iFERENCE IC/91/294

/ c.-i

INTERNATIONAL CENTRE FORTHEORETICAL PHYSICS

MUTUAL SCATTERING OF CARRIERS

IN TWO QUANTUM WIRES

INTERNATIONALATOMIC ENERGY

AGENCY

UNITED NATIONSEDUCATIONAL,

SCIENTIFICAND CULTURALORGANIZATION

Yuri M. Sirenko

MIRAMARE-TRIESTE

IC/91/294

International Atomic Energy Agency

and

United Nations Educational Scientific and Cultural Organization

INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS

MUTUAL SCATTERING OF CARRIERS IN TWO QUANTUM WIRES

Yuri M. Sirenko*

International Centre for Theoretical Physics, Trieste, Italy.

ABSTRACT

We have investigated the momentum and energy exchange between two separated quan-tum wires due to Coulomb mutual scattering. Relevant relaxation frequencies were calculated forall possible combinations of statistics of the gases; dependence of the relaxation frequencies onconcentrations, temperature and distance between the gases was analyzed.

MIRAMARE - TRIESTE

September 1991

Permanent address: The Institute of Semiconductors, Ukranian Academy of Sciences, Kiev, Ukraine.

i. iNikODUCTION

Recently there appealed many theoretical and experimental works

on the Coulomb mutual scattering (CMS) etfect between the electron

gases in the layered heterostructure devices. The problem of CMS was

first proposed by trice who gave the simple analytical evaluation

of energy and momentum transfer between two two-dimensional (2D)

gases, separated by some distance 1. Jacobini and Price z made a

Monte*Carlo simulation ot the energy exchange between the electrons

in two parallel semiconductor layers in the absence of space

quantization.

In the works ot Hoptel and collaborators the first experimental

observation of energy and momentum transfer between electron and

hole plasma moving in the same GaAs quantum well was presented.

Those experiments found the theoretical explanation in Rets. 5. 6

were interaction between quasi-2JJ electron and hole gases occupying

the same quantum well was considered. The influence ot the space

separation between two mutually scattering 20 gases on the energy

and momentum transfer with an account for arbitrary statistics ot

the gases was made by Boiko and Sirenko " .

The problem of coupling between 2D and 3D electron gases was at

first considered by Boiko and Sirenko. They provided both classical

"*' and quantum ' theory calculations of the momentum and energy

transfer between 2D gas and semibounded 3D electron gas due to CMS.

First experimental investigation ot phenomena in coupled transport

between 2D and 3D electron gas layers was carried out bv Solomon <=>'-

al 1O, theoretical interpretation of the observed effects were done

fay Laikhtman and Solomon

Rapid development ot theoretical and experimental investiga-

tions ot transport in quantum wires makes the problem of CMS in ID

systems actual, in this preprint we consider the problem of mutual

scattering between two ID electron gases.

Our treatment is based on dielectric formalism for ID electron

gases proposed in Rets. 12, 13. in this formalism the quantum

kinetic equation for a system ot ID carriers is derived from the

first principles by averaging the equation for fluctuating micro-

scopic density operator and potential. This technique was proposed

by Klimontovich for the problems of 3D multi-component plasma 13. By

taking into account the space quantization in nonuniform media and

arbitrary scattering potential it was applied to 2D (Boiko and

Sirenko 7"°) and ID systems (Boiko, Sirenko and Vasilopoulos12"14 ) .

In deriving the kinetic equation so-called collisionless

approximation for the fluctuations of density operator was used.

This approximation is equivalent to the polarization approximation

in Bogolyubov's method13. This requires certain assumptions to be

hold . Firstly, the electron gas must be weakly nonideal. i.e.

plasma parameter must be small; secondly, the interaction between

two particles at small distances is treated in perturbative manner.

From the kinetic equation momentum and energy balance equations

were derived. The relevant energy and momentum relaxation rates,

with the help of model distribution functions, were expressed in

terms of the dielectric functions of electron gases and equilibrium

media and took a simple form suitable for practical applications.

In Sec. II we formulate the problem and present the basic equa-

tions for the frequencies of momentum end energy relaxation for

mutual scattering of ID gases. In Sec. Ill we give analytical evalu-

ation of the relaxation frequencies for different statistics of the

gases. Remarks and conclusion follow in the last section.

1 12 1 2

uT CT -T D12 1 2

(1)

(2)

Here K and are the friction force between the gases and power.

transferred from first gas to the second,

the carriers of the first gas. Notice that

is effective mass of

andT T

V Ti —Is Tt12 1 21 2

The linearization of R over xt -u and over 1 -~l\H over 1 l\ as

well as definition of relaxation frequencies axe justified ifu << T (see Ref. 12). Here fi<?|7 -T j << r/hw , T and

and hw are actual values of momentum and energy transfer for

collision between the carriers of different gases.

For the calculation of the relaxation frequencies we employ the

following expressions obtained within the dielectric formalism in

Ref. 12

12T n n J w J

13)Im Im

II. BASIC EQUATIONS

We consider the system of two ID gases of carriers (e.g. ID

electron and ID hole gases), scattering each other. We assume that

gases occupy the lowest subbands, exchange effects and intersubband

effects are out of consideration. The diagonal elements of the

density operators of the gases (the latter are denoted by index

i'1.2) are modelled by shifted Fermi-Dirac distribution functions

with effective temperatures T , T^ and drift velocities u , u .

The aim of this paper is to calculate the frequencies of

momentum and energy relaxation u™^ for mutual scattering between

the gases, defined by the equations

Here n^ is a linear concentration of the first sas, A£ q(tu>,(j) is a

dielectric function of the equilibrium ID electron gas (3ee

Appendix).

For the system of two ID gases the dielectric function

has the form

1- + — i - + As &£ I —

The dielectric functions of external system e^, s^ and s^ are

specified by the geometry of the external system (i.e. system

without electron gases being under consideration) and the wave

functions of caixiars in the transverse direction.

Let us specify the geometry of the system. We consider the

external system to be homogeneous and have lattice dielectric

constant s . ID electron gases occupy the quantum wells which adjoin

the s=o plane. The centers of the wells are at the distance L from

each other (see Fig. 1).

S [q11 r.

2 In U/X <? ) , V q « 1

2»T / X <? , X. e? >> 1

(8)

In this case we have (see Ref. 13)

I V

r

rJ

COS ? L d.qy y

* ( 9 > too v oo

Fig.1.

[5)

16)

Form-factors >' and * (we use the notations of Ref. 13) depend on

the form of quantum wells in s- and y- directions. For the simplifi-

cation of further calculations we make two assumptions: i) the width

X of quantum wells in a-direction is smaller than the actual

inverse wave vector- t?'1, then one can take F(<?) = 1; ii) we take para-

bolic potential in y-direction with effective widths K± and x tor

quantum wells, then

exp 2)Z] (7)

4" J cos q L ey

/ 2(9)

where 2 x = X + X^ .

Integral (9) can not be taken explicitly, we present it's

asymptotics;

BU " " / 2 | 2 K | ( X , / 2 ) J ) for X » L ;

for X « L, (10)

) e for / L/q « X </, L ,

One can write another asymptotic form for X<? « l (note, that

for 1DEG on the lowest subband one always has X«? < 1 ):

In(X2 + L*) c* '

LQ « 1 ;

til)

Y 2 n /Lq e ~ L<f* , Lq » X .

Here \=<fi/mto')*'2 , and w' specifies the parabolic confining

potential v j y)=mio' 2yz/2 .. Note, that inequality X;I

<< x4 • \

together with the assumption of gases on the lowest subband automa-

tically implies condition i).

Thus, from Eqa. (5) to (7) we obtain

Eqs. (7) to (11) are obtained for parabolic confining potential

in y-direction. Another reasonable model for confinement in

y-direction is that of square well. In this case instead of Eq. (7)

we have (cf. Ref. 13)

(g ) =- W2 1 - (12)

where W is a width of the i-th quantum well. Then 1/e iq ) s=

In (W. Q ) . and approximation (11) tor 1/£S.(<? ) would

applicable for a model of square well after replacement of x by W

be

In virtue of complicity of the obtained expressions we will

perform an approximate analytical treatment of relaxation frequen-

cies within an accuracy of order unity factor.

ill. ANALYTICAL EVALUATION OF RELAXATION FREQUENCIES

A. Possible cutoffs

Before evaluation of Eq.(3) for relaxation frequencies let us

consider the possible cutoffs in integration over w and Q.

Due to the factors 1m A r in integrand the contribution of

regions <j >> kt and <•> >> ^ are exponentially small (see

Appendix). Here k = (2:n L )1'2,-ti is inverse de Broglie length.

K)being

s12

the factor

v =fik /m. is thermal or Fermi velocity of ID gas,i » L x

is of the order of kinetic energy of carriers, r =n hFermi energy. Secondly, owing to the exponential dependence of c

on L in Eq.(ll) we have a possible cutoff at g " L

(fico/27' )/sinh( hw/27' ) in the integrand leads to cutoff at co ~l/h.

The last possible cutoff is related to perturbation theory tor

interaction at small distances, used in the derivation of Eq . O ) .

Perturbation theory breaks at the distances smaller than r^ - the

minimal distance which two particles with of same charge e, masses

, C can approach. Thus, one mustm. , m and kinetic energies

introduce the cutoff at <i q. , where

m t' +m £1 2 2 1

(13)2 m. +m

e 1 2

(in contrast to aforementioned cutoffs the cutoff at e?A is intro-

duced "deus »x machtna"),

Suitmarizing the possible cutoff values one finds

q - mxn ( k^

to = min ( T

, L

(15)

Now we consider three possible combinations of statistics of

the gases.

U. Two nondegenerate gases

For weakly nonideal gases for the actual values of w and q (

gases is negligible (see Appendix). Therefore we can take

Substituting (A2) into Eq.(3) we arrive at

12

t

1 2

_ r - r

X exp

where

After integration over w we find

12

Tv 7' T

Tl TZ 1 2

(16)

- min < kT, L . <?A > . If in conse-Here the cutoff wave vector

quence with (It) we have cutoff at q^ (this cutoff is not contained

in Eq.(16) but steins from nonperturbative analysis), the upper limit

in Eq.(16) should be changed to <f^.

Then evaluation of expression (16) gives

Eqs. (18) and (19) are in a good agreement with the general

expression (17).

12

T[i-s

1) )2v v r r

tl T2 X 2

(17)

One can see that v m T * <?z . It should be noted that the depe-

ndence of t>mT on cf for mutual scattering of two electron gases

is quite distinct for the different dimensionalities of the gases.

For example, for nondegenerate gases

m.T ,(ID X ID)

m, T

i'm'T(2D X 2D) « Q.

u (2D x 3D) >x (3D X 3D) « In

C. Nondegenerate ti"U and degenerate Ci»23 gases.

Substituting <A2) for Im Ac in Eq.(3) we find

co QO

P dig f dq v2 q1/ 2T l ^

Ime x p f . f „ .

2 )

1j .

( 2 0 )

If r ~7' -r one can rewrite the cutoff wave vector as

= mint L"1. kl=/2mr /h, ), where m ).

In two limiting cases the integration in Eq.(16) can be

performed more accurately.

In the region where ky << i-"1, i?A cutoff at q

essential. Here with logarithmic accuracy one can replace ^

by l/£^2<kT) >• (2/jft) ln(l/^kT). cf. Eq.(ll). Then integration in

Eq.(16) gives

kT is

12T12

cutoff at * i"In the region where L <<k , <?.

al. In virtue of Eq.{10) we take 1 ^ ^ ^

allowing for the integral J* x dxK^(x) = 1/2 we arrive at

(18)

is actu-

>l) and.

For Im As2 one can employ Eq.(A3), then the relaxation

frequencies will be given by the sum of contributions of regions A

and B (cf. Appendix):

m.T m.T n-l.T

A B

Region &. Inserting Im A£-'A>(u,c?) = (<?2<j/h)

Eq.(20) we obtain

12T 3

I-1 I , - , IT12 I(A) F2 Tl I

3 i n h Ur 2

/ 2 T2]

Evaluation of Eq.(21) gives

(21)

e n

u u T 7'Tl T2 1 2

(19)

hu u TF2 Tl 1

2FZ

(22)

10

Here we have used the cutoff wave vector <? = min (27 /hv , t ',

kTi, <?A) ; screening factor is

[, 2 '

It X1 + 2 In [l+l/(\ttl?J J

(23)

screening by the degenerate gas is essential if q < m /£ f,2.

In the particular case where cutoff at c? ' 27' /hv is actual

and screening by the electrons is negligible (i.e. e2m /t*s «

27'2/hvFj « L1, kTt, i?A ) the integration in Eq.(22) can be

performed more accurately (with logarithmic accuracy). With the help

of integral SQ x dx/ sinh x = 7r(3)/2 , where C is Hiemann function,we obtain

12T

(A)7T I (A +I_ ) (fci I £ f w V 2

L 2 J L F Z T 1 1

-(u /vFZ T l

4- v-r •*

(24)

fi. Substituting 1m

2 n v v I- TF2 Tl 2 1

exp "I"(25)

11

Here © (x) % 1 for x « 1, and ®(x) is exponentially small for

x « 1. Factor e( 2kk /<?*'

the cutoff at Q ~ <?A<Approximate integration of Eq.(25) gives

introduced artificially to allow for

12T

to X

( B ) F 2 T l7' i £ (co . 2k ) I

2 1 ' 12 I- FZ '(26)

e -

where « = min (TJti, 2kF2uTl> ; the screening factor is

Under the assumptions of n « 1 (weakly nonideal gas) and r •> i

(degeneracy) the last terra is small, i.e. the contribution of dege-

nerate gas in region B into the screening is small in comparison

with that of lattice. Note, that due to the term i^Uk^.) in the

denominator, the relaxation treaMencies are exponentially small for

L » l/2krj! (cf. (12) ).

2' /h is actual (i.e. 2 /hIn the case where cutoff at

2k v , the opposite case is rather exotic) and 2k « ^., theF2 Tl F2 !\

integration in Eq.(2S) can be performed more accurately. Allowing

for weak dependence of ei2tw,c?) on <J and employing the iden-

tities X^xdx/sinh x = na/4, J~0Dx3dx/sinh x = n4/8 . we find

12T

VJI

1 2 J ( B ) F2 Tl 2

12

n of the gases) the region A of one gas (see

U. T*o nondegenerate gases

The situation where (due to enormous difference in concentrati-

ons n andt

Appendix) overlaps with the region B of the other does not seem to

be realistic. Therefore we consider only the case where the region A

of one gas is disposed in the vicinity of the region A of the other

gas, and the same situation with the regions B. If the regions are

disposed far apart the relaxation frequencies given by Eq.(3) willbe exponentially small <:-' oc e"F'T). Thus, \j"''r=^ ^^'^ u^*•

Regions A.. The relaxation frequencies are not exponentially

small if I>FI * VF2 , or being more accurate, when

\v - v 1/ v < T /F + T /r « 1 .

' Wi FX' Fl " 1 1 2 2

I n t e g r a t i o n o v e r <? and to i n E q . ( 3 ) a r e c u t off a t <?c « min(i- ,

T/hxi , o. ) and fj ~t u (here T = min(T ,T ), v = u i v ). TheF A (. c F 1 2 F Fl FZ

maximal possible contribution of regions A into integral (3),

achieved at u = v , equals to<(max)

12T12 J<A)

Here ^ is given by Eq.(d), where Jac. <<j )j ~ A

B. The contribution of regions B ( -^ 2kFL) into

Eq,(3) is not exponentially small only if

To be more accurate, the condition

is required. The frequenciesm.T are maximal for the exact

equality n - n . The evaluation of Eq.<3> gives

13

m 1 (max)12T

:. (B) •]]X F Ci

6

Here n =n.=n ; the factor ®(2k /g ) allows for possible cutoff

at Q ~<?».

IV. DISCUSSION

In this paper we have investigated the momentum and energy

exchange between two separated quantum wires due to Coulomb mutual

scattering. Relevant relaxation frequencies ware calculated for all

possible combinations of statistics of the gases; dependence of the

relaxation frequencies on concentrations, temperature and distance

between the gases was analyzed.

Some remarks are to be made about the screening of the

interaction between gases. In contrast to the ease of weakly

nonideal ID electron gas scattered by the static potential where the

contribution of screening appears to be insignificant12, in this

problem the screening can be important even for the gases with a

small plasma parameter /see e.g. Eq.(23)/.

Comparing our results with that for mutual scattering of 3D

gases and 2D gases we find that in the case of nondegenerate

gases" in 3D case the cutoff at <? ~ <? eliminates the logarithmic

divergency in the integration at large g (small distances between

scattering particles), in 2D case the relevant relaxation

frequencies are proportional to the first and in ID case to the

second power of cutoff wave vector g .

Due to the strong restrictions on the process of scattering in

ID case imposed by conservation laws in some cases the relaxation

frequencies appeared to be exponentially small <see Sec. III.C and

III.D), Some comments are to be made on this point.

The calculation of the relaxation frequencies was performed on

the base of Eq.(3) which was obtained within polarization

14

approximation - only two-particle interactions (screened by the

rest of the particles) were allowed for. The picture may change

drastically with the allowance for three-particle interactions

(forth order of the perturbation theory, v"1-T « <?") and Eq.(3) mayivt T

become invalid in the cases where v ' are exponentially small.

ACKNOWLEDGMENTS

The author would like to thank Professor Abdus Salam. the

International Atomic Energy Agency and UNESCO for hospitality at the

International Center for Theoretical Physics, Trieste, where this

work was completed. I would also like to thank Professor I.I.Boiko

for discussion of this work.

Appendix

Below we present some expressions for dielectric function

AE(«,Q) of equilibrium ID electron gas. Considering one sort of

particles occupying the lowest subband we can write'2'13

**«•>.*> - ~ Jdk ' - ' ' - • ,

- I - F + 1 0(fti)

where £ =fikV2», /' is a distribution function of ID gas, taken

in Fermi-Dirac form.

For nondegenerate electrons one can find from (AD

1m

Analysis of the real part of Ac for nondegenerate electrons

shows that for w < g-u (a condition appropriate for transport

problems) | Re A£(to,g) | < <? n/T = e r; , where 17 is a plasma para-

meter. For weakly rionideal electron gas ri « 1 , so for w < <ju_ we

have I Re Attto.o) i < e >] « £ , i.e. the screening by theI ' - L L

rons is weak.

eleot-

For the degenerate electron gas at 1=0

?zm f 1

2 ~ i T?1 <? I

In(2k

(2k -<(A3)

sgn

where 3(x<y<z) = 1 for x<y<z, and equals zero otherwise.

Due to the factor (fW22')/sinh(tW2D in Eq.O) only the

region

0 < m <. l/h « i'/i\ gives the contribution to the integrals. In this

case for T •< r we can write a useful approximation

Im

which shows that Im A^ equals to zero outside of two narrow

regions, namely, region A ( u % gvF ), and region B ( Q ^ L.'kF --

16

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17