PHYSICAL REVIEW B, VOLUME 64, 035301
Acoustoelectric current for composite fermions
J. Bergli1 and Y. M. Galperin 1,2,3
1
Department of Physics, University of Oslo, Box 1048 Blindern, N-0316 Oslo, Norway
2
Centre for Advanced Studies, Drammensveien 78, 0271 Oslo, Norway
3
Solid State Division, A. F. Ioffe Physico-Technical Institute, 194021 St. Petersburg, Russia
共Received 16 November 1999; revised manuscript received 24 November 2000; published 13 June 2001兲
The acoustoelectric current for composite fermions in a two-dimensional electron gas 共2DEG兲 close to the
half-filled Landau level is calculated in the random phase approximation. The Boltzmann equation is used to
find the nonequilibrium distribution of composite fermions to second order in the acoustic field. It is shown that
the oscillating Chern-Simons field created by the induced density fluctuations in the 2DEG is important for the
acoustoelectric current. This leads to a violation of the Weinreich relation between the acoustoelectric current
and acoustic intensity. The deviations from the Weinreich relation can be detected by measuring the angle
between the longitudinal and the Hall components of the acoustoelectric current. This departure from the
Weinreich relation gives additional information on the properties of the composite fermion fluid.
DOI: 10.1103/PhysRevB.64.035301
PACS number共s兲: 73.43.⫺f, 71.10.Pm, 73.50.Rb
I. INTRODUCTION
Two-dimensional electron gases 共2DEG’s兲 have been
studied extensively both experimentally and theoretically.
One important experimental technique is to investigate
the interaction of the electron gas with surface acoustic
waves 共SAW兲 propagating along the sample. Because of the
piezoelectric properties of the substrate materials
(GaAs-Alx Ga1⫺x As), the acoustic wave is accompanied by
an electric wave that interacts with the electron gas. A traveling wave of electric field can also be produced by placing a
2DEG sample on the surface of a piezoelectric crystal.
It is well known that both the attenuation and sound velocity are sensitive to changes in the properties of the electron gas.1–8 Less well explored experimentally is the acoustoelectric drag current induced by the acoustic wave. From
the experiments so far, it seems that in some cases this offers
the promise of greater sensitivity than the attenuation or velocity shift measurements, because one measures directly a
small electric current or voltage instead of a small shift in a
large quantity.9 In addition, measurement of the acoustoelectric current gives a direct measure of the intensity of the
electric field created by the acoustic wave, thus enabling determination of the coupling between the SAW and the electron gas, whereas attenuation measurements give only relative intensities.1,2 The theoretical efforts in this direction
have been modest 10,11 and a better understanding is required
to interpret measurements.12 Furthermore, there is some disagreement between the different methods10,11 that has not
been clarified yet.
The properties of a 2DEG in a strong magnetic field have
successfully been described by the composite fermion
model.5,13,14 Especially near even-denominator filling fractions, like ␯ f ⫽1/2, this seems to be a good description. In
this paper, we will calculate the acoustoelectric current for
composite fermions using the Boltzmann equation approach.
This has previously been applied to the calculation of the
conductivity tensor at finite wave vector and frequency.13
The paper is organized as follows. In Sec. II an interaction
between composite fermions and a SAW will be discussed.
0163-1829/2001/64共3兲/035301共9兲/$20.00
The Boltzmann equation is derived and solved in Sec. III,
and the acoustoelectric current is calculated. The resulting
expression is discussed in Sec. IV.
II. INTERACTION BETWEEN COMPOSITE FERMIONS
AND A SURFACE ACOUSTIC WAVE
There are several fields involved in the problem.15 The
real physical fields, those experienced by the electrons, are
the external magnetic field B and a periodic electric field set
up by the acoustic wave.
The
materials
used
to
create
the
2DEG,
GaAs-Alx Ga1⫺x As heterostructures, are piezoelectric. This
means that an acoustic wave propagating on the sample will
create a periodic electric wave that interacts with the composite fermions. To achieve stronger coupling, one sometimes uses a substrate with a larger piezoelectric constant,
and places the heterostructure in close contact with 共but
acoustically isolated from兲 the substrate. The piezoelectric
field induced by the SAW is then able to penetrate the
2DEG.2
It is assumed that the wave is propagating in the x direction, and that the piezoelectric field is in the same direction.
The electric field is then given by the real part of E(r,t)
⫽E0 e ⫺i(⍀t⫺q•r) ⫽⫺ⵜ⌽ with ⌽⫽⌽ 0 e ⫺i(⍀t⫺q•r) . Here ⍀
and q are the SAW frequency and wave vector, respectively;
E0 储 q储 x̂. ⌽ 0 is the amplitude of the screened potential;
it is related to the amplitude of the piezoelectric 共external兲
potential ⌽ ext by the equation 共see Ref. 6兲 ⌽ 0 ⫽ 关 1
⫺ v (q)K 00兴 ⌽ ext , where v (q)⫽2 ␲ / ⑀ effq is the Fourier transform of the Coulomb potential and the response function is
K 00共 q,⍀ 兲 ⫽q
␴m
1
e
v s 共 1⫺i ␴ m / ␴ xx
兲
共1兲
e
with ␴ xx
(q,⍀) the longitudinal electron conductivity, ␴ m
⬅ v s ⑀ eff/2␲ , v s ⫽⍀/q the SAW velocity, and ⑀ eff the effective dielectric constant. The amplitude of the piezoelectric
field is related to the amplitude A of the acoustic wave by the
relation
64 035301-1
©2001 The American Physical Society
J. BERGLI AND Y. M. GALPERIN
⌽ ext⫽Ae 14F 共 qd 兲 / ⑀ ,
PHYSICAL REVIEW B 64 035301
共2兲
where e 14 is the piezoelectric stress constant and F(qd) is
some dimensionless function calculated by Simon.6
When we make the Chern-Simons 共CS兲 transformation,
additional, fictitious, fields are introduced. There will be a
Chern-Simons magnetic field b⫽⫺2 ␾ 0 n tot
e ẑ, where ␾ 0
⫽2 ␲ បc/e is the flux quantum and n tot
is
the
total electron
e
density. Because of the interaction with the piezoelectric
field, there will be an induced density modulation in the electron gas. Therefore, the electron density is conveniently split
into two parts, an average and a fluctuating 共ac兲 part: n tot
e
⫽n e ⫹ ␦ n e . Corresponding to these, there will be an average
and an ac Chern-Simons magnetic field. The average field
will partly cancel the external 共real兲 magnetic field, leaving
the effective field B0* ⫽B(1⫺2 ␯ f ), where ␯ f ⫽ ␾ 0 n e /B is the
Landau level filling factor. For simplicity we assume two
flux quanta to be attached to each electron, appropriate for ␯ f
close to 1/2. The ac component of the Chern-Simons field is
given by bac⫽⫺2 ␾ 0 ␦ n e ẑ. In addition, the motion of the
composite fermions 共CF’s兲 will create an electric ChernSimons field which is given by e⫽ ␳ (ẑ⫻j), where ␳
⫽2 ␾ 0 /ec⫽2h/e 2 . The current is split in two parts, an ac
part jac, which is given by the linear response, and a dc part,
the acoustoelectric current jae, which is given by the second
order response. These give contributions eac and eae to the CS
electric field.
The strength of these fields relative to the perturbing field
E⫽ 兩 E兩 can be estimated from charge conservation. The
equation for conservation of charge is e ⳵ ( ␦ n e )/ ⳵ t⫹ⵜj⫽0,
e
qE x ⫽0. This gives
or, Fourier transformed, ⫺i⍀e ␦ n e ⫹i ␴ xx
e
⫺e ␦ n e ⫽ 共 ␴ xx
/vs兲Ex .
共3兲
Then the force from the fluctuating 共ac兲 component of the
magnetic Chern-Simons field is given by
e
e
4 ␲ v ␴ xx
e v ac
v ␴ xx
b ⫽⫺2 ␾ 0
E ⫽
eE x ,
c
c vs x ␣ c vs
共4兲
where ␣ ⫽e /បc⫽1/137 is the fine structure constant and v
is the composite fermion velocity. In order for the acoustic
wave to be sensitive to changes in the properties of the electron gas, we see from Eq. 共1兲 that the ratio of the conductive
兩 / v s ⬇ ⑀ eff /(2 ␲ ) 共or at
ity to the sound velocity should be 兩 ␴ xx
least not differing by more than one order of magnitude; see
Refs. 1, 9, and 12兲. With ⑀ eff⬇6 共see Ref. 6兲, we have
e
兩 / v s ⬇1. It is then seen that the relative strength of the
兩 ␴ xx
force from the ac Chern-Simons field to the direct force from
the piezoelectric field of the SAW is (4 ␲ / ␣ )( v /c). With a
Fermi velocity of 105 m/s, this factor is of order 1. That is,
the piezoelectric and the CS fields are of comparable importance. For the electric CS field we find for the x component
2
e
e ac
x ⫽⫺ ␳␴ xy 共 q,⍀ 兲 E x .
共5兲
Since static ␴ exy ⬃e 2 /2h for ␯ f ⬇1/2, we see that the ChernSimons field is of the same order as the external electric
field. The y component is similarly
e
e ac
y ⫽ ␳ 兩 ␴ xx 兩 E x ⬇ ␳ v s E x ⫽ 共 4 ␲ / ␣ 兲共 v s /c 兲 E x ,
共6兲
which is smaller than what is given by Eq. 共4兲 by the factor
v s / v . Since the typical sound velocity is 3⫻103 m/s, this
will be small. However, it is acting along the trajectory of the
composite fermions at the points of strong interaction 共see
below兲, and will therefore play an important role.
So far we have expressed the current as a response to the
external field. However, as seen from the composite fermions it is not possible to separate the two fields, and it is more
convenient to consider the response to the total effective
electric field acting on the composite fermions, E⫽E⫹eac.
The latter is described by the composite fermion conductivity tensor ␴ ik (q,⍀). Because the field is not parallel to the
wave vector, this is not a potential field, and we cannot write
it as the gradient of a potential. However, we can consider
the x component Ex ⫽E x ⫹e ac
x ⫽⫺ⵜ x ⌿. In the following we
will consider the response to the effective field Ex acting
upon composite fermions. The only thing that needs to be
changed in order to account for the Chern-Simons field is
then that the induced density modulation must be expressed
in terms of response to the total field,
e ␦ n e ⫽ v s⫺1 共 ␴ xx Ex ⫹ ␴ xy Ey 兲 .
共7兲
The y component of the total field is just the Chern-Simons
field since E y ⫽0, and one has
Ey ⫽ ␳ 共 ␴ xx Ex ⫹ ␴ xy Ey 兲 .
共8兲
Ey ⫽ ␦ Ex ,
共9兲
˜ e i ␩ ⫽ ␳␴ xx / 共 1⫺ ␳␴ xy 兲 .
␦ ⫽ ␳␴
共10兲
b ac⫽⫺ 共 c/ v s 兲 ␦ eEx .
共11兲
From this we get
where
We then have
In the most interesting situation the acoustic wavelength
2 ␲ /q is less than the typical diameter of CF orbits. That is,
the parameter ␬ ⫽qR c is large, where R c is the radius of the
cyclotron orbit. Using the expression for ␴ xy obtained by
Mirlin and Wölfle13 we find in this limit
␳␴ xy ⯝
cos 2 ␬
1
,
␨ sin ␲ 共 ␻ ⫹i ␯ 兲
␨⫽
ប⍀
2m v s2
.
共12兲
Here we have introduced the dimensionless frequency ␻
⫽⍀/ ␻ c and damping ␯ ⫽( ␻ c ␶ ) ⫺1 , where ␻ c ⬀B * is the CF
cyclotron frequency and ␶ is the relaxation time. Substituting
reasonable values ⍀/2␲ ⫽3 GHz, v s ⫽3⫻103 m/s, m
⬇10⫺27 g, and ␯ ⬇1, we obtain a quantity of the order 1 out
of cyclotron resonance, and an enhancement of the order ␬ at
cyclotron resonance, where ␻ ⫽n is an integer. This means
that the factor (1⫺ ␳␴ xy ) ⫺1 may be arbitrary.
In the following we shall use the random phase approximation 共RPA兲, according to which the composite fermions
are considered as noninteracting particles. It is known15,16
035301-2
ACOUSTOELECTRIC CURRENT FOR COMPOSITE FERMIONS
PHYSICAL REVIEW B 64 035301
冕
that if we are to have a consistent Fermi liquid theory we
must also include the Landau interaction parameters. This
will affect the CF mass, and and will possibly give some
additional effects. This is an interesting possibility for further
work.17
Even within the RPA, the problem is not equivalent to the
corresponding problem for electrons in the effective magnetic field B*
0 , because of the fluctuating Chern-Simons
fields associated with the density modulation induced by the
SAW.
where ␤ ⫽ v / v s Ⰷ1. The second order approximation will
have components with frequency 0 and 2⍀. Since we are
only interested in the dc acoustoelectric current, it is sufficient to extract the nonoscillating part of the response. Thus
for the second iteration the equation can be simplified to
III. ACOUSTOELECTRIC CURRENT
Fⵜ p f 共 r,p,t 兲 ⫽C 兵 f 其 .
A. Solution of the Boltzmann equation
The acoustoelectric current will be expressed through the
nonequilibrium distribution function for composite fermions,
f (r,p,t). In this section, we will calculate this function from
the composite fermion Boltzmann equation. As long as the
amplitude of the acoustic wave is sufficiently small we can
treat the induced piezoelectric field as a perturbation. That
means that the Boltzmann equation can be first linearized
and then solved iteratively. Thus, we seek a solution to the
equation
共 ⳵ / ⳵ t⫹vⵜ r⫹Fⵜ p兲 f 共 r,p,t 兲 ⫽C 兵 f 其
共13兲
of the form
f ⫽ f 0 关 ⑀ p⫹e⌿ 共 r,t 兲兴 ⫹e⌿ 共 ⫺ ⳵ f 0 / ⳵ ⑀ 兲 f 1 共 r,t 兲
⫺ 共 e 兩 ⌿ 0 兩 q/2m 兲 f 2 共 r,t 兲 ,
2
2
f 1⫽
d ␾ ⬘ 共 1⫹ ␤ ␦ sin ␾ ⬘ 兲
共17兲
共18兲
⳵f0
v• 共 ẑ⫻jae兲 .
e 兩 ⌿ 0兩 q ␻ c ⳵ ⑀
共19兲
2
Here L̂ 0 ⫽ ⳵ / ⳵ ␾ ⫹ ␯ ,
1
␻c
再冉
cos ␾
冋
˜ sin ␾
⫹ ␳␴
共15兲
The quantity 2 ␲ ␬ has the meaning of the ratio of the CF
cyclotron radius to the acoustic wavelength.
The linearized equation has the periodic solution
␾
2m ␳
L̂ 0 f 2 ⫽S 共 v , ␾ 兲 ⫺
共14兲
共16兲
⫺1
In addition, since we are using complex notation, to get the
dc component from the term proportional to Eⵜ p f 1 we have
to take the complex conjugate of the electric field and divide
by 2. That is, the nonoscillatory component of the product of
two oscillating functions A(r,t)⬅Re 关 A 1 e ⫺i(⍀t⫺q•r) 兴 and
B(r,t)⬅Re 关 B 1 e ⫺i(⍀t⫺q•r) 兴 is equal to Re (A 1 B *
1 /2).
Writing the second order approximation according to Eq.
共14兲 we get the linear equation
S 共 v , ␾ 兲 ⫽⫺
The important scattering mechanism for composite fermions
is scattering by the random magnetic 共Chern-Simons兲 field
which is created by density fluctuations in the electron gas
because of electrostatic potentials from impurities in the doping layer.5 Further, it is known that to ensure particle number
conservation care has to be taken in writing the collision
operator, as emphasized by Mirlin and Wölfle.13 However,
the details of the scattering mechanism are not expected to
change the results in a qualitative way 共see, e.g., Ref. 15兲,
and for simplicity we will use the relaxation time approximation C 兵 f 1 其 ⫽⫺ f 1 / ␶ . This will greatly simplify the calculations.
The Boltzmann equation is most conveniently expressed
in terms of polar coordinates ( v , ␾ ) in the ( v x , v y ) plane,
where v ⫽ 兩 v兩 is the absolute value of the velocity and ␾ is
the angle between the velocity and the x axis. In these coordinates the linearized equation has the form L̂ ␻ f 1 ⫽i ␻
⫹i ␬ ␦ sin ␾, where
L̂ ␻ ⫽ ⳵ / ⳵ ␾ ⫹ ␯ ⫹i 共 ␬ cos ␾ ⫺ ␻ 兲 .
e
␾ ⫹2 ␲
⫻e ( ␯ ⫺i ␻ )( ␾ ⬘ ⫺ ␾ )⫹i ␬ (sin ␾ ⬘ ⫺sin ␾ ) ,
where f 0 is the unperturbed solution 共with E⫽0 but the constant magnetic field included兲. The force must be taken as
F⫽⫺eE⫺ 共 e/c 兲关 v⫻ 共 B0* ⫹bac兲兴 .
i␻
2 ␲ ( ␯ ⫺i ␻ )
⫺
冊冉
册冉
冊冎
⳵ 1
⳵
⫺ sin ␾
⳵v v
⳵␾
⳵ 1
⳵
⫹ cos ␾
⳵v v
⳵␾
冉
˜ ⳵ ⳵f0
␤ ␳␴
Im 兵 e i ␩ f 1 其
v ⳵␾ ⳵⑀
⳵f0
Im f 1
⳵⑀
冊
⳵f0
Im 兵 e ⫺i ␩ f 1 其
⳵⑀
.
冊
共20兲
The last term in the S( v , ␾ ) are the CS contributions. The
periodic solution for f 2 is
f 2共 v , ␾ 兲 ⫽
1
e 2 ␲␯ ⫺1
冕
␾ ⫹2 ␲
␾
d ␾ ⬘S 共 v , ␾ ⬘ 兲 e ␯(␾⬘⫺␾)
⫺
2m ␳
⳵f0
1
2
␲␯
e 兩 ⌿ 0 兩 q ␻ c ⳵ ⑀ e ⫺1
⫻
冕
2
␾ ⫹2 ␲
␾
d ␾ ⬘ v• 共 ẑ⫻jae兲 e ␯ ( ␾ ⬘ ⫺ ␾ ) .
共21兲
This is the solution of the Boltzmann equation to second
order in the perturbation, from which we will now calculate
the acoustoelectric current.
B. Calculation of the direct current
Neither the equilibrium distribution nor the first order perturbation f 1 will give any contribution to the direct current,
so the lowest order contribution is found from the second
order perturbation. Let us for the moment forget the last term
in the expression 共21兲, which comes from the Chern-Simons
035301-3
J. BERGLI AND Y. M. GALPERIN
PHYSICAL REVIEW B 64 035301
field created by the dc acoustoelectric current, and calculate
the current jae
0 from the first term,
再 冎
ae
j 0,x
⫽
j ae
0,y
e 3兩 ⌿ 0兩 2q
2m
⫻
冕
冕
⬁
d vv 2
0
␾ ⫹2 ␲
2␲
0
d␾⬘
␾
冕
d␾
再 冎
cos ␾
˜ Ã 0
J3 ⫽ ␳␴
sin ␾
S 共 v , ␾ ⬘ 兲 e ␯(␾⬘⫺␾)
共 e 2 ␲␯ ⫺1 兲
.
共22兲
The first term of S( v , ␾ ⬘ ), Eq. 共20兲, can be integrated by
parts over v to get
e 3兩 ⌿ 0兩 2q
2m ␻ c
冕
⬁
d vv
0
冕
⳵f0
⳵⑀
2␲
0
d␾
再 冎
cos ␾
F共 ␾ 兲
sin ␾ e
2 ␲␯
⫺1
共23兲
␾ ⫹2 ␲
␾
冋
d ␾ ⬘ 2 cos ␾ ⬘ Im f 1 ⫹sin ␾ ⬘
⳵␾⬘
˜
⫹ ␤ ␳␴
⳵ Im 兵 e ⫺i ␩ f 1 其
⳵␾⬘
⳵ Im 兵 e i ␩ f 1 其
⳵␾⬘
册
e
␯(␾⬘⫺␾)
共24兲
.
Im f 1 ⫽
兺 A ne
n⫽⫺⬁
i␩
Im 兵 e f 1 其 ⫽
,
兺 Ã n e in ␾ ,
n⫽⫺⬁
where the A n and à n will be functions of v . The angular
integrals are then reduced to the six cases
冕
2␲
0
d␾
再 冎 冕
cos ␾
sin ␾
e ⫺␯␾
␾ ⫹2 ␲
␯ 2 ⫹1
␾
再 冎
sin ␾ ⬘
␲ e 3兩 ⌿ 0兩 2q
m ␻ c 共 ␯ 2 ⫹1 兲
兺
i⫽1
where
␳ˆ CS ⫽ ␳
␴ˆ ⫽
␴ 0␯
冉
␯
冕
d vv
0
⳵f0
⳵⑀
jae
0⬅
再 冎
J1 ⫽A 0
再冎
␯
1
共29兲
e 兩 ⌿ 0兩 2q
.
ប ␻ c␳
共30兲
1
⫺1
0
冊
.
冊
n ee 2␶
,
␴ 0⫽
m
,
e ( ␯ ⫹in) ␾ ⬘ ,
共31兲
which gives
共32兲
where
M̂ ⫽ ␴ˆ ␳ˆ CS ⫽
2 ␯ *f
1⫹ ␯
2
冉
1
⫺␯
␯
1
冊
,
⑀F
⫽ ␯ *f .
ប␻c
Here ␯ *f is the filling factor of composite fermions 共the filling factor in the effective field B 0* ).
4
兺 Ji共 v 兲
i⫽1
共25兲
C. Results
The Fourier coefficients A n ,
A n⫽
,
,
0
␯
1⫹ ␯ 2 1
where
ae
j 0,x
j ae
0,y
冉
⫺1
1
⬁
j 0⫽
Ji 共 v F 兲 ,
jae⫽ 共 1⫹M̂ 兲 ⫺1 jae
0 ,
where all combinations of the expressions in the braces are
implied. These can be evaluated directly, and it is found that
only the zeroth, A 0 ⫽Im f 1 , and first Fourier components
will give contributions. The current is then given by
jae
0⫽
.
␦ jae⫽ ␴ˆ eae⫽⫺ ␴␳ˆ CS jae,
cos ␾ ⬘
d␾⬘
␯
Using the Drude conductivity for ␴ we have
⬁
in ␾
⫺1
4
j0
Since Im f 1 and Im 兵 e i ␩ f 1 其 are periodic functions of ␾ they
can be expanded in Fourier series,
⬁
共28兲
,
␯
We must now return to the last item in Eq. 共21兲. Since jae is
a constant, the integral may be evaluated directly and the
resulting equation solved for jae in terms of jae
0 . There is,
however, a simpler and physically more transparent way of
obtaining the same result. We calculate jae
0 as before. Then
ae
ae
ae
ae
we write j ⫽j0 ⫹ ␦ j0 , where ␦ j is the response to the CS
electric field created by the acoustoelectric current,
⳵ Im f 1
˜ sin ␾ ⬘ Im 兵 e ⫺i ␩ f 1 其
⫹2 ␳␴
˜ cos ␾ ⬘
⫺ ␳␴
共27兲
,
1
At low temperatures we may assume ⳵ f 0 / ⳵ ⑀
⫽⫺(m/4 ␲ ប 2 ) ␦ ( ⑀ ⫺ ⑀ F ), and we can perform the integral
over v , which will fix v at v F 共and ␤ at v F / v s ). The final
result is then
where
冕
␯
⫺1
˜ Re à 1
J4 ⫽⫺ ␤ ␳␴
jae
0⫽
F共 ␾ 兲⬅
再冎
再 冎
再 冎
˜ Im à 1
J2 ⫽⫺ ␤ ␳␴
共26兲
1
2␲
冕
2␲
0
d ␾ e ⫺in ␾ Im f 1 ,
共33兲
and à n , given by a similar expression, cannot be evaluated
in closed form for arbitrary ␻ and ␬ . However, in the most
interesting situation the acoustic wavelength appears much
smaller than the CF cyclotron radius, and ␬ Ⰷ1. In this paper
035301-4
ACOUSTOELECTRIC CURRENT FOR COMPOSITE FERMIONS
PHYSICAL REVIEW B 64 035301
FIG. 1. The path of the composite fermions relative to the
acoustic wave. The interaction is efficient only at the turning points
labeled 1 and 2.
we restrict ourselves to this case. It will be shown later that
our calculation method remains valid provided ␯
ⱗ(1/␲ )ln ␬. Then the coefficients A n and à n can be evaluated using the method of stationary phase. The critical points
are approximately ␾ ⫽⫾ ␲ /2, which is consistent with the
physical picture that the CF will interact significantly with
the acoustic wave only when the electron momentum is normal to the direction of propagation of the wave. For other
directions of the momentum, the CF will be subject to a
rapidly oscillating force giving no net contribution 共see Fig.
1兲. It is not convenient to calculate these coefficients directly,
because the expression for Im f 1 is complicated. A simpler
way is to calculate the coefficients
B n⫽
1
2␲
冕
2␲
0
d ␾ e ⫺in ␾ f 1
* )/2i, and similarly for à n .
and then calculate A n as (B n ⫺B ⫺n
The result is 共see the Appendix兲
A 0 ⫽Re
冋
1⫹z 2 ⫹2z 共 sin 2 ␬ ⫺i ␤ ␦ cos 2 ␬ 兲
␤ 共 z 2 ⫺1 兲
à 0 ⫽cos ␩ Re B 0 ⫹sin ␩ Im B 0 ,
à 1 ⫽⫺
冋 册
冋 册
册
,
共34兲
共35兲
冋 册
ze ⫺i ␩
z 2 ⫹1
2i
1
˜ Re
cos 2 ␬ Im 2
⫺ sin ␩ ⫺i ␳␴
␤
z ⫺1 ␤
z 2 ⫺1
˜ sin 2 ␬ Re
⫹2i ␳␴
z2
z 2 ⫺1
,
共36兲
where z⫽e ␲ ( ␯ ⫺i ␻ ) . These expressions are then to be inserted
into the formula 共30兲 for the acoustoelectric current.
In the last expressions we have neglected terms suppressed by factors of ␤ ⫺1 or ( ␤ ␻ ) ⫺1 ⫽ ␬ ⫺1 , even if these
were of larger power in e ␲␯ . This is justified as long as e ␲␯
is smaller than ␬ , or ␯ ⱗ(1/␲ )ln ␬. This inequality sets the
limit of applicability of the stationary phase approximation.
IV. DISCUSSION
The final expressions 共34兲–共36兲 are not very simple, and
we will try to understand how they behave as the external
magnetic field is changed. Also, we will see how this affects
the acoustoelectric current 共30兲.
The Fourier coefficients show two kinds of oscillation:
FIG. 2. The graph shows geometrical oscillations. The first cyclotron resonance is seen as a broad increase in the amplitude
around ␻ ⫽1.
geometric oscillations and cyclotron resonance. We analyze
the behavior of these as functions of increasing dimensionless parameter ␻ . This corresponds to decreasing effective
magnetic field at fixed acoustic frequency ⍀. As the magnetic field changes, the value of ␯ ⫽( ␶ ␻ c ) ⫺1 ⫽˜␯ ␻ will also
change. Here ˜␯ ⫽(⍀ ␶ ) ⫺1 . Consider, for example, A 0 , Eq.
共34兲. First, there are the terms sin 2␬ and cos 2␬, which give
geometric oscillations as for metals.18–20 The oscillations
arise as the difference in the phase of the sound wave at the
two points of the cyclotron orbit where the CF’s interact
efficiently with the sound wave is changing. There is a complete oscillation as the diameter of the cyclotron orbit increases by one wavelength (2 ␬ increases by 2 ␲ ). On top of
this come oscillations from the e i ␲␻ terms. Since ␤ ⬇30
Ⰷ1, these oscillations are much slower. They describe cyclotron resonances, which are determined by the relative
phase of the sound wave as the CF pass through the same
point of the cyclotron orbit at successive revolutions. If the
CF experiences the same phase every time it passes a specific
point, it will resonate, and the interaction will be strong. This
happens when the acoustic frequency is an integer multiple
of the cyclotron frequency, i.e., when ␻ ⫽⍀/ ␻ c ⫽n is an
integer. Finally there will be an overall damping as ␯ is
growing, so that for e ␲␯ Ⰷ1 no interesting behavior is expected.
This behavior is seen in Fig. 2, which shows the expression
cos 2 ␬ Re
z
z 2 ⫺1
⫽cos 2 ␬ Re
冋
e ␲ ( ␯ ⫺i ␻ )
e 2 ␲ ( ␯ ⫺i ␻ ) ⫺1
册
,
which comes from the last term in the numerator of A 0 , as a
function of ␻ for the case ␤ ⫽30 and ˜␯ ⫽0.5. Our approximations are valid for ␯ ⱗ(1/␲ )ln ␬ and ␬ Ⰷ1. For these values of the parameters, this corresponds to the range 0.04
⬍ ␻ ⬍2.8. We will now concentrate on the most realistic
limit of large damping ␯ ⬎1 关remember, however, that we
must have ␯ ⬍(1/␲ )ln ␬ for our stationary phase approximation to be valid兴. In this limit we have 1Ⰶe ␲␯ Ⰶe 2 ␲␯ , and the
Fourier components can be expanded in powers of 兩 z 兩 ⫺1
⫽e ⫺ ␲␯ . To lowest order we get
035301-5
J. BERGLI AND Y. M. GALPERIN
PHYSICAL REVIEW B 64 035301
A 0 ⫽ ␤ ⫺1 ⫹2 ␤ ⫺1 e ⫺ ␲␯ 关 cos ␲␻ cos 2 ␬
˜ e ⫺ ␲␯ cos 2 ␬ sin共 ␲␻ ⫹ ␩ 兲 ,
A 0 ⫽2 ␳␴
⫹ ␤ 兩 ␦ 兩 cos 2 ␬ sin共 ␲␻ ⫹ ␩ 兲兴 ,
˜ e ⫺ ␲␯ cos 2 ␬ sin ␲␻ ,
à 0 ⫽2 ␳␴
à 0 ⫽ ␤ ⫺1 cos ␩ ⫹2 ␤ ⫺1 e ⫺ ␲␯ 关 cos共 ␲␻ ⫺ ␩ 兲 sin 2 ␬
˜.
à 1 ⫽⫺ ␤ ⫺1 sin ␩ ⫺i ␳␴
⫹ ␤ 兩 ␦ 兩 cos 2 ␬ sin ␲␻ 兴 ,
à 1 ⫽⫺ ␤
⫺1
We see that all the coefficients are of the same magnitude,
except that Re à 1 Ⰶ Im Ã. Looking back to the expression
共30兲 for the acoustoelectric current, we see that only one
term is relevant, and we get
˜ ⫹2ie ⫺ ␲␯ 关 兩 ␦ 兩 sin 2 ␬ cos ␲␻
sin ␩ ⫺i ␳␴
⫺ ␤ ⫺1 cos 2 ␬ sin共 ␲␻ ⫺ ␩ 兲兴 .
˜ occurs repeatedly in these
We observe that the factor ␳␴
expressions, and also in the expression 共30兲 for the acoustoelectric current. We will therefore analyze this expression.
To this end we recall the definition 共10兲 of the quantity ␦ and
use the expressions obtained by Mirlin and Wölfle for ␴ xx
and ␴ xy 关Eq. 共8兲 of Ref. 13兴. In the limit ␬ Ⰷ1 we can expand the Bessel functions in asymptotic series in 1/␬ and
obtain ␳␴ xx ⫽⫺i/ ␨ , and ␳␴ xy is given by Eq. 共12兲. This
gives
冋
␦ ⫽⫺i ␨ ⫺
cos 2 ␬
sin ␲ 共 ␻ ⫹i ␯ 兲
册
jae
0 ⫽⫺
M̂ ⫺1 ⫽
共37兲
Substituting reasonable values ␻ /2␲ ⫽3 GHz, v s ⫽3
⫻105 cm/s, m⬇10⫺27 g, we obtain ␨ ⬇1/10. We consider now two cases.
共1兲 At cyclotron resonance, where ␻ ⫽n for an integer n.
In this case we have sin ␲(␻⫹i␯)⫽⫾i sinh ␲␯, the sign being
⫹ or ⫺ when n is even or odd, respectively. In the denominator of Eq. 共37兲 we then have a real contribution from ␨ and
a purely imaginary contribution ⫾i cos 2␬/sinh ␲␯. Thus, we
get
再冎
␯
1
Im à 1 .
1
冉
1
⫺␯
2 ␯ *f ␯
1
冊
,
which gives
jae⫽⫺ j 0
再冎
˜ 0
␤ ␳␴
Im à 1 .
2 ␯ *f 1
共38兲
We see that all the current is in the y direction.
Taking into account all terms we find the ‘‘Hall angle’’
given by
˜ ⫽ 兩 ␦ 兩 ⭐ ␨ ⫺1 ⬇10.
␳␴
tan ␪ H ⫽
At the same time, since sinh ␲⬇10 we will have
jy
jx
⫽⫺
˜ ⭓1/␨ 冑2⬇7.
␳␴
˜ is not oscillating very much. Observe also
We see that ␳␴
that for moderately large ␯ ⬇1 the angle ␩ may show considerable oscillations.
共2兲 Midway between cyclotron resonances, ␻ ⫽n⫹1/2.
Now sin ␲(␻⫹i␯)⫽⫾ cosh ␲␯, still with ⫹ for n even and
⫺ for n odd. In this case, the two terms in the denominator
are both real, which means that for moderate ␯ we can get a
˜ large,
cancellation between the terms, which will make ␳␴
and show large oscillations. Since we have neglected terms
of order 1/␬ we expect these oscillations to be limited by
˜ ⬍ ␬ . Observe also that in this case we have ␩ ⫽⫺ ␲ /2 not
␳␴
oscillating.
˜ Ⰷ1. This means
We remark that in both cases we have ␳␴
⫺ ␲␯
is larger than 1/␤ we
that as long as the damping factor e
can neglect the 1/␤ terms, and simplify the Fourier coefficients further. For ␻ ⬇1 this is the same condition as the
range of validity of the stationary phase approximation. We
obtain
␯ 2 ⫹1
˜
␤ ␳␴
We can now use Eq. 共32兲 to calculate the total acoustoelectric current. The factor ␯ *f , being the ratio of the Fermi energy to the cyclotron energy in the effective magnetic field,
will be large. We can then approximate jae⬇M ⫺1 jae
0 . We
have
⫺1
.
j0
⬇
A 0 ⫹ ␤ 兩 ␦ 兩 Im à 1
兩 ␦ 兩 à 0 ⫹ ␤ 兩 ␦ 兩 Re à 1
␤兩␦兩
2兩␦兩e
⫺ ␲␯
cos 2 ␬ sin ␲␻ ⫺sin ␩
.
共39兲
The two terms in the denominator are of the same magnitude.
It is customary to relate the acoustoelectric current to the
absorption of acoustic energy per area. This is given by the
expression
1
P⫽ 具 Ḣ f 1 典 → 具 Re 兵 Ḣ * f 1 其 典 ,
2
where 具 ••• 典 denotes an average over the period of the acoustic wave, and the replacement indicated by the arrow is because of the use of complex notation.
035301-6
H⫽
关 P⫹ 共 e/c 兲 A兴 2
⫺e⌿
2m
ACOUSTOELECTRIC CURRENT FOR COMPOSITE FERMIONS
is the Hamiltonian, and the vector potential is given by A x
⫽A z ⫽0, A y ⫽⫺ ␦ (c/ v s )⌿. This gives rise to the CS magnetic field, and the solenoidal part (y component兲 of the CS
electric field. We then get
˜ Im à 1 兲 ,
P⫽ 共 ge 2 /2兲 兩 ⌿ 兩 2 ⍀ 共 A 0 ⫺ ␤ ␳␴
where g⫽m/2␲ ប is the density of states. Comparing with
Eq. 共38兲 in the limit ␯ ⬎1 and within the range of applicability of the stationary phase approximation, we then have
the relation between the acoustoelectric current, acoustic attenuation ⌫, and sound intensity I,
2
jae⫽
再 冎
0
␮ tyx
⌫I
.
vs
Here we have introduced the so-called traveling-wave
mobility21,22 defined as ␮ txy ⫽ j ae
y v s /⌫I. In the lowest approximation ␮ tyx ⫽1/␳ en e and n e ⫽g ⑀ F is the composite fermion concentration. Note, however, that the Weinreich relation is only approximate as written here. It is valid only as
long as we consider all the acoustoelectric current to be in
the y direction. By measuring the Hall angle one can determine how large a proportion of the current is turned by the
magnetic field, and then reconstruct the Weinreich relation to
take this into account.
It is instructive to compare the present expression for the
Weinreich relation23 with the one that would be expected in
a normal metal, which can be expressed through the dc electron conductivity
ae
⫽
jnm
再 冎
dc
␮ xx
⌫I
.
␮ dc
yx v s
共40兲
ˆ dc can be found from the electron conductivThe mobility ␮
ity which, in turn, can be expressed in terms of the composite
fermion dc conductivity 共31兲 共see, e.g., Ref. 14兲. As a result,
冉
␯ / ␯ *f
1 1
␮ˆ dc⫽
2 ␳ en e 1/␯ f
⫺1/␯ f
␯ / ␯ *f
冊
PHYSICAL REVIEW B 64 035301
which decreases monotonically with increasing magnetic
field. Our result gives
兩 ␪ H ⫺ ␲ /2兩 ⫽ 共 ␤ 兩 ␦ 兩 兲 ⫺1 共 2 兩 ␦ 兩 e ⫺ ␲␯ cos 2 ␬ sin ␲␻ ⫺sin ␩ 兲 ,
which will show both geometric oscillations and cyclotron
resonance. In the limit ␯ f →1/2 we have ␻ , ␯ Ⰷ1, which
gives 兩 ␦ 兩 ⫽1/␨ and sin ␩⫽⫺1. The Hall angle simplifies then
to
兩 ␪ H ⫺ ␲ /2兩 ⫽ 共 ␤ 兩 ␦ 兩 兲 ⫺1 ⫽បq/2p F .
This limit is outside the range of applicability of our approximation. The result is, however, correct, as can be checked by
a direct calculation for ␯ f ⫽1/2.
By measuring the acoustoelectric current one can use Eq.
共30兲 to determine the amplitude ⌽ 0 of the effective potential
acting upon the composite fermions. As a result, using the
theory6 which relates the effective potential to the coupling
constant, one can determine the coupling constant C between
the piezoelectric field and the composite fermions. Further,
we note that, whereas the non-CS part of the acoustoelectric
current can be expressed in terms of the complex, longitudinal conductivity for electrons through the Weinreich relation,
this is not possible for the CS part. This means that more
information on the system is available if one can measure the
acoustoelectric current compared to the situation where one
measures only attenuation and velocity shifts. For example,
if one measures attenuation and velocity shift, one can detere
,
mine the complex longitudinal electronic conductivity ␴ xx
but in general it is impossible to extract the composite fermion conductivities from this. Measurement of the acoustoelectric current will give one additional relation which enables one to determine one more parameter in the composite
fermion theory.
V. CONCLUSIONS
.
共41兲
Here ␯ f is the electron filling factor, which will be close to
1/2, and ␯ *f is the composite fermion filling factor 共the filling
factor in the effective field B 0* ) which is much greater than
1. Looking at the y components, we see that the two predictions agree at exactly ␯ f ⫽1/2. Since ⌫ is some function of
the magnetic field, it is natural to focus on the traveling-wave
mobility ␮ tyx ⬀ j ae
y /⌫. The normal metal expression would
predict that this quantity should increase linearly with increasing magnetic field. Our result gives instead a constant
value in the region around ␯ f ⫽1/2. In other words, within
our approximations, the mobility ␮ tyx is constant close to
␯ f ⫽1/2. We can do a similar comparison for the Hall angle.
According to Eq. 共39兲 this will be close to ␲ /2, and we
therefore expand around this point. The normal metal prediction would be
e
兩 ␪ H ⫺ ␲ /2兩 ⫽ ␴ xx
/ ␴ exy ⫽ ␯␯ f / ␯ *f ⫽ 共 ␻ ec ␶ 兲 ⫺1 ,
We have shown that, in the cases where the acoustic wave
is sensitive to the properties of the electron gas, the ac magnetic Chern-Simons field created by the induced density fluctuations in the electron gas creates forces that are of strength
comparable to the direct perturbation, and that the same is
true for the x component of the electric Chern-Simons field.
The y component of this field is smaller but plays an important role because it acts along the composite fermion trajectory at the points of strong interaction. At the second order,
necessary for the calculation of the acoustoelectric current,
the Chern-Simons field will be relevant, and we expect a
violation of the Weinreich relation. This violation can be
detected in measurements of the Hall angle. This gives additional information on the composite fermion system not attainable by linear response measurements. Further, the
acoustoelectric current gives a direct measure of the strength
of the piezoelectric field as experienced by the composite
fermions, and may therefore be used to extract the value of
the coupling constant between the acoustic wave and the
composite fermions.
035301-7
J. BERGLI AND Y. M. GALPERIN
PHYSICAL REVIEW B 64 035301
APPENDIX: ASYMPTOTIC EXPANSIONS
The stationary phase approximation is used to extract the
leading term in the asymptotic expansions for both the conductivity and the acoustoelectric current. In general, if we
have a function
F共 ␬ 兲⫽
冕
b
a
f 共 x 兲e
i ␬ g(x)
共A1兲
dx,
Here ␾ ⬘j are the stationary points of g( ␾ ⬘ ),cos ␾⬘j ⫽1/␤ ,
which are in the range ␾ ⭐ ␾ ⬘j ⭐ ␾ ⫹2 ␲ . The integral over ␾
can then be evaluated as
I⫽M
⫻
F 共 ␬ 兲 ⫽F (1) 共 ␬ 兲 ⫹F (2) 共 ␬ 兲 ,
F (1) 共 ␬ 兲 ⫽
冑␬ 兺 冑
2␲
j⫽1
e
⫾i ␲ /4
兩 g ⬙共 x j 兲兩
f 共 x j 兲 e i ␬ g(x j ) ,
where the sum is over all the stationary points of the exponent 关 g ⬘ (x j )⫽0 兴 , and the sign is ⫾ according as g ⬙ (x j )
⭵0.
冋
册
is the contribution from the end points 共see, e.g., Ref. 24兲.
For example to find B 1 we have to evaluate
I⫽
冕
2␲
0
d ␾ e ⫺i ␾ f 1 共 ␾ 兲 .
共A2兲
Inserting the expression 共17兲 for f 1 we get among other
terms the integral
I⫽M
冕
2␲
0
d ␾ cos ␾
冕
␾ ⫹2 ␲
␾
M ⫽i ␻ e ⫺2 ␲ ( ␯ ⫺i ␻ ) ,
⬃
冕 ␾
冑 ␬␲ 兺
␾ ⫹2 ␲
␾
e
2␲
d␾
cos ␾
cos ␾ ⫺1/␤
⫻
e ␯ ( ␾ ⬘ ⫺ ␾ )⫹i ␬ [g( ␾ ⬘ )⫺g( ␾ )]⫺ 共 i ␲ /4兲[sgn(sin ␾ ⬘ )⫺sgn(sin ␾ )]
兺
i, j
⫹
1
␤
j
冕
2␲
0
d␾
i
i
j
i
j
cos ␾
,
cos ␾ ⫺1/␤
共A3兲
The last integral must be expanded in powers of 1/␤ , yielding
冕
2␲
0
␯ ␾ ⬘j ⫹i ␬ g( ␾ ⬘j )⫺ 共 i ␲ /4兲sgn(sin ␾ ⬘ )
d␾
cos ␾
⫽
cos ␾ ⫺1/␤
冕 ␾冉
2␲
d
0
1⫹
冊
1
⫹••• ⫽2 ␲ ,
␤ cos ␾
j
⫺
1
i
e ␯ ␾ ⫹i ␬ g( ␾ ) 关 e 2 ␲ ( ␯ ⫺i ␻ ) ⫺1 兴 .
␬ cos共 ␾ 兲 ⫺1/␤
共A4兲
1
j
where the critical points ␾ i are the same as before. In the
sum over the critical points we can then set ␾ i , ␾ ⬘j equal to
⫾ ␲ /2⫹2 ␲ n in the exponent since ␤ Ⰷ1. Also, critical
points that lie on the boundary of the domain of integration
are given half their normal value. In the second integration,
the contributions from the end points have been neglected, as
they will be of higher order. The sum then is
d e ␯ ␾ ⬘ ⫹i ␬ [sin ␾ ⬘ ⫺ ␾ ⬘ / ␤ ]
2
1
j
共 ••• 兲 ⫽1⫹e 2 ␲ ( ␯ ⫺i ␻ ) ⫹2e ␲ ( ␯ ⫺i ␻ ) sin 2 ␬ .
兺
i, j
Performing first the integral over ␾ ⬘ we get
I ⬘⫽
兺j e ␯ ␾ ⬘⫹i ␬ g( ␾ ⬘)⫺共 i ␲ /4兲sgn(sin ␾ ⬘) ⫹ ␤ 冕0
d ␾ ⬘ e ␯ ( ␾ ⬘ ⫺ ␾ )⫹i ␬ [g( ␾ )⫺g( ␾ ⬘ )] ,
g 共 ␾ 兲 ⫽sin ␾ ⫺ ␾ / ␤ .
d ␾ cos ␾ e ⫺ ␯ ␾ ⫺i ␬ g( ␾ )
2␲M
␬␤
⫽
1 f 共 b 兲 i ␬ g(b) f 共 a 兲 i ␬ g(a)
F (2) 共 ␬ 兲 ⫽
e
⫺
e
i ␬ g ⬘共 b 兲
g ⬘共 a 兲
2␲
0
the leading terms in the asymptotic expansion will be
n
冕
A. Wixforth, J. P. Kotthaus, and G. Weimann, Phys. Rev. Lett.
56, 2104 共1986兲.
2
A. Wixforth, J. Scriba, M. Wassermeier, J. P. Kotthaus, G.
Weimann, and W. Schlapp, Phys. Rev. B 40, 7874 共1989兲.
3
R. L. Willett, R. R. Ruel, K. W. West, and L. N. Pfeiffer, Phys.
Rev. Lett. 71, 3846 共1993兲.
4
R. L. Willett, K. W. West, and L. N. Pfeiffer, Phys. Rev. Lett. 75,
since P 兰 20 ␲ d ␾ /cos ␾⫽0, where P is the principal value.
Combining these results we get one contribution to B 1 , the
rest of the B n being calculated in a similar way.
2988 共1995兲.
B. Halperin, P. A. Lee, and N. Read, Phys. Rev. B 47, 7312
共1993兲.
6
S. H. Simon, Phys. Rev. B 54, 13 878 共1996兲.
7
A. Knäbchen, Y. B. Levinson, and O. Entin-Wohlman, Phys.
Rev. B 54, 10 696 共1996兲.
8
I. L. Drichko, A. M. Diakonov, V. D. Kagan, A. M. Kreshchuk,
5
035301-8
ACOUSTOELECTRIC CURRENT FOR COMPOSITE FERMIONS
G. D. Kipshidze, T. A. Polyanskaya, I. G. Savel’ev, I. Yu.
Smirnov, A. V. Suslov, and A. Ya. Shik, Fiz. Tekh. Poluprovodn. 29, 1306 共1995兲 关Semiconductors 29, 677 共1995兲兴; I. L.
Drichko, A. M. Diakonov, A. M. Kreshchuk, T. A. Polyanskaya,
I. G. Savel’ev, I. Yu. Smirnov, and A. V. Suslov, ibid. 31, 451
共1997兲 关ibid. 31, 384 共1997兲兴; I. L. Drichko and I. Yu. Smirnov,
ibid. 31, 1092 共1997兲 关ibid. 31, 933 共1997兲兴; I. L. Drichko, A. M.
Diakonov, V. D. Kagan, A. M. Kreshchuk, T. A. Polyanskaya, I.
G. Savel’ev, I. Yu. Smirnov, and A. V. Suslov, ibid. 31, 1357
共1997兲 关ibid. 31, 1170 共1997兲兴; I. L. Drichko, A. M. Diakonov, I.
Yu. Smirnov, and A. I. Toropov, ibid. 33, 979 共1999兲 关ibid. 33,
892 共1999兲兴; I. L. Drichko, A. M. Diakonov, V. D. Kagan, I. Yu.
Smirnov, and A. I. Toropov, in Proceedings of the 24th International Conference on Physics of Semiconductors, Jerusalem,
edited by D. Gershow 共World Scientific, Singapore, 1998兲, paper no. V-F-7, PDF file no. 0629.pdf on the CD-ROM; I. L.
Drichko, A. M. Diakonov, I. Yu. Smirnov, Y. M. Galperin, and
A. I. Toropov, Phys. Rev. B 62, 7470 共2000兲.
9
J. M. Shilton, D. R. Mace, V. I. Talyanskii, M. Y. Simmons, M.
Pepper, A. C. Churchill, and D. A. Ritchie, J. Phys.: Condens.
Matter 7, 7675 共1995兲.
10
A. Efros and Y. Galperin, Phys. Rev. Lett. 64, 1959 共1990兲.
11
V. I. Fal’ko, S. Meshkov, and S. Iordanskii, Phys. Rev. B 47,
PHYSICAL REVIEW B 64 035301
9910 共1993兲.
J. M. Shilton, D. R. Mace, V. I. Talyanskii, M. Pepper, M. Y.
Simmons, A. C. Churhill, and D. A. Ritchie, Phys. Rev. B 51, 14
770 共1995兲.
13
A. D. Mirlin and P. Wölfle, Phys. Rev. Lett. 78, 3717 共1997兲.
14
S. H. Simon and B. I. Halperin, Phys. Rev. B 48, 17 368 共1993兲.
15
S. H. Simon, in Composite Fermions, edited by O. Heinonen
共World Scientific, Singapore, 1998兲.
16
A. Stern and B. I. Halperin, Phys. Rev. B 52, 5890 共1995兲.
17
The importance of the Fermi liquid effects for the acoustoelectric
effect was pointed out to us by Ady Stern. We are grateful to
him for a useful discussion.
18
H. Bömmel, Phys. Rev. 100, 786 共1955兲.
19
A. B. Pippard, Philos. Mag. 2, 1147 共1957兲.
20
V. L. Gurevich, Zh. Éksp. Teor. Fiz. 37, 71 共1959兲 关Sov. Phys.
JETP 10, 51 共1960兲兴.
21
H. Fritzsche, Phys. Rev. B 29, 7762 共1984兲.
22
Yu. M. Galperin, Anjun Jin, and B. I. Shklovskii, Phys. Rev. B
44, 5497 共1991兲.
23
G. Weinreich, Phys. Rev. 107, 317 共1957兲.
24
L. Mandel and E. Wolf, Optical Coherence and Quantum Optics
共Cambridge University Press, Cambridge, 1995兲.
12
035301-9