Coulomb drag in intermediate magnetic fields

Coulomb drag in intermediate magnetic fields

A. V. Khaetskii

Institute of Microelectronics Technology, Russian Academy of Sciences, 142432, Chernogolovka, Moscow District, Russia Yuli V. Nazarov

Faculty of Applied Sciences and DIMES, Delft University of Technology, Lorentzweg 1, 2628 CJ Delft, The Netherlands ~Received 31 August 1998!

We theoretically investigated the Coulomb drag effect in coupled two-dimensional electron gases in a wide interval of magnetic field and temperature 1/t!vc!EF/\, T!EF, t being the intralayer scattering time

andvc the cyclotron frequency. We show that quantization of the electron spectrum leads to rich parametric

dependences of the drag transresistance on the temperature and magnetic field. This is in contrast to usual resistance. Small energy scales are found to cut typical excitation energies to values lower than temperature. This may lead to a linear temperature dependence of the transresistance even in a relatively weak magnetic field, and can explain some recent experimental data. We present a mechanism of Coulomb drag when the current in the active layer causes a magnetoplasmon wind and the magnetoplasmons are absorbed by the electrons of the passive layer providing a momentum transfer. We derived general relations that describe the drag as a result of resonant tunneling of magnetoplasmons.@S0163-1829~99!07211-2#

I. INTRODUCTION

When two two-dimensional electron systems are placed in close proximity, then even in the absence of electron tunnel-ing between layers the current in one layer~the active layer! will cause the current in the other ~the passive layer!.1 This phenomenon is known as a frictional drag, and is due to the interlayer Coulomb interaction which causes a momentum transfer from one layer to the other. If no current is allowed in the passive layer, a potential difference develops there to compensate for the frictional interlayer force. The transresis-tance is measured as the ratio between the electric field de-veloped in the passive layer to the current density in the active layer. This drag is of fundamental interest because it can be used as a sensitive probe of the screened interlayer Coulomb interaction and the form of the irreducible polariz-ability functionx(v,q) within the layer.

Until recently, experiments on frictional drag have been done in a zero magnetic field.1In this regime, the theoretical description is well elaborated.2,3 The T2 dependence of the transresistance was explained by the phase-space arguments. A number of experiments on Coulomb drag in magnetic field have been also reported.4,5Most of them are done in the quantum Hall regime (T!\vc.EF, T andvc being tem-perature and cyclotron frequency, respectively! when the screening of the interaction and the polarizability function are determined by the states of the lowermost Landau level. The experiments have shown a strong filling factor depen-dence of the drag resistance. It was strongly enhanced ~com-pared to the B50 case! when the Fermi level lay within the Landau level, and was strongly supressed when the Fermi level was between the Landau levels. A recent theoretical work6,7~which treats the case T,D!\vc.EF, whereD is the width of the Landau level due to disorder! predicted a twin-peak structure of the transresistance as a function of the magnetic field. This is due to the interplay between the screened interlayer interaction and the phase space available

for the interlayer e-e scattering.

In the present work we investigate the drag effect in in-termediate magnetic fields 1/t!vc!EF/\. In contrast to the quantum Hall regime, the electrons occupy many Landau levels. However, these Landau levels remain well resolved since their width D.\

A

v_c/t!\v_c is much smaller than the level spacing.

The electron density of states is strongly distorted in com-parison with its zero-field value. That is why even in the ‘‘classical’’ regime EF@T@\vcand Rc@d(Rc5vF/vc be-ing the cyclotron radius, and d the distance between the lay-ers! the polarization function Imx(v.T,q.1/d), which is responsible for the absorbtion of energy, differs strongly from its zero-field form. As a function of frequency, it con-sists of a series of well-resolved peaks at multiples of cyclo-tron frequency. We will see that this circumstance leads to rich parametric dependences of transresistance on tempera-ture and magnetic field. This is to be contrasted with the usual intralayer resistance that exhibits no anomalies except strongly suppressed Shubnikov–de Haas oscillations, and does not manifest the electron density of states.

In the intermediate magnetic field at temperatures T .\vc, weakly damped boson excitations become impor-tant. Those are magnetoplasmons with energies close to mul-tiples of the cyclotron energy. These excitations provide a mechanism of Coulomb drag in the system. The momentum transfer is provided by magnetoplasmons excited in one layer and absorbed into the other. We have found general relations that allow us to present the magnetoplasmon con-tribution to the drag resistance as the result of the resonant tunneling of magnetoplasmons.

We have assumed a Coulomb mechanism of the drag.2 It has also been shown that the phonons may play an important role in mediating the drag.8They remormalize the electron-electron interaction so that the latter acquires a characteristic v,q dependence. This dependence explains some experi-mentally observed features of the drag. We have not

explic-PRB 59

itly included phonons in our calculations. We believe that a proper account of phononic effects can be achieved by re-placing the bare dielectric constant by its phonon-corrected value. This would not affect the peculiarities that arise from the electron density of states.

The problem we consider in this paper is also interesting because the experimental dependence4 of transresistance in the indicated parameter region has a universal form }B2_T. This temperature dependence is of a highly nontrivial form, because common arguments based on the consideration of the phase space available for the scattering give the T2 de-pendence. We show that small energy scales ~for example, D!T) play the role of characteristic cutoff energies, which leads to a linear temperature dependence of transresistance in a relatively weak magnetic field.

A bird’s eye view of our results is provided in Fig. 1, where we present temperature and magnetic-field exponents of the transresistance in seven distinct parameter regions. We obtain detailed analytical results for the first three regions that correspond to T@\vc. In addition to the slow T and B dependences, the transresistance in the regions IV-VII (T !\vc) exhibits an oscillatory dependence on the inverse

magnetic field ~Shubnikov–de Haas oscillations!. The oscil-lations can be hardly investigated analytically except those in simplest cases. In the present paper we present only analyti-cal results for regions VI and IV, and give estimations of the transresistance in regions V and VII for typical filling fac-tors.

The outline of the paper is as follows. In Sec. II we list the theoretical assumptions we made, and present a method which is essentially the same as in Refs. 2,3,6 and 7. Details of the polarization function and the magnetoplasmon spec-trum are presented in Sec. III. In Sec. IV we give a detailed description of the magnetoplasmon mechanism of the drag. A phenomenological description of the resonant tunneling of magnetoplasmons is elaborated upon in Sec. V. We list our analytical results for high temperatures in Sec. VI. Section VII is devoted to an evaluation of the transresistance at low temperatures.

II. METHOD

In the present paper, we cover the parameter region D !\vc!E05\vF/d. Here D stands for the width of the Landau level. It determines the maximum one-particle den-sity of states in a magnetic field. The ratio between\v_cand

T can be arbitrary.

We assume here that Landau levels acquire width due to scattering by impurities and, following Ref. 9, treat the effect in the self-consistent Born approximation ~SCBA!. This ap-proximation is known to lift the difficulties related to the high Landau-level degeneracy. In this approach, D2 5(2/p)\v_c\/t. This expression forD is valid for a short-range (d-correlated! random potential.

We also assume the Coulomb mechanism of the drag,2so that the dc drag current results from the rectification by the passive layer of the ac fluctuating electric field created by the active one. In diagrammatic language, the transconductance is given by a diagram composed of three-body correlation functions connected by Coulomb interaction lines ~photon propagators!. In Refs. 3 and 7, it was argued that under very general conditions the three-body correlation functions can be expressed in terms of electron polarization functions x1,2(v,q) in each layer.

This yields the following expression for the diagonal ele-ment of the transresistivity tensor @Eq. ~28! of Ref. 7#:

r12xx₅₂ \ 2 2e2 1 n1n2T

E

d2q ~2p!2q 2

E

0 `dv 2p

U

V12~q! E~v,q!

U

2_Imx 1~v,q!Imx2~v,q! sinh2~\v/2T! . ~1!

Here n1 and n2 are electron concentrations in the layers,

V₁₂(q) is the Fourier component of the interlayer Coulomb interaction, and E(v,q) describes ~dynamical! screening of this interaction. Electrostatics gives V₁₂5V(q)exp(2qd) and

V(q)52pe2/eq, e being the bulk dielectric constant, and E~v,q!5@11V~q!x1#@11V~q!x2#2V12

2

~q!x1x2. ~2!

It has been argued7 that Eq.~1! is valid for an arbitrary magnetic field provided q21!l,Rc, l being the mean free path. Our checks confirm that, so we use Eq. ~1! in our calculations in the intermediate magnetic-field regime.

As a reference, we here give the expression for transresis-tance in the absence of a magnetic field for temperature T !E0 ~Refs. 2 and 3!:

FIG. 1. Seven regions of different analytical behavior of the transresistance in the intermediate magnetic-field regime. Note the log-log scale. The first number in each region corresponds to the temperature exponent, and the second number indicates the magnetic-field exponent. The vertical line corresponds to the con-dition (\vc)2.E0D.

r12 xx₅z~3! 64 \ e2 1 n1n2d4 T2a_B2 \2 vF1vF2 . ~3!

Here z(3).1.202 is the Riemann z function, vF1 andvF2 are the corresponding Fermi velocities, and aB is the Bohr radius.

To make use of Eq.~1!, we shall evaluate the polarization functionx(v,q;B). This we do in Sec. III.

We will assume that the layers are macroscopically iden-tical. We keep indices 1 and 2 that label the layers in the formulas solely for the sake of physical clarity. The excep-tion will be the discussion of Shubnikov–de Haas oscilla-tions in regions IV and VI. We allow there for different filling factors in the layers.

III. POLARIZATION FUNCTION AND MAGNETOPLASMONS

We start with the following expression for the imaginary part of the polarization function ~see Appendix A!:

Imx~v,q!5nvc

(

n,m J_n2_2m~qRc!

E

2` 1`de p 3@nF~e!2nF~e1v!#Im Gn r_{~e!Im G} m r_~e1v!. ~4! Here n5m0/p\2 is the two-dimensional thermodynamic density of states in the absence of a magnetic field, nF is the Fermi distribution function, and G_nr is the retarded Green’s function of the electrons in the nth Landau level. In the above formula, we have taken into account that under the conditions considered in this paper (T,\vc!EF), only large Landau-level numbers are important. Thus the bare vertex function is reduced to its quasiclassical form, which is the Bessel function of argument qRc. In the limit D→0 ~no disorder! this expression is equivalent to the semiclassical approximations employed in Refs. 10 and 11.

Expression~4! disregards vertex corrections due to disor-der. This is safe since we always assume thatvFq@1/t and

qRc@1. We will also disregard the rapidly oscillating part of the Bessel function squares at qRc@1, i.e., we assume that

J_m2(qRc)'1/pqRc. We discuss the relevance of this as-sumption in Appendix B.

Using the SCBA expression for Im G ~Ref. 9! in the limit of large n, Im G_nr~e!522 D

A

12

S

e2en D

D

2 Q

F

12

S

e2en D

D

2

G

, ~5! whereQ is the step function anden5(n11/2)\vc, we ob-tain from Eq.~4! that

Imx~v,q!5n4vc p2_D vc qvF 3

(

n,m @nF~en!2nF~en1v!# 3Xim

S

en2em1\v 2D

D

. ~6!

Here we define the dimensionless function Xim(x)[

4 3@(1

1x2_)E(

A

_12x2_)22x2_F(

A

_12x2_{)# for uxu<1, and X}

im(x)

50 otherwise. Functions F(x) and E(x) are complete ellip-tic integrals of the first and second kind, respectively. Note that*₂₁11dx Xim(x)5p2/8. The expression for Imx(v,q) as-sumes different forms depending on the temperature. In the most interesting case T@\v_c and qvF@v ~the latter in-equality is equivalent to T!E0, because the characteristic

q.1/d and the frequency cannot be larger than the

tempera-ture! from Eq. ~6! we obtain

Imx~v,q!5n4vc p2_D v qvFj52`

(

1` X_im

S

v2 jvc 2D

D

. ~7! Since x(v) is an analytical function of v, we can easily obtain the real part from Eq. ~7!. In the vicinity of the jth cyclotron resonance,uv2 jv_cu!v_c, this reads

Rex~v,q!5n1n4vc p2_D v qvF Xre

S

v2 jvc 2D

D

, ~8! Xre~x!5 1 pv.p.

E

21 11d y Xim~y! ~y2x! .

The functions Xre(x) and Xim(x) are plotted in Fig. 2. In the opposite limit of D!T!\v_c and v<2D and

qRc@1 we obtain, from Eq. ~6!,

Imx~v,q!5n4vc p2_T v DXim

S

v 2D

D

1 qRc f_n~12 f_n!, ~9!

where fn51/@11exp(en2m)/T# is the filling factor of the nth Landau level in the layer, and m is the chemical potential.

Finally, at T!D we can set n5m and integrate expres-sion~4! overe in the close vicinity ofm. This gives

Imx54nvc 2 p2_D2 v qvF

S

12~m2en! 2 D2

D

. ~10! Although the magnetoplasmon modes of two-dimensional electron gas have been extensively studied,12 very little at-tention has been paid to their properties at high frequencies in the short-wave limit. Since those are of interest for us, we investigated them in some detail. The dispersion curves of the magnetoplasmon modes in the case of weak damping are

FIG. 2. Functions Xim(x) and Xre(x) determine the shape of

determined by the equation ReE50, E is given by Eq. ~2!. From this equation we obtain (Rex/n)1z₆50, where z₆ 5qaB/2@16exp(2qd)#!1. Using Eq. ~8! for Rex, for each j we obtain two solutions corresponding to two (6) magneto-plasmon modes. The first thing to note is that under our assumptions qaB!1, so that on a large frequency scale z₆ can be safely omitted. The resulting equation Rex50 deter-mines a series of Burstein-like magnetoplasmons11with fre-quencies that in the interesting region of q are close to cy-clotron harmonics jvc. ~see Fig. 3!. If q,q0[ 24jXre(1)vc

2

/p2DvF.0.19jvc/RcD, the root of the disper-sion relation lies beyond the electron adsorption bands, and the magnetoplasmon is not damped. These magnetoplasmons are of no interest for us since they do not ‘‘talk’’ to electrons and thus cannot participate in drag. If q@q₀ magnetoplas-mons lie deep in the electron adsorption band and are strongly damped, so that their conribution to drag cannot be distinguished from the contribution of electron-electron scat-tering. Therefore, we concentrate on a close vicinity of q₀ ~right panel of Fig. 3!.

In this vicinityx(v,q) can be expanded in Taylor series in terms of w5v2 jvc22D and k5q2q0, assuming that

w!D andk!q0; x/n5C2 w 2D 1 k q01iC1

S

w 2D

D

2 Q~2w!. ~11! Here C_1,2are numerical constants characterizing the behav-ior of Xim and Xrenear x51, C1.6.23, and C2.2.19. This determines the dispersion law of magnetoplasmons,

w₆522D C2~z6

1k/q0!, ~12!

and their damping,

G5Q~2w!C1w2

C2D

, ~13!

where z is taken at q5q0. Symmetric and asymmetric modes are split by

dv5D q0aB

C2sinh q0d

. ~14!

We see that G!w!D anddv!D, anddv can be compa-rable toG.

IV. MAGNETOPLASMON CONTRIBUTION

In this section we consider the magnetoplasmon mecha-nism of the Coulomb drag. In the absence of a magnetic field there are two plasmon modes in the double-layer system, one with the electron densities in the two layers oscillating in phase ~the optic mode!, and the other where the oscillations are out of phase~the acoustic mode!.13It was pointed out in Ref. 3 that the drag effect can be greatly enhanced by dy-namical ‘‘antiscreening’’ of the interlayer interaction due to coupled plasmon modes. Since the plasmon modes lie be-yond the T50 particle-hole continuum, temperatures of the order of the Fermi energy are required for a large plasmon enhancement of the drag effect. Only then do the thermally excited electrons and holes with plasmon velocities provide sufficient damping of the plasmon modes, and thus facilitate plasmon interaction with electrons.3

In the case of an intermediate magnetic field, the magne-toplasmons have even better chances to enhance the drag. First there are many modes, and their typical energies are of the order of\vc. Therefore, these modes can be excited at temperatures much lower than the Fermi energy. Second, the magnetoplasmons in our model acquire natural damping: due to the finite Landau-level width, they may lie within the particle-hole continuum. The finite temperature without dis-order does not lead to magnetoplasmon damping, and to the drag effect. This is in contrast to the situation without a magnetic field, where Imx at the plasmon frequency was calculated for collisionless plasma.3

The magnetoplasmon mechanism of the Coulomb drag, when the current in the active layer causes a magnetoplas-mon wind and the magnetoplasmagnetoplas-mons are absorbed by the electrons of the passive layer leading to transfer of the mo-mentum, must be quite general. In this section, we evaluate the magnetoplasmon contribution using Eq.~1!. It turns out that the answer can be expressed through only two quantities for each double plasmon mode: frequency splitting dv and dampingG. Any concrete model would only set specific ex-pressions for G anddv. This clarifies the physical meaning of Eq.~1!.

To prove this, we rederive the result in Sec. V in a phe-nomenological framework.

Let us expand x(v,q) around the frequencyv(q) when Rex50,

x~w,q!5x

8

w1ix

9

, ~15!

w being the frequency deviation. Here x

8

5Re dx/dv@v(q)# and x

9

5Imx@v(q)#. The expression for E is reduced to the form

E5@V2_~q!2V 12 2

~q!#~wx

8

1ix

9

1nz₁!~wx

8

1ix

9

1nz₂!.

~16! Consequently, the integrand in Eq.~1! has a sharp maximum near v(q) as a function ofv. Hence we can now integrate over w in infinite limits.

Equation ~16! determines the magnetoplasmon spectrum, and suggests that mode splittingdv5n(z₂2z₁)/x

8

,

damp-FIG. 3. Electron adsorption in thev,q plane. Left plane: elec-tron adsorption occurs~i! in narrow strips aroundv50 and cyclo-tron resonances~particle-hole continuum! and ~ii! on the magneto-plasmon dispersion curves. Right plane: intersection of the magnetoplasmon dispersion curve and the edge of the strip at a smaller scale. We illustrate the splitting of magnetoplasmons and the level widthG they acquire inside the strip.

ing G52x

9

/x

8

. All factors V12and V can be absorbed into these two quantities. Typical values of w that contribute to the integral are of the order of max(G,dv). The approxima-tion is valid if x and statistical factor sinh2(v/2T) do not

change much in this frequency window, i.e., v(q) @dv, dG/dv!1, and max(G,dv)!T/\.

Provided these conditions are fulfilled, we can reduce the expression for the transresistance to the elegant form

r12xx₅₂ \ 16e2n1n2T

E

d2q q2 ~2p!2_modes

(

1 sinh2„\v~q!/2T… ~dv!2_G @~dv!2_1G2_#. ~17!

Summation over modes means summation over all possible roots of Rex50.

V. RESONANT TUNNELING OF MAGNETOPLASMONS

Here we give another derivation of this formula which clarifies its physical meaning. To describe resonant tunneling of plasmons between the layers, we introduce for each plas-mon mode a density matrixri j5

^

bi

†_b

j

&

. Here i, j51,2 label the layers, and b† and b are boson creation and annihilation operators. In the absence of dissipation, i.e., plasmon emis-sion and absorption, the density matrix obeys the equation

]ri j

]t 5i

(

l ~Hilrl j2ril

Hl j!. ~18!

For identical layers, the diagonal elements of the Hamil-tonian are equal to each other and dissappear from the equa-tion. The nondiagonal element that is responsible for the plasmon tunneling between the layers can be readily ex-pressed in terms of splittingdv between the symmetric and asymmetric plasmon states: H₁₂5H₂₁5dv/2.

The dissipation takes place independently in each layer. It contributes to the time derivative of the diagonal density matrix elements in the following way:

S

]rii

]t

D

_diss5Gi~ni

B_2r

ii!, ~19!

the two terms corresponding to generation and absorption of the plasmons, respectively. The temperature of the Bose dis-tribution function n_iBcorresponds to the electron temperature of the ith layer. The nondiagonal matrix elements aquire a damping equally from both layers

S

]ri j

]t

D

diss

52~Gi1Gj!

2 ri j. ~20!

The system of equations that incorporates both dissipation and resonant tunneling reads as follows:

]r11 ]t 5G1~n1 B_2r 11!1i dv 2 ~r122r21!, ]r22 ]t 5G2~n2 B_2r 22!1i dv 2 ~r212r12!, ]r12 ]t 52 ~G11G2! 2 r121i dv 2 ~r112r22!, wherer215r12*. The stationary solution takes the form

r115n1 B₂ G2 G11G2 dv2_~n 1 B_2n 2 B_! dv2_1G 1G2 . ~21!

The expression forr22is obtained by reverting indices 1 and 2.

We can now evaluate the drag force acting on electrons of each layer by equating it to the momentum flow between the layers. We sum over modes with all possible q, and obtain

F52

(

q \q

S

]rii ]t

D

diss 52

(

q \q G1G2 G11G2 dv2_~n 1 B_2n 2 B_! dv2_1G 1G2 ~22! whereG, nB, anddv may be q dependent.

We assume that the current flows in layer 2 and the drag force in layer 1 are equilibrated by the electric field. The transresistivity is essentially the ratio of this field to that current. The effect of the current is that the n₂B is the equi-librium Bose distribution in the reference frame where the electrons of the second layer are in average at rest, rather than in the laboratory reference frame, so that

n₂B@e~q!#5 f_B@e~q!2\~v_{dri f t}q!# ' fB@e~q!#2\~vdri f tq!

]fB

]e , ~23!

vdri f tbeing the drift velocity.

Substituting Eq.~23! into Eq. ~22!, we obtain

F5\2vdri f t

E

d2q q2 8p2 G1G2 G11G2 dv2 dv2_1G 1G2 ]fB@e~q!# ]e . ~24! The last things to note are that F5en1E and I5en2vdri f t. If we use this and set G15G25G, we reproduce Eq. ~17!.

VI. RESULTS: HIGH TEMPERATURES

In this section we consider the drag resistance at tempera-tures T@\vc, which are sufficiently high to excite magne-toplasmons and electrons in many Landau levels. As we can see in Fig. 1, at high temperature we encounter at least three distinct regions with different temperature and magnetic-field exponents.

First of all, we shall explain why there are so many re-gions. If we compare the imaginary part of the polarization function with and without a magnetic field, we see that their values averaged over frequency intervals larger than\vcare the same

^

(Imx)

&

5

^

@Imx(H50)#

&

. However, Imx50 beyond narrow adsorption bands ~Fig. 3!. This means that within the bands Imx is significantly enhanced in compari-son with its zero-field value, typically by a factor vc/D. Surprisingly enough, the enhancement of Imx can lead both to enhancement and suppression of the drag.

If Imx!n, Rex'n. The denominator E in Eq. ~1! which is responsible for screening of the interlayer potential

is the same as without a magnetic field. We refer to this situation as the normal screening regime. In this regime, the transresistance is enhanced in comparison to its value with-out a magnetic field, since the effect is proportional to

^

(Imx)2

_&

_@

_^

_(Imx)

_&

2_.

_^

_@Imx(H50)#

_&

2_.

Upon further increase of Imx, Rex develops as well, so that bothuRexu and Imx become larger thann. The de-nominator E strongly increases. This efficiently screens out the interlayer interaction and leads to a drastic descrease of the transresistance. This we will call overscreening.

However, E can also decrease with increasing Imx and pass zero. Near this line, the interlayer interaction is greatly increased. This is where the magnetoplasmon contribution dominates. The actual value of the drag effect is thus deter-mined by interplay of these three competing tendencies.

Let us first evaluate the magnetoplasmon contribution. Substituting expressions ~13! and ~14! into Eq. ~17!, we no-tice that the integrand has a sharp extremum neark'0 so we can formally integrate over k in infinite limits. This yields the relation which is valid for all regions,

r12 xx_520.00221 \D e2n1n2d4T

S

aB d

D

3/2 a4

(

j51 ` _~\v j/T!4 sinh2~\vj/2T!

S

a\vj/T sinh~a\vj/T!

D

3/2 , ~25! wherea.0.19(Tv_c/DE₀).

This expression can be simplified further. The region I (\vc!T!E0D/\vc) corresponds toa!1. Since \vc!T, the characteristic values of j in the sum~25! are much larger than unity. Therefore we can convert the sum over j into the integral. Since*0`dx x4sinh22(x/2)516p4/15, we obtain

r12 xx_.20.0003 \ e2n1n2d4 ~\vc!3T4 D3_E 0 4

S

aB d

D

3/2 . ~26!

Region II is defined by inequalities T@E0D/vc@vc. Here a@1. As a result, the characteristic frequencies here v˜

j.E0D/vc!T @see Eq. ~25!#. These frequencies, however, are still much larger than the cyclotron frequency:v˜j@vc. This enables us again to introduce the continuous variable

x5a\vj/T, and convert the sum into the integral:

r12 xx_.20.0095 \ e2_n 1n2d4 T E0

S

a_B d

D

3/2 . ~27!

In region III (vc@

A

E0D), the j51 term dominates the sum. The corresponding expression for the plasmon drag re-sistance is exponentially small:

r12xx_.22.731026 \ e2n1n2d4 ~\vc!9T D9/2_E 0 11/2

S

aB d

D

3/2 3exp@20.28~\vc!2/DE0#. ~28! This is due to the fact that the value of q which is needed to bring the magnetoplasmon pole to the vicinity of the Landau

level is large compared to 1/d. A similar situation occurs in region IV, with the exponential suppression being due to low temperature: r12 xx_.21.1231025 \ e2n1n2d4 ~\vc!8 D3_TE 0 4

S

aB d

D

3/2 3exp~2\vc/T!. ~29!

Below we will see that even the exponentially suppressed magnetoplasmon contribution can efficiently compete with the quasiparticle one.

Now we will estimate the quasiparticle contribution in all three regions. Let us first consider region I (vc!T !E0D/vc). In this parameter interval the characteristic val-ues of \v.T and qd.1. It follows from Eqs. ~7! and ~8! that Rex5nand Imx!n. Thus we are in the normal screen-ing regime. Usscreen-ing the fact that V(q)n@1 ~in other terms,

qaB!1) from Eq. ~1! we obtain

r12 xx_.20.00725\ e2 1 n1n2d4 T2a_B2 \2_v F1vF2 vc D . ~30!

This magnetotransresistance is larger than the zero-field value @Eq. ~3!# by a factor of vc/D@1. This is due to the discreteness of the electron spectrum in the magnetic field when the density of states within the Landau level increases remarkably.

In region II (vc!E0D/vc!T), though the main contri-bution to the drag is due to the magnetoplasmon mechanism, it is instructive to give an estimation of the quasiparticle

contribution. As in region I, q.1/d. The characteristic fre-quency, however, is restricted by value vcut;E0D/vc, which is much smaller than the temperature ~though still much larger than the cyclotron frequency!. The reason for this is the overscreening. For estimations, we approximate Imx near the cyclotron resonance by its value from Eq.~7! at qd.1. This gives Imx.nv/vcut. A good estimation for Rexnear the resonance isnforn@Imxand Imx otherwise. Thus the integrand in Eq. ~1!, (Imx)2/uEu2}(Imx)2/(Re2x 1Im2_x)2_{, achieves a maximum atv}

cut. Since the interval of integration over frequency is effectively smaller than the temperature, the transresistance exhibits a linear temperature dependence:r12

xx_;(\/e2_)(1/n

1n2d4)(aB

2

/d2)(T/E0). Let us compare this with the B50 case. There the linear temperature dependence starts2at T.E0. This is because the absorbtion of energy due to the Landau damping mechanism

is possible only for frequencies smaller thanvF/d. Thermal frequencies are ineffective, because the corresponding phase velocities are larger than vF. In the presence of magnetic field, the role of a cutoff energy is taken by a much smaller value E₀D/v_c. Thus the linear temperature dependence of the transresistance starts at much lower temperatures.

In region III (

A

E0D!\vc!T,),vcut becomes smaller than v_c. There is overscreening at qd.1 for all cyclotron resonances. The contribution of resonances to integral Eq. ~1! is of the order of r12 xx_;(\/e2_)(1/n 1n2d4)(Rc 2 a_B2/ d4)(TD3/vc 4 ).

The main contribution is determined by the frequencies v;D!vc, so that the quasiparticles are created within the same Landau level. For these frequencies we always have the normal screening situation. We can set Rex5n. Imx is determined by the j50 term of Eq. ~7!. We obtain

r12 xx₅₂\ e2 2 p4 1 n1n2

E

d2q q2 ~2p!2 1 sinh2~qd! Ta_B2 D2_R c1Rc2

E

0 2D_dv 2pXim 2

S

v 2D

D

.20.0011 \ e2 1 n1n2d4 vc 2_Ta B 2 DvF1vF2 . ~31!

It is interesting to note that this magnetic and temperature dependence coincides precisely with the observed one4if we assume thatD does not depend on the magnetic field. Indeed, the magnetic dependence ofD is rather weak. These experi-ments were performed in a rather a strong magnetic field, where only a few Landau levels were occupied so that one should not expect quantitative agreement with our calcula-tions. On the other hand, the linear T dependence is remark-able. We believe that in any case this indicates a reduction of typical excitation energies to values much smaller than T, possibly due to overscreening at energies of the order of T.

Now we are in a position to compare the quasiparticle and magnetoplasmon contribution, and thus to set the borders of the gray-shaded regions in Fig. 1. The magnetoplasmon con-tribution dominates throughout region II. In region I we compare expressions ~26! and ~30!. The magnetoplasmon contribution dominates provided T.E0D/\vc(aB/d)1/4. Since experimentally d.aB, this happens in fact close to the border between regions I and II (T.E0D/\vc). The expo-nentially small magnetoplasmon contribution given by Eq. ~28! competes with Eq. ~31! in region III and dominates provided 0.28v_c2/(DE0),ln@(vc 2 /E0D)7/2(d/aB)1/2#. It also dominates in region IV if T/\vc.1/ln@vc 4 d1/2/(E0 2_D2_a B 1/2 )#. The latter condition is obtained by comparing expressions ~29! and ~32!.

VII. RESULTS FOR LOW TEMPERATURES

In this section we present our results for low temperatures

T!\vc. Owing to energy limitations, only the states of the upper partially filled Landau level are involved in the drag. This makes the transresistance sensitive to the concrete value of the filling factor. In addition to the slow dependence on the magnetic field, the drag effect exhibits an oscillatory de-pendence on the inverse magnetic field related to the filling factor. The detailed study of these Shubnikov–de Haas

os-cillations is beyond the limits of the present work, and will be presented elsewhere. Therefore, we provide here analyti-cal results for regions IV and VI only. As to regions V and VII, we present below estimations of the transresistance for typical filling factors rather than detailed analytical results.

In region IV (D!T!\vc!

A

TE0) the situation is most straightforward, since it follows from Eq. ~9! that Imx!n. Thus here we encounter normal screening and may set Rex 5n. For the transresistance we obtain

r12 xx_520.0011\ e2 1 n1n2d4 a_B2 vF1vF2 vc 4 TD 3 fn₁~12 fn₁!fn₂~12 fn₂!, ~32!

fn_1,2being filling factors in the layers.

Even if T@D, we encounter overscreening in region V (

A

TE0!\vc!E0). For estimations, we set Imx;Rex@n and obtain r12xx_;~\/e2_!~1/n 1n2d4!~aB 2_R c1Rc2/d4!~T3/vc 2_D!. ~33! As a consequence of overscreening, the transresistance de-creases rapidly with increasing magnetic field.

At low temperature T!D the integral in Eq. ~1! is con-tributed byv.T. This gives rise to the featureless T2 tem-perature dependence of Eq. ~3! with the coefficient depend-ing on a magnetic field. Region VI (T!D, vc!

A

E0D) again corresponds to normal screening. We take Imx from Eq. ~10! and set Rex5n. This yields

r12xx_520.0031\ e2 1 n1n2d4 a_B2 vF1vF2 T2v_c4 D4

S

12 ~m12en!2 D2

D

3

S

12~m22en! 2 D2

D

, ~34!

wherem1,2are the chemical potentials in the layers. They are related to filling factors fn₁,n₂ by means of

f_n51/21~1/p!

F

~m12en! D

A

12 ~m12en!2 D2 1arcsin~m12en! D

G

. ~35!

For a typical filling factor, the effect is larger than the zero field transresistance by a factor of (vc/D)4.

With increasing magnetic field, we enter the region of overscreening ~region VII, T!D, vc@

A

E0D.) Again we set Rex@n and obtain

r12 xx ;~\/e2_!~1/n 1n2d4!~aB 2 Rc1Rc2/d4!~T2/vc 2_{!, ~36!} that decreases with increasing magnetic field.

ACKNOWLEDGMENTS

This work is part of the research program of the ‘‘Stich-ting voor Fundamenteel Onderzoek der Materie’’ ~FOM!, which is financially supported by the ‘‘Nederlandse Organi-satie voor Wetenschappelijk Onderzoek’’~NWO!. We thank G. E. W. Bauer and L. I. Glazman for useful discussions. A.V.K. acknowledges NWO for its support of his stay in Delft. He is also grateful to F. Hekking for collaboration at the earlier stage of the work, J.T. Nicholls and D.E. Khmelnitskii for useful discussions, and M. Pepper for hos-pitality and financial support at Cavendish Laboratory where part of this work was done.

APPENDIX A

In this appendix we derive Eq. ~4!. The polarization bubble in the Matsubara representation without vertex cor-rections due to disorder can be written as

x~ivn,q!52 2T

S

(

ie_n

(

n,X _m,X

(

₈

Gn~ien!Gm~ien1ivn!~nXueiqW•rWumX

8

!~mX

8

ue2iqW•rWunX!, ~A1! where Gn is the dressed Matsubara Green’s function of the electrons in the nth Landau level, S is the area of a sample,en 5pT(2n11) are fermionic frequencies, and X and X

8

are the quantum numbers which give the position of the Landau oscillator center. Since Green’s functions do not depend on X, we can reduce the above expression to the following:

x~ivn,q!52

T

pl2

(

_ie

n

(

n,m

Gn~ien!Gm~ien1ivn!u fnm~q!u2. ~A2! Here l is the magnetic length, and uf_nm(q)u25e2x(m!/n!)xn2m@L_mn2m(x)#2 for m<n ~the corresponding expression for n ,m can be obtained by interchanging of indices n and m), x5q2_l2_{/2 and L}

n m

are the Laguerre polynomials. Note that for high Landau levels n,m@1, we obtain, using the asymptotics of Laguerre polynomials, ufnm(q)u2.Jn2m

2

(ql

A

n1m11)

.Jn2m

2

(qRc), where we have used that

A

2nl5vF/vc5Rc is the cyclotron radius (n.EF/\vc@1).

The polarization function we need can be obtained from Eq.~A2! by the analytic continuation ivn→v1i01. Writing the sum~A2! overen as a contour integral, and deforming the contour in the standard manner, we obtain

x~v,q!5 1 pl2

(

_n,m Jn2m 2 _~qR c!

E

2` 1`de 2pinF~e!$@Gn r_~e!2G n a_~e!#G m r_~e1v!1G n a_~e2v!@G m r_~e!2G m a_~e!#% 5 1 pl2

(

_n,m Jn2m 2 _~qR c!

E

2` 1`de p nF~e!Im Gn r_~e!@G m r_~e1v!1G m a_{~e2v!#. ~A3!}

Here nF is the Fermi distribution function, and Gr,aare re-tarded and advanced Green’s functions. While deriving Eq. ~A3! we have used that the Bessel function square does not change with the changing of the index sign. Taking the imaginary part of Eq.~A3!, we finally obtain Eq. ~4! of the main text.

APPENDIX B

In this work we disregard rapidly oscillating parts of the polarization function given by Eq. ~4!. This means we

ap-proximate the Bessel function squares in Eq. ~4! in the limit of qRc@1 as Jm 2 (qRc)'1/pqRcrather than J_m2~qRc!' 1 pqRc

F

11cos

S

2qRc2pm2 p 2

DG

, ~B1! which is the mathematically correct expression. Let us ex-plain why.

First let us note that if we take these oscillating parts into account, it would significantly alter our results. In the normal screening regime it would give an extra factor of 32, since the

answer is proportional to (Imx)2. The answer would change even more drastically in the overscreening regime. The point is that the oscillating terms would set the polarization func-tion to ~almost! zero for q corresponding to zeros of the Bessel function. No overscreening would occur near these points and their close vicinity would dominate the drag.

All this would lead to a very sophisticated and extremely unstable picture of the drag effect. Fortunately, we are able to present some arguments that allow one to disregard the oscillating terms in the polarization function.

The physical origin of the oscillating terms can be best understood in the language of semiclassical electron trajec-tories in a magnetic field. A classical trajectrory cannot move from the starting point further than 2Rc. Hence the polariza-tion funcpolariza-tion in the coordinate representapolariza-tion,x(x,x

8

) has a sharp edge atux2x

8

u52Rc. This gives rise to Fourier com-ponents.cos(2qRc).

If the edge is not sharp, the oscillating part is exponen-tially suppressed. The suppression is of the order of exp@2q2(dR)2#, dR being a typical rounding of the edge.

By virtue dR is the typical uncertainty of the coordinate of

the electron which makes half of the Larmor circle. Such an uncertainty can be of quantum-mechanical origin. In this case we estimate dR.

A

Rc/kF!Rc. Another cause of un-certainty may be small-angle scattering by smooth potential fluctuations in the heterostructure. For this case we estimate dR.

A

Rc

3

/lsa, where transport mean free path lsa@Rc. It is interesting to note that scattering on pointlike defects does not contribute to dR providedvct@1.

Now we note that typical q values contributing to the drag resistance are of the order of 1/d. We conclude that the os-cillating part is exponentially suppressed provided d ,

A

Rc/kF or d,Rc

A

Rc/lsa. We assume that at least one of these conditions is fulfilled.

1_{T. J. Gramila, J. P. Eisenstein, A. H. MacDonald, L. N. Pfeiffer,} and K. W. West, Phys. Rev. Lett. 66, 1216 ~1991!; T. J. Gramila, J. P. Eisenstein, A. H. MacDonald, L. N. Pfeiffer, and K. W. West, Surf. Sci. 263, 446 ~1992!; T. J. Gramila, J. P. Eisenstein, A. H. MacDonald, L. N. Pfeiffer, and K. W. West, Phys. Rev. B 47, 12 957~1993!; U. Sivan, P. M. Solomon, and H. Shtrikman, Phys. Rev. Lett. 68, 1196~1992!.

2_{A.-P. Jauho and H. Smith, Phys. Rev. B 47, 4420 ~1993!; L.} Zheng and A. H. MacDonald, ibid. 48, 8203~1993!; A. Kame-nev and Y. Oreg, ibid. 52, 7516~1995!; K. Flensberg, B.Y.-K. Hu, A. P. Jauho, and J. Kinaret, ibid. 52, 14 761~1995!. 3_{K. Flensberg and B.Y.-K. Hu, Phys. Rev. Lett. 73, 3572~1994!;}

Phys. Rev. B 52, 14 796~1995!.

4_{N. P. R. Hill, J. T. Nicholls, E. H. Linfield, M. Pepper, D. A.} Ritchie, A. R. Hamilton, and G. A. C. Jones, J. Phys.: Condens. Matter 8, L557~1996!.

5_{H. Rubel, A. Fischer, W. Dietsche, K. von Klitzing, and K. Eberl,} Phys. Rev. Lett. 78, 1763~1997!; J. P. Eisenstein, L. N. Pfeiffer,

and K. W. West, Bull. Am. Phys. Soc. 42, 486~1997!; N. P. R. Hill, J. T. Nicholls, E. H. Linfield, M. Pepper, D. A. Ritchie, B. Y.-K. Hu, and K. Flensberg, Physica B 249–251, 868~1998!. 6_{M. C. Bonsager, K. Flensberg, B. Y.-K. Hu, and A.-P. Jauho,}

Phys. Rev. Lett. 77, 1366~1996!.

7_{M. C. Bonsager, K. Flensberg, B. Y.-K. Hu, and A.-P. Jauho,} Phys. Rev. B 56, 10 314~1997!.

8_{M. C. Bonsager, K. Flensberg, B. Y.-K. Hu, and A. H.} Mac-Donald, Phys. Rev. B 57, 7085~1998!, and references therein. 9_{T. Ando and Y. Uemura, J. Phys. Soc. Jpn. 36, 959~1974!; T.}

Ando, A. B. Flower, and F. Stern, Rev. Mod. Phys. 54, 437

~1982!.

10_{I. L. Aleiner and L. I. Glazman, Phys. Rev. B 52, 11 296~1995!.} 11_{N. J. M. Horing and M. M. Yildiz, Ann. Phys.~N.Y.! 97, 216}

~1976!.

12_{C. Kallin and B. I. Halperin, Phys. Rev. B 30, 5655~1984!; D.} Antoniou and A. H. MacDonald, ibid. 46, 15 225~1992!. 13_{S. Das Sarma and A. Madhukar, Phys. Rev. B 23, 805~1981!.}