Giant fluctuations of coulomb drag in a bilayer system.
ABSTRACT The Coulomb drag in a system of two parallel layers is the result of electron-electron interaction between the layers. We have observed reproducible fluctuations of the drag, both as a function of magnetic field and electron concentration, which are a manifestation of quantum interference of electrons in the layers. At low temperatures the fluctuations exceed the average drag, giving rise to random changes of the sign of the drag. The fluctuations are found to be much larger than previously expected, and we propose a model that explains their enhancement by considering fluctuations of local electron properties.
arXiv:0704.1242v1 [cond-mat.mes-hall] 10 Apr 2007
Giant Fluctuations of Coulomb Drag in a Bilayer
A. S. Price,1A. K. Savchenko,1∗B. N. Narozhny,2G. Allison,1D. A. Ritchie3
1School of Physics, University of Exeter, Stocker Road, Exeter, EX4 4QL, UK
2The Abdus Salam ICTP, Strada Costiera 11, Trieste I-34100, Italy
3Cavendish Laboratory, University of Cambridge, Madingley Road, Cambridge CB3 0HE, UK
∗To whom correspondence should be addressed: firstname.lastname@example.org.
The Coulomb drag in a system of two parallel layers is the result of electron-
electron interactions between the layers. We have observed reproducible fluc-
tuations of the drag, both as a function of magnetic field and electron concen-
tration, which are a manifestation of quantum interference of electrons in the
layers. At low temperatures the fluctuations exceed the average drag, giving
rise to random changes of the signof the drag. The fluctuations are found to be
much larger than previously expected, and we propose a model which explains
their enhancement by considering fluctuations of local electron properties.
current is driven through the layer and the voltage drop along the layer is measured. In contrast,
Coulomb drag studies are performed on two closely spaced but electrically isolated layers,
where a current I1is driven through one of the layers (active layer) and the voltage drop V2is
measured along the other (passive) layer (Fig. 1). The origin of this voltage is electron-electron
(e-e) interaction between the layers, which creates a ‘frictional’ force that drags electrons in
the second layer. The ratio of this voltage to the driving current RD = −V2/I1(the drag
resistance) is a measure of e-e interaction between the layers. The measurement of Coulomb
drag in systems of parallel layers was first proposed in Ref. (1,2) and later realised in a number
of experiments (3,4,5,6,7) (for a review see Ref. (8)). As Coulomb drag originates from e-e
interactions, it has become a sensitive tool for their study in many problems of contemporary
condensed matter physics. For example, Coulomb drag has been used in the search for Bose-
condensation of interlayer excitons (9), the metal-insulator transition in two-dimensional (2D)
layers (10), and Wigner crystal formation in quantum wires (11).
Electron-electron scattering, and the resulting momentum transfer between the layers, usu-
ally creates aso-called ‘positive’Coulombdrag, where electrons movingintheactivelayer drag
electrons in the passive layer in the same direction. There are also some cases where unusual,
‘negative’ Coulomb drag is observed: e.g. between 2D layers in the presence of a strong, quan-
tising magnetic field (6,7); and between two dilute, one-dimensional wires where electrons are
arranged into a Wigner crystal (11). All previous studies of the Coulomb drag, however, refer
to the macroscopic (average) drag resistance. Recently there have been theoretical predictions
of the possibility to observe random fluctuations of the Coulomb drag (12,13), where the sign
of the frictional force will change randomly from positive to negative when either the carrier
concentration, n, or applied (very small) magnetic field, B, are varied.
Drag fluctuations originate from the wave nature of electrons and the presence of disorder
(impurities) in the layers. Electrons travel around each layer and interfere with each other, after
collisions with impurities, over the characteristic area ∼ L2
ϕ, where Lϕis the coherence length
(Fig. 1). This interference is very important for conductive properties of electron waves. For
example, the interference pattern is changed when the phase of electron waves is varied by a
smallmagneticfield, producinguniversalconductancefluctuations(UCF) seeninsmallsamples
with size L ∼ Lϕ. There is, however, a significant difference between UCF and the fluctuations
of the drag resistance. The former are only a small correction to the average value of the
conductance: in our experiment the single-layer resistance fluctuates by ∼ 200 mOhm around
an average resistance of approximately 500 Ohm. In contrast, the drag fluctuations, although
small in absolute magnitude (∼ 20 mOhm) are able to change randomly, but reproducibly the
sign of the Coulomb drag between positive and negative. Surprisingly, we have found that
these fluctuations of the Coulomb drag, observed at temperatures below 1 K, are four orders of
magnitude larger than predicted in Ref. (12).
Our explanation of the giant drag fluctuations takes into account that, unlike the UCF, the
drag fluctuations are not only an interference but also fundamentally an interaction effect. In
conventional drag structures the electron mean free path l is much larger than the separation d
between the layers, and therefore large momentum transfers ¯ hq between electrons in the layers
become essential. According to the quantum mechanical uncertainty principle, ∆r∆q ∼ 1,
electrons interact over small distances ∆r ≪ l when exchanging large values of momentum
(Fig. 1). As a result the local properties of the layers, such as the local density of electron states
(LDoS), become important in the interlayer e-e interaction. These local properties at the scale
∆r ≪ l exhibit strong fluctuations (14) that directly manifest themselves in the fluctuations of
the Coulomb drag.
The samples used in this work are AlGaAs-GaAs double-layer structures, in which the car-
rier concentration of each layer can be independently controlled by gate voltage. The two GaAs
quantum wells of the structure, 200˚ A in thickness, are separated by an Al0.33Ga0.67As layer
of thickness 300˚ A. Each layer has a Hall-bar geometry, 60µm in width and with a distance
between the voltage probes of 60µm (15).
Figure 2 shows the appearance of the fluctuations in the drag resistivity, ρD, at low temper-
atures. At higher temperatures, the drag resistance changes monotonically with both T and n:
the insets to Fig. 2 show that ρDincreases with increasing temperature as T2and decreases
with increasing passive-layer carrier concentration as nb
2, where b ≈ −1.5. These results are
consistent with existing experimental work on the average Coulomb drag (4,16).
Figure 3A shows a zoomed-in view of the reproducible fluctuations as a function of n2.
These fluctuations result in an alternating sign of the drag, which is demonstrated in the inset
to Fig. 3 where the temperature dependence of the drag is shown at two different values of n2.
The drag is seen first to decrease as the temperature is decreased, but then become either in-
creasingly positive or increasingly negative, dependent upon n2. The reproducible fluctuations
of the drag resistivity have also been observed as a function of magnetic field (Fig. 3B). For a
fixed temperature, the magnitude of the drag fluctuations as a function of n2is roughly the same
as that as a function of B.
The theory of Ref. (12) calculates the variance of drag fluctuations in the so-called diffusive
regime, l < d. In this case the drag is determined by global properties of the layers, aver-
aged over a region ∆r ≫ l. The expected variance of drag fluctuations (at low T when the
fluctuations exceed the average) in the diffusive regime is
D? ≈ Ae4
where σD ≈ ρD/(ρ1ρ2), and ρ1and ρ2are the active and passive layer resistivities, respec-
tively; ET(L) is the Thouless energy, ET(L) = ¯ hD/L2, D is the diffusion coefficient; τϕis
the decoherence time; κ is the inverse screening length; A = 4.9 × 10−3and g = h/(e2ρ) is
the dimensionless conductivity of the layers. Using the parameters of our system, this expres-
sion gives a variance of ∼ 6 × 10−11µS2, which is approximately eight orders of magnitude
smaller than the variance of the observed drag fluctuations. The fluctuations in ρDhave been
measured in two different samples, and their variance is seen to be similar in magnitude and
T-dependence, confirming the discrepancy with the theoretical prediction (12).
The expected fluctuations of the drag conductivity share the same origin as the UCF in
the conventional conductivity: coherent electron transport over Lϕin the layers prior to e-e
interaction between the layers (Fig. 1). For this reason we have compared the drag fluctuations
with the fluctuations seen in the single-layer resistivity of the same structure (Fig. 3B, inset),
which have shown the usual behaviour (17). We estimate the expected variance of the single-
layer conductance fluctuations using the relation ?∆σ2
xx? = (e2/h)2(LT/L)2, where LT =
¯ hD/kBT is the thermal length (17). This expression produces a value of 0.8µS2, which is in
(the correlation field, ∆Bc) is similar to that of the UCF (15), indicating that both depend upon
the same Lϕand have the same quantum origin.
To address the question of the discrepancy between the magnitude of drag fluctuations in
theory (12) and our observations, we stress that the theoretical prediction for the variance, Eq. 1,
was obtained under the assumption of diffusive motion of interacting electrons, with small
interlayer momentum transfers, q ≪ 1/l. As the layers are separated by a distance d, the
e-e interactions are screened at distances ∆r > d. Therefore, in all regimes the maximum
momentum transfers are limited by q < 1/d. In the diffusive regime, l < d, this relation also
means that q < 1/l, that is, interlayer e-e interactions occur at distances ∆r > l and involve
scattering by many impurities in the individual layers. In the opposite situation, l ≫ d, the
transferred momenta will include both small and large q-values: q < 1/l and 1/l < q < 1/d.
Wehaveseen thatsmallq cannot explainthelargefluctuationsofthedrag (12), andso arguethat
it is large momentum transfers with q > 1/l which give rise to the observed effect. In this case
the two electrons interact at a distance ∆r that is smaller than the average impurity separation
and, therefore, it is the local electron properties of the layers which determine e-e interaction.
In Ref. (14) it is shown that the fluctuations of the local properties are larger compared to those
of the global properties that are responsible for the drag in the diffusive case.
A theoretical expression for the drag conductivity is obtained by means of a Kubo formula