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). · 3.66 Impact Factor
Metal clusters, quantum dots and trapped
atoms – from single-particle models to
M. Manninenaand S.M.Reimannb
aNanoscience Center, Department of Physics, FIN-40014 University of Jyv¨ askyl¨ a,
bMathematical Physics, Lund Institute of Technology, SE-22100 Lund, Sweden
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
Preprint submitted to Elseviertoday
arXiv:cond-mat/0703292v1 [cond-mat.mes-hall] 12 Mar 2007
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 . Spherical metal
clusters , 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. . (For a review, see ). 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 . 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 .
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.
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
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
in question. Sometimes we want to use the occupation number representation
imay be three-, two- or one-dimensional depending on the system
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 . 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 . 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” . 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 .
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
different oscillation frequencies ωialong different directions) is simply
2) + ?ω2(n2+1
2) + ?ω3(n3+1
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
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
6 102036 12
1535705 15 30
radial states disappears, and other degeneracies occur. A famous example is
the Wood-Saxon potential,
VWS(r) = −
1 + e(R−r)/a, (4)
frequently used in nuclear physics (where the spin-orbit interaction of the
nucleons further splits the shells ). 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 , 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 , 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 .
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 , 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 , lattice defects 
and impurities in metals . The first application to metal clusters was made
by Martins et al. , 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 .
In the density functional Kohn-Sham method the electrons more in an effective
eff(r) = −eφ(r) + Vσ
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 . 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 . 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
8π?0R3(3R2− r2) if r ≤ R
if r > R .
Note that the potential inside the sphere is harmonic and can be written as
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 .
Ekardt  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.  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.  corrected the pure
jellium results by including ion pseudo-potentials via first-order perturbation
theory, in a similar fashion than Lang and Kohn  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 . 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 .
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 . This
was predicted prior to the success of the jellium model by Geguzin , who
studied highly symmetric cub-octahedral Na13clusters. For free clusters, how-
ever, deformation wins over Hund’s rule and removes both the degeneracy and
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 , 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 .
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 , or faceted structures with shapes determined by
the Wulff polyhedra . 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
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 . The clusters showed the
electronic shell structure as well as deformation, as determined by the splitting
of the plasmon peak . 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 .
To model cluster deformations, Clemenger  was the first to apply the Nils-
son model familiar from nuclear physics . 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 , was developed by Reimann et al. , and applied to triaxial ge-
ometries by Yannouleas and Landmann  as well as Reimann et al. [51,52].
It could explain nearly quantitatively the stabilities and deformation of small
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 .
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 . In the ul-
timate limit, the energy is minimized, when the background density equals
the electron density – as suggested by Manninen already in 1986 . In this
so-called ’ultimate jellium model’ (UJM) , 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 .
Remarkably, the results obtained from the UJM for deformations are very
close to those of ab initio calculations for sodium , as shown in Fig. 2 and
Koskinen et al.  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.  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 .
The robustness of the shape on the specific model was further studied by
Manninen et al. , who showed that deformations of the UJM are in very
good agreement with results of the ’ultimate’ tight-binding model: the H¨ uckel
Fig. 1. Constant-density surfaces of ’ultimate jellium’ clusters with up to 22 elec-
trons. After Koskinen et al., see Ref.  for details and scales).
model for clusters .
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 ), 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.  analyzed systematically the UJM ground-state shapes for quasi two-
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. .
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
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-
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. .
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.  (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-
ever, that in 2D, a triangular cavity has precisely these magic numbers ,
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. .
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 . Na20, and Mg20 are not tetrahedra, but
Au20seems to be . The experimental abundance spectrum of Mg shows a
maximum at Mg35 – but so far, there is no evidence that its geometry is
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. .
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. . 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., ).
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
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
and the center-of-mass motion separates out exactly (Kohn’s theorem ).
This means that in the ideal case (in zero magnetic field) there is only one
dipole absorption peak, as seen in experiments .
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 . 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.  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. . 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,
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 . For details see Refs. .
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. , addition energies for 14 different quantum dot
structures, all similar to the device used in the earlier work by Tarucha et
al. , 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 ,
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  as
well as the more recent Attaccalite–Moroni–Gori-Giorgi–Bachelet (AMGB)
 functionals for exchange-correlation. As the AMGB functional depends