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Surface distortion effects on quantum dot helium Mario Encinosa, Babak Etemadi* Department of Physics, Florida A&M University, Tallahassee, FL 32707-4800, USA Received 4 September 1998; accepted 12 November 1998

Abstract The Schro¨dinger equation for a quantum mechanical particle constrained to a surface includes a potential term dependent on surface curvature. We use differential forms to derive this term and employ Monge representations for two surfaces to obtain specific expressions for the potential. We calculate the first order perturbative effect of this potential on the ground state energy of model quantum dot helium. We find that the energy shift can be sensitive to the detailed shape of the surface distortion. This dependence arises from the Coulomb repulsion between the electron pair, which causes each electron to preferentially sample (or not sample) regions where physical curvature leads to comparatively large values of the distortion potential. 1999 Elsevier Science B.V. All rights reserved. PACS: 3.65.!w; 71.10.!w; 73.20. Dx Keywords: Quantum dots; Surface distortion; Helium

1. Introduction Current research in the field of quantum nanostructures has compelled physicists to examine tractable models of few fermion, reduced dimensionality systems [1—3]. There now exists a considerable body of literature in this subject and many open problems remain. Devices are now fabricated on scales where the geometry of the object can induce quantum effects. Here we investigate one of these possible effects. It was first shown by daCosta [4,5] that within certain limits a quantum mechanical particle constrained to a surface experiences a potential depen-

* Corresponding author.

dent on the mean and Gaussian curvatures of the surface [6]. This result has also been derived with field theoretic and path integral methods [7—9]. In a previous paper [10] differential forms were used to reproduce daCosta’s results. There we showed with toy models that for reasonable surface distortions the potential could yield nonnegligible shifts to the ground state energy. In this paper we extend those calculations to a two electron model usually referred to in the literature as quantum dot helium (QDHe) [1]. Clearly the Coulomb potential is the major new feature of the two electron model. The single electron dot, in any sensible model that confines the electron will tend to have the electron wavefunction concentrated near the center of the dot. In the case of QDHe the Coulomb repulsion term in the

0921-4526/99/$ — see front matter 1999 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 9 9 ) 0 0 0 4 2 - 3

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two-body relative coordinate r produces different behavior. This paper is organized as follows. In Section 2 we write the model Hamiltonian of QDHe and outline the procedure for its solution. We also give the unperturbed ground state energy and wavefunction used in this work. Section 3 gives a more detailed exposition than in Ref. [10] of how the distorton potential » is derived with differential forms. In Section 4 the Monge forms for the surfaces and the potentials derived from them are shown. Section 5 gives results for the ground state energy shifts as a function of surface shapes and parameters. Section 6 is reserved for conclusions and suggestions for futher investigation.

2. Model quantum dot helium The energy shift calculations in Section 5 are perturbative so a model for an undistorted (flat surface) system is needed. The model Hamiltonian is taken as D 1 D H"! ! # mH Xr 2mH 2mH 2 1 e # mH Xr# 2 e"r !r "

(1)

r #r , r"r !r , R" 2

p 1 H" # kHXr, 2kH 2 e »" . er

(3)

The spectrum and eigenfunctions of H and 0 H are identical in form (we reserve upper case letters for CM quantities and lower case for relative coordinate quantities) H "NM2" X(2N#M#1)"NM2 and 0 H "nm2" X(2n#m#1)"nm2. The basis set expansion is

(R) c(m; n ,n) (r) . LK ,+ L The coordinate space eigenfunctions are products of exponentials and Laguerre polynomials. They may be found in standard texts [11] or in Ref. [1]. The matrix elements of (3) can be evaluated in closed form (r, R"NM; n m2"

e nm er

"(2

e j e

n! (n#m)!

n! (n#m)!

LY L n#m n#m ; n!l n!l JY J (!1)J>JY C(m#l#l#) ; l!l! (p

with j " /(4XmH. This expression allows the Hamiltonian to be solved by matrix methods. For the parameters quoted above diagonalization of a 6;6 matrix gives E "6.667 meV for the ground state energy and a truncated ground state relative coordinate wavefunction

p #p , P" 2

mH kH" , MH"2mH. 2 These transformations in Eq. (1) give H"H #H #» , 0

1 P # MHXR, H " 0 2MH 2

nm

a form standard in the literature [3]. In this paper we use GaAs parameters mH"0.067m , e"12.4 and X"1.6 meV. This Hamiltonian is solved by diagonalization. The relative and center of mass quantities are defined as

p"p !p ,

where

(2)

"0 0 2"C "0 02#C "1 02.

(4)

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363

With C "0.84 and C "!0.54 this wavefunc tion contains 0.997 of the full six-state relative wavefunction normalization; we consider this to be an adequate approximation for our purposes. The full state is a product of Eq. (4) with the center of mass ground state wavefunction.

3. Derivation of Vd with differential forms In what follows we borrow heavily from Flanders [6]; however several other excellent sources on differential forms exist [12,13]. Given a surface described by x with Monge form [14] x(o, )"o cos i#o sin j#S(o)k

(5)

application of the exterior derivative operator d to Eq. (5) yields dx"p e #p e with

(6)

do 1 p " , p "o d , Z" , Z (1#S M e "(cos , sin , S ) Z; e "(!sin , cos , 0), M (7) S "jS/jo . dx is a vector that at any point on the M surface R lies in the tangent plane of the surface at that point. p and p are the one-forms on this surface. Near R (Fig. 1) r"x#q e . Application of d to r results in dr"dx#dq e #q de , where

dr"dq

jr "p dq G jq G G G

it follows that p "a (q) dq . Once the one forms G G G p are known the a are known. For q "o, q "

G G and q , (1#q k ) , a "o(1#q k ), a "1 a " (9) Z with k "!S Z/o, k "!S Z. M MM The Laplacian is identified with d(*d) f"* fp p p . For three dimensional spatial forms

(8)

e "(!S cos ,!S sin , 1)Z. M M We will eventually take p "dq . In general (with summation convention assumed) 1 jr e" G a (q) jq G G and since

Fig. 1. Generic surface described by Monge vector x and local normal e3.

*pG"pHpI with i, j, k cyclic. Hence * fp p p "a a a * f dq dq dq so 1 j aa j H I * f" a a a jq a jq G H I G G G with the a given in Eq. (9). The Schro¨dinger equaG tion near the surface for a particle of mass m is then

jW ! *W#»(q)W"i

. 2m jt

(10)

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We consider the situation where a constraining force everywhere normal to the surface becomes large enough to restrict the particle to the surface. In this q P0 limit we anticipate that the full con figuration space wavefunction will decouple into surface and normal parts, or as discussed in Ref. [15] fast and slow variables

4. Monge parameterization

W(o, , q )Ps (o, )s (q ). R L A consistent relation in obtained by insisting the norms satisfy

S(o)"» cos p

"W"FdS dq ""s " dS"s " dq R L with

(11)

F"1#2q H#qK, K"k k , H" (k #k ) as the surface is approached. Insertion of Eq. (11) into Eq. (10) gives in obvious notation

s

js ! * "i . 2m (F (F jt

Our model surface Monge forms are characterized by

S(o)"» exp !

o!R , a

o!R a

.

(17) (18)

S"S(o) restricts the surface to having azimuthal symmetry. As in Ref. [10] this restriction is primarily for computational simplicity. These choices for S(o) should not be taken as literal models of surface distortion. However, they are motivated by past and recent developments in the fabrication of nanostructures [16,17].

(12)

After some algebra the Hamiltonian separates into surface and normal parts

js ! * s ! (H!K)s "i R , R R R 2m jt 2m

j js ! s #» (q )s "i L . L L 2m jq L jt The term

» "! (H!K) 2m

(13) (14) Fig. 2. The distortion potential » for the Gaussian surface. » and a are set to 2.0 and 20.0 As , respectively.

(15)

in Eq. (13) is the distortion potential. The extension to the two-particle problem is simple. We have F "(1#2q H#qK)(1#2q H#qK) so that » becomes the sum of two one body terms, » (r , r )"» (r )#» (r ). (16) We note in passing that should the particle be allowed to move in the q and q directions » will contain terms that couple these coordinates.

Fig. 3. The distortion potential » for the cosinusoidal surface form. » and a are set to 1.0 and 50.0 As , respectively.

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365

Table 1 Surface parameter

» (As )

a (As )

Gaussian surface Cosinusoidal surface

2.0 1.0

20.0 50.0

A direct application of the results in the last section yields potentials plotted in Figs. 2 and 3 for values of » and a given in Table 1. These poten tials appeared in Ref. [10] and are plotted again here for completeness. Fig. 4. The ground state energy shift *E for the Gaussian surface as a function of a for three values of R.

5. Results for the ground state energy shifts In Ref. [10] we showed that the contribution to the energy shift from the modification of the Laplacian on a curved surface was small compared to the potential energy contribution. We take the energy shift then as *E"1W "» (r , r )"W 2 (19) with » is given by Eqs. (15) and (16) and W given by the product of (4) with the center of mass ground state wavefunction. This reduces to *E"["C "# "C "#C C ] t !()C C t # "C "t , where

z o t " dz zI\» (z) exp ! , z" . I 2 j for k"1, 2, 3. (We drop the center of mass label on j below.) The lower limit of these integrals is zero for the cosinusoidal surface and 0.05j for the gaussian. This truncation to exclude the origin for the gaussian form is an artifact of the model (there is a cusp at the origin for R not zero) and can be circumvented by numerically smoothing the gaussian near the origin. We expect for both surfaces that *E is a monotonically decreasing function of a. We show this to be the case in Figs. 4 and 5 (note all energy shifts are negative). For smaller a, meaning greater curvature,

Fig. 5. The ground state energy shift *E for the cosinusoidal surface as a function of a for R"320 As . Plots for values of R indicated in Fig. 4 would not differ from this curve on these axes.

Fig. 4 clearly demonstrates the dependence of the energy shift on the position of the peak gaussian distortion. In Fig. 5 we plot only the R"0 result; this curve is insenstive to the value of R for reasons we will discuss below. In Fig. 6 we show *E for the gaussian surface as a function of the shift parameter R. This figure clearly illustrates the interplay of the curvature with the Coulombic interaction and the one body nature of » . For each electron 1o2" 2.4 j; when R is moved to a position on the dot near this value (320 As here) each electron samples the region where » is comparatively large. In Fig. 7 *E is given as a function of R for the cosinusoidal

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Fig. 6. *E for the Gaussian surface as a function of R for three values of the surface parameter a.

the curvature of the surface. The mutual repulsion of the electrons from the Coulomb potential plays an important role in the magnitude of the shift. A suitable displacement of the region of curvature can resonate with the electron wavefunction to produce either near zero or maximum effect. Clearly our results were obtained in the limit of ideal normal confinement of a particle to the surface of a device. How well this limit approximates a physical situation likely depends on the detailed properties of the structure. Correcting to include finite excursions along q is currently a direction of investigation. The experimental verification of this effect is an open question. Consider two nanostructures of equal dimensions. If one is slightly distorted with respect to the other, the normal confining potential » (q ) should produce approximately equivalent effect on both systems. What remains is the surface part of the Hamiltonian; this in principle allows for isolating the effect.

Acknowledgements

Fig. 7. *E for the cosinusiodal surface as a function of R for two values of the surface parameter a.

Mario Encinosa would like to acknowledge the Center for Nonlinear and Nonequilibrium Aeroscience (CeNNAs) for partial financial and computing support.

References surface. Here the effect is not pronounced. The energy shift for this shape is composed of a contribution from each oscillation; shifting the distortions on the surface still roughly maintains the same number of potential centers the wavefunction samples.

6. Conclusions In this paper we have extended our previous work on distortion effects to model quantum dot helium. We demonstrated that for surfaces with mild distortions there exist nonnegligible shifts to the ground state energy of the system arising from

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