PHYSICAL REVIEW B 15 FEBRUARY 2000-IVOLUME 61, NUMBER 7

Semiclassical perturbation theory for two electrons in aD-dimensional quantum dot

B. A. McKinney and D. K. WatsonDepartment of Physics and Astronomy, University of Oklahoma, Norman, Oklahoma 73019

~Received 17 February 1999; revised manuscript received 8 November 1999!

Dimensional perturbation theory is applied to the two-electronD-dimensional quantum dot, obtaining accu-rate values for the ground- and excited-state energies. The expansion parameter is 1/k, wherek5D12u l u, Dis the effective spatial dimensionality of the quantum dot environment, andl is the relative-motion angularmomentum quantum number. In this method, no approximations are made in the treatment of correlation.Analytic approximations for ground- and excited-state energies are obtained from the zeroth- and first-orderterms of the perturbation expansion; thus, constituting a semiclassical approach to the quantum dot from aperturbation formalism. Using this analytic form of the energy, parametrized byD, the effects of the effectivequantum dot dimensionality on the energy spectra may be investigated. Systematic corrections are made to thesemiclassical approximation by adding higher-order perturbation terms. The method described may be ex-tended to obtain analytic approximations to the ground-state energy of the many-electronD-dimensionalquantum dot Hamiltonian by truncating the 1/D expansion to low order.

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I. INTRODUCTION

In recent years, various theoretical approaches have bemployed to calculate the approximate energy spectraquantum dot systems, systems in which correlation andchange effects have been found to play a significant rolethe early development of quantum dot theory, the approacused were ‘‘exact’’ diagonalization1,2 and Hartree-Fock.1,3–5

More recent approaches include semiclassical6,7 and pertur-bation methods8,9 proposed with the goals of including corelation and maintaining accuracy for many-electron dwhile simultaneously reducing computational expense.

A variation of the dimensional perturbation method todescribed in this paper has been applied to two-dimensisemiconductor systems like magnetoexcitons10 and hydro-genic donor states in a magnetic field.11 This approach isknown as the shifted 1/N expansion whereN is the numberof spatial dimensions. In this method, the energy is expanin inverse powers ofN2a where the shift parametera isdetermined by requiring that the first-order energy correctvanish.

The effective dimensionality of the quantum dot enviroment depends on the relative size of the confinement lenscales in each of the three spatial dimensions. The effecdimensionality may be one, two, or three; or for some qutum dots, the experimental energy spectra may be bettescribed by the energy obtained via a model Hamiltonian ispace with some noninteger effective dimensionality tocount for anisotropy in the quantum dot interactions. Tconcept of fractional dimensionality has been appliedsemiconductor systems such as excitons in anisotropiconfined quantum well structures in order to account foreffective medium and the anisotropy of the interactions.12–14

An advantage of dimensional perturbation theory is thatobtain analytic energy approximations in terms of the paraeterD.

Dimensional perturbation theory15,16 has been successfully applied to atomic systems in which correlation is sinificant. In this paper, we present a perturbation approac

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the two-electron quantum dot with expansion parame1/D12u l u, where D is the effective dimensionality of thequantum dot environment andl is the angular momentumquantum number for the relative motion of the electrons. Tanisotropy of the quantum dot interactions is modeled bHamiltonian in an isotropicD-dimensional space whereD isa measure of the degree of anisotropy and may be integenoninteger. In theD52 andD53 limits of this generalizedHamiltonian, we regain the usual isotropic Hamiltoniansplane polar and spherical coordinates, respectively.

The zeroth- plus first-order terms of the energy series pvide an accurate analytic approximation to the energy. Inzeroth-order calculation, we take the infinite-D limit of theHamiltonian, which results in an effective potential. Thzeroth-order problem is then reduced to a simple minimition calculation of the effective potential that turns out tosimilar in form to WKB and other semiclassical approximtions of theD52, two-electron quantum dot.6,7 This similar-ity is not by fortuity because the infinite-D limit is itself aclassical limit.

When a higher degree of accuracy than that providedthe analytic approximation is desired, our method allowssystematic improvement by including higher-order terms;partial sums of the resulting energy series are calculateding straight summation. We compare our results with exnumerical results obtained by directly integrating the Sch¨-dinger equation using a Noumerov method forD52 andD53. In Sec. V, we briefly discuss extensions of this pertbation method including the extension to the many-electD-dimensional quantum dot.

II. FORMALISM

To illustrate the dimensional perturbation method, wconsider a two-electron dot within the effective-mass aproximation where the two electrons, each with effectimassm* , move in a medium with dielectric constante. Be-fore considering the more general quantum dot with effectdimensionality 0,D<3, we will work with the more re-

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PRB 61 4959SEMICLASSICAL PERTURBATION THEORY FOR TWO . . .

strictedD53 case with a spherical harmonic confining ptential with frequencyvo . In three dimensions, this systeis described by the Hamiltonian,

H52\2

2m* ~¹121¹2

2!11

2m* vo

2~r 121r 2

2!1e2

eur12r2u.

~1!

We define

a* 5e\2

m* e2, ~2!

l o5A \

m* vo, ~3!

wherea* is the effective Bohr radius andl o is the charac-teristic size of the dot. Lettingr i→a* r i , yields the Hamil-tonian:

H52¹122¹2

211

4g2~r 1

21r 22!1

2

ur12r2u, ~4!

where the energy is in units of effective rydbergsR* andg52(a* / l o)2. The parameterg describes the relative magnitude of the confinement energy and Coulombic enescales. Later we investigate quantum size effects on ourcuracy by varyingg21/2.

The Hamiltonian of Eq.~4! can be separated into centeof-mass and relative-motion pieces as

H5Hc.m.1H rel ~5!

with

Hc.m.521

2¹c.m.

2 11

2g2R2 ~6!

and

H rel522¹ rel2 1

1

8g2r 21

2

r, ~7!

where R5 12 (r11r2), ¹c.m.5¹11¹2 , r5r12r2, and ¹ rel

5 12 (¹12¹2). This essentially reduces the problem to tw

one-particle equations. In the rest of this section andfollowing, our main focus will be on calculating the relativemotion energyErel by applying a perturbation approachthe Hamiltonian of Eq.~7! in arbitrary dimension.

Next we explicitly generalize the Schro¨dinger equation toD dimensions where r becomes the radius ofD-dimensional sphere withD21 remaining angles. Usingthe procedure where the Laplacian is generalized toD di-mensions, but the potential terms retain their thrdimensional form, we obtain the following Schro¨dingerequation:

H 22F 1

r D21

]

]r S r D21]

]r D1LD21

2

r 2 G11

8g2r 21

2

r J C~r !

5ErelC~r !, ~8!

where LD212 is a generalized angular momentum opera

depending onD21 angles with eigenvalues2u l u(D1u l u

yc-

e

-

r

22).17 Substituting these eigenvalues and introducingradial Jacobian factor in a transformation of the wave fution, F(r )5r (D21)/2C(r ) to eliminate the first derivativeterms, we find

H 22F ]2

]r 21~D21!~D23!

4r 2 2u l u~D1u l u22!

r 2 G1

1

8g2r 21

2

r J F~r !5ErelF~r !. ~9!

We now define an expansion parameterk as

k5D12u l u, ~10!

and introducing the dimensionally-scaled variables

r 5k2r , g5k3g, e rel5k2Erel , ~11!

we obtain a dimensionally-scaled Schro¨dinger equation thathas a finite energy asD→`:

H 22d2S ]2

] r 2D 1124d13d2

2r 21

1

8g2r 21

2

rJ F~ r !

5e relF~ r !, ~12!

whered51/k is treated as a continuous parameter. HenceDmay be noninteger as well as integer by this dimensiocontinuation. One may find similar dimensional scalingsatomic systems in Refs. 18 and 19.

One may perform a similar dimensional scaling of tcenter-of-mass Hamiltonian, Eq.~6!. The eigenenergies cabe obtained exactly and are found to be

Ec.m.~N,L,D !5S 2N1uLu1D

2 Dg, ~13!

whereN and L are the radial and orbital quantum numbefor the center-of-mass motion andD is the effective dimen-sionality of the quantum dot. The total energy isE5Ec.m.1Erel.

III. PERTURBATION EXPANSION

To obtain the zeroth-order energy approximation for trelative-motion energy, we begin by taking the infinitdimension limitD→` ~i.e., d→0) of Eq.~12!. In this limit,the derivative terms of the kinetic energy drop out of tHamiltonian reducing the calculation of the zeroth-order eergy e` to finding the minimum of an effective potentiaVeff :

e`5Veff~ r m!. ~14!

Veff takes the form

Veff~ r !51

2r 21

1

8g2r 21

2

r, ~15!

andr m is the smallest positive root of the quartic polynomia

g2r m4 28r m2450. ~16!

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4960 PRB 61B. A. MCKINNEY AND D. K. WATSON

The zeroth-order energy is already in good agreemwith exact values forg51 and g50.05 for states withn50 ~see Tables I and II!. Our effective potential has thsame form as those found in other semiclassical approxitions of two-electron dots for D52. Klama andMishchenko6 propose a simple model within the WKB context called the harmonic approximation that results in an alytic approximation to the energy. They achieve this byplacing the WKB potential in the Schro¨dinger equation withits Taylor expansion about its minimum up to harmonicder. This results in a linear oscillator spectrum. Also withthe WKB context, Garcı´a-Castela´n, Choe, and Lee7 obtainanalytic results by approximating the true potential curvematching two half parabolas at the minimum of the WKeffective potential.

In contrast, we systematically improve upon our semiclsical, zeroth-order approximation by adding the next highorder term of the energy series. The perturbation seriegenerated by defining a scaled displacement coordinatex byr 5 r m1d1/2x and expanding the wave function and energy

F~x!5(j 50

`

f j~x!d j /2, ~17!

e rel5e`1d(j 50

`

e2 jdj . ~18!

The first-order equation ind is harmonic:

H 22]2

]x2 11

2veff

2 x21vooJ fo~x!5eofo~x!, ~19!

eo5S n11

2D2veff1voo , ~20!

TABLE I. Comparison of the results of our analytic zerotorder, first-order, and numerically calculated sixth-order appromations to the exact energies forg51 andD52. The energy stateis specified in the first column by the quantum numbers (n,l ;N,L).Total energy is in units of effective rydbergs.

(n,l ;N,L) Exact Zero order First order Sixth order

~0,0;0,0! 3.3242 2.8998 3.1575 3.3188~0,1;0,0! 3.8279 3.6784 3.7929 3.8279~0,0;0,1! 4.3242 3.8998 4.1575 4.3188~0,2;0,0! 4.6437 4.5641 4.6306 4.6436~0,1;0,1! 4.8279 4.6784 4.7929 4.8278~1,0;0,0! 5.1568 2.8998 5.0150 5.1456~0,0;1,0! 5.3242 4.8998 5.1575 5.3188~0,3;0,0! 5.5432 5.4924 5.5369 5.5432~0,2;0,1! 5.6436 5.5641 5.6306 5.6436~1,1;0,0! 5.7439 3.6784 5.7224 5.7439~0,1;1,0! 5.8279 5.6784 5.7929 5.8278~1,0;0,1! 6.1568 3.8998 6.0150 6.1456~0,0;1,1! 6.3242 5.8998 6.1575 6.3188~0,4;0,0! 6.4782 6.4423 6.4746 6.4782~1,2;0,0! 6.5957 4.5641 6.5882 6.5957~0,2;1,0! 6.6436 6.5641 6.6306 6.6436

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-

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-r-is

s

fo~x!5S veff

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hnF S veff

2 D xG , ~21!

where

voo522

r m2

, ~22!

veff2 5Veff9 ~ r m!5

3

r m4

11

4g21

4

r m3

, ~23!

and thehn are harmonic oscillator eigenfunctions. This firsorder correction represents normal-mode vibrations aboutclassical minimum. The analytic, semiclassical approximtion of the energy to first order can then be compactly writas

Erel~n,l ,D !'d2e`1d3F S n11

2D2veff1vooG , ~24!

where n is the harmonic quantum number for the relatimotion, and we have used the relations in Eq.~11! to convertthe energy units to unscaled effective rydbergs.Erel is param-etrized explicitly byn and implicitly by l and D. Equation~24! may then be evaluated at the physically relevant spadimension that best describes a particular quantumnamely, 0,D<3.

The infinite set of differential equations for thef j (x) andthe e2 j are computed using a linear algebraic method texpands thef j (x) in terms of the harmonic-oscillator functions hn and represents the displacement coordinatex as amatrix in the harmonic-oscillator basis. A recursion relatiyields the wave function and energy coefficients.20

TABLE II. Comparison of the results of our analytic zerothorder, first-order, and numerically calculated sixth-order appromations to the exact energies forg50.05 andD52. The energystate is specified in the first column by the quantum numb(n,l ;N,L). Total energy is in units of effective rydbergs.

(n,l ;N,L) Exact Zero order First order Sixth order

~0,0;0,0! 0.2963 0.2624 0.2896 0.2963~0,1;0,0! 0.3062 0.2856 0.3028 0.3062~0,0;0,1! 0.3463 0.3124 0.3396 0.3463~0,2;0,0! 0.3311 0.3177 0.3293 0.3311~0,1;0,1! 0.3562 0.3356 0.3528 0.3562~1,0;0,0! 0.3853 0.2624 0.3778 0.3853~0,0;1,0! 0.3963 0.3624 0.3896 0.3963~0,3;0,0! 0.3644 0.3550 0.3633 0.3644~0,2;0,1! 0.3811 0.3677 0.3793 0.3811~1,1;0,0! 0.3968 0.2856 0.3939 0.3968~0,1;1,0! 0.4062 0.3856 0.4028 0.4062~1,0;0,1! 0.4353 0.3124 0.4278 0.4353~0,0;1,1! 0.4463 0.4124 0.4396 0.4463~0,4;0,0! 0.4025 0.3955 0.4018 0.4025~1,2;0,0! 0.4240 0.3177 0.4227 0.4240~0,2;1,0! 0.4311 0.4177 0.4293 0.4311

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PRB 61 4961SEMICLASSICAL PERTURBATION THEORY FOR TWO . . .

IV. RESULTS AND DISCUSSION

In the special case ofD52, we compare our analyticzeroth-order, first-order, and numerically calculated sixorder approximations for ground- and excited-state enerto exact values obtained by numerical integration ofSchrodinger equation. In Table I, we show this comparisfor g51, which is an intermediate regime where neitherconfinement energy nor the Coulombic energy scale donates the other. Table II makes the same comparison fog50.05, which is a regime where the repulsive interactenergy becomes more dominant and the dot becomes laFor all three approximations in each table, we add the excenter-of-mass energy, Eq.~13!, to the relative-motion en-ergy to obtain the total-energy approximation. Forg51, theanalytic results of dimensional perturbation theory to fiorder are in good agreement with exact calculations wrelative error less than 5% for states withu l u50 and less than1% for states withu l u.0. Forg50.05, the analytic results tofirst order agree with exact calculations with relative erless than 2.3% for states withu l u50 and less than 1.1% fostates withu l u.0. By sixth order, all energies are convergto at least seven digits forg50.05 while some of the energies are slightly less accurate forg51.

Effects of the harmonic quantum numbern, are notbrought into the calculation until first order by Eq.~20!.Tables I and II illustrate this fact by the marked improvment of the first-order over the zeroth-order approximatfor states withn.0. For states withn50, the zeroth-orderresults are already in good agreement with exact valumaking higher-order terms only minor corrections. Tableand II also show that the accuracy of the zeroth-order pturbation starting point improves as the magnitude ofrelative-motion angular momentum quantum numberu l u in-creases. This is understood by noting that increasingu l u de-creases the size of the perturbation parameterd. For the samereason, we expect the accuracy to improve as the effecdimensionality increases.

In order to demonstrate the flexibility of this method wirespect to the quantum dot dimensionality, we compelectron-electron interaction energiesEe-e(n,l ,D) for D52andD53. The eigenenergies for the relative motion withothe electron-electron interaction may be obtained exactly

Eo~n,l ,D !5S 2n1u l u1D

2 Dg. ~25!

The electron-electron interaction energies may then betained simply by subtracting out this relative-motion enerwithout the electron-electron interaction from the furelative-motion energyErel(n,l ,D):

Ee-e~n,l ,D !5Erel~n,l ,D !2Eo~n,l ,D !. ~26!

In Fig. 1, we show the behavior for the 1s, 2p, and 3dstates in two and three dimensions asg increases. We usenotation conventions from atomic physics where the prinpal quantum numbers are given by (n1u l u11) and the azi-muthal quantum numbers,u l u50,1,2,3, . . . , aredesignatedby s,p,d, f , . . . . Figure 1 is in excellent agreement with thenergy ordering predicted by other calculations done se

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rately for D52 and D53, ~Ref. 21! and shows that theelectron-electron interaction energy is proportional tog1/2.

V. CONCLUSIONS

We have obtained an analytic approximation, paraetrized byD, for the ground- and excited-state energies otwo-electron quantum dot with an effective dimensionalD. D is a measure of the degree of anisotropy in the quandot interactions or confinement. Our zeroth-order approximtion is the infinite-D ~or infinite-u l u) limit of thedimensionally-scaled Schro¨dinger equation, Eq.~12!, whichresults in an effective potential reminiscent of semiclassWKB approaches.6,7 This is to be expected because the prameter,k, acts as an effective mass so that the electrbecome infinitely heavy asD or u l u approaches infinity. Thisin turn causes the derivative terms of the kinetic energybecome zero and the infinitely heavy electrons to remstationary at the bottom of the effective potential. The effetive potential evaluated at its minimum provides a gozeroth-order analytic starting point for the perturbation epansion because the minimumr m is sensitive to the interplaybetween the confining energy and the repulsive electronteraction energy viag. We then systematically improve upothis classical approximation by adding the semiclassical fiorder harmonic correction, which is analogous to vibratioof a Wigner molecule.

Dimensional perturbation theory~DPT! is applicable tothe entire range ofg, unlike perturbation methods that usthe confinement strength or the Coulomb strength as theturbation parameter or methods that use a basis set thatadvantage of a high or low limit ofg. For all values ofg, theanalytic semiclassical results of DPT perform extremely wfor u l u.0 excitations because in our scaling the kinetic eergy scales as the inverse square of the angular momenFor u l u50 states, analytic semiclassical results are extremaccurate for typical dots in the mesoscopic regime (g,1)because the kinetic energy scales linearly withg.22 Not sur-prisingly, the domain of validity ofg for different values ofl is the same as that of the WKB approximation, which c

FIG. 1. Comparison ofEe-e(n,l ,D) vs g for D52 andD53 forstates 1s, 2p, and 3d. The solid lines indicateD52 and dashedlines indicateD53.

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4962 PRB 61B. A. MCKINNEY AND D. K. WATSON

be determined in terms of either the de Broglie wavelenor the classical momentum.7 The domain of validity of thesemiclassical perturbation expansion forl 50 may be ex-tended to very largeg by applying sophisticated summatiomethods such as Pade´ approximants to the asymptotic seriehowever, even for quantum dot systems where the kineand potential-energy scales are comparable (g'1) the ana-lytic semiclassical results continue to work surprisingly wsuggesting that DPT is a particularly robust method for treing quantum dot systems. These results also suggestDPT would be well suited for application to a quantum dotan external magnetic field. We are currently pursuing textension of DPT.

Another advantage of this semiclassical method isready extension to systems with many degrees of freedallowing one, for example, to obtain an analytic approximtion to the ground and excited states of theN-electron quan-tum dot of effective dimensionalityD. One of the ways weare approaching this problem is similar to the largdimension limit described above for theN52 electron dot.We outline the procedure as follows. We begin by writiout the full N-electron quantum dot Hamiltonian igeneralized-D form and dimensionally scale it as we hadone for the two-electron dot. In the classical infinite-Dlimit, the derivative terms of the kinetic energy become zeand we assume a symmetric geometry for the electrons ineffective potential minimum. We would then be able to otain an analytic zeroth-order approximation of this newfective potential by assuming the electrons are equivalThis classical result could then be improved upon by appeing the semiclassical harmonic correction term that repsents normal-mode vibrations about the symmetric equlent geometry. The normal modes may be obtainedapplying the FG matrix method to the N-electronHamiltonian.23

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A semiclassical technique that uses correlated ‘‘pocstate’’ basis functions and the WKB approximation has beused to study the low-lying excitation spectrum for quantudots with two or more electrons.24 Although the method isdifferent, many of the ideas are similar to the arbitraryNperturbation method described above. The total potentialdouble well, and each basis function of the finitdimensional ‘‘pocket state’’ basis is strongly localized atpotential minimum becoming smaller further from the minmum. The WKB method is used to compare the magnituof off-diagonal Hamiltonian matrix elements, which descricorrelated tunneling between different arrangements of thNelectrons. For sufficiently low-electron densities~or smallgin the language of this paper!, only one tunneling integradominates exponentially. As the quantum dot electronscome more Wigner moleculelike, this method becomes maccurate.

The large-D limit is analogous to the prequantum valenmodels of Lewis and Langmuir. In the large-D limit of theN-electron quantum dot, the electrons become frozen iLewis structure. The first-order harmonic oscillations aLangmuir vibrations about the Lewis structure, or, in codensed matter terminology, vibrations about the Wignstructure. This semiclassical perturbation method wouldamenable to the calculation of correlation energies, andprospects for a treatment of the many-electron quantumare promising based on successful applications to maelectron atomic systems.25

ACKNOWLEDGMENTS

B. A. M. would like to thank J. C. Carzoli for manyhelpful discussions on scaling methods. This work was sported by the Office of Naval Research Grant NN00014-96-1-1029.

ce

m

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