PHYSICAL REVIEW B 97, 245308 (2018)

Vorticity-induced negative nonlocal resistance in a viscous two-dimensional electron system

A. D. Levin,1 G. M. Gusev,1 E. V. Levinson,1 Z. D. Kvon,2,3 and A. K. Bakarov2,3

1
Instituto de Física da Universidade de São Paulo, 135960-170, São Paulo, SP, Brazil
2
Institute of Semiconductor Physics, Novosibirsk 630090, Russia
3
Novosibirsk State University, Novosibirsk 630090, Russia

(Received 10 April 2018; revised manuscript received 6 June 2018; published 25 June 2018)

We report nonlocal electrical measurements in a mesoscopic size two-dimensional (2D) electron gas in a GaAs
quantum well in a hydrodynamic regime. Viscous electric flow is expected to be dominant when electron-electron
collisions occur more often than the impurity or phonon scattering events. We observe a negative nonlocal
resistance and attribute it to the formation of whirlpools in the electron flow. We use the different nonlocal
transport geometries and compare the results with a theory demonstrating the significance of hydrodynamics in
mesoscopic samples.

DOI: 10.1103/PhysRevB.97.245308

I. INTRODUCTION As a consequence, for an electronic fluid, a negative voltage

drop occurs across the strip in close proximity to the current
It is generally believed that, in the absence of disorder,
probes. Figure 1(b) illustrates the nonlocal, vicinity transport
a many-body electron system resembles the viscous flow.
geometry, where the current is injected in the left lateral and
Hydrodynamic characteristics can be specially enhanced in a bottom contacts, while the voltage drop occurs near the current
pipe flow setup, where the mean free path for electron-electron injection region. This geometry has been proposed in [9–11],
collision lee is much shorter than the sample width W , while the and the model clearly demonstrates the formation of whirlpools
mean free path due to impurity and phonon scattering l is larger in the hydrodynamic flow, yielding a negative nonlocal signal
than W . Viscosity is characterized by momentum relaxation in in transport measurements [11]. Note that the swirling features
the fluid and, in narrow samples, occurs at the sample boundary. can be observed only in the nonlocal configuration.
Calculation of the shear viscosity η is a difficult task because it The nonlocal vicinity effect has been studied experimentally
requires knowledge of particle interactions on the scale of l [1]. in an ultraclean graphene sheet [11]. It has been demonstrated
It has been predicted that the resistivity of metals in that the nonlocal signal undergoes a sign change from positive,
the hydrodynamic regime is proportional to electron shear at low temperatures, to negative, above elevated temperatures,
viscosity η = 14 vF2 τee , where vF is the Fermi velocity and τee that is associated with whirlpool emergence in the hydro-
is the electron-electron scattering time τee = lee /vF [2–6]. dynamic regime. Near room temperature, the signal again
This dependency could lead to interesting properties. For undergoes a sign change because the Ohmic contribution starts
example, resistance decreases with the square of temperature to dominate the vicinity response at high T . Note that such
ρ ∼ η ∼ τee ∼ T −2 , the so called Gurzhi effect, and with the dramatic experimental appearance of hydrodynamic features
square of sample width ρ ∼ W −2 . The negative differential in nonlocal transport has not been accompanied by observation
resistance has been observed previously in GaAs wires, which of the Gurzhi effect in local transport. Moreover, the transversal
has been interpreted as the Gurzhi effect due to heating by the nonlocal geometry (Fig. 1) has not been studied experimentally
current [7]. A remarkable manifestation of the hydrodynamic with respect to possible vorticity effects. Other materials, such
effect is a swirling feature in the flow field, referred to as a as GaAs quantum wells, have a particular interest because
vortex. The vorticity can drive the current against an applied they possess the highest mobility over wide temperature
electric field and generate backflow near the current injection ranges. It is also worthwhile to extend the theoretical approach
region, which can be detected in the experiment as a negative [8–11] to a two-dimensional electron in GaAs with a parabolic
voltage drop [8]. A different transport measurement setup energy spectrum, which is different from the linear spectrum
has been proposed for the identification of viscosity related in graphene.
features in the hydrodynamic regime [8–11]. A series of updated theoretical approaches has been pub-
When fluid flows along a pipe, a quadratical velocity profile lished recently [12–15], providing additional possibilities to
is formed, which leads to the Gurzhi effect, and can be detected determine the viscosity from magnetotransport measurements,
from the anomalous temperature and sample width depen- which can be used for comparison with nonlocal measure-
dence, as is mentioned above. For illustration we modeled ments.
the Poiseuille flow for a two-dimensional neutral fluid. Figure In this study we measure the nonlocal resistance in meso-
1(a) shows the configuration, which has been proposed in [8], scopic GaAs quantum well systems. We determine all relevant
and where the current is injected across the sample between electron viscous parameters from the longitudinal magne-
vertical probes. In this geometry, one can see the vortex or toresistance in a wide temperature range, which provides an
whirlpools in the liquid flow outside of the main current path. estimate of the nonlocal signal, and compare it with

2469-9950/2018/97(24)/245308(7) 245308-1 ©2018 American Physical Society

LEVIN, GUSEV, LEVINSON, KVON, AND BAKAROV PHYSICAL REVIEW B 97, 245308 (2018)

FIG. 1. Sketch of the different transport setup measurements,

showing a velocity flow profile. (a) Nonlocal transport setup, proposed
in [8]. (b) Nonlocal (vicinity) transport setup, proposed in [8–11].

experimental results. A good qualitative agreement between

experimental and simulated data has been obtained.
FIG. 2. Top: Image of the Hall-bar device. Top right: Zoomed
II. NEGATIVE GIANT MAGNETORESISTANCE, Hall-bar bridge. Temperature dependent magnetoresistance of a GaAs
EXPERIMENT, AND DISCUSSION quantum well in a Hall bar sample W = 5 μm. Thick lines are
examples illustrating magnetoresistance calculated from Eqs. (1) and
Our samples are high-quality, GaAs quantum wells with (2) for different temperatures: 4.2 K (red), 14 K (green), 19 K (blue),
a width of 14 nm. Parameters characterizing the electron 26 K (magenta), and 37.1 K (black).
system are given in Table I. We present experimental results
on Hall-bar devices designed in two different configurations.
Design I consists of three 5 μm wide consecutive segments consecutive segments of different length (2, 7, 2 μm), and
of different length (10, 20, 10 μm), and eight voltage probes. eight voltage probes. The measurements were carried out in
Figure 2(top) shows the image of a typical multiprobe Hall a VTI cryostat, using a conventional lock-in technique to
device I. Design II is also a Hall bar with three 2 μm wide measure the longitudinal ρxx resistivity with an ac current of
0.1–1 μA through the sample, which is sufficiently low to avoid
overheating effects. Five Hall bars from two different wafers
TABLE I. Parameters of the electron system in a mesoscopic were studied.
samples at T = 1.4 K. Parameters are defined in the text. Longitudinal magnetoresistance has been studied in pre-
vious research for different configurations of the current and
W ns vF l l2 η voltage probes [16]. Before analyzing the nonlocal effect and
(μm) (1011 cm2 ) (107 cm/s) (μm) (μm) (m2 /s) in order to make this analysis more complete, we present
5 9.1 4.1 40 2.8 0.3 the results of measurements of the longitudinal magnetore-
2 6.0 3.3 20.6 1.4 0.12 sistivity ρxx (B). Figure 2(a) shows ρxx (B) as a function of
magnetic field and temperature. One can see two characteristic

245308-2
VORTICITY-INDUCED NEGATIVE NONLOCAL … PHYSICAL REVIEW B 97, 245308 (2018)

features: a giant negative magnetoresistance (∼400%–600%)

with a Lorentzian-like shape, and a pronounced temperature
dependence on zero field resistance. In the hydrodynamic
approach, the semiclassical treatment of the transport describes
the motion of carriers when the higher order moments of the
distribution function are taken into account. The momentum
relaxation rate 1/τ is determined by electron interaction with
phonons and static defects (boundary). The second moment
relaxation rate 1/τ2,ee leads to the viscosity and contains
the contribution from the electron-electron scattering and
temperature independent scattering by disorder [12,13]. It has
been shown that conductivity obeys the additive relation and
is determined by two independent parallel channels: the first
is due to momentum relaxation time and the second due to
viscosity [12,13]. This approach allows the introduction of the
magnetic field dependent viscosity tensor and the derivation of
the magnetoresisivity tensor [12–15]:
 
τ 1
ρxx = ρ0bulk 1 + ∗ , (1) FIG. 3. Relaxation rate 1/τ2 as a function of the temperature
τ 1 + (2ωc τ2,ee )2
obtained by fitting the theory with experimental results W = 5 μm.
+6ls )
where ρ0bulk = m/ne2 τ , τ ∗ = W (W12η , viscosity η = Thick black line is Eq. (2), thin black line is Eq. (3), dashes are
1 2 Eq. (4).
v τ
4 F 2,ee
.
We also collect the equations for relaxation rates separately:
1 T2 1 fitting parameters: τ (T ), τ ∗ (T ), and τ2,ee (T ). Figure 3 shows
= AFeeL 2
+ , (2) the dependencies of 1/τ2,ee (T ) extracted from the comparison
τ2,ee (T ) [ln(EF /T )] τ2,0
of the magnetoresistance shown in Fig. 2 and Eq. (1). We
where EF is the Fermi energy, and the coefficient AFeeL be can compare the temperature dependence of τ2,ee1(T ) with theoretical
expressed via the Landau interaction parameter, however, it predictions given by Eqs. (2)–(4) and present the results of such
is difficult to calculate quantitatively (see discussion in [12]). comparison in Fig. 3. The following parameters are extracted:
The relaxation rate τ2,01
is not related to the electron-electron 1/τ2,0 = 1.45 × 1011 s, AFeeL = 0.9 × 109 s−1 K−2 , A0ee = 1.3,
10 −1 −2
collisions, since any process responsible for relaxation of the Adaee = 2.0 × 10 s K . All theoretical curves demonstrate
second moment of the distribution function, even scattering by reasonable agreement within experimental uncertainty. Hence,
static defect, gives rise to viscosity [12]. A logarithmic factor is these mechanisms lead to nearly equivalent results and cannot
also present in the expression for quantum lifetime of weakly be unambiguously distinguished based only on the temperature
interacting 2D gas due to electron-electron scattering [17]: dependence of the relaxation time. Note that analysis of the
nonlocal effect, considered below, does not depend on the
h̄ T 2 [ln(2EF /T )] h̄
= A0ee + , (3) relaxation mechanism.
τ0,ee (T ) EF τ2,0 In addition, we extract the temperature dependence of
where A0ee is a numerical constant of the order of unity. Note, the moment scattering rate and determine parameters Aph =
however, that since the relaxation time τ0,ee is related to the 109 s K−1 and τ0 = 5 × 10−10 s, which are correlated with
kinematic of the electron-electron collisions, Expression (2) previous studies [17,18]. Relaxation time τ ∗ (T ) depends on
is more convenient and it is preferable to use. Finally, it has the τ2,ee (T ) and boundary slip length ls . Comparing these
been shown that due to the disorder assisted contribution to values, we find that ls = 3.2 μm < L, and, therefore, in our
the relaxation rate of the second moment of the distribution case, it is appropriate to use diffusive boundary conditions.
function, the expression is rewritten as Table I shows the mean free paths l = vF τ , l2 = vF τ2,ee , and
viscosity, calculated with parameters extracted from the fit of
1 1 experimental data.
da ee T +
= Ada 2
, (4)
τ2,ee (T ) τ2,0
where the coefficient Ada
ee depends on the disorder type and its
III. EXPERIMENT: NONLOCAL RESISTANCE
strength [12]. The moment relaxation rate is expressed as In this section we focus on the nonlocal configurations
1 1 because such geometry facilitates the observation of current
= Aph T + , (5) whirlpools. Figure 4 shows the transport in a nonlocal setup,
τ τ0
where the current is injected across the strip between probes
where Aph is the term responsible for the phonon scattering
4 and 8. The voltage drop is measured between probes 5
[18,19], and τ10 is the scattering rate due to static disorder (not and 7. Below we refer to it as C1 configuration. Poiseuille
1
related to the second moment relaxation rate τ2,0 ). flow for a two-dimensional liquid is presented in Fig. 1(a).
We fit the magnetoresistance curves in Fig. 2 and Note, however, that 2D charged liquid displays pronounced
the resistance in zero magnetic field with the three ballistic transport behavior. One can see strong oscillations

245308-3
LEVIN, GUSEV, LEVINSON, KVON, AND BAKAROV PHYSICAL REVIEW B 97, 245308 (2018)

FIG. 4. Nonlocal transport signal versus magnetic field for FIG. 5. T dependence of the nonlocal signal for different sample
different temperatures W = 5 μm. The dots represent results for the configuration. Solid lines show the calculations from Eq. (6) for
billiard model. x = 10 μm (W = 5 μm) and x = 5 μm (W = 2 μm). Dashes: T
dependence of the ballistic peak at B = 0.017T .
in weak magnetic fields due to geometrical resonance effects
considered in the semiclassical billiard model [20,21]. We in configuration C2, and the results are displayed in Fig. 6.
perform numerical simulations of the electron trajectories in Note, however, that in contrast to configuration C1, the bend
ballistic structures. The results of theses simulations (black resistance reveals a strong negative resistance peak near zero
dots) are compared to the experimental data. We observe magnetic field [21,22]. This peak may mask the negative
an agreement with experimental data only at low magnetic nonlocal signal due to viscosity, and detailed comparison is
field. Although the position of the resistance peaks at higher required to examine the significance of the hydrodynamic
magnetic field coincide with calculations, the negative peak effect at low and high temperatures. Figure 7 presents the
has a much smaller value, and the positive peak is wider than results of the nonlocal resistance temperature measurements in
that obtained from the billiard model. Figure 4 also shows the configuration C2 in zero magnetic field. One can see that the
evolution of the nonlocal magnetoresistance with temperature. signal dramatically drops to zero in the W = 5 μm sample, and
One can see that all oscillations are smeared out by temperature resistance changes sign at high temperature in the W = 2 μm
and magnetoresistance at high temperature has a parabolic sample. We also used a similar voltage measurement setup,
shape. Remarkably, the nonlocal resistance at B = 0 is positive where the current is injected between probes 1 and 8 and the
at low temperatures, in accordance with the billiard model voltage is measured between probes 4 and 5 (referred to as
calculations, and then it changes sign and becomes negative configuration C3). The nonlocal resistance in configuration
at higher temperatures (Fig. 5). Figure 6 shows the transport C3 at zero magnetic field is shown in Fig. 7 for both samples
in a nonlocal setup, where the current is injected between designs.
probes 1 and 8 and the voltage is measured between probes
5 and 6 (referred to as configuration C2). The Poiseuille
IV. THEORY AND DISCUSSION
flow for a two-dimensional liquid is presented in Fig. 1(b).
As in configuration C1, one can see strong oscillations due As has been shown in the previous section the viscosity
to the geometrical resonance effect. Note that the ballistic leads to the incorporation of an extra relaxation mechanism
transport in this configuration is very well established and [12–15] in zero magnetic field: ρ = ρ0bulk (1 + ττ∗ ). The domi-
studied previously in numerous publications [20,21]. In cross nant viscous contribution to resistivity corresponds to the small
junction geometry, it was denominated as bend resistance [21]. ratio between relaxation of the second moment of the electron
We also perform the classical simulations for the transport distribution function and the first moment τ ∗ /τ  1.

245308-4
VORTICITY-INDUCED NEGATIVE NONLOCAL … PHYSICAL REVIEW B 97, 245308 (2018)

FIG. 7. T dependence of the nonlocal signal for different sample

configurations. Thin solid lines show the calculations from Eq. (7) for
FIG. 6. Nonlocal transport signal versus magnetic field for differ-
x = 3 μm (W = 5 μm) and x = 1.5 μm (W = 2 μm).
ent temperatures W = 5 μm. The dots represent results for the billiard
model.
to the current injector probe [10]. However, the value of Dη
affects the spatial extension of the whirlpools, therefore, a high
Comparative analysis between nonlocal geometries C1 viscosity system facilitates observation of the negative vicinity
and C2 demonstrates a qualitative difference. Crucially, the resistance for a voltage detector placed at a large distance
experimental observation of swirling features depends on the from the current injection probe. Moreover, the ballistic effect
parameters that affect the spacial distribution of the two- may induce the negative vicinity signal [19] and, therefore,
dimensional potential inside the viscous charge flow. The requires more careful qualitative analysis. In the previous
first parameter is the boundary slip length ls . The boundary section we show the temperature dependence of low field
no-slip conditions correspond to the ideal hydrodynamic case magnetoresistance as well as the electrical resistivity over a
of diffusive boundaries with ls = 0. It has been shown that temperature range extending from 1.7 to 40 K and obtain
the negative nonlocal signal is robust to boundary conditions variation of the viscosity time with temperature. We use this
[10]. For example, the Gurzhi effect disappears for free surface data to estimate the nonlocal signal in our samples. The models
boundary conditions (ls = ∞), while whirlpools in hydro- [8–10] predict negative nonlocal resistance in configuration C1
dynamic electron flow, and the resulting negative nonlocal at the distance x = π x/W from the main current path in the
response, do exist. The second parameter which drastically limits of free surface boundary conditions (ls = ∞) in zero
affects whirlpool behavior is the vorticity diffusion length magnetic field:
√  
Dη = ητ . Figure 8 represents the temperature dependence of  2
characteristic lengths in a W = 5 μm sample. Previous studies ln[tanh2 (x/2)] Dν cosh(x)
RNL = −ρ0
C1
+ 4π . (6)
have not investigated whether typically developed current π W sinh2 (x)
whirlpools show sensitivity to the geometry and confinement
effect [8–10]. However, the careful inspection of theoretical In contrast to configuration C1, the results for vicinity
results [9] reveals that geometry C1 exhibits the occurrence geometry can be simplified only in the limit where the distance
of whirlpools only above the threshold value of Dη = 0.225W between the current injection probe is infinite:
(Fig. 8). The vicinity geometry C2, which is shown in Fig. 1(b),   2 
by contrast, allows the formation of current whirlpools for ρ0 ln[4T ] x Dν 1
RNL = −
C2
− +π , (7)
arbitrary small values of Dη , but only in very close proximity 2 π W W T

245308-5
LEVIN, GUSEV, LEVINSON, KVON, AND BAKAROV PHYSICAL REVIEW B 97, 245308 (2018)

FIG. 8. The characteristic parameters as a function of the temper- FIG. 9. The absolute value of the nonlocal resistance for two
ature for the sample with width W = 5 μm. The whirlpool threshold configurations as a function of the distance from the injector electrode
is indicated by the dashes. T = 4.2 K, parameters are determined from local magnetoresistance
measurements.

where T = sinh2 (x/2). Figure 9 shows the nonlocal resis-

tances in both configurations as a function of distance between Figure 7 shows the calculations from Eq. (7). Note that the
voltage probe and current injector x calculated from Eqs. (6) analytical formula has been derived under several assumptions
and (7) with parameters independently extracted from the local and we can apply the formula just for the evaluation of the
magnetoresistance measurements at T = 4.2 K. For visualiza- upper limits of the signal. Figure 7 presents the results of
tion of the data in the negative range, we used an absolute such calculations. One can see that the predicted signal agrees
log scale. We observe that the magnitudes of nonlocal signals with experimental data for x = 3 μm (W = 5 μm) and x =
exhibit a universally exponential decay with distance from the 1.5 μm (W = 2 μm), which roughly correspond to the distance
current injector. Note that the nonlocal resistance is much between the centers of the probes. Note that, in a realistic
stronger for geometry C1. The advantage of configuration sample, the width of the probes is comparable with the sample
C1 is that the ballistic contribution is positive and, therefore, width W , while the theory considers x  W , also indicating
it can be unambiguously discriminated from the negative the approximate character of the calculation. We may conclude
viscous contribution. The calculated temperature dependence here that geometry C1 exhibits a direct relation between the
of RNLC1
is shown in Fig. 5 for x = 10 μm (W = 5 μm) and negative signal and formation of the current whirlpools. In
x = 5 μm (W = 2 μm), which roughly correspond to the geometries C2 and C3, negative nonlocal resistance follows
distance between the center of the probes. Note that the ballistic the hydrodynamic predictions up to 30 K, however, it is
contribution to the transport also depends on the tempera- very likely that the ballistic contribution is comparable or
ture due to the thermal broadening of the Fermi distribution bigger than the hydrodynamic one at low temperatures. Above
function and scattering by the phonons. A rough estimate of 30 K we observe a positive signal, which disagrees with
the nonlocal ballistic resistance temperature dependence for both ballistic and hydrodynamic predictions. We attribute this
L < l may be obtained using the formula RNL ∼ exp(−L/ l), behavior to approaching the condition Dη = 0.225W . Note
where L is the distance between probes [23]. Figure 5 shows that the observation of negative vicinity nonlocal resistance in
the T dependence of the ballistic peak at B = 0.017T . One graphene [11] requires more careful inspection of the ballistic
can see a rapid decrease of the peak with temperature. contribution. Moreover, the condition Dη = 0.225W is not
Therefore, the negative nonlocal resistance in zero field and fully completed (see also discussion in [10]), therefore, our
at high temperature can be attributed only to hydrodynamic observation of the negative nonlocal resistance in geometry
effects. C1 provides more clear evidence of current vortices. It is
We also compare predictions for configurations C2 and C3 important to note that the transport signatures of the viscosity
with experimental results. Note that we normalized ballistic in the nonlocal effect are correlated in our samples with other
resistance for the peak value at B = 0.008T (Fig. 6), which observations, such as a giant longitudinal magnetoresistance
we found more reliable, since this peak weakly depends on the and the Gurzhi effect [16].
boundary conditions and sample geometry [20]. The residual
contribution at zero magnetic field could be due to viscous
V. SUMMARY AND CONCLUSIONS
effects. In general, the ballistic contribution alone can explain
the temperature dependence in zero field, below 20 K, without In conclusion, we have studied nonlocal transport in a
taking into account the viscous term. Above T = 20 K, bal- mesoscopic two-dimensional electron system in terms of
listic contribution should be exponentially small (see Fig. 5). viscosity of the fluids. In contrast to the Ohmic flow of the

245308-6
VORTICITY-INDUCED NEGATIVE NONLOCAL … PHYSICAL REVIEW B 97, 245308 (2018)

particles, viscous flow can result in a backflow of the current and low magnetic field transport described by hydrodynamic
and negative nonlocal voltage. We have measured voltage in theory.
different arrangements of current and voltage contacts and
found a negative response, which we attributed to the formation ACKNOWLEDGMENTS
of current whirlpools. Nonlocal viscosity-induced transport
is strongly correlated with observations of the Gurzhi effect The financial support of this work by the Russian Science
Foundation (Grant No.16-12-10041), FAPESP (Brazil), and
CNPq (Brazil) is acknowledged.

[1] S. Conti and G. Vignale, Phys. Rev. B 60, 7966 (1999). [12] P. S. Alekseev, Phys. Rev. Lett. 117, 166601 (2016).
[2] R. N. Gurzhi, Sov. Phys. Usp. 11, 255 (1968); R. N. Gurzhi, [13] T. Scaffidi, N. Nandi, B. Schmidt, and A. P. Mackenzie, and J. E.
A. N. Kalinenko, and A. I. Kopeliovich, Phys. Rev. Lett. 74, Moore, Phys. Rev. Lett. 118, 226601 (2017).
3872 (1995). [14] L. V. Delacretaz and A. Gromov, Phys. Rev. Lett. 119, 226602
[3] M. Dyakonov and M. Shur, Phys. Rev. Lett. 71, 2465 (1993). (2017).
[4] M. I. Dyakonov and M. S. Shur, Phys. Rev. B 51, 14341 (1995). [15] F. M. D. Pellegrino, I. Torre, and M. Polini, Phys. Rev. B 96,
[5] M. Dyakonov and M. Shur, IEEE Trans. Electron Devices 43, 195401 (2017).
380 (1996). [16] G. M. Gusev, A. D. Levin, E. V. Levinson, and A. K. Bakarov,
[6] A. O. Govorov and J. J. Heremans, Phys. Rev. Lett. 92, 026803 AIP Adv. 8, 025318 (2018).
(2004). [17] Z. Qian and G. Vignale, Phys. Rev. B 71, 075112 (2005).
[7] L. W. Molenkamp and M. J. M. de Jong, Solid-State Electron. [18] J. J. Harris, C. T. Foxon, D. Hilton, J. Hewett, C. Roberts, and
37, 551 (1994); M. J. M. de Jong and L. W. Molenkamp, Phys. S. Auzoux, Surface Science 229, 113 (1990).
Rev. B 51, 13389 (1995). [19] T. Kawamura and S. Das Sarma, Phys. Rev. B 45, 3612 (1992).
[8] L. Levitov and G. Falkovich, Nat. Phys. 12, 672 (2016). [20] C. W. J. Beenakker and H. van Houten, Phys. Rev. Lett. 63, 1857
[9] I. Torre, A. Tomadin, A. K. Geim, and M. Polini, Phys. Rev. B (1989).
92, 165433 (2015). [21] M. L. Roukes, A. Scherer, S. J. Allen, Jr., H. G. Craighead,
[10] F. M. D. Pellegrino, I. Torre, A. K. Geim, and M. Polini, Phys. R. M. Ruthen, E. D. Beebe, and J. P. Harbison, Phys. Rev. Lett.
Rev. B 94, 155414 (2016). 59, 3011 (1987).
[11] D. A. Bandurin, I. Torre, R. K. Kumar, M. B. Shalom, A. [22] G. Timp, H. U. Baranger, P. de Vegvar, J. E. Cunningham, R. E.
Tomadin, A. Principi, G. H. Auton, E. Khestanova, K. S. Howard, R. Behringer, and P. M. Mankiewich, Phys. Rev. Lett.
Novoselov, I. V. Grigorieva, L. A. Ponomarenko, A. K. Geim, 60, 2081 (1988).
and M. Polini, Science 351, 1055 (2016). [23] Y. Hirayama and S. Tarusha, Appl. Phys. Lett. 63, 2366 (1993).

245308-7