m. koskinen, m. manninen and s. m. reimann- hund’s rules and spin density waves in quantum dots

8/3/2019 M. Koskinen, M. Manninen and S. M. Reimann- Hunds Rules and Spin Density Waves in Quantum Dots

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VOLUME 79, NUMBER 7 P H Y S I C A L R E V I E W L E T T E R S 18 AUGUST 1997

Hunds Rules and Spin Density Waves in Quantum Dots

M. Koskinen and M. Manninen Department of Physics, University of Jyvskyl, P.O. Box 35, FIN-40351 Jyvskyl, Finland

S. M. ReimannNiels Bohr Institute, DK-2100 Copenhagen, Denmark

(Received 17 March 1997)Spin density functional theory is used to calculate the ground state electronic structures of circularparabolic quantum dots. We find that such dots either have a spin configuration determined byHunds rule or make a spin-density-wave-like state with zero total spin. The dependence of thespin-density-wave amplitudes on the density of the two-dimensional electron gas is studied. [S0031-9007(97)03740-X]

PACS numbers: 75.30.Fv, 71.10.Ca, 73.20.Dx

Semiconductor technology now allows the fabricationof quantum dots being so tiny that they contain only afew electrons. Usually, such dots are formed by lateralconfinement of a high-mobility two-dimensional electron

gas (2DEG) in a semiconductor heterostructure. Theirelectronic properties are determined by the interplay ofthe external confinement and the electron-electron inter-actions, manifesting a quantum-mechanical many-particleproblem (see, e.g., Ref. [1] for exact diagonalization stud-ies and Ref. [2] for a mean-field approach, to mentiononly a few from a broad field of research). The proper-ties of such small dots strongly depend on the number ofconfined electrons, and the situation is quite similar to thedifferent properties of the first elements of the periodictable, why quantum dots now often are called artificialatoms [3].

Recently, Tarucha et al. [4] have developed a verticallyconfined quantum dot, where they could experimentallyshow that for weak or zero magnetic fields the electronicstructure of a small circular dot containing up to 20electrons is mostly determined by the subsequent fillingof shells, obeying Hunds rules as in atoms.

Motivated by their experimental study, we performedspin-density functional calculations for such circular, par-abolic quantum dots containing up to N 46 electrons.

Complicated magnetic structures and excited states ofthe 2DEG have been obtained earlier in the presence ofan external magnetic field [5 7].

To our own surprise we found that quantum dots have

a rich variety of different magnetic structures in theground state, even without an external magnetic field.As one would expect from the knowledge of atomicphysics, Hunds first rule dominates for the smallest sizes.However, some dots have zero total spin, but exhibita space-dependent spin polarization, a so-called spin-density wave (SDW) [8].

At very low temperatures the electrons in the 2DEGare confined to the lowest subband, and it is thus suf-ficient to consider them as being bound laterally in the

x-y plane. For such small dots as studied here, we thenmake the frequently used [1,2] approximation that theexternal potential is harmonic, V mv2x2 1 y22.In such a parabolic dot the single-particle electron lev-

els form a pronounced gross-shell structure, the magicshells corresponding to electron numbers 2, 6, 12, 20,30, . . . . The exact degeneracy of the shells of the 2Doscillator, however, is reduced by the electron-electroninteractions. The situation in the case of open shellsis analogous to that in atoms. For example, if a shellis half filled, the spins align according to Hunds rules.This causes the empty states with opposite spin to behigher in energy and removes the degeneracy of the Fermilevel. In the case of atoms the ionization potential as afunction of the atomic number has maxima at half-filledp shells. The experiments of Tarucha et al. [4] for para-bolic quantum dots show a quite similar behavior: A half-filled shell shows a maximum in the addition energies as amanifestation of Hunds first rule.

For the electronic structure calculations for N electronsin the parabolic dot, we apply density functional theory[9,10] and treat the exchange-correlation part of theelectron-electron interactions in the local spin-density [11]approximation. To be more specific, we solve the single-particle Kohn-Sham [10] equations

2h2

2m=2x

1 Vseffx

ci,s x ei,sci,sx , (1)

where x x, y and the index s accounts for the spin(

"or

#). The effective Kohn-Sham potential consists of

the external harmonic confinement, the Hartree potentialof the electrons, and the functional derivative of the localexchange-correlation energy

Exc

Zdx nxexcnx, zx , (2)

where n is the electron density and z n" 2 n#n thespin polarization. For the exchange-correlation energy ofthe homogeneous 2D electron gas, we use the parame-trized form of Tanatar and Ceperley [12] for nonpolarized

0031-90079779(7)1389(4)$10.00 1997 The American Physical Society 1389


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(z 0) and ferromagnetic (z 1) cases. For interme-diate polarizations, following the work of von Barth andHedin [11] as well as Perdew and Zunger [13], which isfrequently used for electronic structure calculations in 3Dsystems, one can write

excn, z excn, 0 1 fz excn, 1 2 excn, 0 .

(3)

The polarization dependence fz in 2D [14] is then

fz 1 1 z32 1 1 2 z32 2 2

232 2 2. (4)

In order to obtain the electron densities which minimizethe total energy functional En",n#, the Kohn-Shamequations are solved self-consistently. To avoid anysymmetry restrictions for the wave functions, we use aplane-wave basis. To find the ground state of all thepossible spin configurations, the iterative solution of theKohn-Sham equations was started with different formsand depths of the initial potential for the spin up and down

densities [15]. This assures that one is not trapped in alocal minimum, but with a high probability can separatethe electronic ground-state configuration from the lowestexcited states.

The results will be given in effective atomic units withRy me42 h24pe0e2 and a

B h24pe0em

e2,where m is the effective mass and e the dielectricconstant. The results can then be scaled to the actualvalues for typical semiconductor materials.

The calculations are done for different values of thedensity parameter rs, which approximately correspond tothe average particle density in the dot, n0 1pr2s .For the external parabolic confinement, which actuallydetermines the average particle density, we then use v2 e24pe0em

r3spN.

We have first calculated ground-state and isomeric elec-tronic structures for quantum dots corresponding to theequilibrium density of the two-dimensional electron gaswith rs 1.51a

B. The results for dots with even elec-tron numbers N are summarized [16] in Table I. Dotswith 2, 6, 12, 20, and 30 electrons correspond to magicconfigurations of a 2D harmonic well and have a par-ticularly large Fermi gap. Consequently, the system isfully paramagnetic and the ground-state total spin in allthese cases is zero. For nonclosed shells, however, inmost of the cases the total spin [17] is determined byHunds rule, which maximizes the spin for orbital de-generacy. From Table I we see that dots with N4, 8,10, 14, 18, 22,28, and 32 electrons have total spinS 1 in the ground state, whereas for dots with 16 and26 electrons ground states with a total spin S 2 werefound. For larger dots we found up to three different spinisomers, being little higher in energy than the ground-state configuration. As an example, we show in the firstrow of Fig. 1 the spin down and spin up ground-state

TABLE I. Total spin S of the ground states and some low-energy spin isomers for rs 1.51a

B. States with a S 0 spin-density wave are labeled SDW, whereas nonzero total spinsaccording to Hunds rules are labeled with H.

Number of Ground Excited DEelectrons state state(s) (mRy*)

2 04 1 H

6 08 1 H 0 SDW 23.010 1 H 0 SDW 4.6212 014 1 H16 2 H 0 SDW 18.1

1 H 20.20 SDW 20.7

18 1 H 0 SDW 11.0520 022 1 H 0 SDW 9.0624 0 SDW 2 H 0.7226 2 H 0 SDW 2.75

0 SDW 6.46

28 1 H 0 SDW 1.9830 032 1 H 0 SDW 7.0634 0 SDW 2 H 0.73

FIG. 1. Spin down and spin up densities n#,n" and normalizedpolarization zx, y for the ground state ( first row) and excitedstates (lower rows) for a dot with N 16 electrons andrs 1.51a

B. The maximum amplitude for the polarizationcorresponds to zx, y 0.23.

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densities for N 16, together with the normalized po-larization z n" 2 n#n0. In this case the total spinis S 2. It is intriguing to see that z shows a pro-nounced radial oscillation, which means that the excessspin is not homogeneously distributed over the whole dotregime. This effect was also seen for other sizes Nwhenthe ground state had nonzero total spin. It reminds one ofthe spin inversion states found by Gudmundsson et al. [6]

for finite magnetic fields.For many of the excited states at rs 1.51a

B the totalspin is zero. A priori one would in these cases expect thatthe system is fully unpolarized. But now looking againat Fig. 1, where the three lower rows show the densitiesand polarizations of the three different spin isomers forN 16, we see that the electronic structure is morecomplicated. Both the second and fourth rows of Fig. 1show isomeric S 0 states, being slightly higher in energythan the ground state (see Table I). Their space-dependentspin polarization shows apparent spatial oscillations, whichremind one of the phenomenon of spin-density waves [8] inthe 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 givenspatial region in the dot.

Studying these effects further, the next surprise is thatin larger dots this rather peculiar electronic structurewith total spin S 0 gets even lower in energy thanthe Hund state. For N 24 and N 34, the groundstate has spin zero, but is associated with a SDW statewhich has a lower energy than the uniform state. (Itsshape is similar to the one shown in Fig. 3 below forrs 5.0a

B.) The general possibility of such ground-stateconfigurations with uniform or nearly uniform electrondensity, but a nonuniform density of spin magnetization,was first discussed by Overhauser [8]. He stated thatthe nonmagnetic state must become unstable with respectto SDW formation at low densities, whereas at higher,metallic densities, it seems unlikely that the SDW stateis of enhanced stability. The energy balance, however, isvery delicate. Overhauser pointed out [8] that when theSDW states are stable, they are energetically only slightlylower than the nonmagnetic state.

On the basis of the results discussed above it is nowinteresting to study how the SDW states depend on thedensity of the two-dimensional electron gas. We thuscalculated for selected electron numbers the ground stateas a function of the strength v of the confining harmonicpotential, corresponding to rs values from 0.25a

B to 5.0a

B.Figure 2 shows the maximum amplitude of the nor-

malized polarization z as a function of rs for dots withN 24, 34, and 46 electrons, where the SDW was foundto 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

FIG. 2. Maximum amplitudes of zx, y of the ground-stateSDW for N 24, 34, and 46 as a function ofrs.

rs does not generally follow the trend suggested fromthis figure: For some N , 46, the SDW sets in at

a higher critical value of rs, but with a comparableamplitude. From our calculations we find that even inmagic configurations, which are fully paramagnetic withtotal spin zero at low rs values, for sufficiently large rsthe SDW formation sets in. In the case of the filled shellsN 12, 20, and 30 a SDW state was obtained for verylarge rs * 5a

B. Increasing rs still further, this SDW stategets more pronounced, in a similar way than the exampleswith N 24, 34, and 46 discussed above.

Finally, we note that the occurrence of a SDW isrelated to a change in the pattern of the single-particlelevels. As an example, we show in the bottom of Fig. 3the Kohn-Sham single-particle spectra of N 34 at adensity corresponding to rs 5.0aB, both for the ground-state SDW with S 0 and the excited state with totalspin S 2, being 1.9 mRy higher in energy. The thirdspectrum shows the result of a LDA calculation for theunpolarized case, which gives a 2.13 mRy higher energythan the SDW state. Note that for a SDW, there is alwaysone pair of degenerate spin up and spin down orbitals, andthe Fermi gap of the SDW state is much higher than in theS 2 case or within LDA. In quite a similar way to thespontaneous shape deformations of nuclei with nonclosedshells [18], the SDW opens a large energy gap at theFermi surface, leading to a more stabilized electronicstructure in the dot.

In conclusion, we have found that in finite quantumdots a static spin-density-wave-like state occurs even atrather high densities of the two-dimensional electron gas.

For many open-shell systems we found that the SDWstate with zero spin has a higher energy than the groundstate. In other cases the SDW is the ground state. Alsofor nonzero total spin a strong spatial dependence of thespin polarization was found. The amplitude of the SDWstrongly depends on the size of the Fermi gap: The

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FIG. 3. Upper part: Polarization zx, y for N 34 at rs 5a

B. Left: Ground state with S 0. Right: Excited statewith 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 5

a

B. The numbers in the spectra indicate thedegeneracies of the occupied single-particle states. The shorterlines indicate the lowest unoccupied states. Left: Hunds casewith S 2. Middle: SDW ground state with S 0. Right:LDA result, unpolarized. The total width of the spectrum isabout 73.5 mRy.

inset of the SDW-like state for the magic configurationsoccurs at much higher rs values than for nonclosedshells. Recent calculations have shown that the SDW-likestates are rather stable against distortions of the externalconfinement. In a real quantum dot, the deformationsof the effective confinement (caused, for example, by

quantum point contacts) will thus lead to a pinning of thepolarization. Rather peculiar properties of such quantumdots could be expected.

We would like to thank J. Helgesson, P. E. Lindelof,and B. R. Mottelson for helpful discussions. This workwas partially financed by the Studienstiftung des deut-schen Volkes, the BASF AG, the Academy of Finland,and CNAST. S. M. R. thanks the University of Jyvskylfor its hospitality.

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No. Cond-Mat 9701079, 1997.[15] The initial potentials for both spins were chosen to be of

square well type, with different depths V",#

0 1 6 hV0.Choosing h 0.0 and h 0.3 was sufficient to obtainthe unpolarized and partly polarized (Hunds rules) cases.Different radial forms R",#m f R01 1 e cosmf werechosen for m, e 0, 0, 3, 0.3, 4, 0.4, and 6, 0.6,where R"m and R

#m are twisted against each other by an

angle u pm.[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 leftover in the highest orbital, or Hunds rule, which alsooccurs for even N.

[17] In this formalism, the wave function is a single Slaterdeterminant of the Kohn-Sham single-particle wave func-tions. Only in cases where spin up and down space wavefunctions are exactly the same, the Slater determinant isthe eigenstate of the S2 operator with S Sz , but gener-ally this is only approximately true.

[18] . Bohr and B.R. Mottelson, Nuclear Structure (Ben-jamin, New York, 1975), Vol. II.

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m. koskinen, m. manninen and s. m. reimann- hund’s rules and spin density waves in quantum dots

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