Polarisability of a confined multisubband electron gas with exchange and correlation interactions

Microelectronics Journal 36 (2005) 778–785 www.elsevier.com/locate/mejo

Polarisability of a confined multisubband electron gas with exchange and correlation interactions H. Rodrı´guez-Coppolaa,*, F. Garcı´a-Molinerb, J. Tutor-Sa´nchezc a

Departamento de Fı´sica Teo´rica, Facultad de Fı´sica, Universidad de La Habana, Vedado 10400, La Habana, Cuba b Universitat ‘Jaume I’, Campus Carretera de Borriol, 12080 Castello´n de la Plana, Spain c Instituto de Materiales y Reactivos para la Electro´nica, Universidad de la Habana, Vedado 10400, La Habana, Cuba Available online 4 January 2005

Abstract The Singwi–Sjo¨lander–Tosi–Lang approximation to the dielectric response of a 3D electron gas was designed to include exchange and correlation effects. This paper presents the formal extension to a low dimensionality, confined electron gas having a multisubband spectrum of one electron states. The quasi 1D case (quantum wire) is explicitly considered. It is seen that the analytical properties of the polarisability change quite significantly when manybody interactions are switched on. q 2004 Elsevier Ltd. All rights reserved. Keywords: Dielectric constant; SSTL-approximation; Quantum well wire; Exchange and correlation PACS: 73.21.Hb; 71.45.Gm; 77.22.Ej; 68.65.La

1. Introduction This article is concerned with the dielectric response of an electron gas confined in low dimension (D!3) quantum heterostructures, where the electronic states have a momentum of dimension D and 3D discrete quantum numbers. Assuming one starts from a bulk one band model, the wavefunctions are then free electron like in D dimensions, with an amplitude which depends on the 3D confinement coordinates. Moreover, the spectrum of electronic states is then multisubband. For vanishing D-dimensional momentum, the discrete 3D quantum numbers give the bottom levels of the subbands. For the quantum dot, with DZ0, there are only sublevels, with no free degree of motion, but we shall refer generally to a multissuband spectrum for simplicity. Thus, the dielectric response function differs from that of the 3D case because the quantisation scheme differs, * Corresponding author. Tel.: C53 7 8788 956x533. E-mail addresses: [email protected] (H. Rodrı´guez-Coppola), [email protected] (F. Garcı´a-Moliner), [email protected] (J. TutorSa´nchez). 0026-2692/$ - see front matter q 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.mejo.2004.12.001

the wavefunctions are also different and the spectrum is then multisubband, with populated and empty subbands. The dielectric function is then a matrix and this raises practical and formal questions related to its inversion. These problems were discussed in [1]. Having framed the problem in the appropriate quantisation scheme, the next question concerns the approximation one makes in the dynamical theory of the response of the electron gas. The multisubband extension of the RPA dielecric function was given in [2]. Results obtained for various situations showed that the contributions of a—sometimes sizeable—number of empty subbands may sometimes be significant in practice [2,3]. Then, the multisubband extension of the Hubbard dielectric function was derived and again significant contributions of empty states were found in practice [4]. The purpose of the work here described is to go beyond the Hubbard approximation, which includes only exchange interactions. More specifically, we obtain the multisubband version of the Singwi–Sjo¨lander–Tosi–Lang dielectric function [5], which includes exchange and correlation. This is different from other approaches variously employed to study correlation effects in the dielectric response of

H. Rodrı´guez-Coppola et al. / Microelectronics Journal 36 (2005) 778–785

confined systems [6–13]. The paper is organised as follows. The local field correction factor (LFCF) for a multisubband system is derived in Section 2, the dielectric function is discussed in Section 3 and the practical use of the theory thus developed is illustrated in Section 4 with an application to the quasi 1D case of a quantum wire. Final comments are made in Section 5.

779

selfenergy terms and (b) for the equation of motion of the (a,a 0 ) elements we retain only the terms in (a,a 0 ), On combining the r.h.s. of (1) with its Hermitean conjugate in order to obtain a Hermitean equation of motion, as required in order to ensure that the eigenvalues are real, we obtain KðZuÞ2 r^ ð1Þ a;a 0 ðq; uÞ Z g2a;a 0 ðp; qÞC^ a;a 0 ðp; s; qÞ C K^ ee1 ðp; qÞ C K^ ee2 ðp; qÞ; with

2. Local field correction factor In Mahan’s approach [14] the local field correction factor (LFCF), in the SSTL approximation, is derived from the equation of motion for the time derivative of the density matrix, as this brings into the analysis the correlation between elements of the density matrix. We now set out to develop this approach for the multisubband case. The density matrix operator is denoted r^ð0Þ for the ^ r^ ð0Þ C r^ ð1Þ for the noninteracting electron gas and rZ interacting electron gas. Let a denote the quantum number—or numbers—labelling the discrete energy levels of the confined system (one for a quantum well, two for a quantum wire, tree for a quantum dot) and q the wavevector associated with free motion (2D for the well, 1D for the wire, none for the dot). Then, in linear response theory, after time Fourier transform, ^ ^ ^ ð1Þ KðZuÞ2 r^ð1Þ a;a 0 ðq; uÞ Z ½H; ½H; ra;a 0 ðq; uÞ:

(2) where D^ L ðp; k; q;s; s

Þ Z a^ †pCq;s;a1 a^ †kKq;s0 ;a2 a^ k;s0 ;a3 a^ p;s;a4 ;

e2 ; Lha1 ; a2 ; a3 ;a4 ; z Z 2L3o ð VL ðqÞhfa1 ; a4 ja2 ; a3 gq Z d2 rd2 r0 Ko ðqjr Kr0 jÞFL ðr; p0 Þ; FL ðr; p0 Þ Z f a1 ðrÞf a2 ðr0 Þfa3 ðrÞfa4 ðr0 Þ:

Ga;a 0 Z Ea0 K Ea :

Z2 q2 Z2 p$q C ; m 2m ð5Þ

K^ ee1 is the interaction term in the RPA, namely K^ ee1 ðp; qÞ Z Z2 u2a;a 0 ðqÞhr^ ð0Þ a;a 0 ðqÞi:

(6)

while K^ ee2 contains the rest of the interactions beyond the RPA and    e2 Z2 q2 ðKÞ K^ ee2 ðp; qÞ Z Va;a 0 ðqÞ Ga;a 0 C 2L30 2m X Z 2 q2 ! ½Sa;a 0 ðq C q 0 Þ K 1 K fa; aja 0 ; agq m q0 9   = X q0 ! ð7Þ ½Sa;a0 ðq C q 0 Þ K 1 hr^ ð0Þ 0 ðqÞi: ; a;a q q0 The analysis leads to the introduction of the quantities

L;s;s0 p;k;q

0

ga;a0 ðp; qÞ Z ½Ea0 ðp C qÞ K Ea ðpÞ Z Ga;a 0 C

(1)

The analysis presently to be developed can be equally carried out for any confined heterostructure. The general form of the results is the same but some details depend on the reduced dimension D. To fix ideas we shall carry out the process for DZ1 (quantum wire). The Hamiltonian of the quasi 1D electron gas is then X XX H^ Z Ea ðpÞa^ †p;s;a a^ p;s;a Cz VL ðqÞD^ L ðp; k;q; s; s0 Þ; a;p;s

(4)

ð3Þ

L is a quantisation length, analogous to the usual quantisation box in 3D. The first term of (2) is the unperturbed Hamiltonian of the electron gas and the subbands are the Ea ðpÞ. As in Mahan’s analysis [14], in the development of the double commutator (a) we discard

ðGÞ Va;a 0 ðqÞ

Z ½fa 0 ; a 0 ja 0 ; agq Gfa; aja; a 0 gq 

(8)

If we stop at the RPA, then we obtain the dispersion relation for the intersubband plasmons sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 

ffi e2 Na;a0 Ga;a0 1 ðCÞ q2 ðKÞ Va;a 0 ðqÞ 2 C Va;a 0 ðqÞ ; uPa;a 0 ðqÞ Z 2 2p30 2m Z Na;a 0 Z ðNa C Na 0 Þ and, for the intrasubband case sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi e2 q2 fa; aja; agq Na uPa;a ðqÞ Z ; 2p30 m

ð9Þ

(10)

where Na is number of particles in the subband a. The evaluation of the terms appearing in the expansion of the double commutator requires the use of the equality 0 ð0Þ 0 ^ hr^ ð0Þ a;a 0 ðq C q Þra 0 ;a ðKq Þi 0 0 0 0 Z hrð0Þ a;a 0 ðqÞi½ðNa C Na ÞdqCq Z0 C Sa;a ðq C q Þ K 1:

This is proved in Appendix A, where it is seen that K^ ee2 contains the static structure factor (SSF) Sa,a 0 (q), which accounts for the particle correlation. The last step of the analysis is to take the statistical averages of the equations involving operators.

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780

On working out (7) we encounter the sum 1 X ½S 0 ðq C q 0 Þ K 1 2L q0 a;a Z

1 4p

ðN KN

dq 0 ½Sa;a 0 ðq C q 0 Þ K 1:

(11)

By using the relationship between SSF and the pair distribution function [14] we obtain 1 X ðN C Na 0 Þ (12) ½Sa;a0 ðq C q 0 Þ K 1 Z K a 2L q0 4 Let us define     1 X q0 1 U~ a;a 0 ðqÞ Z : ½Sa;a0 ðq C q 0 Þ K 1 2L q0 Na C Na 0 q Then we obtain )  2 ( ðKÞ Va;a 0 ðqÞGa;a0 Z2 q2 ðCÞ e C Va;a 0 ðqÞU~ a;a ðqÞ ðNa Kee2 Z K 30 4 m

(13) It is in order to remark that an object like Kee2 is the real quantity associated to K^ ee2 after combination with its Hermitean conjugate. Finally, the sum of all—real—interaction terms is Kee1 ðqÞ C Kee2 ðqÞ Z Z2 u2Pa;a 0 ðqÞf1 K Ga;a0 ðqÞghr^ ð0Þ a;a 0 ðqÞi;

(14)

where Ga;a0 ðqÞZ

ðNa CNa0 Þ ðZuPa;a0 Þ2 K

3S ðq; u; r; r 0 Þ Z dðr K r 0 Þ X K Lm ðq; rÞRSmm ðq; uÞSm ðr 0 Þ;

 ðKÞ Va;a 0 ðqÞ 4

Ga;a0 C

where RSmm ðq; uÞ, the polarisability, is in this case—SSTL approximation RSmm ðq; uÞ Z

Pmm ðq; uÞ : 1 C Pmm ðq; uÞHmS ðqÞ

Z2 q 2m

Pmm(q,u) is the mth diagonal matrix element of the RPA polarisability and (19)

Now, consider the integral {a,a 0 ja 0 ,a} defined in (3). With the change of label (a1,a2)/m, (a3,a4)/n this becomes {mjn} which, in the new dual basis is Ð (20) fmjngq Z drSm ðrÞLm ðq; rÞ hbmn ðqÞ: The last equality defines the matrix elements of a matrix b. Furthermore, we define a diagonal matrix R of elements Rmn hRSm dmn , with the RSm equal to the elements given in (18). Then, the inverse of (17) is [1–4]

Z dðr K r 0 Þ C

X

Lm ðq; rÞBSmn ðq; uÞSn ðr 0 Þ;

(21)

mn

(15)

where, in compact matrix notation, BS ¼ ½RK1 K bK1 :

This is the Local Field Correction Factor (LFCF) in the multisubband version of the SSTL approximation to include exchange and correlation [5,14]. For a0 Za it reduces to ~ a;a ðqÞ: Ga;a ðqÞZ4pU

(18)

0 3K1 S ðq; u; r; r Þ

2

 Z2 q2 ðCÞ ~ 0 V ðqÞ U ðqÞ : 0 a;a m a;a

(17)

m

HmS ðqÞ Z fa; a 0 ja 0 ; agq GSa;a 0 ðqÞ:

C Na 0 Þhr^ ð0Þ a;a 0 ðqÞi:

"

but it is denumerable, so it can be ordered in a chosen sequence [1]. The same holds in the case of m, which stands for four discrete numbers. In the end, by adopting an ordering criterion, the matrices involved in the algebra are cast as simple second rank matrices, which is a convenient device in practice. Also, for the matrix representation we change to the dual, biorthogonal basis of long ðLÞ and short ðSÞ range functions defined in [1–4]. The general form of the dielectric function is then

(16)

3. The dielectric function and its inverse As in previous studies [1–4] of the multisubband dielectric function in lower approximations—namely, RPA and Hubbard (H)—we condense two labels (a,a 0 ), associated to a single particle excitation in one label m. For a quantum wire, each a stands for two discrete quantum numbers. Thus we may face a doubly infinite set,

(22)

As in the quasi 2D case, but with the position vector r normal to the—cylindrical—boundary instead of the coordinate z normal to the—plane—boundary, it is easily seen that the matrix representing d(rKr 0 ) in the dual basis is bK1 and finally, in compact vector and matrix notation 0 K1 K1 3K1 S ðq; u; r; r Þ Z Lðq; rÞfb ðqÞ C ½R ðq; uÞ

K bðqÞK1 g$Sðr 0 Þ: The evaluation of the dielectric function 3 then involves a selfconsistent process: e depends on GSa;a 0 (15), this depends on U~ a;a 0 (13), this depends on the SSF Sa;a0 and this, in turn, has a general relationship with Imf3K1 g. This relationship, wellknown in 3D [14], must be established anew for the multisubband case of low dimensionality systems.

H. Rodrı´guez-Coppola et al. / Microelectronics Journal 36 (2005) 778–785

To this effect we consider the response to some external potential Ve of the interacting electron gas having the dielectric function just derived. By subtracting Ve from the total screened potential, the induced potential is, from the above results, X V S ðq; u; rÞ Z Lt ðq; rÞBtn ðq; uÞVne ðq; uÞ; (23) t;n

where Vne ðq; uÞ

e

Z drV ðq; u; rÞSn ðrÞ:

(24)

Taking again the scalar product of (23) with Sn ðrÞ, we have X bst ðqÞBtn ðq; uÞVne ðq; uÞ: (25) VsS ðq; uÞ Z tn

On the other hand, the induced potential is related to the induced charge through the corresponding Poisson equation and this relationship can be expressed as X bst ðqÞhr^ ð1Þ (26) VsS ðq; uÞ Z 4pe2 t ðq; uÞi: t

This must equal (25), hence ( ) X X e 2 ð1Þ bst Btn ðq; uÞVn ðq; uÞ K 4pe hr^t ðq; uÞi Z 0; t

n

(27) whence hr^ ð1Þ t ðq; uÞi

By taking real (principal value) and imaginary parts, we obtain the equalities X  e2 Wt 2Zun0

P ðZuÞ2 K ðZun0 Þ2 n )   (X 1 Z Btn ðq; uÞbnt ðqÞ (32) Re 4pe2 bmm ðqÞ n and

Ð

X 1 Z Btn ðq; uÞVne ðq; uÞ: 4pe2 n

781

X n

)  (X 1 Wnt dn Z Btn ðq; uÞbnt ðqÞ : Im 4pe2 bmm ðqÞ n 

(33) ðN 0

Integration of (33) for positive frequencies yields X X X 2 du Wnt dn Z Wnt Z jhnjr^ð0Þ† t ðqÞj0ij n

n

^ð0Þ Z hF0 jr^ ð0Þ† t ðqÞr t ðqÞjF0 i

n

Z St ðqÞ;

whence the desired formula ( ) ðN X 1 St ðqÞ Z 2 du Im Btn ðq; uÞbnt ðqÞ : 4e pbtt ðqÞ 0 n (34) This completes the analysis of the elements needed for a selfconsistent calculation of the dielectric function in the SSTL approximation for the multisubband spectrum.



(28)

The vanishing of each term separately follows from the fact that bK1 is the matrix representing the delta function in the complete dual basis. The induced particle density can be obtained in the standard way from the equation of motion for the density matrix in an external potential. This yields X e hr^ ð1Þ Wnt Gn0 ðuÞ; (29) t ðq; uÞi Z Vt ðq; uÞ n

with 

1 1 K Gn0 ðuÞ Z ; Zu K Zun0 C iZh Zu C Zun0 C iZh (30) where h is the customary infinitesimal. By equating (28) and (29) we find ( ð X Wnt Gn0 ðuÞSt ðrÞ d2 rV e ðq; u; rÞ ! n

)  X 1 K Btn ðq; uÞSn ðrÞ Z 0; 4pe2 n which of course is independent of Ve, as it must.

ð31Þ

4. Example of application: effect of correlation on the polarisability of a quantum wire The changes in the polarisability of a quantum wire were studied elsewhere [3,4] in the RPA and Hubbard approximations. In order to illustrate the effects of exchange interactions it suffices to study the first matrix element (m, m) with mZ1, which in turn corresponds to the intrasubband excitation (aZ11)/(aZ11) where (aZ11) denotes the two ordinary quantum number of the one electron states for qZ0. We shall study here the same system in order to see what further effects the correlation interactions introduce. The system under study is a GaAs wire (m*Z0.066m0, 30Z ˚ , dyZ150 A ˚ ) with 10.9) of rectangular section (dxZ200 A infinite barriers. In the selfconsistent process required for the SSTL calculation we used the fact that the SSF is the Fourier transform of the PDF, which in Appendix A is evaluated in the Hartree–Fock approximation. This yields a simple expression for the integral needed for the LFCF. We then obtain, for the intrasubband case   ð 2q N sinkFa z 2 Ua;a ðqÞ Z dz sin qzSi½kFa z; (35) p 0 kFa z where z is the 1D coordinate along the wire axis and kFa is the 1D Fermi wavevector for the corresponding subband.

H. Rodrı´guez-Coppola et al. / Microelectronics Journal 36 (2005) 778–785

782

Table 1 Values of characteristic parameters pertaining to the first intrasubband excitation for the same quantum wire (text) with three different values of the population n1D (cmK1) 5

7.08!10 4.06!105 1.38!105

[F

r11/aB

rh1/aB

1.5 1.16 1.02

1.78 3.02 8.78

0.31 0.52 1.51

First column: [F Z EF =E11 . Second column: particle number per unit length of the wire. Third column: 1D mean distance between the light quasiparticles (see text) in units of the Bohr radius. Fourth column: The same for the heavy quasiparticles.

The ground state of the aZ11 subband is denoted x11. In Table 1 we give some parameters for the wire referred to above with three different values of the electron population. As this decreases, we expect exchange and correlation effects to decrease. The first column gives [F Z EF =E11 , a parameter entering naturally the analysis, which decreases with decreasing population. The measure of the latter is given in the second column (n1D: number of electrons per unit wire length). The Q1D case was studied elsewhere [3] by means of a picture in terms of light (l) and heavy (h)— formal—quasiparticles. We give, in the present approximation, the 1D radius of the exchange and correlation hole in units of the Bohr radius for the light (rl: third column) and heavy (rh: fourth columns) quasiparticles. As expected, as n1D decreases, both rl and rh increases, which means that the electrons stay apart due to increased manybody interactions.

Fig. 1. Local field correction factor (LFCF) for the first subband, assumed to be the only occupied subband, with [F Z EF =E11 Z 1:16. All calculations, in this and the following figures, are for the quantum wire described in the text. The figure compares the results for the H (dashed line) and SSTL (full line) approximations.

Furthermore, also as expected [3], these affect more strongly the light quasiparticles. Fig. 1 compares the wavevector dependence of the LFCF (fixed u) with [FZ1.16, the intermediate of the three cases considered, for the Hubbard and SSTL approximations. The effect of correlation is quite significant and it shows that the exchange and correlation hole (SSTL) is appreciably smaller than the exchange hole (H). We now discuss the behaviour of Re{R11} as a function of q for fixed u. The next two figures focus on the long wave range and compare the result for the RPA, H and SSTL approximations. In all cases there are two peaks, of which the first can be interpreted in terms of the light quasiparticles and the second in terms of the heavy ones [3]. In Fig. 2, with [FZ1.5, we have (Table 1) that (r0/aB)!1, so we expect the RPA to be a fair approximation. Indeed, the three results are very close, appearing to be practically indistinguishable. However, while the peak positions are the same, there are analytical differences, a fact which already appeared in previous studies of the H approximation [4]. The first peak corresponds to a logarithmic singularity in the RPA and a finite maximum in both, the H and SSTL cases, while the second peak corresponds to a logarithmic singularity (RPA) a simple pole (H) and a finite minimum (SSTL), respectively. Fig. 3 displays the results of the three approximations in the range of the first (light quasiparticle) peak and shows the effect of changing the electron concentration by comparing case (a), [FZ1.16 with case (b), [FZ1.02. As in Fig. 2, the logarithmic singularity of the RPA is changed into a finite peak in the H and SSTL approximations but the results demonstrate that the manybody effects grow in importance

Fig. 2. LFCF for the one occupied subband (1,1) and [FZ1.5. Dotted line: RPA. Dashed line: H. Full line: SSTL.

H. Rodrı´guez-Coppola et al. / Microelectronics Journal 36 (2005) 778–785

Fig. 3. The same in Fig. 2 for (a) [FZ1.16 and (b) [FZ1.02 and Zu=E11 Z 2:5.

as the population decreases. On the other hand, the growing importance of the correlation interactions is to push back the results closer to the RPA in the peak and immediately beyond. Fig. 4 concentrates on the static limit (uZ0) for the same systems as in Fig. 3. It is found in the calculations that, as u decreases, the two peaks grow further apart while the intensity of the first one decreases. For uZ0, the light quasiparticle hole peak disappears and the heavy quasiparticle hole appears at qZ2kF1. The numerical values of Re{R11} appear to be very close for the three

783

Fig. 4. Static case (uZ0). Otherwise as in Fig. 3.

approximations, but the significant difference concerns the analytical behaviours. In the RPA, the static polarisability is a logarithmic function of an argument which, in terms of the quantities EG m ðqÞ defined in [3] is

W1 ðq; 0Þ Z

E1ðKÞ ðqÞ E1ðCÞ ðqÞ

!2



1 K ð2kF1 =qÞ Z 1 C ð2kF1 =qÞ

2

:

(36)

This vanishes at qZ2kF1 and produces the logarithmic singularity characteristic of the RPA.

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784

With a LFCF, we obtain instead, that R1 ðq; 0Þ x

ð2a=qÞ½lnj1 K ð2kF1 =qÞj K lnj1 C ð2kF1 =qÞj ; 1 C ð2a=qÞH1 ðqÞ½lnj1 K ð2kF1 =qÞj K lnj1 C ð2kF1 =qÞj

where aZe2m*/30p-2. In the H approximation, H1(q) is fairly constant (Fig. 1). Let qp denote the value of q for which the denominator of (37) vanishes. Then, to a good approximation, we can put H(q)ZH(qp) in the range of interest, where we seek the vanishing of the said denominator, i.e. 1C

2ab ½lnj1 K ð2kF1 =qp Þj K lnj1 C ð2kF1 =qp Þj Z 0: qp

(37)

Then we find qpx2kF1. At this value of q, a pole appears in (37), as seen in Fig. 4. On the other hand, in the SSTL approximation the LFCF stays definitely below that for the H approximation (Fig. 1) and the denominator of (37) does not vanish, so the singularity at 2kF1 disappears and only a finite minimum remains. Now, it was suggested in [3] that this singularity indicated a Fro¨hlich–Peierls transition associated with the appearance of a gap in the quasiparticle spectrum due only to electron–electron interactions, without invoking any external agent like a phonon field. Since exchange interactions also result in a singularity, one might enquire as to whether this is or not eliminated by manybody interactions. The present calculation shows that it is when correlation is switched on.

5. Final comments The SSTL approximation was originally designed to include exchange and correlation effects in the dielectric response of a homogeneous 3D electron gas. The theoretical analysis here presented provides its formal extension to the case of a confined, low D(!3), inhomogeneous, multisubband electron gas. The analysis has been developed for a quantum wire (DZ1) but the same routine holds, with appropriate changes of details, for DZ2 (e.g. a quantum well) or DZ0 (quantum dot), as has been stressed in previous studies of the RPA and Hubbard approximations. By comparison of the results obtained in the three successive applications to the study of the analytical properties of R11, we can ascertain the consequences of the manybody interactions. Lower approximations than SSTL may be reasonably adequate for some problems, but we find that simply including exchange may induce to error concerning the possibility of a Fro¨hlich–Peierls singularity due only to electron–electron interactions. Indeed, inclusion of correlation effects shows that it is not possible. It can be easily seen that the same holds for a multisubband case.

Indeed, the case of one occupied subband has been here chosen as the most immediate and simple example of application. The real purpose of the analysis is actually to open the way for the study of more involved situations where intersubband effects may have significant consequences, as experience with the lower approximations has shown. The study of such problems, including the occupation of more than one subband, requires quite substantial calculations and is in our immediate programme for further work.

Acknowledgements We are very grateful to Pilar Jime´nez for her unvaluable help in the preparation of the paper. Most of this work was carried out while one of the authors (H R–C) was enjoying the hospitality and support of the University Jaume I. This help is greatly appreciated.

Appendix A. Static structure factor, pair distribution function and correlation of elements of the density matrix in a multisubband quasi 1D electron gas We start by writing down the density matrix operator [14] in terms of coordinates and labels adequate to Q1D case: ^ 2 ; r1 Þ Z rðr

1 X X iðk$z2Kk0 $z1 Þ 0 ;e fa 0 ðr2 Þfa ðr1 Þa^ †k0 sa 0 a^ ksa : L a;a 0 p;k;s

We are interested in rZr 0 . Then " # X 1 X iq$z ð0Þ ^ Z rðrÞ fa 0 ðrÞfa ðrÞ e r^a;a 0 ðqÞ ; L q a;a 0 where r^ ð0Þ a;a 0 ðqÞ Z

X

a^†pCq;s;a 0 a^p;s;a :

(A.1)

(A.2)

p;s

Since the system is uniform in the z direction, we can write qa;a 0 X

dðz K zj Þ Z

jZ1

1 X iq$z ð0Þ e r^ a;a 0 ðqÞ; L q

(A.3)

where qa;a0 Z ðNa C Na 0 Þ=2 and Na is the number of electrons in the subband a. A quantisation length—hence Na finite- is everywhere used which in the end disappears from the results, as is ordinarily done in 3D. Some interesting properties required for the analysis are r^ ð0Þ a;a 0 ðKqÞ Z

qa;a 0 X

eKiq$zj ;

(A.4)

jZ1 0 hF0 jr^ ð0Þ a;a 0 ðqÞjF0 i Z Na da;a dq;0

(A.5)

H. Rodrı´guez-Coppola et al. / Microelectronics Journal 36 (2005) 778–785

and † ½r^ ð0Þ a;a 0 ðqÞ

785

and in the second case Z r^ ð0Þ a 0 ;a ðKqÞ:

(A.6)

and the static structure factor (SSF) is defined, analogously to the 3D case, as ^ð0Þ† qa;a 0 dq;0 C Sa;a0 ðq; tÞ Z hF0 jr^ð0Þ a;a 0 ðq; tÞra;a 0 ðq; 0ÞjF0 i:

(A.7)

By following the same routine as in [14] we obtain X 1 hF0 j (A.8) Sa;a 0 ðqÞ K 1 Z n^pCqsa 0 n^ psa jF0 i: qa;a 0 ps Now, the SSF is the Fourier transform of the pair distribution function (PDF). On adapting the 3D relationship between S and g [14] to the quasi 1D case we write ð ðna C na 0 Þ N Sa;a 0 ðqÞ K 1 Z dz½ga;a 0 ðzÞ K 1eiqz ; (A.9) 2 KN where na is the linear electron concentration in the subband a and we have used the equality ð ðn C na 0 Þ N qa;a 0 dq;0 Z a dzeiqz : (A.10) 2 KN The essential difference with the 3D case is that in the multisubband situation it is necessary to introduce in the analysis all possible intra and intersubband PDFs. We recall that in the 3D development of the SSTL approximation one uses the Hartree–Fock equality   X  jl ðr1 Þ jl ðr2 Þ 2 V2 gðr1 ; r2 Þ Z det (A.11)  ;  jl0 ðr1 Þ jl0 ðr2 Þ  NðN K 1Þ ll0 where l is the set of quantum numbers labelling a one electron state. On adapting this to the quasi 1D case and setting r1Zr2Zr we have gðr; z1 ; z2 Þ Z

L2z1 NðN K 1Þ X f2a ðrÞf2a 0 ðrÞNa Na 0 ga;a 0 ðz1 ; z2 Þ; !

(A.12)

aa 0

where we have defined:  ikz  e 1 cs X 1  det 0 ga;a 0 ðz1 ; z2 Þ Z Na Na 0 k;k 0 ;s;s0  eik z1 c 0 s

2 eikz2 cs   : 0 eik z2 cs0  (A.13)

This is the sum of two contributions, one for antiparallel spins and one for parallel spins. In the first case 2 X 2 Na Na 0 1 gAa;a 0 ðz1 ; z2 Þ Z (A.14) Z 1Z Na Na 0 k;k 0 Na Na 0 2 4

gPa;a 0 ðz1 ; z2 Þ Z

0 2 X ½1 K
(A.15)

which yields gPa;a 0 ðz1 ; z2 Þ Z



1 sinxa sinxa 0 1K ; 2 xa xa 0

(A.16)

where xaj Z kFaj jzj. The total PDF is the sum of (A.15) and (A.16). It is easily seen that, for jz1Kz2j/0, ga,a 0 (0) equals 1/2, which means that the two electrons can get arbitrarily close but only in one of the two possibilities, i.e. with antiparallel spins. With this we can finally evaluate the required elements of the density matrix 0 ð0Þ 0 ^ Ca;a 0 ðq; q 0 Þ Z hr^ ð0Þ a;a 0 ðq C q Þra 0 ;a ðKq Þi:

(A.17)

Proceeding as in 3D, we obtain Ca;a 0 ðq; q 0 Þ Z h

qa;a 0 X jZ1

0

eiðqCq Þzj

qa;a 0 X

0

eKiq zk i

kZ1

0 0 0 0 Z hr^ð0Þ a;a 0 ðqÞi½qa;a dqCq ;0 C Sa;a ðq C q Þ K 1:

(A.18)

References [1] J. Ferna´ndez-Velicia, F. Garcı´a-Moliner, V.R. Velasco, J. Phys. A 28 (1995) 391. [2] J. Ferna´ndez-Velicia, F. Garcı´a-Moliner, V.R. Velasco, Phys. Rev. B 53 (1996) 2034. [3] J. Tutor-Sa´nchez, H. Rodrı´guez-Coppola, S. Rodrı´guez-Romo, J. Phys. Condens. Matter 10 (1998) 9999. [4] H. Rodrı´guez-Coppola, J. Tutor-Sa´nchez, F. Garcı´a-Moliner, Phys. Scripta 63 (2001) 342. [5] K.S. Singwi, M.P. Tosi, R.H. Land, A. Sjo¨lander, Phys. Rev. B 176 (1968) 589. [6] S. Das Sarma, E.H. Hwang, Phys. Rev. B 54 (1996) 1936. [7] W.I. Friesen, B. Bergersen, J. Phys. C: Solid State Phys. 13 (1980) 6627. [8] L. Calmels, A. Gold, Phys. Rev. B 52 (1995) 10841. [9] L. Calmels, A. Gold, Phys. Rev. B 56 (1997) 1762. [10] L. Calmels, A. Gold, Phys. Rev. B 57 (1998) 1436. [11] B. Tanatar, Phys. Lett. A 239 (1998) 300. [12] E. Demirel, B. Tanatar, Solid State Commun. 110 (1999) 51. [13] L. Wendler, R. Haupt, R. Pechstedt, Phys. Rev. B 43 (1990) 14669. [14] D.G. Mahan, Many-Particle Physics, Plenum Press, New York, 1990.