VOLUME 79, NUMBER 7
PHYSICAL REVIEW LETTERS
18 AUGUST 1997
Hund’s Rules and Spin Density Waves in Quantum Dots
M. Koskinen and M. Manninen
Department of Physics, University of Jyväskylä, P.O. Box 35, FIN-40351 Jyväskylä, Finland
S. M. Reimann
Niels Bohr Institute, DK-2100 Copenhagen, Denmark
(Received 17 March 1997)
Spin density functional theory is used to calculate the ground state electronic structures of circular
parabolic quantum dots. We find that such dots either have a spin configuration determined by
Hund’s rule or make a spin-density-wave-like state with zero total spin. The dependence of the
spin-density-wave amplitudes on the density of the two-dimensional electron gas is studied. [S00319007(97)03740-X]
PACS numbers: 75.30.Fv, 71.10.Ca, 73.20.Dx
Semiconductor technology now allows the fabrication
of quantum dots being so tiny that they contain only a
few electrons. Usually, such dots are formed by lateral
confinement of a high-mobility two-dimensional electron
gas (2DEG) in a semiconductor heterostructure. Their
electronic properties are determined by the interplay of
the external confinement and the electron-electron interactions, manifesting a quantum-mechanical many-particle
problem (see, e.g., Ref. [1] for exact diagonalization studies and Ref. [2] for a mean-field approach, to mention
only a few from a broad field of research). The properties of such small dots strongly depend on the number of
confined electrons, and the situation is quite similar to the
different properties of the first elements of the periodic
table, why quantum dots now often are called “artificial
atoms” [3].
Recently, Tarucha et al. [4] have developed a vertically
confined quantum dot, where they could experimentally
show that for weak or zero magnetic fields the electronic
structure of a small circular dot containing up to 20
electrons is mostly determined by the subsequent filling
of shells, obeying Hund’s rules as in atoms.
Motivated by their experimental study, we performed
spin-density functional calculations for such circular, parabolic quantum dots containing up to N ­ 46 electrons.
Complicated magnetic structures and excited states of
the 2DEG have been obtained earlier in the presence of
an external magnetic field [5–7].
To our own surprise we found that quantum dots have
a rich variety of different magnetic structures in the
ground state, even without an external magnetic field.
As one would expect from the knowledge of atomic
physics, Hund’s first rule dominates for the smallest sizes.
However, some dots have zero total spin, but exhibit
a space-dependent spin polarization, a so-called spindensity wave (SDW) [8].
At very low temperatures the electrons in the 2DEG
are confined to the lowest subband, and it is thus sufficient to consider them as being bound laterally in the
0031-9007y97y79(7)y1389(4)$10.00
x-y plane. For such small dots as studied here, we then
make the frequently used [1,2] approximation that the
external potential is harmonic, V ­ mp v 2 sx 2 1 y 2 dy2.
In such a parabolic dot the single-particle electron levels form a pronounced gross-shell structure, the magic
shells corresponding to electron numbers 2, 6, 12, 20,
30, . . . . The exact degeneracy of the shells of the 2D
oscillator, however, is reduced by the electron-electron
interactions. The situation in the case of open shells
is analogous to that in atoms. For example, if a shell
is half filled, the spins align according to Hund’s rules.
This causes the empty states with opposite spin to be
higher in energy and removes the degeneracy of the Fermi
level. In the case of atoms the ionization potential as a
function of the atomic number has maxima at half-filled
p shells. The experiments of Tarucha et al. [4] for parabolic quantum dots show a quite similar behavior: A halffilled shell shows a maximum in the addition energies as a
manifestation of Hund’s first rule.
For the electronic structure calculations for N electrons
in the parabolic dot, we apply density functional theory
[9,10] and treat the exchange-correlation part of the
electron-electron interactions in the local spin-density [11]
approximation. To be more specific, we solve the singleparticle
[10] equations
∏
∑ Kohn-Sham
h̄2 2
s
(1)
2 p =x 1 Veff sxd ci,s sxd ­ ei,s ci,s sxd ,
2m
where x ­ sx, yd and the index s accounts for the spin
(" or #). The effective Kohn-Sham potential consists of
the external harmonic confinement, the Hartree potential
of the electrons, and the functional derivative of the local
exchange-correlation energy
Z
dx nsxdexc s nsxd, z sxddd ,
(2)
Exc ­
where n is the electron density and z ­ sn" 2 n# dyn the
spin polarization. For the exchange-correlation energy of
the homogeneous 2D electron gas, we use the parametrized form of Tanatar and Ceperley [12] for nonpolarized
© 1997 The American Physical Society
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VOLUME 79, NUMBER 7
PHYSICAL REVIEW LETTERS
(z ­ 0) and ferromagnetic (z ­ 1) cases. For intermediate polarizations, following the work of von Barth and
Hedin [11] as well as Perdew and Zunger [13], which is
frequently used for electronic structure calculations in 3D
systems, one can write
exc sn, z d ­ exc sn, 0d 1 fsz d fexc sn, 1d 2 exc sn, 0dg .
(3)
The polarization dependence fsz d in 2D [14] is then
fsz d ­
s1 1
z d3y2
1 s1 2
23y2 2 2
z d3y2
22
.
(4)
In order to obtain the electron densities which minimize
the total energy functional Efn" , n# g, the Kohn-Sham
equations are solved self-consistently. To avoid any
symmetry restrictions for the wave functions, we use a
plane-wave basis. To find the ground state of all the
possible spin configurations, the iterative solution of the
Kohn-Sham equations was started with different forms
and depths of the initial potential for the spin up and down
densities [15]. This assures that one is not trapped in a
local minimum, but with a high probability can separate
the electronic ground-state configuration from the lowest
excited states.
The results will be given in effective atomic units with
Ryp ­ m p e4 y2h̄2 s4pe0 ed2 and aBp ­ h̄ 2 s4pe0 edymp e2 ,
where mp is the effective mass and e the dielectric
constant. The results can then be scaled to the actual
values for typical semiconductor materials.
The calculations are done for different values of the
density parameter rs , which approximately correspond to
the average particle density in the dot, n0 ­ 1ysprs2 d.
For the external parabolic confinement, which actually
determines the average
particle density, we then use v 2 ­
p
2
p
3
e ys4pe0 em rs Nd.
We have first calculated ground-state and isomeric electronic structures for quantum dots corresponding to the
equilibrium density of the two-dimensional electron gas
with rs ­ 1.51aBp . The results for dots with even electron numbers N are summarized [16] in Table I. Dots
with 2, 6, 12, 20, and 30 electrons correspond to “magic”
configurations of a 2D harmonic well and have a particularly large Fermi gap. Consequently, the system is
fully paramagnetic and the ground-state total spin in all
these cases is zero. For nonclosed shells, however, in
most of the cases the total spin [17] is determined by
Hund’s rule, which maximizes the spin for orbital degeneracy. From Table I we see that dots with N ­
4, 8, 10, 14, 18, 22, 28, and 32 electrons have total spin
S ­ 1 in the ground state, whereas for dots with 16 and
26 electrons ground states with a total spin S ­ 2 were
found. For larger dots we found up to three different “spin
isomers,” being little higher in energy than the groundstate configuration. As an example, we show in the first
row of Fig. 1 the spin down and spin up ground-state
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18 AUGUST 1997
TABLE I. Total spin S of the ground states and some lowenergy spin isomers for rs ­ 1.51aBp . States with a S ­ 0 spindensity wave are labeled SDW, whereas nonzero total spins
according to Hund’s rules are labeled with H.
Ground
state
Excited
state(s)
DE
(mRy*)
2
4
6
8
10
12
14
16
0
1
0
1
1
0
1
2
0 SDW
0 SDW
23.0
4.62
18
20
22
24
26
1
0
1
0
2
0
1
0
0
SDW
H
SDW
SDW
18.1
20.2
20.7
11.05
28
30
32
34
1H
0
1H
0 SDW
0
2
0
0
0
SDW
H
SDW
SDW
SDW
9.06
0.72
2.75
6.46
1.98
0 SDW
2H
7.06
0.73
Number of
electrons
H
H
H
H
H
H
H
SDW
H
FIG. 1. Spin down and spin up densities n # , n" and normalized
polarization z̃ sx, yd for the ground state (first row) and excited
states (lower rows) for a dot with N ­ 16 electrons and
rs ­ 1.51aBp . The maximum amplitude for the polarization
corresponds to z̃ sx, yd ø 0.23.
VOLUME 79, NUMBER 7
PHYSICAL REVIEW LETTERS
densities for N ­ 16, together with the normalized polarization z̃ ­ sn" 2 n# dyn0 . In this case the total spin
is S ­ 2. It is intriguing to see that z̃ shows a pronounced radial oscillation, which means that the excess
spin is not homogeneously distributed over the whole dot
regime. This effect was also seen for other sizes N when
the ground state had nonzero total spin. It reminds one of
the spin inversion states found by Gudmundsson et al. [6]
for finite magnetic fields.
For many of the excited states at rs ­ 1.51aBp the total
spin is zero. A priori one would in these cases expect that
the system is fully unpolarized. But now looking again
at Fig. 1, where the three lower rows show the densities
and polarizations of the three different spin isomers for
N ­ 16, we see that the electronic structure is more
complicated. Both the second and fourth rows of Fig. 1
show isomeric S ­ 0 states, being slightly higher in energy
than the ground state (see Table I). Their space-dependent
spin polarization shows apparent spatial oscillations, which
remind one of the phenomenon of spin-density waves [8] in
the bulk. The spin-density-wave-like states in the finite 2D
system (which in the following are called “SDW states”)
associate a certain preferred spin direction with a given
spatial region in the dot.
Studying these effects further, the next surprise is that
in larger dots this rather peculiar electronic structure
with total spin S ­ 0 gets even lower in energy than
the Hund state. For N ­ 24 and N ­ 34, the ground
state has spin zero, but is associated with a SDW state
which has a lower energy than the uniform state. (Its
shape is similar to the one shown in Fig. 3 below for
rs ­ 5.0aBp .) The general possibility of such ground-state
configurations with uniform or nearly uniform electron
density, but a nonuniform density of spin magnetization,
was first discussed by Overhauser [8]. He stated that
the nonmagnetic state must become unstable with respect
to SDW formation at low densities, whereas at higher,
metallic densities, it seems unlikely that the SDW state
is of enhanced stability. The energy balance, however, is
very delicate. Overhauser pointed out [8] that when the
SDW states are stable, they are energetically only slightly
lower than the nonmagnetic state.
On the basis of the results discussed above it is now
interesting to study how the SDW states depend on the
density of the two-dimensional electron gas. We thus
calculated for selected electron numbers the ground state
as a function of the strength v of the confining harmonic
potential, corresponding to rs values from 0.25aBp to 5.0aBp .
Figure 2 shows the maximum amplitude of the normalized polarization z̃ as a function of rs for dots with
N ­ 24, 34, and 46 electrons, where the SDW was found
to be the ground state.
It can be clearly seen that, depending on the size N,
there is a critical value of rs , where the SDW sets in,
and then rises its amplitude with increasing rs . Note,
however, that the N dependence of the critical value of
18 AUGUST 1997
FIG. 2. Maximum amplitudes of z̃ sx, yd of the ground-state
SDW for N ­ 24, 34, and 46 as a function of rs .
rs does not generally follow the trend suggested from
this figure: For some N , 46, the SDW sets in at
a higher critical value of rs , but with a comparable
amplitude. From our calculations we find that even in
magic configurations, which are fully paramagnetic with
total spin zero at low rs values, for sufficiently large rs
the SDW formation sets in. In the case of the filled shells
N ­ 12, 20, and 30 a SDW state was obtained for very
large rs * 5aBp . Increasing rs still further, this SDW state
gets more pronounced, in a similar way than the examples
with N ­ 24, 34, and 46 discussed above.
Finally, we note that the occurrence of a SDW is
related to a change in the pattern of the single-particle
levels. As an example, we show in the bottom of Fig. 3
the Kohn-Sham single-particle spectra of N ­ 34 at a
density corresponding to rs ­ 5.0aBp , both for the groundstate SDW with S ­ 0 and the excited state with total
spin S ­ 2, being 1.9 mRyp higher in energy. The third
spectrum shows the result of a LDA calculation for the
unpolarized case, which gives a 2.13 mRyp higher energy
than the SDW state. Note that for a SDW, there is always
one pair of degenerate spin up and spin down orbitals, and
the Fermi gap of the SDW state is much higher than in the
S ­ 2 case or within LDA. In quite a similar way to the
spontaneous shape deformations of nuclei with nonclosed
shells [18], the SDW opens a large energy gap at the
Fermi surface, leading to a more stabilized electronic
structure in the dot.
In conclusion, we have found that in finite quantum
dots a static spin-density-wave-like state occurs even at
rather high densities of the two-dimensional electron gas.
For many open-shell systems we found that the SDW
state with zero spin has a higher energy than the ground
state. In other cases the SDW is the ground state. Also
for nonzero total spin a strong spatial dependence of the
spin polarization was found. The amplitude of the SDW
strongly depends on the size of the Fermi gap: The
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VOLUME 79, NUMBER 7
PHYSICAL REVIEW LETTERS
FIG. 3. Upper part: Polarization z̃ sx, yd for N ­ 34 at rs ­
5aBp . Left: Ground state with S ­ 0. Right: Excited state
with S ­ 2. The scales are the same as in Fig. 1 above.
Lower part: Single-particle spectra for N ­ 34 at the same
density rs ­ 5aBp . The numbers in the spectra indicate the
degeneracies of the occupied single-particle states. The shorter
lines indicate the lowest unoccupied states. Left: Hund’s case
with S ­ 2. Middle: SDW ground state with S ­ 0. Right:
LDA result, unpolarized. The total width of the spectrum is
about 73.5 mRyp .
inset of the SDW-like state for the magic configurations
occurs at much higher rs values than for nonclosed
shells. Recent calculations have shown that the SDW-like
states are rather stable against distortions of the external
confinement. In a real quantum dot, the deformations
of the effective confinement (caused, for example, by
quantum point contacts) will thus lead to a pinning of the
polarization. Rather peculiar properties of such quantum
dots could be expected.
We would like to thank J. Helgesson, P. E. Lindelof,
and B. R. Mottelson for helpful discussions. This work
was partially financed by the Studienstiftung des deutschen Volkes, the BASF AG, the Academy of Finland,
and CNAST. S. M. R. thanks the University of Jyväskylä
for its hospitality.
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18 AUGUST 1997
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[15] The initial potentials for both spins were chosen to be of
",#
square well type, with different depths V0 ­ s1 6 hdV0 .
Choosing h ­ 0.0 and h ­ 0.3 was sufficient to obtain
the unpolarized and partly polarized (Hund’s rules) cases.
Different radial forms Rm",# sfd ­ R0 f1 1 e cossmfdg were
chosen for fm, eg ­ f0, 0g, f3, 0.3g, f4, 0.4g, and f6, 0.6g,
where Rm" and Rm# are twisted against each other by an
angle u ­ pym.
[16] Odd particle numbers were left out because in most of
the cases, one gets a net spin of 0.5 or 1.5, respectively,
corresponding either to the trivial case of one spin left
over in the highest orbital, or Hund’s rule, which also
occurs for even N.
[17] In this formalism, the wave function is a single Slater
determinant of the Kohn-Sham single-particle wave functions. Only in cases where spin up and down space wave
functions are exactly the same, the Slater determinant is
the eigenstate of the Ŝ 2 operator with S ­ Sz , but generally this is only approximately true.
[18] Å. Bohr and B. R. Mottelson, Nuclear Structure (Benjamin, New York, 1975), Vol. II.