Fermi-liquid effects in the transresistivity in quantum Hall double layers near ν = 1/2
Natalya A. Zimbovskaya
Department of Physics and Astronomy, St. Cloud State University,
720 Fourth Avenue South, St. Cloud, MN 56301, USA;
Urals State Mining University, Kuibysheva Str. 30, Yekaterinburg, Russia, 620000
(Dated: November 4, 2018)
Here, we present theoretical studies of the temperature and magnetic field dependences of the
Coulomb drag transresistivity between two parallel layers of two dimensional electron gases in
quantum Hall regime near half filling of the lowest Landau level. It is shown that Fermi-liquid
interactions between the relevant quasiparticles could give a significant effect on the transresistivity,
providing its independence of the interlayer spacing for spacings taking on values reported in the
experiments. Obtained results agree with the experimental evidence.
PACS numbers: 71.27.+a 73.43.-f
During the last decade double-layer two-dimensional
electron gas (2DEG) systems were of significant interest
due to many remarkable phenomena they exhibit, includ-
ing so called Coulomb drag. In Coulomb drag experi-
ments two 2DEGs are arranged close to each other, so
that they can interact via Coulomb forces. A current I
is applied to one layer of the system, and the voltage VD
in the other nearby layer is measured, with no current
allowed to flow in that layer. The ratio −VD/I gives a
transresistivity ρD which characterizes the strength of
the effect. The physical interpretation of the Coulomb
drag is that momentum is tranferred from the current
carrying layer to the nearby one due to interlayer inter-
actions [1, 2, 3].
It was shown theoretically [4, 5] and confirmed with
experiments [5] that the transresistivity between two
2DEGs in quantum Hall regime at one half filling of
the lowest Landau level for both layers is proportional
to T 4/3 (T is the temperature of the system) which is
quite different from the temperature dependence of ρD
in the absence of the external magnetic field applied to
2DEGs. This temperature dependence of the drag at
ν = 1/2 originates from the ballistic contribution to
the transresistivity. The latter reflects the response of
the two-layer system to the driving disturbance of finite
wave vector q and finite frequency ω when considering
scales are smaller than the mean free path l of electrons
(ql ≫ 1) , and times are shorter than their scattering
time τ (ωτ ≫ 1) [6].
In further experiments [7] the Coulomb drag was mea-
sured between 2DEGs where the layer filling factor was
varied around ν = 1/2. The transresistivity was re-
ported to be enhanced quadratically with ∆ν = ν− 1/2.
It was also reported that the curvature of the enhance-
ment depended on temperature but it was insensitive to
both sign of ∆ν and distance d between the layers. The
present work is motivated with these experiments of [7].
We calculate the transresistivity between two layers of
2DEGs subject to a strong magnetic field which provides
ν close to 1/2 for both layers.
We start from the well-known expression [1, 3] which
relates the Coulomb drag transresistivity to density-
density components of the polarization in the layers
Π(1)(q, ω) and Π(2)(q, ω) :
2(2π)2
(2π)2
sinh2(h̄ω/2T )
∣U(q, ω)
ImΠ(1)(q,ω)ImΠ(2)(q,ω). (1)
Here, U(q, ω) is the screened interlayer Coulomb inter-
action, and electron densities in the layers are supposed
to be equal (n1 = n2 = n).
Within the usual Composite Fermion (CF) approach
[8] a single layer polarizability describes that part of the
density-current electromagnetic response which is irre-
ducible with respect to the Coulomb interaction. Adopt-
ing for simplicity the RPA, we obtain the following ex-
pression for the 2× 2 polarizability matrix:
Π−1 = (K0)−1 + C−1. (2)
Here, the matrix K0 gives the response of noninteract-
ing CFs and C is the Chern-Simons interaction matrix.
Assuming for certainty the wave vector q to lie in the
”x” direction we have:
. (3)
Starting from the expression (2) we arrive at the fol-
lowing results for the density-density response function
Π00(i)(q, ω) :
Π00(i)(q, ω) = Π(i)(q, ω)
00(i)
(q,ω)
8iπh̄
K001(i)(q,ω)−
∆(i)(q,ω)
. (4)
http://arxiv.org/abs/0704.0246v1
∆(i)(q,ω) = K
00(i)(q,ω)K
11(i)(q,ω) +
K001(i)(q,ω)
Within the RPA response functions included in Eqs. (4),
(5) are simply related to components of the CF conduc-
tivity tensor σ̃ [8]:
xx(q,ω)
00(i)
(q,ω)
00(i)
(q,0)
σ̃(i)yy (q,ω) = −
K011(i)(q,ω)−K
11(i)(q,0)
σ̃(i)xy = −σ̃
K001(i)(q,ω). (6)
To proceed we calculate the components of the CF
conductivity at ν slightly away from 1/2. In this case
CFs experience a nonzero effective magnetic field Beff =
B−B1/2. We concentrate on the ballistic contribution to
the transresistivity, so we need asymptotics for the rel-
evant conductivity components applicable in a nonlocal
(ql ≫ 1) and high frequency (ωτ ≫ 1) regime. Cor-
responding expressions for σ̃ij were obtained in earlier
works [8]. However, these results are not appropriate for
our analysis for they do not provide a smooth passage to
the Beff → 0 limit at finite q. Due to this reason we
do not use them in further calculations. To get a suitable
approximation for the CF conductivity we start from the
standard solution of the Boltzmann transport equation
for the CF distribution function. This gives us the fol-
lowing results for the CF conductivity components for a
single layer [9]:
σ̃αβ =
(2πh̄)2
dψvα(ψ) exp
′′)dψ′′
′) exp
′′)dψ′′
(ψ′ − ψ)(1− iωτ)
dψ′. (7)
Here, m∗, Ω are the CF effective mass and the cyclotron
frequency at the effective magnetic field Beff ; ψ is the
angular coordinate of the CF cyclotron orbit. Now we
carry out some formal transformations of this expression
(7) following the way proposed before [9, 10]. First, we
expand the CF velocity components vβ(ψ
′) in Fourier
series:
vkβ exp(ikψ
′). (8)
Substituting this expansion (8) into (7) we obtain:
(2πh̄)2
dψvα(ψ) exp(ikψ)
ikΩ− iω +
+ iqvx(ψ)
vx(ψ +Ωθ
′)− vx(ψ)
dθ (9)
where θ = (ψ′ − ψ)/Ω.
Then we introduce a new variable η which is related
to the variable θ as follows:
ikΩ− iω +
+ iqvx(ψ)
vx(ψ +Ωθ
′)− vx(ψ)
dθ′, (10)
and we arrive at the result:
σ̃αβ =
im∗e2
(2πh̄)2
vα(ψ) exp(ikψ)
ω + i/τ − kΩ− qvx(ψ +Ωθ)
dψ. (11)
Under the conditions of interest ωτ ≫ 1, ql ≫ 1, and
also assuming that the filling factor is close to ν = 1/2,
so that qvF ≫ Ω (vF is the CFs Fermi velocity), the
variable θ is approximately equal to ητ(1 + iql cosψ +
ikΩτ − iωτ)−1. Taking this into account and expanding
the last term in the denominator of (11) in powers of Ωθ
we obtain:
qvx(ψ +Ωθ)
≈ qvx(ψ) + ηΩqτ(1 + iql cosψ + ikΩτ − iωτ)−1
(Ωτ)2(1 + iql cosψ + ikΩτ − iωτ)−2
. (12)
Substituting this asymptotic expression into (9) we can
calculate first terms of the expansions of relevant compo-
nents of the CF conductivity in powers of the small pa-
rameter (qR)−1 where R = vF /Ω is the CF cyclotron
radius. Within the ”collisionless” limit 1/τ → 0 we
have:
σ̃xx = −N
1− δ2
(1− δ2)5
2(qR)2
1− δ2
; (13)
σ̃yy = N
1− δ2 + iδ +
2(qR)2
(1− δ2)5
(1− δ2)3
; (14)
σ̃xy = iN
1− δ2
(1− δ2)3
. (15)
Here, N = m∗/2πh̄2 is the density of states at the
CF Fermi surface, and δ = ω/qvF . Using these re-
sults we can easily get approximations for the functions
αβ(i)
(q, ω) (α, β = 0.1) and, subsequently, the de-
sired density-density response function given by (4). It
was shown [3] that the integral over ω in the expres-
sion for ρD (1) is dominated by ω ∼ T, and the ma-
jor contribution to the integral over q in this expres-
sion comes from q ∼ kF (T/T0)1/3, where kF is the
Fermi wave vector and the scaling temperature T0 is de-
fined below. Therefore we get an estimate for δ, namely
δ ∼ (T/µ)(T0/T )1/3, where µ is the chemical potential
of a single 2DEG included in the bilayer. For the param-
eter T0 taking on values of the order of room tempera-
ture, δ is small compared to unity at low temperatures
(T ∼ 1K).
Here, we limit ourselves to the case of two identical
layers (Π(1) = Π(2) ≡ Π) . For δ ≪ 1 we obtain the
approximation:
Π00(q, ω)
− 8πih̄ωkF
1 + 2(kFR)
(qR)−2
Here, dn/dµ is the compressibility of the ν = 1/2 state
which is defined as [3]:
≡ Π00(q → 0; ω → 0) =
8πh̄2
. (17)
This differs from the compressibility of the noninteracting
2DEG in the absence of an external magnetic field (the
latter equals N). The difference in the compressibility
values is a manifestation of the Chern-Simons interaction
in strong magnetic fields.
In following calculations we adopt the expression used
in the work [3] for the screened interlayer potential
U(q,ω), namely:
U(q,ω) =
Vb + Ub
1 + Π(q,ω)(Vb + Ub)
Vb − Ub
1 + Π(q,ω)(Vb − Ub)
where Vb(q) = 2πe
2/ǫq and Ub(q) = (2πe
2/ǫq)e−qd
are Fourier components of the bare Coulomb potentials
for intralayer and interlayer interactions, respectively,
and ǫ is the dielectric constant. Substitung (18) into
(1) and using our result (16) for Π(q, ω) we can present
the transresistvity in the ”ballistic” regime as:
ρD = ρD0 + δρD. (19)
Here, the first term ρD0 is the transresistivity at ν = 1/2
when the effective magnetic field is zero, and the sec-
ond term gives a correction arising in a nonzero effective
magnetic field (away from ν = 1/2 ). As it was to be
expected, our expression for ρD0 coincides with the al-
ready known result [3]:
ρD0 =
Γ(7/3)ζ(4/3)
with T0 =
πe2nd/ǫ
(1 + α), and
2πe2d
. (21)
The leading term of the correction δρD at low tempera-
tures (T/T0) ≪ 1 can be writen as follows:
δρD =
(kFR)2
ρD0∆ν
+ 4a2
(∆ν)2.
Here, the dimensionless positive constant a2 can be ap-
proximated as:
sinh2 y
y4/3 cosh2 y
dy. (23)
We have to remark that our result (23) cannot be
used in the limit T → 0. Actually, this expression pro-
vides a good asymptotic form for the coefficient a2 when
(TkF l/µ)
1/3 ≥ 1.5. Assuming that the mean free path
is of the order of 1.0µm as in the experiments [11] on
dc magnetotransport in a single modulated 2DEG at ν
close to 1/2, and using the estimate of [7] for the electron
density n = 1.4 × 1015m−2, we obtain that the expres-
sion (23) gives good approximation for a2 when T/µ is
no less than 10−2 .
It follows from our results (19), (22) that transresis-
tivity ρD enhances nearly quadratically with ∆ν when
the filling factor deviates from ν = 1/2. The linear in
∆ν term is also present in the expression for δρD. This
causes an asymmetric shape of the plot of Eq. (22) rel-
ative to ∆ν = 0. However, this asymmetry is not very
significant for the linear term is smaller than the last
term on the right hand side of (22). This difference in
magnitudes is due to different temperature dependences
of the considered terms. The first term including the lin-
ear in (kFR)
−1 correction is proportional to (T/T0)
whereas the second one is proportional to (T/T0)
2/3 and
predominates at low temperatures. So, the magnetic field
dependence of the transresistivity near ν = 1/2 matchs
that observed in the experiments (See Fig. 1).
Keeping only the greatest term in (22), the ratio
ρD/ρD0 can be presented in the form:
= 4β(∆ν)2 + 1. (24)
FIG. 1: Scaled drag resistivity versus ∆ν at T = 0.6; lowest
dashed curve is the plot of Eq. (22) at m∗ = 4mb; A0 = 15,
and remaining curves present experimental data of [7];
Here, the coefficient β equals:
Γ(7/3)ζ(4/3)
. (25)
This coefficient is proportional to the curvature of the
plot of Eq. (22) assuming that the first term is neglected.
The curvature reveales a strong dependence on tempera-
ture whose character also agrees with experiments of [7]
as it is shown in Fig. 2.
A striking feature in the experimental results is that
they appear to be nonsensitive to the distance between
the 2DEGs. Sets of data corresponding to samples with
different interlayer spacings dA = 10nm and dB =
22.5nm fall on the same curve. This concerns both mag-
netic field dependence of the transresistivity and tem-
perature dependence of the parameter β . Results of the
present analysis provide a possible explanation for this
feature. It follows from (20)–(25) that the dependence
of ρD of the interlayer spacing is completely included in
the characteristic temperature T0 which is defined with
Eq. (21). The above quantity is nearly independent of
the interlayer separation d when the parameter α takes
on values larger that unity. Estimating the parameter
α as it is given by Eq. (21), we obtain that the con-
dition α > 1 could be satisfied for small values of the
compressibility of the ν = 1/2 state. However, within
the RPA the effective mass of CFs coincides with the
single electron band mass mb which takes on the value
mb ≈ 0, 07me for GaAs wells (me is the mass of a free
FIG. 2: Temperature dependence of the coefficient β−1 for
interlayer distances d = 10nm (upper curve) and d =
22.5nm (lower curve) compared to the summary of experi-
mental curvature at both spacings [7]
electron). Using this value to estimate the compressibil-
ity as it is introduced by Eq. (17) we get α ≈ 0.44. This
is too small to provide insensitivity of the coefficient β
determined by Eq. (25) to the interlayer distance for in-
terlayer spacings reported in the experiments [3]. The
above discrepancy could be removed taking into account
Fermi liquid interactions among quasiparticles (CFs). To
include Fermi liquid effects into consideration we write
the renormalized polarizability Π∗ in the form [8]:
Π∗−1 = Π−1 + F(0) + F(1). (26)
Here, Π is the polarizability of noninteracting CFs de-
fined with Eq. (2), and the remaining terms present con-
tributions arising due to Fermi liquid interaction in the
CF system. Only contributions from the first and great-
est two terms in the expansion of the Fermi liquid in-
teraction function in Legendre polynomials ( f0 and f1 ,
respectively) are kept in Eq. (26) to avoid too lengthy
calculations. Matrix elements of the 2× 2 matrices F(0)
and F(1) equal:
F(0) =
F(1) =
m∗ −mb
m∗ −mb
Within the Fermi liquid theory the effective mass m∗
is related to the ”bare” mass mb as follows:
2πh̄2
1 +A1
. (28)
Using these expressions (26)–(28) and carrying out cal-
culations within the relevant limit δ ≪ 1, we obtain that
the expression for the density-density response function
for a single layer keeps the form given by Eq. (16) where
the compressibility dn/dµ is replaced with the quantity
dn∗/dµ renormalized due to the Fermi liquid interaction:
8πh̄2
8πh̄2
. (29)
For strongly correlated quasiparticles this renormaliza-
tion may significantly reduce the compressibility of the
CF liquid, and, consequently, increase the value of the
parameter α. It is usually assumed [3, 8] that the Fermi
liquid renormalization of the effective mass significantly
changes its value: m∗ ∼ 5 − 10 mb. This gives for the
Fermi liguid coefficient A1 values of the order of 10. Us-
ing this estimate, and substituting our renormalized com-
pressibility (29) into the expression (21) we arrive at the
conclusion that dn∗/dµ is low enough for the condition
α > 1, to be satisfied when the Fermi liquid parameter
A0 ≡ f0/2πh̄
2 takes on values of the order of 10 − 100.
This conclusion does not seem an unrealistic one for it
is reasonable to expect A0 to be of the order or greater
than the next Fermi liquid parameter A1. We obtain a
reasonably good agreement between the plot of our Eq.
(22) and the experimental results, using A0 = 15 and
A1 = 3 (m
∗ = 4mb). (Fig. 1).
Our results for temperature dependence of β−1 also
agree with the results of experiments [7]. The up-
per curve in Fig.2 corresponds to the double-layer sys-
tem with with smaller interlayer spacing dA = 10nm
which gives T0 = 487K, and the lower curve exhibits
the temperature dependence of β−1 for greater spacing
dB = 22.5nm (T0 = 587K). The curves do not coincide
but they are arranged rather close to each other.
Finally, the results of the present analysis enable us
to qualitatively describe all important features observed
in experiments of [7] on the Coulomb drag slightly away
from one half filling of lowest Landau levels of both in-
teracting 2DEG. They also give us grounds to treat these
experimental results as one more evidence of strong Fermi
liquid interaction in the CF system near one half filling
of the lowest Landau level. The above interaction pro-
vides a significant reduction of the compressibility of the
CF liquid and a consequent enhancement in the screen-
ing length in single layers. Essentially, the parameter α
characterizes the ratio of the Thomas–Fermi screening
length in a single 2DEG at ν = 1/2 and the separation
between the layers [3]. When α > 1, intralayer interac-
tions predominate those between the layers which could
be the reason for low sensitivity of the bilayer to changes
in the interlayer spacing. It is likely that here is an expla-
nation for the reported nearly independence of the drag
on the interlayer separation [7]. We believe that at larger
distances between the layers the dependence of the tran-
sresistivity of d could be revealed in the experiments.
At the same time the results of [7] give us a valuable
opportunity to estimate a strength of Fermi liquid inter-
actions between quasiparticles at ν = 1/2 state which is
important for further studies of such systems.
Acknowledgments: The author thank K.L. Haglin and
G.M. Zimbovsky for help with the manuscript.
[1] L. Zheng and A.H. MacDonald, Phys. Rev. B 48, 8203
(1993).
[2] A. Kamenev and Y. Oreg, Phys. Rev. B 52, 7516 (1995).
[3] I. Ussishkin and A. Stern, Phys. Rev. B 56, 4013 (1997).
[4] S. Sakhi, Phys. Rev. B 56, 4098 (1997);
[5] M.P. Lilly, J.P. Eisenstein, L.N. Pfeiffer and K.W. West,
Phys. Rev. Lett. 80, 1714 (1998).
[6] When the external driving disturbance applied to one of
the layers is of small q,ω (ql ≪ 1, ωτ ≪ 1) the tran-
sresisitivity is dominated with the diffusion contribution,
and new effects could emerge (See e.g. F. von Oppen,
S.H. Simon and A. Stern, Phys. Rev. Lett. 87, 106803
(2001) and references therein).
[7] M.P. Lilly, J.P. Eisenstein, L.N. Pfeiffer and K.W. West,
cond-mat/9909231.
[8] B.I. Halperin, P.A. Lee and N. Read, Phys. Rev. B 47,
7312 (1993); S.H. Simon and B.I. Halperin, Phys. Rev.
B 48, 17368 (1993).
[9] N.A. Zimbovskaya and J.L. Birman, Phys. Rev. B, 60
16762 (1999).
[10] N.A. Zimbovskaya, ”Local Geometry of the Fermi
Surface and High-Frequency Phenomena in Metals”,
Springer-Verlag, New York, 2001.
[11] R.L. Willett, K.W. West, and L.N. Pfeiffer Phys. Rev.
Lett. 83, 2624 (1999).
http://arxiv.org/abs/cond-mat/9909231

  Here, we present theoretical studies of the temperature and magnetic field
dependences of the Coulomb drag transresistivity between two parallel layers of
two dimensional electron gases in quantum Hall regime near half filling of the
lowest Landau level. It is shown that Fermi-liquid interactions between the
relevant quasiparticles could give a significant effect on the
transresistivity, providing its independence of the interlayer spacing for
spacings taking on values reported in the experiments. Obtained results agree
with the experimental evidence.

Fermi-liquid effects in the transresistivity in quantum Hall double layers near ν = 1/2
Natalya A. Zimbovskaya
Department of Physics and Astronomy, St. Cloud State University,
720 Fourth Avenue South, St. Cloud, MN 56301, USA;
Urals State Mining University, Kuibysheva Str. 30, Yekaterinburg, Russia, 620000
(Dated: November 4, 2018)
Here, we present theoretical studies of the temperature and magnetic field dependences of the
Coulomb drag transresistivity between two parallel layers of two dimensional electron gases in
quantum Hall regime near half filling of the lowest Landau level. It is shown that Fermi-liquid
interactions between the relevant quasiparticles could give a significant effect on the transresistivity,
providing its independence of the interlayer spacing for spacings taking on values reported in the
experiments. Obtained results agree with the experimental evidence.
PACS numbers: 71.27.+a 73.43.-f
During the last decade double-layer two-dimensional
electron gas (2DEG) systems were of significant interest
due to many remarkable phenomena they exhibit, includ-
ing so called Coulomb drag. In Coulomb drag experi-
ments two 2DEGs are arranged close to each other, so
that they can interact via Coulomb forces. A current I
is applied to one layer of the system, and the voltage VD
in the other nearby layer is measured, with no current
allowed to flow in that layer. The ratio −VD/I gives a
transresistivity ρD which characterizes the strength of
the effect. The physical interpretation of the Coulomb
drag is that momentum is tranferred from the current
carrying layer to the nearby one due to interlayer inter-
actions [1, 2, 3].
It was shown theoretically [4, 5] and confirmed with
experiments [5] that the transresistivity between two
2DEGs in quantum Hall regime at one half filling of
the lowest Landau level for both layers is proportional
to T 4/3 (T is the temperature of the system) which is
quite different from the temperature dependence of ρD
in the absence of the external magnetic field applied to
2DEGs. This temperature dependence of the drag at
ν = 1/2 originates from the ballistic contribution to
the transresistivity. The latter reflects the response of
the two-layer system to the driving disturbance of finite
wave vector q and finite frequency ω when considering
scales are smaller than the mean free path l of electrons
(ql ≫ 1) , and times are shorter than their scattering
time τ (ωτ ≫ 1) [6].
In further experiments [7] the Coulomb drag was mea-
sured between 2DEGs where the layer filling factor was
varied around ν = 1/2. The transresistivity was re-
ported to be enhanced quadratically with ∆ν = ν− 1/2.
It was also reported that the curvature of the enhance-
ment depended on temperature but it was insensitive to
both sign of ∆ν and distance d between the layers. The
present work is motivated with these experiments of [7].
We calculate the transresistivity between two layers of
2DEGs subject to a strong magnetic field which provides
ν close to 1/2 for both layers.
We start from the well-known expression [1, 3] which
relates the Coulomb drag transresistivity to density-
density components of the polarization in the layers
Π(1)(q, ω) and Π(2)(q, ω) :
2(2π)2
(2π)2
sinh2(h̄ω/2T )
∣U(q, ω)
ImΠ(1)(q,ω)ImΠ(2)(q,ω). (1)
Here, U(q, ω) is the screened interlayer Coulomb inter-
action, and electron densities in the layers are supposed
to be equal (n1 = n2 = n).
Within the usual Composite Fermion (CF) approach
[8] a single layer polarizability describes that part of the
density-current electromagnetic response which is irre-
ducible with respect to the Coulomb interaction. Adopt-
ing for simplicity the RPA, we obtain the following ex-
pression for the 2× 2 polarizability matrix:
Π−1 = (K0)−1 + C−1. (2)
Here, the matrix K0 gives the response of noninteract-
ing CFs and C is the Chern-Simons interaction matrix.
Assuming for certainty the wave vector q to lie in the
”x” direction we have:
. (3)
Starting from the expression (2) we arrive at the fol-
lowing results for the density-density response function
Π00(i)(q, ω) :
Π00(i)(q, ω) = Π(i)(q, ω)
00(i)
(q,ω)
8iπh̄
K001(i)(q,ω)−
∆(i)(q,ω)
. (4)
http://arxiv.org/abs/0704.0246v1
∆(i)(q,ω) = K
00(i)(q,ω)K
11(i)(q,ω) +
K001(i)(q,ω)
Within the RPA response functions included in Eqs. (4),
(5) are simply related to components of the CF conduc-
tivity tensor σ̃ [8]:
xx(q,ω)
00(i)
(q,ω)
00(i)
(q,0)
σ̃(i)yy (q,ω) = −
K011(i)(q,ω)−K
11(i)(q,0)
σ̃(i)xy = −σ̃
K001(i)(q,ω). (6)
To proceed we calculate the components of the CF
conductivity at ν slightly away from 1/2. In this case
CFs experience a nonzero effective magnetic field Beff =
B−B1/2. We concentrate on the ballistic contribution to
the transresistivity, so we need asymptotics for the rel-
evant conductivity components applicable in a nonlocal
(ql ≫ 1) and high frequency (ωτ ≫ 1) regime. Cor-
responding expressions for σ̃ij were obtained in earlier
works [8]. However, these results are not appropriate for
our analysis for they do not provide a smooth passage to
the Beff → 0 limit at finite q. Due to this reason we
do not use them in further calculations. To get a suitable
approximation for the CF conductivity we start from the
standard solution of the Boltzmann transport equation
for the CF distribution function. This gives us the fol-
lowing results for the CF conductivity components for a
single layer [9]:
σ̃αβ =
(2πh̄)2
dψvα(ψ) exp
′′)dψ′′
′) exp
′′)dψ′′
(ψ′ − ψ)(1− iωτ)
dψ′. (7)
Here, m∗, Ω are the CF effective mass and the cyclotron
frequency at the effective magnetic field Beff ; ψ is the
angular coordinate of the CF cyclotron orbit. Now we
carry out some formal transformations of this expression
(7) following the way proposed before [9, 10]. First, we
expand the CF velocity components vβ(ψ
′) in Fourier
series:
vkβ exp(ikψ
′). (8)
Substituting this expansion (8) into (7) we obtain:
(2πh̄)2
dψvα(ψ) exp(ikψ)
ikΩ− iω +
+ iqvx(ψ)
vx(ψ +Ωθ
′)− vx(ψ)
dθ (9)
where θ = (ψ′ − ψ)/Ω.
Then we introduce a new variable η which is related
to the variable θ as follows:
ikΩ− iω +
+ iqvx(ψ)
vx(ψ +Ωθ
′)− vx(ψ)
dθ′, (10)
and we arrive at the result:
σ̃αβ =
im∗e2
(2πh̄)2
vα(ψ) exp(ikψ)
ω + i/τ − kΩ− qvx(ψ +Ωθ)
dψ. (11)
Under the conditions of interest ωτ ≫ 1, ql ≫ 1, and
also assuming that the filling factor is close to ν = 1/2,
so that qvF ≫ Ω (vF is the CFs Fermi velocity), the
variable θ is approximately equal to ητ(1 + iql cosψ +
ikΩτ − iωτ)−1. Taking this into account and expanding
the last term in the denominator of (11) in powers of Ωθ
we obtain:
qvx(ψ +Ωθ)
≈ qvx(ψ) + ηΩqτ(1 + iql cosψ + ikΩτ − iωτ)−1
(Ωτ)2(1 + iql cosψ + ikΩτ − iωτ)−2
. (12)
Substituting this asymptotic expression into (9) we can
calculate first terms of the expansions of relevant compo-
nents of the CF conductivity in powers of the small pa-
rameter (qR)−1 where R = vF /Ω is the CF cyclotron
radius. Within the ”collisionless” limit 1/τ → 0 we
have:
σ̃xx = −N
1− δ2
(1− δ2)5
2(qR)2
1− δ2
; (13)
σ̃yy = N
1− δ2 + iδ +
2(qR)2
(1− δ2)5
(1− δ2)3
; (14)
σ̃xy = iN
1− δ2
(1− δ2)3
. (15)
Here, N = m∗/2πh̄2 is the density of states at the
CF Fermi surface, and δ = ω/qvF . Using these re-
sults we can easily get approximations for the functions
αβ(i)
(q, ω) (α, β = 0.1) and, subsequently, the de-
sired density-density response function given by (4). It
was shown [3] that the integral over ω in the expres-
sion for ρD (1) is dominated by ω ∼ T, and the ma-
jor contribution to the integral over q in this expres-
sion comes from q ∼ kF (T/T0)1/3, where kF is the
Fermi wave vector and the scaling temperature T0 is de-
fined below. Therefore we get an estimate for δ, namely
δ ∼ (T/µ)(T0/T )1/3, where µ is the chemical potential
of a single 2DEG included in the bilayer. For the param-
eter T0 taking on values of the order of room tempera-
ture, δ is small compared to unity at low temperatures
(T ∼ 1K).
Here, we limit ourselves to the case of two identical
layers (Π(1) = Π(2) ≡ Π) . For δ ≪ 1 we obtain the
approximation:
Π00(q, ω)
− 8πih̄ωkF
1 + 2(kFR)
(qR)−2
Here, dn/dµ is the compressibility of the ν = 1/2 state
which is defined as [3]:
≡ Π00(q → 0; ω → 0) =
8πh̄2
. (17)
This differs from the compressibility of the noninteracting
2DEG in the absence of an external magnetic field (the
latter equals N). The difference in the compressibility
values is a manifestation of the Chern-Simons interaction
in strong magnetic fields.
In following calculations we adopt the expression used
in the work [3] for the screened interlayer potential
U(q,ω), namely:
U(q,ω) =
Vb + Ub
1 + Π(q,ω)(Vb + Ub)
Vb − Ub
1 + Π(q,ω)(Vb − Ub)
where Vb(q) = 2πe
2/ǫq and Ub(q) = (2πe
2/ǫq)e−qd
are Fourier components of the bare Coulomb potentials
for intralayer and interlayer interactions, respectively,
and ǫ is the dielectric constant. Substitung (18) into
(1) and using our result (16) for Π(q, ω) we can present
the transresistvity in the ”ballistic” regime as:
ρD = ρD0 + δρD. (19)
Here, the first term ρD0 is the transresistivity at ν = 1/2
when the effective magnetic field is zero, and the sec-
ond term gives a correction arising in a nonzero effective
magnetic field (away from ν = 1/2 ). As it was to be
expected, our expression for ρD0 coincides with the al-
ready known result [3]:
ρD0 =
Γ(7/3)ζ(4/3)
with T0 =
πe2nd/ǫ
(1 + α), and
2πe2d
. (21)
The leading term of the correction δρD at low tempera-
tures (T/T0) ≪ 1 can be writen as follows:
δρD =
(kFR)2
ρD0∆ν
+ 4a2
(∆ν)2.
Here, the dimensionless positive constant a2 can be ap-
proximated as:
sinh2 y
y4/3 cosh2 y
dy. (23)
We have to remark that our result (23) cannot be
used in the limit T → 0. Actually, this expression pro-
vides a good asymptotic form for the coefficient a2 when
(TkF l/µ)
1/3 ≥ 1.5. Assuming that the mean free path
is of the order of 1.0µm as in the experiments [11] on
dc magnetotransport in a single modulated 2DEG at ν
close to 1/2, and using the estimate of [7] for the electron
density n = 1.4 × 1015m−2, we obtain that the expres-
sion (23) gives good approximation for a2 when T/µ is
no less than 10−2 .
It follows from our results (19), (22) that transresis-
tivity ρD enhances nearly quadratically with ∆ν when
the filling factor deviates from ν = 1/2. The linear in
∆ν term is also present in the expression for δρD. This
causes an asymmetric shape of the plot of Eq. (22) rel-
ative to ∆ν = 0. However, this asymmetry is not very
significant for the linear term is smaller than the last
term on the right hand side of (22). This difference in
magnitudes is due to different temperature dependences
of the considered terms. The first term including the lin-
ear in (kFR)
−1 correction is proportional to (T/T0)
whereas the second one is proportional to (T/T0)
2/3 and
predominates at low temperatures. So, the magnetic field
dependence of the transresistivity near ν = 1/2 matchs
that observed in the experiments (See Fig. 1).
Keeping only the greatest term in (22), the ratio
ρD/ρD0 can be presented in the form:
= 4β(∆ν)2 + 1. (24)
FIG. 1: Scaled drag resistivity versus ∆ν at T = 0.6; lowest
dashed curve is the plot of Eq. (22) at m∗ = 4mb; A0 = 15,
and remaining curves present experimental data of [7];
Here, the coefficient β equals:
Γ(7/3)ζ(4/3)
. (25)
This coefficient is proportional to the curvature of the
plot of Eq. (22) assuming that the first term is neglected.
The curvature reveales a strong dependence on tempera-
ture whose character also agrees with experiments of [7]
as it is shown in Fig. 2.
A striking feature in the experimental results is that
they appear to be nonsensitive to the distance between
the 2DEGs. Sets of data corresponding to samples with
different interlayer spacings dA = 10nm and dB =
22.5nm fall on the same curve. This concerns both mag-
netic field dependence of the transresistivity and tem-
perature dependence of the parameter β . Results of the
present analysis provide a possible explanation for this
feature. It follows from (20)–(25) that the dependence
of ρD of the interlayer spacing is completely included in
the characteristic temperature T0 which is defined with
Eq. (21). The above quantity is nearly independent of
the interlayer separation d when the parameter α takes
on values larger that unity. Estimating the parameter
α as it is given by Eq. (21), we obtain that the con-
dition α > 1 could be satisfied for small values of the
compressibility of the ν = 1/2 state. However, within
the RPA the effective mass of CFs coincides with the
single electron band mass mb which takes on the value
mb ≈ 0, 07me for GaAs wells (me is the mass of a free
FIG. 2: Temperature dependence of the coefficient β−1 for
interlayer distances d = 10nm (upper curve) and d =
22.5nm (lower curve) compared to the summary of experi-
mental curvature at both spacings [7]
electron). Using this value to estimate the compressibil-
ity as it is introduced by Eq. (17) we get α ≈ 0.44. This
is too small to provide insensitivity of the coefficient β
determined by Eq. (25) to the interlayer distance for in-
terlayer spacings reported in the experiments [3]. The
above discrepancy could be removed taking into account
Fermi liquid interactions among quasiparticles (CFs). To
include Fermi liquid effects into consideration we write
the renormalized polarizability Π∗ in the form [8]:
Π∗−1 = Π−1 + F(0) + F(1). (26)
Here, Π is the polarizability of noninteracting CFs de-
fined with Eq. (2), and the remaining terms present con-
tributions arising due to Fermi liquid interaction in the
CF system. Only contributions from the first and great-
est two terms in the expansion of the Fermi liquid in-
teraction function in Legendre polynomials ( f0 and f1 ,
respectively) are kept in Eq. (26) to avoid too lengthy
calculations. Matrix elements of the 2× 2 matrices F(0)
and F(1) equal:
F(0) =
F(1) =
m∗ −mb
m∗ −mb
Within the Fermi liquid theory the effective mass m∗
is related to the ”bare” mass mb as follows:
2πh̄2
1 +A1
. (28)
Using these expressions (26)–(28) and carrying out cal-
culations within the relevant limit δ ≪ 1, we obtain that
the expression for the density-density response function
for a single layer keeps the form given by Eq. (16) where
the compressibility dn/dµ is replaced with the quantity
dn∗/dµ renormalized due to the Fermi liquid interaction:
8πh̄2
8πh̄2
. (29)
For strongly correlated quasiparticles this renormaliza-
tion may significantly reduce the compressibility of the
CF liquid, and, consequently, increase the value of the
parameter α. It is usually assumed [3, 8] that the Fermi
liquid renormalization of the effective mass significantly
changes its value: m∗ ∼ 5 − 10 mb. This gives for the
Fermi liguid coefficient A1 values of the order of 10. Us-
ing this estimate, and substituting our renormalized com-
pressibility (29) into the expression (21) we arrive at the
conclusion that dn∗/dµ is low enough for the condition
α > 1, to be satisfied when the Fermi liquid parameter
A0 ≡ f0/2πh̄
2 takes on values of the order of 10 − 100.
This conclusion does not seem an unrealistic one for it
is reasonable to expect A0 to be of the order or greater
than the next Fermi liquid parameter A1. We obtain a
reasonably good agreement between the plot of our Eq.
(22) and the experimental results, using A0 = 15 and
A1 = 3 (m
∗ = 4mb). (Fig. 1).
Our results for temperature dependence of β−1 also
agree with the results of experiments [7]. The up-
per curve in Fig.2 corresponds to the double-layer sys-
tem with with smaller interlayer spacing dA = 10nm
which gives T0 = 487K, and the lower curve exhibits
the temperature dependence of β−1 for greater spacing
dB = 22.5nm (T0 = 587K). The curves do not coincide
but they are arranged rather close to each other.
Finally, the results of the present analysis enable us
to qualitatively describe all important features observed
in experiments of [7] on the Coulomb drag slightly away
from one half filling of lowest Landau levels of both in-
teracting 2DEG. They also give us grounds to treat these
experimental results as one more evidence of strong Fermi
liquid interaction in the CF system near one half filling
of the lowest Landau level. The above interaction pro-
vides a significant reduction of the compressibility of the
CF liquid and a consequent enhancement in the screen-
ing length in single layers. Essentially, the parameter α
characterizes the ratio of the Thomas–Fermi screening
length in a single 2DEG at ν = 1/2 and the separation
between the layers [3]. When α > 1, intralayer interac-
tions predominate those between the layers which could
be the reason for low sensitivity of the bilayer to changes
in the interlayer spacing. It is likely that here is an expla-
nation for the reported nearly independence of the drag
on the interlayer separation [7]. We believe that at larger
distances between the layers the dependence of the tran-
sresistivity of d could be revealed in the experiments.
At the same time the results of [7] give us a valuable
opportunity to estimate a strength of Fermi liquid inter-
actions between quasiparticles at ν = 1/2 state which is
important for further studies of such systems.
Acknowledgments: The author thank K.L. Haglin and
G.M. Zimbovsky for help with the manuscript.
[1] L. Zheng and A.H. MacDonald, Phys. Rev. B 48, 8203
(1993).
[2] A. Kamenev and Y. Oreg, Phys. Rev. B 52, 7516 (1995).
[3] I. Ussishkin and A. Stern, Phys. Rev. B 56, 4013 (1997).
[4] S. Sakhi, Phys. Rev. B 56, 4098 (1997);
[5] M.P. Lilly, J.P. Eisenstein, L.N. Pfeiffer and K.W. West,
Phys. Rev. Lett. 80, 1714 (1998).
[6] When the external driving disturbance applied to one of
the layers is of small q,ω (ql ≪ 1, ωτ ≪ 1) the tran-
sresisitivity is dominated with the diffusion contribution,
and new effects could emerge (See e.g. F. von Oppen,
S.H. Simon and A. Stern, Phys. Rev. Lett. 87, 106803
(2001) and references therein).
[7] M.P. Lilly, J.P. Eisenstein, L.N. Pfeiffer and K.W. West,
cond-mat/9909231.
[8] B.I. Halperin, P.A. Lee and N. Read, Phys. Rev. B 47,
7312 (1993); S.H. Simon and B.I. Halperin, Phys. Rev.
B 48, 17368 (1993).
[9] N.A. Zimbovskaya and J.L. Birman, Phys. Rev. B, 60
16762 (1999).
[10] N.A. Zimbovskaya, ”Local Geometry of the Fermi
Surface and High-Frequency Phenomena in Metals”,
Springer-Verlag, New York, 2001.
[11] R.L. Willett, K.W. West, and L.N. Pfeiffer Phys. Rev.
Lett. 83, 2624 (1999).
http://arxiv.org/abs/cond-mat/9909231
