Electron wave functions on T2 in a static magnetic field of arbitrary direction

ARTICLE IN PRESS

Physica E 28 (2005) 209–218 www.elsevier.com/locate/physe

Electron wave functions on T 2 in a static magnetic field of arbitrary direction Mario Encinosa Florida A&M University, Department of Physics, 205 Jones Hall, Tallahassee, FL 32307, USA Received 10 February 2005; accepted 9 March 2005 Available online 31 May 2005

Abstract A basis set expansion is performed to find the eigenvalues and wave functions for an electron on a toroidal surface T 2 subject to a constant magnetic field in an arbitrary direction. The evolution of several low-lying states as a function of field strength and field orientation is reported, and a procedure to extend the results to include two-body Coulomb matrix elements on T 2 is presented. r 2005 Elsevier B.V. All rights reserved. PACS: 03.65.Ge; 73.21.b Keywords: Torus; Magnetic field; Wave functions

1. Introduction Quantum dots with novel geometries have spurred considerable experimental and theoretical interest because of their potential applications to nanoscience. Ring and toroidal structures in particular have been the focus of substantial effort because their topology makes it possible to explore Ahranov–Bohm and interesting transport phenomena [1–4]. Toroidal InGaAs devices have been fabricated [5–8] and modelled, [9] and toroidal carbon nanotube structures studied by several groups [4,10,11]. E-mail address: [email protected]fl.gov.

This work is concerned with the evolution of one-electron wave functions on T 2 in response to a static magnetic field in an arbitrary direction. The problem of toroidal states in a magnetic field has been studied with various levels of mathematical sophistication. Onofri [12] has employed the holomorphic gauge to study Landau levels on a torus defined by a strip with appropriate boundary conditions and Narnhofer has analyzed the same in the context of Weyl algebras [13]. Here, the aim is to do the problem with standard methodology: develop a Schro¨dinger equation inclusive of surface curvature, evaluate the vector potential on that surface, and proceed to diagonalize the resulting Hamiltonian matrix.

1386-9477/$ - see front matter r 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physe.2005.03.009

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As noted in Ref. [14], ideally one would like to solve the N-electron case, but the single particle problem is generally an important first step, and while the N electron system on flat and spherical surfaces has been studied [15–20], the torus presents its own difficulties. In an effort to partially address this issue, the evaluation of Coulombic matrix elements on T 2 is also discussed here. This paper is organized as follows: in Section 2, the Schro¨dinger equation for an electron on a toroidal surface in the presence of a static magnetic field is derived. In Section 3, a brief exposition on the basis set employed to generate observables is presented. Section 4 gives results. Section 5 develops the scheme by which this work can be extended to the two-electron problem on T 2 , and Section 6 is reserved for conclusions.

The Schro¨dinger equation with the minimal prescription for inclusion of a vector potential A is  2 1 _ r þ qA C ¼ EC. (9) H¼ 2m i The magnetic field under consideration will take the form B ¼ B1 i þ B0 k,

(10)

which by symmetry comprises the general case. In the Coulomb gauge, the vector potential Aðy; fÞ ¼ 1 2 B r expressed in surface variables reduces to Aðy; fÞ ¼ 12½B1 ðW sin f cos y þ a sin2 y sin fÞh þ ðB0 W  B1 a sin y cos fÞ/ þ B1 ðF sin f sin y  a cos y sin y sin fÞn ð11Þ

W ¼ R þ a cos y,

(2)

with n ¼ / h. The normal component of A contributes a quadratic term to the Hamiltonian but leads to no differentiations in the coordinate normal to the surface as per Eq. (8). There is a wealth of literature concerning curvature effects when a particle is constrained to a two-dimensional surface in three-space [21–38], including some dealing with the torus specifically [39], but the scope of this work will remain restricted to study of the Hamiltonian given by Eq. (9). The Schro¨dinger equation (spin splitting will be neglected throughout this work) is more simply expressed by first defining

q ¼ cos fi þ sin fj.

(3)

a ¼ a=R,

2. Formalism The geometry of a toroidal surface of major radius R and minor radius a may be parameterized by rðy; fÞ ¼ W ðyÞq þ a sin yk

(1)

with

The differential of Eq. (1) dr ¼ a dyh þ W df/

F ¼ 1 þ a cos y, (4)

with h ¼  sin yq þ cos yk yields for the metric elements gij on T 2 gyy ¼ a2 , gff ¼ W 2 .

gN ¼

p_ , q

t0 ¼

g0 , gN

t1 ¼

g1 , gN

(6)

and 1 q 1 q þ/ . a qy W qf

g1 ¼ B1 pR2 ,

(5)

The integration measure and surface gradient that follow from Eqs. (5) and (6) become pffiffiffi 1 2 (7) g dq dq ! aW dy df

r¼h

g0 ¼ B0 pR2 ,

(8)

e¼

2mEa2 , _2

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after which Eq. (9) may be written  2 q a sin y q a2 q2 þ  F qy F 2 q2 f q2 y   t 1 a3 q sin y cos f þ i t 0 a2  qf F q þ iat1 sin fða þ cos yÞ qy   t20 a2 F 2 t21 a2 F 2 a2 sin2 y 2  sin f þ  4 4 F2  3 t0 t1 a F sin y cos f C ¼ eC, þ 2 ) H t C ¼ eC.

starting from N 0 ¼ 12 for positive parity states and N 1 ¼ 1 for negative parity states. The Knth basis state is attained by appending azimuthal eigenfunctions onto the GS functions described above, 1 X C cKm Kn ðy; fÞ ¼ pffiffiffiffiffiffi 2p m ! um ðyÞ ð18Þ einf . vm ðyÞ ð12Þ (13)

To proceed with a basis set expansion, Gram–Schmidt (GS) functions orthogonal over the integration measure F ¼ 1 þ a cos y must be generated. Fortunately, it is possible to construct such functions almost trivially. The method for doing so has been described elsewhere [40], so only the salient results will be presented below. The t1 ¼ 0; y ! y invariance of H t suggests that the solutions of the Schro¨dinger equation be split into even and odd functions, and the primitive basis set can be taken to possess this property: 1 vn ðyÞ ¼ pffiffiffi sin½ny. p

The GS functions will take the form ! X um ðyÞ

cKm cK ðyÞ ¼ vm ðyÞ m

(14)

(15)

with the cKm given by (momentarily supressing the parity superscripts [41]) cKm ¼ ðÞKþm N K ðN K1 bN K1 Þ ðN K2 bN K2 Þ    ðN m bN m Þ

ð16Þ

and the normalization factors N K determined from N 2kþ1 ¼

1 1  b2 N 2k

The matrix ¯ ¯ jH t jK ni H pq ¯ n¯ n ¼ hK n tKK

3. Calculational scheme

1 un ðyÞ ¼ pffiffiffi cos½ny; p

211

qffiffi

(17)

(19)

is then easily constructed since the matrix elements can all be written in closed form (see the Appendix), and the eigenvalues and eigenvectors determined here with a 30-state expansion for each y-parity. The ordering convention adopted for the states was taken as þ þ   Cþ 0;2 ; C0;1 ; . . . ; C5;2 ; C1;2 ; . . . ; C6;2

yielding a Hamiltonian matrix blocked schematically into ! H þþ H þ . H þ H 

4. Results Rather than present a large number of tables conveying little useful information per unit page length, the focus will be on indicating how some low-lying states evolve as a function of magnetic field strength for two distinct orientations. Some remarks will also be made regarding the general trend seen for higher excited states. Here the ratio a ¼ a=R was set to 12 as a compromise between smaller a where the states tend toward decoupled ring functions and larger a which are less likely to be physically realistic. Fig. 1 illustrates the evolution of the energy eigenvalue for five low-lying states as a function of t0 with t1 ¼ 0. The states are all distinct and are labelled in the caption. Not shown are values trivially obtained from the nB0 splitting arising

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212

0.8

3 2.5

0.6

2 0.4





1.5 1

0.2

0.5 1

2

-0.5


0

3

4

0.6 0.5

*

0.4 0.3 0.2 0.1 2

3 

2

-0.2

Fig. 1. e as a function of t0 for five low-lying states. Diamonds correspond to the jnK i ¼ j00þ i state, stars to j  10þ i, squares to j  20þ i, triangles to j01 i and circles to j01þ i.

1

1

5

4

5

6

Fig. 2. Evolution of C CF for the j00þ i state given for t0 ¼ 0 (thin line), t0 ¼ 2:5 (medium line) and t0 ¼ 5:0 (thickest line).

from B0 ¼ 0; na0 degeneracy. It is interesting that level crossings with attendant movement toward a ground state with different Kn occurs near integer values of t0 , though it is not immediately clear if this is of real significance. It is also of interest to show the sensitivity of the dependence of C CF on field strength. Fig. 2 shows that even for moderate field values (t0 ¼ 5 corresponds to a field of 1.3 T for a torus with R ¼ 50 nm) the large effective flux as compared to atomic or molecular dimensions causes substantial modification to C CF in the ground state. The results given in Figs. 1 and 2 were for a field configuration that did not mix azimuthal basis states. To investigate an asymmetric case, let t0 ¼ 0 and vary t1 wherein no field threads the torus.

3

4

5


0

Fig. 3. e as a function of t1 for five low-lying states. Diamonds correspond to the jnK i ¼ j00þ i state, stars/squares to j  10þ i, and triangles/circles to j  20þ i.

Fig. 3 is analogous to Fig. 1 as described above with the notable exception that the n splitting is non-trivial (hence fewer distinct states are shown), and level crossings occur at larger values. There is no plot analogous to Fig. 2 for larger values of t1 with t0 ¼ 0 because f dependence increases rapidly in the eigenstates even for small t1 ; a contour plot is preferable. Figs. 4 and 5 show contour plot results for two states at three field strengths. Note that there is slightly more dependence in y when t1 ¼ 0 for the state displayed in Fig. 6 than in Fig. 5; the integration measure acts to cancel the angular variation of the state displayed in Fig. 5.

5. Extension to Coulomb integrals The two-electron problem on T 2 is complicated by the inability (at least by the author) to find a transformation that decouples the relative electron motion from their center of mass motion as is easily done on R2 [42]. The obvious transformations do not lead to any advantage over the method adopted by workers long used in atomic and molecular physics, which is to evaluate the two-body matrix elements (supressing spin indices and physical constants) Z Z Fi ðr1 ÞFj ðr2 ÞV ðr1 ; r2 Þð1  P12 Þ Fk ðr1 ÞFl ðr2 Þ d3 r1 d3 r2

ð20Þ

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F*

6 4 2

2

1 0.75 0.5 0.25 0 0

4



213

6 4 

2 1.5 1 0.5 0 0



2 6

0

2 

4

1 0.75 0.5 0.25 0 0

0

6

1

2

2 

4 6 0

0.75 0.5 0.25 0 0

6 4 

4

F*

F*

6

2

2 4



1 0.75 0.5 0.25 0 0

0

6 4 2

2 4

F*



1



F*

6

6 0 Fig. 4. Sequential evolution of the C CF ¼ F j00þ i state on T 2 for t0 ¼ 0, t0 ¼ 2:5 and t0 ¼ 5:0.

0.75 0.5 0.25 0 0

6 4 2

2 



F*

M. Encinosa / Physica E 28 (2005) 209–218

4 6

0

Fig. 5. Sequential evolution of the C CF ¼ F j  10þ i state on T 2 for t1 ¼ 0, t1 ¼ 2:5 and t1 ¼ 5:0. The ground state variation in y is partially cancelled by the integration measure.

with V ðr1 ; r2 Þ ¼ 4p

X

1 rlo 2L þ 1 rlþ1 4

L;M Y LM ðy1 ; f1 ÞY LM ðy2 ; f2 Þ.

ð21Þ

Eq. (20) can be adopted on T 2 subject to some peculiarities which are due to the restriction of r1 ; r2 to a surface. Eq. (20) on T 2 with the notation

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employed in Section 3 becomes Z 2p Z 2p ... CPn1 ðy1 ; f1 ÞCQn2 ðy2 ; f2 Þ 0

change term proceeds similarly; only the densities need modification.

0

V ðr1 ; r2 Þð1  P12 ÞCRn3 ðy1 ; f1 ÞCSn4 ðy2 ; f2 Þ F ðy1 ÞF ðy2 Þ dy1 dy2 df1 df2 .

6. Conclusions ð22Þ

Consider the direct term; in terms of a spherical coordinate system centered at the middle of the torus ðr sin ys cos f; r sin ys sin f; r cos ys Þ, r sin ys ¼ R þ a cos y,

(23)

r cos ys ¼ a sin y,

(24)

r¼

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R2 þ a2 þ 2aR cos y

(25)

and defining rIJ  cI ðyÞcJ ðyÞ

(26)

gives for Eq. (22) after some manipulation X ðL  MÞ! ðdMn1 þn3 dMþn2 n4 Þ ðL þ MÞ! LM Z 2p Z 2p rPQ ðy1 ÞrRS ðy2 Þ 0

0

PLM ðys1 ðy1 ÞÞPLM ðys2 ðy2 ÞÞF ðy1 ÞF ðy2 Þ

ðR2 þ a2 þ 2aR cos yo ÞL=2 ðR2 þ a2 þ 2aR cos y4 ÞðLþ1Þ=2

dy1 dy2 .

ð27Þ

The arguments of the PLM are evaluated with   R þ a cos yi ysi ¼ arctan . (28) a sin yi To evaluate the integral, care must be taken with the 4; o character of the radial factor in the integrand. One way to proceed is as follows: 1. Fix y1 ¼ 0. At this point r1 is at its maximum; integrate the integrand of Eq. (27) numerically over dy2 from ½0; 2p by some suitable method to attain a value labelled by, say, G 0 ðy1 Þ. 2. Set y1 ¼ d. Integrate dy2 with r4 ¼ r1 , ro ¼ r2 over the interval ½d; 2p  d, then set r4 ¼ r2 and ro ¼ r1 from ½2p  d; d. This is G d ðy1 Þ. 3. Repeat the second step until the entire interval around the toroidal cross section is covered. A table [G 0 ðy1 Þ; G d ðy1 Þ; G 2d ðy1 Þ; . . . results that can then be integrated numerically. The ex-

This work presents a method to calculate the spectrum and wave functions for an electron on T 2 in an arbitrary static magnetic field. Aside from the character of the solutions and numerical data, perhaps the main result of this paper has to do with the ease with which an arbitrarily large number of GS states can be trivially generated. Because every physical interaction on T 2 can eventually be expressed as a periodic function on T 2 , matrix elements for the interaction may then be evaluated in closed form; hence the only restriction to doing any problem on T 2 is matrix inversion. The procedure employed here to generate observables lends itself to easy incorporation of an arbitrary number of surface delta functions or other type of potential. This is important because on a closed nanotube, in contrast to a macroscopic crystal, there are a relatively small number charge carriers, so the continuum approximation may break down for smaller torii. Clearly, the magnetic field treated here will not be sufficient to comprise the general case as soon as any sort of lattice structure breaking azimuthal symmetry is imposed on the torus. However, the extension is simple to implement in Eq. (12), requiring only a few more terms. It would also be interesting to see the extension of the static case discussed here to a time-dependent laser control problem of the type in Ref. [43]. Some remarks should be made regarding the curvature potential V C well known to workers in the field of quantum mechanics on curved surfaces. It was shown in Ref. [44] that a full three-dimensional treatment of the problem of a particle near, but not necessarily restricted to T 2 , yields a spectra consistent with inclusion of V C added to the two-dimensional surface Hamiltonian. Here, the potential could not be included without substantially increasing the scope and complexity of the problem undertaken. It was shown in Ref. [45] that the inclusion of a vector

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potential precludes a separation of variables into surface and normal degrees of freedom; A added to the Schro¨dinger equation requires solving coupled differential equations in the surface and normal variables, or if a basis set expansion is employed, a much more complicated procedure to generate three-dimensional GS states.

h a ðDmþn  Dmn Þ þ ðDmþnþ1 þ Dmnþ1 2i ðA1Þ þ Dmþn1 þ Dmn1 Þ ,

ðDmþn1 þ Dmnþ1  Dmn1 Þ,



cKm ¯ cKn ðnÞDn¯ n

h a ðDmþn þ Dmn Þ þ ðDmþnþ1 þ Dmnþ1 2i ðA4Þ þ Dmþn1 þ Dmn1 Þ ,

pffiffiffiffiffiffiffiffiffiffiffiffiffi 1  a2  1 f ðaÞ ¼ a

2 2  þ t a K¯ n¯  0 F 2 K þ n 4 K K¯ X pt20 a2 X c ¯ cKn Dn¯ n 4 m¼0 n¼0 Km  1 P1 Dmn þ ðP2 ðDmþn1 þ Dmnþ1 þ Dmn1 Þ 2

¼

and define dJ;K  DJK .

m¼0 n¼0

K K¯ X X m¼0 n¼0

P3 ¼ 32a2 ,

Each operator in Eq. (9) will connect either only like parity states or opposite parity states; no single operator will do both. The matrix elements that connect like positive parties are 2

 þ q þ ¯ K n¯ 2 K n qy K K¯ X X 2 ¼p cKm ¯ cKn ðn ÞDn¯ n

ðA3Þ

 q þ K¯ n¯ it0 a2 K þ n qf ¼ pt0 a2

P2 ¼ 3a þ 34a3 ,

a3 , 4

2 2  þ a q þ ¯ K n¯ 2 2 K n F qf

½f nþm ðaÞ þ f jnmj ðaÞ,

P1 ¼ 1 þ 32a2 ,

P4 ¼

ðA2Þ

K¯ X K a2 p X 2 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffi cKm ¯ cKn ðn ÞDn¯ n 1  a2 m¼0 n¼0

Appendix This appendix gives closed-form expressions for the matrix elements needed to construct the matrix ¯ ¯ jH t jK ni. First let H pq ¯ n¯ n ¼ hK n tKK

 q þ þ a ¯ K n¯  sin y K n F qy K K¯ X ap X ¼ c ¯ cKn nDn¯ n 2 m¼0 n¼0 Km

Acknowledgements The author would like to thank B. Etemadi for useful discussions and F. Sales-Mayor for a careful reading of the text.

215

þ P3 ðDmþn2 þ Dmnþ2 þ Dmn2 Þ

 þ P4 ðDmþn3 þ Dmnþ3 þ Dmn3 ÞÞ ,

2 2  þ t a K¯ n¯  1 F 2 sin2 f K þ n 4 ¼

K K¯ X pt21 a2 X c ¯ cKn 4 m¼0 n¼0 Km

ðA5Þ

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  1 1 1 Dn¯ n  Dn¯nþ2  Dn¯n2 2 4 4  1 P1 Dmn þ ðP2 ðDmþn1 þ Dmnþ1 þ Dmn1 Þ 2

¼ pt0 a2



K¯ X K pt21 a4 X c ¯ cKn Dn¯ n 8 m¼0 n¼0 Km h a Dmþn þ Dmn þ ðDmþnþ1 þ Dmnþ1 4 þ Dnmþ1 þ D1mn Þ 1  ðDmþnþ2 þ Dmnþ2 þ Dnmþ2 þ D2mn Þ 2 i a  ðDmþn3 þ Dmnþ3 þ Dmn3 þ D3mn ÞÞ . 4 ðA7Þ

¼

þ P3 ðDmn1 þ Dnm2  D2mn Þ

 þ P4 ðDmn1 þ Dnm2  D3nm ÞÞ ,



ðA9Þ

2 4   t1 a 2  ¯ sin y K n K n¯  4 K K¯ X pt21 a4 X d ¯ d Kn Dn¯ n 8 m¼1 n¼1 Km h a Dmn þ ðDmnþ1 þ Dnmþ1  D1mn Þ 4 1 þ ðDmnþ2  Dmn2 þ D2mn Þ 2 i a þ ðDmnþ3  Dmn3 þ D3mn ÞÞ . ðA14Þ 4

¼

2 2   a q  ¯ K n¯ 2 2 K n F qf K K¯ X a2 p X 2 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffi d Km ¯ d Kn ðn ÞDn¯ n 1  a2 m¼1 n¼1

½f nþm ðaÞ  f jnmj ðaÞ,

2 2   t a K¯ n¯  1 F 2 sin2 f K  n 4 K K¯ X pt21 a2 X d ¯ d Kn 4 m¼1 n¼1 Km   1 1 1 Dn¯ n  Dn¯nþ2  Dn¯n2 2 4 4  1 P1 Dmn þ ðP2 ðDmn1 þ Dnm1  D1mn Þ 2 þ P3 ðDmn2 þ Dnm2  D2mn Þ  þ P4 ðDmn3  Dnm3 þ D3mn ÞÞ , ðA13Þ

ðA8Þ

K K¯ X pX d ¯ d Kn n 2 m¼1 n¼1 Km

Dn¯ n ðDmnþ1  Dnmþ1 Þ,

ðA12Þ

¼

 a q   ¯ K n¯  sin y K n F qy ¼a

K K¯ X pt20 a2 X d ¯ d Kn Dn¯ n 4 m¼1 n¼1 Km

 1 P1 Dmn þ ðP2 ðDmn1 þ Dnm1  D1mn Þ 2

m¼1 n¼1

ðA11Þ

2 2   t a K¯ n¯  0 F 2 K  n 4 ¼

The negative to negative terms are 2

  q  ¯ K n¯ 2 K n qy K¯ X K X 2 ¼p d Km ¯ d Kn ðn ÞDn¯ n i a Dmn þ ðDmnþ1 þ Dmn1 Þ , 2

d Km ¯ d Kn ðnÞDn¯ n

h a Dmn þ ðDmþnþ1 þ Dmnþ1 2 i þ Dmþn1 þ Dmn1 Þ ,

ðA6Þ

2 4  þ t a K¯ n¯  1 sin2 y K þ n 4

h

K¯ X K X m¼1 n¼1

þ P3 ðDmþn2 þ Dmnþ2 þ Dmn2 Þ

 þ P4 ðDmþn3 þ Dmnþ3 þ Dmn3 ÞÞ ,

 q  K¯ n¯ it0 a2 K  n qf

ðA10Þ

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The matrix elements connecting different parties are

 t 1 a3 q  þ ¯ sin y cos f K n K n¯ i qf F ¼

K¯ X K pt1 a3 X c ¯ d Kn ðnÞðDn¯ nþ1 þ Dn¯ n1 Þ 4 m¼0 n¼1 Km

½Dmþnþ1 þ Dnmþ1  Dmnþ1  Dmþn11 , ðA15Þ

 q  þ ¯ K n¯ iat1 sin fða þ cos yÞ K n qy 3

¼ t1 a p 

K¯ X K X

cKm ¯ d Kn ðnÞðDn¯nþ1  Dn¯n1 Þ

m¼0 n¼1

3a ð1 þ a2 Þ ðDmþn þ Dmn Þ þ 4 4

ðDmþnþ1 þ Dmnþ1 þ Dmþn1 þ Dmn1 Þ

 a þ ðDmþnþ2 þ Dmn2 þ Dmþn2 þ Dmn2 Þ , 4

 t 0 t 1 a3 sin y cos fF K  n K¯ n¯ 2

ðA16Þ

þ

K¯ X K t 0 t 1 a3 p X ncKm ¯ d Kn 4 m¼0 n¼1    1 a2 1þ ðDn¯nþ1  Dn¯n1 Þ 2 2

¼

ðDmþn1  Dmþnþ1 þ Dmn1  Dmnþ1 Þ a þ ðDmþn2  Dmþnþ2 þ Dmn2  Dmnþ2 Þ 2 a2 þ ðDnþmþ1  Dmþn1 þ Dmnþ1  Dmn1 8  þ Dnþm3  Dmþnþ3 þ Dnm3  Dnmþ3 Þ . ðA17Þ The negative to positive elements are obtained by interchanging all indices, or equivalently, by taking their transpose. References [1] Y.Y. Chou, G.-Y. Guo, L. Liu, C.S. Jayanthi, S.Y. Wu, J. Appl. Phys. 96 (2004) 2249. [2] W. Tian, S. Datta, Phys. Rev. B 49 (1994) 509.

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[39] M. Encinosa, L. Mott, Phys. Rev. A 68 (2003) 014102. [40] M. Encinosa, arXiv:physics/0501161, Appl. Math. Comp., submitted. [41] It simplifies the discussion to let parity here refer to the even/oddness of the y-basis functions; labeling the true quantum mechanical parity will of course depend on the f ! f þ p transformation.

[42] D. Pfannkuche, R.R. Gerhardts, P. Maksym, V. Gudmundsson, Physica B 189 (1993) 6. [43] Y. Pershin, C. Piermarocchi, arXiv:cond-mat/ 0502001. [44] M. Encinosa, L. Mott, B. Etemadi, arXiv:quant-ph/ 0409141, Phys. Scr., accepted for publication. [45] M. Encinosa, R. O’Neal, arXiv:quant-ph/9908087.