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RESISTIVITY OF A DISORDERED
TWO-DIMENSIONAL ELECTRON GAS UNDER MAGNETIC FIELD
S. Ying, I.C. da Cunha Lima
To cite this version:
S. Ying, I.C. da Cunha Lima. RESISTIVITY OF A DISORDERED TWO-DIMENSIONAL ELEC-
TRON GAS UNDER MAGNETIC FIELD. Journal de Physique Colloques, 1984, 45 (C5), pp.C5-515-
C5-518. �10.1051/jphyscol:1984576�. �jpa-00224197�
JOURNAL DE PHYSIQUE
Colloque C5, suppl6ment au n04, Tome 45, avril 1984 page C5-515
RESISTIVITY OF A DISORDERED TWO-DIMENSIONAL ELECTRON GAS UNDER MAGNETIC FIELD-.
+ r S.C. Ying and I.C. da Cunha Lima
Dept. of Physics, Brown U n i v e r s i t y , Providence, R.I., U.S.A.
* ~ n s t i t u t o de Pesquisas Espaciais, INPE/CNPq, S . J . C a p o s , BraziZ and Dept
.
of Physics, Brown University, Providence, R . I . , U . S.A.Rdsumd
-
On obtient la moyenne sur les impuretds de la rssistivitd d'un gaz d'dlec- trons sous un champ magnstique avec la technique de la fonction mdmoire-opdrateurs de projection de Mori. L'hamiltonien est transforms en coordonndes relatives et de centre-de-masse. La fonction mgmoire est exprimge en termes de la fonction de cor- rdlation force-force.Abstract
-
The impurity averaged resistivity for an electron gas under external mag- netic field is obtained directly by using Mori's memory function-projector technique.The Hamiltonian is transformed into center-of-mass and relative coordinates. The memory function is expressed in terms of the force-force correlation function.
The transport properties of a quasi-two-dimensional electrons in superlattice inter- faces and inversion layers under a strong D.C. magnetic field have been studied ex- tensively. The existing theoretical works focus on the Hall and longitudinal con- ductivities while the experimentally measured quantities are actually p and p
XX XY' It is not clear whether the direct inversion of the conductivity tensor g, which has been averaged over all impurity configurations, yield the appropriate quantity for comparison with the experiments. In this paper we show that the impurity averaged resistivity for an interacting electron gas under external magnetic field can be obtained directly by using Mori's projection operator technique.
We start with the Hamiltonian for an electron gas under an external field given by the vector potential i A, and in the presence of impurities whose scattering potential is given by the Fourier transform u(q):
-+ + -+ -+ -+
H I . ( i ) 1 2 + ( r - r . u(q)exp[iq.(r.-R1)l (1) 2m I C
i> j J q.j.1 J
Transforming the Hamiltonian above into center-of-mass and relative coordinates 111,
where N is the total number of electrons, i i P and R are the momentum and coordinate of the center-of-mass,
G'
is the relative coordinate of the electron and M=Nm. Thej
second term of the Hamiltonian contains the Coulomb interaction among electrons which depends only on their relative coordinates. The conductivity tensor can be
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1984576
C5-516 JOURNAL DE PHYSIQUE
with
Q (t) = -ie(t) < [fla(t),n6(0)l- >
a6 ( 4 )
-f -f
If we choose the gauge A(R) = (O,HX,O) we have IIx = -ia/aX and II = -ia/aY
+
MwcXY
where w = eH/mc is the cyclotron frequency. In terms of the Laplace transform X36 (z) defined by
+m
6 (z) = i / exp(izt) < [fla(t),IIB(0)]- >
,
Im z > 0 (5)the conductivity can be expressed as i~e' ie2 oaB(~) = 6aB -
& xaB(')
Next we will use Mori's memory function-projector operator technique [ 2 ] and obtain the expressions for the matrix elements of the resistivity. Let us define the momentum-momentum correlation function following Forster's notation
It is easy to show that its Laplace transform C (z) can be related to
x
(z) ina 6 aB
Eq. (5) by
x a ~ (z) = iBzC (z)
+
X ~ ~ ( O + ~ O + )a6 (8)
Bringing Eq.(8) into Eq.(6), and noticing that the conductivity remains finite as w -f 0, we obtain
x
(OiiO+
) = M6aB anda0
Next we choose Il and I1 as the primary variables to express the projection oper-
x Y
ators P and Q = 1-P:
P = M
1 n 1 + 1 n >ay 1 I
Y (10)
The dynamics of + TI is given by
" = --Wny
-
Uxwhere -U is the generalized force acting on the center-of-mass due to scattering by impurities, i.e.,
u a u l a ~ ~
= I iq e~p[i~.~~]u(q)p(;) (11) qiR+ -f + +
where p(q) is the density operator, p(q) = Cexp(iq-r.). The correlation function
j J
$(z) obeys the equation [ Z ] :
[z;-Q+i;(z)] $ 2 ) = i B -1
x
,I,
where B = (kgT) -1 and Q and are given by
and L is the Liouville operator L$ = [ H , $ ] -
Using the commutation relations for Il and Eq.(l3a) we obtain -f-
and <U
"
IIl B > = 0 for any a and 6.
Except for the exclusion of the zero total momentum state, we assume that the dynam- ics of the generalized force is governed by the full Liouville operator,
exp(itQL) Iua> = exp(itL) / U > = /ua(t) >
Expanding the operator (z - QLQ) -1 we obtain
Now we can follow the steps done between Eqs.(7) and (8) and, if we define the force-force correlation function as
we obtain
M ( w + i ~ ) = - - 2
+
a B Nmw
The calculation of the longitudinal conductivity through an averaged memory func- tion needs hardly justifiable approximations concerning the inverse of a configur- ational average over impurities. However, if we invert Eqs.(9) and (12), and later perform the average, we obtain the resistivity:
imz im
- + (2)
=
- 3
6aG + S ' c t B Ne aB (19) where the bar represent the configurational average. For low impurity concentra- tion a standard approximation leads to 111(20)
C5-5 18 JOURNAL DE PHYSIQUE
where ni is the impurity density and S(q,w) is the retarded density-density + propagator
-f +m
+ +
S(q,w) = -i ie(t)< [~(q,t), P(-q,O)l->
-m
Then we have, finally,
and
'L -R
where ll (w) =
iR
(w) - naB(0).a B aB
'L
Ting et a1 [I] have calculated If (a) for a 3-D electron gas. Their first term,
XX
corresponding to the lowest order diagram in a perturbation expansion, is enough to generate Drude's result in Eq.(22). Higher order terms have to be handled with care since zero momentum states for the electron system have been projected out through the approximation involved in Eq.(15).
In the case of a 2-D interacting electron gas we can follow basically similar steps used by Houqhton et a1 [4] in their calculation of the conductivity. There they calculated o($,w) using diagrammatic expansion based on Landau quasi-particle Green's function, and in the limit q + 0. In our case, however, the summation on
->
q that appears in Eq.(20) gives a different selection of the diagrams involved in the force-force correlation function. That part will be presented in a future publication.
'%ark
partially supported by a grant from CNPq(Brazil)/NSF(USA) under contract 88119267+ ~ ~ ~ ~ ~ / ~ u l b r i ~ h t Scholar; later under a CNPq post-doctoral fellowship.
[I] Ting C. S., Ying S. C. and Quinn J. J., Phys. Rev.
B14
(1976) 4439.[2] Forster, D. in Hydrodynamic Fluctuations, Broken Symmetry and Correlation Functions, The Benjamin/Cummings Publishing Co., Inc. (1975).
[3] Gotze W. and Wolfle P., Phys. Rev. (1972) 1226.
[4] Houghton, A., Senna J. R., and Ying S. C., Phys. Rev. B25 (1982) 2196.
