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