roughly twice as much volume. Finally, for this artificial molecule, there is no evidence from Figs. 2b and 7 for the occupation of anti-symmetric states for N ~< 12, i.e. the first 12 electrons all occupy the symmetric states and are delocalized. This might look inconsistent with a single-particle picture as the symmetric and anti-symmetric states can only be distinguished in the presence of quantum mechanical coupling, and both sets of lateral states have an identical single-particle energy specmnn with a characteristic confining energy of A~o0. Putting electrons consecutively into the symmetric states costs much single-particle excitation energy. For example, the complete filling of electrons in the second shell costs 4/zco0, which is much greater than As AS- However, besides As AS, there are a number of important interaction effects that determine the filling of electrons. These effects are the exchange effects within either the symmetric states or within the anti-symmetric states, are between symmetric and anti-symmetric states, screening effects, and direct Coulomb repulsion. Coulomb repulsion favours the filling of laterally delocalized electrons, i.e. p-type electrons rather than s-type electrons, so the consecutive filling of electrons in to the symmetric states is favoured. This consecutive filling is also promoted by the screening effect since it reduces the lateral confinement energy with increasing N. These three factors help to explain the experimental data for this strongly coupled dot device. The exchange effect between the symmetric and anti-symmetric states favours the filling of electrons in to the anti-symmetric states. However, this effect can be weaker than the other effects described above. Exact calculations incorporating many-body interactions are necessary for more detailed arguments. These are underway and will be discussed elsewhere.
5. Conclusions We have studied the atomic-like properties of a single disk shaped dot and the molecular-like properties of two vertically coupled disk-shaped dots. For the
119
single dot the addition energy spectrum for the few electron ground states at B = 0 T and in the presence of a magnetic field induced transitions in the ground states and these compare well to the exact calculations of many-body states in an artificial atom. For the strongly coupled double dot device the few electron ground states show properties similar to those of a single dot, indicating that the first few electrons only occupy the symmetric states and are delocalized throughout the whole system.
Acknowledgements We thank R.J. van der Hage, M.W.S. Danoesastro, Y. Kervennic, J.E. Mooij, S.K. Nair, L.L. Sohn, and N. Uesugi for help and discussions. Part of the work is supported by the Dutch Foundation for Fundamental Research on Matter (FOM). L.P.K. is supported by the Royal Netherlands Academy of Arts and Sciences (KNAW).
References [1] M. Reed, Scientific American 268 (1993) 118; M. Kastner, Physics Today 46 (1993) 24; R.C. Ashoori, Nature 379
(1996) 413. [2] See for a review: Proe. Advanced Study Institute on Mesoscopic Electron Transport, Curacao, June 1996, Series E, Kluwer, Dordrecht, 1997. [3] S. Tarucha, D.G. Austing, T. Honda, R.J. van der Hage, L.P. Kouwenhoven, Phys. Rev. Lett. 77 (1996) 3613. [4] L.P. Kouwenhoven, T.H. Oosterkamp, M.W.S. Danoesastro, M. Eto, D.G. Austing, T. Honda, S. Tarucha, Science 278 (1997) 1788. [5] D.G. Austing, T. Honda, S. Tarucha, Semiconductor Sci. Technol. 11 (1995) 212. [6] S. Tarucha, D.G. Austing, T. Honda, R.J. van der Hage, L.P. Kouwenhoven, Jpn. J. Appl. Phys. 36 (1997) 3917. [7] Y. Tokura, L.P. Kouwenhoven, D.G. Austing, S. Tarucha, Physica B 246-247 (1998) 83. [8] This ratio of Q/~o~ is for example 1.5 for a parabolic potential with ~co0= 5 meV. It is assumed to be 1 in our previous paper (see Ref. [4]). [9] Y. Tanaka, H. Akera, J. Phys. Soc. Japan 66 (1997) 15. [10] See for example:J.J. Palacios,L. Martin-Moreno,G. Chiappe, E. Louis, C. Tejedor, Phys. Rev. B 50 (1994) 5760; See for more references the review by N.F. Johnson, J. Phys.: Condens. Matter 7 (1995) 965. [11] V. Fock, Z. Phys. 47 (1928) 446; C.G. Darwin, Proc. Cambridge Phil. Soc. 27 (1930) 86.
323
120
S. Tarucha et al. / Physica E 3 (1998) 112-120
[12] See for theory: M. Wagner, U. Merkt, A.V. Chaplik, Phys. Rev. B 45 (1992) 1951; P. Hawrylak, Phys. Rev. Lett. 71 (1993) 3374; Sr162for experiment: B. Su, V.J. Goldman, J.E. Cunningham, Phys. Rev. B 46 (1992) 7644; R.C. Ashoori et al., Phys. Rev. Lett. 71 (1993) 613; T. Schmidt et ai., Phys. Rev. B 51 (1995) 5570.
[ 13] D.G. Austing, T. Honda, Y. Tokura, S. Tarucha, Jpn. J. Appl. Phys. 34 (1995) 1320. [14] D.G. Austing, T. Honda, K. Muraki, Y. Tokura, S. Tarucha, Physica B 249-251 (1998) 152.
324
VOLUME 66, NUMBER 21
PHYSICAL
REVIEW
LETTERS
27MAY 1991
Electron Pinball and Commensurate Orbits in a Periodic Array of Scatterers D. Weiss, (I)'(2) M. L. Roukes, (t) A. Menschig, (3) P. Grambow, (2) K. yon Klitzing, (2) and G. Weimann (4) ~ Red Bank, New Jersey 07701 (2)Max Planck lnstitut f~r Festkdrperforschung, D-7000 Stuttgart 80, Germany (3)IV Physikalisches lnstitut der Universit~t Stuttgart, D-7000 Stuttgart 80, Germany (4)Walter-Schottky lnstitut der Technische Unit~rsitiit Miinchen, D-8046 Garching, Germany (Received 15 November 1990) We have introduced an artificial array of scatterers into a macroscopic two-dimensional conductor nearly devoid of intrinsic defects. This generates pronounced structure in the magnetoresistanee, anomalous low-field Hall plateaus, and a quenching of the Hall effect about B - 0 . Our calculations show that the predominant features in the data arise from commensurate classical orbits impaled upon small groups of the imposed scatterers. PACS numbers: 73.50.Dn, 72.10.Fk Generally, impurity scattering is considered to be a stochastic process. Electrons collide with defects located randomly throughout a conductor. In this Letter, electron transport is studied in an unusual situation: We lithographically impose a periodic lattice of strong scatterers upon a relatively defect-free two-dimensional electron gas (2DEG). We accomplish this by etching an array of microscopic holes into a high-mobility 2DEG conductor. Introduction of this strong spatially modulated potential leads to dramatic commensurability effects at low temperatures in an applied magnetic field. Pronounced structure is manifested in the magnetoresistance at low B. We find that the predominant features can be explained classically, but many interesting, and anomalous, properties of this system appear beyond the scope of simple electron-orbit analysis. Recently, it has been observed that magnetoresistance oscillations periodic in lIB emerge at low magnetic fields when a high-mobility 2DEG is subjected to a weak periodic I D potential, i This phenomenon is attributed to the formation of Landau bands due to the soft lateral superlattice potential. 2-4 Preliminary work in 2DEG systems involving strong periodic potentials indicates that entirely different behavior is to be expected. 5-9 We shall explore this in detail below. We fabricate samples from high-mobility GaAsAIGaAs heterojunctions. At 4 K, before patterning, these have carrier densities between ns--2.2x 10 I= and 3.0x10 II cm -2 and mobilities from po---0.56x10 6 to 1.2x10 6 cm2/Vs. The corresponding transport mean free path, Io-m*vrpo/e, ranges between 4.4 and 9.6 pro. Here, vt is the Fermi velocity and m* the effective mass. A periodic (square) lattice of scatterers is introduced by etching an array of holes (lithographic diameters dlith with period a) into a 100-pro-wide Hall bar patterned from the 2DEG by conventional techniques (Fig. 1, insets). The periodic array is defined by electron-beam lithography and transferred into the 2DEG by dry etching. I~ Note that although 10>>a, the Hall bar itself is macroscopic; its dimensions are large compared to 10. The device geometry allows comparison of the resistivity (P,,x) and the Hall resistance (Pxy) from both 2790
patterned and unpatterned segments of the same sample. The imposed array of "antidots" dramatically affects transport at low B (Fig. 1). The enhanced Pxx ( B - O ) reflects reduced mobility at low B ~ ' - 4 • 104 cm2/Vs), i.e., a mean free path (1'--0.34/~m) comparable to a, the spacing between the imposed scatterers. New peaks at low 8 are accompanied by (nonquantized) steps in Px, and the Hall effect is quenched about B - 0 . Arrows in Fig. i, which closely correlate with these features, mark field positions where the normalized cyclotron radius, r c - r c [ a , equals ~- and ~. When rc < ~, Pxx drops
FIG. !. (a) Magnetoresistance and (b) Hall resistance measured in patterned (solid line) and unpatterned (dashed line) sample segments at 1.5 K after brief illumination. In the patterned segment, ns (determined from Shubnikov-de Haas oscillations) is ---8% higher (n, -2.4x 10 ==cm -2). Top insets of (a) and (b): Electron micrograph of the "antidot" array (300 nm period) and a sketch of the sample geometry. Bottom inset of (b): Magnification of the quench in px~,about B-O.
O 1991 The American Physical Society
325
9P H Y S I C A L
VOLUME 66, NUMBER 21
.
.
.
REVIEW. .
quickly, quantum oscillations commence, and pxy begins to display accurately quantized plateaus. In this (quartturn Hall) regime traces from patterned and unpatterned segments become essentially identical. This suggests that the intrinsic mobility is preserved after patterning. Magnetoresistance curves from three Samples (Table I) are compared in Fig. 2. In traces with smaller zerofield resistance a progressively greater number of peaks in Px~, and steps in px~., become resolved. Their emergence is controlled by two parameters: the (normalized) antidot cross section, d - d / a , and ns. nn Here, the effectit,e cross section of the extrinsic scatterers, d - d , t h + 2 Xlde~, involves the depletion length 12 lde~, itself dependent upon ns. Brief illumination of the samples at low temperature enhances ns via persistent photoconductivity add reduces /deC. Figure 2 and Table I sugg~est richer low field structure emerges for small values of d. Sample 3* ( d - - ~ ) exhibits the largest sequence of new Pxx peaks and Pxy plateaus. At each peak, rc can he associated with a commensurate orbit encircling a specific number, n, of "antidots" (Fig. 2, inset). This observation motivates the explanation we present below. The striking features described above occur at low B, in a regime where electron orbits encompass a large number of flux quanta. In this field regime, Landau quantization is suppressed in unpatterned samples when T>_ 1.5 K [Fig. I(a)], while the microstructure-induced anomalies continue to be manifested up to temperatures T - - 5 0 K. This suggests that a classical description involving commensurate orbits, but not involving orbit quantization, might account for the predominant strutture. At low B, when thermal broadening of the Landau levels is significant (ksT > hoJc), magnetotransport is described by the Drude model. In an ideal unpatterned 2DEG, with B applied normally, carriers perform cyclotron orbits with radius r~ -vF/~oc and angular frequency oJ~-eBIm*. For r scattering terminates the motion before a full orbit is completed. Here t is the intrinsic momentum relaxation time, reflecting interactions between electrons and, e.g., intrinsic impurities, phonons, etc. For co~f>> 2x, despite the circular trajectories, bulk current flows in the conductor since orbits drift with velocity v ~ - E , I B in the Hall field E , established. TABLE i. Parameters for samples of Fig. 2. Asterisks denote samples after brief illumination at low T. Sample I I9 2 2* 3 3*
10-:In, (cm - 2)
1.4 2.4 Depleted 2.0 2.8 3.7
a
(nm) 300 300 200 200 300 300
d.th (nm)
60 60 70 70 40 40
LETTERS
.
27 MAYI991
"
.
,,
,
Within this classical pi~ u ~ the m agnetofesistance is B independent, pxx-m*/nse2tlPo, and "the Hall resistance rises lineady, pxy-B/n:e~RoB (Re is the Hall coefficient). "To understand magnetotransport at low B in the patterned samples we envision transport as involving three distinct "pools" of carriers: p/nned orbits, drifting orbits, and scattered orbits. In a patterned sample each contingent contributes to the total resistivity, which is obtained from the inverted sum of the individual conductivity tensors. . ' The resistivities (pp, Pd, Ps) and Hall coefficients (Rp, Rd, Rs) for the pinned, drifting, and scattered carriers each depend upon (normalized) magnetic field, I/~c. We evaluate these making the simplifying assumption that the imposed potential rises quickly near each antidot, while remaining essentially flat in between. In this approximation, valid when I d ~ a - - d n a h (and especially relevant for the case of sample 3"), electrons interact with the fabrication-imposed electrostatic potential only in the immediate locale of each antidot--eisewhere carriers move freely in the applied fields. Pinned orbits, within this simple picture, remain localized about their orbit centers and cannot contribute to transport; hence p p - c o and R e - 0 . Nonetheless, they play a central role in the story since they remove a fraction fp(rc) of carriers from the transport process, n3 As in a pinball game, the scattered orbits constitute the col5--
l
t
9
oo'.
A -
9
_ ./.
3 .
~
.
V.
.
.
.
9
. . .. . ..
2
~
0
'
,., . , , t . ~ o
X X
~
0
0.4
0.8
1.2
1.6
2.0
BIT)
I'
(nm)
.
d~"
100 0.7 i + 0.06 340 0.48 + 0.09 99. . . . 120 0.47 -t- 0.03 600 0.39 _ 0.08 720 0.33 +--.0.08
"Estimate from decay of pxx at high B: fs--" 0 when fc _< (l -,~)12.
FIG. 2. Low-B anomalies from samples of three different heterojunctions (see Table I). Arrows mark &-2'- for each trace. Illumination of sample 1 (I - - I*) increases n, by only a small factor (--!.7), whereas p,,x(B-O) drops almost by a factor of 5. This indicates that !,~ and, consequently, d are reduced after illumination. For smaller d more structure in pxx becomes resolved. Peaks in trace 3* can be ascribed to commensurate orbits with n - I, 2, 4, 9, and 21, as sketched in the inset (for d - " and average Fc-0.5, 0.8, 1.14, !.7, and 2.53, respectively). -2791
326
V O L U M E 66, N U M B E R
21
PHYSICAL
REVIEW
FIG. 3. One-eighth of the real-space zone diagram constructed for one specific value of ~:c (0.44) for a square lattice of cross section d - ~. Pinned, scattering, and drifting orbits have centers within regions marked p, s, and d, respectively. The fraction of area within each zone directly determines that contingent's density. The zone d* comprises skipping orbits, assumed here to be a subset of the drifting contingent (see Ref. 14). Orbit centers within s* precess about an antidot, and then scatter in the zone s. (b) Orbit densities vs ~:c. Fine lines demark average ~ values of orbits impaled upon n - I, 2, 4, 9, I0, i 6, and 21 antidots. lection of paths (arcs) leading between pairs of antidots. At low B, where ic is large, it is primarily these uncompleted "orbits" which carry current through the conductor. This fraction of carriers fs(rc) scatters with an effective relaxation time r' - (~ - I + ~ t ) - t. Here, 9- ! and re~ t are the intrinsic and extrinsic rates. Transport coefficients for scattered carriers are thus ps-poT/f st' and Rs -Ro/fs. At high B, however, it is drifting orbits, involving the fraction f#(lc) of carriers, which dominate transport. These behave as if in an unpatterned sample; for them Pd "Po/f# and Rs--Ro/fd. t4 These extensions of the Drude picture to describe a periodic lattice of scatterers yield (normalized) expressions for the total resistivity,
(1)
-s176 po
(~,f,+~fd)2+~,2(l _fp)2~:~2
'
and Hall coefficient,
Ro
z'2fs+ z2fd+ z'2fl--fP)r176 (z'fs+zfd)2+t~ --fp)2OO2cT2"
(2)
These explicitly relate transport coefficients to t~-
2792
27 M A Y 1991
LETTERS .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
, f
.
.
.
.
.
.
.
.
.
dependent orbit densities and scattering rates. At low B [specifically, for ic > ( , , / 2 - d ) / 2 ] drifting orbits vanish, hence f s " 1-fp, and transport involves only the scattered orbits: p--, ps and R---* Rs. For intermediate to high B the orbit densities can be ascertained by a simple geometric construction, specific for a given rc [e.g., Fig. 3(a) where rc -0.44]. This delineates zones within the real-space unit cell associated with each contingent of electrons, based on orbit-center positions. As a:c increases, this zone diagram becomes increasingly complex--a heirarchy of pinned orbits, which surround progressively greater numbers of antidots, unfolds, t5 In this regime geometric analysis becomes impractical and we resort to straightforward numerical calculation of the f ' s . ,6 In Fig. 3(b) orbit densities calculated for d - ~- are seen to display pronounced commensurability effects. For larger values of d both calculations I~ and experiments (Fig. 2) show that structure for n > I becomes suppressed. As a first approximation, we may assume that the effective mean free path of scattered electrons, l''vFr', is rc independent. This has been confirmed by calculations of I' for the case of complete memory loss after a single antidot collision; these shall be described elsewhere. 17 For sample 3* with d - - ~-, the intrinsic mean free path is [mvFr/a--Io/a-.-33, while pxx at low B indicates I~ - 2 . 4 (Table I). The calculations show ! ~ is nearly constant and featureless for these same values of [ and d when rc is in the relevant range 1.2 < rc < 20. In this range I~ saturates at a value ---2.54, in close agreement with experiment. We compare calculated results to experimental data in Fig. 4. Remarkable similarities are evident: n - 1 , 2, and 4 commensurability effects are prominent in both Pxx and Pxy. In the experimental trace, surprisingly, we find clearly resolved n - 9 and 21 features. Our calculations predict these to be quite weak, even for smaller d. We attribute their enhancement in real samples to the finite potential gradient between antidots. This should act to "guide" electrons around the antidots, permitting deviations from strictly circular trajectories (as assumed in the model) and enhancing fp. For large d (Table I) this potential gradient becomes significant and our simple model is inapplicable. The peak at S:c-~ ~ in samples 1" and 3 is strongly forbidden for strictly circular orbits when d ~ ~-. Despite the more complicated dynamics of such samples, our simple approach elucidates the origin of the dominant structure observed. Well-resolved commensurability features emerge when the intrinsic mobility is maintained between the imposed scatterers and the lattice is "open" (small d). In Refs. 8 and 9 it appears that these conditions (respectively) were not achieved. Recent results from a hexagonal "antidot" array, ~ previously not fully explained, can also be accounted for by our simple model. Quenching of the Hall effect is seen in the data at low B, yet is absent from our calculated traces. Experiments
327
VOLUME66, NUMBER 21
PHYSICAL
REVIEW
LETTERS
27 MAY 1991
petted in transport. 2~ Manifestations of this irregular spectrum have recently been obtained from samples with weak periodic potentials.21 We thank R. R. Gerhardts, A. Forchel, and O. A l e r hand for helpful discussions, S. Koch for assistance with mK measurements, and M. Rick and E. Vasiliadou for technical contributions.
FIG. 4. Comparison between the simple model (top) and experiment (bottom). Calculated curves are obtained assuming a constant.effective mean free path for extrinsically scattered carriers, I'/a--2.4, and an intrinsic value 1 / a ~ 3 3 (values taken from experiment). Features attributed to n - I, 2, and 4 pinned orbits are denoted. Very weak temperature dependence is observedtexperimentally for 50 mK < T < 4.2 K. in mesoscopLc junctions, is and subsequent theory, 19 indicate this is a classical phenomenon requiring a component of specular reflection from the boundaries. Both demonstrate that strictly zero, even negative, low-B Hall coefficients can result. The extensions of the Drude model developed here implicitly involve a relaxation-time approximation for extrinsic collisions, valid when scattering from the antidots is diffuse. This clearly precludes accounting for phenomena, such as quenching, involving correlated multiple reflections. The agreement of our model with experiment emphasizes that low-B commensurability effects originate from a different mechanism. Our experiments verge on the quantum regime. For example, data from sample 2* with the smallest a (Fig. 2) show three weak oscillations near 0.4 T. These features, not reproduced by our classical model, are separated by a field (-~0.1 T) corresponding to addition of one flux quantum through the unit cell. Quantum behavior should clearly emerge with further reduction of the lattice constant. In this realm, the energy spectrum is known to be self-similar, and exotic consequences are ex-
ID. Weiss et al., Europhys. Lett. $, 179 (1989). :R. R. Gerhardts, D. Weiss, and K. von Klitzing, Phys. Rev. Lett. 62, 1173 (1989). 3R. W. Winkler, J. P. Kotthaus, and K. Ploog, Phys. Rev. Lett. 62, 1177 (1989). 4E. S. Aires et al., J. Phys. Condens. Matter 1, 8252 (1989). 5M. L. Roukes and A. Scherer, Bull. Am, Phys. Soc. 34, 633 (1989); A. Scherer and B. P. van der Gaag, Proc. SPIE 1284, 149 (1990). 6C. G. Smith et al., J. Phys. Condens. Matter 2, 3405 (1990). 7H. Fang and P. J. Stiles, Phys. Rev, B 41, 10171 (1990). SK. Ensslin and P. M. Petrofl', Phys. Rev. B 41, 12307 (l~o). 9J. P. Kotthaus, in Granular Nanoelectrom'cs (Plenum, London, 1990). l~ Weiss et al., Appl. Phys. Lett. (to be published). liAr low B, p.... is only weakly dependent upon p0 since long 10 implies r' is dominated by r~ (see text). 12A. Scherer and M. L. Roukes, Appl. Phys. Lett. 55, 377 (1989); T. Demel et al., ibid. 53, 2176 (1988). f3For impaled orbits, the repulsive potential at each antidot provides a local restoring force against drift induced by electric fields. The "removal" of an electron from transport requires a long pinned orbit lifetime, re, > a/vF-,- re~,obtained when po is preserved between antidots and EH is small. 14At high B, a drifting electron encountering an antidot will briefly skip about its periphery. Its orbit center then precesses, stepwise, to the back side where it is ultimately freed, once again, to drift away. Here, we assume a short dwell time and equate skipping to drifting orbits. 151n related work, T Geisel et al., Phys. Rev. Lett. 64, 1581 (I 990), describe chaotic classical dynamics. 16At each value of Fc we calculate fp as the fraction of completed electron orbits with centers on a grid of ---105 sites within the real-space unit cell [M. L. Roukes (unpublished)]. For Fc> (,J2-- d)/2, f d - 0 , and hence f, - ! - f p . tTRoukes (Ref. 16). tSM. L. Roukes et al., Phys. Rev. Lett. 59, 3011 (1987); 64~,' !!54 (1990). J9C. W. J. Beenakker and H. van Houten, Phys. Rev. Lett. 63, 1857 (1989). 20p. G. Harper, Proc. Phys. Soc. London A 68, 874 (1955); M. Ya. Azbei, Zh. Eksp. Teor. Fiz. 46, 929 (1964) [Soy. Phys. JETP 19, 634 (1964)]; D. R. Hofstadter, Phys. Rev. B 14, 2239 (1976). 2tR. R. Gerhardts, D. Weiss, and U. Wulf, Phys. Rev. B 43, 5192 (1991).
2793
328
Journal of the Physical Society of Japan Vol. 65, No. 3, March, 1996, pp. 811-817
On the Mechanism of Commensurability Oscillations in Anisotropic Antidot Lattices Kazuhito TSUKAGOSHI, Masaru HARAGUCHI, Sadao TAKAOKAand Kazuo MURASE Department of Physics, Faculty of Science, Osaka University, Toyonaka, Osaka 560
(Received October 19, 1995) IVe investigate the transport properties in various antidot lattices. It is found that the fundamental peak on resistivity is determined by the conductivity perpendicular to the current flow direction in the case of smaller diagonal component of conductivity tensor than the off-diagonal one. Moreover, the peaks of conductivity are explained by the runaway trajectory, on which the electrons diffuse along the antidot array like as skipping over the antidots when the cyclotron diameter of electron corresponds to the antidot period, and the conductivity along the antidot period is enhanced. Tiffs is the origin of the fundamental peak in the commensurability oscillations. KEYWORDS: GaAs/AIGaAs heterostructure, ballistic transport, magnetoresistance, periodic potential, commensurability oscillations, chaos
w
standing point of chaos theory, Baskin et al. have also calculated the classical electron trajectories in an antidot lattice with hard wall. 22) They calculate the diffusion coefficient (D~x), and find that D ~ is enhanced by the contribution of the "runaway trajectories" at 2P~ = a. They argue that the oscillation peaks of Dx~ coincide the peaks of measured resistance in thc antidot lattice. From the quantum mechanical approach, Silberbaucr and RSssler, and Ishizaka et al. have calculated the conductivity tensor numerically by thc selfconsistent Born approximation and have had thc good agrcement between the calculated rcsults and the experimental ones. 25, 26) However, these approaches have becn performed only in square antidot lattice. In a rectangular antidot lattice, Nagao has simulated the electron motions and has showed that diagonal conductivity tensots depend on the lattice anisotropy and it also affects the resistivity tensors. TM In this description, we consider in more detail the relation between the commensurability oscillations and the current flow direction in anisotropic antidot lattices where the fluctuations either parallel or perpendicular to thc current flow direction are introduced into antidot locations. We also investigate various rectangular or isosceles triangular lattices. Finally, we introduce a model for the commensurability oscillations in the relatively high magnetic fields where the diagonal component of the experimental conductivity is nmch smaller than the off-diagonal one.
Introduction
During the past few years, electron transport in lateral antidot superlattices has been attracting much attention.l_ -31) In the high-mobility two-dimcnsional electron gas (2DEG) with antidots which arc the high potential pillars, the electron mean frec path of homogeneous 2DEG is much larger than antidot pcriod. In this system, various phenomena, e.g., the AhaxonovBohm (AB) effect, 2'3) the Al'tshuler-Aronov-Spivak (AAS) effect, 4,5) and the negative magnetoresistance due to the weak localization by thc spccular scattering at antidot walls, 29"31) have been observed. The low field magnetotransport dominated by geometric effects is also observed: Some maxima in the magnetoresistance appear. 1,6-2~ The magnetic ficlds of the peaks almost corrcspond to the cyclotron motions (cyclotron diameter=2Rc) encircling some antidots. The oscillations are called "commensurability oscillations", which have been intuitively interpreted by the trapped electrons on pinned electron orbits around some antidots. 6, 7) However, there is no reason that the electron can stay near a n t i d o t for a long time because the potential of antidot pillar repulses the electrons. This pinned electron model can not express the peak behavior in an anisotropic antidot lattice where the peak magnetic fields depend on the current flow direction, 8-11) because in the pinned electron model the electrons should be trapped around the antidots irrespective of the current flow direction. Theoretically, the commensurability oscillations have been investigated by the calculation of classical dynamics or fully quantum-mechanics. Fleischmann et a/. 21) have solved the classical equations of electron motion and have applied the linear response relation 32) to the square antidot lattice. The result of the calculation suggests that the commensurability oscillations are not caused by the varying of pinned electrons, but by correlations within the chaotic trajectories. From the same
w
Experiment
Our devices were fabricated from a wafer of GaAs/A1GaAs heterostructure. The electrons were confined to a 2DEG about 60 nm beneath the surface. From the Hall measurement in the unpatterned 2DEG region, the carrier density and the electron mean free path were deduced to be 5.2 x 1011 cm -2 and about 12 pm, respectively, at 1.5 K and under the weak illumination. 811
329
812
Kazuhito
TSUKAGOSHI et o2.
After the writing of the dots by electron beam (EB) lithography on the PolyMetylMethacrylate (PMMA) resist which was coated on the mesa-etched Hall bar, the patterns were transferred to the heterostructure by the bombardment using 1 kV Ar ions. T h e etched hole depth was about 15 nm. Since the accelerated Ar ions penetrated well beyond 100 nm from the sample surface, many defects were introduced, and electrons were trapped t h e r e ? 3"~s) Thus, the areas of 2DEG under the etched holes were depleted and "antidots" were constructed. The images of the sample surface by atomic force microscope (AFM) are shown in Figs. l ( a ) - l ( c ) . Although the circumferences of the holes are not perfect circle and individual shapes in the sample surface differ, the boundary of the depleted area in 2DEG may be ;smoother than the surface hole, because the 2DEG plane is located at several ten nano-meter beneath the holes and the depletion layer spreads around the antidot. However, it is desirable t h a t the peaks due to the commensurability oscillations are compared in one sample, because the peak height is sensitive to the antidot diameter e, 11) and it is easily changed by slight change of the fabrication condition. Thus, we made two or five antidot regions in the same device and measured them under the same condition. The magnetoresistance was measured by an ac resistance bridge at 15 Hz. The peaks due to the commensurability oscillations did not depend on the current level between 30 nA and 1 #A. The insensitivity to the current level shows that our experimental results are in the category of the linear response to the external electric field.
w
R e s u l t s a n d Discussion
3.1
Fluctuation of antidot location due to disorder
Figure 1 shows the experiment about the fluctuation effect of antidot location due to disorder. A fundamental unit cell is a square (Fig. l(a)). The period (a) of the antidot cell is 0.85 #m and the designed antidot diameter is 0.15 #m. We introduce the fluctuations to all directions of antidot location 16) in the Gaussian distribution (standard deviation = a). In the disordered antidot lattice, a is one-quarter of the lattice period (Fig. l(b)). Antidot areas are 170 # m x 150 #m. The magnetoresistances of the ordered lattice, disordered lattice, and unmodified 2DEG region are shown in Fig. l(d). The 2DEG region shows only Shubnikov-de Haas (SdH) oscillations. At the lower magnetic fields (0 ,,~ 3 kG), both resistances of antidot lattices are much higher t h a n t h a t of 2DEG. These resistances decrease and shows SdH oscillations at higher magnetic fields where the cyclotron diameter becomes smaller than the period. In the magnetoresistance of the ordered lattice (a=0), the peaks due to the commensurability oscillations clearly appear at the corresponding magnetic fields to the encirclement orbit around one or four antidots (lower inset in Fig. l(d)). In the disordered lattice (a = 0.25a), the peaks vanish, although enough space for the electrons to encircle the antidot exist (Fig. l(b)). In comparison with both resistances, the peaks due to the commensurability oscillations in the ordered lattice superimpose on the broad
(Vol. 65,
Fig. 1. (a)-(c) Atomic force microscope (AFM) images of sample surface. Ordered square lattice (a = 0 (a)) and disordered lattice (a = 0.25a (b)). Fundamental period (a) is 0.85 #m and antidot diameter is about 0.15 ~m. (c) Enlarged image of typical hole for an antidot. (d)Magnetoresistances of the ordered (a=0) and disordered (a = 0.25a) antidot lattice, and unpatterned region (2DEG). The schematic pinned electron orbits (lower inset) correspond to the two maxima (1 and 4) in the magnetoresistance. Upper inset: Sample layout.
background which is the magnetoresistance in disordered lattice. In the disordered lattice, the electrons collide randomly with many antidots and stray there. So, the commensurate peaks do not appear at the commensurate condition. This situation is the same in the ordered antidot lattice out of the commensurate conditions: the resistances of ordered and disordered lattices are the same at off-commensurate magnetic field. As a result, no commensurability oscillations are observed in the absence of the ordering of antidot location. Moreover, we investigate the directional dependence of the fluctuation with respect to the current flow direction. The fluctuation is introduced either in the Xdirection (X-disorder) or in the Y-direction (Y-disorder) (Figs. 2(a) and 2(b)). The X-direction is parallel to the current flow direction. The period is 1 #m and the designed diameter is 0.15 #m. The value of a is 0, 0.1a, or 0.25a. The magnetoresistances of these antidot lattices are shown in Fig. 2(c). In the ordered antidot lattice (a=0), two peaks due to the commensurability oscil-
330
1996)
Commensurability Oscillations in Anisotropic Antidot Lattices
Fig. 2. Schematic view of one-directional disordered lattices (a)X- or (b)Y-disorder. (c) Magnet . . . . ist . . . . . . f the ordered lattice (a=O), X-disordered lattices (a = O.la(X), 0.25a(X)), and Y-disordered lattices (a = O.la(Y), 0.25a(Y)). Lower inset shows the replotted traces of above magnetoresistances in a series of fluctuated directions.
813
several rectangular antidot lattices. In Fig. 3(b), ax is changed from 0.8 # m to 3 # m at a y = l # m (Fig. 3(a1)). The principal peak, at which the cyclotron m o t i o n is commensurate with the circumference around one antidot, does not shift in spite of varying ax. A bold line shows a magnetic field predicted from 2Rc=1 # m ( = a u ) , which agrees with the peak position. However, in the 90 ~ rotated antidot lattices, in which ax is 1 # m and a u varies between 0.8 # m and 3 # m (Fig. 3(a-2)), the commensurate peak shifts drastically to low fields with increasing ay (Fig. 3(c)). In the rectangular lattice, the peak magnetic fields can be predicted from 2Re = a,~. In the triangular antidot lattices (Fig. 3(d)), however, the peak magnetic fields are not determined only by a v. In Fig. 3(e), a u is l #m and ax varies (Fig. 3(d-1)). The main peak does not shift for ax = 0 . 8 6 6 - 3 #m. In these three antidot lattices, the nearest neighbor distances between antidots are the same at a~ = 1 # m and peak appears at 2R~ = ay = 1 #m. In antidot lattices of ax = 0.683 # m and 0.5 #m, the peak shifts to higher magnetic fields. At ax = 0.5 #m, the antidot cell is a square and the nearest neighbor distance is not the length of the diagonal (a u = 1 #m) but of the side ( 1 / y ~ #m), because of the relation ay > x / a ~ + (ay/2) 2. The results indicate t h a t the nearest neighbor distances determine the commensurate condition. These effective distances are presented by bold lines in Fig. 3(d-1). Similarly, in the lattices where az is 1 # m and a u varies between 1 # m and 3 # m (Fig. 3(d-2)), the nearest neighbor distance is also effective in the emergence of the principal peak (Fig. 3(f)). In the triangular lattices, the principal peak appears in the condition that the cyclotron diameter is commensurate to the nearest neighbor distance.
3.3
lations clearly appear. It is surprising that the clear peaks are observed in the X-disordered lattices (0.1a(X), 0.2ha(X)) in spite of the fact that the all-directional fluctuation with a = 0.25a suppresses the peak as shown in Fig. 1. We also find that the peak amplitude depends on the fluctuations direction. The peaks in the Xdisordered lattices (0.1a(Z), 0.2ha(X)) are higher than those in the Y-disordered lattices (0.1a(Y), 0.2ha(Y)). The replotted traces in the lower inset of Fig. 2(c) show the difference more clearly. Because the X-direction and the Y-direction are originally equivalent in the square lattice and the X-disordered lattice and Y-disordered lattice are essentially equivalent under 90 ~ rotation, the commensurability oscillations of the two lattices must be the same in the light of the pinned electron model where the current flow direction is not considered. 6) Thus, we focus on the anisotropy of the antidot latticc as follows.
3.2
Anisotropic antidot lattice with rectangular or isosceles triangular cell To investigate the role of antidot arrays along the X- and Y-direction, we made several antidot lattices with rectangular (Fig. 3(a)) or isosceles triangular cell (Fig. 3(d)). The ordered 9000 antidots are arranged in each lattice. Figures 3(5) and 3(c) shows the magnetoresistances in
Dependence on current flow direction in rotated rectangular antidot lattice In the next experiment, we investigate the dependence on the current flow direction in the rectangular lattice. The antidot lattices are composed of an unit cell (2 # m x I #m) where the longer side of a fundamental cell ( a = o) is parallel to the current flow direction. The lattices are rotated in five tilted angles (0=0-90 ~) as shown in the inset of Fig. 4. There are 9000 antidots in each lattice. In the magnetoresistance of the antidot lattice at 0=0, the peaks due to the commensurability oscillations appear when the cyclotron diameter (2Re) is 1 # m or 2 # m as shown by vertical lines. Each cyclotron diameter corresponds to the integral multiple of the lattice period of the shorter side. Bold lines in Fig. 4 show the calculated magnetic fields corresponding to 2Rc=1 # m and 2 #m. Except for the antidot lattice of 0= 90 ~ the peak of 2Re= 1 # m appears at all angles, but decreases with increasing 0. Under a = 90 ~ peak magnetic field is determined by 2 # m period which is perpendicular to the current follow direction. 3.4 Antidot array number dependence In order to examine the dependence of the peak of the commensurability oscillations on the lattice size, we change the antidot array number (N) along the Xdirection (--the current flow direction). If the electrons
331
814
(Vol. 65,
Kazuhito TSUKAGOSHI et al.
(a-l)
(a-2) (oarrmt)
lr!ls~
lHsq
$0~/"Z-
~8
t'~176
~'L/A
0 ~ / ~
~~~2 '
,C'1
i
~
401" 2R~--Itmf"|VVV~20t. --r-~
~
oT.m-r.. ....
(d) Isosceles triangular cell (d-l) (d-2) 9 ,0.5
200
t-/~ 1
ire '.'~
~.5~m
12
'lhlc.lequil-t~al 160
1.5 ~
.0.~ br
100
~'~120
8
~,q.25
uimsie 2
a"~2.06 I
40:
o
o.~L~ ~
1~.g46
80
,
l~.l'-
ffD.707
ic) l
2 r - ~ * a ~ ~ 80 Ell.41 "~ 311 ~
~ ~f3.04
I/1.80 ~"
40 0
0
(~tm)
2
4 6 0 Magnetic Field (kG)
2
4
6
Fig. 3. Severalantidot lattices with rectangular (a) or isosceles triangular cell (d). In the lattice with rectangular cell, ay is 1 pm (a-l), or az 1 /~m (a-2). Similarly, in isosceles triangular cell, ay is 1 ~m (b-l), or ax 1 /~m (b-2). Bold lines show the effective distances for the main peaks. Magnetoresistances of various rectangular (b)(c), a n d i s o s c e l e s triangular lattices (e)(f). The bold line shows the magnetic field of 2Rr = 1 t~m (b), 2Re = a u (c), and 2Rc calculated from the nearest neighbor distance between the antidots (e)(f). The curveshave been offset for clarity (e)(f).
traveling complexly, whose motions are chaotic, are effective at the commensurate peak of resistance, the peak height is expected to strongly depend on N. Figure 5 shows the magnetoresistances with several N ( N = 1, 2, 5, 10 and 50). The antidot number in the Y-direction is 150 per array. There are three antidot arrangements of the fundamental cell: square (Fig. 5(a)), triangle (Fig. 5(b)) and rectangular arrangement (Fig. 5(c)). At N - 1, three types are coincident. Zero-field resistances (R o) are shown in the insets. The resistance /to is in linear proportion to N. At N = 50 in square lattice (Fig. 5(a)), some peaks due to the commensurability oscillations appear, and their heights decrease with decreasing N. However, even at N = 1, the peak does not vanish. The tendency is similar in other lattices (Figs. 5(5) and 5(c)). In single antidot array, the complex trajectory is almost absent. In the single array, following the suggestion by Baskin et al., 22) the "runaway trajectory" is only considered as the chaotic trajectories. The electrons on the runaway trajectory skip over the antidot array and diffuse away when 2Rc corresponds to the antidot period. In rectangular lattice of N - 50 (Fig. 5(c)), complex trajectory is also not so considerable, because the separation between the adjacent antidot arrays are enough large in comparison with the elec-
tron cyclotron diameter at the main peak (2Rc = 1 #m). In the square and the triangular lattice of N = 50, however, we cannot distinguish the contributions to the commensurability oscillations from the runaway trajectories or the complexly traveling trajectories, and also cannot distinguish the two runaway trajectories along X- or Ydirection. 3.5
C o n d u c t i v i t y tensor in antidot lattice
The electron stream responds to the externally applied electric field linearly through the conductivity tensor. If we get the conductivity of the system, we can imagine the electron stream in the antidot lattice. Thus, we convert the experimental resistivity tensors into the conductivity tensor 12) as shown in Fig. 6. In a rectangular antidot lattice (a~ = 1 /zm, a u = 0.8 #m, d ~ 0.1 #m), the anisotropy effect is clearly observed (Fig. 6(a)). Conductivity tensors a ~ , ayy, and axy (Fig. 6(5)) are calculated by using the standard formula for the homogeneous two-dimensional conductor from the experimental resistivity tensors px~, pyy, and P~u; i.e. cr~ = P y ~ / ( P ~ P u u + pzy2), etc. The conductivity azz has some structures on a background which decreases with increasing magnetic fields. If the background is determined by the pure classical effect in a ho-
332
Commensurability Oscillations in Anisotropic Antidot Lattices
1996)
i
120 _
:x k - -
because it depends on two f i t t i n g parameters. However, another structures undoubtedly stem from axx. The pronounced peak of z~axx appears at the vicinity of the peak of p~y (-~ 3 kG). In the same way, the trend of A%y coincides with that of px~. This can be explained as follows, As shown in Fig. 6(b), cr~y is much larger than ax~ and ay~ near the main peaks of px~ and Pry- Because of the anisotropy of the rectangular lattice, we get the relation Px~ = cr~y/(azzayy + azy 2) -.~ ayy/axy 2. In the same way, py~ is approximated by axx/az~ 2. From these relations, Pxz (or pyy) mainly depends on ayy (or axx). T h a t is, in the case of axy >> crxx and ayy, the peak of the resistivity must be determined by the diagonal conductivity perpendicular to the current flow direction. Near B = 3 kG in Fig. 6, the magnetic field of the pronounced peak of Pzz (or pyy) corresponds to a peak of ayy (or a ~ ) . Following the consideration by Baskin et al., in which the "runaway trajectory" is effective at the commensurability peaks, 22) the peak o f a y y (or crzx) can be explained. When the cyclotron motion is commensurate with the period perpendicular to the current flow direction, the electrons on the runaway trajectories diffuse along the Y-direction (Fig. 7(a)). Thus, the conductivity ayy increases. At this condition, axz is not enhanced. On the other hand, at 2Re = ax where the electrons are guided along the X-direction (Fig. 7(b)), the peak of a ~ appears and the peak of pyy is observed. Here, we emphasize that it is i m p o r t a n t t h a t the electrons diffuse along the antidot arrays perpendicular to the current flow direction. Recently, Nagao has performed a numerical simulation on a rectangular antidot lattice and has showed that above consideration is appropriate for the commensurability oscillations in antidot lattice. TM The calculation by Fleischmann et al. have showed t h a t the pinned electrons are not effective. TM Their results agree with our consideration because the electrons on runaway trajectory do not stay at an antidot for a long time. The electrons traveling complexly in the antidot
2j
60~
45 ~
90~
100
80
~
40
20
0
2
4
815
6
Magnetic Field (kG) Fig. 4. Magnetoresistance of various tilted antidot lattices from fundamental rectangular one (2 /~m x 1 /zm). Vertical dotted lines show the calculated magnetic fields corresponding to 2Re -- I /zm and 2 pm. Inset: Schematic view of sample. The X-direction is the current flow direction. Frames show the typical antidot regions.
mogeneous 2DEG, it may be given by O'zx o ---- a/(1 +/3B2). p a r a m e t e r "a" and "if' are determined by fitting procedure w i t h the experimental conductivity at zerofield and t h a t at B ~- 4.5 kG where the deviation of p ~ from the Hall resistance of 2DEG region is little. To see the structures of a~z, we subtract azz~ from a~z. A peak of Z~a~ ( = a ~ - a ~o ) at B ..~ 0.8 kG may be trivial The
,~I:.'9 9
1.2!
9
9
9
9 9 9
9
9
9
1.0
1.0 0.8
l-
-4
~v 0 I0 20_N3040 50
-2
0
2
_i[_
4-4
.v 0 I0 20 N30 40 50
-2
0
2
1~av-O"']'O"~
4-4
-2
0
2
4
Magnetic Field (kG) Fig. 5.
Magnetoresistance of various line numbers of antidot arrays (N). The line number N is 1, 2, 5, 10, or 50. Fundamental cell
of the antidot lattice is square (a), triangle (b) or rectangular (c). Typical arrays of the lattices and the defined lengths are shown in upper inset. Lower inset: Zero-field resistance (R0) of each antidot lattice.
333
816
K a z u h i t o TSUKAGOSHI et al.
0.15
2Rc=I run 0.81~n
,
,
'
~ i
,
0.6
0.05
--o,J
L i
i
j O'IL.-.~'...--~ ! '~
Magnetic Field(kG) Fig. 6. (a) Magnetoresistivy Pzx and Puv, and Hall resistivity pz~ in rectangular lattice (1 /~mx0.8/~m, d,,, 0.1 #m). (b) Solid lines show the conductivity axx, ayy, and axy, and dotted lines show classical magnetoconductivities a~ and auu~ (c) Different conductivities (zSazz = axz - a~ Zla~v = avu-ayv). 0 (d) Solid lines are resistivities re-constructed from az,~,/a~, or ax~/a~,, and dotted lines a r e pxz or p x x . T w o v e r t i c a l d o t t e d lines a r e drawn a t 2Rr = 1 / ~ m a n d 0.8 # m .
(a) 2Rr "~ a
~
9
(b) 2Rc=ax O'xx O'xx ~III
I (~yye
I
I
~ :
L aX
this condition, Schuster et al. have suggested that the pinned orbits may be dominant at the peaks of the commensurability oscillations. TM In our early stage of this investigation, from the analogy between the commensurability oscillations and the magnetic focusing effect, 3e'3s) we had considered that the origin of both effect was almost the same. 13,14) In the samples of refs. 36 and 37, some narrow wire channels are arranged in parallel. The magnetic focusing effect occurs when the electrons emitted from one wire are brought back to another wire. In this case, because the wire channels are narrow and long compared with the antidot lattice, the total resistance (Rxx) of the system is so high that these cases may obey the case of axu << axx. Thus, this situation of the electrons seems to be "pinned". However, the case of ref. 38 may be the same as the commensurability oscillations since the resistance P~x is small. Thus, in spite of the resemblance of the peak emerging condition of the resistivity between in the magnetic electron focusing effect and in the commensurability oscillations, the mechanism differs. Here, we note that only conductivity tensor can clearly distinguish two cases: e.g., the runaway or pinned electron. Our model based on the electron diffusion on the runaway trajectory can explain all experimental results described in previous paragraphs. In rectangular lattices, the peak position is determined only by ay (Figs. 3(a)-3(c)). When the rectangular lattice is tilted, the peak, which appears at 2 R c = s h o r t e r period, decreases with increasing tilted angle (Fig. 4), because the contribution of the electron diffusion to auy decreases in the tilted antidot arrays. In the same way, in the triangular lattices, the main peak fields are determined by the nearest neighbor distance in the lattice (Figs. 3 ( d ) - 3 ( f ) ) , because the commensurate condition is determined by the electrons diffusing along the nearest neighbor distance between the antidots and its component perpendicular to the current flow direction contributes to the commensurability oscillations. In the one-directionally disordered lattices (Fig. 2), the lengths between the antidots along Y-direction are varied more by Y-directional disorder than by X-directional disorder, because Y-directional disorder changes these lengths more directly (Fig. 2). Thus, the peaks in X-disordered systems are larger than those in Y-disordered systems in this case. w
'
II
Fig. 7. Schematic electron trajectories in rectangular lattice. Trajectory I is pinned orbit, II complexly traveling, and III runaway one. At 2Re = a~ (a), conductivity o'u~ is enhanced. In the same way, at 2P~ = ax (b), conductivity o'xx is enhanced.
lattice contribute only to the background peak as shown in the disordered lattice in Fig. 1. However, in the case of axu << axx and auy, that is, when resistivity px, is too high, the resistivity P~x is proportional to 1~axe. At
(Vol. 65,
Conculsion
In the case of axy >> axx and ayu, it is found that the fundamental peak on resistivity is determined by the conductivity perpendicular to the current flow direction. Moreover, the peaks of conductivity are explained by the runaway trajectory, on which the electrons diffuse along the antidot array like as skipping over the antidots. This is the origin of the fundamental peak in the commensurability oscillations. Acknowledgements We are grateful to T. Nagao for useful discussions, and for useful suggestions from the simulations of the electron transport in the periodic potential, to K. Gamo for the
334
Commensurability Oscillations in Anisotropic Antidot Lattices
1996)
use of facilities of micro-fabrication, and to S. Wakayama and K. Oto for help in the experiment. We thank H. Okada of Sumitomo Electric Industry Co., Ltd. for providing high quality GaAs/AIGaAs heterostructures. One of the authors (K. T.) wishs to thank JSPS Research Fellowships for financial support. This work is partially s u p p o r t e d by a G r a n t - i n - A i d for Scientific Research on P r i o r i t y A r e a from the M i n i s t r y of E d u c a t i o n , Science and C u l t u r e .
1) E. S. Aires, P. H. Beton, M. Henini, L. Eaves, P. C. Main, O. H. Hughes, G. A. Toombs, S. P. Beaumont and C. D. W. Wilkinson: J. Phys. C 1 (1989) 8257. 2) F. Nihey and K. Nakamura: Physics B 184 (1993) 398. 3) R. Schuster, K. Ensslin, D. Wharam, S. Kfihn, J. P. Kotthaus, G. Bfhm, W. Klein, G. Trgnkle and G. Weimann: Phys. Rev. B 49
(1994) 8510.
4) G. +M. Gusev, Z. D. Kvon, L. V. Litvin, Yu. V. Nastaushev, A. K. Kalagin and A. I. Toropov: J. Phys. C 4 (1992) L269. 5) F. Nihey, S. W. Hwang and K. Nakamura: Phys. Rev. B 51 (1995) 4649. 6) D. Weiss, M. L. Roukes, A. Menschig, P. Grambow K. yon Klitzing and G. Weimann: Phys. Rev. Lett. 66 (1991) 2790. 7) J. Takahaxa, T. Kakuta, T. Yamashiro, Y. Takagaki, T. Shiokawa~ K. Gamo, S. Namba, S. Takaoka and K. Murase: Jpn. J. Appl. Phys. 30 (1991) 3250. 8) R. Schuster, K. Ensslin, J. P. Kotthaus, M. Holland and C. Stanley: Phys. Rev. B 47 (1993) 6843. 9) D. Weiss, K. Richter, E. Vasiliadou and G. Liitjering: Surf. Sci. 305 (1994) 408. 10) R. Schuster and K. Ensslin: Festk6rperprobleme 34 (1994) 195, Springer, Beh'in. 11) K. Tsukagoshi, S. Wakayama, K. Oto, S. Takaoka, K. Murase and K. Gamo: Superlatt. and Microstruct. 16 (1994) 295. 12) R. Schuster, G. Ernst, K. Ensslin, M. Entin, M. Holland, G. B6hm and W. Klein: Phys. Rev. B 50 (1994) 8090. 13) K. Tsukagoshi, S. Wakayama, K. Oto, S. Takaoka, K. Murase and K. Gamo: Phys. Rev. B 52 (1995) 8344. 14) K. Tsukagoshi, M. Haxaguchi, K. Oto, S. Takaol~, K. Murase and K. Gaxno: Jpn. J. Appl. Phys. 34 (1995) 4335. 15) J. Takahara, A. Nomura, K. Gamo, S. Takaoka, K. Murase and H. Ahmed: Jpn. J. Appl. Phys. 34 (1995) 4325. 16) G. M. Gusev, P. Basmaji, Z. D. Kvon, L. V. Litvin, Yu. V. Nastaushev and A. I. Toropov: J. Phys. C 6 (1994) 73.
817
17) A. Lorke, J. P. Kotthaus and K. Ploog: Phys. Rev. B 44 (1991) 3447. 18) G. Berthold, J. Smoliner, V. P~sskopf, E. Gornik, G. B6hm and G. Weimann: Phys. Rev. B 45 (1992) 11350. 19) G. Berthold, J. Smoliner, V. Rosskopf, E. Gornik, G. B6hm and G. Weimann: Phys. Rev. B 47 (1993) 10383. 20) G. M. Gusev, P. Basmaji, D. L Lubyshev, L. V. Litvin, Yu. V. Nastaushev and V. V. Preobrazhenskii: Phys. Rev. B 47 (1993) 9928. 21) R. Fleischmann, T. Geisel and R. Ketzmerick: Phys. Rev. Lett. 68 (1992) 1367. 22) E. M. Baskin, G. M. Gusev, Z. D. Kvon, A. G. Pogosov and M. V. Entin: Pis'ma Zh. Eksp. Teor. Fiz. 55 (1992) 649 [JETP Lett. 55 (1992) 678 ]. 23) T. Nagao: J. Phys. Soc. Jpn. 64 (1995) 4097. 24) R. Fleischmann, T. Geisel and R. Ketzmerick: Europhys. Lett. 25 (1994) 219. 25) H. Silberbauer and U. R6ssler: Phys. Rev. B 50 (1994) 11911. 26) S. lshizaka, F. Nihey, K. Nakamura, J. Sons and T. Ando: Jpn. J. Appl. Phys. 34 (1995) 4317. 27) H. Xu: Phys. Rev. B 50 (1994) 12254. 28) 1. V. Zouzulenko, F. A. Maao and E. H. Hauge: Phys. Rev. B 51 (1995) 7058. 29) G. M. Sundaram, N. J. Bassom, R. J. Nicholas, G. J. Rees, P. J. Heard, P. D. Prewett, J. E. F. Frost, G. A. C. Jones, D. C. Peacock and D. A. Ritchie: Phys. Rev. B 47 (1993) 7348. 30) Y. Chen, R. J. Nicholas, G. M. Sundaram, P. J. Heard, P. D. Prewett, J. E. F. Frost, G. A. C. Jones, D. C. Peacock and D. A. Ritchie: Phys. Rev. B 47 (1993) 7354. 31) C. T. Liang, C. G. Smith, J. T. Nicholls, IL J. F. Hughes, M. Pepper, J. E. F. Frost, D. A. Ritchie, M. P. Grimshaw and G. A. C. Jones: Phys. Rev. B 49 (1994) 8518. 32) R. Kubo: J. Phys. Soc. Jpn. 12 (1957) 570. 33) Y. Yuba, T. Ishida, K. Gamo and S. Namba: J. Vac. Sci. Technol. B 6 (1988) 253. 34) H. F. Wong, D. L. Green, T. Y. Liu, D. G. Lishan, M. Bellis, E. L. Hu, P. M. Petroff, P. O. Holtz and J. L. Merz: J. Vac. Sci. Technol. B 6 (1988) 1906. 35) N. G. Stoffel: J. Vac. Sci. TechnoL B 10 (1992) 651. 36) K. Nakamura, D. C. Tsui, F. Nihey, H. Toyoshima and T. Itoh: Appl. Phys. Lett. 56 (1990)385. 37) F. Nihey, K. Nakamura, M. Kuzuhara, N. Samato and T. Itoh: Appl. Phys. Lett. 57 (1990) 1218. 38) K. Nakazato, R. I. Hornsey, R. J. Blaikie, J. R. A. Cleaver, H. Ahmed and T. J. Thornton: Appl. Phys. Lett. 60 (1992) 1093.
335
VOLUME 71, NUMBER 23
PHYSICAL
REVIEW
LETTERS
6 DECEMBER 1993
How Real Are Composite Fermions? W. Kang, H. U Stormer, L. N. Pfeiffer, K. W. Baldwin, and K. W. West AT& T Bell Laboratories, Murray Hill, New Jersey 07974 (Received 14 September 1993) According to recent theories, a system of electrons at the half-filled Landau level can be transformed to an equivalent system of composite fermions at zero effective magnetic field. In order to test for these new particles, we have studied transport in antidot superlattiees in a two-dimensional electron gas. At tow magnetic fields electron transport exhibits well-known resonances at fields where the classical cyclotron orbit becomes commensurate with the antidot lattice. At v-- ~- we observe the same dimensional resonances. This establishes the semiclassical behavior of composite fermions. PACS numbers: 73.40.Hm
During the past decade two-dimensional electron systems at low temperature and high magnetic field have repeatedly surprised us with exotic electron correlation phenomena. The quantum liquids of the fractional quantum Hall effect (FQHE) [1-3], the still enigmatic electron crystal [4] at very low filling factors, and the vanishing and reappearance of certain quantum Hall states in double layer electron systems [5-7] all reflect the dominance of electron-electron interaction at very high magnetic fields. Recently the nature of the electronic interaction at the half-filled Landau level has received much attention. There is now mounting evidence for a novel particle called a 'teomposite fermion" [8] that plays a crucial role in the physics of two-dimensional (2D) electron systems in the lowest Landau level. In this paper we present resuits of an experiment that demonstrates the semiclassical motion of such a particle. The significance of the physics at the half-filled Landau level was foreshadowed in exceptional electrical transport and surface acoustic wave anomalies exactly at v - ~ - . At this filling fraction Jiang et al. [9] observed a deep minimum in the magnetoresistivity Pxx that persisted to unusually high temperatures, exhibiting a temperature dependence distinctly different from the neighboring FQHE states. Surface acoustic wave (SAW) experiments by Willett et al. [10] at v - ~- revealed attenuation and velocity changes that were opposite to those observed in the regime of the FQHE liquids. Independently, the hierarchical model of the FQHE that orders the various odd-denominator states at v - p / q ( p - i n t e g e r , q - o d d integer) had come under increasing criticism. Starting from the Laughlin liquids [11] condensed from electrons at v - I / m and v - = l - I / m , the higher order FQHE states are derived from lower order states as Laughlin states of fractionally charged quasiparticles [12,13]. In particular, the two prominent series of liquids at v - ' p / ( 2 p +_. 1) represent a succession of parental and daughter states starting from v -= ~ and v and converging towards v "= ~-. However, questions were raised [14] regarding the density of quasiparticles and their apparent noninteracting nature. Jain proposed an innovative model for these series of 3850
FQHE liquids based on hypothetical particles which he termed composite fermions [14]. The liquids of the FQHE are then derived as the integral quantum Hall effect of such composite fermions. These composites consist of an even number of magnetic flux quanta bound to an electron as a result of strong electron-electron interaction which had been contemplated earlier in a related context [15-19]. In a seminal paper Halperin, Lee, and Read [20] used a Chern-Simons gauge field construction to transform the state at exactly v - ~ - to a mathematically equivalent State of composite fermions with a well defined Fermi surface at vanishing magnetic field. Like magic, the magnetic field (two flux quanta per electron at v ~ ~-)is incorporated into the particles themselves and the resulting composite fermions move in an apparently vanishing external magnetic field. The composite particles of these theories have been able to account for several of the previously puzzling features in the vicinity of v-- ~r. The electronic transport anomaly at v - ~- is explained in terms of the appearance of a metallic state and the suppression of electron localization [21]. The SAW data at this filling factor are interpreted as wave vector dependent relaxation of the composites that make up the Fermi sea [20] and the width of the SAW anomaly is consistent with the size of the postulated Fermi surface [22]. The tunneling experiment at v'= ~- between pairs of 2D electron systems by Eisenstein, Pfeiffer, and West [23] also finds an explanation in terms of the Fermi liquid at the half-filled Landau level [24]. And finally, the Halperin-Lee-Read theory provides a very natural interpretation of the observation by Du et al. [25] that the size of the energy gaps of the main sequence of FQHE states at v - p / ( 2 p +_ !) increases linearly with the magnetic field deviation from B I/2 at v - ~-. It simply reflects the linearly increasing "'Landau-level splitting" of composite fermions exposed to an effective magnetic field Beer- B -- B 1/2. Although there is considerable experimental support for composite fermions, it is natural to wonder just how real these particles are. It seems unsatisfactory to simply regard them as a convenient mathematical construct. In fact, it would be far more satisfying if we could detect a
0031-9007/93/71 (23)/3850(4)$06.00 9 1993 The American Physical Society
336
VOLUME
71,
NUMBER
PHYSICAL
23
REVIEW
semiclassical aspect of the composites. Since the v - ~ state is proposed to be largely equivalent to a metal at zero magnetic field, one wonders whether experiments that usually reveal the semiclassical motion of electrons could not be performed on these new particles. Experiments that come to mind are transverse focusing [26] and transport through surface gratings [27,28] or antidot superlattices [29]. These experiments are performed in the ballistic regime where the electronic mean free path is larger than the characteristic length scale of the experiment and the electronic transport can be treated semiclassically. Electronic transport through antidot superlattices yields particular strong dimensional resonances. The resistivity of the patterned two-dimensional electron gas shows a sequence of strong peaks at low magnetic field. A simple geometrical construct reveals that the resonances occur when the classical cyclotron orbit, r c - m * • [m* is the effective mass, t,F is the Fermi velocity, kr-(2Jrne) I/2 is the Fermi wave vector, and ne is the electron density], encircles a specific number of antidots. The inset in Fig. ! illustrates the configurations for s - i , 4, and 9 dots. According to a simple electron "pinball" model [29], at these magnetic fields the orbit is minimally scattered by the regular dot pattern, electrons get "'pinned," and transport across the sample is impeded. In a more sophisticated model the resistivity peaks arrive from the correlation of chaotic, classical trajectories [30]. We have used such an antidot superlattice to first establish the semiclassical behavior of electrons around B - 0 and then probe the equivalent semiclassical behavior of these bizarre composite particles around v - ~ - . One of the particularly telling signatures for the compos-
'
I
:,"~-:~:
3 I
i-
213
~
i~
1
3I
~o
0
0
5
10
15
B(tesla) FIG. !. Comparison of the magnetoresistancc R~x of the bulk two-dimensional electron gas (lower trace) and the Rxx of the d-600 nm period antidot superlattice (upper trace) at T-300 inK. The bulk Rx~ was measured in a rectangular strip containing approximately three squares. The antidot superlattice was nearly a square. The fractions near the top of the figure indicate the Landau-level filling factor. Inset: Schematic of commensurate orbits encircling s - l, 4, and 9 antidots.
LETTERS
6 DECEMBER 1993
ite fermions we expect to find is their modified resonant field as compared to the resonances of the usual electrons. Not only should the peaks around B - 0 reoccur symmetrically around v - ~ - but their spacing should differ by exactly a factor of , ~ between electrons and composites. This is due to the spin alignment of the fermions, which increases their Fermi velocity t'F by a factor of ~ as compared to the unpolarized electron systems [20]. The antidot superlattice was fabricated on a high quality modulation-doped GaAs/AIGaAs heterostructure with electron density n e - l . 4 5 x 1 0 jl cm -2 and mobility - 7 . 8 x l06 cm2/Vsec prior tO processing. The antidot superlattice was initially defined as an array of holes by standard electron-beam lithography on a ! 25 x 125 #mZ area. The holes were then transferred to the 2D electron gas via reactive ion etching, producing cylindrical holes with minimal undercutting. The antidot region was defined as a bridge between two large two-dimensional electron gases by photolithography. The period of antidots ranged from d - 5 0 0 to d - 7 0 0 nm with the dot sizes of 100-200 nm. The electron-beam ekposure~and the etch depth were varied to optimize the size of the wfllestablished antidot oscillations for electrons around B - 0 . All experiments were performed in a i 5 T magnet with the sample immersed in pumped 3He at 300 mK,following a brief illumination from a light emitting diode. Because of a slight density gradient across the 2 in. GaAs wafer, the density in the superlattice varied slightly between samples from !.45 to 1.52x 10 II cm-2. Figure I shows the magnetoresistance Rxx of a d - 6 0 0 nm period antidot superlattice in comparison with Rxx of the unprocessed bulk part of the sample, devoid of dots. The exceptional quality of the sample is demonstrated by two clear sequences of FQHE states in the bulk Rx~ around half filling, reaching filling factors as high as v - ~ . A similarly pronounced series of FQHE states is observed in the upper trace through the antidots. However, a striking difference in Rx~ is apparent near B - 0 , v - ~-, and v - ~-. The bulk sample clearly exhibits local minima at these field positions, whereas the resistance in the antidot trace shows overall maxima with peaks of varying strength superimposed at the same fields. The features around B - 0 are clearly identifiable as the wellknown dimensional resonances of the electrons followed by the appearance of Shubnikov-de Haas oscillations in higher magnetic field. The peaks around v - ~ - are the sought after dimensional resonances of composite fermions. A direct comparison between electron and composite resonances is made in Fig. 2. Figure 2 shows four sets of Rxx data taken around B--0 and v - ~ - on four different specimens in the absence of a superlattice (a) and with three different antidot superlattices of periodicity 700, 600, and 500 nm [(b)-(d)]. The origin, B - 0 , resides at the center of the figure and the lower traces in each section represent the electron transport data taken for positive and negative magnetic fields. The well'established electron dimension3851
337
VOLUME 71, NUMBER 23 9 0.8
PHYSICAL
I
'
"
'
"
REVIEW
I
-(a)bulk
0.0
:~:
__
3o 4 2
..
0-
I
-1
:
-
:
0
I
1
B(Lesla) FIG. 2. Expanded views of the magnetoresistances near v - ~ and B--0 for (a) bulk, (b) 700 nm, (c) 600 nm, and (d) 500 nm period antidot superlattices. The v-~- results are shown as upper traces in each figure. They have been shifted to zero and the field scale has been divided by ~/2 for comparison. The vertical scale reflects the resistance for the v-~- traces. The electron traces have been multiplied by the factor shown in the figure. The dashed cun,e in (c) shows simulated smearing by Fourier filtering of the'electron trace.
al resonances are clearly visible for all periods [(b)-(d)]. They are of course absent in the unprocessed specimen of Fig. 2(a). The top trace of each section represents Rxx around v - ~ - . To shift the traces down to B - 0 we first define the exact magnetic field position B i/2 at half filling from nearby, well-established FQHE features around v - ~ - and translated to B - 0 . Concomitantly, the magnetic field scale is compressed by a factor of ~ to account for the expected difference in kr between electrons and composites, in comparing the dimensional resonances around B - 0 with those around v - ~ - we observe excellent agreement between the strong s - ! features of the electrons and the peaks in the respective traces at half filling. This is compelling evidence that these magnetoresistance peaks around v - ~ - are due to antidot dimensional resonances of composite fermions. Although the peak positions of the fermion resonance are in good agreement with the expected value, the width tends to be much broader than the electron resonance and 3852
LETTERS
6 DECEMBER 1993
the higher order peaks are not observed. This is probably due to the difference in mean free path between electrons and fermions. Since the v - ~ - state is equivalent to a metal at zero magnetic field, "resistivity" and "mobility" measurements can be made for the composite fermions. Van der Pauw measurements on the unprocessed sample at v-, ~- yield a resistivity of 350 n/t~ equivalent to a mobility p - 1.23 x 10 s cm2/V sec. The corresponding fermion mean free path, I-h.f2kFp/e, is approximately 2 pro. This contrasts sharply with the electron mean free path of --50 pm at zero magnetic field. Thus, it should be no surprise that the fermion features are so much broader than the electron features and that higher order peaks are absent. To simulate such a smearing of the fermion resonance, we plot in Fig. 2(c) the electron data as a dashed trace after Fourier filtering it and removing frequencies greater than i kG. The resulting resemblance with the composite resonance is very striking. A more detailed analysis of the shape of the peaks around v.- ~- is beyond the scope of this paper. However, we would like to point out an intriguing asymmetry. In all our data we find the lower magnetic field peaks, for both field directions, to exceed "the higher field peak by as much as a factor of l0 [see Fig. 2(d)]. The origin of, this effect remains unclear. Furthermore, not only are the s - 1 peaks present around v - ~-, the overall resistance behavior around B - 0 is reflected in the transport around v - ~ - in t h e unprocessed as well as the processed sampies. This shows the similarity of transport in the extended region around B - 0 and v - ~ - . On the other hand, the temperature dependence of both phenomena differs considerably. While the electron resonances around B - 0 are known [29] to persist up to temperatures as high as 40 K, we observe composite fermion resonances to disappear above --- I K. Finally to further quantify our findings, we plot in Fig. 3 the main peak positions for the electron resonance and the fermion resonance as a function of inverse lattice period, lid. These axes were chosen since the s--1 resonances occur at B-m*t,t:/de==ttkt:/de and at Bc~-.,l~hkF/de for electrons and fermions, respectively. To correct for the slight differences in the density between samples (---3%), the peak positions have been normalized with respect to the density of the d - 5 0 0 nm antidot sample. The solid (dashed) line indicates the calculated electron (fermion) peak positions using k~-(2xne) !/2. The slopes of the two lines differ exactly by a factor of ,f2", reflecting the spin polarization of the fermion system. Good agreement between the experimental and the calculated peak positions further reinforces the existence of dimensional resonance of composite fermions. Another interesting feature of our data is the fermion resonance at v - ~. We expect the antidot oscillations to occur also in higher Landau and spin levels. The Fermi wave vector for composite fermions at v - , n + 89 is reduced by a factor of v-I/2 as compared to kF of the Fermi surface at B - 0 . In fact, a peak can be seen in Fig. 1
338
VOLUME 71, NUMBER 23
PHYSICAL '1
4
i
-
'
.~1
2
REVIEW
/ f /
..~4c
-2 -4
0.0
1!o
21.0 1/d(~m-')
""3.0
FIG. 3. Main resonance positions around B - O (circles)and around v - ~ (squares) as a function of inverse antidot period I/d. The vertical scale represents the external magnetic field for the electron case and the effective magnetic field Br -Bit2 for the composites. The peak positions have been corrected for the small density differences between the samples. The solid (dashed) lines show the calculated electron (fcrmion) peak positions. The slopes of the lines differ by ,J2.
at v - 89 although the s - I doublet remains unresolved. While our experiments were performed in a static geometry, related geometrical resonances are also expected in S A W experiments when the wave vector of the surface phonons becomes commensurate with the classical cyclotron orbit [20]. Such resonances have been observed by Willett et al. [31]. In summary, we have observed dimensional resonances of new particles at v - ~ - filling factor of a Landau level. The resonances scale by exactly a factor of v~" between traditional electron resonances and those of the new composite particles as expected from their spin polarization. The observation of the resonances and their appropriate scaling demonstrates the semiclassicai motion of composite fermions and suggests that in transport experiments around v - ~ , these new particles, in many aspects, behave like ordinary electrons. It is remarkable that the complex electron-electron interaction in the presence of a magnetic field can be described in such simple, semiclassical terms. We would like to thank R. L. Wiilett for discussions and assistance with the e-beam lithography during earlier phases of the experiment.
[1] D. C. Tsui, H. L. Stormer, and A. C. Gossard, Phys. Rev. Lett. 48, ! 559 (1982). [2] The Quantum Hall Effect, edited by R. E. Prange and S. M. Girvin (Springer-Verlag, New York, 1990). [3] T. Chakraborty and P. Pietilainen, The Fractional Quan-
LETTERS
6 DECEMBER 1993
turn Hall Effect (Springer-Verlag, New York, 1988). [4] See, for example, Proceedings of the 9th International Conference on Electronic Properties of Two-Dimensional Systems [Surf. Sci. 263 (1992)]. [5] G. S. Bocbinger, H. W. Jiang, L. N. Pfeiffer, and K. W. West, Phys. Rev. Lctt. 64, 1793 (1990). [6] Y. W. Such, L. W. Engel, M. B. Santos, M. Shayegan, and D. C. Tsui, Phys. Rev. I.,r 68, 1379 (1992). [7] J. P. Eiscnstein, G. S. Boebinger, L. N. Pfeiffer, K. W. West, and Song He, Phys. Rev. Lett. 68, 1383 (1992). [8] The new particles are termed "'composite fermions" (Ref. [! 4]) as well as "Chcrn-Simons gauge transformed fermions" (Ref. [20]). [9] H. W. Jiang, H. L. Stormer, D. C. Tsui, L. N. Pfeiffer, and K. W. West, Phys. Rev. B 40, 12013 (1989). [10] R. L. Willett, M. A. Paalanen, R. R. Ruel, K. W. West, L. N. Pfeiffer, and D. J. Bishop, Phys. Rev. Lett. 65, !12 (1990). [I l] R. B. Laughlin, Phys. Rev. Left. 50, 1395 (1983). [12] F. D. M. Haldanr Phys. Rcv. Lett. 51,605 (1983). [13] B. !. Hall~rin, Phys. Rev. Lctt. 52, 1583 (1984); 52, 2390(E) (1984). [14] J. K. Jain, Phys. Rev. Lett. 63, 199 (1989); Phys. Rev. B 40, 8079 (1989); 41, 7653 (1990). [15] S. M. Girvin and A. H. MacDonald, Phys. Rev. Lett. 58, 1252 (1987). [16] R. B. Laughlin, Phys. Rev. Lett. 60, 2677 (1988). [17] S. C. Zhang, H. Hanson, and S. A. Kivelson, Phys. Rev. Lett. 62, 82 (I 989). [ 18] N. Read, Phys. Rev. Lett. 62, 86 (! 989). [19] A. Lopez and E. Fradkin, Phys. Rev. B 44, 5246 (1991). [20] B. l. Halperin, P. A. Ler and N. Read, Phys. Rev. B 47, 7312 (1993). [21] V. Kalmeyer and S. C. Zhang, Phys. Rev. B 46, 9889 (1992). [22] R. L. WilieR, R. R. Ruel, M. A. Paalanen, K. W. West, and L. N. Pfeiffer, Phys. Rev. B 47, 7344 (1993). [23] J. P. Eisenstein, L. N. Pfeiffer, and K. W. West, Phys. Rev. Lctt. 69, 3804 (1992). [24] S. He, P. M. Platzman, and B. i. Halperin, Phys. Rev. Lett. 70, 777 (1993). [25] R. R. Du, H. L. Stormer, D. C. Tsui, L. N. Pfeiffer, and K. W. West, Phys. Rev. Lctt. 70, 2944 (1993). [26] H. van Houten, C. W. J. Bcenakker, J. G. Williamson, M. E. I. Brockaart, P. H. M. van Loosdrecht, B. J. van Wees, .I.E. Mooij, C. T. Foxon, and .I.J. Harris, Phys. Rev. B 3 9 , 8556 (1989). [27] R. R. Gerhardts, D. Weiss, and K. yon Klitzing, Phys. Rev. Lctt. 62, 1173 (1989). [28] R. W. Winkler, J. P. Kotthaus, and K. Pioog, Phys. Rev. Lett. 62, I 173 (I 989). [29] D. Weiss, M. L. Roukes, A. Menschig, P. Grambow, K. yon Klitzing, and G. Weimann, Phys. Rev. Lctt. 66, 2790 (1991). [30] R. Fleischmann, T. Geisel, and R. Ketzmerick, Phys. Rev. Lett. 68, 1367 (i 992). [31] R. L. Willett, R. R. Ruel, K. W. West, and L. N. PfeilTcr, preceding Letter, Phys. Rev. Lett. 71, 3846 (1993).
3853
339
VOLUME66, NUMBER 12
PHYSICAL
REVIEW
LETTERS
25 MARCH 1991
Collective Excitations in Antidots K. Kern, D. Heitmann, P. Grambow, Y. H. Zhang, and K. Ploog
Max-Planck-lnstitut fiir Festkfrperforschung, Heisenbergstrasse 1, D-7000 Stuttgart 80, Federal Republic of Germany (Received i 9 October 1990) Antidot structures have been prepared by etching arrays of 100-nm holes into a two-dimensional electron gas of Gaxlm-,,As quantum wells. In the far-infrared response we observe the unique collectiveexcitation spectrum of antidots. It consists of a high-frequency branch which starts, in a magnetic field B, with a negative B dispersion and then increases in frequency with B. A second low-frequency branch corresponds at high B to edge magnetoplasmons which circulate around the holes. For small B this branch approaches the cyclotron frequency, where the electrons perform classical cyclotron orbits around the holes. PACS numbers: 72.15.Rn,73.20.Dx, 73.20.Mf
With today's highly sophisticated submicron lithography it has ~ e possible to prepare very small lateral structures starting from two-dimensional electron systems (2DES) in semiconductors. The ultimate limits are quantum dots, artificial "atoms" which contain only a very small number of electrons on discrete energy levels. t-7 A reversed structure with respect to dots is "antidots," where "holes" are "punched" into a 2DES. There are already several investigations on the transport properties of antidots, s'9 in particular in search for incommensurability, phenomena and the Hofst~idter butterfly in magnetic fields. We have investigated the far-infrared (FIR) excitation spectrum in perpendicular magnetic fields B and observed unique collective excitations which, to our knowledge, have so far not been observed or theoretically predicted. In particular, we observe two branches. A high-frequency resonance exhibits a negative B dispersion at small magnetic fields B and then increases in frequency with B. A second branch at lower frequency corresponds at high B to an edge magnetoplasmon mode which circulates around the circumference of the hole. This low-frequency branch approaches the cyclotron resonance (CR) frequency r at small B where the classical cyclotron orbit is comparable to the hole radius. This indicates that the electrons perform a classical cyclotron orbit around the holes. Exchange of oscillator strength indicates a coupling between the two modes. Antidot samples have been prepared by deep-mesa etchingt~ starting from 2DES in modulation-doped Gaxlnt-xAs/Alylnt-yAs single quantum wells. All samples were grown lattice matched ( x - 0 . 4 8 , y - 0 . 4 7 ) on semi-isolating InP substrates by molecular-beam epitaxy. Typical growth sequences and conditions are discussed in Refs. 11 and 12. A photoresist grid mask was prepared by a holographic double exposure and arrays of holes with typical diameters 2 r g - 1 0 0 - 3 0 0 nm, were etched 100 nm deep, i.e., through the active layer, into the buffer. The period in both lateral directions was 1618
a - 300-400 nm. A scanning electron micrograph of the antidot structure of sample ( b ) i s shown in Fig. 1(a). The Ga,,Inl-xAS system has the advantage of a small effective mass (m* ~0.042m0 at the band edge and x - 0 . 4 7 ) and, as we have shown in Ref. 11, a very small lateral edge depletion width w~r (estimated < 30 nm) at the etched sidewalls of the holes. Thus the "electronic" radius r e - r t +Wde~ of the hole is not much larger
FIG. 1. (a) Scanning electron micrograph taken at an angle of 45~ from an antidot array [sample (b)] with a period a - 3 0 0 nm and holes of diameter 2rs -100 nm etched into a Gaxlnt-xAs/Alylnt-yAs single quantum well. (b) Sketch of the antidot structure. Hatched areas are holes punched into a 2DES. The motion of individual electrons is shown schematically for the high-frequency mode (~o~.) at high magnetic fields, and for the low-frequency mode at low (~ot--)and at high magnetic field (~oh-).
O 1991 The American Physical Society
340
VOLUME66, NUMBER 12
PHYSICAL
REVIEW
than the etched geometrical radius r z. The twodimensional charge density Ns in the samples was varied via the persistent photoeffect and was determined in situ from Shubnikov-de Haas oscillations in quasi-de microwave transmission, i I We will see later that the small effective mass, the small values of re, and also the relatively high values of N, which can be realized in the Gaxln=-xAs system shift the important features of the antidot excitation spectrum to high frequencies which makes them easier to observe as compared, e.g., to the AIGaAs/GaAs system. FIR transmission experiments have been performed in a superconducting magnet cryostat which was connected to a Fourier-transform spectrometer. The spectral resolution was set to 0.5 c m - I . The temperature was 2.2 K. Experimental transmission spectra of unpolarized FIR radiation in perpendicular magnetic fields B for the antidot sample (a) with period a - 3 0 0 nm, hole diameter 2rz - 2 0 0 nm, and carrier density Ns - g x 10 II cm -2 are shown in Fig. 2. We have found that the spectra do not depend on the polarization direction of linearly polarized radiation. Several resonances with different B dispersions are observed. The dispersions and the amplitudes of the resonances are plotted in Fig. 3. The excitation spectrum consists mainly of two modes, a high- and a low-frequency branch labeled ca+ and ca-, respectively. The high-frequency resonances ca+ start for 8 - 0 at ca+0-94 c m - ! . They first decrease in frequency and then increase with B, approaching at high B the CR frequency cac of the 2D sample. The low-frequency branch ca- starts at small B at ~-cac. It then bends down and exhibits for higher B values a negative B dispersion. Figure 3(c) demonstrates that with increasing B, oscillator strength is transferred from the ca- branch, which has a large amplitude at small B, to the ca+ branch, which increases in intensity with B. A very similar behavior has
LETTERS
25 MARCH 1991
been observed on several antidot samples which have been prepared starting from heterostructures and symmetrically or asymmetrically modulation-doped quantum wells with different spacer layers. In Fig. 3(b) we show the dispersion of sample (b) which has an antidot lattice with the same period a - 3 0 0 nm but holes with a smaller diameter of only 2 r t - 1 0 0 nm [see Fig. l(a)] and a
:. . -
,
,
,
l
~
,
i
,
't . . . .
i"'
i ...... '
2o0 : tso
ID ID
,~
100
i
50
-
......
(a)
-
-.
""" / ' " " CR " ,~..~.:~...
i
"/" *
I
I
!
-~
'i
'
L0_
I
I .... 1
I
I
I
I
I
I
'i
|
I
i
I
!'
!'
1
'
I
I
I L,
I[ ....... i
""
."
w+
~x x ~ ~x x xx ~x x x ~
. 0
i
LO+ *
--
I
"
o~o~
-
&
0I
1,50 --
x x~ ~lo(x xx~
. 9
X
x
-
.
o
..'
(b)
CR
1o0 _ 0
X~ ~X X X~ ~ICXX X)I(~(X X XX ~:X X X X X X X
, !
........ I
-
!
I
I
"
i"
I
I
i
t
i
LO;X.,x
i+
I
i
t
i
* .
....... i
I
'
I
I ....
I ....
|
I
I .....
i
~o..O (~+o__,, lX---x--x ''x''x-W+ X
i
~" 95
i
i
I
i
I
I
I
i
I
i
i
i
i
t~_ CR W+
i
i
i
!
i
i
i
(~
\____/_~::
./~x/X/x/X/'~~176 i
o
i
2
I
1
4
I
x'X'-x~. I
6
I
[
8
1
i
10
I
! I
I
12
.... "
14
Magnetic Field B (T) 11~1
0
I
I
50
I
I
I
I
I
I
I
100
Wavenumbers
I
I
I
I
I
1,50
I
I
I
I
200
(cm - 1 )
FIG. 2. Normalized transmission spectra of unpolarized FIR radiation through the antidot sample (a) with hole diameter 2rt - 2 0 0 nm and period a - 3 0 0 nm at various magnetic fields B.
FIG. 3. (a),(b) Experimental dispersions of the high-frequency (co+) and the low-frequency branch (m-) in the antidot system of sample (a) (a-300 nm, 2rs --200 nm) and sample (b) (a-300 nm, 2rt-100 nm), respectively. CR labels the weak resonances near the CR position. Dashed-dotted lines in (a) are the calculated dispersions for dot structures [Eq. (l)] with different o~owhich have been fitted to either the high- or the low-frequency modes at high 8. (c) Amplitude of the antidot excitation for sample (a) (x) and for sample (b)
(o). 1619
341
VOLUME66, NUMBER 12
PHYSICAL
REVIEW
higher density of N s - 2 . 5 • 1012 r -2. The frequencies of both branches are increased significantly (~o+o-150 cm - t ) and the maximum frequency (55 cm - I ) of the co- mode is shifted to a higher B value (B _max~ 8 - 9 T) as compared to Fig. 3(a). This unique excitation spectrum of antidots has so far not been observed or theoretically predicted. We interpret our observations in the following way. At high B the dispersion of both branches resembles the excitation spectrum of quantum dots. 3-7"13'14 For the excitation spectrum of a dot, using the model of a 2D disk with radius r and density Ns, the resonance frequencies are given according to Fetter 14 by co• - [e~+ (a~J2) 21 I/2 + mJ2, (i) a ~ " N , e Z/2m * e=frr ,
where e d is the effective dielectric function of the surrounding medium. At B - 0 dots have only one resonance peak at a)o which splits with increasing B into two modes. The resonances of the higher branch increase in frequency with B and approach ~c. The resonances of the lower branch decrease in frequency with increasing B and represent at high B an edge magnetoplasmon mode, i.e., a collective mode where the individual electrons perform skipping orbits along the circumference inside the dot. For antidots the individual electrons perform skipping orbits along the circumference outside the hole. We have sketched these orbits for high B, co~, in Fig. 1(b). For high B there is not much difference between a dot or an antidot system since here the edge magnetoplasmon frequency is only governed by the circumference of the structure. Thus, also for antidots, we find that with decreasing B the resonances of the low-frequency branch ~ - first increase in frequency. But then, in contrast to dots, starting at a certain magnetic field B m-=x, the re.sonances in antidots decrease in frequency and approach coc. This means that the orbits of the a~_ mode become larger and eventually the electrons can perform classical cyclotron orbits rc-(2xNs)ff2h/eB around the hole. Thus the collective edge-magnetoplasmon excitation gradually changes into a classical CR-like excitation. We have estimated the value Bc-(2xNs)l/2h/ers where the classical cyclotron radius rc becomes equal to the radius of the holes rt and find for the sample (a) shown in Fig. 3(a) Bc--1.4. T (or, including 25-nm depletion width, Bc-1 T). This is indeed the regime where the resonance frequency is close to O~c and the edge magnetoplasmon mode has changed to a classical CR motion around the hole. For sample (b) with the smaller hole diameter 2rg - 1 0 0 nm and the larger N~ we have a larger value of Bc-5 T (3.3 T including 25-nm edge depletion). Indeed, we observe in Fig. 3(b) that the resonances of the ~o_ branch are shifted to higher frequencies as compared to Fig. 3(a) and at B - 3 T the o)mode is very close to the CR. We can observe this up1620
LETTERS
25 M A R C H 1991
ward shift of Bc and B-m~ also directly on one and the same sample if we increase Ns via the persistent photocffect. This strongly supports our interpretation. The high-frequency mode which increases in intensity with increasing B represents at small B a plasmon type of collective excitation of all electrons. A unique behavior is that at small B this resonance shows a weak, but distinct, negative B dispersion which was observed on all our samples where we were able to evaluate the resonance position down to B ~-0. In dots, a positive B dispersion is found? -7 and confined local plasmon oscillations in wire structures start without and then exhibit a positive B dispersion (e.g., Ref. 10). Another interesting feature as shown in Fig. 3(c) is that with increasing B the oscillator strength is transferred from the e0- to the co+ mode, which clearly indicates a coupling between the two modes. At higher fields the w+ branch approaches ~0c and represents, as denoted in Fig. I ( b ) b y e ~ , a CR type of excitation in the region between the holes. This is supported by the observation that we find different values of e~o for the eJ+ (48 c m - i ) and the m - branch (68 cm - I ) if we fit the high-B regime with the dot dispersion (1). This reflects the fact that the confining diameter is, in principle, different for the a~- and the co+ mode. For the co- mode it is determined by the hole diameter 2rz [200 nm for sample (a), 100 nm for sample (b)], and for the oJ+ mode approximately by a diameter between four neighboring holes (220 and 320 nm). Thus we expect different values for o ~ Within the remaining space we can only briefly discuss several further important findings. On all our samples we observe, weakly as compared to the ~0- mode (see the 3 T curve in Fig. 2), a resonance at the position of the CR in the large gap between the a~- and the oJ+ mode [see Figs. 3(a) and 3(b)]. The intensity of this mode is larger on samples with larger 2DES regions with respect to hole regions. So we believe that the CR mode is an intrinsic feature of the mode spectrum of antidots. Another noteworthy experimental fact is that we observe, especially pronounced on sample (b) in Fig. 3(b), an anticrossing of the m+ branch with 2coc, the harmonic of the CR. We attribute this resonant interaction as arising from nonlocal interaction, i.e., from the Fermi pressure, since it is very similar in strength (which is deduced from the amount of the resonance splitting) to that observed for 2D magnetoplasmons in homogeneous 2DES. 15 It is interesting to note that such nonlocal effects have so far not been observed in isolated quantum wires (e.g., Refs. 10 and 11) and that nonlocal interaction occurs in a very different form, i.e., as a resonant coupling of different magnetoplasmon modes and not at 2~Oc, in dot structures. 5 We so far do not know whether nonlocal effects are also the origin of the slight oscillations of the e0+ resonances which we observe for increasing B near the origin [see Fig. 3(a)]. It is also tempting
342
VOLUME66, NUMBER 12
PHYSICAL
REVIEW
to speculate that this arises from incommensurability effects between the classical CR orbit and the geometry of the antidot array. 9 However, this interpretation needs further confirmation. In conclusion, we have investigated the excitation spectrum of antidots. It consists of two branches where the high-frequency branch starts at small B as a collective plasmon excitation with, surprisingly, a negative B dispersion. With increasing B the high-frequency branch approaches the CR. The resonances of the lowfrequency branch start at small B as a classical CR excitation with orbits around the holes and approach with increasing B collective edge-magnetoplasmon modes where the individual electrons perform skipping orbits around the circumference of the holes. ~We thank E. Vasiliadou and C. Lange for expert help in the etching of the samples and the characterization by scanning electron microscopy. We gratefully acknowledge financial support by the Bundesministerium fiir Forschung und Technologie.
IM. A. Reed, J. N. Randall, R. J. Aggarwal, R. J. Matyi, T. M. Moore, and A. E. Wetsel, Phys. Rev. Lett. 60, 535 (1988). 2W. Hansen, T. P. Smith, III, K. Y. Lee, J. A. Brum, C. M.
LETTERS
25 MARCH 1991
Knoedler, J. M. Hong, and D. P. Kern, Phys. Rev. Lett. 62, 2168 (19~9). 3Ch. Sikorski and U. Merkt, Phys. Rev. Lett. 62, 2164 (1989). 4C. T. Liu, K. Nakamura, D. C. Tsui, K. Ismail, D. A. Antoniadis, and H. I. Smith, Appl. Phys. Lett. 55, 168 (1989). ST. Demel, D. Heitmann, P. Grambow, and K. Ploog, Phys. Rev. Lett. 64, 788 (I 990). 6A. Lorke, J. P. Kotthaus, and K. Ploog, Phys. Rev. Lett. 64, 2559 (1990). 7For a recent review on the FIR response of dots, see U. Merkt, in Advances in Solid State Physics, edited by U. R6ssler (Vieweg, Braunschweig, 1990), Vol. 30, p. 77. 8K. Ensslin and P. M..Petroff, Phys. Rev. B 41, 12307 (1990). 9D. Weiss, K. yon Klitzing, and K. Ploog, Surf. Sci. 229, 88 (1990). i~ Demei, D. Heitmann, P. Grambow, and K. Ploog, Phys. Rev. B 38, 12732 (1988). I IK. Kern, T. Demei, D. Heitmann, P. Grambow, K. Ploog, and M. Razeghi, Surf. Sci. 229, 256 (1990). 12y. H. Zhang, D. S. Jiang, R. Cingolani, and K. Pioog, Appl. Phys. Lett. 56, 2195 (1990). t3V. Fock, Z. Phys. 47, 446 (1928). 14A. L. Fetter, Phys. Rev. B 32, 7676 (1985). 15E. Batke, D. Heitmann, J. P. Kotthaus, and K. Ploog, Phys. Rev. Lett. 54, 2367 (1985).
1621
Quantum dots
343
Subject Index
absorption -coefficient 65, 70, 74 - e n e r g y 60, 82,279-281,288 intensity 80-81,279-281,285,288 - spectroscopy 127 - spectrum 31, 65, 94 absorption strength 28 - integrated 28, 30 addition energy 14, 22, 39, 40, 48-50, 104-106 addition spectrum 22, 45, 50, 99, 229 - o f double quantum dots 99 analytic solutions 63-65 anisotropic antidot lattice 114 anisotropic parabolic potentials 10, 100 anisotropic quantum dots 10, 33, 100 angular momentum - of the ground state 177 - selection rules 80 transition 50 anticrossings 258-259, 287 antidot arrays 85, 110 - rectangular 114 square 112-113 antidot lattice 110 - p e r i o d 111 antidot potentials 109, 112,120 antidots 82, 109, 111, 119 correlations in 119-122 - optical spectroscopy 116 -
-
-
-
artificial atoms 7, 23, 176 artificial molecules 97, 321 aspect ratio 120 azimuthal quantum number 13, 92, 103, 218, 238 ballistic regime 1 basis states 55-56 Bernstein modes 29 bulk magnetoplasmon mode bulk mode 81
82
canonical angular momentum 187 capacitance spectroscopy - conventional 19-21, 156-159 - single-electron 21-26 - of self-assembled dots 127, 301, 304 capacitance-voltage (CV) characteristics 302,305 center-of-mass (CM) coordinate 61, 88, 178, 237 center-of-mass (CM) motion 61 charge density 107, 181,252 chaotic system 79 chaotic trajectories 112 charging diagrams 38 - in double-dot system (DDS) 9798 charging effects 35-38 charging energy 29, 35, 270
T. Chakraborty
344 chemical potential 10 Chern-Simons gauge field 115 classical orbits 172-173 classical cyclotron orbit 336, 341 coherent tunneling mode 98 collective excitations in antidots 339 commensurability effects 326 commensurability oscillations 109-111, 328-334 commensurate orbits 325 conductance peaks 10, 37, 40, 226 conductance peak positions 10, 39, 226 conductance spectroscopy 44 conductivity tensor 331 confinement potential 12, 61, 64-65, 74, 77, 88, 94, 100, 106, 280 triangular 106 constant interaction m o d e l 25, 229, 267 coulomb blockade 36, 41,226 oscillations 38 regime 273-274 coulomb charging energy 29, 126-127 - o f s shell 307 - in p and d shell 306-307 coulomb-coupled dot-pair 74 coulomb diamond diagrams 38, 99 coulomb energy 58, 60, 62, 226 coulomb integral 54 coulomb interaction 53, 56, 62, 65-66, 73-74, 85 coulomb island 225 coulomb matrix elements 54 coulomb oscillations 36, 270, 315 coulomb staircase 46 coupled quantum dots 73-75, 261 current density 107 current oscillations 317 current-voltage characteristics 154-155 current-voltage staircase 46-48 cyclotron frequency 12, 238 -
-
-
cyclotron radius 110, 115, 119, 219 cyclotron resonance 32, 219 density of states (DOS) 2-3, 196, 302, 310 - local 123 - thermodynamic 19 deep-mesa-etched quantum dots 26-27, 33 degenerate states 10, 48, 79, 195 diamagnetic shift 295-296 diamond diagrams 38, 99 differential capacitance 10, 306 differential conductance 15, 38, 47, 272275 dimensional resonances 336 dipole approximation 17, 178, 258 dipole matrix elements 17-18, 26, 258 dipole modes 28-33 dipole operators 18, 61,278 dipole transitions 28, 74, 103, 218 dipole transition energy 28-33, 74, 258259 discrete energy levels 3, 19-20, 90 disordered antidot lattice 329-330 double-barrier heterostructures 14, 4445, 47 double quantum dots 60, 75, 96-100 edge channels 160 edge mode 32, 81, 85, 279 edge magnetoplasmons (EMP) 83-84 effective filling factor 178 effective mass 12, 78, 104, 217, 307 effective Rydberg 164 electric dipole approximation 278 electric dipole moment 209 electric dipole transition energy 259 electrochemical potential 35, 230, 270, 317 electron-hole separation 285
Quantum dots electron turnstile 40-42 electronic heat capacity 67, 178 elliptical quantum dots 33, 100-105 emission lines 291-292 energy gap 120 energy level diagram 306 energy levels of self-assembling dots 303 energy spectrum single-electron 12-15, 158 many-particle 57-59 - d o t with an impurity 78-80 evolution of energy levels 9, 11,164-165 exact diagonalization method 55-58, 77 exchange effect 320 excitation spectroscopy 319 excitation spectrum of antidots 340 excitons in a quantum dot 87-94 far-infrared (FIR) - absorption spectrum 31, 65 - absorption energy 60 - spectroscopy 28-33, 94, 116-118 Fermi wavelength 110 Fermi wavevector 115, 336 filling factor 21,178 - Landau level 24 field-effect confined quantum dots 26-27 Fock-Darwin diagram 14 Fock-Darwin levels (FDL) 12, 258 - index 54 - state 241 fractional quantum Hall effect (FQHE) 4-5, 177 - regime 71 - i n antidots 114-116, 119-122 gaussian impurity potential 77, 278 gaussian orthogonal ensemble 108 gaussian unitary ensemble 108 generalized Kohn theorem 60-61 ground state angular momentum 177
345 half-filled Landau level 31, 117, 335 half-filled shells 10, 106 harmonic confinement 158, 171 harmonic oscillator potential 168, 218, 237 Hartree approximation 10, 70 Hund's rule 10, 315, 318 hydrogenic exciton 87, 283 incompressibility 25 interacting quantum dots 55-56 - basis states 55-56 - energy spectra 57-59 - matrix elements 53-55 interaction matrix elements 284 Kondo effect in quantum dots Kondo resonance peak 131
130-131
ladder operators 64 Landau-level degeneracy 195 Larmor frequency 168 Laughlin state 119 Laughlin wave function 252 magic angular momentum 66-67, 96 magic numbers 59-60, 243, 251 - o r i g i n of 60, 242, 250, 254-255 magnetic length 12, 229, 251 magnetic moment 174, 187, 239 magnetic dipole moment 210 magnetic quantum dots 130 magnetization 67, 233-236, 243 - of non-interacting electrons 68, 235 - operator 233 magnetoluminescence - energy 70 - spectroscopy 91 magnetoplasmon dispersion 30-32
T. Chakraborty
346 magnetoplasmon resonance 32 magnetoresistance 110, 111-116 oscillations 109, 115,324 - peaks 112 model interaction 64 multiexciton complexes 93 -
nanostructures 1 natural quantum dots 90 negative differential resistance 271 negative differential conductance 38, 275 nonintegrability 107 nonlinear transport effects 270 one dimensional interferometer 160 optical absorption 80-81 - energy 82, 288 - intensity 80-84, 90, 285, 288 - spectrum 116 optical spectroscopy 27-34 - of antidots 116-119 - on single quantum dots 91 optical transition 26-34 oscillator strength 26 overlap matrix elements 71, 73 pair-correlation function 63, 253-254 parabolic confinement potential 12,318 anisotropic 10, 100 deviations from 61 - energy spectrum 13-14 parabolic quantum dot 16-18, 34 photoluminescence spectra 127-128, 293 of single InAs clusters 298-299 pinned orbits 112, 114, 325 photon-assisted tunneling (PAT) 43 photonic molecules 130 photonic quantum dots 130 Poisson distribution 108 -
-
-
Poisson equation 10, 65, 181 polarization linear 17 - circular 17 potential contours 11, 183 probability density 107 probability distributions 243-244, 247 -
quantum box 9, 164 - evolution of energy levels 9, 164165 quantum confinement to zero dimension 153 quantum corrals 122-126 quantum dot arrays 19, 28 quantum-dot helium 62-63 quantum dot laser 128, 313-314 quantum dot molecules 96-100 - electronic states 321 - Coulomb diamonds in 321 quantum dot pair 73-75, 258 quantum dot stadium 107-108 single-electron states 107 quantum-Hall dots 70-73 quantum numbers - angular m o m e n t u m 18, 77, 176, 306, 320 principal 18, 92, 103 radial 13, 77, 218, 238, 320 quantum point contact (QPC) 35, 160, 269 quantum ring 84-87 -
-
-
rectangular antidot lattice i14, 330-332 relative coordinates 61, 88 relative motion 61 repulsive scatterer in a dot 76-87, 278 - energy spectrum 78-80 - optical absorption spectrum 8087
Quantum dots resistivity tensor 331 resonance frequency 31-32, 83, 94-95, 218 resonance peaks 32 resonance positions 27, 33, 87, 219,222, 262 resonant anticrossing 28, 221 resonant tunneling 9, 14, 45 runaway trajectory 113, 332-333 selection rules 27, 103, 209 spin 72 self-assembled quantum dots (SAQD) 126-128, 293, 297, 301,304, 313 - energy levels 303 self-consistent (SC) model 230 self-organized quantum rings 86-87 shell filling 10, 26, 48-50, 99, 106 shell structure 9, 48, 315 signature of quantum chaos 107 single-electron capacitance spetroscopy (SEES) 21-26, 265 single-electron -states 15-17, 77, 107, 176, 207 -energies 13, 56, 170,176,207,218 - spectrum 13-14, 158 -tunneling (SET) 35, 39, 274 single-electron transport (SETR) 36 - in a double-dot system 98 single-electron wavefunction 15-17, 251, 258 single-particle matrix elements 18, 71, 278 single-particle states 320 singlet-triplet crossing 266 singlet-triplet transition 23, 46, 70, 80, 238 singular gauge transformation i15
347 skipping orbits 117 spectral function 71-72 spectral weight 72 spin blockade 95-96 spin oscillations 237 spin selection rule 72 spin transitions 23, 46-47, 69-70, 122, 236 square antidot arrays 110-113 square antidot lattice 332 square-well dots 65 Stark effect 70 symmetric gauge 12, 77, 88, 100, 107, 181,193, 237, 278, 284 threshold current 128, 310-311 - density 314 tilted-field effects 94-95 - resonance frequencies 94 transition - amplitude 26 - frequency 27 matrix elements 258 - p r o b a b i l i t y 17, 209, 258 transmission spectra 28, 262-263, 340 transport effective mass 308 transport regime 3 8 transport spectroscopy 34-40, 225 triangular antidot lattice 332 tunneling barrier 22, 37, 42, 44, 74, 97 tunneling probability 71 tunneling rate 14, 24-25, 71-73 tunneling spectroscopy 38, 47, 123 vector potential 12, 64, 77, 88,100, 107, 193, 237, 250, 278, 284 vertically coupled dots 60, 75, 96, 98-99 vertical quantum dots 9, 44-50 vertical transport 8 wetting layer 127
T. Chakraborty
348 Wigner crystal 9 Wigner lattice 9, 165 Wigner lattice 9, 165
Zeeman energy 64, 104, 238 Zeeman splitting 28, 293 zero-dimensional states 91,160-163,289 lasing through 313-314 zero-dimensional system 3, 156 in narrow quantum wells 289 -
Zeeman bifurcation 19 Zeeman effect 92,293
-
