Temperature dependence of Coulomb drag between finite-length quantum
  wires
arXiv:0704.0515v2  [cond-mat.mes-hall]  16 Jul 2007
Temperature dependence of Coulomb drag between finite-length quantum wires
J. Peguiron,1 C. Bruder,1 and B. Trauzettel1
Department of Physics and Astronomy, University of Basel, Klingelbergstrasse 82, 4056 Basel, Switzerland
(Dated: July 2007)
We evaluate the Coulomb drag current in two finite-length Tomonaga-Luttinger-liquid wires cou-
pled by an electrostatic backscattering interaction. The drag current in one wire shows oscillations
as a function of the bias voltage applied to the other wire, reflecting interferences of the plasmon
standing waves in the interacting wires. In agreement with this picture, the amplitude of the current
oscillations is reduced with increasing temperature. This is a clear signature of non-Fermi-liquid
physics because for coupled Fermi liquids the drag resistance is always expected to increase as the
temperature is raised.
PACS numbers: 71.10.Pm,72.10.-d,72.15.Nj
Coulomb drag phenomena in coupled one-
dimensional (1D) electron systems have been
investigated quite extensively in the past [1–10]. The
interest has mainly been driven by the fact that
Coulomb drag, i.e. the electrical response of one wire
as a finite bias is applied to the other wire, seems to
be an ideal testing ground for Tomonaga-Luttinger-
liquid (TLL) physics in nature. This is because
both inter-wire and intra-wire Coulomb interactions
substantially modify transport properties such as
the average current and the current noise. On the
experimental side, there have been a few works, some
of which have claimed to have observed TLL behavior
in the drag data [11–13]. Recently, Yamamoto and
coworkers have measured Coulomb drag in coupled
quantum wires of different lengths and found peculiar
transport properties that depend, for instance, on the
asymmetry of the two wires [14]. This experiment is
the major motivation for our work.
We analyze theoretically the Coulomb drag current
of two electrostatically coupled quantum wires using
the concept of the inhomogeneous TLL model [15–17].
This model is known to capture the essential physics of
an interacting 1D wire of finite length coupled to non-
interacting (Fermi liquid) electron reservoirs. Within
this framework, we are able to study finite-length and
finite-temperature effects and therefore to make quali-
tative contact with the experimental setup of Ref. [14].
Since the Coulomb interaction varies between the wire
regions and the lead regions, charge excitations feel
the interaction difference at the boundaries and are
known to exhibit Andreev-type reflections [17]. We
show that these reflections play a crucial role in the
Coulomb drag setup illustrated in Fig. 1. Further-
more, we show that the quantum interference phe-
nomena associated with the Andreev-type reflections
considerably modify the drag current. This is particu-
larly interesting as far as the temperature dependence
of the drag current is concerned. For Fermi-liquid sys-
tems, it is well known that the drag resistance should
always increase as the temperature is raised [6]. In
our setup instead, the drag current at a fixed drive
bias can either increase or decrease as a function of
temperature. It crucially depends on the interference
pattern due to finite-length effects. This is a clear
signature of non-Fermi-liquid physics which could be
observed in the double-wire setup of Ref. [14].
The system considered consists of two in-
teracting parallel wires (j = <,>) of fi-
nite length L< (shorter wire) and L> (longer
wire) connected to non-interacting semi-infinite 1D
leads (Fig. 1) and is described by the Hamiltonian
j=<,>
H0j +H
+HC. The intra-wire inter-
action is modelled through a TLL description [15–17]
H0j =
Π2j +
g2j (x)
(∂xΦj)
with the piecewise constant interaction parame-
ter gj(x) = gj < 1 in the wire region |x| <
Lj/2 and gj(x) = 1 in the non-interacting lead re-
gions |x| > Lj/2. The Fermi velocity vFj , the in-
teraction strength gj, and the wire length Lj set the
frequency ωLj = vFj/gjLj of the collective plasmonic
excitations hosted in each wire. A voltage eVj =
interwire
coupling
wire <
wire >
FIG. 1: (color online). The system under consideration.
Each interacting wire of length Lj [gray area, interaction
parameter gj(|x| < Lj/2) = gj < 1] is connected to a pair
of non-interacting leads [gj(|x| > Lj/2) = 1]. The region
of backscattering inter-wire interaction (red dashed box)
extends over the length of the shorter wire. A voltage V is
applied between the leads connected to the drive wire (here
the longer wire j = >) and the backscattering-induced
current Idr in the drag wire (here the shorter wire j = <)
is investigated.
http://arxiv.org/abs/0704.0515v2
0 1 2 3
V / 2πV
a) g=0.1
b) g=0.25
c) g=0.5
d) g=0.75
e) g=1
FIG. 2: Drag current as a function of drive voltage for
identical wires at zero temperature (solid curves). The
inter-wire interaction strength ranges from strongly inter-
acting (g = 0.1) to non-interacting (g = 1) for the dif-
ferent curves (with I
dr = eλ
2α4g/~2ωL and VL =
~ωL/e). The dashed curves show the dominant contri-
bution ∝ V 4g−2 [given in Eq. (10)] for g = 0.1, 0.25, 0.5.
L − µ
R applied to the leads is described by
HVj = −
dx µj(x)∂xΦj(x), (2)
with the piecewise constant electro-chemical potential
µj(x) =
L for x < −Lj/2,
0 for |x| < Lj/2,
R for x > Lj/2
with µ
L = −µ
R . This model is expected to cap-
ture the essential physics of a quantum wire coupled
smoothly to electron reservoirs (with typical smooth-
ing length Ls) as long as Lj=<,> ≫ Ls ≫ λF , where
λF is the electron Fermi wavelength [18, 19]. Fi-
nally, we include an inter-wire backscattering inter-
action over the length L< of the shorter wire,
HC = λBS
∫ L</2
−L</2
dx cos{
4π[Φ<(x)−Φ>(x)]}. (4)
This term includes the contribution of the density-
density interaction which is most relevant to Coulomb
drag [1, 2] when the Fermi wave-vectors of both wires
are similar in magnitude, i.e. kF< ≈ kF> [22].
In the following, we choose to apply a voltage V> =
V to the longer wire (µ>L = −µ
R = eV/2, drive wire)
and none V< = 0 to the shorter wire (µ
L = µ
R = 0,
drag wire). The average current in the wires may then
be written as I< = Idr and I> =
V − Idr. In our
model, the two currents I< and I> always flow in the
same direction, which is due to momentum conserva-
tion. This is known as positive Coulomb drag.
In order to get an expression for the drag cur-
rent Idr, we follow the formalism used in [18] in the
case of a single wire with an impurity. We consider
the situation of weak inter-wire coupling. To second
order in λBS, we obtain
Idr = I
∫ 1−R
drjdr(r, R), (5)
with the normalization I
dr = eλ
2ωL<,
where α< = ωL</ωc is the ratio between the plas-
mon frequency of the shorter wire and a cutoff fre-
quency ωc, of the order of the wire bandwidth. The
plasmon frequency defines a voltage VL = ~ωL</e
and a temperature TL = ~ωL</kB. It is conve-
nient to introduce corresponding dimensionless volt-
age u = V/VL and temperature θ = T/TL. The inte-
grand in (5),
jdr(r, R) =
eiuτ − e−iuτ
× exp
4πC<(r, R; τ) + 4πC>
, (6)
involves the parameters l = L>/L<, p = g>/g<,
and q = vF>/vF<. The correlation function Cj =
CGSj + C
j of each wire can be decomposed in a
zero-temperature and a finite-temperature contribu-
tion given by [18]
CGSj (r, R; τ) = −
αj + i(τ − sr − 2k)
αj + i(−2k)
|2k+1|
αj + i(τ − sR− 2k − 1)
[α2j + (r − sR− 2k − 1)2]1/2
, (7)
CTFj (r, R; τ) = −
sinch[πθ(τ − sr − 2k)]
sinch[πθ(−2k)]
|2k+1|
sinch[πθ(τ − sR− 2k − 1)]
sinch[πθ(r − sR− 2k − 1)]
, (8)
with γj = (1− gj)/(1 + gj) and sinchx = (sinhx)/x. It is to be noted that the expression for the drag cur-
3rent Idr does not depend on whether the drive wire is
the longer wire or the shorter one due to the symme-
try of our model. In the following, we present results
obtained by numerical evaluation of the triple integral
involved in (5) and (6) and discuss several analytical
approximations.
First we set the temperature to zero and consider
identical wires (l = p = q = 1, thus we drop the wire
index j). The drag current shows non-monotonous
behavior and oscillations with period ∼ 2π~ωL/e as a
function of the bias voltage (Fig. 2). It decays at large
voltages for g < 1/2 whereas it increases for g > 1/2.
Thus, we obtain qualitatively the same behavior as in
a dual Coulomb drag setup where a drive current is
applied and a drag voltage is measured [2]. Similar os-
cillations as a function of voltage have been predicted
in the context of two coupled fractional quantum Hall
line junctions [20].
An analytic approximation can be derived in the
limit u = eV/~ωL ≫ 1, that is for high voltages
or long wires. In Eq. (7), the terms proportional
to γ|m| account for contributions from plasmon exci-
tations reflected |m| times inside the wire. When the
wire length is much longer than other relevant length
scales, the contribution without any reflection m = 0
becomes dominant and yields the integrand
jdr(r, R) ∼
Γ(2g)
)2g−1/2
J2g−1/2(ur), (9)
where Γ(z) denotes the Gamma function and Jν(z)
the Bessel function of order ν. The resulting expres-
sion for the drag current, which involves hypergeo-
metric functions, underestimates the amplitude of the
oscillations with respect to the exact numerical re-
sult. However, the behavior at large u, governed by
the dominant contribution
Idr ∼
2Γ2(2g)
)4g−2
, (10)
shows good agreement in the appropriate parameter
regime (dashed curves in Fig. 2). Since we do per-
turbation theory in λBS, the relation Idr ≪ (e2/h)V
has to hold. In the large u regime, this means
(λBSL/~ωc)
2(ωL/ωc)(eV/~ωc)
4g−3 ≪ 1.
Now we consider the situation where the two wires
have different lengths. The qualitative behavior of the
drag current does not change for increasing length ra-
tio l = L>/L<, but the peak positions get shifted to
lower voltages (Fig. 3). Here, neglecting plasmon re-
flections in the correlation function of the wires leads
again to the expression (9), which is independent of l.
This fact explains why the drag current does not
change appreciably as a function of l and indicates
that the peak shifts observed result from plasmon re-
flections inside the wires. Our studies are the first to
analyze the effect of an asymmetry in the length on
Coulomb drag phenomena in coupled quantum wires
which is of recent experimental relevance [14].
 0  0.5  1  1.5  2
Idr/Idr
L>/L<V/2πVL
Idr/Idr
FIG. 3: (color online). Drag current as a function of the
bias voltage and of the length ratio of the wires (with
dr = eλ
2ωL< and VL = ~ωL</e).
We now discuss in detail the temperature depen-
dence of the drag current. For clarity, we consider
again symmetric wires (l = 1) [23]. The oscilla-
tions of the drag current as a function of bias voltage
get washed out with increasing temperature (Fig. 4).
This behavior is consistent with the picture which at-
tributes the oscillations to interferences of the plas-
mon excitations of the wires. Thus, for bias voltages
such that the drag current is close to a maximum at
zero temperature, one observes a decrease of the drag
current with increasing temperature, whereas the op-
posite behavior can be observed close to a minimum of
the zero-temperature drag current for strong interac-
tions. This behavior is in stark contrast to the linear
temperature dependence predicted for Coulomb drag
between 1D Fermi-liquid conductors [4] and therefore
a clear signature of TLL physics. Note that our Fig. 4
bears significant resemblance to Fig. 9 of Ref. [12].
A good approximation of the drag current can be
obtained for temperatures much larger than the tem-
perature associated with the plasmon frequency, θ =
kBT/~ωL ≫ 1, by neglecting contributions from plas-
mon reflections in the wires. Then, we obtain the
dominant contribution
Idr ∼
dr (2πθ)
4g−2 sinh
g + iu
4Γ2(2g)
Taking the limit of low bias voltage in this result,
Idr ∼
θ≫1,u
πΓ4(g)
4Γ2(2g)
(2πθ)4g−3u, (12)
we recover the power-law dependence T 4g−3 of the lin-
ear conductance predicted by renormalization group
analysis [5]. At large bias voltage u ≫ θ ≫ 1, we
recover the zero-temperature result (10).
In the case g = 1/2, Eq. (11) as well as the con-
tribution to next order in θ−1 can be brought into a
0 1 2 3
V / 2πV
0 1 2 3
g=0.25
g=0.6
FIG. 4: Temperature dependence of the drag current for
identical wires with interaction strength g = 0.25. The
solid curves (labelled a - e) are evaluated for tempera-
tures given by T/TL = kBT/~ωL = 0, 0.5, 1, 1.5, 5, respec-
tively (with I
dr = eλ
2α4g/~2ωL and VL = ~ωL/e).
The dashed curve shows the high-temperature limit (11)
for T/TL = 5. The inset shows a similar plot for weaker
intra-wire interaction g = 0.6 where the oscillations are
less pronounced.
compact analytic form
Idr ∼
, (13)
where Ψ′(z) = d2 ln Γ(z)/dz2 denotes the trigamma
function. The first term is Eq. (11) evaluated
at g = 1/2 (dotted curve in Fig. 5), and the second
one is the dominant correction, which takes values
within [−I(0)dr /θ, 0] (included in the dashed curve in
Fig. 5). This illustrates that the first-order approx-
imation already yields a nice qualitative description
for the full numerical result, which makes us confident
that Eq. (11) is a good high-temperature approxima-
tion also for g 6= 1/2.
In summary, we have analyzed two coupled quan-
tum wires that exhibit both a finite intra-wire interac-
tion and a finite inter-wire interaction. We have taken
into account finite-length effects within the inhomoge-
neous TLL model that is known to capture the essen-
tial physics of quantum wires coupled to Fermi-liquid
reservoirs. We have investigated how an asymmetry
in the lengths of the wires changes the drag current
and we have predicted a rich temperature dependence
of the drag current that shows clear signatures of non-
Fermi-liquid physics.
We would like to thank F. Dolcini, H. Grabert,
M. Kindermann, Y. Nazarov, S. Tarucha, and M. Ya-
mamoto for interesting discussions. This work
was supported by the Swiss NSF and the NCCR
Nanoscience.
0 1 2 3 4 5 6 7 8
V / 2πV
exact numerical result [Eq. (5)]
dominant term [Eq. (11)]
dominant term + first correction [Eq. (13)]
g=0.5, T/T
=5g=0.5, T/T
FIG. 5: Drag current at high temperature (T/TL = 5)
for identical wires with interaction strength g = 0.5 (solid
curve), where I
dr = eλ
2α4g/~2ωL and VL = ~ωL/e.
The dotted curve shows the dominant contribution in the
high-temperature limit [first term in (13)], the dashed
curve includes the first correction as well [both terms
in (13)].
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