Counterflow of electrons in two isolated quantum point contacts
V.S. Khrapai,1, 2 S. Ludwig,1 J.P. Kotthaus,1 H.P. Tranitz,3 and W. Wegscheider3
Center for NanoScience and Department für Physik, Ludwig-Maximilians-Universität,
Geschwister-Scholl-Platz 1, D-80539 München, Germany
Institute of Solid State Physics RAS, Chernogolovka, 142432, Russian Federation
Institut für Experimentelle und Angewandte Physik,
Universität Regensburg, D-93040 Regensburg, Germany
We study the interaction between two adjacent but electrically isolated quantum point contacts
(QPCs). At high enough source-drain bias on one QPC, the drive-QPC, we detect a finite electric
current in the second, unbiased, detector-QPC. The current generated at the detector-QPC always
flows in the opposite direction than the current of the drive-QPC. The generated current is maximal,
if the detector-QPC is tuned to a transition region between its quantized conductance plateaus and
the drive-QPC is almost pinched-off. We interpret this counterflow phenomenon in terms of an
asymmetric phonon-induced excitation of electrons in the leads of the detector-QPC.
PACS numbers: 73.23.-b, 73.23.Ad, 73.50.Lw
The state of a confined quantum system is modified
by interactions with an external field (or with exter-
nal sources of energy). In semiconductor nanostruc-
tures the energy and quasi-momentum of electrons act-
ing as probe are strongly influenced by the environment,
e. g. via electron-electron or electron-phonon interaction.
If driven out of equilibrium, Coulomb forces establish
the local equilibrium within the electron system whereas
electron-phonon interactions dominate the energy ex-
change with the environment [1]. Drag experiments in
semiconductor nanostructures provide a tool to study the
effect of external electrons or phonons onto a probe elec-
tron system.
Current drag between parallel two-dimensional (2D)
electron layers has been investigated in GaAs/AlGaAs
bilayer systems. At small interlayer separations, ob-
servations are consistent with the Coulomb drag phe-
nomenon [2]. At larger separations virtual-phonon ex-
change has been invoked to explain the data [3]. A neg-
ative sign of a current drag between 2D and 3D electron
gases in GaAs was explained by the Peltier effect [4]. At
high filling factors in a perpendicular magnetic field a
sign change of the longitudinal drag between parallel 2D
layers was found as a function of the imbalance of the
electron density in the two layers [5, 6].
Interactions between two lateral quantum wires in
GaAs have been investigated in Ref. [7]. The observed
frictional drag, strongly oscillating as a function of the
one-dimensional (1D) subband occupation, was inter-
preted in terms of Coulomb interaction between two
Luttinger liquids. Recently, the observation of negative
Coulomb drag between two disordered lateral 1D wires in
GaAs in perpendicular magnetic fields was reported [8].
Here we report on a novel interaction effect between
two neighboring quantum point contacts (QPCs), em-
bedded in mutually isolated electric circuits. When a
strong current is flowing through the partially transmit-
ting drive-QPC, we detect a small current in the sec-
ond, unbiased, detector-QPC. The detector current flows
in the opposite direction of the drive current and shows
a nonlinear dependence on the source-drain bias of the
drive-QPC. It oscillates as a function of the detector-
QPC transmission. We suggest an explanation of this
counterflow phenomenon in terms of asymmetric phonon-
induced excitation of ballistic electrons in the leads of the
detector-QPC.
Our samples are prepared on a GaAs/AlGaAs het-
erostructure containing a two-dimensional electron gas
90 nm below the surface, with an electron density of
nS = 2.8 × 10
11 cm−2 and a low-temperature mobility
of µ = 1.4 × 106 cm2/Vs. An AFM micrograph of the
split-gate nanostructure, produced with e-beam lithogra-
phy, is shown in the left inset of Fig. 1. The negatively
biased central gate C divides the electron system into two
separate circuits, and prevents leakage currents between
them. Two QPCs are defined on the upper and lower
side of the central gate, respectively, by biasing gates 8
and 3. Other gates are grounded if not stated otherwise.
The right inset of Fig. 1 shows a sketch of the coun-
terflow experiment. We use separate electric circuits for
the (upper) drive-QPC and (lower) detector-QPC. A dc
bias voltage, Vdrive, is applied to the left lead of the drive-
QPC, while the right lead is grounded. A current-voltage
amplifier with an input voltage-offset of about 10 µV is
connected to the right lead of the detector-QPC. Its left
lead is always maintained at the same offset potential
in order to assure zero voltage drop across the detector-
QPC. In both circuits, a positive sign of the current cor-
responds to electrons flowing to the left. For differential
counterflow conductance measurements, the drive bias is
modulated at a frequency of 21 Hz and the resulting ac
current component in the detector circuit is measured
with lock-in detection in the linear response regime. All
measurements are performed in a dilution refrigerator at
an electron temperature below 150 mK. The experimen-
tal results are the same if detector and drive QPC are
interchanged.
First, we characterize the QPCs using a standard dif-
ferential conductance measurement. Figure 1 displays
the differential conductances of both QPCs in linear re-
http://arxiv.org/abs/0704.0132v2
drive
-0.7 -0.5 -0.3
FIG. 1: Conductance of the drive-QPC (dashed line) and the
detector-QPC (solid line) in the linear response regime as a
function of respective gate voltages V8 and V3. Symbols on
the detector-QPC curve mark the V3 values used for coun-
terflow conductance measurement presented in Fig. 2b. Left
inset: AFM micrograph of the metal gates on the surface
of the heterostructure (bright tone). Crossed squares mark
contacted 2DEG regions. The scale bar equals 1 µm. Right
inset: sketch of the counterflow measurement. The directions
of currents are shown for the case of Vdrive > 0.
sponse, measured as a function of the respective gate
voltage V3, or V8. At low gate voltages, the QPCs are
pinched-off and the conductance is close to zero. With
increasing gate voltage, 1D channels successively open
up [9]. For both QPCs we observe three conductance
plateaus approximately quantized in units of G0 = 2e
With high bias spectroscopy [10] we find the spacing
between the two lowest subbands to be approximately
4 meV (3 meV) for the drive (detector) QPC. The half-
width of the energy window for opening a 1D subband is
∆ ≈ 0.5 meV in both QPCs.
Having characterized the QPCs, we turn to counter-
flow measurements. Fig. 2a shows the dc counterflow
current, Icf, through the detector-QPC and the differ-
ential counterflow conductance, gcf ≡ dIcf/dVdrive, as a
function of the bias on the drive-QPC. Here, the drive-
QPC is tuned to nearly half a conductance quantum
Gdrive = G0/2, while the detector-QPC is in the pinch-off
regime (i.e. the lowest 1D subband bottom is well above
the Fermi level) with Gdet ≃ 10 GΩ
−1. Surprisingly, for
|Vdrive| & 1 mV, a finite current is observed in the un-
biased detector circuit. The direction of Icf is opposite
to that of the drive-QPC current Idrive. The dc coun-
terflow current is a threshold-like, nearly odd function of
Vdrive. Correspondingly, the differential counterflow con-
ductance is negative and a nearly even function of Vdrive.
The sign of gcf expresses a phase shift of π between the
applied ac modulation of Vdrive and the detected ac com-
ponent of the counterflow current.
Figures 2c and 2d show the absolute value of Icf for the
nearly pinched-off detector as a function of the voltage
on gate 8, which tunes the drive-QPC transmission. The
-4 -2 0 2 4
-0.04
-0.02
-0.7 -0.6 -0.5 -0.4
drive
 (mV)
(d) G
FIG. 2: (a) - Icf and gcf for the nearly pinched-off detector-
QPC as a function of Vdrive. (b) - gcf measured for a set of
Gdet values marked by according symbols in Fig. 1. (c,d)
- Absolute value of Icf as a function of the drive-QPC gate
voltage V8, for Vdrive = ±2.25 mV (c) and Vdrive = ±4 mV(d).
Also shown is the drive-QPC’s conductance in linear response
(c) and its differential conductance at Vdrive = ±4 mV (d).
Solid (dotted) lines correspond to Vdrive < 0 (> 0). In (a),(b)
gates 7 and 9 are grounded, while in (c),(d) V7 = V9 = −0.4 V.
The drive bias modulation used to measure gcf is 92 µV rms.
corresponding drive-QPC differential conductance curves
are also shown. For not too high Vdrive (Fig. 2c), a non-
zero counterflow current is only detected in the region
between pinch-off and the first conductance plateau of
the drive-QPC. For higher Vdrive (Fig. 2d) Icf increases
superlinearly with Vdrive at its maximum and remains
finite at higher gate voltages V8. Since the source bias
effects the potential distribution near the constriction,
the nonlinear 1/2 conductance plateau of the drive-QPC
shifts when changing Vdrive [11]. This causes the shift of
the extrema on Fig. 2d as well as the asymmetry of gcf
in Fig. 2a when reversing the bias.
We proceed to study the counterflow effect in the
-0.6 -0.5 -0.4 -0.3
-0.10
-0.05
-0.65 -0.60 -0.55
=17 kΩ
=17 kΩ
FIG. 3: (a)- gcf as a function of the detector-QPC gate volt-
age V3. Filled symbols correspond to the gcf measured at a
finite external resistance Rext = 17 kΩ, while open symbols
show the corrected counterflow conductance Rext = 0 (see
text). Also shown are the transmission functions Tn(1 − Tn)
for the three lowest 1D subbands of the detector-QPC (dashed
lines), scaled to fit the corrected data. During the gcf mea-
surement the drive bias is modulated with a 230 µV rms signal
about the mean value Vdrive = +2.05 mV. (b) - Normalized gcf
(symbols as in (a)) and transmission function of the lowest 1D
detector-QPC subband 4T0(1−T0) (dashed line) as a function
of V3. The scale bar shows a gate voltage interval correspond-
ing to a change of the 1D subband energy by 0.5 meV. Inset:
Sketch of possible scattering processes of nonequilibrium elec-
trons and holes at a partially transmitting detector-QPC.
regime of a more opened detector-QPC. Figure 2b plots
gcf [12] as a function of Vdrive for several values of Gdet
between 0 and G0 (marked with the same symbols in
Fig. 1). The qualitative appearance of gcf(Vdrive) is in-
dependent of Gdet. However, the amplitude of gcf is a
strongly non-monotonic function of the detector trans-
mission. The counterflow conductance reaches its maxi-
mum for Gdet ≈ G0/2 and decreases rapidly with further
increasing Gdet. Note that the absolute value of gcf is
small, corresponding to a maximal ratio of the counter-
flow and drive currents |Icf/Idrive| . 10
In Fig. 3a gcf is plotted as a function of V3, controlling
the detector transmission. Vdrive and V8 are adjusted for
maximal gcf and kept fixed. Confirming the trend seen in
Fig. 2b, the measured gcf (solid symbols) strongly oscil-
lates with increasing V3 and displays three pronounced
maxima before the detector-QPC is fully opened. The
position of the n-th maximum (n = 0,1,2) is close to the
value of V3, where Gdet/G0 ≃ n + 0.5 (Fig. 1). Here,
the energy EnS of the bottom of the n-th 1D subband
-4 -2 0 2 4
��������������
drive
 (mV)
FIG. 4: Drive bias dependence of the counterflow current
through the pinched-off detector-QPC for the drive-QPC
formed with gate 6 (dotted line) or gate 10 (solid line). The
detector-QPC conductance is about Gdet = 5 GΩ
−1. The
drive-QPCs are tuned to provide the maximal effect. Insets:
sketches of the two counterflow measurements. The directions
of currents are shown for the case of Vdrive > 0.
of the detector-QPC aligns with the Fermi level of the
leads EnS ≃ EF. In contrast, gcf is close to zero for fully
transmitting 1D channels (Gdet/G0 ≃ n+1). The over-
all magnitude of gcf decreases with increasing V3, hence
Gdet. This is caused by a finite series resistance Rext of
the external circuit, which results in a measured gcf lower
than the case for an ideal ammeter [13]. The corrected
counterflow conductance, gidealcf ≡ gcf · (1 + Rext · Gdet),
corresponding to Rext = 0, is shown in Fig. 3a with open
symbols. The corrected maxima are roughly equal in size
and symmetric. Moreover, the shape of the n-th maxi-
mum compares quite well with the corresponding func-
tion of the equilibrium transmission Tn(1−Tn), extracted
from the detector conductance data Tn ≡ Gdet/G0 − n
(dashed lines in Fig. 3a).
In Figure 3b we plot the normalized gcf and the trans-
mission function 4T0(1− T0) on a logarithmic scale near
the detector pinch-off. In the pinch-off regime (i.e. for
T0 ≪ 1) the transmission probability of a QPC is ex-
pressed as T0(E) ∝ exp([E − E
S]/∆) [11]. Here E is
the kinetic energy of current carrying electrons and ∆
is the half-width of the energy window for opening a 1D-
subband. The energy E0S of the detector-QPC is con-
trolled by gate 3 via E0S ∝ −|e|V3. This explains a nearly
exponential drop of the transmission function with de-
creasing V3 (Fig. 3b). In contrast, the measured gcf
drops considerably slower and remains finite even where
the detector-QPC is already pinched-off in equilibrium.
This experimentally observed excess contribution of the
normalized gcf versus T0(EF ) signals that the counter-
flow current carrying electrons are excited well above the
Fermi level. Converting the shift in gate voltage (see the
bar in Fig. 3b) to energy, we find a characteristic excita-
tion energy of E∗ ≈ 0.5 meV. This is consistent with a
recently reported 1 meV bandwidth excitation provided
by the drive-QPC for electrons in a nearby double-dot
quantum ratchet [14].
Next we study the counterflow effect between spatially
shifted QPCs. Figure 4 shows Icf through the nearly
pinched-off detector-QPC as a function of the bias on the
drive-QPC, which is formed either with gate 10 or gate
6, while gate 8 is now grounded (Fig. 1). Despite the
shift of the drive-QPC position relative to the detector-
QPC by about 300 nm, the odd drive bias dependence
of the counterflow current found in Fig. 2 is practically
preserved. This indicates that the excitation of electrons
in one of the leads of the detector-QPC is not restricted
to the close vicinity of the drive-QPC.
The oscillations of the counterflow conductance gcf in
Fig. 3 are reminiscent of thermopower oscillations that
have been investigated on individual QPCs [15, 16]. This
suggests that Icf is caused by an energetic imbalance
across the detector-QPC. If the bottom of the n-th 1D-
subband of the detector-QPC is well separated from the
Fermi-energy in comparison to the characteristic excita-
tion energy, i. e. if |EnS −EF| ≫ E
∗, this subband is either
fully transmitting (Tn(E) = 1) or closed (Tn(E) = 0). In
both cases electrons (holes) excited by E∗ above (below)
EF are equally transmitted and gcf = 0. In contrast,
if EnS ≃ EF excited electrons are more likely transmit-
ted than excited holes (see inset of Fig. 3b), resulting in
gcf 6= 0.
The energetic imbalance across the detector-QPC we
propose to be caused by phonon-based energy transfer
from the drive-QPC. The excess energy of carriers in-
jected across the drive-QPC is mainly relaxed by emis-
sion of acoustic phonons. We consider the drive-QPC in
the non-linear regime near pinch-off where µS − µD ≫
∆ and the transmission probability is strongly energy-
dependent (the source and drain leads are defined so that
their chemical potentials satisfy µS > µD). In this case
electrons injected into the drain lead have an excess en-
ergy of about e|Vdrive| ≡ µS−µD whereas the source lead
remains essentially in thermal equilibrium [17]. Hence
acoustic phonons are predominantly generated in the
drain lead of the drive-QPC. Because of this asymme-
try electron-hole pairs are excited preferentially in the
adjacent lead of the detector-QPC [18]. This gives rise
to Icf directed opposite to the current through the drive-
QPC (and gcf < 0). The data in Fig. 2 clearly show, that
the counterflow effect is only observed in the non-linear
regime of the drive-QPC.
For a rough estimate we consider injected electrons
with a momentum relaxation time of 60 ps limited by
elastic scattering and an energy relaxation time of
1 ns [19, 20]. Assuming isotropic phonon emission we
estimate an energy transfer ratio which can account for
the observed value of Icf/Idrive within one order of mag-
nitude.
In summary, the current in a strongly biased drive-
QPC generates a current flowing in the opposite direc-
tion through an adjacent unbiased detector-QPC. This
counterflow current is maximal in between the conduc-
tance plateaus of the detector-QPC. The effect is most
pronounced near pinch-off of the drive-QPC, where it
behaves strongly non-linear. We interpret the results in
terms of an asymmetric phonon-based energy transfer.
The authors are grateful to V.T. Dolgopolov,
A.W. Holleitner, C. Strunk, F. Wilhelm, I. Favero,
A.V. Khaetskii, N.M. Chtchelkatchev, A.A. Shashkin,
D.V. Shovkun and P. Hänggi for valuable discussions
and to D. Schröer and M. Kroner for technical help. We
thank the DFG via SFB 631, the BMBF via DIP-H.2.1,
the Nanosystems Initiative Munich (NIM) and VSK the
A. von Humboldt foundation, RFBR, RAS, and the pro-
gram ”The State Support of Leading Scientific Schools”
for support.
[1] V. F. Gantmakher and Y. B. Levinson, in Carrier Scat-
tering in Metals and Semiconductors (North-Holland,
Amsterdam, 1987)
[2] T.J. Gramila et al., Phys. Rev. Lett. 66, 1216 (1991)
[3] T.J. Gramila et al., Phys. Rev. B 47, 12957 (1993);
H. Rubel et al., Semicond. Sci. Technol. 10, 1229 (1995)
[4] B. Laikhtman et al., Phys. Rev. B. 41, 9921 (1990)
[5] X.G. Feng et al., Phys. Rev. Lett. 81, 3219 (1998)
[6] J.G.S. Lok et al., Phys. Rev. B 63, 041305 (2001)
[7] P. Debray et al., J. Phys.: Condens. Matter 13, 3389,
(2001); P. Debray et al., Semicond. Sci. Technol. 17, R21,
(2002)
[8] M. Yamamoto et al., Science 313, 204, (2006)
[9] B.J. van Wees et al., Phys. Rev. Lett. 60, 848 (1988);
D.A. Wharam et al., J. Phys. C 21, L209 (1988)
[10] A. Kristensen et al., Phys. Rev. B 62, 10950 (2000)
[11] L.I. Glazman, A.V. Khaetskii JETP Lett. 48 591 (1988)
[12] For increasing Gdet the noises in the detector circuit in-
crease, making the dc measurements very difficult.
[13] The input resistance of the I-V amplifier, the ohmic con-
tacts and wiring resistances result in Rext = 17 kΩ. The
validity of the above formula has been checked by apply-
ing an additional 47 kΩ resistor in series to Rext.
[14] V.S. Khrapai et al., Phys. Rev. Lett. 97, 176803 (2006)
[15] L.W. Molenkamp et al., Phys. Rev. Lett. 68, 3765 (1992);
H. van Houten et al., Semicond. Sci. Technol. 7, B215
(1992)
[16] A.S. Dzurak et al., J. Phys.: Condens. Matter 5, 8055,
(1993)
[17] A. Palevski et al., Phys. Rev. Lett. 62, 1776 (1989); for
asymmetric heat production in 3D point contacts see
U. Gerlach-Meyer, H.J. Queisser Phys. Rev. Lett. 51,
1904 (1983)
[18] |Icf| is reduced for Vdrive < 0 (> 0) and the drive-QPC
shifted to the lh (rh) side of the detector-QPC (Fig. 4).
This is understood in terms of absorption of phonons in
both leads of the detector-QPC.
[19] B.K. Ridley, Rep. Prog. Phys 54, 169 (1991)
[20] A.A. Verevkin et al., Phys. Rev. B 53, R7592 (1996)

  We study the interaction between two adjacent but electrically isolated
quantum point contacts (QPCs). At high enough source-drain bias on one QPC, the
drive QPC, we detect a finite electric current in the second, unbiased,
detector QPC. The current generated at the detector QPC always flows in the
opposite direction than the current of the drive QPC. The generated current is
maximal, if the detector QPC is tuned to a transition region between its
quantized conductance plateaus and the drive QPC is almost pinched-off. We
interpret this counterflow phenomenon in terms of an asymmetric phonon-induced
excitation of electrons in the leads of the detector QPC.

Counterflow of electrons in two isolated quantum point contacts
V.S. Khrapai,1, 2 S. Ludwig,1 J.P. Kotthaus,1 H.P. Tranitz,3 and W. Wegscheider3
Center for NanoScience and Department für Physik, Ludwig-Maximilians-Universität,
Geschwister-Scholl-Platz 1, D-80539 München, Germany
Institute of Solid State Physics RAS, Chernogolovka, 142432, Russian Federation
Institut für Experimentelle und Angewandte Physik,
Universität Regensburg, D-93040 Regensburg, Germany
We study the interaction between two adjacent but electrically isolated quantum point contacts
(QPCs). At high enough source-drain bias on one QPC, the drive-QPC, we detect a finite electric
current in the second, unbiased, detector-QPC. The current generated at the detector-QPC always
flows in the opposite direction than the current of the drive-QPC. The generated current is maximal,
if the detector-QPC is tuned to a transition region between its quantized conductance plateaus and
the drive-QPC is almost pinched-off. We interpret this counterflow phenomenon in terms of an
asymmetric phonon-induced excitation of electrons in the leads of the detector-QPC.
PACS numbers: 73.23.-b, 73.23.Ad, 73.50.Lw
The state of a confined quantum system is modified
by interactions with an external field (or with exter-
nal sources of energy). In semiconductor nanostruc-
tures the energy and quasi-momentum of electrons act-
ing as probe are strongly influenced by the environment,
e. g. via electron-electron or electron-phonon interaction.
If driven out of equilibrium, Coulomb forces establish
the local equilibrium within the electron system whereas
electron-phonon interactions dominate the energy ex-
change with the environment [1]. Drag experiments in
semiconductor nanostructures provide a tool to study the
effect of external electrons or phonons onto a probe elec-
tron system.
Current drag between parallel two-dimensional (2D)
electron layers has been investigated in GaAs/AlGaAs
bilayer systems. At small interlayer separations, ob-
servations are consistent with the Coulomb drag phe-
nomenon [2]. At larger separations virtual-phonon ex-
change has been invoked to explain the data [3]. A neg-
ative sign of a current drag between 2D and 3D electron
gases in GaAs was explained by the Peltier effect [4]. At
high filling factors in a perpendicular magnetic field a
sign change of the longitudinal drag between parallel 2D
layers was found as a function of the imbalance of the
electron density in the two layers [5, 6].
Interactions between two lateral quantum wires in
GaAs have been investigated in Ref. [7]. The observed
frictional drag, strongly oscillating as a function of the
one-dimensional (1D) subband occupation, was inter-
preted in terms of Coulomb interaction between two
Luttinger liquids. Recently, the observation of negative
Coulomb drag between two disordered lateral 1D wires in
GaAs in perpendicular magnetic fields was reported [8].
Here we report on a novel interaction effect between
two neighboring quantum point contacts (QPCs), em-
bedded in mutually isolated electric circuits. When a
strong current is flowing through the partially transmit-
ting drive-QPC, we detect a small current in the sec-
ond, unbiased, detector-QPC. The detector current flows
in the opposite direction of the drive current and shows
a nonlinear dependence on the source-drain bias of the
drive-QPC. It oscillates as a function of the detector-
QPC transmission. We suggest an explanation of this
counterflow phenomenon in terms of asymmetric phonon-
induced excitation of ballistic electrons in the leads of the
detector-QPC.
Our samples are prepared on a GaAs/AlGaAs het-
erostructure containing a two-dimensional electron gas
90 nm below the surface, with an electron density of
nS = 2.8 × 10
11 cm−2 and a low-temperature mobility
of µ = 1.4 × 106 cm2/Vs. An AFM micrograph of the
split-gate nanostructure, produced with e-beam lithogra-
phy, is shown in the left inset of Fig. 1. The negatively
biased central gate C divides the electron system into two
separate circuits, and prevents leakage currents between
them. Two QPCs are defined on the upper and lower
side of the central gate, respectively, by biasing gates 8
and 3. Other gates are grounded if not stated otherwise.
The right inset of Fig. 1 shows a sketch of the coun-
terflow experiment. We use separate electric circuits for
the (upper) drive-QPC and (lower) detector-QPC. A dc
bias voltage, Vdrive, is applied to the left lead of the drive-
QPC, while the right lead is grounded. A current-voltage
amplifier with an input voltage-offset of about 10 µV is
connected to the right lead of the detector-QPC. Its left
lead is always maintained at the same offset potential
in order to assure zero voltage drop across the detector-
QPC. In both circuits, a positive sign of the current cor-
responds to electrons flowing to the left. For differential
counterflow conductance measurements, the drive bias is
modulated at a frequency of 21 Hz and the resulting ac
current component in the detector circuit is measured
with lock-in detection in the linear response regime. All
measurements are performed in a dilution refrigerator at
an electron temperature below 150 mK. The experimen-
tal results are the same if detector and drive QPC are
interchanged.
First, we characterize the QPCs using a standard dif-
ferential conductance measurement. Figure 1 displays
the differential conductances of both QPCs in linear re-
http://arxiv.org/abs/0704.0132v2
drive
-0.7 -0.5 -0.3
FIG. 1: Conductance of the drive-QPC (dashed line) and the
detector-QPC (solid line) in the linear response regime as a
function of respective gate voltages V8 and V3. Symbols on
the detector-QPC curve mark the V3 values used for coun-
terflow conductance measurement presented in Fig. 2b. Left
inset: AFM micrograph of the metal gates on the surface
of the heterostructure (bright tone). Crossed squares mark
contacted 2DEG regions. The scale bar equals 1 µm. Right
inset: sketch of the counterflow measurement. The directions
of currents are shown for the case of Vdrive > 0.
sponse, measured as a function of the respective gate
voltage V3, or V8. At low gate voltages, the QPCs are
pinched-off and the conductance is close to zero. With
increasing gate voltage, 1D channels successively open
up [9]. For both QPCs we observe three conductance
plateaus approximately quantized in units of G0 = 2e
With high bias spectroscopy [10] we find the spacing
between the two lowest subbands to be approximately
4 meV (3 meV) for the drive (detector) QPC. The half-
width of the energy window for opening a 1D subband is
∆ ≈ 0.5 meV in both QPCs.
Having characterized the QPCs, we turn to counter-
flow measurements. Fig. 2a shows the dc counterflow
current, Icf, through the detector-QPC and the differ-
ential counterflow conductance, gcf ≡ dIcf/dVdrive, as a
function of the bias on the drive-QPC. Here, the drive-
QPC is tuned to nearly half a conductance quantum
Gdrive = G0/2, while the detector-QPC is in the pinch-off
regime (i.e. the lowest 1D subband bottom is well above
the Fermi level) with Gdet ≃ 10 GΩ
−1. Surprisingly, for
|Vdrive| & 1 mV, a finite current is observed in the un-
biased detector circuit. The direction of Icf is opposite
to that of the drive-QPC current Idrive. The dc coun-
terflow current is a threshold-like, nearly odd function of
Vdrive. Correspondingly, the differential counterflow con-
ductance is negative and a nearly even function of Vdrive.
The sign of gcf expresses a phase shift of π between the
applied ac modulation of Vdrive and the detected ac com-
ponent of the counterflow current.
Figures 2c and 2d show the absolute value of Icf for the
nearly pinched-off detector as a function of the voltage
on gate 8, which tunes the drive-QPC transmission. The
-4 -2 0 2 4
-0.04
-0.02
-0.7 -0.6 -0.5 -0.4
drive
 (mV)
(d) G
FIG. 2: (a) - Icf and gcf for the nearly pinched-off detector-
QPC as a function of Vdrive. (b) - gcf measured for a set of
Gdet values marked by according symbols in Fig. 1. (c,d)
- Absolute value of Icf as a function of the drive-QPC gate
voltage V8, for Vdrive = ±2.25 mV (c) and Vdrive = ±4 mV(d).
Also shown is the drive-QPC’s conductance in linear response
(c) and its differential conductance at Vdrive = ±4 mV (d).
Solid (dotted) lines correspond to Vdrive < 0 (> 0). In (a),(b)
gates 7 and 9 are grounded, while in (c),(d) V7 = V9 = −0.4 V.
The drive bias modulation used to measure gcf is 92 µV rms.
corresponding drive-QPC differential conductance curves
are also shown. For not too high Vdrive (Fig. 2c), a non-
zero counterflow current is only detected in the region
between pinch-off and the first conductance plateau of
the drive-QPC. For higher Vdrive (Fig. 2d) Icf increases
superlinearly with Vdrive at its maximum and remains
finite at higher gate voltages V8. Since the source bias
effects the potential distribution near the constriction,
the nonlinear 1/2 conductance plateau of the drive-QPC
shifts when changing Vdrive [11]. This causes the shift of
the extrema on Fig. 2d as well as the asymmetry of gcf
in Fig. 2a when reversing the bias.
We proceed to study the counterflow effect in the
-0.6 -0.5 -0.4 -0.3
-0.10
-0.05
-0.65 -0.60 -0.55
=17 kΩ
=17 kΩ
FIG. 3: (a)- gcf as a function of the detector-QPC gate volt-
age V3. Filled symbols correspond to the gcf measured at a
finite external resistance Rext = 17 kΩ, while open symbols
show the corrected counterflow conductance Rext = 0 (see
text). Also shown are the transmission functions Tn(1 − Tn)
for the three lowest 1D subbands of the detector-QPC (dashed
lines), scaled to fit the corrected data. During the gcf mea-
surement the drive bias is modulated with a 230 µV rms signal
about the mean value Vdrive = +2.05 mV. (b) - Normalized gcf
(symbols as in (a)) and transmission function of the lowest 1D
detector-QPC subband 4T0(1−T0) (dashed line) as a function
of V3. The scale bar shows a gate voltage interval correspond-
ing to a change of the 1D subband energy by 0.5 meV. Inset:
Sketch of possible scattering processes of nonequilibrium elec-
trons and holes at a partially transmitting detector-QPC.
regime of a more opened detector-QPC. Figure 2b plots
gcf [12] as a function of Vdrive for several values of Gdet
between 0 and G0 (marked with the same symbols in
Fig. 1). The qualitative appearance of gcf(Vdrive) is in-
dependent of Gdet. However, the amplitude of gcf is a
strongly non-monotonic function of the detector trans-
mission. The counterflow conductance reaches its maxi-
mum for Gdet ≈ G0/2 and decreases rapidly with further
increasing Gdet. Note that the absolute value of gcf is
small, corresponding to a maximal ratio of the counter-
flow and drive currents |Icf/Idrive| . 10
In Fig. 3a gcf is plotted as a function of V3, controlling
the detector transmission. Vdrive and V8 are adjusted for
maximal gcf and kept fixed. Confirming the trend seen in
Fig. 2b, the measured gcf (solid symbols) strongly oscil-
lates with increasing V3 and displays three pronounced
maxima before the detector-QPC is fully opened. The
position of the n-th maximum (n = 0,1,2) is close to the
value of V3, where Gdet/G0 ≃ n + 0.5 (Fig. 1). Here,
the energy EnS of the bottom of the n-th 1D subband
-4 -2 0 2 4
��������������
drive
 (mV)
FIG. 4: Drive bias dependence of the counterflow current
through the pinched-off detector-QPC for the drive-QPC
formed with gate 6 (dotted line) or gate 10 (solid line). The
detector-QPC conductance is about Gdet = 5 GΩ
−1. The
drive-QPCs are tuned to provide the maximal effect. Insets:
sketches of the two counterflow measurements. The directions
of currents are shown for the case of Vdrive > 0.
of the detector-QPC aligns with the Fermi level of the
leads EnS ≃ EF. In contrast, gcf is close to zero for fully
transmitting 1D channels (Gdet/G0 ≃ n+1). The over-
all magnitude of gcf decreases with increasing V3, hence
Gdet. This is caused by a finite series resistance Rext of
the external circuit, which results in a measured gcf lower
than the case for an ideal ammeter [13]. The corrected
counterflow conductance, gidealcf ≡ gcf · (1 + Rext · Gdet),
corresponding to Rext = 0, is shown in Fig. 3a with open
symbols. The corrected maxima are roughly equal in size
and symmetric. Moreover, the shape of the n-th maxi-
mum compares quite well with the corresponding func-
tion of the equilibrium transmission Tn(1−Tn), extracted
from the detector conductance data Tn ≡ Gdet/G0 − n
(dashed lines in Fig. 3a).
In Figure 3b we plot the normalized gcf and the trans-
mission function 4T0(1− T0) on a logarithmic scale near
the detector pinch-off. In the pinch-off regime (i.e. for
T0 ≪ 1) the transmission probability of a QPC is ex-
pressed as T0(E) ∝ exp([E − E
S]/∆) [11]. Here E is
the kinetic energy of current carrying electrons and ∆
is the half-width of the energy window for opening a 1D-
subband. The energy E0S of the detector-QPC is con-
trolled by gate 3 via E0S ∝ −|e|V3. This explains a nearly
exponential drop of the transmission function with de-
creasing V3 (Fig. 3b). In contrast, the measured gcf
drops considerably slower and remains finite even where
the detector-QPC is already pinched-off in equilibrium.
This experimentally observed excess contribution of the
normalized gcf versus T0(EF ) signals that the counter-
flow current carrying electrons are excited well above the
Fermi level. Converting the shift in gate voltage (see the
bar in Fig. 3b) to energy, we find a characteristic excita-
tion energy of E∗ ≈ 0.5 meV. This is consistent with a
recently reported 1 meV bandwidth excitation provided
by the drive-QPC for electrons in a nearby double-dot
quantum ratchet [14].
Next we study the counterflow effect between spatially
shifted QPCs. Figure 4 shows Icf through the nearly
pinched-off detector-QPC as a function of the bias on the
drive-QPC, which is formed either with gate 10 or gate
6, while gate 8 is now grounded (Fig. 1). Despite the
shift of the drive-QPC position relative to the detector-
QPC by about 300 nm, the odd drive bias dependence
of the counterflow current found in Fig. 2 is practically
preserved. This indicates that the excitation of electrons
in one of the leads of the detector-QPC is not restricted
to the close vicinity of the drive-QPC.
The oscillations of the counterflow conductance gcf in
Fig. 3 are reminiscent of thermopower oscillations that
have been investigated on individual QPCs [15, 16]. This
suggests that Icf is caused by an energetic imbalance
across the detector-QPC. If the bottom of the n-th 1D-
subband of the detector-QPC is well separated from the
Fermi-energy in comparison to the characteristic excita-
tion energy, i. e. if |EnS −EF| ≫ E
∗, this subband is either
fully transmitting (Tn(E) = 1) or closed (Tn(E) = 0). In
both cases electrons (holes) excited by E∗ above (below)
EF are equally transmitted and gcf = 0. In contrast,
if EnS ≃ EF excited electrons are more likely transmit-
ted than excited holes (see inset of Fig. 3b), resulting in
gcf 6= 0.
The energetic imbalance across the detector-QPC we
propose to be caused by phonon-based energy transfer
from the drive-QPC. The excess energy of carriers in-
jected across the drive-QPC is mainly relaxed by emis-
sion of acoustic phonons. We consider the drive-QPC in
the non-linear regime near pinch-off where µS − µD ≫
∆ and the transmission probability is strongly energy-
dependent (the source and drain leads are defined so that
their chemical potentials satisfy µS > µD). In this case
electrons injected into the drain lead have an excess en-
ergy of about e|Vdrive| ≡ µS−µD whereas the source lead
remains essentially in thermal equilibrium [17]. Hence
acoustic phonons are predominantly generated in the
drain lead of the drive-QPC. Because of this asymme-
try electron-hole pairs are excited preferentially in the
adjacent lead of the detector-QPC [18]. This gives rise
to Icf directed opposite to the current through the drive-
QPC (and gcf < 0). The data in Fig. 2 clearly show, that
the counterflow effect is only observed in the non-linear
regime of the drive-QPC.
For a rough estimate we consider injected electrons
with a momentum relaxation time of 60 ps limited by
elastic scattering and an energy relaxation time of
1 ns [19, 20]. Assuming isotropic phonon emission we
estimate an energy transfer ratio which can account for
the observed value of Icf/Idrive within one order of mag-
nitude.
In summary, the current in a strongly biased drive-
QPC generates a current flowing in the opposite direc-
tion through an adjacent unbiased detector-QPC. This
counterflow current is maximal in between the conduc-
tance plateaus of the detector-QPC. The effect is most
pronounced near pinch-off of the drive-QPC, where it
behaves strongly non-linear. We interpret the results in
terms of an asymmetric phonon-based energy transfer.
The authors are grateful to V.T. Dolgopolov,
A.W. Holleitner, C. Strunk, F. Wilhelm, I. Favero,
A.V. Khaetskii, N.M. Chtchelkatchev, A.A. Shashkin,
D.V. Shovkun and P. Hänggi for valuable discussions
and to D. Schröer and M. Kroner for technical help. We
thank the DFG via SFB 631, the BMBF via DIP-H.2.1,
the Nanosystems Initiative Munich (NIM) and VSK the
A. von Humboldt foundation, RFBR, RAS, and the pro-
gram ”The State Support of Leading Scientific Schools”
for support.
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both leads of the detector-QPC.
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