7 Giant Fluctuations of Coulomb Drag in a Bilayer
System
A. S. Price,1 A. K. Savchenko,1∗ B. N. Narozhny,2 G. Allison,1 D. 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: a.k.savchenko@ex.ac.uk.
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 sign of 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.
In conventional measurements of the resistance of a two-dimensional (2D) layer an electrical
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 I1 is driven through one of the layers (active layer) and the voltage drop V2 is
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
http://arxiv.org/abs/0704.1242v1
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 a so-called ‘positive’ Coulomb drag, where electrons moving in the active layer 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 ∼ L2ϕ, where Lϕ is 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
small magnetic field, producing universal conductance fluctuations (UCF) seen in small samples
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 h̄q 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 Å in thickness, are separated by an Al0.33Ga0.67As layer
of thickness 300 Å. 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 ρD increases with increasing temperature as T
2 and decreases
with increasing passive-layer carrier concentration as nb
, 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 n2 is 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
〈∆σ2D〉 ≈ A
ET (L)τϕ ln κd
g4h̄(κd)3
, (1)
where σD ≈ ρD/(ρ1ρ2), and ρ1 and ρ2 are the active and passive layer resistivities, respec-
tively; ET (L) is the Thouless energy, ET (L) = h̄D/L
2, D is the diffusion coefficient; τϕ is
the decoherence time; κ is the inverse screening length; A = 4.9 × 10−3 and g = h/(e2ρ) is
the dimensionless conductivity of the layers. Using the parameters of our system, this expres-
sion gives a variance of ∼ 6 × 10−11 µS2, which is approximately eight orders of magnitude
smaller than the variance of the observed drag fluctuations. The fluctuations in ρD have 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 Lϕ in 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 〈∆σ2xx〉 = (e
(LT /L)
2, where LT =
h̄D/kBT is the thermal length (17). This expression produces a value of 0.8µS
2, which is in
good agreement with the measured value of 0.6µS2. The typical ‘period’ of the drag fluctuations
(the correlation field, ∆Bc) is similar to that of the UCF (15), indicating that both depend upon
the same Lϕ and 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.
We have seen that small q cannot explain the large fluctuations of the drag (12), and so argue that
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
analysis (18, 19, 20, 21) (detailed description in supporting text). For a qualitative estimate,
three factors have to be taken into account: (i) the inter-layer matrix elements of the Coulomb
interaction Dij ; (ii) the phase space (the number of electron states available for scattering); and
(iii) the electron-hole (e-h) asymmetry in both layers. Point (iii) takes into account that in a
quantum system the current is carried by both electron-like (above the Fermi surface) and hole-
like (below the Fermi surface) excitations. If they were completely symmetric with respect to
each other, then the current-carrying state of the active layer would have zero total momentum
and thus no drag effect would be possible. The physical quantity that measures the degree of e-h
asymmetry is the non-linear susceptibilityΓ of the 2D layer. Theoretically, the drag conductivity
is represented in terms of the non-linear susceptibilities of each layer and dynamically screened
interlayer Coulomb interaction Dij(ω) as σD ∝
dωD12(ω)Γ2(ω)D21(ω)Γ1(ω) (indices 1 and
2 correspond to the two layers) (18, 12). The e-h asymmetry appears in Γ as a derivative of
the density of states ν and the diffusion coefficient D: Γ ∝ ∂ (νD) /∂µ, and it is this quantity
that is responsible for the fact that drag fluctuations can exceed the average. As Dν ∼ g and
the typical energy of electrons is the Fermi energy, EF , we have ∂(νD)/∂µ ∼ g/EF for the
average drag. The typical energy scale for the interfering electrons, however, is ET (Lϕ) (17),
which is much smaller than EF and therefore a mesoscopic system has larger e-h asymmetry.
Under the condition of large momentum transfer between the layers, fluctuations in Γ are
similar to the fluctuations of the LDoS, which can be estimated as δν2 ∼ (ν2/g) ln (max(Lϕ, LT )/l)
(14). Also, the interaction in the ballistic regime can be assumed to be constant, Dij ≈ −1/νκd,
as q is limited by q ≤ 1/d. Finally, to average fluctuations of the drag over the sample with
size L we should divide it into coherent patches of size Lϕ that fluctuate independently and thus
decrease the total variance: 〈∆σ2D〉 = 〈∆σ
D(Lϕ)〉 (Lϕ/L)
2. If kBT > ET (Lϕ), fluctuations
are further averaged on the scale of ∼ kBT , and therefore the variance is suppressed by an
additional factor of ET (Lϕ)/kBT . Combining the above arguments we find
〈∆σ2D〉 = N
g2h̄2(κd)4
(kBT )
E2T (Lϕ)
l4L2ϕ
, (2)
where N is a numerical coefficient.
Compared to the diffusive situation (Eq. 1) the fluctuations described by our model are
greatly enhanced. The difference between Eqs. 2 and 1 comes from the fact that in the ballistic
regime electrons are not scattered by impurities between events of e-e scattering. Large momen-
tum transfers correspond to short distances, and thus in the ballistic regime drag measurements
explore the local (as opposed to averaged over the whole sample) non-linear susceptibility. This
leads to the appearance of three extra factors in Eq. 2: (i) the factor l4/d4 (which is also present
in the average drag in the ballistic regime – see Ref. (18)); (ii) the phase space factor T/ET
(which appears due to the fact that interaction parameters Dij are now energy-independent);
and (iii) the extra factor g2 due to fluctuations of the local non-linear susceptibility. Local fluc-
tuations are enhanced since the random quantity Γ is now averaged over a small part of the
ensemble, allowing one to detect rare impurity configurations.
Our model not only explains the large magnitude of the fluctuations, but also predicts a
non-trivial temperature dependence of their magnitude. The latter comes from the change in
the temperature dependence of Lϕ (22): at low temperatures, kBTτ/h̄ ≪ 1, the usual result is
Lϕ ∝ T
−1/2, while for kBTτ/h̄ > 1 the temperature dependence changes to Lϕ ∝ T
−1 (23).
Consequently, the temperature dependence of the variance of the drag fluctuations is expected
to change from T−1 at low T , to T−4 at high T . This temperature dependence is very different
from the T -dependence of drag fluctuations in the diffusive regime, 〈∆σ2D〉 ∝ T
−1. To test the
prediction of Eq. 2, the T -dependence of 〈∆σ2D〉 has been analysed (Fig. 4). The variance is
calculated in the limits of both the diffusive τϕ (solid line, τ
ϕ ∝ T ) and ballistic τϕ (dashed
line, τ−1ϕ ∝ T
2), using N ≃ 10−4. In fitting the drag variance we have found τϕ to agree
with theory to within a factor of two (15), which is typical of the agreement found in other
experiments on determining τϕ (24). (The single-layer values of τϕ found from our analysis of
the UCF agree with theory to within a factor of 1.5.) Thus, the temperature dependence of the
observed drag fluctuations strongly supports the validity of our explanation.
We have observed reproducible fluctuations of the Coulomb drag and demonstrated that they
are an informative tool for studying wave properties of electrons in disordered materials, and
the local properties in particular. Contrary to UCF which originate from quantum interference,
fluctuations of drag result from an interplay of the interference and e-e interactions. More the-
oretical and experimental work is required to study their manifestation in different situations.
For instance, similarly to the previous extensive studies of the evolution of UCF with increasing
magnetic field, such experiments can be performed on the fluctuations of drag. One of the im-
portant developments in the field of Coulomb drag fluctuations can be their study in quantising
magnetic fields, including the regimes of integer and fractional quantum Hall effects.
References
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8. A. G. Rojo, J. Phys.: Condens. Matter 11, R31 (1999).
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13. N. A. Mortensen, K. Flensberg, A.-P. Jauho, Phys. Rev. B 65, 085317 (2002).
14. I. V. Lerner, Phys. Lett. A 133, 253 (1988).
15. The details of the structures, measurements and the model are available as supporting ma-
terial on Science Online.
16. M. Kellogg, J. P. Eisenstein, L. N. Pfeiffer, K. W. West, Solid State Communications 123,
515 (2002).
17. B. L. Altshuler, P. A. Lee, R. A. Webb, eds., Mesoscopic Phenomena in Solids (North-
Holland, New York, 1991).
18. A. Kamenev, Y. Oreg, Phys. Rev. B 52, 7516 (1995).
19. L. Zheng, A. H. MacDonald, Phys. Rev. B 48, 8203 (1993).
20. A.-P. Jauho, H. Smith, Phys. Rev. B 47, 4420 (1993).
21. K. Flensberg, B. Y.-K. Hu, A.-P. Jauho, J. M. Kinaret, Phys. Rev. B 52, 14761 (1995).
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23. B. L. Altshuler, A. G. Aronov, Electron-Electron Interactions in Disordered Systems, A. L.
Efros, M. Pollak, eds. (North-Holland, Amsterdam, 1985).
24. C. W. J. Beenakker, H. van Houten, Solid State Physics, H. Ehrenreich, D. Turnbull, eds.
(Academic Press Limited, 1991).
25. Authors thank I.L. Aleiner, M. Entin, I.L. Lerner, A. Kamenev, and A. Stern for numerous
helpful discussions.
Supporting Online Material
www.sciencemag.org
Materials and Methods
SOM text
Figs. S1 to S3
Fig. 1. Schematic showing the origin of the drag signal V2 induced by the current I1.
The fluctuations of the drag arise from the interference of electron waves in each layer,
before the two electrons take part in the interlayer interaction.
0.5 1.0 1.5 2.0
 (1011cm-2)
T2 (K2)
 (1011 cm-2)
Fig. 2. Drag resistivity as a function of passive-layer carrier concentration for different
temperatures: T = 5, 4, 3, 2, 1, 0.4, and 0.24 K, from top to bottom. Inset (A): ρD as a
function of T2. Inset (B): ρD as a function of n2, with n1 = 1.1× 1011 cm−2; dashed line
is a n−1.5
0.5 1.0
0 10 20
20 30
-0.07
 (1011cm-2)
B (mT)
 T (K)
Fig. 3. (A) Drag resistance measured at low temperatures as a function of passive-
layer concentration; T = 1, 0.4, and 0.24 K, from top to bottom. Inset: ρD as a function
of T for two values of n2 denoted by the dotted lines in Fig. 3A; solid line is the expected
2 dependence of the average drag. (B): ρD as a function of B; T = 0.4, 0.35, and 0.24
K, from top to bottom. (Graphs for higher T are vertically offset for clarity.) Single-layer
concentration for each layer is 5.8× 1010cm−2. Inset: The UCF of the single-layer, with
an average background resistance of 500 Ohm subtracted.
Fig. 4. The variance of the drag conductivity fluctuations (squares) plotted against
temperature. The solid and dashed lines are calculated using Eq. 2 with the diffusive
and ballistic asymptotes of τϕ, respectively. Inset: τϕ extracted from the correlation
magnetic field of the single-layer fluctuations, plotted against temperature.

  We have observed reproducible fluctuations of the Coulomb 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 which explains their
enhancement by considering fluctuations of local electron properties.

7 Giant Fluctuations of Coulomb Drag in a Bilayer
System
A. S. Price,1 A. K. Savchenko,1∗ B. N. Narozhny,2 G. Allison,1 D. 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: a.k.savchenko@ex.ac.uk.
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 sign of 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.
In conventional measurements of the resistance of a two-dimensional (2D) layer an electrical
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 I1 is driven through one of the layers (active layer) and the voltage drop V2 is
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
http://arxiv.org/abs/0704.1242v1
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 a so-called ‘positive’ Coulomb drag, where electrons moving in the active layer 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 ∼ L2ϕ, where Lϕ is 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
small magnetic field, producing universal conductance fluctuations (UCF) seen in small samples
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 h̄q 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 Å in thickness, are separated by an Al0.33Ga0.67As layer
of thickness 300 Å. 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 ρD increases with increasing temperature as T
2 and decreases
with increasing passive-layer carrier concentration as nb
, 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 n2 is 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
〈∆σ2D〉 ≈ A
ET (L)τϕ ln κd
g4h̄(κd)3
, (1)
where σD ≈ ρD/(ρ1ρ2), and ρ1 and ρ2 are the active and passive layer resistivities, respec-
tively; ET (L) is the Thouless energy, ET (L) = h̄D/L
2, D is the diffusion coefficient; τϕ is
the decoherence time; κ is the inverse screening length; A = 4.9 × 10−3 and g = h/(e2ρ) is
the dimensionless conductivity of the layers. Using the parameters of our system, this expres-
sion gives a variance of ∼ 6 × 10−11 µS2, which is approximately eight orders of magnitude
smaller than the variance of the observed drag fluctuations. The fluctuations in ρD have 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 Lϕ in 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 〈∆σ2xx〉 = (e
(LT /L)
2, where LT =
h̄D/kBT is the thermal length (17). This expression produces a value of 0.8µS
2, which is in
good agreement with the measured value of 0.6µS2. The typical ‘period’ of the drag fluctuations
(the correlation field, ∆Bc) is similar to that of the UCF (15), indicating that both depend upon
the same Lϕ and 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.
We have seen that small q cannot explain the large fluctuations of the drag (12), and so argue that
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
analysis (18, 19, 20, 21) (detailed description in supporting text). For a qualitative estimate,
three factors have to be taken into account: (i) the inter-layer matrix elements of the Coulomb
interaction Dij ; (ii) the phase space (the number of electron states available for scattering); and
(iii) the electron-hole (e-h) asymmetry in both layers. Point (iii) takes into account that in a
quantum system the current is carried by both electron-like (above the Fermi surface) and hole-
like (below the Fermi surface) excitations. If they were completely symmetric with respect to
each other, then the current-carrying state of the active layer would have zero total momentum
and thus no drag effect would be possible. The physical quantity that measures the degree of e-h
asymmetry is the non-linear susceptibilityΓ of the 2D layer. Theoretically, the drag conductivity
is represented in terms of the non-linear susceptibilities of each layer and dynamically screened
interlayer Coulomb interaction Dij(ω) as σD ∝
dωD12(ω)Γ2(ω)D21(ω)Γ1(ω) (indices 1 and
2 correspond to the two layers) (18, 12). The e-h asymmetry appears in Γ as a derivative of
the density of states ν and the diffusion coefficient D: Γ ∝ ∂ (νD) /∂µ, and it is this quantity
that is responsible for the fact that drag fluctuations can exceed the average. As Dν ∼ g and
the typical energy of electrons is the Fermi energy, EF , we have ∂(νD)/∂µ ∼ g/EF for the
average drag. The typical energy scale for the interfering electrons, however, is ET (Lϕ) (17),
which is much smaller than EF and therefore a mesoscopic system has larger e-h asymmetry.
Under the condition of large momentum transfer between the layers, fluctuations in Γ are
similar to the fluctuations of the LDoS, which can be estimated as δν2 ∼ (ν2/g) ln (max(Lϕ, LT )/l)
(14). Also, the interaction in the ballistic regime can be assumed to be constant, Dij ≈ −1/νκd,
as q is limited by q ≤ 1/d. Finally, to average fluctuations of the drag over the sample with
size L we should divide it into coherent patches of size Lϕ that fluctuate independently and thus
decrease the total variance: 〈∆σ2D〉 = 〈∆σ
D(Lϕ)〉 (Lϕ/L)
2. If kBT > ET (Lϕ), fluctuations
are further averaged on the scale of ∼ kBT , and therefore the variance is suppressed by an
additional factor of ET (Lϕ)/kBT . Combining the above arguments we find
〈∆σ2D〉 = N
g2h̄2(κd)4
(kBT )
E2T (Lϕ)
l4L2ϕ
, (2)
where N is a numerical coefficient.
Compared to the diffusive situation (Eq. 1) the fluctuations described by our model are
greatly enhanced. The difference between Eqs. 2 and 1 comes from the fact that in the ballistic
regime electrons are not scattered by impurities between events of e-e scattering. Large momen-
tum transfers correspond to short distances, and thus in the ballistic regime drag measurements
explore the local (as opposed to averaged over the whole sample) non-linear susceptibility. This
leads to the appearance of three extra factors in Eq. 2: (i) the factor l4/d4 (which is also present
in the average drag in the ballistic regime – see Ref. (18)); (ii) the phase space factor T/ET
(which appears due to the fact that interaction parameters Dij are now energy-independent);
and (iii) the extra factor g2 due to fluctuations of the local non-linear susceptibility. Local fluc-
tuations are enhanced since the random quantity Γ is now averaged over a small part of the
ensemble, allowing one to detect rare impurity configurations.
Our model not only explains the large magnitude of the fluctuations, but also predicts a
non-trivial temperature dependence of their magnitude. The latter comes from the change in
the temperature dependence of Lϕ (22): at low temperatures, kBTτ/h̄ ≪ 1, the usual result is
Lϕ ∝ T
−1/2, while for kBTτ/h̄ > 1 the temperature dependence changes to Lϕ ∝ T
−1 (23).
Consequently, the temperature dependence of the variance of the drag fluctuations is expected
to change from T−1 at low T , to T−4 at high T . This temperature dependence is very different
from the T -dependence of drag fluctuations in the diffusive regime, 〈∆σ2D〉 ∝ T
−1. To test the
prediction of Eq. 2, the T -dependence of 〈∆σ2D〉 has been analysed (Fig. 4). The variance is
calculated in the limits of both the diffusive τϕ (solid line, τ
ϕ ∝ T ) and ballistic τϕ (dashed
line, τ−1ϕ ∝ T
2), using N ≃ 10−4. In fitting the drag variance we have found τϕ to agree
with theory to within a factor of two (15), which is typical of the agreement found in other
experiments on determining τϕ (24). (The single-layer values of τϕ found from our analysis of
the UCF agree with theory to within a factor of 1.5.) Thus, the temperature dependence of the
observed drag fluctuations strongly supports the validity of our explanation.
We have observed reproducible fluctuations of the Coulomb drag and demonstrated that they
are an informative tool for studying wave properties of electrons in disordered materials, and
the local properties in particular. Contrary to UCF which originate from quantum interference,
fluctuations of drag result from an interplay of the interference and e-e interactions. More the-
oretical and experimental work is required to study their manifestation in different situations.
For instance, similarly to the previous extensive studies of the evolution of UCF with increasing
magnetic field, such experiments can be performed on the fluctuations of drag. One of the im-
portant developments in the field of Coulomb drag fluctuations can be their study in quantising
magnetic fields, including the regimes of integer and fractional quantum Hall effects.
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Supporting Online Material
www.sciencemag.org
Materials and Methods
SOM text
Figs. S1 to S3
Fig. 1. Schematic showing the origin of the drag signal V2 induced by the current I1.
The fluctuations of the drag arise from the interference of electron waves in each layer,
before the two electrons take part in the interlayer interaction.
0.5 1.0 1.5 2.0
 (1011cm-2)
T2 (K2)
 (1011 cm-2)
Fig. 2. Drag resistivity as a function of passive-layer carrier concentration for different
temperatures: T = 5, 4, 3, 2, 1, 0.4, and 0.24 K, from top to bottom. Inset (A): ρD as a
function of T2. Inset (B): ρD as a function of n2, with n1 = 1.1× 1011 cm−2; dashed line
is a n−1.5
0.5 1.0
0 10 20
20 30
-0.07
 (1011cm-2)
B (mT)
 T (K)
Fig. 3. (A) Drag resistance measured at low temperatures as a function of passive-
layer concentration; T = 1, 0.4, and 0.24 K, from top to bottom. Inset: ρD as a function
of T for two values of n2 denoted by the dotted lines in Fig. 3A; solid line is the expected
2 dependence of the average drag. (B): ρD as a function of B; T = 0.4, 0.35, and 0.24
K, from top to bottom. (Graphs for higher T are vertically offset for clarity.) Single-layer
concentration for each layer is 5.8× 1010cm−2. Inset: The UCF of the single-layer, with
an average background resistance of 500 Ohm subtracted.
Fig. 4. The variance of the drag conductivity fluctuations (squares) plotted against
temperature. The solid and dashed lines are calculated using Eq. 2 with the diffusive
and ballistic asymptotes of τϕ, respectively. Inset: τϕ extracted from the correlation
magnetic field of the single-layer fluctuations, plotted against temperature.
