denis ullmo et al- landau fermi-liquid picture of spin density functional theory: strutinsky...

8/3/2019 Denis Ullmo et al- Landau Fermi-liquid picture of spin density functional theory: Strutinsky approach to quantum dots

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Landau Fermi-liquid picture of spin density functional theory:

Strutinsky approach to quantum dots

Denis Ullmo,1,2 Hong Jiang,1,3 Weitao Yang,3 and Harold U. Baranger11Department of Physics, Duke University, Durham, North Carolina 27708-0305, USA

2Laboratoire de Physique Thorique et Modles Statistiques (LPTMS), 91405 Orsay Cedex, France3Department of Chemistry, Duke University, Durham, North Carolina 27708-0354, USA

(Received 21 January 2004; revised manuscript received 26 May 2004; published 8 November 2004)

We analyze the ground-state energy and spin of quantum dots obtained from spin density functional theory

(SDFT) calculations. First, we introduce a Strutinsky-type approximation, in which quantum interference is

treated as a correction to a smooth Thomas-Fermi description. For large irregular dots, we find that the

second-order Strutinsky expressions have an accuracy of about 5% of a mean level spacing compared to the

full SDFT and capture all the qualitative features. Second, we perform a random matrix-theory/random-plane

wave analysis of the Strutinsky SDFT expressions. The results are statistically similar to the SDFT quantum

dot statistics. Finally, we note that the second-order Strutinsky approximation provides, in essence, a Landau

Fermi-liquid picture of spin density functional theory. For instance, the leading term in the spin channel is

simply the familiar exchange constant. A direct comparison between SDFT and the perturbation theory derived

universal Hamiltonian is thus made possible.

DOI: 10.1103/PhysRevB.70.205309 PACS number(s): 73.21.La, 73.23.Hk, 05.45.Mt, 71.10.Ay

I. INTRODUCTION

Semiconductor quantum dots are now routinely obtainedusing electrostatic gates or etching processes to pattern atwo-dimensional electron gas formed in some heterostructure(typically GaAs / AlGaAs).1 Many of their sometimes sur-prising properties are now reasonably well understood froma qualitative or statistical point of view,1,2 and it is now re-alistic to think about using these quantum dots for somespecific purpose, such as spin filtering,3 current or spinpumping,4 or in the setting of quantum information.5

In this context, it becomes important to go beyond a

qualitative or statistical description, and to develop tools ableto predict quantitatively the properties of a specific quantumdot for given parameters. For isolated or weakly connecteddotsthe Coulomb blockade regimethe Coulomb interac-tion between electrons plays an important role and has to betaken into account properly.

Here, we have in mind relatively large quantum dots,6,7

with an electronic density high enough that Wigner crystal-lization is not an issue. A method of choice to address inter-action effects is, therefore, the density functional approach,8,9

which has been widely used in theoretical modeling of quan-tum dots. For small quantum dots, Ref. 10 provides a com-prehensive review of results obtained by this technique, aswell as comparison with quantum Monte Carlo calculationsand a discussion of the Wigner crystal regime.

For many situations of interest, it is necessary to describecorrectly the spin degree of freedom. We want therefore toconsider more specifically a spin density functional, whereeach density of spin nr, with =, corresponding tomajority and minority spins, is treated as an independentvariable. How this can be achieved in practice for a dot con-taining up to four hundred electrons, and for an rs parameteras high as 4, has been demonstrated in a series of papers. 1113

One striking feature of these calculations, however, is thatthe qualitative picture which emerges is somewhat unex-

pected in view of previous results. Indeed, within a statisticalapproach and assuming the classical dynamics within thenanostructure is sufficiently chaotic, one can model the wavefunctions in the quantum dot using random matrix theory(RMT). If furthermore the Coulomb interaction is treatedwithin the random phase approximation (RPA), it is possibleto derive various statistical quantities,1417 such as the distri-bution of spacing between Coulomb Blockade conductancepeak, or the probability of occurrence of nonstandard spins[that is, not zero (not one-half) for even (odd) particle num-ber]. It turns out, for instance, that the spin density functionalcalculations give a larger number of high spins than waspredicted within RMT plus RPA modeling.11 Such discrep-ancies could originate from a variety of causes, ranging fromthe statistical behavior of the Kohn-Sham wave functions toan intrinsic failure of one or the other approach. The goal ofthis paper is to clarify this issue.

For this purpose, we need a way to understand, or toorganize, the numbers obtained from the full-fledged spindensity functional theory (SDFT). This can be done using theStrutinsky approximation to DFT discussed in Ref. 18, up tostraightforward modifications to include the spin variable.

The Strutinsky approach is an approximation scheme, de-veloped in the late 1960s in the context of nuclear physics,in which the interference (or shell) effects are introducedperturbatively.19,20 It has been used since then in many sub-fields of physics,21 including the calculation of the bindingenergy of metal clusters.22 Standard treatments are usuallylimited to first-order corrections; in this case, it is similar tothe Harris functional23 familiar to the DFT literature. Here, incontrast, we use the second-order approximation19,20,24 de-veloped in Ref. 18. We shall see that the second-orderStrutinsky scheme turns out to be extremely accurate in somecircumstances. Furthermore, even when it is less precise,which happens in conjunction with the occurrence of spincontamination in the SDFT calculations, it still provides aqualitatively correct statistical description.

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The second-order Strutinsky corrections can be cast in avery natural form:18 in fact, they amount to taking into ac-count the residual (screened) interactions between quasipar-ticles in a Landau Fermi-liquid picture. They are thus ame-nable to treatment by the same RMT approach as was usedpreviously for RPA, providing an analogue of the universalHamiltonian in the case of SDFT. Since within the Strutin-sky approximation the quantum dot properties are relatively

transparent, we will then be in position to discuss the differ-ence between SDFT results and those obtained from RMTplus RPA.

The paper is organized as follows. In Sec. II, we brieflyreview the Strutinsky approximation as it applies to spin den-sity functional theory. In Sec. III, we consider more specifi-cally the local density approximation from this perspective,and in particular the screened potential that it implies. Sec.IV covers in detail the specific case of a model quantum dotwith quartic external confining potential. This model is usedto investigate the accuracy of the Strutinsky approximationfor various electron densities. In Sec. V, we discuss how theFermi-liquid picture emerging from the Strutinsky approxi-

mation scheme can be used, in conjunction with a randomplane wave model of the wavefunctions, to analyze the peakspacing and spin distributions resulting from the SDFT cal-culation. This framework also makes it possible to discussthe mechanism of spin contamination. Finally, in the lastsection we come back to the original question motivating thiswork and make use of what we understand from the Strutin-sky approximation to discuss the discrepancies betweenSDFT calculations and RPA plus RMT predictions.

II. STRUTINSKY APPROXIMATION FOR SPIN DENSITY

FUNCTIONAL THEORY

In the spin density functional description of a quantumdot, one considers a functional of both spin densitiesnr , nr

FKSn,n = TKSn

,n + Etotn,n, 1

where =, correspond to majority and minority spins,respectively. In this expression, the second term is an explicitfunctional of the densities

Etotn,n drnrUextr

+ drdrnrvintr rnr + Excn,n,2

where nr = nr + nr, Uextr is the exterior confiningpotential, and the precise form of the exchange correlationterm Excn

, n is to be discussed in more detail in Sec. III.vintr , r is the electron-electron interaction kernel. Thepresence of metallic gates can be taken into account by in-cluding an image term in addition to the bare Coulomb in-teraction

vintr r =e2

r r

e2

r r2 + 4zd2, 3

where zd is the distance between the top confining gate andtwo-dimensional electron gas. The image term in the inter-action kernel greatly reduces classical Coulomb repulsionbetween electrons at a distance larger than zd so that theelectron density far from the boundary is quite uniform. Thebare Coulomb interaction can be recovered by letting zd go toinfinity.

The kinetic energy term TKSn, n, on the other hand, is

expressed in terms of the auxiliary set of orthonormal func-

tions i

, i = 1 , ,N, such that nri=1

Nir2 as

TKSn,n =

2

2m

=,i=1

N

ir2dr . 4

From the density functional Eq. (1), the ground-state en-ergy of the quantum dot containing N,N particles of spin, is obtained as

EKSN,N = FKSnKS ,nKS , 5

where the Kohn-Sham densities nKS r , nKS

r minimizeFKS under the constraint given by the number of particles ofeach spin. This in practice is equivalent to solving the Kohn-Sham equations

22m

2 + UKS rir = iir, i = 1, . .. ,N 6

with the spin-dependent self-consistent potential

UKS r =

Etot

n

r

nKS ,nKS

. 7

In this section, we shall give a brief description of thesecond-order Strutinsky approximation as it applies to spindensity functional theory. Up to the introduction of the spinindices, the derivation of this approximation follows exactlythe same lines as the spinless case discussed in detail in Ref.18. We shall therefore not reproduce it here, but rather try toemphasize what exactly are the assumptions made in deriv-ing the approximation, and then merely write down the ex-pression we shall use in the following sections.

A. Generalized Thomas-Fermi approximation

The basic idea of the Strutinsky energy correction methodis to start from smooth approximations ETF and nTFr, to theDFT energy EKS and electronic density nKSr. Then, fluctu-ating terms are added perturbatively as an expansion in thesmall parameter

nr = nKSr nTFr. 8

In the original work of Strutinsky19,20 various ways of con-structing the smooth approximation have been considered.The most natural choice for nTF

r turns out to be the solu-tion of the Thomas-Fermi equation

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FTF

nn,n = TF

9

coupled with ETF =FTFnTF r , nTF

r. The generalizedThomas-Fermi functional is defined as

FTFn,n = TTFn

,n + Etotn,n, 10

with Etot

n, n given by Eq. (2). It differs from the originalspin density functional only in that the quantum-mechanicalkinetic energy TKS, Eq. (4), is replaced by an explicit func-tional of the density. For two-dimensional systems this takes

the form TTFn, n =TTF

0n, n +TTF1n, n with

TTF0n,n =

1

2N0 drnr2 + nr2 , 11

TTF1n,n =

8N0 dr n2

n+

n2

n , 12

where N0 = m/2 the density of states for two-dimensional systems and a dimensionless constant takenhere to be 0.25. Such a choice for the kinetic energy func-tional correctly takes into account the Pauli exclusion prin-ciple, and thus that an increase in kinetic energy is requiredto put many particles at the same space location, but fails toinclude fluctuations associated with quantum interference.

To illustrate this, let us for a short while assume = 0, i.e.,

only the first term TTF0

of the Thomas-Fermi kinetic energy istaken into account. Then, one can show that the Thomas-Fermi density fulfills the self-consistent equation

nTF r = nUTF

r, 13

where the Thomas-Fermi potential is defined, as in Eq. (7),by

UTF r =

Etot

nrnTF

,nTF 14

and

nUTF r = dp

2dTF p

2

2m UTF

r 15is the Weyl part of the density of states of a system of inde-pendent particles evolving under the potential UTF

r ( isthe Heaviside step function, d=2 is the dimensionality, andthe chemical potentials TF

are chosen so as to fulfill theconstraints on the total number of particles with spin and

). From its definition, nUTF r [and as a consequence

nTF r] is a smooth function, in the sense that it can change

appreciably only on the scale on which Uextr varies, butcannot account for the quantum fluctuations of the density onthe scale of the Fermi wavelength.

The Weyl approximation is, however, only the leadingterm in a semiclassical expansion of the smooth part of thedensity of states, and higher-order corrections in can be

added in a systematic way. The standard way21 to choose TTF1

is such that Eq. (12) holds, but with an approximation to thesmooth density of states nUTF

r which includes both the

Weyl term (15) and the first corrections. For two-dimensional systems, however, the corrective term computedfrom this prescription turns out to be zero, while the presence

of TTF1

is actually useful in smoothing the Thomas-Fermidensity near the boundaries of the classically allowed region.We have therefore used the phenomenological Weiscker-like term21 (12) which plays a similar role.

B. Strutinsky correction terms

The practical implementation for the Strutinsky schemecan be summarized as follows. The first step consists in solv-ing the generalized Thomas-Fermi equation (9), which de-fines a zeroth-order approximation for the ground-state en-ergy ETF as well as an approximation to the density ofparticles nTF

r. From this latter quantity, one derives foreach spin =, the effective potential UTF

r through Eq.(14).

The second step consists in solving the Schrdinger equa-tions (again for each spin)

2

2m

2 + UTF rir = iir, i = 1, . .. ,N.

16

Therefore, while the Kohn-Sham equations are both quantummechanical in nature and self-consistent, here all the self-consistency is left at the classical-like level of the Thomas-Fermi equation, and the quantum mechanical wave interfer-ence aspect is taken into account without self-consistency.One obtains in this way a new density

nr = 1

N

i2r . 17

It can be seen18 that nTF r is by construction a smooth ap-

proximation to nr [this is basically the content of Eq.(13)]. Therefore

nosc r nr nTF

r 18

can be considered as the oscillating part of n, and will de-scribe the short scale variations of the density associatedwith quantum interference.

Once the eigenfunctions and eigenvalues of Eq. (16) areknown, corrections to the Thomas-Fermi energy can beadded perturbatively

EKS

ETF

+ E1 + E2. 19

The first-order correction turns out to be simply the oscillat-ing part of the one particle energy18,21

E1 = E1posc = E1p E

1p 20

with

E1p = i=1

N

i+

i=1

N

i 21

the one particle energy and

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E1p = TTFnTF ,nTF

+ =,

drUTF rnTF r 22its smooth part.

The second-order correction E2 can be expressed intwo different ways.18 The first one is the intermediate result

E2 =1

2 ,=,drdrn

osc

r

V

bare

,

r,r

n

r

23

which is expressed in term of the bare interaction

Vbare,r,r =

2Etot

nrnrnTF

,nTF 24

but involves the a priori unknown densities n,r. Thesecond form

E2* =1

2

,=,

drdrnosc rVscreened, r,rnosc r

25

can be computed entirely in terms of the known [once Eq.(16) has been solved] densities nosc

,r, but requires the useof the screened interaction

Vscreened, r,r =

=,

dr2TTFn2

+ Vbare1,r,r

2TTFn2

Vbare,

r,r. 26

(The matrix inversion here should be taken with respect toboth the spatial position and the spin indices.) Note that inEq. (26) we can use

2TTF0

nrnr=

2

N0,r r 27

and, neglecting terms involving derivatives of the Thomas-Fermi density,

2TTF1

nrnr=

4N0

,

nTF r

rr r. 28

We stress that since the first of these second-order expres-sions, (23), involves the n,r which are in principle un-known, only the second form, Eq. (25), should be regarded

as the genuine second order Strutinsky result. It turns outhowever that in the practical implementation of Eq. (25), weshall in the following perform a further approximation tosimplify the numerical implementation. Note this is not,however, because of the matrix inversion in Eq. (26)forthe case we shall consider, the screening length is muchsmaller than the size of the system, allowing this inversion tobe done analytically. Rather, the resulting screened potentialwill turn out to be diagonal neither in the position nor in themomentum representation (see next section), in contrast to

Vbare, which is the sum of two parts, one diagonal in the

position and the other in the momentum representation.To simplify the numerical implementation of Eq. (25), we

shall therefore in Sec. IV use this equation with an approxi-mate Vscreened which is diagonal in the momentum represen-tation, thus inducing additional errors that are not intrinsic tothe Strutinsky scheme. We shall see that even with this ad-ditional approximation, the error on the fluctuating part ofthe total energy will be of the order of 5% of a one particle

mean level spacing.For most of our discussion, this is small enough that we

do not need a better approximation of Vscreened. What we willget in this way is however an upper bound on the precisionof the Strutinsky approximation. A lower bound, on the otherhand, can easily be obtained with the use of the intermediateexpression (23). Indeed we shall also implement the fullKohn-Sham calculation and therefore we will actually knownr. This lower bound on the error will turn out to beuseful for instance to discuss issues related to spin contami-nation. We shall therefore in Sec. IV compare the full DFTresults to both the first form (23) and the second form (25) ofthe Strutinsky approximation. In the figure labeling, the first

one will be referred to as ST and the second one as ST.*

C. Applicability of second-order Strutinsky expressions

Although we shall not reproduce here the derivation ofEq. (23) (see again Ref. 18 for details), the condition ofapplicability of this equation can be understood easily bycomparing the Kohn-Sham equations (6) with the ones de-fined by the Thomas-Fermi potential UTF

r, Eq. (16). Wesee there that the Kohn-Sham wave functions i

and theirapproximations i

are defined through Schrdinger equa-tions that differ only by a difference in the potential term

Ur = UKS r UTF r =, drVbare

,r,rn

r

29

with Vbarer , r defined by Eq. (24).The main approximation in the derivation of Eq. (23) is

that the Ur can be taken into account by a second-orderperturbative calculation. In general, this implies that the non-diagonal matrix elements Uij

= iUj

ij shouldtypically be smaller than the mean level spacing betweenone particle energies. More specifically, since the only rel-evant errors are ones modifying the Slater determinantformed by the occupied orbitals, the actual condition is that

the matrix element between the first unoccupied orbital andthe last occupied orbital is smaller than the spacing betweenthese level, that is,

N+1 UN

N+1 N = , . 30

A good accuracy, on the scale of the one particle mean levelspacing, of the Strutinsky approximation in the form (23) isequivalent to the above condition.

Now, Eq. (30) again involves nr, which is unknown.It is therefore not possible to prove rigorously that it shouldbe fulfilled. It is possible, however, to show the consistency


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of such an assumption. If we assume that (i) iUj

for all pairs of orbitals i ,j and =,, so that U

can be treated perturbatively and (ii) the oscillating parts ofthe Kohn-Sham densities nKSosc

r do not differ signifi-cantly from nosc

r, implying that nrnr + nosc r,

then the following can be shown.

a Ur = =, drVscreenedr,rn

oscr 31

which immediately yields Eq. (25) from Eq. (23).b Uij

2 /2 and nKSosc r nosc

r2/nr2are both negligible for a large dot for which the chaotic mo-tion in the potential UTF

allows modeling of the wave func-tions in terms of random plane waves (being of order ln g/gwith g the dimensionless conductance of the dot).

III. THE SCREENED INTERACTION IN THE LOCAL

DENSITY APPROXIMATION

The second order Strutinsky corrections, (23) or (25), in-volve the bare and screened interactions (24) and (26). In thissection, we shall consider the particular form these interac-tions take for the exchange correlation term we use to per-form the actual SDFT calculations, namely, the local spindensity approximation

Excn,n drnrxcnr,r , 32

where r = n n/n is the polarization of the electron gasand xc is the exchange plus correlation energy per electronfor the uniform electron gas with polarization . We further-more use Tanatar and Ceperleys form of

xcat =0 and 1,25

and the interpolation xcn , =xcn , 0 +fxcn , 1xcn , 0 for arbitrary polarization, with f = 1 +

3/2

+1 3/2 2/23/2 2. (This functional form is such thatthe result would be exact if only the exchange term wasconsidered.) Recently, Attaccalite et al.26 parametrized aLSDA exchange-correlation functional form based on moreaccurate quantum Monte Carlo calculations (see, e.g., Ref.27) that include spin polarization explicitly. We have, how-ever, checked that for the quantities in which we are inter-ested here, this functional introduces only minor modifica-tions; we shall therefore use Tanatar-Ceperleys form for ourdiscussion.

From the expression of the functional Excn, n, the bare

and screened interaction potentials Eq. (24) and (25), neededfor the evaluation of the second-order Strutinsky corrections,are easily computed. The bare interaction is the sum of twoterms Vbare = Vcoul+ Vxc. The Coulomb interaction

Vcoulr,r = vintr r 1 11 1

33is independent of both the density and spin. The matrix struc-ture here refers to the spin indices and . The exchangecorrelation term is local and can be expressed as

Vxcr,r = 2a0e2r r

vanr,r vbnr,rvbnr,r vanr, r

, 34where va and vb are obtained from the partial derivative ofxcn , [defined in Eq. (32)]. In all numerical calculationswe shall keep entirely the dependence of va and vb on thepolarization . However, this latter will usually be relativelysmall, and va and vb contain already second derivatives ofthe functional Exc with respect to the polarization. We shalltherefore proceed assuming van , van , 0 and vbn ,vbn , 0. The dependence of these functions on the param-eter rs =a0

2n1/2, with a0 =2/me2 the Bohr radius, is

shown in Fig. 1.Turning now to the screened interaction Vscreenedr , r, it

is useful to switch to the variables R , l =r+ r/2, r r. The Fourier transform of VbareR , l with respect to lreads

VbareR,q = 2e2a0vcq1 11 1 2a0e2

vaR vbRvbR vaR ,

35

where

vcq 1

a0q1 e2zdq 36

(again, the pure Coulomb case is recovered by letting thedistance to the top gate zd go to infinity). This can by furthersimplified by diagonalizing the matrix in the spin indices: the

eigenvectors are the charge channel c=+/2 and the

spin channel s

=

/2. The eigenvalues are

cbareR,q =

2

N02vcq vaR + vbR 37

in the charge channel and

sbareR,q =

2

N0vbR vaR 38

in the spin channel [note 2a0e2 = 2 /N0].

Because the screening length is short on the scale forwhich the smooth part of the electronic density varies appre-

FIG. 1. (Color online) The functions van , 0 and vbn , 0, whichdefine the functional second derivative ofExcn

, n [see Eq. (34)],as a function of the parameter rs = 1 /a02n.


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ciably, the operator 2TTF/n2 + Vbare can be inverted by

treating the variable R as a parameter, appearing thus diag-onal in the q representation. One obtains in this way

VscreenedR,q = c + s/2 c s/2c s/2 c + s/2 39in terms of the eigenvalues

cR,q =2

N0

2vcq vaR + vbR1 + gq2vcq vaR + vbR

40

for the charge channel and

sR,q =2

N0

vbR vaR

1 + gqvbR vaR41

for the spin channel. The R dependence ofc and s arisesfrom vanR and vbnR, and therefore from the localvalue of the density (that is, of the parameter rs) at the loca-tion the interaction is taking place. Furthermore, the function

gq = 1 + q2

8nTFR1

42

would just be 1 in the absence of the correction TTF1

to theThomas-Fermi kinetic energy term; it prevents effectivescreening to take place for large momenta.

IV. THE GATED QUARTIC OSCILLATOR MODEL

To evaluate the accuracy of the Strutinsky approximationscheme, we consider a two-dimensional model system forwhich the electrons are confined by a quartic potential

Uextx,y = ax4b

+ by4 2x2y2 + x2y y2xr . 43The role of the various parameters of Uextx ,y is the follow-ing: a controls the total electronic density (i.e., the parameterrs), and therefore the relative strength of the Coulomb inter-action; allows one to place the system in a regime wherethe corresponding classical motion is essentially chaotic; fi-nally, b and have been introduced to lower the symmetryof the system. In the following sections, we use [a1] to de-note the parameter value a = 101ER/a0

4 (with ER = e2/ 2a0)

and [a4] for a = 104ER/a04, which at N=120200 corre-

spond to rs 0.3 and 1.3, respectively. We use =0.53 and=0.2 in our calculations unless specified otherwise. In ad-dition to this two-dimensional potential, we assume the ex-

istence of a metallic gate some distance zd away from the 2Delectron gas whose purpose is to cut off the long-range partof the Coulomb interaction. This gate is placed sufficientlyfar from the electrons to prevent the formation of a densitydeficit in the center of the potential well without modifyingqualitatively the quantum fluctuation. In practice, we take zdabout 0.75a0 for [a1] and 2.5a0 for [a4].

28

A. Electronic densities

For any set of parameters defining the potential (43), andfor any number of up and down electrons N,N, we can

compute the Kohn-Sham energies EKSN,N and densities

nKS r following the approach described in detail in Ref. 12.

For the Strutinsky approximation, the only part which pre-sents some degree of difficulty is actually the Thomas-Fermicalculation, for which we have developed a new conjugate-gradient method which turns out to be extremely efficient.29

Once nTF,r are known, the effective potentials UTF

,rand the corresponding densities n,r follow immediately.

Figure 2 shows the densities nTFr, nKSr, nr, andnoscr for the ground state with N=200 electrons N=N=100 of the gated quartic oscillator system with parameter[a4], corresponding to an interaction parameter of rs =1.3.Noting, in particular, the difference of scale between the up-per and lower panels, one can observe that the Thomas-Fermi density already is a very good approximation to theexact one, and that nr is a small oscillating correction, ofthe same magnitude as the oscillating part of nr. Apparentalso on this figure is the fact that the largest errors are locatedat the boundary of the dot where corrections to the Weyldensity of higher order in are the largest. To make thismore visible, we plot in Fig. 3 the densities nr and noscralong a cut on a diagonal of the confining potential for twosets of parameters. This makes it clear that noscr is an os-cillating function only in the interior of the dot, but has aproportionally large secular component at the boundary. As a

consequence, choosing correctly the term TTF1

of the Thomas-Fermi kinetic energy term is actually important to obtaingood accuracy.30 We have therefore determined the param-eter =0.25 of this functional by imposing that E1p

oscN os-cillates around zero, rather than having a significant meanvalue. As an illustration, we also plot in Fig. 3 the samequantities but for a calculation where a value =1 has beenused for the Thomas-Fermi kinetic energy correcting term.We see that this increases significantly the error in the

Thomas-Fermi density at the boundary, reducing the effec-tiveness of the Strutinsky approximation.

B. Ground-state energies

With the expressions (35) and (39) of the bare andscreened interactions, and using the known eigenvalues andeigenfunctions for the Schrdinger equations (6) and (16),the Strutinsky approximation for the total energy, includingthe order one Eq. (20) and order two Eqs. (23) and (25)corrections, can be computed for any N,N. As for the fullspin density calculations, the ground-state energy for a giventotal number of electron N is then obtain as the minimum

over N,N with N+N=N of these energies.Let us now consider these ground-state energies for a

choice of parameters [a1], such that the coefficient rs =0.3 isstill smaller than one and thus the Coulomb interaction is notvery large compared to the kinetic energy of the particles.Figure 4 displays the corresponding difference betweenStrutinsky and Kohn-Sham ground-state energies, in meanlevel spacing units, for a number of electrons ranging from50 to 200. For the upper panel, ESTN is obtained from theintermediate expression (23) using the bare interaction andrequiring the knowledge of the exact n,. For the lower


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panel, EST*N is obtained from Eq. (25), which involves theapproximate screened interaction, but only the knowledge of

nosc,r.

The first observation that can be made on that figure isthat the second form of the Strutinsky approximation appearssubstantially less accurate than the first one. As mentioned

earlier, this is, however, probably due to the fact that, be-cause our code was devised to handle only two body inter-actions that were diagonal either in position or in momentumrepresentation, we had to use in that calculation an approxi-mation where, for the screened interaction, the local value ofthe parameter rsr =1 /nTFr has been replaced by itsmean value.

Indeed, a second feature immediately visible on Fig. 4 isthe presence of a net trend in the energy differences betweenKohn-Sham and Strutinsky calculations. This secular term isrelated to the nonoscillating component of noscr and nrvisible on the lower panels of Figs. 2 and 3 at the boundaryof the quantum dots. This can be checked by using a lessaccurate Thomas-Fermi approximation (e.g., with =1 for

the correcting term of the kinetic energy TTF1

), and noticingthat this secular term in the deviation increases noticeablywhile the fluctuations remain less affected. Since the totaldensity of electrons at the boundary of the dot is significantlylower than its average value, it is relatively natural that inour approach, this secular deviation is made significantlyworse in the second form of our approximation.

The secular deviation is not completely negligible interms of the mean level spacing. The relevant scale for thesmooth part of the ground energies is, however, the charging

FIG. 2. (Color online) Particle density for the parameters [a4]

for a system of N=200 electrons N=N=100. From top to bot-tom: nTFr, nKSr, nr, and noscr. Note that nTF is a smoothapproximation to nKS and that nr and noscr are very similar.

FIG. 3. (Color online) Diagonal cut of particle densities nr(solid) and noscr (dashed), in units of the average Thomas-Fermidensity nTF inside the dot. Upper panel: parameters [a1] rs 0.3.Lower panel parameters [a4] rs 1.3. The dot contains N=200electrons N=N=100. The thin dashed lines correspond tonoscr from a less accurate choice of the Thomas-Fermi kineticenergy term, namely, =1.0 instead of 0.25.


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energy, compared to which these secular corrections are ex-tremely small. To focus on the fluctuating part, we thereforeremove the secular term (i.e., the straight lines in Fig. 4),obtaining in this way Fig. 5.

Let us consider first the upper panel, where the interme-diate result Eq. (23) has been used. We observe that thefluctuating part of the error is usually extremely small, typi-cally of order a percent of a mean level spacing, and that thisis mainly an oscillation between odd and even number of

particles in the system. Nevertheless, in a few circumstancessignificantly larger deviations are observed, with a magni-tude typically 5% of a mean level spacing and a sign whichis always negative.

To understand the origin of these larger deviations, it isuseful to correlate them with the occurrence of spin contami-nation, that is to situations where the SDFT calculationsbreak the spin rotation symmetry. Since the actual spin is asomewhat ill-defined quantity in a spin density calculation,we need, however, first to specify what we understand bythis. Indeed, in spin density functional theory the differenceN

N

can be interpreted as twice the component Sz

of thequantum dot total spin. Another quantity that can be easily

computed is the mean value SS + 1 of the operator S2 for theSlater determinant formed by the Kohn-Sham orbitals i

, i= 1 , ,N, which can be expressed as SS + 1 = SzSz + 1+S with the spin contamination S given by31

S = N i,jN

ij

2. 44

From this expression, one sees that if all occupied orbitalsare identical to the corresponding orbitals, S =0, and it ispresumably reasonable to interpret NN /2 as the sys-tems total spin. However, as soon as different-spin orbitalsstart to differ, S can take any positive value smaller than N,signaling that, at least, there is some ambiguity in the assess-ment of the total spin of the system. In Fig. 6 we have plottedthe ground-state spin contamination SN as a function ofthe particle number. For [a1], the spin contamination is usu-ally negligible, except in some few cases where Ss of orderone half or so are encountered.

Coming back to the Strutinsky approximation, we can usethe information from Fig. 6 to exclude in Fig. 5 the groundstates with significant spin contamination. The remainingpoints correspond to the dotted symbols in this figure. In the

FIG. 4. (Color online) Difference between Strutinsky and Kohn-

Sham ground-state energies, in units of the mean level spacing, as a

function of the number of electrons, for the high density dot [a1]

rs 0.3. Upper panel: ESTN obtained from Eq. (23). Lower

panel: EST*N obtained from Eq. (25). In both cases, the differenceshows small fluctuations (a few percent) about a linear secular trend

(dashed line is fit).


Sham ground-state energies as in Fig. 4 but now with the secular

trend removed. Solid: all ground states. Dots: ground states without

significant spin contamination. Note the excellent agreement in the

case of EST, obtained with the intermediate result Eq. (23) (upper

panel)of order 1%when spin contamination is not present. In

the case of the genuine Strutinsky approximation EST* [i.e., using

Eq. (25)], the agreement is still very good (lower panel).

FIG. 6. Spin contamination SN as a function of the numberof particles. The spin contamination is infrequent at high density

([a1], upper panel), but becomes frequent and substantial for rs 1.3 ([a4], lower panel).


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upper panel, we see that there is almost a one to one corre-spondence between larger errors and spin contamination.

Turning to the lower panel in Fig. 5, we see that the fur-ther approximations in treating the screening used in evalu-ating Eq. (25) do degrade the accuracy of the ground-stateenergy somewhat. Still, the genuine Strutinsky result (25)gives the fluctuating part of the energy to within a few per-cent of the mean level spacing. For the spin contamination,no particular correlation is seen, presumably again because

the overall agreement is slightly spoiled by the approxima-tion we made for the screened Coulomb interaction.

In lower density (larger rs) more realistic dots modeled bythe parameter set [a4], spin contamination in KS groundstates is much more pronounced, as shown in the lower panelof Fig. 6it is, in fact, always significant. In conjunction,Fig. 7 shows that the accuracy of both the intermediate ex-pression (23) and the genuine Strutinsky approximation [i.e.,using Eq. (25)] becomes worse. As at higher density, themain error is a secular trend: in the case of Eq. (25) it attainsa value of several mean level spacings, due presumably tothe approximations made in treating the screening. After re-moving the secular deviation, however, the fluctuation in the

errors of both forms of the Strutinsky approximation for theground-state energy is still quite small: the r.m.s. is 0.05when Eq. (23) is used and 0.06 when Eq. (25) is. Thus forcharacterizing the fluctuating part of the ground-state energy,the genuine Strutinsky approximation is nearly as good asthe intermediate result (23), a property we expect to remainvalid at larger rs.

C. Coulomb Blockade peak spacings and spin distribution

In the previous subsection, we have considered the accu-racy of the Strutinsky approximation for individual ground-

state energies. We found that as long as no significantamount of spin contamination is present in the SDFT calcu-lation, the Strutinsky result provides an excellent approxima-tion when the intermediate expression (23) is used, and agood, though slightly degraded one, when applying the genu-ine Strutinsky approximation (25). In this latter case, it isprobable that the additional errors come mainly from theneglect of the local density dependence in the screened in-

teraction rather than to the Strutinsky approximation itself.We shall come back to the issue of spin contamination in thenext section, and sketch an extension of the theory thatwould make it suitable to deal with the spin contaminatedcase.

Before doing so, however, we shall address another ques-tion, namely, how well, even in the case where a one-to-onecomparison of ground-state energies can imply an error of afraction of mean level spacing, are the statistical propertiesof the quantum dots described within the Strutinsky approxi-mation. For instance, we have in mind the distribution ofground-state spin SzN or of ground-state energy second dif-ference sN =EKSN+ 1 +EKSN 1 2EKSN+ 1. This lat-

ter quantity is accessible experimentally by measuring thespacing between conductance peaks in the Coulomb Block-ade regime, and will be referred to below as the peak spac-ing.

In Fig. 8, both peak spacing and spin distributions areplotted for two interaction strength regimes ([a1] and [a4])using either Kohn-Sham results or one or the other forms ofthe Strutinsky approximation. In the small rs case, the agree-ment is naturally excellent, but we see that even for thehigher rs case, both forms of the Strutinsky approximationgive a fairly good approximationcertainly they provide aqualitatively correct description.

V. FERMI-LIQUID PICTURE

It is not possible to develop a real Landau Fermi-liquidtheory for quantum dots because the mesoscopic fluctuationsprevent any Taylor expansion of the free energy in terms ofoccupation number. (Landau theory basically assumes thatthe excitation energies are the smallest energy scales of theproblem, which is clearly not true for mesoscopic systemsbecause of variation on the scale of the mean level spacing.)However, discussing what we may call a Landau Fermi-liquid picture, in the sense that the low-energy physics isdescribed by a renormalized weak interaction, is still some-thing meaningful.

In that sense, what Eqs. (19)(26) express is that SDFT, inthe limit where the Strutinsky approximation scheme is ac-curate, is equivalent to a Landau Fermi liquid picture, wherequasiparticles with spin =, evolve in the effective po-tential UTF

and interact through a residual weak interaction

Vscreened, r , r that can be taken into account as a perturba-

tion. The only unusual feature is the absence of an exchange-like contribution to the total energy. Indeed the main role ofthe exchange correlation functional Excn, n is to make theinteraction between same spin particles different from theone between opposite spins.


Sham ground-state energies, in units of the mean level spacing, as a

function of the number of electrons, for the rs 1.3 dot [a4]. Upperpanel: ESTN; lower panel: EST*N. The linear secular trend(dashed line is a fit) is now significantly larger than for [a1] (com-

pare with Fig. 4), but the fluctuation about this trend remains small

(of order 5%).


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Since moreover we have chosen the confining potential insuch a way that the classical motion within our model quan-tum dot is chaotic, we know that we can use a statistical

description of the eigenlevels and eigenstates of HTF= p2/ 2m + UTFr in terms of random matrix theory (RMT)and random plane wave (RPW) modeling. We are thereforein the position to follow the line of reasoning in Ref. 15 to

analyze the SDFT calculation. We shall do this in this sectionfirst to get some understanding of the peak spacing and spindistributions obtained, and in a later stage to address themechanism of spin contamination.

A. Universal Hamiltonian form

To model the statistical properties obtained from theSDFT calculations, let us assume that the Thomas-Fermidensity across the dot has variation small enough that we cantake the parameter rs as a constant. We furthermore imposenTF r = nTF

r = nTFr/ 2 [nTF rdr might then be half inte-

ger for odd N, which is not a problem since at the Thomas-Fermi level the quantization of particle number is not playingany role] and write the second order Strutinsky correction as

E2 =1

2,

i,j

fifjMi,j, E2 , 45

with fi=0,1 the occupation number of orbital i with spin ,

E2 =1

2,

drdrnTF rVscreened, r,rnTFr 46and

Mi,j, = drdrir2Vscreened, r,rjr2. 47

For a chaotic system, it can be shown that the fluctuations

of the Mi,j, are small (variance of order ln g/g2) and that

their mean values are given by

M

i,j

,

=

V

screened

,

q = 0

+

T

ij V

screened

,

FC/A ,

48

where A is the area of the dot, T is 2 here since time rever-sal symmetry is preserved (but would be 1 if it were broken),and

Vscreened, FC =

1

2

0

2

dVscreened, 21 + cos kF

49

is the average of the screened interaction over the Fermicircle (note a0kF=2 /rs).

Note, however, that the screened interaction (26) is de-rived under the assumption that the oscillating part of the

density integrates to zero, so that the total displaced chargeproviding the screening also does. Between the reference S=0 configuration (the TF case) and the higher S ones, thetotal number of electrons is, of course, conserved, but not thenumber for each spin. It should be kept in mind, therefore,that the q =0 component of the density cannot be screened;this can be included simply by setting sq = 0 s

bareq= 0.

The screened interaction matrix is characterized by its ei-genvalues (40) and (41). In Fig. 9, we thus plot the rs depen-dence of these quantities averaged over the Fermi circle, andcompare them to their bare counterparts.

A few remarks are in order concerning this figure. First,we see that, because of the divergence of the Coulomb inter-action at small q, screening has a drastic effect for the chargechannel.32 Screening is less dramatic in the spin channel, butcan still be an order one effect as rs increases.

Furthermore, while the screening decreases the absolutestrength of the interaction in the charge channel, it actuallyincreases it in the spin channel. Indeed, since the interactionin the spin channel coming from SDFT is attractive, thecharges in the bulk of the Fermi sea will, as long as this doesnot involve too much kinetic energy, move so as to increasethe spin polarization. Finally, we note that for the value ofthe parameter that we use, the effect of the first correc-

FIG. 8. (Color online) Spin and peak spacing distributions for

the cases [a1] (left column) and [a4] (right column). Solid: even N;

dashed: odd N. The statistics are obtained for N=120200

with , =0.53,0.2 , 0.565,0.2 , 0.6,0.1 , 0.635,0.15 and0.67,0.1. From top to bottom: Kohn-Sham, intermediate [Eq.

(23)], and genuine [Eq. (25)] Strutinsky approximation. Agreementbetween the three methods is excellent, of course, for [a1]. But even

for [a4] where individual energies may be in error, the agreement of

the distributions is very good.


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tion TTF1

on the screened interaction is very small in thecharge channel, and only slightly larger in the spin channel.

If we neglect the fluctuations of the Mi,j,, Eqs. (39), (45),

and (48) imply that E2 is just a function of the number ofparticles N=N+N in the dot and the z component Sz=NN /2 of the ground state the spin,

E2N,Sz =1

2TNcFC + s

bareSz

2 T

2cFC sFCSz.

50

The main value of this expression is how it compares to theuniversal Hamiltonian form,33,34 and we shall come back tothis point in the discussion section. Already we can see, how-ever, that it contains almost all the information necessary tounderstand qualitatively the ground-state spin distributions.Indeed, looking at Fig. 10, which shows the differenceE2N, Sz + 1 E

2N, Sz as a function of rs for sev-eral values of Sz, we see that for rs 0.85, the interaction

energy gain and one particle energy cost of forming a tripletare equal on average, and therefore triplets should become asprobable as singlets. In the same way, spin 3/2 becomes asprobable as 1/2 at rs 1.8, and spin 2 as probable as 1 atrs 2.8.

To have a more precise idea of the whether the random-plane-wave model captures the main physics, we can followthe approach of Ref. 15 and make a simulation of the peakspacing and spin distributions. We use GOE distributed en-ergy levels for the first-order correction (20), and take the

second-order correction in the form (45) with the Mi,j, in-

dependent variables with mean (48) and a variance whichcan be computed using the method of Appendix A of Ref. 15:

varMi,j

2 =

32

kFL2

2/kFL

22/kFL dx

x

1

4 x2vx

vx + ijvx + v0

+ v4 x2 , 51

where vq/kF N0Vscreened q /2. That the random-

plane-wave model correctly describe the wave function sta-tistics is illustrated in Fig. 11 where the analytic expressions(48) and (51) are compared, as a function of rs, to the resultfrom the actual eigenfunctions derived from Eq. (16). Theanalytic expression for the mean is expected to be quite re-liable and hence the good agreement. The variance (51) isless accuratebecause of the cutoff used, for instanceandso we consider the agreement in Fig. 11 quite good.

FIG. 9. (Color online) Average over the Fermi circle of the

eigenvalues of the screened and bare SDFT interactions [in units of

2 /N0] as a function of rs = a02n1/2. Dark: charge channel.

Lighter color (green online): spin channel. Dashed: bare interaction

c,sbare (with a cutoff of the momentum at q 1 /L in the charge

channel). Solid: screened interaction c,s with =0.25. Thin dot-dashed: same but with =0. Since the q dependence of s is en-

tirely due to the TTF1

correction to the Thomas-Fermi kinetic energy

functional, it therefore disappears in this latter case. The interaction

in the charge channel is, of course, dramatically decreased by

screening; in contrast, screening increases the magnitude of the in-

teraction in the spin channel.

FIG. 10. (Color online) Mean value of the gain in interaction

energy E2Sz + 1 E2Sz, as computed from Eq. (50),

(thick lines) and of the mean one-particle energy cost (thin horizon-

tal lines) associated with flipping the spin of one particle in the

quantum dot. Solid: Sz =0; dot-dashed Sz =1/2; Dashed: Sz = 1.

FIG. 11. (Color online) Comparison between the analytical

RPW predictions [Eqs. (48) and (51)] and numerical calculations of

the mean and variance of Mijs. The wave functions used in thenumerical calculations are eigenfunctions of the effective Thomas-

Fermi potential with N=200 in the quartic oscillator system. Lines

(points) correspond to analytical (numerical) results, with the solid

(circle) for Mi,j,, short dashed (square) for Mi,j

,, long dashed (up-

triangle) for Mi,i,, and dot dashed (down triangle) for Mi,i

,.


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Figure 12 displays the peak spacing and spin distributionfor rs =0.3 (corresponding to [a1]) and rs =1.3 ([a4]) comingfrom a simulation in which the fluctuations of level spacing

and of the Mi,j, are included. We see that the qualitative

behavior observed in Fig. 8and in particular the lowerpanelis very well reproduced. Thus the RMT/RPW ap-

proach using the LSDA interaction is successful in compari-son with the full SDFT calculation.

B. Spin contamination

As we saw in the previous sections, the statistical proper-ties of the model quantum dots obtained from the full SDFTcomputation are, at least up to rs 1.3, well reproduced bythe various forms of the Strutinsky approximation. However,we also saw that spin contamination, when present in theSDFT calculations, was actually degrading the accuracy ofthe Strutinsky approximation on a case by case basis. Indeed,by construction, the Strutinsky approximation as we pre-

sented it cannot involve any spin contamination. Spin con-tamination is a way, in the SDFT calculations, to lower theenergy of the system without changing the total z componentof the spin Sz by having different wave functions for the and orbitals. However, the eigenstates implicit in the

Strutinsky approach are almost identical to those of HTF, i,

and the i are nearly independent of the spin.

In this section, we shall discuss how this spin contamina-tion mechanism could be understood within this Strutinskyscheme. Rather than trying to consider the most general situ-ation, we will limit ourselves to the case of even number ofparticle N, and a ground-state z component spin equal tozero.

Let HeffnTF = 2/ 2m2 + UTF r be the Thomas-FermiHamiltonian defining the orbitals i

[see Eqs. (14) and (16)].What we have done is to construct an approximate solutionof the SDFT equations (5) as nr = j=1

N jr2 plus some

screening charge. In this respect, an important point was thatthe resulting potential change U given by Eq. (31) wassuch that the matrix element i

Uj for ij was neg-

ligible, to first order in 1/g.However, finding an approximate solution of the Kohn-

Sham equation implies only that one has an extremum of thespin density functional, but not necessarily an absolute mini-

mum. As pointed out earlier, modifications of the wavefunc-tions change the electronic density only if they mix occupiedand unoccupied orbitals. Therefore, when searching for anew extremum of the spin density functional, with somechance to actually get the true minimum, a natural choice isto mix the last occupied orbital M

with the first unoccupiedone M+1

(for M=N/2=N=N).Let us therefore look for approximations i

to the KS

wavefunctions defined by i=i for iM and

M = cos M + sin

M+1 ,

M+1 = sin M + cos

M+1 , 52

with possibly a different value of the angle for the twospins =,. For these wave functions, the Thomas-FermiHamiltonian has a matrix element

M HTFM+1

= cos sin M+1 M 53

in terms of the Thomas-Fermi energies M+1 and M.This change in the wave functions produces a modifica-

tion of the densities n

= M

2

2

which, once screeningis taken into account, itself generates a perturbation potential

Ur = =,

drVscreened, r,rnr . 54(The modification of the other wavefunctions, except forscreening this term, does not play a role here.) The self-consistent condition for the angles , is that

M HTFM+1

+ M UrM+1

= 0.In order to find matrix elements of U, let us consider

for a moment the case =. Equation (52) implies that

M = cos M

+ sin M+1 55

and, therefore,

MUM+1

= 2 cos sin I 56

with

I= drdrMrM+1 rVscreened, r,rMrM+1 r Vscreened

, FC/A , 57

where the last equality applies in lowest order in 1 /g, that is,neglecting the fluctuations.

Noting finally that the same reasoning can be applied tothe orbitals, we find that to get an extremum for the spindensity functional the angles and should obey

cos sin M+1 M = 2 cos sin

V,screenedFC/A 58

and the analogous equation with and interchanged. Ob-vious solutions are == 0, (= 0, =/ 2), (

=/2, = 0), and ==/ 2. The first one correspondsto the standard S = 0 (non-spin-contaminated) solution; theother ones involve promotion of particles from the last occu-pied orbital to the first unoccupied one, and so are obviously

FIG. 12. (Color online) Peak spacing distributions for the RMT/

RPW model corresponding to rs =0.3 (left) and 1.3 (right). Solid:

even N, dashed: odd N. Inset: corresponding spin distribution.


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of higher energy. However, other solutions may exist.Clearly they should fulfill

cos sin = cos sin , 59

that is, up to an irrelevant multiple of phase

= 60

and, therefore,

cos 2= M+1 M

4Vscreened, FC/A

. 61

It can be checked that whenever the condition (61) can be

fulfilled [i.e., when M+1 M4Vscreened, FC], the corre-

sponding extremum has an energy smaller by a quantitysin22sFC cFC/A with respect to the noncontami-nated configuration. In this situation, the spin contaminatedSz =0 state will be favored (but its energy still needs to becompared to the lowest energy state with Sz = 1).

VI. DISCUSSION

To summarize our findings, we have seen that up to rsvalues of order one (in practice, 1.3 here), the Strutinskyapproximation yields a ground state total energy with fluctu-ating part reliable up to typically 5% of the mean level spac-ing. Furthermore, these errors can be related to the occur-rence of spin contamination in the SDFT calculation, whichcannot be reproduced by the Strutinsky scheme (in its sim-plest form). Indeed, as discussed in the last section, this lattergives, by construction, an approximation to an extremum ofthe Kohn-Sham functional for which the and orbitals arenearly identical, but not necessarily an approximation to the

true minimum.Nevertheless, the qualitative properties of the peak spac-ing and spin distribution are correctly reproduced by theStrutinsky approximation (at least up to rs of order one, forwhich spin contamination does not appear to change drasti-cally the distributions). For a chaotic confining potential thismakes it possible to use the modeling in terms of randommatrix theory (for the energy levels) and random planewaves (for the eigenstates) introduced in Ref. 15. Within thisRMT/RPW modeling, and in the limit of large dots for whichfluctuations of the residual interaction term are small, themain features of the peak spacing and spin distributions canbe understood as arising from the interplay between one par-ticle level fluctuations and the spin dependence of the meanresidual interaction term [see Eq. (50)]

ESDFT2 N,Sz = s

bareSz

2 T

2cFC sFCSz

+ term depending on N only . 62

It is useful to compare this expression, as well as thecorresponding distributions based on RMT/RPW modelingsuch as those in Fig. 12 (which actually take into account thefluctuations of the residual interaction term), to what is ob-tained following the more traditional route to the analysis of

peak spacing and spin distributions for chaotic quantumdots.14,15,17 In these earlier approaches, the ground-state en-ergy of the quantum dot would, in a way very similar way toEq. (19), be described as the sum of a large nonfluctuating

classical-like term, a one-particle energy contribution com-puted for some effective confining potential, and finally a

residual interaction term ERPA2

. This latter would, however,be understood as originating from a weak interactionVRPAr r which in the random phase approximation canbe shown to be just the RPA screened potential,14,17 butshould be understood more properly as the residual interac-tion between quasiparticles in Landau Fermi-liquid spirit. Weshall thus refer to this latter description as the RPA ap-proach, although this is slightly inappropriate.

While the Strutinsky approximation to SDFT gives a re-sidual interaction which can be understood as a first-orderperturbation (without exchange) in terms of the spin-dependent potential (26), in contrast one has in the RPAapproach a residual interaction arising from the perturbativecorrections in some VRPAr , r (including both direct andexchange) as well as possibly higher-order terms which turnout to be important for time-reversal invariant systems (theCooper series). Under this assumption and following exactlythe same analysis as the one leading to Eq. (62), one gets forthe mean residual interaction term in the RPA approach

ERPA2 N,Sz = JSSzSz + 1 + T 1Sz

+ term depending on N only 63

(again T= 2 but would be 1 if time-reversal invariance werebroken).

In Eq. (63), the parameter JS is equal to VqFC, where

the Fermi circle average is defined by Eq. (49) and VRPAqis the Fourier transform of VRPAr r. More properly how-ever, one should understand JS as being related to Fermi

liquid parameter f0a

through

JS/ = f0a. 64

In first-order perturbation theory, would be equal to JS, butscreening associated with higher order terms in the Cooperchannel somewhat reduces this value.35,36 For mesoscopic

FIG. 13. (Color online) Same as Fig. 12(b), but with a residual

interaction modeled by a perturbative calculation in the interaction

potential VRPAr r whose parameters are set by Eqs. (64) and(65).


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ballistic systems, an analysis following the lines of Ref. 37suggests that

=Js

1 +JS

lnkFL

, 65

with kF the Fermi momentum and L the typical size of thesystem. We shall assume this in the following discussion,bearing, in mind that this is true only up to logarithmicaccuracy.

The remarkable point here is that sbarers/ is actually

the same thing as f0ars, in the sense that the Landau Fermi-

liquid parameter f0ars can be interpreted as the second de-

rivative with respect to the polarization, at fixed total density,of xc the exchange correlation energy per particle of the

uniform electron gas. This implies that the term quadratic inSz of Eqs. (62) and (63) actually do correspond.

As a consequence if we compare Fig. 12(b), which isobtained from a RMT/RPW simulation with the interactioncorresponding to the spin density functional, with Fig. 13,obtained in the same way but with an RPA-likeinteraction,38 the differences in spin polarization and in odd/even asymmetry for the peak spacing distribution can bealmost entirely associated with the different linear Sz terms inEqs. (62) and (63).

To get some further insight into the difference betweenthe two approaches, let us consider Table I which shows thespin distributions for an interaction parameter rs 1.3 andvarious approximations discussed in this paper. Comparingdifferent pairs of lines gives a sense of the importance of thevarious issues. For instance, comparing the second line withthe third gives an idea of how accurate the RMT/RPW modelis for the statistical properties of the real energy levels andeigenfunctions of the quartic oscillator system, since bothlines are based on the same spin-dependent interaction (26).The first line compared to the second, on the other hand, is ameasure of how well the Strutinsky scheme approximates thefull SDFT calculation. Presumably, most of the differencebetween these two lines can be associated with the presenceof spin contamination in SDFTit is in some sense a mea-

sure of the effectiveness of spin contamination in loweringthe total spin of the system. Going further down the table, thedifference between the third and the two last lines is a mea-

sure of the impact of different linear terms in Sz in Eqs. (62)and (63), without screening the Cooper channel for thefourth line and with a screened Cooper channel [according toEq. (65)] for the last line of the table.

From Table I, it appears that within the accuracy of theRMT/RPW modeling, which seems to be around 5%, theSDFT result is compatible with an RPA-like approach if theCooper channel is not screened. In other words, the fact thatsFC cFC/2 is more negative that Js =s

bareproducing

higher spinsis compensated by the effect of spincontaminationwhich favors lower spins. As seen in Ref.13, this compensation between the two effects seems to existalso for higher values of rs. On the other hand, spin contami-nation is not a sufficiently strong effect to compensate for theabsence of screening of the Cooper channel.

It remains to decide which of the two approaches is themore correct. This, in the end, can only be addressed bycomparison with exact calculations for quantum dots (e.g.,quantum Monte Carlo). One argument that may be consid-ered is that in the presence of a time-reversal breaking term(i.e., T= 1), general symmetry considerations impose thatthe mean value of the residual interaction term is a functionof SS + 1, but not independently of S2 and S. Expressions(62) clearly do not fulfill this constraint, while Eq. (63) does.Since, however, spin contamination seems to compensate forthis difference it might just be that for time-reversal nonsym-metric systems, SDFT and RPA basically agree. On the other

hand, the screening of the Cooper channel does not seem tobe reproduced by the SDFT calculations, and this might bethe cause of the higher spin found in this approach.

ACKNOWLEDGMENTS

We thank G. Usaj for several valuable conversations andM. Brack for helpful comments regarding corrections. Thiswork was supported in part by the NSF (Grant No. DMR-0103003).

TABLE I. Ground-state spin probability for various model introduced in this paper. The two first lines

correspond, respectively, to the SDFT calculation and Strutinsky approximation [using Eq. (25)] for the

quartic oscillator systems introduced in Sec. IV. The statistics are built from the ground-state spin of dots

containing 100 to 200 electrons, for a few confining potential corresponding to an interaction parameter rs 1.3. The three last lines are the results of RMT/RPW modeling for 150 electron and the same value of theinteraction parameter rs, and, respectively, the interaction derived from SDFT (third line), the RPA-like

interaction using a unscreened Cooper channel, i.e., such that Eq. (64) applies but =JS (fourth line) and the

RPA-like interaction using a screened Cooper channel, i.e., such that Eqs. (64) and (65) apply (fifth line).

model S = 0 S = 12 S = 1 S = 32 S = 2

SDFT 0.42 0.03 0 .76 0.03 0 .54 0.03 0 .23 0.03 0 .03 0.01

ST* 0.34 0.03 0.74 0.03 0.61 0.03 0.25 0.03 0.04 0.01

ST* / RPW 0.28 0.68 0.62 0.30 0.09

RPA/RPW (unscreened Cooper) 0.48 0.84 0.49 0.16 0.03

RPA/RPW (screened Cooper ) 0.58 0.89 0.40 0.11 0.02


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1 L. P. Kouwenhoven, C. M. Marcus, P. L. McEuen, S. Tarucha, R.

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S. M. Reimann and M. Manninen, Rev. Mod. Phys. 74, 1283(2002), and references therein.

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026806 (2003).12 H. Jiang, H. U. Baranger, and W. Yang, Phys. Rev. B 68, 165337

(2003).13 H. Jiang, D. Ullmo, W. Yang, and H. U. Baranger, Phys. Rev. B

69, 235326 (2004).14 Y. M. Blanter, A. D. Mirlin, and B. A. Muzykantskii, Phys. Rev.

Lett. 78, 2449 (1997).15 D. Ullmo and H. U. Baranger, Phys. Rev. B 64, 245324 (2001).16 G. Usaj and H. U. Baranger, Phys. Rev. B 63, 184418 (2001);

66, 155333 (2002).

17 I. L. Aleiner, P. W. Brouwer, and L. I. Glazman Phys. Rep. 358,309 (2002).

18 D. Ullmo, T. Nagano, S. Tomsovic, and H. U. Baranger, Phys.

Rev. B 63, 125339 (2001).19 V. M. Strutinsky, Nucl. Phys. A122, 1 (1968).20 M. Brack, J. Damgaard, A. S. Jensen, H. C. Pauli, V. M. Strutin-

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Wesley, Reading, MA, 1997).22 C. Yannouleas and U. Landman, Phys. Rev. B 48, 8376 (1993).23 J. Harris, Phys. Rev. B 31, 1770 (1985).24 This second order Strutinsky approximation has to be distin-

guished from second order approximations made on the density

functional itself, such as in M. Elstner, D. Porezag, G. Jung-

nickel, J. Elsner, M. Haugk, T. Frauenheim, S. Suhai, and G.

Seifert, Phys. Rev. B 58, 7260 (1998).25 B. Tanatar and D. M. Ceperley, Phys. Rev. B 39, 5005 (1989).26 C. Attaccalite, S. Moroni, P. Gori-Giorgi, and G. B. Bachelet,

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and R. M. Nieminen, Phys. Rev. B 67, 205327 (2003).28 For the higher density case [a1], it was actually necessary to place

the gate quite close to the electron gas to see any effect on the

density, even on the classical scale. The image charge associated

with the gate is at a distance 2z0 which is, however, still larger

than the screening length within the electron gas.29 H. Jiang and W. Yang, J. Chem. Phys. 121, 2030 (2004).30 Note, however, that the precise choice of does not affect in any

significant way the qualitative feature of the Strutinsky approxi-

mation. We have checked that using different value for this pa-

rameter might slightly degrade the accuracy of the approxima-

tion, but that the quantities we are most interested in (peak

spacing fluctuations, spin distributions) are essentially un-

changed.31 J. Wang, A. D. Becke, and V. H. Smith, Jr., J. Chem. Phys. 102,

3477 (1995).32 Actually c

barefc would be divergent if we did not use a cutoff on

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JETP 57, 889 (1983).

37 D. Ullmo, H. U. Baranger, K. Richter, F. von Oppen, and R. A.Jalabert, Phys. Rev. Lett. 80, 895 (1998).

38 For this simulation, the fluctuation of the residual interaction term

also depends on the precise way the Cooper channel contribu-

tion is screened. Figure 12 corresponds to an evaluation of the

variance of these fluctuations for which we have assumed no

screening, and is therefore rather an upper bound. The fluctua-

tions being in any case rather small, their magnitude does not

change the peak spacing distribution drastically (their main ef-

fect is to broaden the sharp peak for odd spacings), and a more

detailed treatment of the effect of screening of the Cooper chan-

nel on the fluctuations of the residual interaction term should not

change significantly the picture.


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denis ullmo et al- landau fermi-liquid picture of spin density functional theory: strutinsky...

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