# Metal clusters, quantum dots and trapped atoms -- from single-particle models to correlatio

**ABSTRACT** In this review, we discuss the electronic structure of finite quantal systems on the nanoscale. After a few general remarks on the many-particle physics of the harmonic oscillator -- likely being the most studied example for the many-body systems of finite quantal systems, we discuss properties of metal clusters, quantum dots and cold atoms in traps. We address magic numbers, shape deformation, magnetism, particle localization, and vortex formation in rotating systems.

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**ABSTRACT:**We report results for the ground state energies and wave functions obtained by projecting spatially unrestricted Hartree Fock states to eigenstates of the total spin and the angular momentum for harmonic quantum dots with $N\leq 12$ interacting electrons including a magnetic field states with the correct spatial and spin symmetries have lower energies than those obtained by the unrestricted method. The chemical potential as a function of a perpendicular magnetic field is obtained. Signature of an intrinsic spin blockade effect is found.Physical review. B, Condensed matter 07/2007; 77(3). · 3.77 Impact Factor

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Metal clusters, quantum dots and trapped

atoms – from single-particle models to

correlation

M. Manninenaand S.M.Reimannb

aNanoscience Center, Department of Physics, FIN-40014 University of Jyv¨ askyl¨ a,

Finland

bMathematical Physics, Lund Institute of Technology, SE-22100 Lund, Sweden

Abstract

In this review, we discuss the electronic structure of finite quantal systems on the

nanoscale. After a few general remarks on the many-particle physics of the harmonic

oscillator – likely being the most studied example for the many-body systems of fi-

nite quantal systems, we turn to the electronic structure of metal clusters. We

discuss Jahn-Teller deformations for the so-called ’ultimate’ jellium model which

assumes a complete cancellation of the electronic charge with the ionic background.

Within this model, we are also able to understand the stable electronic shell struc-

ture of tetrahedral (3D) or triangular (2D) cluster geometries, resembling closed

shells of the harmonic confinement, but for Mg clusters being “doubly-magic” as

the electronic shells occur at precisely twice the atom numbers in the close-packed

tetrahedra. Taking a turn to the physics of quantum dot artifical atoms, we discuss

the electronic shell structure of the quasi two-dimensional, harmonically confined

electron gas. Between the clear shell closings, corresponding to the magic numbers

in 2D, Hund’s rule acts, maximizing the quantum dot spin at mid-shell. After a brief

excursion to multicomponent quantum dots and the formation of Wigner molecules,

we turn to finite quantal systems in strong magnetic fields or, equivalently, electron

droplets that are set highly rotating. Working within the lowest Landau level, we

draw the analogy between magnetic fields and rotation, commenting on the forma-

tion of the so-called maximum density droplet (MDD) and its edge reconstruction

beyond the integer quantum Hall regime. Formation and localization of vortices

beyond the MDD, as well as electron localization at extreme angular momenta, are

discussed in detail. Analogies to the bosonic case, and the systematic build-up of

the vortex lattice of a rotating Bose-Einstein condensate at high angular momenta,

are drawn. With our contribution we wish to emphasize the many analogies that

exist between metallic clusters, semiconductor artificial atoms, and cold atoms in

traps.

Preprint submitted to Elseviertoday

arXiv:cond-mat/0703292v1 [cond-mat.mes-hall] 12 Mar 2007

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1 Introduction

One common feature of most small quantum-mechanical systems is the dis-

creteness of the quantum states. In systems with high symmetry the single-

particle energy levels are degenerate, which may lead to shell structure. This

is known to happen in free atoms, but also in nuclei [1]. Spherical metal

clusters [2], where the particles move in a spherically symmetric mean field,

provide another example. In semiconductor quantum dots with circular sym-

metry, shell structure was observed by conductance spectroscopy by Tarucha

et al. [3]. (For a review, see [4]). The ’universality’ of shell structure bridges

these fields of physics. However, there are also fundamental differences: In

atoms and in quantum dots, the fixed external potential dominates, leading

to Hund’s rule with maximum spin at mid-shell to resolve the degeneracy of the

spherical confinement. The valence electrons in metal clusters, or the neutrons

and protons of a nucleus, however, move in a mean-field potential determined

solely by the particle dynamics. To resolve degenaracies for non-closed shells,

metal clusters and nuclei exhibit spontaneous shape deformation, while atoms

and quantum dots do not. Consequently, the often used name ’artificial atom’

is well suited for semiconductor quantum dots, but would be misleading for

free metal clusters.

Many properties of metal clusters can be calculated by using so-called ab ini-

tio electronic structure calculations and molecular dynamics. These computa-

tional results often are in very good agreement with experimental data – as,

for example, in photoemission spectroscopy. They can pin-point the detailed

ground-state geometries of particular cluster sizes [5]. However, many overall

features can even be understood using simple models [2,6]. This also holds for

semiconductor quantum dots, where often, simple single-particle models have

been very successful [4].

The purpose of this (brief, and by no means complete) review is to summarize

the simple models, their advantages and limitations in describing overall prop-

erties of metal clusters and quantum dots, and to draw analogies between these

finite quantal systems to the more recently emerging field of cold (bosonic or

fermionic) atom gases in traps.

Let us begin by looking at the simple, but relevant many-particle physics

of the harmonic oscillator. These results are then applied to understand the

jellium model for metal clusters and electronic states in quantum dots. The

universality of deformation is shortly described using simple models. We finally

turn to a comparsion of quantum dots at strong magnetic fields, and weakly

interacting bosonic systems that are set rotating.

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2 Many-particle physics in harmonic oscillator

The harmonic oscillator confining a single-particle is solved in about all text

books of quantum mechanics. However, adding more particles immediately

makes it more challenging to describe the system theoretically, and new inter-

esting phenomena appear. The many-body Hamiltonian is then written as

H =

?

i

?

−?2

2m∇2

i+1

2mω2

0r2

i

?

+

?

i?=j

v(|ri− rj|), (1)

where ω0is the frequency of the confining harmonic oscillator and v(r) the

interparticle two-body interaction. The position vector ri and the Laplace

operator ∇2

in question. Sometimes we want to use the occupation number representation

and write

imay be three-, two- or one-dimensional depending on the system

H =

?

i,σ

?ic+

i,σci,σ+

?

{i,σ}

vi1,i2,i3,i4c+

i1,σ1c+

i2,σ2ci4,σ4ci3,σ3,(2)

where c+and c are the normal creation and annihilation operators (as here, for

fermions), and ?iis the single-particle energy of the form ?i= ?ω0(ni+ d/2),

d being the dimension of the system. Most conveniently, one uses the single-

particle states of the confining harmonic oscillator as a basis. It is important to

note that even if the spin index appears in this formulation of the Hamiltonian

(as a summation index), here we consider only spinless interactions, i.e. the

Hamiltonian is, as obvious from Eq. (1), independent of the spin.

The perhaps most important feature of a harmonic confinement is, that the

center-of-mass motion separates from the internal motion, regardless of the

interaction between the particles. This can easily be shown for both classical

and quantum systems [7]. As a consequence, the selection rule for the dipole

oscillations only allows the center-of-mass excitation. In the case of simple

metal clusters and quantum dots this is the plasmon resonance, with energy

?ω0, where ω0is the frequency of the harmonic confinement [6]. In connection

with two-dimensional quantum dots [8,9,10,11,12], the effect of the separation

of the center-of-mass motion was earlier often referred to as “Kohn’s theo-

rem” [13]. In the case of atomic nuclei, the related excitation is called the

“giant resonance”, where the proton and neutron distributions oscillate with

respect to each other [1].

Another important property of the harmonic confinement is the separation

of the spatial coordinates from the center-of-mass motion (or single-particle

motion). This means that the level structure in the most general case (with

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different oscillation frequencies ωialong different directions) is simply

?n1,n2,n3= ?ω1(n1+1

2) + ?ω2(n2+1

2) + ?ω3(n3+1

2).(3)

For spherically symmetric potentials, labelling the energy levels by their ra-

dial and angular momentum indices, one obtains the harmonic energy shells

given in Table 1. Including the spin degree of freedom with a factor of two,

the “magic numbers” of the harmonic oscillator in three dimensions occur

at particle numbers 2,8,20,40,70,..., and at 2,6,12,20,30,42,... in two di-

mensions. In a non-harmonic potential, the additional degeneracy of different

Table 1

Shell structure of three and two-dimensional (3D and 2D) harmonic oscillators. g

is the degeneracy of the shell and N the cumulatice number of states without spin

degeneracy.

3D2D

shell

levelsg

N

2N

g

N

2N

11s

112112

21p

348236

32s1d

6 102036 12

4

2p1f

10 204041020

53s2d1g

1535705 15 30

63p2f1h

21561126 2142

radial states disappears, and other degeneracies occur. A famous example is

the Wood-Saxon potential,

VWS(r) = −

V0

1 + e(R−r)/a, (4)

frequently used in nuclear physics (where the spin-orbit interaction of the

nucleons further splits the shells [1]). Single-particle states in this potential

are filled following the sequence 1s,1p,1d,2s,2p,1f, etc. . The Woods-Saxon

potential Eq. (4) is a good approximation for the mean-field potential in metal

clusters [14], with energetically dominant magic numbers at 2, 8, 20, 40, 58,

92, etc. Note, however, that the first few magic numbers, as here 2,8 and 20,

are mainly determined by the angular momentum degeneracy and are nearly

independent of the radial shape of the spherical potential.

In the 2D harmonic confinement the noninteracting electron states can be

solved analytically also in the presence of a magnetic field [4], the resulting

single-particle states being the so-called Fock-Darwin states [15,16,17]. Con-

sequently, the harmonic confinement has been widely utilized when studying

the quantum Hall effect in finite systems [18].

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3Jellium model of metal clusters

Many properties of simple metals, like alkalis, alkali earths and even aluminum

can be explained as properties of the interacting, homogeneous electron gas.

The role of the ions is then merely to keep the electron gas together at its

equilibrium density. In the jellium model [20], inside the metal the charge of

the ions is smoothened out and replaced by a homogeneous background charge

with the same density as the electron gas. At the surface, the background

charge goes abruptly to zero. The electron density is usually described by

the density parameter rsdefined as the radius of a sphere (in units of Bohr

radius) containing one electron: 4πr3

density of the electrons. The density functional method in the local density

approximation (LDA) is ideally suited for the jellium model which naturally

has a smooth and slowly varying electron density.

s/3 = 1/n0, with n0 being the number

This approach was first used to study metal surfaces [19], lattice defects [21]

and impurities in metals [22]. The first application to metal clusters was made

by Martins et al. [23], who studied the size variation of the ionization energy.

Similar work had been successful for calculating the work function of planar

surfaces of alkali metals [24].

In the density functional Kohn-Sham method the electrons more in an effective

mean-field potential

Vσ

eff(r) = −eφ(r) + Vσ

xc(r), (5)

where φ is the total electrostatic potential of the background charge and elec-

tron density distribution, and Vxcis the exchange-correlation potential which

depends locally on the electron density and spin polarization [25]. In the case

of a spherical jellium cluster the effective potential resembles a finite potential

well with rounded edge. It can be well approximated by the above mentioned

Woods-Saxon potential (Eq. 4). The spherical jellium model suggests that, like

in free atoms, the ionization potential is largest for the magic clusters, and at

a minimum when only one electron occupies an open shell. The experimental

results, however, show a much richer structure as a function of the cluster size,

which in alkali metals is dominated by a marked odd-even staggering [2]. The

reason behind are shape deformations, as described in the next section.

In the spherical jellium model for metal clusters, the the background charge is

a homogeneously charged sphere of radius R =

caused by this sphere is

3√Nrs. The (external) potential

Vsphere(r) =

−

−Ne2

Ne2

8π?0R3(3R2− r2) if r ≤ R

4π?0r

if r > R .

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Note that the potential inside the sphere is harmonic and can be written as

Vsp(r) =1

2mω2

spr2, (6)

where

ω2

sphere=

e2

4π?0r3

sm=n0e2

3?0m=ω2

p

3

(7)

is the square of the plasmon frequency of a metallic sphere. Since the electrons

only slightly spill out from the region of the harmonic potential, the plasmon is

the dominating dipole absorption mechanism for spherical jellium clusters [6].

Ekardt [26] used the spherical jellium model in connection with the time-

dependent density functional theory to study optical absorption. He found that

the anharmonicity of the background potential caused fragmentation of the

single plasmon peak to a distribution of close-lying absorption peaks. Similar

work was subsequently done by several groups, using the RPA method [6,27].

Koskinen et al. [28] used shell-model methods from nuclear physics to try

to solve the electronic structure and photo-absorption of the jellium clusters

beyond the mean-field approach. For up to eight electrons, they could diago-

nalize the many-electron Hamiltonian nearly exactly. Already for 20 electrons,

however, the configuration interaction method showed a much too slow con-

vergence as a function of the size of the basis set (in fact, it was shown that the

error in the correlation energy was ∝ E−3/2

sult agreed with those of the RPA calculations. For positively charged jellium

spheres, the fragmentation of the plasmon peak disappears as the confinement

of the electrons becomes harmonic [29,28].

cut−off). For eight electrons, their re-

Historically, it is interesting to note that Martins et al. [23] corrected the pure

jellium results by including ion pseudo-potentials via first-order perturbation

theory, in a similar fashion than Lang and Kohn [20] had done for metal sur-

faces. While for planar surfaces the correction had only a minor effect (in alkali

metals), it became dominating for large metal clusters, and completely dimin-

ished the effects of the electronic shell structure of the pure jellium sphere. A

similarly large effect of the pseudopotential correction was observed for large

voids in metals, shown to be due to the low-index surfaces present in spherical

systems cut from an ideal lattice [31]. The notion of the possible importance of

the lattice potential made the theoreticians cautious in making too strong pre-

dictions of the applicability of the jellium model to real metal clusters [32,33],

until the magic numbers of alkali metal clusters were observed [34].

The degeneracy of the open shell clusters should lead to Hund’s rules like

in the case of free atoms. In the spherical jellium models the clusters with

open shells should have a large total spin and magnetic moment [32]. This

was predicted prior to the success of the jellium model by Geguzin [35], who

studied highly symmetric cub-octahedral Na13clusters. For free clusters, how-

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ever, deformation wins over Hund’s rule and removes both the degeneracy and

magnetism [36].

We conclude that the simple spherical shell structure explains well the magic

numbers in the experimental mass spectra of sodium clusters. It lies behind

the so-called super shells [37,38] observed in alkali metal clusters [39], as well

as the importance of the collective plasmon resonance.

The simple jellium model also accounts for some properties of noble metals.

Recently, it has been observed that even in gold clusters some features can be

explained most easily with arguments based on the jellium model [40].

4Deformed jellium

The similarity of small nuclei and simple metal clusters is not limited to magic

numbers and to the existence of the plasmon-type giant resonance, but extends

even to the internal deformation of the system. It is clear that the smallest

clusters can be viewed as well-defined molecules with a geometry determined

by the atomic configurations. Quantum chemistry can be used to character-

ize the ground state and spectroscopic properties of clusters with only a few

atoms. For larger clusters (N > 10) the early theories assumed spheres cut

from a metal lattice [30], or faceted structures with shapes determined by

the Wulff polyhedra [41]. In reality, however, the clusters exhibit geometries

very different from these ideal structures. Many metals form icosahedral clus-

ters [42,43]. Jahn-Teller deformations are important even in quite large clus-

ters, as manifested, for example, by the odd-even staggering of the ionization

potential [2].

In the early cluster beam experiments, the temperature of the clusters was

lowered only by evaporative cooling. The resulting cluster temperatures were

so high that the clusters were most likely liquid [44]. The clusters showed the

electronic shell structure as well as deformation, as determined by the splitting

of the plasmon peak [45]. In fact, the super-shell structure could only been

seen in liquid sodium clusters. Solid clusters formed icosahedral structures

which governed the abundance and ionization potential spectra [46].

To model cluster deformations, Clemenger [47] was the first to apply the Nils-

son model familiar from nuclear physics [1]. He was able to explain qualitative

features of the abundance spectrum of sodium clusters, including the observed

odd-even staggering. A more general model, based on the Strutinsky-model of

nuclei [48], was developed by Reimann et al. [49], and applied to triaxial ge-

ometries by Yannouleas and Landmann [50] as well as Reimann et al. [51,52].

It could explain nearly quantitatively the stabilities and deformation of small

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sodium clusters.

4.1 Ultimate jellium model

The simplest way to include deformation in the jellium model is to assume

the uniform background charge density to be a spheroid, or an ellipsoid [53].

The model explains qualitatively the splitting of the plasmon peak and the

size dependence of the ionization potential of alkali metal clusters. However,

the optimal deformation shape determined by the electronic structure is not

an ellipsoid, but a more generally shaped jellium background [54]. In the ul-

timate limit, the energy is minimized, when the background density equals

the electron density – as suggested by Manninen already in 1986 [55]. In this

so-called ’ultimate jellium model’ (UJM) [56], the density of the background

is not fixed, but in a large cluster adjusts itself to correspond to r2≈ 4.2 a0, a

value close to the equilibrium electron density in sodium. (Here, a0is the Bohr

radius). The ground-state densities for clusters with N = 2 to 22 electrons are

shown as constant density surfaces in Fig. 1. Clearly, the magic numbers at

small N, here for 2, 8, and 20 electrons, correspond to spherical symmetry of

the freely deformable ’ultimate jellium’ droplet. Off-shell, however, the shapes

of the clusters exhibit breaking of axial and inversion symmetries. In general,

the resulting ground-state geometries are far from ellipsoidal. Clusters which

lack inversion symmetry, are very soft against odd-multipole deformations [56].

Remarkably, the results obtained from the UJM for deformations are very

close to those of ab initio calculations for sodium [57], as shown in Fig. 2 and

Fig. 3.

Koskinen et al. [59] applied the UJM to determine the shape deformations

of small nuclei. Their method gave rather good agreement with experimen-

tal results, and surprisingly, nearly exactly the same geometries as for the

electron-gas jellium. H¨ akkinen et al. [58] studied further the idea of this ’uni-

versal deformation’ and found that in the LDA, density functional theory

predicts similar deformations for all small fermion clusters.

This shape universality can be easily understood in systems where the parti-

cles move in a mean field caused by the particles themselves. When the num-

ber of the particles is small, there is only a small number of single-particle

states which determine the shape. For example, for four particles, only the

1s and, say, 1pxstates are filled. Consequently, the shape is prolate along the

x-direction. This corresponds to the basis of the Nilsson model [1].

The robustness of the shape on the specific model was further studied by

Manninen et al. [60], who showed that deformations of the UJM are in very

good agreement with results of the ’ultimate’ tight-binding model: the H¨ uckel

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Fig. 1. Constant-density surfaces of ’ultimate jellium’ clusters with up to 22 elec-

trons. After Koskinen et al., see Ref. [56] for details and scales).

model for clusters [61].

For nuclei, the simple universal model only needs two parameters, the bulk

modulus and the average binding energy per nucleon (the first term in the so-

called mass formula [1]), to give good quantitative approximations to the de-

formation parameters and even excitation energies of shape isomers, as shown

in Fig. 4.

4.2Triangles and tetrahedra

The jellium model has also been aplied to quasi two-dimensional clusters, as

for example in the early studies by Kohl et al. [62,63]. A physical realiza-

tion of two-dimensional clusters could be sodium clusters on an inert surface,

or even two-dimensional electron-hole liquids in semiconductors. Reimann et

al. [64] analyzed systematically the UJM ground-state shapes for quasi two-

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Fig. 2. Comparison of shapes of UJM clusters (left) to those of DFT-LDA molec-

ular-dynamics methods (right), for Na-6 (upper panel) and Na-14 (lower panel).

In all cases the outer surface shown corresponds to the same particle density. Blue

spheres represent the ions. From Ref. [58].

Fig. 3. The three radii of the anionic sodium clusters along the principal axis, plotted

vs the number of atoms in the cluster. Down-triangles, circles, and up-triangles

correspond to Rmin, Rmiddle, and Rmax, respectively. (a) Radii corresponding to

the ground-state geometry of ab initio calculations, (b) thermally averaged radii

from room-temperature simulations, (c) radii calculated within the UJM. From

Ref. [57].

dimensional sodium clusters. Contours of the self-consistent ground-state den-

sities of these two-dimensional fermion droplets are shown in Fig. 5, calculated

for a 2D layer thickness of 3.9a0. The shape systematics reveals that for elec-

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Fig. 4. Left: Shape parameter a20 for fermion clusters from 5 to 40 particles cal-

culated with the UJM (black dots connected with solid line) compared to the ex-

perimental results for even-even nuclei (stars). Right: Excitation energies of linear

isomers calculated with the UJM for nuclei (open circles) and compared to the

experimental results (black dots). From Ref. [60].

Fig. 5. Contours of the self-consistent ground-state densities of two-dimensional

UJM clusters for electron numbers N ≤ 2 ≤ 34, calculated for a 2D layer thickness

of 3.9a0. From Ref. [64] (see this Ref. for details).

tron numbers 6, 12, 20, and 30 the 2D clusters have triangular shape. Initially,

this result appeared puzzling, as these shell closures correspond to those of

the circular two-dimensional harmonic oscillator, and one should thus expect

azimuthal symmetries of the ground-state densities. The explanation was, how-

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ever, that in 2D, a triangular cavity has precisely these magic numbers [65],

and only in the large-N limit, the increased surface tension at the corners

makes the oscillator shells more stable. In 2D, the shell closings are rather

weak, with favorable energy minima (gaps at the fermi level) appearing mainly

in the small-N limit. Given the freedom of unrestricted shape deformations, a

pronounced odd-even staggering appears in the ground-state energies, as seen

in Fig. 6. Incidentally, these shell fillings for the triangular geometries (without

Fig. 6. Ground-state energies per electron of two-dimensional clusters, as a func-

tion of cluster size N. (The kinetic energy contribution in z-direction, tz, was sub-

tracted). The inset shows the self-consistent single-particle Kohn-Sham energies for

even particle numbers. From Ref. [64].

spin-degeneracy) equal precisely the number of atoms forming a close-packed

triangle. In fact, the same holds in three dimensions: at small N, thetrahedral

shell structure is prefered [66,64], with magic numbers at N = 2,8,20,40,70

and 112. These numbers correspond precisely to twice the numbers of atoms in

a close-packed thetrahedral cluster geometry (see Fig. 7). One should expect

that the compact tetrahedral geometry at an electronic magic number sta-

bilizes theses clusters. However, first principles calculations have shown that

this is not generally the case. Mg10has an overall tetrahedral shape, but is not

a perfect tetrahedron [67]. Na20[68], and Mg20[67] are not tetrahedra, but

Au20seems to be [69]. The experimental abundance spectrum of Mg shows a

maximum at Mg35[70] – but so far, there is no evidence that its geometry is

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Fig. 7. Left: Ground-state shape of the UJM for 112 electrons. Right: A possible

structure of a Mg35cluster with 70 valence electrons. From Ref. [64].

a tetrahedron like the one shown in Fig. 7. The above results suggests that

trivalent metals on an oxide or graphite surface could favor triangular shapes.

In fact, advances in the experimental realization of surface-supported planar

clusters have been recently reported by Chiu et al. [71]. They found magic

numbers in quasi two-dimensional Ag clusters grown on Pb islands, and stud-

ied the transition from electronic to geometric shell structure.

We finally mention that high stability of tetrahedral shapes has also been

discussed in nuclear physics [72,73], predicting tetrahedral ground states for

some exotic nuclei around110Zr (see Schunck et al., [74]).

5Semiconductor quantum dots

Generally speaking, a quantum dot is a system where a small number of elec-

trons are confined in small volume in all three spatial directions. It can be,

for example, a three-dimensional atomic cluster or a two-dimensional island of

electrons formed by external gates in a semiconductor heterostructure [75,4].

In this review, we shall only consider two-dimensional semiconductor quan-

tum dots. Most often they are formed from AlGaAs-GaAs layered structures,

where a low-density 2D conduction electron gas is formed in the AlGaAs layer.

The quantum dot is formed by removing the electrons outside the dot region

with external gates (lateral dot), or by etching out the material outside the

dot region (vertical dot). In both cases, the resulting confining potential is,

to a good approximation, harmonic. The underlying lattice of the semicon-

ductor material can be taken into account by using an effective mass for the

conduction electrons, and a static dielectric constant, reducing the Coulomb

repulsion.

The resulting generic model for a semiconductor quantum dot is a 2D harmonic

oscillator with interacting electrons. This in fact is like a 2D jellium model,

with the simplification that now the harmonic confinement has infinite range

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Page 14

and the center-of-mass motion separates out exactly (Kohn’s theorem [13]).

This means that in the ideal case (in zero magnetic field) there is only one

dipole absorption peak, as seen in experiments [76].

Conductance spectroscopy can be used on one single dot. The dot is weakly

connected to leads and the current is measured as a function of the gate voltage

which determines the chemical potential and thus the number of electrons in

the dot [4]. When the electron number in the dot is large, the energy of an

additional electron can be estimated from the capacitance C of the dot, as

∆E = e2/C. The resulting conductance then shows equidistant peaks as a

function of the gate voltage.

When the number of electrons is small, the individual single electron levels in

the dot become important and their shell structure can be seen in the con-

ductance spectrum. Tarucha et al. [3] were the first to successfully determine

the shell structure of circular quantum dots. Their result is shown in the lower

panel of Fig. 8, where the second derivative of the total energy of the dot is

plotted as a function of the number of electrons, N. For comparison, the corre-

sponding result of the LSDA calculation for electrons in a harmonic oscillator

is included, too.

Fig. 8. Second derivative of the total energy of electrons in a quantum dot as a func-

tion of the number of electrons. The magic numbers are shown. The experimental

result is from Tarucha et al. [3]. The upper panel shows the calculated total spin.

The density functional Kohn-Sham method for semiconductor quantum dots

usually assumes that i) the system is two-dimensional, ii) only the conduction

electrons are considered, with an effective mass m∗and their Coulomb inter-

action screened by the static dielectric function ? of the material in question,

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Page 15

and iii) they move in a harmonic confinement mω2

is used for the spin-dependent exchange-correlation energy, derived from the

functionals for the 2D electron gas [77]. For details see Refs. [4].

0r2. A local approximation

Shell structure with main shell fillings (magic numbers) at N = 2,6,12,20

appears very clearly in the addition energy differences ∆2(N). (See Fig. 8).

Furthermore, like in free atoms, due to Hund’s rule at mid-shell the total spin

is maximal. This means that (just like for the spherical jellium model discussed

above), any half-filled shell shows as a weak ’magic’ number, with increased

stability. This is clearly seen in Fig. 8 where the second derivative of the total

energy shows maxima at N = 4 and N = 9 in addition to the clear peaks at

the filled shells, N = 2,6, and 12. Figure 8 also shows the calculated total spin

as a function of the number of electrons in the dot. The self-consistent data

appear to agree very nicely with the experimental data. However, we notice

that this agreement becomes worse with increasing N, showing very clear

deviations between theory and experiment after the third shell, i.e. around

N = 20. Another series of experimental data, was later published by the same

group in 2001. In Ref. [78], addition energies for 14 different quantum dot

structures, all similar to the device used in the earlier work by Tarucha et

al. [3], were analyzed. Strong variations in the spectra were reported, very

clearly differing from device to device and seemingly indicating that each of

these vertical quantum dots indeed has its own properties: a comparison to the

theoreticalle expected shell structures needs to be taken with care. Progress

with vertical quantum dots was achieved more recently, where few-electron

phenomena could be studied by tunneling spectroscopy through quantum dots

in nanowires [79,80].

The self-consistent electronic structure calculations for quantum dots for some

electron numbers showed internal symmetry-breaking of the spin-density [81],

leading to a static “spin-density wave” (SDW). Figure 9 shows, as an ex-

ample, the intriguing ground-state spin polarization for a quantum dot with

six electrons. For not too small densities of the electron gas, i.e. rs ≤ 6a∗

this quantum dot still has a closed-shell configuration, with S = Sz = 0.

This result is obtained from SDFT. The total density obtained by the SDFT

method is circularly symmetric, with zero net polarization (S = 0). However,

the spin polarization (which equals the difference between the spin densities

n↑− n↓normalized by the total density, n↑+ n↓), in standard SDFT breaks

the azimuthal symmetry of the confinement, showing a regular spin structure.

Fig. 9 shows this very clearly for the example of a six-electron quantum dot

at rs= 4a0. Both spin-up and spin-down densities exhibit three clear bumps,

which are twisted against each other by an angle of π/3. This resembles very

much an antiferromagnet-like structure, with alternating up- and down spins,

on a ring. Such states were obtained both with the Tanatar-Ceperley [77] as

well as the more recent Attaccalite–Moroni–Gori-Giorgi–Bachelet (AMGB)

[82] functionals for exchange-correlation. As the AMGB functional depends

0,

15

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