Ground state correlations in quasi-one-dimensional electron and charged Bose systems

ALKAN PHYSICS LETTERS

. L, 3 (3), pp. 135 - 142 (1995) 1 July 1995

G�OUND STATE CORRELATIONS IN QUASI-ONE-DIMENSIONAL

ELECTRON AND CHARGED BOSE SYSTEMS

E. H. TURECi and B. TANATAR Department of Physics, Bilkent University,

Bilkent, 06533 Ankara, TURKEY.

(received 17 April 1995 ; accepted 23 May 1995)

Abstract. - Ground-state properties of quantum confined quasi-a d'

(Q1D) systems in which the charge carriers are electrons or charged b ne- 1!'1e!ls1onal

Th · b · th' k osons 1s ·

gated. e charge earners are assumed to em a zero !c ness xy-plane w'th 1nvesti-confinement potential. We assume that the charge earners are in their I 1 a harmonic and employ many-body techniques such as the Hartree-Fock approximation owest subband, approximation (RPA), and the self-consistent scheme of Singwi et al. t� ra:idorn-phase ground-state properties of this model system. Similar calculations are also c culate the thin the Hubbard approximation for a double-layer Q1D electron system in th Performed wi­

limit. e no-tunneling

,'·INTRODUCTION

111.ecent advances in the fabrication technology have made possible the realizat · _[ \�imensional

(Q1D)

on

_l]

r.

�ts have been detected_ with promi�ing possibilities for device applications.esQ many novel

(�ms have been the subject of extensive theoretical study because of the enh lD electron ('ttelation effects in lower dimensions

[2, 3, 4].

c

�;centrated on

·r\cts

!� '

S In Ql0

1�·

.. 1'he ground state properties of charged-particle systems obeying quantum 8

!� a major problem in many-body theory. Although a vast amount of lit tat1stics have

i . _{( I} . _{1· I} eratur .

, arged-ferm1ons

e.g.,

,�arged bosons, mainly because they have no simple experimental realizatio een devoted to :�arged Bose gas is of interest because of its intrinsic simplicity and its relation�- However, the

��l>orta.nt ele�tron gas. Recently, the importance of disorder for the experirnen

°

. 'Perconductmg films led to the suggestion that a 20 charged Bose fluid · ts on ultra-tl ·

136

_{BALKAN PHYSICS LETTERS}

There has bee

a lot of i

terest in the one-

imensional electron systems from the �a-called

Lutti

ger liqui

point of view [8] which ·uses a linear energy-momentum relatior.. Her� we take

the Fermi liquid approach for the electro

s as argue

by Hu an

Das Sarma (4]. Experimental

evide

ce from light scatteri

g measureme

ts [9] support this point of view for quantum wires

as occur in semiconductor structures.

In this paper we stu

y the grou

state properties of a QID system in which the charge

carriers are either electrons or charge

boso

s, with the aim of applying stan<lard many-body

tech

iqu

i

a comparative ma

er. For the case of electro

s, we assume the partides are all

confined to the low

t subba

d. Begi

ni

g from the Hartree-Fock (for electrons) and random­

phase approximatio

(RPA), we employ the self-consistent appro

ch of Singwi d al. [10)

(STLS) to inv

tigate the ground state e

ergy a

various correlation functions. STLS scheme

was applie

to QID electro

s usi

g differe

t interaction mo

els [2, 11] an

to charged bosons

with a contact i

teraction (12]. As an i

teresti

g application of the STLS method, we also

co

sider a double-layer QI D system, a

calculate the groun

state energy as a function of the

I

,,

layer separatio

dista

ce with the

o tu

eling assumption within the Hubbar

approximation.

�

Correlations i

double-layer electron systems have been calculated mai

ly"for 20 systems and

li

qua

tum-wells [13, 14] with i

teresting results. The ·aPA is k

own to

escribe the dielectric

!

properti

of i

teracting particles quite well i

the high-

e

sity limit. It is believe

that RPA

\J

becomes less accurate as the dime

sionality of the system is reduce

. As a computatienally

.

i'•.1.

ma

ageable formulatio

, the STLS method gives remarkably good results for the correlation

_�

energy in 20. Our aim is to check the extent to which STLS approach

escribes the ground

�

state properti

of Q 1 D systems.

The r

t of this paper is orga

ized as follows. In the

ext sectio

, we

escribe the model

use

QID system, an

outline the calculation of excha

ge-correl

tion energies of electron and

charged Bose systems. I

sectio

3, we present an

discuss our results.

2. MODEL AND THEORY

Semico

ucti

g QlD systems are realize

experimentally by several groups i11

ifferent sha­

pes. In our model system, the charge .carriers, bosons or fermions, are assume

to occupy a zero

thick

ess zy.. plane with a harmo

ic co

fi

eme

t pote

tial [3, 14] in they-direction so that the

subband e

ergies are

=

O(n+l/2) where n

=

0, 1, 2, ··.,

O

escribes the !:3tre

gth of the

confini

g pote

tial. Furthermore, we make the assumptio

that the partic.1 es occupy the lowest

subban

, i

the case of electrons. For a system of particles co

fi

ed to one dimeusio

we define

the average i

terparticle dista

ce a , such that the ID de

sity of particles is given by n

=

1/2a.

As the scali

g unit for our system, we choose the effective Bohr radius aB -= foh

/(me

) where

to is the backgrou

ielectric constant an

m is the effective ban

ma..'-.s. The

imension!�

e

sity parameter is

_efi

_e

_{by r}

a/as

=

_{1r/(4kFas), where kF is the Fermi wave vector}

for the corresponding noni

teracti

g Fermi system which is relate

to the particle density by

n

=

_{2kF/'lr_· We shall use as to scale le}

_{gths an}

_{effective ry}

_{bergs (R}

y

=

1/(2mat)) to

scale e

ergies. The effective {2oulomb i

teraction is give

by (14] V{q) = (e

_/to)

_F(q),

_where

F(q) = exp(q

_/20)/(o(q

_{/20), 'in which /(o(z) is the. mo}

_ifie

_{Bessel function of the first kind}

an

_.q

nd O are expresse

in units of as a

R

y

, respettively. The interaction e

ergy (p.;

particle) of the charge

particle system is

.. fl. TURECI et al. _{: GROUND STATE CORRECTIONS 137}

E

(

N

= ;

lo

dq F(q)[S(q) - 1],

(1)

,here S( _�ptibility

q)

S(q)

= -

f

d

x(q,

).

1r

lo

(2)

fbe dynamic susceptibility is expressed in a general way as

(

Xo(q,

)

X q,

)

=

_{1 - V(q)[l - G(q)]xo(q,w)}

₍₃₎

IJ.ere

xo(q,w)

fac-G(q) describes the short-range correlations m the system. Takmg_G == O amounts to the :dom-phase approximation (RPA), whereas G

=

q)

An approximation scheme to calculate

G(q),

r

1

G(q)=-

-

dk-

_{q V(q)}

-(S(q-k)-1].

(4)

S'nce

S(q),

_S

to co t u ar

ap-prv.,. ns ruct G(q) in

electron and charged Bose systems are �lso pos�ible [15J. _.

For charged boson systems, the non-mteractmg response function is given by ( w) _

2n(q

/2m)

xo

_{- ( . )2 ( 2/2 )2 .}

The density response function x( q,

w)

espression

q

susceptibility yields [6 ,7], · dynamic

S(q)

=

[1

2F(q)[l -

raq

₍₅₎

Similar to the electron system, G

=

of STLS S(q) and

G(q)

system.·

QID

1 38 BA LKAN PHYSICS UTTERS

a distance

d,

ment in the lateral direction renders a double-layered quantum wire system. Denoting by 1 and 2 the different layers, the interaction energy is given by

E 2 /00

N

_Jo

dq

(q)[S1 1 (q)

(6 )

where Fu(q) is the same as F(q) for a single-layer system, and inter-Jaye� Couiomb interaction ·

lS

The static structure factors Sij(q) arr evaluated from the dynamic response funct ions [13]

_ l ₍ [XO ( q , W )] - I - ef> I 1 (

q)

:

-q,

in which ,/Ji;(q) = Vij(q) [1 - Gi; (q)]. The RPA is recovered if the local-fidd factors are set

to zero,

viz.,

=

.energy of a double-layer system is similar to those of two-component plasma, first considered

by Vashishta

et al.

3� RESULTS AND DISCUSSION

Experimentally realized semiconducting quantum wires typical1y hav<' rarrin densities n

-105 - _{101 cm- •. We choose a GaAs material (dielectric constant c0 � 1 3) ! o present our}

numerical calculations, so that the density parameter r3 r.J 0.05 - 5. We -1 lso n .... sume that the

electrons occupy only the lowest subba11d, although generalization of our forn1,1.lism to higher sub bands is possible [5]. We use the same parameters for Ql D chargeJ b, ,�ons !or comparison. Figure 1 shows the interaction energy (per particle) for Q 1 D electro1 1 and charged boson

systems with harmonic potential parameter

n

=

we consider the electron system. Shown by the dashed line is the Hartree Fock <>r the exchange energy. Dotted and solid lines indicate the correlation energies in RPA :wd llubbard appro­

ximation, respectively. Similar correlation energies for the charged hos• 1 1 1 syst ,·m is illustrated

in Fig. 1 b. We observe that for the same Coulomb potential model, ti,,· ek, tron correlation energy is lower than that for charged bosons. although the many-body , · ,rrert ions to the RPA are roughly the same for both systems.

In Fig. 2, we display the static structure fact.or S(

q)

energies. RPA ( dotted lines) and self-consistent STLS approximation : solid lines) to S(q) in

a Q lD elec_tron system is depicted in Fig. 2a for 7'8 = 0.5 (lower cun · s) and 7•3

=

curves). Similar results of

S(q)

STLS scheme were reported'··by Borges el

al.

_S(q)

Q 1

D

In the calculation of ground-state energy of a double-layer Q I D . :(·ctron systems, rat.her than employing the full STLS scheme, we resort to the Hubbard appr. xirnat.ion for the former

E. H. TU RECI et al. : G RO U N D STATE CO RRECTIONS

is

B

G

r3

loo

_{k . q �;(k) HF}

]

Gii (q)

= - -

0 ...

- 1

(a)

0=5 Ry

-

-2

-3

-4

0.0

0.2

0.6

0.8

1.0

0

- 1

(b}

0=5 Ry

-

-2

-3

-4

0.0

0.2

0.6

0.8

1.0

Fig. I . - ( a) I nteract.ion energy per particle for a QJ D electron system in a confining potential. Solid , dotted, and dashed lines are calculated using Ga (q) ( Hubbard ) , G

=

=

1 39

(9)

The Hubbard approximation takes only the exchange into account where the Pauli hole is used for electrons in the same layer,. and Coulomb correlations are not included. Since

sr{

=

the

for a double-layer Q 1 D electron system in Fig. 3. The density dependence of

E

N

i

�

Fig. 3a for RPA (dotted line) and Hubbard approximation (solid line). We assume the density ..

of electrons on each layer is the same, and inter-layer distance is taken as d = q_{8 . Similar}

to _{the case of dou ble-layer 20 electron systems [13], magnitude of the RPA energy remains}

larger than that of H ubbard approximation. This is a consequence of RPA dielectric func­ tion's not incorporating the short-range correlations of electrons. Since we have not attempted

140

BALKAN PHYSICS LETTERS

slightly, and the difference is small suggesting the need of better approxim:itions. In

Fig. 3b,

the i nteraction energy per particle obtained from the Hubbard approximatian (sohd line) and the RPA (dotted line) are shown as a function of the layer separation. Th'..· energy attains 8 constant value for large separation distances which is about twice that for two independent layers. This implies that correlations between layers are important only for smal l distances.

1 .2 ... ---�---1 .0

0.8

0.4

0.2

0=5 Ry

(a)

0

4

10

qaB

0.8

'?i.f

o .. a

0=5 Ry

0

4

qaB

Fig.2. - (a) The static structure factor S(q) for elec­

trons. Solid and dotted lines indicate Hubbard appro­

ximation and

RPA,

(b)

bosons.

We have applied the self-consistent scheme of Singwi

et al.

state energies of Q l D electron and charged Bose systems. Corrections imroduced by local­

field factors to the interaction energy are approximately the same for

:Joth

approximation is used to investigate the density dependence of ground ::tatf' ,·nergy of a-�ouble­ layer Ql D electron system. We find that, to obtain reliable result�· beyc ,id the RPA, it is neces�ary t.o develop better approximations such as the full STLS scl: · · me. .\lthough we have

presented results of ground-stat.e energy calculations in Q I D system� for � :

=

t.

z

Ry

···

- 1 0

- 12 ___.__.__...._.__.__..___.__._.._.__._._.._.._.__,_._.__.__.

0.0

0.2

0.6

0.8

1.C

-6 ,_ ______ __,...,. ____ __,._. __________ __

Ry

-8

- 10

---

---·

·

- 12 ._.._._...,.,..._ ... _,_...._ ... _,_..._�_.__._..__.

0.0

0.2

0.4

0.6

0.8

1 .0

Fig.3. - (a) Interaction energy per particle as a func­ tion of electron density for a double-layer QI D elec­ tron system. Solid and dotted lines are for Hubbard approximation and RPA, respectively. The layer se­

paration is d

=

tion of layer separation, at r$

=

lines are for Hubbard approximation and RPA, re­ spectively.

Acknowledgements. - E. H . T. gratefully acknowledges a TUBITAK scholar h'

s lp.

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141

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