Electronic structure of quantum dots - USPmacbeth.if.usp.br/~gusev/QuantumDots.pdfquasi-one-dimensional rings, deformed dots, and dot molecules are considered. CONTENTS I. Introduction

REVIEWS OF MODERN PHYSICS, VOLUME 74, OCTOBER 2002

Electronic structure of quantum dots

Stephanie M. Reimann

Mathematical Physics, Lund Institute of Technology, 22100 Lund, Sweden

Matti Manninen

Department of Physics, University of Jyväskylä, 40351 Jyväskylä, Finland

(Published 26 November 2002)

The properties of quasi-two-dimensional semiconductor quantum dots are reviewed. Experimentaltechniques for measuring the electronic shell structure and the effect of magnetic fields are brieflydescribed. The electronic structure is analyzed in terms of simple single-particle models,density-functional theory, and ‘‘exact’’ diagonalization methods. The spontaneous magnetization dueto Hund’s rule, spin-density wave states, and electron localization are addressed. As a function of themagnetic field, the electronic structure goes through several phases with qualitatively differentproperties. The formation of the so-called maximum-density droplet and its edge reconstruction isdiscussed, and the regime of strong magnetic fields in finite dot is examined. In addition,quasi-one-dimensional rings, deformed dots, and dot molecules are considered.

CONTENTS

I. Introduction 1284A. Shell structure 1284B. ‘‘Magic numbers’’ in finite fermion systems 1286

II. Quantum Dot Artificial Atoms 1287A. Fabrication 1287B. Coulomb blockade 1288C. Probes of single-electron charging 1291

1. Gated transport spectroscopy 12922. Single-electron capacitance spectroscopy 12923. Transport through a vertical quantum dot 1293

III. Addition Energy Spectra 1293A. Many-body effects in quantum dots 1294B. Density-functional method 1295C. Parabolic confinement 1296D. Addition energy spectra described by mean-field

theory 1296E. Reproducibility of the experimental addition

energy spectra 1298F. Oscillator potential with flattened bottom 1298G. Three-dimensionality of the confinement 1299H. Triangular quantum dots 1300I. Elliptic deformation 1300J. Self-assembled pyramidal quantum dots in the

local spin-density approximation 1302IV. Internal Electronic Structure 1302

A. Classical electron configurations 1303B. Spin polarization in the local spin-density

approximation 13041. Circular and elliptic quantum dots 13042. Quantum wires and rings 1305

a. Quantum wires 1305b. Quantum rings 1305

3. Artifacts of mean-field theory? 1306C. Mean-field versus exact diagonalization 1306D. Quasi-one-dimensional systems 1308

1. One-dimensional square well 13082. Quasi-one-dimensional quantum rings 1309

E. Quantum Monte Carlo studies 13101. Hund’s rule 13102. Wigner crystallization 1311

0034-6861/2002/74(4)/1283(60)/$35.00 128

V. Magnetic Fields: Addition Energy Spectra and aSingle-Particle Approach 1311

A. Harmonic oscillator in a magnetic field 13121. Fock-Darwin spectrum and Landau bands 13122. Constant-interaction model 1312

B. Measurements of addition energy spectra 13131. Tunneling spectroscopy of vertical dots 13132. Gated transport spectroscopy in magnetic

fields 13143. B-N phase diagram obtained by single-

electron capacitance spectroscopy 13154. B-N phase diagram of large vertical

quantum dots 1315VI. Role of Electron-Electron Interactions in Magnetic

Fields 1316A. Exact diagonalization for parabolic dots in

magnetic fields 1316B. Electron localization in strong magnetic fields 1319

1. Laughlin wave function 13192. Close to the Laughlin state 13193. Relation to rotating Bose condensates 13204. Localization of electrons and the Laughlin

state 13215. Localization and the many-body spectrum 13226. Correlation functions and localization 1323

VII. Density-Functional Approach for Quantum Dots inMagnetic Fields 1324

A. Current spin-density-functional theory 1324B. Parametrization of the exchange-correlation

energy in a magnetic field 1325C. Ground states and addition energy spectra

within the symmetry-restricted current-spin-density-functional theory 13251. Angular momentum transitions 13252. Addition energy spectra in magnetic fields 1326

D. Reconstruction of quantum Hall edges in largesystems 13261. Edge reconstruction in current-spin-density-

functional theory 13272. Phase diagram 1327

E. Edge reconstruction and localization inunrestricted Hartree-Fock theory 1328

©2002 The American Physical Society3

1284 S. M. Reimann and M. Manninen: Electronic structure of quantum dots

F. Ensemble density-functional theory andnoncollinear spins 1328

VIII. Quantum Rings in a Magnetic Field 1329A. Electronic structure of quantum rings 1329B. Persistent current 1331

IX. Quantum Dot Molecules 1331A. Lateral-dot molecules in the local spin-density

approximation 1333B. Vertical double dots in the local spin-density

approximation 1334C. Exact results for vertical-dot molecules in a

magnetic field 1334D. Lateral-dot molecules in a magnetic field 1335

X. Summary 1335Acknowledgments 1336References 1336

I. INTRODUCTION

Low-dimensional nanometer-sized systems have de-fined a new research area in condensed-matter physicswithin the last 20 years. Modern semiconductor process-ing techniques allowed the artificial creation of quantumconfinement of only a few electrons. Such finite fermionsystems have much in common with atoms, yet they areman-made structures, designed and fabricated in thelaboratory. Usually they are called ‘‘quantum dots,’’ re-ferring to their quantum confinement in all three spatialdimensions. A common way to fabricate quantum dots isto restrict the two-dimensional electron gas in a semi-conductor heterostructure laterally by electrostaticgates, or vertically by etching techniques. This creates abowl-like potential in which the conduction electronsare trapped. In addition to the many possibile techno-logical applications, what makes the study of these ‘‘ar-tificial atoms’’ or ‘‘designer atoms’’ (Maksym andChakraborty, 1990; Chakraborty, 1992, 1999; Kastner,1992, 1993; Reed, 1993; Alivisatos, 1996; Ashoori, 1996;McEuen, 1997; Kouwenhoven and Marcus, 1998; Gam-mon, 2000) interesting are the far-reaching analogies tosystems that exist in Nature and have defined paradigmsof many-body physics: atoms, nuclei, and, more recently,metallic clusters (see, for example, the reviews by Brack,1993 and de Heer, 1993) or trapped atomic gases [seethe Nobel lectures by Cornell, 2001, Ketterle, 2001, andWieman, 2001, and, for example, the reviews by Dalfovoet al., 1999, and Leggett, 2001 and the recent book byPethick and Smith (2002)]. Quantum dots added an-other such paradigm. Their properties can be changed ina controlled way by electrostatic gates, changes in thedot geometry, or applied magnetic fields. Their techno-logical realization gave access to quantum effects in fi-nite low-dimensional systems that were largely unex-plored.

After the initial success in the fabrication and controlof mesoscopic semiconductor structures, which are typi-cally about one hundred nanometers in size and confineseveral hundred electrons, many groups focused on thefurther miniaturization of such devices. A breakthroughto the ‘‘atomic’’ regime was achieved with the experi-mental discovery of shell structure in fluctuations of the

Rev. Mod. Phys., Vol. 74, No. 4, October 2002

charging energy spectra of small, vertical quantum dots(Tarucha et al., 1996): the borderline between the phys-ics of bulk condensed matter and few-body quantumsystems was crossed. Much of the many-body physicsthat was developed for the understanding of atoms ornuclei could be applied. In turn, measurements on arti-ficial atoms yielded a wealth of data from which a fun-damental insight into the many-body physics of low-dimensional finite fermion systems was obtained(Kouwenhoven, Austing, and Tarucha, 2001). With fur-ther progress in experimental techniques, artificial atomswill continue to be a rich source of information onmany-body physics and undoubtedly will hold a few sur-prises.

The field of nanostructure physics has been growingrapidly in recent years, and much theoretical insight hasbeen gained hand in hand with progress in experimentaltechniques and more device-oriented applications. Re-viewing the whole, very broad and still expanding fieldwould be an almost impossible task, given the wealth ofliterature that has been published in the last decade. Wethus restrict this review to a report on the discovery ofshell structure in artificial atoms (with a focus on well-controlled dots in single-electron transistors) and sum-marize aspects of theoretical research concerning theelectronic ground-state structure and many-body physicsof artificial atoms. (A review of the statistical theory ofquantum dots, with a focus on chaotic or diffusive elec-tron dynamics, was recently provided by Alhassid,2000.) In our analysis of shell structure, we shall beguided by several analogies to other finite quantal sys-tems that have had a major impact on both theoreticaland experimental research on quantum dots: the chemi-cal inertness of the noble gases, the pronounced stabilityof ‘‘magic’’ nuclei, and enhanced abundances in the massspectra of metal clusters.

A. Shell structure

The simplest approach to a description of finite quan-tal systems of interacting particles is based on the ideathat the interactions, possibly together with an externalconfinement, create an average ‘‘mean field,’’ which, onan empirical basis, can be approximated by an effectivepotential in which the particles are assumed to moveindependently. This a priori rather simple idea forms thebasis of Hartree, Hartree-Fock, and density-functionaltheories. The last, with its many extensions, providespowerful techniques for electronic structure calculationsand is nowadays applied extensively in many differentareas of both physics and chemistry (see the Nobel lec-tures by Kohn, 1999, and Pople, 1999).

The distribution of single-particle energy levels of themean-field potential can be nonuniform, and bunches ofdegenerate or nearly degenerate levels, being separatedfrom other levels by energy gaps, can occur. Such group-ing of levels or the formation of shells (as shown sche-matically in Fig. 1) is a consequence of both the dimen-sionality and the symmetry of the mean-field potential.A high degree of symmetry results in a pronounced levelbunching (Brack et al., 1972). This level bunching ismanifested in many of the physical properties of finite,

1285S. M. Reimann and M. Manninen: Electronic structure of quantum dots

quantal many-fermion systems, such as, for example,their stability, ionization energies, chemical reactivity, orconductance.

The density of single-particle states at the Fermi sur-face is of particular importance for the stability of thesystem. If it is at a minimum, the particles occupy stateswith a smaller energy on average, and consequently thesystem is more bound: shell filling leads to particularlystable states. If a shell is not filled, however, the systemcan stabilize itself by spontaneously breaking its internalsymmetry. For atomic nuclei, for example, such a spatialdeformation of the mean field was confirmed by ananalysis of rotational spectra (see Bohr and Mottelson,1975, and the review by Alder et al., 1956). More re-cently, similar effects were observed from plasmon reso-nances of metallic clusters, as reviewed by Brack (1993),and de Heer (1993).

In a finite quantal system of fermions, the stabilitycondition is that there be no unresolved degeneracy atthe Fermi surface. This condition is met for certain num-bers of confined particles, for which a degenerate shellwith a large energy gap to the next unoccupied shell canbe filled. We illustrate this with the example of an aniso-tropic harmonic-oscillator confinement in two dimen-sions (x ,y),

V~x ,y !512

m* v2S dx21 1d y2D , (1)as an empirical mean-field potential in which a numberN of fermions with an effective mass m* are assumed tomove independently. The ratio d5vx /vy with frequen-cies vx5vAd and vy5v/Ad defines the ratio of semi-axes of the ellipse equipotentials. Imposing the con-straint v25vxvy conserves their area with deformation.The corresponding single-particle energy spectrum,

«nx ,ny~d!5\vF S nx1 12 DAd1S ny1 12 D YAdG , (2)

FIG. 1. Schematic illustration of the bunching of single-particle states (shell structure) in a finite fermion system. Thebinding energy is lower if the single-particle level density at theFermi energy has a minimum. After Brack et al., 1972.


is shown as a function of deformation d in Fig. 2 (left).In the isotropic case d51, one clearly recognizes the(N011)-fold degeneracy for a principal quantum num-ber N05nx1ny50,1,2, . . . . By filling the states withnoninteracting fermions, respecting the Pauli principle,and including spin degeneracy with a factor of 2, one canreach closed shells for a sequence of N52,6,12,20, . . .particles. For these configurations, particular stability isreached, as the degeneracy of the shell is resolved andthe density of states is minimal at the Fermi energy.Adding one more electron to a closed shell would resultin single occupancy of an orbit belonging to the nexthigher shell, and the system would be less stable. In thecase of open shells, however, the degeneracy can be low-ered by deformation and an energetically more favor-able configuration can be reached. (This phenomenon isknown as the Jahn-Teller effect; Jahn and Teller, 1937.)In particular, for noncircular shapes, subshells with de-generacies comparable to the nondeformed case can oc-cur, leading to a pronounced stability at the correspond-ing deformation (Geilikman, 1960; Wong, 1970). Figure2 (right) shows the total energies (nx ,ny«nx ,ny(d). Wesee that depending on the number of confined particlesand deformation, cusps and minima in the total energyoccur for d.1 at frequency ratios where more pro-nounced subshells are formed. As is obvious from theshell patterns in the single-particle spectra, configura-tions with N52, 6, or 12 particles have the lowest ener-gies in the isotropic case, while for N54, 8, or 10, theenergy can be lowered by deformation.

Despite the simplicity of this example, it containssome of the basic features of a finite, fermionic quantumsystem: the occurrence of shells and the correspondingstability for closed-shell configurations, and the ten-

FIG. 2. Effects of deformation. Left: Single-particle states of atwo-dimensional anisotropic harmonic oscillator as a functionof deformation (d.1). Degeneracies in the isotropic case (d51) lead to closed shells for N52,6,12,20, . . . (for noninter-acting fermions, including spin degeneracy) and subshells oc-cur at frequency ratios d5q/p for integer q ,p . Right: Totalenergies (nx ,ny«nx ,ny of the anisotropic harmonic oscillator forN52, 4, 6, 8, 10, and 12 noninteracting particles as a functionof deformation.


FIG. 3. Shell structure and‘‘magic numbers’’ in finite fer-mion systems. Upper left,atomic ionization energies;lower left, separation energiesof atomic nuclei (after Bohrand Mottelson, 1975); upperright, abundance spectra of me-tallic clusters (counting rate inarbitrary units), (after Knightet al., 1984); lower right, differ-ences in the chemical potentialm(N11)2m(N) of disk-shaped quantum dots; inset, de-vice setup, schematic (fromTarucha et al., 1996).

dency to spontaneously break the symmetry of the meanfield in the case of open shells. Subshell closings enhancethe stability of the broken-symmetry states compared tothe nondeformed, open-shell system.

B. ‘‘Magic numbers’’ in finite fermion systems

The Periodic Table, with the appearance of its eightgroups of elements showing similar chemical properties,is the most widely appreciated example of shell struc-ture. Atomic shells are most strikingly seen in the pro-nounced maxima of the ionization energies of neutralatoms for certain atomic numbers Z52,10,18, . . . , cor-responding to the noble gases He, Ne, Ar, . . . (see up-per left panel of Fig. 3). The spherical symmetry of thevery rigid confinement of the electrons caused by thestrong Coulomb potential of the nucleus results in largedegeneracies at the midshell regions. These shells arethen populated according to Hund’s rules: due to thePauli principle and the repulsive Coulomb interaction,the spin is maximized for half filled orbitals. (Note thatin nuclei the interaction is attractive, and consequentlythe spins are paired off.)

In nuclei, the separation energy (i.e., the energy that isrequired to remove a nucleon from the nucleus) of neu-trons and protons shows distinct steps for certain par-ticle numbers, originating from the shell structure (lowerleft panel of Fig. 3). These steps are very similar to theabrupt decrease of the atomic ionization potentials forelectron numbers that exceed the atomic shell closingsby one.

Parametrizing an average mean-field potential and in-cluding spin-orbit coupling, Goeppert-Mayer (1949) andHaxel, Jensen, and Suess (1949) could formulate a shell


model that successfully explained the ‘‘magic numbers’’of nucleons for which particular stability was observed.

In the early 1980s, finite-size clusters of atoms at-tracted much interest: they provided a link between thephysics of single atoms or the smallest molecules, andthe solid. Knight et al. (1984) succeeded in fabricatingclusters consisting of from a few up to hundreds of alkaliatoms by supersonic expansion of a mixture of metalatoms and a carrier gas through a tiny nozzle. Conden-sation of droplets and subsequent evaporation of singleatoms until equilibrium was reached produced stableclusters that could then be counted and size selected.The anomalies in the mass abundance spectra, i.e., thecounting rates of clusters with a given number of atomsper cluster, are striking: for certain numbers of atoms,one observes an enhanced stability of the cluster. Theupper right panel of Fig. 3 shows the cluster countingrate (in arbitrary units) as a function of the number ofatoms per cluster: pronounced maxima are observed forclusters with 2, 8, 20, 40, and 58 atoms. This reminds usof the magic numbers in nuclei mentioned above. Ametal cluster can be described in a simple model thatassumes that the delocalized valence electrons experi-ence a homogeneous positively charged background(‘‘jellium’’) of the atom ions. This approach has longbeen used in solid-state physics to describe, for example,metal surfaces (Lang and Kohn, 1970; Monnier and Per-dew, 1978) or voids in metals (Manninen et al., 1975;Manninen and Nieminen, 1978). Indeed, the jelliummodel of metals provided an explanation for the en-hanced stability of clusters with specific sizes. Density-functional calculations for electrons confined by aspherical jellium of sodium ions (Martins et al., 1981,1985; Hintermann and Manninen, 1983; Beck, 1984;Chou et al., 1984; Ekardt, 1984) actually had suggestedthe enhanced stability of closed-shell configurations forsodium clusters with N52, 8, 20, 34(40), or 58 atoms


shortly before their experimental discovery in 1984. Forlarge cluster sizes, theoretical and experimental evi-dence for a supershell structure, i.e., a beating patternthat envelopes the shell oscillations, was found in 1990(Nishioka, Hansen, and Mottelson, 1990; Pedersen et al.,1991). Much experimental and theoretical effort was de-voted in the following years to detailed studies of theelectronic and geometric structure of metallic clustersand their physical and chemical properties. (For a re-view of shell structures in metallic clusters, see Brack,1993, de Heer, 1993, and the recent monograph byEkardt, 1999).

In beautiful analogy to atoms, nuclei, or clusters, shellstructure can also be observed in the conductance spec-tra of small semiconductor quantum dots. As an ex-ample, the inset to Fig. 3 (lower right panel) schemati-cally shows the device used by Tarucha et al. (1996): inan etched pillar of semiconducting material, a small,quasi-two-dimensional electron island is formed be-tween two heterostructure barriers. The island can besqueezed electrostatically by applying a voltage to themetallic side gate that is formed around the verticalstructure. The dot is connected to macroscopic voltageand current meters via the source and drain contacts.Measuring the current as a function of the voltage onthe gates at small source-drain voltage, one observescurrent peaks for each single electron subsequently en-tering the dot (see Sec. II.B). The spacing between twosubsequent current peaks is proportional to the differ-ence in energy needed to add another electron to a dotalready confining N particles. This quantity is plotted inFig. 3 (lower right panel) for two different dots withdiameters D50.5 mm and D50.44 mm and shows largeamplitudes at electron numbers N52, 6, and 12. Indeed,these numbers correspond to closed shells of a two-dimensional harmonic oscillator. As we shall see, the ad-ditional structures at the midshell regions are a conse-quence of spin alignment due to Hund’s rules, in analogyto the atomic ionization spectra (Zeng, Goldman, andSerota, 1993; Tarucha et al., 1996; Franceschetti andZunger, 2000).

II. QUANTUM DOT ARTIFICIAL ATOMS

Quantum dots constitute an excellent model system inwhich to study the many-body properties of finite fermi-onic systems. Without attempting to review the manyexperimental techniques that have been developed, weprovide in this section a brief introduction to the fabri-cation of these man-made structures (Sec. II.A). A sub-stantial amount of information on the electronic proper-ties of quantum dots is drawn from conductancemeasurements. Here the discrete nature of the electroncharge manifests itself as a Coulomb blockade. This im-portant feature is discussed in Sec. II.B, and more de-tails are given in Sec. II.C regarding the different typesof experimental setups for studying the level spacing.


A. Fabrication

The development in the early 1970s of superlatticestructures (Esaki and Tsu, 1970; Chang et al., 1973) andthe demonstration of carrier confinement in reduced di-mensions by electron and optical spectroscopy in GaAs-AlGaAs quantum wells (Chang, Esaki, and Tsu, 1974;Dingle, Gossard, and Wiegmann, 1974; Esaki andChang, 1974) were of crucial importance for further de-velopments in semiconductor physics. With the trend to-ward miniaturizing electronic devices, systems based ona quasi-two-dimensional electron gas (which can form inheterostructures, quantum wells, or metal-oxide semi-conductor devices; see Ando, Fowler, and Stern, 1982)attracted much attention. By applying metallic gate pat-terns or etching techniques, it became possible to furtherrestrict a two-dimensional electron gas to geometries inwhich the carriers are confined to a ‘‘wire’’ (i.e., a quasi-one-dimensional system) or a ‘‘dot,’’ where the carriermotion is restricted in all three spatial directions (i.e., a‘‘zero-dimensional’’ system).

Experiments on quantum wires like those, for ex-ample, reported in the very early work of Sakaki (1980),led to further investigations of the localization and inter-action effects in one-dimensional systems (Wheeleret al., 1982; Thornton et al., 1986). For the fabrication ofzero-dimensional artificial atoms and the search for ex-perimental evidence of energy quantization, various ap-proaches were taken in the beginning.1 Regarding theobservation of energy quantization, Reed et al. (1988)performed pioneering experimental studies on etchedheterostructure pillars. Figure 4 shows a scanning elec-tron micrograph of these dot structures, which had elec-tric contacts on their top and bottom, respectively.

1See, among others, Smith et al., 1987, 1988; Hansen et al.,1989, 1990; Sikorski and Merkt, 1989; Demel et al., 1990;Lorke, Kotthaus, and Ploog, 1990; Silsbee and Ashoori, 1990;Meurer, Heitmann, and Ploog, 1992.

FIG. 4. Scanning electron micrograph showing etched quan-tum dots. (The white bars have a length of 0.5 mm.) Inset,schematic picture of a single dot structure. After Reed et al.,1988.


(A schematic drawing of the double-barrier heterostruc-ture is shown as an inset in Fig. 4.) Measuring thecurrent-voltage characteristics of single dots, Reed et al.(1988) reported evidence that electron transport indeedoccurred through a discrete spectrum of quantum states.

Single-electron tunneling and the effect of Coulombinteractions in asymmetric quantum pillars was also dis-cussed by Su, Goldman, and Cunningham (1992a,1992b). Guéret et al. (1992) built an etched double-barrier vertical-dot structure, surrounded by a metallicand a separately biased Schottky gate, that allowed avariable control of the lateral confinement. In additionto avoiding edge defects and allowing for a rathersmooth confinement of the electrons, with this deviceone can control the effective size of the quantum dot byvarying the voltage on the vertical gate. Despite theseefforts, it was not until 1996 when, with a rather similarsetup (see the inset to the lower right panel in Fig. 3),Tarucha et al. could obtain for the first time very clearexperimental evidence for energy quantization and shellstructure on a truly microscopic level. We shall return tothese measurements and their theoretical analysis lateron.

Another method frequently used to create quantumconfinement in a semiconductor heterostructure is thelithographic patterning of gates, i.e., the deposition ofmetal electrodes on the heterostructure surface. An ex-ample is shown in Fig. 5, here for an inverted GaAs-AlGaAs heterostructure. Application of a voltage to thetop gate electrodes confines the electrons of the two-dimensional electron gas that is formed at the interfacebetween the different semiconductor materials (see Mei-rav, Kastner, and Wind, 1990).

Other examples of the creation of quantum dots arethe selective and self-assembled growth mechanisms ofsemiconducting compounds (Petroff et al., 2001). In theStranski-Krastanow process (Stranski and von Krast-anow, 1939), a phase transition from epitaxial structureto islands with similar sizes and regular shapes takesplace, depending on the misfit of the lattice constants(strain) and the growth temperature. For a descriptionof the self-organized growth of quantum dots at the sur-faces of crystals we refer the reader to the monographby Bimberg, Grundmann, and Ledentsov (1999). The

FIG. 5. Lateral device structure. Left, schematic drawing of alateral device structure; right, scanning electron micrograph ofthe sample. From Meirav, Kastner, and Wind, 1990.


growth conditions determine the form of self-assembleddots, which, for example, can be pyramidal, disk shapedor lens shaped (Marzin et al., 1994; Petroff and Den-baars, 1994; Grundmann et al., 1995; Notzel et al., 1995).Drexler et al. (1994), Fricke et al. (1996), Miller et al.(1997), and Lorke and Luyken (1997, 1998) probed theground states and electronic excitations of small self-assembled quantum dots and rings by far-infrared andcapacitance spectroscopy. Double layers of verticallyaligned quantum dots were investigated by Luyken et al.(1998). A theoretical analysis of the few-electron statesin lens-shaped self-assembled dots compared well withthe experimental results of Drexler et al. (1994) andshowed that the calculated charging and infrared ab-sorption spectra reflect the magnetic-field-induced tran-sitions between different states of interacting electrons(Wójs and Hawrylak, 1996). Ullrich and Vignale (2000)were the first to provide time-dependent spin-density-functional calculations of the far-infrared density re-sponse in magnetic fields and were able to reproduce themain features of the far-infrared spectroscopy measure-ments by Fricke et al. (1996) and Lorke et al. (1997).Fonseca et al. (1998) performed an analysis of theground states of pyramidal self-assembled dots withinspin-density-functional theory, as discussed briefly inSec. III.J.

Quantum dots and quantum wires can also be fabri-cated by the so-called cleaved-edge overgrowth (Pfeifferet al., 1990; see also Wegscheider, Pfeiffer, and West,1996 and Wegscheider and Abstreiter, 1998). Much ex-perimental and theoretical work has concentrated onoptical excitations, as summarized in the monograph byJacak, Hawrylak, and Wójs (1998). The latter work alsoprovides a comprehensive review on studies of excitonsin quantum dots.

B. Coulomb blockade

Electron transport through a quantum dot is studiedby connecting the quantum dot to surrounding reser-voirs. The fact that the charge on the electron island isquantized in units of the elementary charge e regulatestransport through the quantum dot in the Coulombblockade regime (Kouwenhoven and McEuen, 1999).Here the transport between the reservoirs and the dotoccurs via tunnel barriers, which are thick enough thatthe transport is dominated by resonances due to quan-tum confinement in the dot (Tanaka and Akera, 1996).This requires a small transmission coefficient throughthe barriers, and thus the tunnel resistance has to belarger than the quantum resistance h/e2. If the dot isfully decoupled from its environment, it confines a well-defined number N of electrons. For weak coupling, de-viations due to tunneling through the barriers are small,leading to discrete values in the total electrostatic en-ergy of the dot. This energy can be estimated by N(N21)e2/(2C), where C is the capacitance of the dot.Thus the addition of a single electron requires energyNe2/C , which is discretely spaced by the charging en-ergy e2/C . If this charging energy exceeds the thermal


energy kBT , the electrons cannot tunnel on and off thedot by thermal excitations alone, and transport can beblocked, which is referred to as a Coulomb blockade(Averin and Likharev, 1986, 1991; Grabert and Devoret,1991; see also Kouwenhoven, Marcus, et al., 1997, Kou-wenhoven, Oosterkamp, et al., 1997, and Kouwenhovenand McEuen, 1999).

Single-electron charging effects in electron tunnelingwere first studied by transport measurements on thinfilms of small metallic grains (Gorter, 1951; Giaever andZeller, 1968; Lambe and Jaklevic, 1969; Zeller and Gi-aever, 1969). In 1975, Kulik and Shekter pointed out thatin a double-junction system, the current through a smallgrain at low bias voltages is blocked by the charge on theisland, whereas the differential conductance can vary pe-riodically at a higher bias. The Coulomb blockade andthe Coulomb ‘‘staircase’’ were observed by Kuzmin andLikharev (1987) and by Fulton and Dolan (1987) forgranular systems and thin-film tunnel junctions, respec-tively. Single-electron charging effects were further in-vestigated for one-dimensional arrays of ultrasmall tun-nel junctions by Kuzmin et al. (1989) and Delsing et al.(1989a, 1989b). Scott-Thomas et al. (1989) found peri-odic variations in the conductance of a narrow disor-dered channel in a Si inversion layer, for which vanHouten and Beenakker (1989) suggested an interpreta-tion in terms of single-electron charging effects. Theirexplanation was based on the assumption that chargedimpurities along the narrow channel would form a par-tially isolated segment, and the conductance oscillationsshould arise from its sequential, quantized charging.Groshev (1990) argued that the experiment by Reedet al. (1988) mentioned above can be better understoodby taking Coulomb charging effects into account. To beable to observe the conductance oscillations in a morecontrolled way, Meirav, Kastner, and Wind (1990) sug-gested the construction of a narrow channel by a litho-graphically defined gate structure, as shown in Fig. 5.The two barriers define the coupling of the channel to itssurroundings. Figure 6 shows the result of the experi-

FIG. 6. An example of the first measurements of Coulombblockade as a function of the gate voltage, observed for a lat-erally confined narrow channel. The figure is taken from Meir,Wingreen, and Lee, 1991 and goes back to the experimentalwork of Meirav, Kastner, and Wind, 1990. (a),(c) regions ofCoulomb blockade; (b) conductance region, as schematicallyillustrated in Fig. 7.


ment, in which the conductance of the double-barrierchannel was measured as a function of the gate voltageat different temperatures (see also Meir, Wingreen, andLee, 1991). One can see how the Coulomb blockade af-fects transport: clear peaks, equidistantly spaced, areseparated by regions of zero conductance. For an earlyreview on conductance oscillations and the related Cou-lomb blockade, see Kastner, 1992.

The possibility of forming quantum dots by gates on aheterostructure was later used by many different groups,and measurements of Coulomb blockade spectra fordots of various sizes and geometries were analyzed. Werestrict our discussion here to the most elementary argu-ments needed to understand the basic features in theexperimental spectra and—for the sake of simplicity—discuss the Coulomb blockade mechanism only on aqualitative level. For a more thorough analysis, see, forexample, Meir, Wingreen, and Lee (1991) and the recentreview by Aleiner, Brouwer, and Glazman (2001) onquantum effects in Coulomb blockade.

Following Kouwenhoven and McEuen (1999), the up-per panel of Fig. 7 schematically illustrates an electronisland connected to its environment by electrostatic bar-riers, the so-called source and drain contacts, and a gateto which one can apply a voltage Vg . (In this example,the quantum dot is formed by the positively chargedback gate, in contrast to Fig. 5, where negatively chargedgates surrounded a region in which the dot is formed.)

The level structure of the quantum dot connected tosource and drain by tunneling barriers is sketched sche-matically in Figs. 7(a)–(c). The chemical potential insidethe dot, where the discrete quantum states are filled withN electrons [i.e., the highest solid line in Figs. 7(a)–(c)],equals mdot(N)5E(N)2E(N21), where E(N) is thetotal ground-state energy (here at zero temperature).When a dc bias voltage is applied to the source s and thedrain d , the electrochemical potentials ms and md aredifferent, and a transport window ms2md52eVsd opensup. In the linear regime the transport window 2eVsd issmaller than the spacing of the quantum states, and onlythe ground state of the dot can contribute to the conduc-tance. By changing the voltage on the back gate, one canachieve an alignment of mdot(N11) with the transportwindow [Fig. 7(b)], and electrons can subsequently tun-nel on and off the island at this particular gate voltage.This situation corresponds to a conductance maximum,as marked by the label (b) in Fig. 6. Otherwise transportis blocked, as a finite energy is needed to overcome thecharging energy. This scenario corresponds to zero con-ductance as marked by the labels (a) and (c) in Fig. 6.The mechanism of discrete charging and discharging ofthe dot leads to Coulomb blockade oscillations in theconductance as a function of gate voltage (as observed,for example, in Fig. 6): at zero conductance, the numberof electrons on the dot is fixed, whereas it is increased byone each time a conductance maximum is crossed. (Ifthe gate voltage is fixed but the source-drain voltage isvaried instead, the current-voltage characteristic showscurrent steps occurring at integer multiples of the


FIG. 7. Single-electron transport in a quantum dot: upper panel, setup for transport measurements on a lateral quantum dot, afterKouwenhoven and McEuen (1999); lower panel, (a)–(c) schematic picture of the level structures for single-electron transport(courtesy of A. Wacker). The solid lines represent the ionization potentials where the upper equals mdot(N), whereas the dashedlines refer to electron affinities, where the lowest one equals mdot(N11). The gate bias increases from (a) to (c) [Color].

single-electron charging energy threshold; see, for ex-ample, Kouwenhoven et al., 1991.)

The distance between neighboring Coulomb peaks isthe difference between the (negative) ionization poten-tial I(N)5E(N21)2E(N) and the electron affinityA(N)5E(N)2E(N11) of the artificial atom (Kastner,1993). It equals the difference in the electrochemical po-


tentials of a dot confining N11 and N electrons, i.e., thesecond differences of the corresponding total ground-state energies E(N):

D2~N !5mdot~N11 !2mdot~N !

5E~N11 !22E~N !1E~N21 !. (3)


Following Kouwenhoven et al. (1991; Kouwenhoven andMcEuen, 1999), in the simple, constant-capacitancemodel (Silsbee and Ashoori, 1990; McEuen et al., 1991,1992) it is assumed that the difference between thechemical potential of a dot confining N or N11 elec-trons can be approximated by the differences of thesingle-particle energies D«5«N112«N , plus the single-electron charging energy e2/C :

D2~N !'D«1e2/C . (4)

As the capacitance of the dot increases and thus leads toa reduced charging energy, the addition energy (the en-ergy required to add an electron to the system) de-creases with increasing number of confined electrons.From Eq. (4) we see that when e2/C@kBT@D« , quan-tum effects can be neglected, and the Coulomb blockadeoscillations are periodic in e2/C . (This, for example, wasthe case for the Coulomb oscillations shown in Fig. 6,measured for a fairly large sample in the mesoscopicregime.)

For a larger transport window 2eVsd , i.e., in the non-linear transport regime, additional structures in the Cou-lomb blockade occur: the excitation spectrum leads to aset of discrete peaks in the differential conductancedI/dVsd (Johnson et al., 1992; Wies et al., 1992, 1993;Foxman et al., 1993). Plotting the positions of thesepeaks as a function of Vsd and the gate voltage, oneobserves a characteristic diamond-shaped structure,which reveals information about the ground and excitedstates; see Fig. 8. (The details of these structures dependon the particular experimental setup; a further descrip-tion is given, for example, by Kouwenhoven, Marcus,et al., 1997 and Kouwenhoven and McEuen, 1999.) Inthe linear regime, i.e., Vsd'0, we observe Coulombblockade as described above. Upon increasing Vsd , ex-cited states (though usually only a few low-lying ones)become accessible and additional transport channels areprovided (see Fig. 7). The vertical gap (Fig. 8) reflectsthe charging energy causing the Coulomb blockade. Be-tween successive diamonds touching at the Vsd50 axis,the electron number increases from N to N11. Similar

FIG. 8. Peak positions in the differential conductance dI/dVsdas a function of Vsd (converted to an energy by some factoreb) and gate voltage (see Foxman et al., 1993) for a quantumdot structure as shown in Fig. 5 above.


observations were reported on vertical quantum dotstructures, which allowed a more detailed analysis of theground and excited states in magnetic fields (see Kou-wenhoven, Oosterkamp, et al., 1997, and Sec. VI).

Studying the ground and excited spectra of a quantumdot by linear and nonlinear magnetoconductance mea-surements, Stewart et al. (1997) demonstrated thatstrong correlations exist between the quantum dotenergy-level spectra of successive electron numbers inthe dot. They observed a direct correlation between theith excited state of an N-electron system and the groundstate of the (N1i)-electron system for i&4. This wassurprising, as a notable absence of spin degeneracy anddeviations from the simple single-particle picture are ex-pected due to the interactions of the particles.

C. Probes of single-electron charging

For metallic systems, signatures of quantum effects inthe Coulomb blockade spectra were discussed at anearly stage by Korotkov et al. (1990), Averin andLikharev (1991), Beenakker (1991), and Meir et al.(1991). More recent experiments were performed on ul-trasmall and very clean metallic nanoparticles (Ralph,Black, and Tinkham, 1997; Davidović and Tinkham,1999). For a recent review, see von Delft and Ralph,(2001).

For semiconductor quantum dots, different experi-mental techniques for probing single-electron chargingeffects on single dots or arrays of dots have been ap-plied, allowing a detailed spectroscopic study of theground and excited states of individual artificial atoms.Infrared and optical spectroscopy was reported for ar-rays of quantum dots (Sikorski and Merkt, 1989; Ba-wendi, Steigerwald, and Brus, 1990; Meurer, Heitmann,and Ploog, 1992). Brunner et al. (1992) were among thefirst to apply optical spectroscopy to individual dots. Thefirst capacitance measurements of quantum dots werereported by Smith et al. (1988). Single-electron capaci-tance spectroscopy has been applied both to arrays(Hansen et al., 1989; Silsbee and Ashoori, 1990; Ashoori,Silsbee, et al., 1992) and to individual quantum dots(Ashoori, Stormer, et al., 1992).

From a microscopic approach to electron tunnelingthrough a quantum dot based on a Hartree-Fock calcu-lation, Wang, Zhang, and Bishop (1994) inferred that,for a small number of electrons, the Coulomb oscilla-tions are nonperiodic and become periodic only in thelarge-N limit. Such irregularities in the single-electrontransport spectra of quantum dots were observed experi-mentally rather early (see Ashoori et al., 1993, andSchmidt et al., 1995). However, sample-dependent inho-mogeneities probably inhibited a very clear observationof shell structure in these pioneering experiments. Forgated vertical-dot structures, Tarucha et al. (1995, 1996)and Austing et al. (1996) were the first to demonstratevery clearly the electronic shell structure of small verti-cal dots.


The methods of gated transport spectroscopy, single-electron capacitance spectroscopy, and single-electrontransport through a vertical quantum dot are brieflytreated in what follows, before we turn to a theoreticalanalysis of the electronic structure of quantum dots.Many of the experimental data can be well understoodwithin a model that assumes the quantum dot to be iso-lated from its environment.

1. Gated transport spectroscopy

A measurement of the Coulomb blockade peaks as afunction of gate voltage for a lateral gate structure wasshown in Fig. 6 above. As this dot was rather large,quantization effects were negligible and the Coulombblockade peaks were equidistant. The sensitivity of‘‘gated transport spectroscopy’’ (Ashoori, 1996) on Cou-lomb blockade and quantum level splittings was demon-strated by McEuen et al. (1991, 1992) for systems confin-ing ;100 electrons. For transport measurements onsmall lateral quantum dots in both the linear and thenonlinear regime, Johnson et al. (1992) observed a com-bined effect of zero-dimensional quantum states andsingle-electron charging. Oscillations in the conductancemeasured on a lateral gated quantum dot with circularshape, formed electrostatically by gates on top of aGaAs-AlGaAs heterostructure, were later reported byPersson et al. (1994), Persson, Lindelof, et al. (1995), andPersson, Petterson, et al. (1995). Here the dot was fairlylarge, confining between about 600 and 1000 electrons.While the discreteness of the quantum states could notbe observed directly from the measurements, the pro-nounced oscillations in the conductance could be relatedto regular shell patterns in the quantum density ofstates, modeled by a simple single-particle model for acircular disk confinement with ideally reflecting walls. Asemiclassical interpretation of these experiments interms of periodic orbits was suggested by Reimann et al.(1996; see also Brack and Bhaduri, 1997). For metallicclusters, a similar analysis had been performed earlier byNishioka, Hansen, and Mottelson (1990). They pre-dicted a supershell structure, as we briefly mentioned inthe Introduction: a beating pattern in the density ofstates is superposed on the individual shell oscillations(Balian and Bloch, 1970, 1971, 1972). Such a supershellstructure was experimentally discovered in mass abun-dance spectra of metallic clusters (Pedersen et al., 1991),but has not yet been observed in two-dimensional quan-tum dot systems (Bøggild et al., 1998).

If the Fermi wavelength is small compared to both thedevice dimensions and the phase coherence length, andif the mean free path exceeds the dot size, the electronmotion in the dot becomes ballistic. Such dots can betreated like electron billiards, where the electrons moveas classical particles but do carry phase information. In-terference effects of trajectories scattered from the con-fining walls of microstructures with irregularly shapedboundaries have been compared to disordered materi-als, in which the electron motion is diffusive and quan-tum interference adds to the classical conductance.These studies of mesoscopic transport, quantum inter-


ference, and classically chaotic systems constitute a largefield of research in themselves. We refer the reader toBeenakker and van Houten (1991) and Altshuler, Lee,and Webb (1991) for reviews of mesoscopic phenomenain semiconductor nanostructures. The recent review byAlhassid (2000) discusses statistical theories in the me-soscopic regime. A description of chaos in ballistic nano-structures was further provided by Baranger and West-ervelt (1998).

Large electron numbers are rather typical for lateralquantum dot structures: it is very difficult to decreasethe number of confined electrons to below about 20(Kouwenhoven, Marcus, et al., 1997), as the tunnel bar-riers formed by the depletion potential become too largefor observation of a current. Progress was made onlyrecently, when Ciorga et al. (2000) demonstrated that bydesigning specially formed gates on a GaAs-AlGaAsheterostructure, one can create lateral quantum dotseven in the few-electron regime.

2. Single-electron capacitance spectroscopy

Ashoori, Störmer, et al. (1992), and Ashoori et al.(1993) applied capacitance spectroscopy to a GaAs tun-nel capacitor containing a microscopic region for chargeaccumulation (see Fig. 9, upper panel) in which thecharge could be varied from zero to thousands of elec-trons. The structure consisted of a thin AlGaAs layer(forming a tunnel barrier), followed by a layer of GaAsforming the quantum well, and a thick layer of AlGaAs

FIG. 9. Single-electron capacitance spectroscopy. Upper panel,scheme of the quantum dot device used in single-electron ca-pacitance measurements by Ashoori, Störmer, et al. (1992),Ashoori et al. (1993), and Ashoori (1996). The arrow indicatesthe tunneling of an electron back and forth between the dotand the bottom electrode in response to a (periodic) voltageapplied to the top gate. Lower panel, capacitance measured asa function of the gate voltage Vg . From Ashoori, 1996.


that acted as an insulator, prohibiting the tunneling ofelectrons to the top electrode. A circular GaAs disk wasgrown on top of this AlGaAs layer, right above thequantum well, and the wafer was covered with a topelectrode. This capacitor was shaped so that when a gatevoltage was applied, electrons were confined in theGaAs quantum well. The quantum dot formed betweenthe two electrodes (top and bottom) was close enough tothe bottom electrode that single electrons could tunnelon and off. Single-electron capacitance spectroscopymeasures the capacitance signal due to the tunneling ofa single electron into the dot, which induces charge onthe top electrode. Detection of this charge makes it pos-sible to accurately determine the gate voltage at which asingle electron can enter the dot.

Figure 9 shows a plot of the capacitance versus gatebias. With increasing positive bias on the top plate, elec-trons tunnel subsequently into the dot. The spacing be-tween the peaks is approximately constant, similar towhat was observed by Meirav, Kastner, and Wind (1990)for larger, lateral quantum dots. For smaller bias, how-ever, the data of Ashoori, Störmer, et al. (1992) andAshoori et al. (1993) show irregularities: the distancesbetween subsequent peaks are increased and are non-uniform. These deviations from the equidistant Cou-lomb blockade spectra can be attributed to energy quan-tization in the dot structure, as we shall explain later.

3. Transport through a vertical quantum dot

Despite the early success in obtaining very clear Cou-lomb blockade spectra (Kastner, 1992), for a rather longtime it remained a challenge to fabricate dots so regularand clean that clear signals of energy quantization andshell structure in the small-N limit could be observed.

In vertical dots (as already described briefly in Secs.I.B and II.A above) one uses thin heterostructure barri-ers that are only very weakly affected by the gate poten-tial. Such dots were the most promising candidates forachieving a truly quantum-mechanical confinement of asmall number of electrons. Their fabrication, however, isdifficult, and until about 1996, transport measurementson such structures were reported by only a few groups(Reed et al., 1988; Dellow et al., 1991; Goodings et al.,1992; Guéret et al., 1992; Kolagunta et al., 1995).Tarucha et al. (1996) worked with a gated vertical quan-tum dot structure (as schematically shown earlier in theinset to Fig. 3, lower right panel). The quantum dot wasmade from a double-barrier heterostructure by etchingtechniques, and the electron puddle was located be-tween two heterostructure barriers that separated itfrom the outside environment. A metallic Schottky gatewas wrapped around the circular etched pillar close tothe dot region. Tunnel junctions allowed Tarucha et al.to vary the electron number N by applying a negativevoltage Vg to the Schottky gate. The energy gap be-tween conductance and valence band could be reducedby includingly indium in the well. The bottom of theconduction band was then below the Fermi level of thecontacts, i.e., electrons could accumulate in the dot even


if no voltage was applied. This made it possible to studyelectron transport even at very small bias voltages. Thegate changed the effective diameter of the island. In theexperiment, the current flowed vertically through thedot in response to a small dc voltage applied betweenthe contacts (i.e., in the linear transport regime assketched in Fig. 7). By measuring this conductance as afunction of the gate voltage Vg , Tarucha et al. (1996)could observe clear Coulomb oscillations, correspondingto a one-by-one increase in the number of confined elec-trons in the dot each time a Coulomb blockade peak wascrossed. The period of the Coulomb peaks showed avery pronounced N dependence: the spacings betweenthe second and third, sixth and seventh, and twelfth andthirteenth peaks were larger than the spacings betweenthe neighboring peaks (Fig. 10). This becomes evenmore clear when looking at the addition energy differ-ences D2(N), which are proportional to the spacings be-tween the Coulomb blockade peaks, as shown in Fig. 3.[Note that Tarucha et al. (1996) used a scaling factor forconverting gate voltages to energies.] Pronouncedmaxima at N52, 6, and 12 can be seen in D2(N) fordevices with a diameter of 0.5 and 0.44 mm.

The vertical quantum dot structure has a diameterthat is about ten times larger than its thickness in thevertical direction. To a good approximation one can as-sume that the motion of the electrons in the z directionis frozen, i.e., only the ground state is occupied, and thedot can be well approximated by a smoothly confinedcircular electron island in two dimensions (x ,y). It isobvious that the maxima in the spacings of the conduc-tance peaks or in the addition energy differences D2(N)are related to shell structure. In fact, the numbers coin-cide with the lowest closed shells of the two-dimensionalharmonic oscillator confinement in the isotropic case,N52,6,12, . . . (see Fig. 2 for d51). Keeping in mindthe discussion of atomic ionization spectra, we shouldfurthermore expect that as a consequence of Hund’srules the pronounced structures at the midshell regionsare related to spin alignment. These issues are discussedin the next section.

III. ADDITION ENERGY SPECTRA

The independent-particle model provides an intuitiveunderstanding of the shell structure. At low electron

FIG. 10. Coulomb blockade oscillations in the linear regime asa function of gate voltage, measured on the vertical gatedquantum dot structure by Tarucha et al. (1996). The quantumdot is estimated to have a diameter of 0.5 mm. The measure-ment was performed at 50 mK.


densities or in strong magnetic fields, however, oneshould turn to more sophisticated models for an accu-rate description.

Hartree-Fock calculations give a first estimate of ex-change effects. The correlation energy can be includedwithin density-functional theory (DFT), while keepingthe description on a mean-field level. This restriction canbe overcome by numerical, ‘‘exact’’ diagonalization ofthe many-body Hamiltonian (the configuration-interaction method), in which one must necessarily re-strict the Hilbert space to a finite basis (i.e., the ‘‘exact’’method is not truly exact), or by quantum Monte Carlomethods. All these methods have been applied to a va-riety of different systems such as circular or deformedquantum dots, rings, and quantum wires.

After some general remarks in Sec. III.A, we turn to abrief discussion of spin-density-functional theory in Sec.III.B. Most authors consider a parabolic confinement forthe external quantum dot potential, which is explainedin Sec. III.C. After setting the stage, results for additionenergy spectra and spin configurations in quantum dotsare discussed in Secs. III.D, III.H, and III.I; these resultscan be directly related to the experimental results ad-dressed in the previous section.

A. Many-body effects in quantum dots

The electrons in the quantum dot belong to the con-duction band of the semiconductor. The conduction-electron density is low, the mean electron-electron dis-tance being of the order of 10 nm. Consequently effectsdue to the underlying lattice and to interaction with thevalence and core electrons can be taken into accountusing the effective-mass approximation: the conductionelectrons in the quantum dot form a separate interactingelectron system with an effective mass m* , and theirmutual Coulomb interaction is screened with the staticdielectric constant of the semiconductor in question.

The many-body Hamiltonian H of a quantum dot, de-coupled from its environment, is usually written as thesum of a single-particle part (kinetic and potential con-tributions) and the two-body part, describing the Cou-lomb interaction between the electrons confined in thedot:

H5(i51

N S pi22m* 1V~ri! D 1(i,jN e2

4pee0uri2rju. (5)

Here, m* is the effective electron mass and e is the di-electric constant of the corresponding background mate-rial. [The energy and length units are frequently given ineffective atomic units, with the effective rydberg Ry*5m* e4/2\2(4pee0)

2 or hartree Ha* 52 Ry* and the ef-fective Bohr radius aB* 5\

2(4pee0)/m* e2, as this allows

a scaling to the actual values for typical semiconductormaterials. For GaAs, aB* 59.8 nm and Ry* 56 meV.]

Interacting electrons confined in a two-dimensionalharmonic trap form a seemingly simple many-bodyproblem: if the number of electrons is not too large,standard methods can be applied. Bryant (1987) was the


first to point out the importance of electron correlationsthat give rise to many intriguing properties of quantumdots. For a two-electron system in a long and narrowrectangular box, he studied theoretically the continuousevolution from single-particle-level structure to a regimein which the electron-electron interactions dominate anda Wigner crystal can form (see also Sec. IV.A). A quan-tum dot confining two electrons (the so-called quantumdot helium; Pfannkuche, Gerhardts, et al., 1993) is thesimplest example for which the eigenstates and spectraof the two-particle Schrödinger equation can be ob-tained analytically (Taut, 1994; see also El-Said, 1996;González, Quiroga, and Rodriguez, 1996; Dineykhanand Nazmitdinov, 1997). For realistic interactions likethe Coulomb repulsion, analytic solutions for N.2 areimpossible to obtain. Only for some modified forms ofthe interparticle interaction, which, however, are ofmore academic interest, have exact analytic solutionsbeen obtained; see, for example, Johnson and Payne,1991; Quiroga, Ardila, and Johnson, 1993; Johnson andQuiroga, 1994, 1995.)

The traditional way to attack a correlated few-electron problem (Bryant, 1987; Maksym andChakraborty, 1990; Pfannkuche, Gerhardts, et al., 1993,Pfannkuche, Gudmundsson, et al., 1993) is to applyconfiguration-interaction methods, which are also fre-quently used in quantum chemistry. (For a detailed de-scription of the exact diagonalization method for quan-tum dots, see Chakraborty, 1999.) In many cases,however, we have to face the drawback that numericaldiagonalization methods are applicable only to fairlysmall numbers of electrons at not too low densities. Ifenough configurations, i.e., linear combinations of Slaterdeterminants made up from the single-particle basisstates, are included in the calculation, the solution con-verges to the exact result and both ground and low-lyingexcited states are obtained with rather high accuracy.The advantage of exact diagonalization methods is that,in addition to the ground-state energy and wave func-tion, all low-lying excitations are computed with essen-tially no extra cost or reduction of accuracy. This is im-portant, since the excitation spectrum can also provideinsight into the electronic structure of the ground state.On the experimental side, transport spectroscopy can beused to identify the quantum numbers of excited statesand offers a direct link between theory and experiment(Kouwenhoven, Oosterkamp, et al., 1997; see Sec.VI.A).

Configuration-interaction calculations have been par-ticularly useful for a description of dots in strong mag-netic fields (see Sec. VI), where restrictions of Hilbertspace allow us to obtain accurate results for dots confin-ing up to about ten electrons (Maksym and Chakraborty,1990; Hawrylak and Pfannkuche, 1993; Pfannkuche,Gudmundsson, and Maksym, 1993; Yang, MacDonald,and Johnson, 1993; Palacios et al., 1994).

An alternative to the configuration-interactionmethod is the use of quantum Monte Carlo calculations,which have been performed for quantum dots in mag-netic fields [see Bolton, 1994a, 1994b (fixed-node quan-tum Monte Carlo), Harju et al., 1999, and Pederiva, Um-


rigar, and Lipparini, 2000 (diffusion quantum MonteCarlo)] as well as in the low-density regime [see Eggeret al., 1999 (variational quantum Monte Carlo) and Secs.VI.A and IV.E].

B. Density-functional method

The mean-field approach offers considerable simplifi-cation: it models a many-body system by noninteractingparticles confined in an average potential into which theinteractions are incorporated. To study large electronsystems in the presence of correlations, density-functional theory in the self-consistent formulation ofKohn and Sham (1965) provides a particularly powerfultool. (For reviews see, for example, Jones and Gunnars-son, 1989; Dreizler and Gross, 1990; Gross, Runge, andHeinonen, 1991.) Density-functional theory is based onthe theorem by Hohenberg and Kohn (1964) and itsgeneralization by Levy (1979): the exact ground-stateenergy of a many-body system is a unique functional ofthe electron density n(r). Its variation with respect tothe density yields an absolute energy minimum for thetrue-ground state density. Initially, DFT was developedin a spin-independent formalism. Effects of spin polar-ization (as they can occur in open-shell atoms or systemswith broken spin symmetry, such as ferromagnets) werelater incorporated by von Barth and Hedin (1972). Thisso-called spin-density-functional theory (SDFT) relieson the assumption that orbital currents give only a neg-lible contribution to the energy functional. A further ex-tension of the theory to include gauge fields was laterformulated by Vignale and Rasolt (1987, 1988) and be-came known as the current-spin-density-functionaltheory (CSDFT). This method, which was frequently ap-plied to the description of artificial atoms in magneticfields, is further described in Sec. VII.

In SDFT, the total energy is a functional of thespin-up and spin-down densities ns(r), where s5(↑ ,↓)labels the spin. Equivalently, we can use the total densityn(r)5n↑(r)1n↓(r) and the spin polarization z(r)5@n↑(r)2n↓(r)#/n(r). By minimizing this functionalone obtains the well-known Kohn-Sham equations,

S 2 \22m* ¹21VKS ,s@n ,z# Df i ,s~r!5e i ,sf i ,s~r!, (6)with the effective potential

VKS ,s@n ,z#5V~r!1e2

4p«0«

3E dr8 n~r8!ur2r8u 1 dExc@n ,z#dns~r! (7)consisting of the external potential, the Hartree contri-bution, and the variational derivative of the exchange-correlation energy. Because the mean-field potentialVKS depends on the single-particle wave functions ordensities, the equations have to be solved self-consistently by iteration. For a finite system with non-uniform density n(r) one often makes the assumption


that locally, the exchange-correlation energy per particlecan be approximated by that of the corresponding infi-nite system at constant density. We label this energy byexc„n(r),z(r)… and write

ExcLSDA@n ,z#5E dr n~r!exc„n~r!,z~r!…. (8)

[The most frequently used parametrizations of exc in thelocal spin-density approximation (LSDA) are discussed,for example, by Dreizler and Gross (1990).] For the two-dimensional electron gas, Tanatar and Ceperley (1989)provided a parametrized form of exc as a Padé form fornonpolarized (z50) and ferromagnetic (z51) cases,obtained by a fit to a small set of numerical Monte Carlodata for a few discrete values of the electron density. Forintermediate polarizations, a standard practice is the adhoc assumption that the unknown polarization depen-dence of the correlation energy can be adopted from theanalytical expression for the exchange energy. [This ap-proach has been implemented for the electron gas inthree dimensions by von Barth and Hedin (1972) andPerdew and Zunger (1981).] For the interpolation, onewrites

exc~n ,z!5exc~n ,0!1f~z!@exc~n ,1!2exc~n ,0!# , (9)

with the polarization dependence (in the two-dimensional case)

f~z!5~11z!3/21~12z!3/222

23/222. (10)

It should be emphasized that the mean-field equationsgiven in Eq. (6) in principle need to be solved in a geo-metrically unrestricted scheme, i.e., the symmetry of thesolution should not be constrained by the symmetry ofthe confinement. In the quantum dot literature, manyauthors choose to simplify the solution of the Kohn-Sham equations by imposing axial symmetry. We refer tothis scheme in the following discussion as the restricted(spin) density-functional approach. A more general so-lution requires unrestricted symmetries in both the spa-tial and the spin parts of the single-particle wave func-tions. We shall return to this point later when discussingthe broken-symmetry ground states in the mean-fielddescription. We note that the degree to which the re-stricted DFT scheme gives a reliable approximation tothe exact ground-state energies and densities dependson the average electron density in the dot. In two dimen-sions for high densities n , i.e., small values of theWigner-Seitz parameter rs51/Apn , the single-particlepart of the Hamiltonian Eq. (5) dominates over the in-teractions. In this case, the solutions of the restrictedDFT scheme compare well with those found in the un-restricted approach. Broken-symmetry solutions origi-nating from spatial deformation of the mean field occurfor larger values of rs , i.e., in the correlated regime.Here, the energies usually are lower than those of thesymmetry-restricted DFT approach, reflecting the gainin correlation energy.

From the Hohenberg-Kohn theorem we know thatthe ground state is characterized by the single-particleground-state density n , which can be determined varia-


tionally by minimizing the total energy functional. Thisargument applies to the lowest state of a given symmetry(Gunnarsson and Lundqvist, 1976). Starting the self-consistent iterations of the Kohn-Sham equations withdifferent initial configurations in general leads to a set ofconverged solutions out of which we have to identify thetrue ground state. (Note that in order to achieve a reli-able scan of the potential-energy surface, it is essentialto start with a rather large set of initial conditions.) Theaddition energy differences D2(N) are then computedfrom the Kohn-Sham ground-state energies after Eq.(3). In this connection, it is useful to note that Koop-mans’s theorem (Koopmans, 1933) connects the negativeof the ionization energy with the highest occupiedsingle-particle level. This theorem was originally formu-lated for the Hartree-Fock approximation and was latergeneralized to density-functional theory by Schulte(1974, 1977) and Janak (1978).

Capelle and Vignale (2001) have recently shown thatin the spin-dependent formalism the effective potentialsare not always unique functionals of the spin densities.While this notion is important for many applications ofSDFT, it does not invalidate the use of LSDA as anapproximate method for calculating ground-state prop-erties using local approximations based on the homoge-neous electron gas.

The time-dependent version of DFT can be used tostudy collective excitations. For applications to quantumdots, see, for example the work by Serra and Lipparini(1997), Serra et al. (1998, 1999), Lipparini and Serra(1998), and Ullrich and Vignale (2000).

C. Parabolic confinement

Kumar, Laux, and Stern (1990) determined the effec-tive single-particle confinement for a square-shapedquantum dot of the type shown in Fig. 5 in a self-consistent Hartree approach, where the electrostaticconfinement was incorporated by a self-consistent solu-tion of the combined Hartree and Poisson equations.They found that in the limit of small particle numbers,the effective confinement can have a symmetry veryclose to circular, even if the confinement was formed bya square-shaped metallic gate pattern. On the basis oftheir work, the simple isotropic harmonic oscillator wasadopted as the standard quantum dot model potentialfor electronic structure calculations, in both exact diago-nalization studies and mean-field approaches. We usedthis model above when we identified oscillatorlike shellstructure and magic numbers in the addition energyspectra (see Sec. I.A).

In many cases, measurements of far-infrared absorp-tion spectra on ensembles of quantum dots (see, for ex-ample, Sikorski and Merkt, 1989; Demel et al., 1990;Lorke, Kotthaus, and Ploog, 1990; Meurer, Heitmann,and Ploog, 1992) correspond to those of a noninteractingsystem. In the view of the generalized Kohn theorem(Kohn, 1959, 1961; Brey et al., 1989; Yip, 1991), this fur-ther supports the assumption of parabolic confinement:the center-of-mass motion separates out and the only


possible dipole excitation is the center-of-mass excita-tion. Kohn’s theorem implies that the effects of electron-electron interactions in a quantum dot can be observedby far-infrared spectroscopy only if the anharmonicity ofthe confinement is sufficiently strong (Gudmundssonand Gerhardts, 1991; Darnhofer and Rössler, 1993;Pfannkuche et al., 1994; Gudmundsson, Braatas, et al.,1995).

Much of the theoretical work modeling the additionenergy spectra for small, parabolic quantum dots hasbeen performed under the assumption that the finitethickness of a quantum dot, typically much smaller thanthe lateral extension of the electrostatic confinement,can be neglected. As a compromise between realisticsimulation and numerical feasibility (Macucci, Hess, andIafrate, 1993), one can separate the effective confine-ment into an in-plane and perpendicular part, V5V(x ,y)1V(z). With the assumption that in the z di-rection only the ground state is occupied, the solutioncan then be restricted to the (x ,y) plane. For the 2Dparabolic confinement V(x ,y)5 12 mv

2(x21y2), if wekeep the oscillator parameter v constant, the electrondensity in the dot increases with N . Experimentally,however, if one makes the voltage on the (side or top)gates less negative, the effective confinement strength vdecreases. At the same time, the number of electrons inthe dot increases. Thus we may also consider keepingthe average electron density in the dot constant, i.e., fix-ing the density parameter rs and varying v with N . In-deed, for vertical dots it turns out that as N increases,the confinement weakens so that the particle densitytends to a constant (Austing, Tokura, et al., 1999). Theaverage value of the electron-density parameter rs forthe circular quantum dot sample studied by Taruchaet al. (1996) is estimated to be between 1.3aB* and 1.4aB* .This value is close to the equilibrium value of the two-dimensional electron gas, rs51.5aB* . For a constant av-erage density the N dependence of the oscillator param-eter v can be approximated by

v25e2

4p«0«m* rs3AN

(11)

(Koskinen et al., 1997).

D. Addition energy spectra described by mean-fieldtheory

Macucci, Hess, and Iafrate (1993, 1995) extended thework by Kumar, Laux, and Stern (1990), including theexchange and correlation contributions within density-functional theory. As a consequence of the degeneraciesintroduced by the symmetry of the dot confinement,they observed a shell-like grouping of the values of thechemical potentials m(N). Extensive density-functionalcalculations were performed by Stopa (1993, 1996) forstudying the Coulomb blockade in 2D dots containing50–100 electrons. Fujito, Natori, and Yasunaga (1996)used an unrestricted Hartree-Fock approach to studyboth the effects of electron spin and the vertical extent


of the trapping potential. Oscillations in the dot capaci-tance caused by shell structure clearly demonstrated theoccupation of single-particle levels in accordance withHund’s rules. The unrestricted Hartree-Fock methodwas also applied to a calculation of the addition energyspectra of cylindrical quantum dots by Szafran, Ad-amowski, and Bednarek (2000).

Macucci, Hess, and Iafrate (1997) determined the ad-dition energy differences D2(N) [see Eq. (3)] for 2Dquantum dots confining up to 24 electrons. They workedin the symmetry-restricted DFT formalism and ne-glected the spin degree of freedom. Figure 11 shows theaddition energy spectra obtained in the local-density ap-proximation (LDA) for material parameters m*50.0648 m0 (where m0 is the bare electron mass) ande512.98. Macucci et al. assumed a parabolic confine-ment for dot radii r


For closed-shell configurations with N52, 6, and 12, alarge energy gap between the highest occupied and thelowest unoccupied Kohn-Sham single-particle levels isfound. The dot is nonmagnetic with ‘‘total’’ spin zero(which in Kohn-Sham theory implies that it has equalspin-up and spin-down electron densities). The system-atic development of the total spin S is plotted in theupper panel of Fig. 12. Comparing the two panels, thefine structure at midshell (i.e., the peaks at N54 andN59) is clearly associated with maximized spin (Zeng,Goldmann, and Serota, 1993; Tarucha et al., 1996; Hi-rose and Wingreen, 1999; Reimann, Koskinen, Koleh-mainen, et al., 1999; Austing et al., 2001). This is ex-pected, as Hund’s first rule implies that the electronspins align up to half filling of a degenerate shell. Thetotal energy is lowered, as exchange energy is gained bythe maximized spin. In the large-N limit, deviationsfrom the above picture (Hirose and Wingreen, 1999) area simple consequence of the increasing nonparabolicityof the mean-field potential (see Fig. 11).

At relatively large electron densities, the 2D SDFTaddition energy spectra compare rather well to resultsobtained in the unrestricted Hartree-Fock formalism(Yannouleas and Landman, 1999). The Hartree-Fockformalism was also applied to spherical quantum dots, inwhich the 3D spherical confinement yields the lowestclosed shells at N52,8,20 (Bednarek, Szafran, and Ad-amowski, 1999). These magic numbers were also ob-served in metallic clusters (Sec. I.B and Fig. 3, upperright panel). However, as the jellium clusters are notrigidly confined, their physical behavior at midshell isdifferent: Jahn-Teller deformation is often energeticallymore favorable than spin alignment due to Hund’s ruleat maintained spherical symmetry.

E. Reproducibility of the experimental addition energyspectra

We have seen above that the unrestricted spin-density-functional formalism with the quantum dot elec-tron density as the only fitting parameter indeed seemsto provide a fairly accurate description of the additionspectra. However, we notice from Fig. 12 that the agree-ment between the theoretical data and the experimen-tally measured addition energies becomes worse with in-creasing electron number N . Comparing the peakstructures in the third and fourth shells, i.e., betweenN56, N512, and 20, the theoretical and experimentalvalues D2(N) show very clear deviations. Recently a se-ries of experimental addition energy spectra for 14 dif-ferent structures was published (Matagne et al., 2001),with diameters between 0.44 and 0.6 mm, similar to thevertical quantum dot device used in the earlier work byTarucha et al. (1996). This work very clearly revealsstrong variations in the spectra from device to device:While all structures show the first shell at N52, only71% of them show shells at both N52 and 6, 64% atN52, 6, and 12, and 21% at N52, 6, 12, and 20. In viewof these recent data, which seems to indicate each singlequantum dot has its own properties, one should be cau-


tious about a quantitative comparison between theoryand experiment like that was in Fig. 12. An explanationfor the disagreement at larger shell fillings could, forexample, be either nonparabolicity of the confining po-tential or the unavoidable inaccuracies in device fabrica-tion that randomly disturb the perfect circular symmetry.

F. Oscillator potential with flattened bottom

The effect of deviations from a pure parabolic exter-nal confinement on D2(N) in the midshell regions wasfurther analyzed by Matagne et al. (2001), taking intoaccount components of the electrostatic quantum dotconfinement in the vertical direction. They solved a 3DPoisson equation for a charge model that included thedoped material above and below the quantum dot in acylindrical configuration. The resulting contributions ef-fectively add an anharmonicity ;lr4 to the potential.Such a perturbation does not affect the single-particlestates of the lowest two shells. In the third shell, how-ever, where the single-particle states un ,m&5u1,0& andu0,62& (with radial and azimuthal quantum numbers nand m) are degenerate in the nonperturbed system, thestates u0,62& with nonzero angular momentum areshifted downwards2 by an energy difference El . Mat-agne et al. (2001) discuss the addition energy spectra forthe three cases Ex.El , Ex;El , and Ex,El , whereEx labels the exchange energy between two spin-parallelelectrons. The chemical potential values m(N)5E(N11)2E(N) were obtained within SDFT in a mannersimilar to that used by Nagaraja et al. (1999; see alsoJovanovic and Leburton, 1994).

Figures 13 and 14 summarize the results of Matagneet al. (2001) for the filling of the third shell, i.e., betweenN56 and N512. For Ex.El (see upper panel of Fig.13) the situation is similar to that of a purely harmonicexternal confinement. When all single-particle states inthe third shell are exactly degenerate, the shells arefilled according to the sequence ((m)((sz)521/2→21→03/2→21→21/2→00 when proceeding from the secondshell at N56 to the third shell at N512 (see Fig. 14).

The addition energy spectra in Fig. 13 are consistentwith the addition energies and spin fillings in Fig. 12above. If the degeneracy between the u0,62& and u1,0&states is lifted by a small difference in energy (see Fig.14, middle panel), the shell-filling sequence changes to((m)((sz)521/2→01→03/2→21→01/2→00. For verysmall splitting the addition energy spectrum does notchange significantly. For larger splitting, at midshell N59 the spin remains at its maximum value (Hund’srule). The enhanced energy difference between the

2Note the similarity to the Nilsson model in nuclear physics,in which phenomenological l2 term for angular momentum land a spin-orbit contribution, which we may neglect here, weresubtracted from the oscillator Hamiltonian to flatten the bot-tom of the harmonic well: states with higher single-particleangular momenta are shifted to smaller energies (Nilsson,1955; Bohr and Mottelson, 1975).


u0,62& and u1,1& states, however, leads to a reduction inD2(9) and to a higher value of the addition energy atN58 (Ex;El , Fig. 13, middle panel). For an evenlarger energy splitting between the u0,62& and u1,0&states, maximum spin alignment is first reached for thesubshell formed by the u0,62& states. The u1,0& state isfilled subsequently (Fig. 14, lower panel). Thus the ad-dition energies show a small maximum at N58 and aneven smaller one at N510 (Ex,El , Fig. 13, lowerpanel). Matagne et al. (2001) concluded that maximumspin alignment at midshell does not guarantee the occur-rence of corresponding maxima in the addition energyspectra. They also pointed out that for Hund’s first ruleto apply, the states do not necessarily need to bequasidegenerate (see also Austing et al., 2001).

G. Three-dimensionality of the confinement

The actual thickness of a quantum dot relative to itslateral extent depends on both the fabrication methodand the applied gate voltages. Experimentally, for typi-cal quantum dots the depth in the growth direction ofthe heterostructure material is about one order of mag-nitude smaller than the lateral extent. Thus, neglecting a

FIG. 13. Addition energy differences for different magnitudesof El compared to Ex , where Ex labels the exchange energybetween two spin-parallel electrons and El is the energy shiftdue to an anharmonicity 2lr4 of the confining potential. AfterMatagne et al., 2001.


possible extension of the electron cloud in the z direc-tion, the above description of quantum dots in two di-mensions seemed justified. Within this approximation,the addition energy spectra from both exact diagonaliza-tion and mean-field approaches compare well with theexperimental results. [This also holds for the depen-dence of D2(N) on magnetic fields; see Sec. V.] How-ever, as pointed out by several authors, such a compari-son may be obscured by an adjustment of the averageelectron density, which (as it defines the strength of theCoulomb interactions between the particles) can mask3D effects (Fujito et al., 1996; Maksym and Bruce, 1997;Nagaraja et al., 1997; Lee et al., 1998; Rontani et al.,1999a, 1999b; Jiang et al., 2001; Pi, Emperador, et al.,2001).

Rontani et al. (1999a, 1999b) compared the exact en-ergies and pair correlations for a two-electron quantumdot to results of Hartree-Fock and a single-site Hubbardmodel. They showed that the differences between theseapproximate approaches and the exact solution are re-duced in the three-dimensional case. This is due to thefact that the 2D description artificially enhances thestrength of the Coulomb and exchange matrix elements.

Bruce and Maksym (2000) used the exact diagonaliza-tion method to study the effects of the three-dimensionality of the confinement potential in a realisticquantum dot (here with N53), including the screeningof the electrons by the metallic gates.

FIG. 14. Schematic single-particle configurations un ,m& andoccupation sequences in the third shell from N57 to N512.After Matagne et al., 2001.


The most important 3D effect in small dots seems tobe the change in the effective Coulomb interaction be-tween the electrons. However, while important for a de-tailed quantitative description, this effect seems not tolead to qualitatively new features when compared to thestrictly 2D model. This holds as long as the dot is quasi-two-dimensional in the sense that only the lowest single-particle state perpendicular to the dot plane is occupied.In this review we concentrate mainly on the strictly 2Dcase, which has turned out to be surprisingly rich anddifficult. When comparing with experiments, the readershould note, however, that the parameters (such as elec-tron density, confinement strength, etc.) providing agood agreement with experiment might be slightly dif-ferent in a 3D model.

H. Triangular quantum dots

Ezaki et al. (1997, 1998a, 1998b) determined the addi-tion energy spectra from numerical diagonalization ofthe full many-body Hamiltonian, Eq. (5) (excitationsalong the z direction of the confinement were ignored).In addition to a circular oscillator, they considered a tri-angular deformation of the confinement,

V~r ,f!512

m* v2r2S 11 27 cos~3f! D , (12)with polar coordinates (r ,f). For material parametersm* 50.065me , «512.9 (corresponding to values be-tween InAs and GaAs), and an oscillator shell spacing\v53 meV, the addition energy spectra obtained fromthe many-body ground-state energies E(N) are shownin Fig. 15.

In the single-particle model, the lowest shells of bothcircular (spherical) and triangular (tetrahedral) geom-etries in 2D (3D) are similar (Hamamoto et al., 1991;Brack et al., 1997; Reimann et al., 1997; Reimann, Koski-nen, Helgesson, et al., 1998). Thus it is not surprisingthat at relatively large densities, the addition energyspectra for both symmetries differ only very little. Inboth cases, shell closings are found at N52 and N56,with additional maxima at N54 and 9 due to spin po-larization.

FIG. 15. Addition energy differences and density distributionfor (a) circular and (b) triangular quantum dots with threeelectrons. The length unit is l0520 nm. From Ezaki et al.,1997.


Naturally, for a confinement with circular symmetry,the density distribution obtained from the ‘‘exact’’ wavefunction, as shown in the upper right panel of Fig. 15,has azimuthal symmetry. Triangular deformation breaksthis symmetry, and localization of electrons due to theirCoulomb repulsion becomes visible in the electron den-sity: Three electrons confined by a triangular potentiallocalize in the corners of the triangle. [Similar resultswere reported by Creffield et al. (1999), who studied twointeracting electrons in polygonal quantum dots by exactdiagonalization techniques. They observed a transitionfrom a weakly correlated charge distribution for smalldots to a strongly correlated Wigner molecule. See alsoJefferson and Häusler (1997). A further discussion ofWigner crystallization in parabolic quantum dots is givenin Sec. IV.A.]

I. Elliptic deformation

For a rectangular dot structure like that displayed inthe upper right panel of Fig. 16, one should expect theeffective lateral confinement to have an elliptic shape: atthe corners, the electrostatic potential is rounded off,provided that the number of electrons in the dot is nottoo large. The left panel of Fig. 16 shows the experimen-tal addition energy changes D2(N) for deformed quan-tum dots with estimated ratios of side lengths L/S51.375 [curve (b)], 1.44 [curve (c)] and 1.5 [curve (d)].We notice that the shell structure for circular dot shape[curve (a) in Fig. 16] has been smeared out in the de-formed case: for the rectangular dot structures [curves(b)–(d)], no prominent maxima of D2(2,6,12) are found.

FIG. 16. Measured addition energy differences D2(N) as afunction of electron number N for different rectangular quan-tum dot structures with estimated length/side ratios: (b) L/S51.375; (c) 1.44; and (d) 1.5. The different curves are offset by3 meV and the results for the circular dot (a) are shown forcomparison. From Austing, Sasaki, et al., 1999.


Exact numerical diagonalization methods (Ezakiet al., 1997, 1998a, 1998b; Maksym, 1998; Zhu, 2000), aswell as spin-density-functional methods (Lee et al., 1998,2001; Reimann, Koskinen, Helgesson, et al., 1998, Rei-mann, Koskinen, Lindelof, et al., 1998; Austing, Sasaki,et al., 1999; Hirose and Wingreen, 1999; Reimann,Koskinen, Kolehmainen, et al., 1999) predict significantmodifications of the addition energy spectra, accompa-nied by transitions in the spin states with increasing de-formation. One can model such a rectangular quantumdot by an anisotropic oscillator potential V5 (1/2) m* (vx

2x21vy2y2) with deformation-dependent

frequencies vx5vAd and vy5v/Ad [see Eq. (1)]. Weimpose the constraint v25vxvy , which is equivalent toconserving the area of the quantum dot with deforma-tion. For d51, the dot shape is circular, whereas d.1corresponds to an elliptical shape. The oscillator fre-quency v is approximated by Eq. (11). Minimizing theKohn-Sham energy density functional in a manner simi-

FIG. 17. Calculated addition energy differences D2(N) for el-liptic quantum dots with deformation d, obtained within thelocal spin-density approximation. The curves are offset by 1meV. There is an additional offset of 1 meV between (f) and(g). The different symbols correspond to the different spins, asdefined in the figure. The inset shows D2(4) versus d. Betweend51.2 and d51.3, a transition between a spin-triplet and aspin-singlet state occurs. From Austing, Sasaki, et al., 1999.


lar to that described in Sec. III.D above allows us todetermine the addition energy spectra D2(N) for vari-ous values of the deformation d, as displayed in Fig. 17[where the circular shell structure in local spin-densityapproximation is included again in panel (a) for com-parison]. By deforming the confinement slightly, one canstill identify the shell closures at N52, 6, and 12, butwith significantly suppressed amplitudes: even a verysmall deviation from perfect circular symmetry, wherethe single-particle level degeneracies are lifted by just asmall amount, can have a very noticeable effect onD2(N). For d>1.2, the circular shell structure is com-pletely eliminated. At certain deformations one expectsaccidental degeneracies, leading to subshells (see Fig. 2),for which the sequence of magic numbers differs fromthe circular case. At d52, for example, shell closuresoccur at 2, 4, 8, 12, and 18. Compared to circular sym-metry, however, the reduced separation between thesubshells makes any shell structure less clear to observe.A systematic comparison between curves (b)–(d) in Fig.16 and curves (b)–(h) in Fig. 17 fails: although the ex-perimental data of curve (b) in Fig. 17 partly resemblethe theoretical addition energies for deformations be-tween d51.1 and d51.3, the data of curves (c) and (d)do not compare well to the SDFT values for d.1.3. Ford>2 we observe a tendency to odd-even oscillations.

The SDFT calcul

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Electronic structure of quantum dots - USPmacbeth.if.usp.br/~gusev/QuantumDots.pdfquasi-one-dimensional rings, deformed dots, and dot molecules are considered. CONTENTS I. Introduction