Quantum Dots-Tapash Chakraborty

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 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 T Chakraborty Figure 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 + 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 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 2m*ky+E x+E~, j = 1, 2, , 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, , 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 ~., - 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 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 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 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 (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 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 T Chakraborty 0.3 el PS ~ o0 I / o / ~ e O ~1 ~ i - - / i I oO i ' iI - i 0.2 o - , e 1! l l - - e o = = = ," i / II II / - o , _, = = : _ - - - - - - - - - _ - -e , , - - - - - - : : : : : - _- -_- - el ii ii " ,' ,' ' _ el oO , , " ,-, ," ,,;, -" "" e0 / ,/ r ," ,,,," i i / ol / eO i i i i i1 iI ' - .-/ i1 / ' " , ,, / o iI II ,, / " i I n l 0.I / iI , ~ -/ 11/ ,,~ ;,; i -" = ,/'/ ' ,, ,' ,' e i ' , -" / o0 -' =:::::~ ol " el " - e0 e0 (a) (b) (c) (d) (e) F i g u r e 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) 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, 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 - 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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 / ( 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 / ( 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 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 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 - and then probe the equivalent semiclassical behavior of these bizarre composite particles around v - ~ - One of the particularly telling signatures for the compos- ' I :,"~-:~: I i- 213 ~ i~ 3I ~o 0 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 antidots LETTERS 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 - 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 x jl cm -2 and mobility - 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 - 0 to d - 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 - All experiments were performed in a i 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 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 - 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 - , 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 - 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 Figure shows four sets of Rxx data taken around B 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 - , 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 0.8 PHYSICAL I ' " ' " REVIEW I -(a)bulk 0.0 :~: 3o 0- I -1 : - : I B(Lesla) FIG Expanded views of the magnetoresistances near v - ~ and B 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 - 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 - 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 - 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 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 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 - peaks present around v - ~-, the overall resistance behavior around B - 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 - and v - ~ - On the other hand, the temperature dependence of both phenomena differs considerably While the electron resonances around B - 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 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 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 - 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 - In fact, a peak can be seen in Fig 338 VOLUME 71, NUMBER 23 PHYSICAL '1 i - ' ~1 REVIEW / f / ~4c -2 -4 0.0 1!o 21.0 1/d(~m-') ""3.0 FIG 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 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 , 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 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 - , y - ) 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 r g - 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 - ) 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 (a) Scanning electron micrograph taken at an angle of 45~ from an antidot array [sample (b)] with a period a - 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 - 0 nm, hole diameter 2rz - 0 nm, and carrier density Ns - g x 10 II cm -2 are shown in Fig We have found that the spectra 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 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 - 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 - 0 nm but holes with a smaller diameter of only r t - 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 I I I I I I 'i | I i I !' !' ' I I I L, I[ i "" " w+ ~x x ~ ~x x xx ~x x x ~ i LO+ * I " o~o~ - & 0I 1,50 x x~ ~lo(x xx~ X x - o ' (b) CR 1o0 _ 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 I I x'X'-x~ I I [ i 10 I ! I I 12 " 14 Magnetic Field B (T) 11~1 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 - ) FIG Normalized transmission spectra of unpolarized FIR radiation through the antidot sample (a) with hole diameter 2rt - 0 nm and period a - 0 nm at various magnetic fields B FIG (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 (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 - • 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~ - 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 - 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 - 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 - 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 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 We so far 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 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 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 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 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 wetting layer 127 T Chakraborty 348 Wigner crystal 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 - ...vi T Chakraborty like to express my gratitude to Professor Peter Fulde for his continuing support and... 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... expectation that the miniaturization will lead to new type of electronic (and optoelectronic) devices T Chakraborty Figure The evolution (shown schematically) of a (a) three-dimensional electron systems