INSTITUTE OF PHYSICS PUBLISHING
NANOTECHNOLOGY
Nanotechnology 12 (2001) 610–613
PII: S0957-4484(01)26848-2
Nonlinear absorption of surface acoustic
waves by composite fermions
J Bergli1 and Y M Galperin1,2,3
1
2
3
Department of Physics, University of Oslo, PO Box 1048 Blindern, N-0316 Oslo, Norway
Solid State Division, A F Ioffe Physico-Technical Institute, 194021 St Petersburg, Russia
Centre for Advanced Studies, Drammensveien 78, 0271 Oslo, Norway
E-mail: jbergli@fys.uio.no
Received 14 June 2001, in final form 7 July 2001
Published 27 November 2001
Online at stacks.iop.org/Nano/12/610
Abstract
Absorption of surface acoustic waves by a two-dimensional electron gas in a
perpendicular magnetic field is considered. The structure of such a system at
the filling factor ν close to 1/2 can be understood as a gas of composite
fermions. It is shown that the absorption at ν = 1/2 can be strongly
nonlinear, while small deviation from 1/2 will restore the linear absorption.
Study of nonlinear absorption allows one to determine the force acting upon
the composite fermions from the acoustic wave at turning points of their
trajectories.
1. Introduction
Two-dimensional electron gases (2DEGs) have been intensely
studied in the last years. One of the experimental techniques
used to investigate the properties of this system is interaction
with a surface acoustic wave (SAW) [1, 2]. We shall consider
the 2DEG in the fractional quantum Hall regime where a strong
magnetic field is applied normal to the plane of the 2DEG. As
is well known, close to half filling of the lowest Landau level
the system will exhibit a metallic phase. This phase can be
described in terms of a new type of quasiparticle, called composite fermions. The theory of composite fermions, as formulated in the language of Chern–Simons field theory in [3], has
successfully explained the acoustic properties of the electron
gas [2, 4–6]. So far, however, greatest attention, both experimentally and theoretically, has been given to the linear response
regime, which is appropriate for low-intensity acoustic waves.
From the study of ultrasonic absorption by electrons in metals
it is known [7–9] that very interesting nonlinear effects can
be observed. It is therefore natural to study nonlinear effects
in the context of SAW absorption by the composite fermion
metallic state. In particular we are interested in the fact that
when nonlinear absorption occurs in a metal a weak external
magnetic field will restore the linear absorption [10].
2. Nonlinear acoustic absorption
First we need to understand how the acoustic wave interacts
with the electron gas. Two mechanisms are usually considered,
0957-4484/01/040610+04$30.00
© 2001 IOP Publishing Ltd
the deformational and the piezoelectric interactions. In the
typical materials used to create the 2DEG the deformational
interaction can be neglected, and we consider only the
piezoelectric field of the wave. Letting the x-axis point in the
direction of sound propagation this is then given by E (x, t) =
E0 sin ξ = −∇ with = 0 cos ξ . Here ξ = qx −ωt is the
wave coordinate, ω is the SAW frequency and E0 q x̂. By
we mean the screened electrostatic potential. The relation
between this and the external electrostatic potential of the SAW
will be discussed in detail in section 5.
The sensitivity to external magnetic fields was explained
for the case of electrons and bulk acoustic waves in the
following way [7]. If the wavelength of the acoustic wave
is much smaller than the mean free path of the electrons
(q −1 , where q is the wavevector of the acoustic wave),
the electrons will traverse many periods of the wave before
being scattered. If the electron moves in such a way that the
component of its velocity in the direction in which the acoustic
wave is propagating is very different from the sound velocity, it
will experience a rapidly oscillating force. Consequently, the
interaction between this electron and the acoustic wave will
be weak. Since the electron velocity (Fermi velocity) is much
larger than the sound velocity, this will be the case for most of
the electrons. Only a small group, called the resonant group,
will have their velocities in the sound propagation direction
matched to the sound velocity, and thus interact strongly with
the wave. The linear absorption, which is obtained in the
limit of an acoustic wave of small (infinitesimal) amplitude,
is determined by these resonant electrons. Consider a finite-
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610
Nonlinear absorption of surface acoustic waves by composite fermions
4. Kinetic equation
Untrapped
Untrapped
Trapped
B=0
(a)
Trapped
B=0
(b)
Figure 1. The trapping of resonant electrons in the absence (a) and
in the presence (b) of an external magnetic field B.
amplitude wave. Some of the resonant electrons will then be
trapped in the valleys of the potential of the acoustic wave
(see figure 1(a)). The trapped electrons, moving in complete
synchrony with the wave, will not contribute to the absorption,
and the absorption decreases. This leads to an amplitudedependent absorption coefficient, i.e. to nonlinear absorption.
Consider now the effect of an externally applied magnetic field.
The electrons will feel an extra force, which will act to remove
the electrons from the trapped group (see figure 1(b)). Thus, we
expect the linear absorption to be restored by the application of
a magnetic field. The important point is that the magnetic force
needed to restore the linear absorption gives a direct measure
of the trapping force from the acoustic wave. That is, from
the strength of the field needed to restore linear absorption we
can infer the strength of the force from the acoustic wave on
the electrons. These effects were observed for electrons in
semiconductors in [8, 9].
3. Composite fermions
Let us recall the main facts of the Chern–Simons theory for the
composite fermions [3]. The Chern–Simons transformation
mapping the electron system to an equivalent system of
composite fermions can be described as attaching an even
number of fictitious flux quanta of Chern–Simons magnetic
field to each electron. In the mean-field approximation
the composite fermions will experience an effective field
B0∗ = B − mφ0 n0 , where B is the external field, m is the
number of attached flux quanta, φ0 = h/e is the flux quantum
and n0 is the electron density. If m = 2, the effective
field will vanish if the Landau level filling factor ν = 1/2.
The perturbation by the acoustic wave will induce a density
modulation in the electron gas, n = n0 + δn. This will lead
to an oscillating component of the Chern–Simons magnetic
field, so that the total effective field will be B ∗ = B0∗ + bac
with bac = −2φ0 δn. In addition, the motion of the electrons
will drag with it the attached flux, and by Faraday’s law induce
a Chern–Simons electric field, eac = (2φ0 /e) [ẑ × j ]. The
y-component of this field is found from the x-component of
the current, which can be related to the density modulation
by the equation for conservation of charge. Assuming the
density modulation to be harmonic, δn = (δn)0 cos ξ , we
obtain eyac = 2φ0 vs δn. We shall later see that this assumption
is justified. As explained in [12] can we combine the xcomponent of eac with the piezoelectric field of the acoustic
wave. The corresponding total potential will be denoted .
The main objective of this work is to determine the trapping
of the electrons by the combined action of these fields, which
are not present in the previously studied electron problem.
In [13] we used the Boltzmann equation to calculate
the nonequilibrium distribution function, f , of composite
fermions, from which the absorption was found. Here we just
repeat the main results. The Hamiltonian is
H = (P + eA)2 /2m − e,
(1)
where P is the canonical momentum (the kinematic
momentum is p = P + eA), A is the vector potential.
The vector potential consists of two parts. One emerges
from the static external effective magnetic field B ∗ , and one
from the ac Chern–Simons field that is created by the SAWinduced density modulation.
The magnetic field is then bac =
2
−2
2φ0 [n0 − (2π h̄)
d P f ].
It is convenient to split the distribution function as f =
f0 (H )+f1 where f0 is the Fermi function. Then the Boltzmann
equation for f1 is
∂f1 /∂t + ∇P H ∇r f1 − ∇r H ∇P f1 + f1 /τ
= − (∂H /∂t)(∂f0 /∂H ).
(2)
Here we use the relaxation time approximation −f1 /τ for
the collision operator, which significantly simplifies the
calculations. It should be noted that the Hamiltonian (1) is
written in terms of the ac Chern–Simons magnetic field bac .
The latter must be expressed through the density modulation
as an integral over the distribution function. The Boltzmann
equation (2) is then in reality a complicated integro-differential
equation for the non-equilibrium distribution function. It is
easy to show, however, that the main contribution to the density
modulation comes from the equilibrium part f0 (H ), so that in
calculating f1 we can approximate the density modulation with
δn(0) coming from f0 (H ). Indeed, using the fact that in the
whole region of acoustic amplitudes e "F , where "F is
the Fermi energy, we can then expand f0 (H ) around the point
H = p2 /2m. The lowest-order term, δn(0) , is estimated as
δn(0) = −e(2π h̄)−2 d2 p (∂f0 /∂H )|H =p2 /2m = ge.
(3)
Here g = m/2π h̄2 is the density of states per spin (as usual,
we assume the 2DEG to be fully spin polarized). Then we can
solve equation (2) for f1 with the assumption that δn = δn(0) ,
and come back to show that the non-equilibrium correction
coming from f1 is small compared with δn(0) . This will then
justify our assumption of a harmonic density perturbation.
We shall first consider the case B ∗ = 0. Proceeding to
the solution, we note that in the resonant region vy ≈ vF and
vx ≈ s vF . The magnetic force then points mainly in
the x-direction, and the main part of this will be given by
bac vF . This may be combined with the potential to give
the effective potential % such that −∇% = (±vF bac + E)x̂,
the sign being + for particles with vy > 0 and − for vy < 0.
Using the approximation δn ≈ δn(0) and assuming the electric
potential to have the form = 0 cos ξ we can find the explicit
expression for %, %(ξ ) = %0 ψ(ξ ), where
√
(4)
ψ(ξ ) = cos(ξ ∓ θ ),
% 0 = 0 1 + α 2 ,
α = 2mvF /qh̄,
θ ≡ arctan α.
(5)
611
J Bergli and Y M Galperin
The equation for f1 can then be written as
s(∂f1 /∂ξ ) + ψ (∂f1 /∂s) + af1 = aU,
(6)
s = (vx − vs )/v̄,
(7)
with
v̄ 2 = e%0 /m,
U = −τ eω(∂%/∂ξ )(∂f0 /∂H ),
a = (q v̄τ )−1 .
(8)
The dimensionless parameter a has a clear physical meaning.
Indeed, v̄ is just a typical velocity of the particles trapped in
the potential %(ξ ), while ω0 ≡ q v̄ is their typical oscillation
frequency. Since each scattering event rotates the particle
momentum and leads to its escape from the resonant group,
nonlinear behaviour exists only if ω0 τ 1, or a 1. Thus
a is the main parameter responsible for nonlinear behaviour.
Solving the Boltzmann equation (6) by the method of
characteristics we find the distribution function from which
we calculate the absorption using the expression
d2 p
P =
Ḣ f ,
(9)
(2πh̄)2
where · · · denotes average over the period of the acoustic
wave. For the case ν = 1/2, B ∗ = 0 we find, after rather
tedious calculations, the result
P = C(e%0 /2π)2 (vs /vF ) gωa,
(10)
where C ∼ 1 is some numerical factor that can be found from
numerical integration.
From the solution of (6) we also find the correction to the
density modulation coming
2from the nonequilibrium distri(1)
bution
function
δn
=
d p f1 /(2πh̄) ∼ age%0 vs /vF ≈
√
a 1 + α 2 vs /vF δn(0) . Thus we can neglect it as long as
aαvs /vF < 1. The same suppression also occurs
in calculating
the y-component of the current jy = e d2 p vy f1 /(2πh̄)2 ,
which gives the x-component of the CS electric field. This
means that no higher harmonics will arise in the total potential, and that, in fact, the CS part will become smaller and
smaller compared with the real electric potential. That is, in
the limit a → 0 we have = .
Including in (2) a nonzero effective magnetic field, the
Boltzmann equation is transformed to the form
s(∂f1 /∂ξ ) + [b − ψ (ξ )](∂f1 /∂s) + af1 = aU,
b = vF B ∗ /q%0 .
(11)
The physical interpretation of the parameter b is the ratio of
the magnetic force from the external magnetic field to the force
from the modified electrostatic potential %(ξ ). Solving this
equation we find that as expected the absorption is restored to
the value of the linear absorption. The condition for this is that
b > 1, and the field necessary for this is B0∗ Bc = q%0 /vF .
5. Discussion
We can now see how the various fields contribute to the trapping
of the composite fermions. For electrons we would have 0
appearing in place of %0 in the expression for B√
c . Assuming
a 1√and setting 0 = 0 we have %0 = 0 1 + α 2 , the
factor 1 + α 2 describing the trapping effect of the oscillating
612
Chern–Simons magnetic field bac . Inserting reasonable values,
m = 10−30 kg, vF = 105 m s−1 , vs = 3 × 103 m s−1 and
//2π = 3 × 109 GHz, we obtain α ≈ 50. We see that the
effect of the Chern–Simons field is to considerably enhance the
efficiency of the acoustic wave in trapping composite fermions,
and consequently that the effective magnetic field necessary to
restore linear absorption will be correspondingly larger. A way
to check the above concept is first to reach nonlinear behaviour
at B = B1/2 , then restore the linear behaviour by changing the
magnetic field by a quantity Bc , without changing the SAW
intensity.
We also want to discuss the dependence of the critical
field Bc on the amplitude of the acoustic wave. In order to
understand this we must also consider the screening of the
external piezoelectric field by the electron gas. That this can
be nontrivial in the regime of strong nonlinearity can be seen
by the example of absorption of transverse electromagnetic
waves by an electron gas discussed in [11], which shows
clear parallels to the present case because of the transverse
Chern–Simons electric field. In the present case, however,
the situation is simpler, and we shall show that the screening is
satisfactorily described in the linear approximation. Screening
in a 2DEG has been discussed previously by Simon [4], with
whose results we shall compare ours.
The induced density modulation is to leading order, as
we have seen, δn(0) = ge. The induced electrostatic
potential is then I = qe δn(0) = q1 ge2 . Denoting the
external piezoelectric field by E , so that = E + I
we have
1
E .
=
1 − q1 ge2
From [4] we have the result
=
1
1−
iσxx
vs
E ,
and comparing the two we find σxx = −ie2 g qω2 . This agrees
with the results of Mirlin and Wölfle [6] in the appropriate
limit (vF vs and qvF τ 1).
Considering
the most relevant case of α 1 so that
√
%0 = 1 + α 2 0 ≈ α0 we obtain that the critical magnetic
0
. This
field will approach the constant value Bc = 2m
h̄
could be tested experimentally by measuring the critical field
as a function of q (changing q is not straightforward, but
can be achieved by using higher-harmonic generation on the
interdigital transducers, or multiple parallel transducers on the
same sample).
The fact that the screening is linear also renders the
situation similar to that considered for electrons in [10], and we
can write the same interpolation formula as was obtained there
2 ∼ a20 + b20 .
(12)
This applies as long as a, b 1.
In conclusion, we have shown how the effective coupling
between the SAW and the composite fermions can be found by
measuring the restoration of the linear absorption by a magnetic
field. Also, the trapping effect of the Chern–Simons magnetic
field can be studied by determining the critical field Bc as a
function of q.
Nonlinear absorption of surface acoustic waves by composite fermions
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