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Magnetoconductance of a disordered interacting 2-D electron gas
2(}N4
Physica l(I t) & I I()B (19<"42) 2()S4 21)~ N o r t h - H ~ l h i n d Puhlishing ('onlpan\
MAGNETOCONDUCTANCE A. H O U G H T O N ,
OF A D I S O R D E R E D INTERACTING 2-D ELECTRON GAS
J.R. S E N N A and S.('. Y I N G
Department of Physics', Brown Univers'itv, Proridence, Rhode Island 02912, USA
We calculate the corrections to the resistance R and Hall resistance RH of a two-dimensional disordmcd elecironi<_ system due to interactions in the strong field limit ~o~< ~oF, ,~'l 1. where localisation effects are suppressed. We find that -br~, 0. With the result that (aR,,URn)/(~SR/R) : 2/( 1 (~o
In this work we study the effect of strong magnetic fields w< < e~<, e v r > 1 (here r is the ehtstic scattering time, e~: the F e r m i energy a n d w< the cyclotron f r e q u e n c y ) on a disordered, interacting, t w o - d i m e n s i o n a l electronic system. In this field r a n g e as it is k n o w n that Iocalisation effects [ 1. 2] are suppressed by modest m a g n e t i c fields ~o
are the s c r e e n e d i n t e r a c t i o n V~(q, w ) and h e n c e the
polarisability
the
l I ( q , oJ).
particle-hole
diffusion p r o p a g a t o r DH(q, w), a n d the impurity r e n o r m a l i s e d vertices of the C o u l o m b i n t e r a c t i o n I'H(q,~o), which are easily found when DH is k n o w n . T h e polarisability of the i n t e r a c t i n g sys-
(1~eFt) (r/rin~ia~t), ri,,el~, is the inelastic scattering time, we may focus on e l e c t r o n - e l e c t r o n inter-
action only, t r e a t i n g i m p u r i t y scattering by the c o n v e n t i o n a l d i a g r a m m a t i c t e c h n i q u e of the weak
2( ¸
~
~,
;
scattering limit. T h e i n t e r a c t i o n , which is t a k e n to be the d y n a m i c a l l y s c r e e n e d C o u l o m b i n t e r a c t i o n , is t r e a t e d in lowest order.
) <
-
,
?-
(b)
T h e central q u a n t i t y is the m a g n e t o c o n d u c tivity t e n s o r o': the p r e s e n t t e c h n i q u e , in addition to e x t e n d i n g p r e v i o u s calculations [2] to
,
higher m a g n e t i c fields, allows s i m u l t a n e o u s calculation of both the l o n g i t u d i n a l o-.... a n d transverse o-~ conductivities. T h e F e y n m a n d i a g r a m s c o n t r i b u t i n g to the c o n d u c t i v i t y t e n s o r are shown in fig. 1. T h e s e diagrams are g e n e r a t e d in a c o n s e r v i n g a p p r o x i m a t i o n from the e x c h a n g e c o n t r i b u t i o n to the electron self-energy. It can be shown that the c o n t r i b u t i o n s of diagrams (a), (b) and (c) to both o-<,, and cr~ exactly cancel each other. Furthermore. the Aslamazov-Larkin diagrams, (f) a n d (g), do not c o n t r i b u t e to the conductivity. It r e m a i n s t h e r e f o r e to d e t e r m i n e the c o n t r i b u t i o n s of d i a g r a m s (d) a n d (e). T h e i m p o r t a n t c o m p o n e n t s of these diagrams )37S-4363/82/(R)(t()-01)li0/$()2.75 © 1982 N o r t h - H o l l a n d
(ci
s,
"l~
•7,#/
:t'
=~
i [
,,~
~-
i
(e)
/
('?
(g) Figure
]
b.
A . Houghton et al. / Magnetoconductance of a disordered interacting 2-D electron gas
tem in the presence of disorder is given by
where ~b, are harmonic oscillator functions, e, = (n + 1/2)htoc, and a = 1/mto¢. Impurity scattering is included via a relaxation time which is energy dependent,
0o
H(x, x', w) = ~ i
de (f(e + w ) - f ( e ) )
× El(x, x'; e + to, e ) + N l ( e v ) ,
(1)
here N1(eF) is the density of states of a single spin species and El(x,x', e +to, e) is the propagator of a particle with energy e + ~ and a hole with energy e:
H(x, x', e + to, e) = (G++~(x, x ' ) G ; ( x ' , x ) ) .
2085
(2)
In eq. (2) G + and G - are the advanced and retarded single particle G r e e n ' s functions, respectively, and ( ) indicates average over disorder. In the weak scattering limit H satisfies the Dyson equation
1/r(e)
= 27rNl(E)u 2 ,
(6)
and inversely proportional to the density of states which oscillates as a function of magnetic field. In the strong field limit, since all quantities are evaluated at the Fermi surface, all the field dependence of the theory is retained if this relationship is kept in mind [4]. The bare polarisability H°(q, to) is found from eqs. (2) and (5) as
II°(q, w) = p ~ IF,,,(q)pD,,, ,
(7)
nn'
where
El(x, x', to) : r/°(x, x', to) q.. "//2 E /-/0(x' XI' to)//(Xl' X', to).
F,,,(q) = f dx~* eiqx~b,(x),
Xl
(3) D,,, = 1/(e + w - e, + i/2r)(e - e, - i/2r), The particle-hole diffusion propagator is defined by the Dyson equation
D(x, x', to) = u28x,x' + u 2 ~'~ H°(x, xl, to)D(xb x', to).
(4)
the Landau level degeneracy p = 1/27ra and q =
~/ q~ + q~. Expanding for small (q, w) and performing the sum over Landau levels we find
Xl
H°(q, to) = 27rNl(ev)r[1 + iwr - DHq2r], The basic building block of both quantities is the bare " b u b b l e " El°(x, x', to). We work in the Landau gauge, with a uniform magnetic field in the z direction chosen perpendicular to the two-dimensional system. The eigenstates of the noninteracting system are labelled by a Landau level index n and a wave vector k in the y direction. The G r e e n ' s functions are given by
G±(x, x', e) = ~ dp,(x + ka)ck*(x' + k a ) eit(y_y,) k,n e -- e, + i/2r
(5)
(S)
where the diffusion constant DH
D n = evr/m (1 + (to¢r) 2)
(9)
reduces to the usual result in the weak field limit, but oscillates as a function of field via r. As noted, to determine the corrections to the conductivity &rxx and 8O-xy due to interaction we need only consider diagrams (d) and (e) of fig. 1. The most important result is that the contribution of these diagrams to the transverse conductivity, and hence of interactions to the transverse
A. Houghton et al. / Magnetoconductance of a disordered interacting 2-D electron gas
conductivity, is identically zero. F u r t h e r m o r e , in the longitudinal conductivity, most of the magnetic field d e p e n d e n c e cancels having a result of the same form as the weak field case e= l o g , 0 , ~$o~,, 277.2h
(1(~)
where ,0 = ~ r . W h e n H a r t r e e and spin splitting effects are included [2, 3]
60-,,
2~-5h [1 - F] log ~),
(11)
F is field d e p e n d e n t [31. T h e quantities directly m e a s u r e d by experiment are the magnetoresistance, R - P .... and the Hall constant, RH = px,./H. In the strong field limit considered here, localisation effects are completely suppressed and we need only take into account interaction effects. As 80.~, (~ we easily find 6R/R(, : -[m/(2rr=hnr)]( l
F)[1
(w~r):] log *).
aR/Ru = -[2m/(27rehnr)]( l - F) log/).
(12)
N o t e the c h a n g e in sign of a R / R , , at ,0~r -- 1. T h e ratio of these two quantities 6RH/R(~ _ 2 6R/R ( I - (o),.r)2) "
be included, the situation is m o r e complicated. In this case the zero field limit ~)f the ratio (6RH/R'h)/(6R/R) becomes (2 - 2 F ) / ( 2 - F ) which varies from the value l when F = 0 (in the limit k v / K - - , z , K is the inverse T h o m a s - F e r m i screening length) to the value zero for F I ( k v / K - , 01, For small but finite fields, the localisation and spin splitting contribution to the Hartree term do not have a pure l o g . 0 or l o s T form. A formula such as eq. (13) is less useful in this case than a direct fit of 6R or 6RH with the k n o w n d e p e n d e n c e on both t e m p e r a t u r e and magnetic field. W e should note that recent experiments [5, 6] appear to give (6RH/Ru)/(6R/R) close to 2 in the weak field limit; however, these experiments are d o n e in the region of field strengths where both Iocalisation and interaction are expected to contribute. and care is needed to interpret the data.
Acknowledgements This work is supported by the Materials Research L a b o r a t o r y at Brown University funded t h r o u g h the National Science Foundation. the National Science F o u n d a t i o n u n d e r grant no. NSF DMR-2{1321 and by the Office of Naval Research.
(13)
References which is close to 2 for o)~.r<~ I. H o w e v e r , the ratio eq. (13) diverges as ~ocr a p p r o a c h e s 1 and then changes sign. It also oscillates as a function of field because of the intrinsic field d e p e n d e n c e of r. Detailed m e a s u r e m e n t of the field depend e n c e of these quantities would be an important test of the interactions plus weak scattering theory. At very weak magnetic fields where Iocalisation effects are still present and both effects must
II1 E. Abrahams. P.W ,knderson. D.('. l,icciardelh~ and T.V. Ramakrishnam Phys. Rev. I_ett, 39 (1~t77) I1(~7. 121 B.L. Altuschuler, D. Khmel'nitzkii, A.I. I,arking and P.A. l~ee. Phys. Rev. B 22 (D)NI)) 5142. [3] P.A. t,ce and T.V. Ramakrishnan, Bell [.aboratorics prcprint (1974). [41 T. Ando, J. Phys. Soc. Jap. 37 (1~74) ~322H, I51 D.J. Bishop. D.('. Tsui and R . C Dynes, Phys. Rev. 1 ell 44 (It)80) 1153. [e,] M..I. tJren. R.A. Daniel and M. Pepper, .I. Phys. ( 13 (198(I) l_~)SS.
Physica l(I t) & I I()B (19<"42) 2()S4 21)~ N o r t h - H ~ l h i n d Puhlishing ('onlpan\
MAGNETOCONDUCTANCE A. H O U G H T O N ,
OF A D I S O R D E R E D INTERACTING 2-D ELECTRON GAS
J.R. S E N N A and S.('. Y I N G
Department of Physics', Brown Univers'itv, Proridence, Rhode Island 02912, USA
We calculate the corrections to the resistance R and Hall resistance RH of a two-dimensional disordmcd elecironi<_ system due to interactions in the strong field limit ~o~< ~oF, ,~'l
In this work we study the effect of strong magnetic fields w< < e~<, e v r > 1 (here r is the ehtstic scattering time, e~: the F e r m i energy a n d w< the cyclotron f r e q u e n c y ) on a disordered, interacting, t w o - d i m e n s i o n a l electronic system. In this field r a n g e as it is k n o w n that Iocalisation effects [ 1. 2] are suppressed by modest m a g n e t i c fields ~o
are the s c r e e n e d i n t e r a c t i o n V~(q, w ) and h e n c e the
polarisability
the
l I ( q , oJ).
particle-hole
diffusion p r o p a g a t o r DH(q, w), a n d the impurity r e n o r m a l i s e d vertices of the C o u l o m b i n t e r a c t i o n I'H(q,~o), which are easily found when DH is k n o w n . T h e polarisability of the i n t e r a c t i n g sys-
(1~eFt) (r/rin~ia~t), ri,,el~, is the inelastic scattering time, we may focus on e l e c t r o n - e l e c t r o n inter-
action only, t r e a t i n g i m p u r i t y scattering by the c o n v e n t i o n a l d i a g r a m m a t i c t e c h n i q u e of the weak
2( ¸
~
~,
;
scattering limit. T h e i n t e r a c t i o n , which is t a k e n to be the d y n a m i c a l l y s c r e e n e d C o u l o m b i n t e r a c t i o n , is t r e a t e d in lowest order.
) <
-
,
?-
(b)
T h e central q u a n t i t y is the m a g n e t o c o n d u c tivity t e n s o r o': the p r e s e n t t e c h n i q u e , in addition to e x t e n d i n g p r e v i o u s calculations [2] to
,
higher m a g n e t i c fields, allows s i m u l t a n e o u s calculation of both the l o n g i t u d i n a l o-.... a n d transverse o-~ conductivities. T h e F e y n m a n d i a g r a m s c o n t r i b u t i n g to the c o n d u c t i v i t y t e n s o r are shown in fig. 1. T h e s e diagrams are g e n e r a t e d in a c o n s e r v i n g a p p r o x i m a t i o n from the e x c h a n g e c o n t r i b u t i o n to the electron self-energy. It can be shown that the c o n t r i b u t i o n s of diagrams (a), (b) and (c) to both o-<,, and cr~ exactly cancel each other. Furthermore. the Aslamazov-Larkin diagrams, (f) a n d (g), do not c o n t r i b u t e to the conductivity. It r e m a i n s t h e r e f o r e to d e t e r m i n e the c o n t r i b u t i o n s of d i a g r a m s (d) a n d (e). T h e i m p o r t a n t c o m p o n e n t s of these diagrams )37S-4363/82/(R)(t()-01)li0/$()2.75 © 1982 N o r t h - H o l l a n d
(ci
s,
"l~
•7,#/
:t'
=~
i [
,,~
~-
i
(e)
/
('?
(g) Figure
]
b.
A . Houghton et al. / Magnetoconductance of a disordered interacting 2-D electron gas
tem in the presence of disorder is given by
where ~b, are harmonic oscillator functions, e, = (n + 1/2)htoc, and a = 1/mto¢. Impurity scattering is included via a relaxation time which is energy dependent,
0o
H(x, x', w) = ~ i
de (f(e + w ) - f ( e ) )
× El(x, x'; e + to, e ) + N l ( e v ) ,
(1)
here N1(eF) is the density of states of a single spin species and El(x,x', e +to, e) is the propagator of a particle with energy e + ~ and a hole with energy e:
H(x, x', e + to, e) = (G++~(x, x ' ) G ; ( x ' , x ) ) .
2085
(2)
In eq. (2) G + and G - are the advanced and retarded single particle G r e e n ' s functions, respectively, and ( ) indicates average over disorder. In the weak scattering limit H satisfies the Dyson equation
1/r(e)
= 27rNl(E)u 2 ,
(6)
and inversely proportional to the density of states which oscillates as a function of magnetic field. In the strong field limit, since all quantities are evaluated at the Fermi surface, all the field dependence of the theory is retained if this relationship is kept in mind [4]. The bare polarisability H°(q, to) is found from eqs. (2) and (5) as
II°(q, w) = p ~ IF,,,(q)pD,,, ,
(7)
nn'
where
El(x, x', to) : r/°(x, x', to) q.. "//2 E /-/0(x' XI' to)//(Xl' X', to).
F,,,(q) = f dx~* eiqx~b,(x),
Xl
(3) D,,, = 1/(e + w - e, + i/2r)(e - e, - i/2r), The particle-hole diffusion propagator is defined by the Dyson equation
D(x, x', to) = u28x,x' + u 2 ~'~ H°(x, xl, to)D(xb x', to).
(4)
the Landau level degeneracy p = 1/27ra and q =
~/ q~ + q~. Expanding for small (q, w) and performing the sum over Landau levels we find
Xl
H°(q, to) = 27rNl(ev)r[1 + iwr - DHq2r], The basic building block of both quantities is the bare " b u b b l e " El°(x, x', to). We work in the Landau gauge, with a uniform magnetic field in the z direction chosen perpendicular to the two-dimensional system. The eigenstates of the noninteracting system are labelled by a Landau level index n and a wave vector k in the y direction. The G r e e n ' s functions are given by
G±(x, x', e) = ~ dp,(x + ka)ck*(x' + k a ) eit(y_y,) k,n e -- e, + i/2r
(5)
(S)
where the diffusion constant DH
D n = evr/m (1 + (to¢r) 2)
(9)
reduces to the usual result in the weak field limit, but oscillates as a function of field via r. As noted, to determine the corrections to the conductivity &rxx and 8O-xy due to interaction we need only consider diagrams (d) and (e) of fig. 1. The most important result is that the contribution of these diagrams to the transverse conductivity, and hence of interactions to the transverse
A. Houghton et al. / Magnetoconductance of a disordered interacting 2-D electron gas
conductivity, is identically zero. F u r t h e r m o r e , in the longitudinal conductivity, most of the magnetic field d e p e n d e n c e cancels having a result of the same form as the weak field case e= l o g , 0 , ~$o~,, 277.2h
(1(~)
where ,0 = ~ r . W h e n H a r t r e e and spin splitting effects are included [2, 3]
60-,,
2~-5h [1 - F] log ~),
(11)
F is field d e p e n d e n t [31. T h e quantities directly m e a s u r e d by experiment are the magnetoresistance, R - P .... and the Hall constant, RH = px,./H. In the strong field limit considered here, localisation effects are completely suppressed and we need only take into account interaction effects. As 80.~, (~ we easily find 6R/R(, : -[m/(2rr=hnr)]( l
F)[1
(w~r):] log *).
aR/Ru = -[2m/(27rehnr)]( l - F) log/).
(12)
N o t e the c h a n g e in sign of a R / R , , at ,0~r -- 1. T h e ratio of these two quantities 6RH/R(~ _ 2 6R/R ( I - (o),.r)2) "
be included, the situation is m o r e complicated. In this case the zero field limit ~)f the ratio (6RH/R'h)/(6R/R) becomes (2 - 2 F ) / ( 2 - F ) which varies from the value l when F = 0 (in the limit k v / K - - , z , K is the inverse T h o m a s - F e r m i screening length) to the value zero for F I ( k v / K - , 01, For small but finite fields, the localisation and spin splitting contribution to the Hartree term do not have a pure l o g . 0 or l o s T form. A formula such as eq. (13) is less useful in this case than a direct fit of 6R or 6RH with the k n o w n d e p e n d e n c e on both t e m p e r a t u r e and magnetic field. W e should note that recent experiments [5, 6] appear to give (6RH/Ru)/(6R/R) close to 2 in the weak field limit; however, these experiments are d o n e in the region of field strengths where both Iocalisation and interaction are expected to contribute. and care is needed to interpret the data.
Acknowledgements This work is supported by the Materials Research L a b o r a t o r y at Brown University funded t h r o u g h the National Science Foundation. the National Science F o u n d a t i o n u n d e r grant no. NSF DMR-2{1321 and by the Office of Naval Research.
(13)
References which is close to 2 for o)~.r<~ I. H o w e v e r , the ratio eq. (13) diverges as ~ocr a p p r o a c h e s 1 and then changes sign. It also oscillates as a function of field because of the intrinsic field d e p e n d e n c e of r. Detailed m e a s u r e m e n t of the field depend e n c e of these quantities would be an important test of the interactions plus weak scattering theory. At very weak magnetic fields where Iocalisation effects are still present and both effects must
II1 E. Abrahams. P.W ,knderson. D.('. l,icciardelh~ and T.V. Ramakrishnam Phys. Rev. I_ett, 39 (1~t77) I1(~7. 121 B.L. Altuschuler, D. Khmel'nitzkii, A.I. I,arking and P.A. l~ee. Phys. Rev. B 22 (D)NI)) 5142. [3] P.A. t,ce and T.V. Ramakrishnan, Bell [.aboratorics prcprint (1974). [41 T. Ando, J. Phys. Soc. Jap. 37 (1~74) ~322H, I51 D.J. Bishop. D.('. Tsui and R . C Dynes, Phys. Rev. 1 ell 44 (It)80) 1153. [e,] M..I. tJren. R.A. Daniel and M. Pepper, .I. Phys. ( 13 (198(I) l_~)SS.