# Magnetohydrodynamics and electro-electron interaction of massless Dirac
fermions.

D. A. Khudaiberdiev,1,3 G. M. Gusev,2 E. B. Olshanetsky, 1 Z. D. Kvon,1,3 and
N. N. Mikhailov 1,3 1Institute of Semiconductor Physics, Novosibirsk 630090,
Russia  2Instituto de Física da Universidade de São Paulo, 135960-170, São
Paulo, SP, Brazil  3Novosibirsk State University, Novosibirsk 630090, Russia

###### Abstract

The magnetotransport properties of massless Dirac fermions in a gapless HgTe
quantum well are investigated. In samples with narrow channels, a large
negative magnetoresistance with a Lorentzian profile is observed, which is
interpreted as a manifestation of electron viscosity due to electron-electron
interaction. Comparison of experiment with theory yields the shear stress
relaxation time of the Dirac fermions caused by electron-electron scattering.

††preprint: Phys.Rev.Research

In many cases, for the description of phenomena in solid-state structures, the
approximation of non-interacting electrons turns out to be inapplicable and
the electron-electron (e-e) interaction, which is one of the oldest problems
in solid state physics, acquires a key role quinn ; chaplik . The reduction of
dimensionality (D) from 3D to 2D sharply enhances the importance of e-e
interaction effects quinn2 . Moreover, it has been discovered recently that
relaxation of perturbations of various types in 2D Fermi system, could be both
quantitatively and qualitatively different from the quasiparticle lifetime
principi ; narozhny ; alekseev .

In the so called hydrodynamic regime of electron transport the shear stress
relaxation time $\tau_{2,ee}$ is responsible for shear viscosity which
describes the friction between adjacent layers of liquid moving with different
velocities. The index 2 in the e-e scattering time subscript $\tau_{2,ee}$
means that the viscosity coefficient is determined by the relaxation of the
second harmonic of the distribution function alekseev . The hydrodynamic
regime requires $l/l_{2,ee}\gg 1$ and $l_{2,ee}/W\ll 1$, where $l=v_{F}\tau$
is the electron transport mean free path related to momentum relaxation time
($\tau$) brought about by scattering on defects and phonons, $v_{F}$ is the
Fermi velocity, $W$ is the channel width, and $l_{2,ee}=v_{F}\tau_{2,ee}$ is
the mean free path for shear viscosity relaxation gurzhi -raichev . In the
presence of the perpendicular magnetic field B the shear viscosity becomes a
tensor depending on B, which leads to giant negative magnetoresistance with a
Lorentzian profile in narrow channel devices alekseev1 . It is possible to
extract shear viscosity and shear stress relaxation time from the comparison
of the experimental data and the theory. This has been preformed in Refs.
gusev1 ; gusev2 ; levin ; gusev3  for high-mobility GaAs quantum wells with
parabolic spectrum. The study of the temperature dependence of viscosity
allows one to distinguish between the electron hydrodynamics and ballistic
transport raichev  and explain the difference between e-e scattering in a
single well and in a bilayer gusev4 .

Figure 1: (Color online) Schematic of the transistor and the top view of the
sample. Resistivity $\rho$ as a function of gate voltage measured for sample
sections of different size, $T=4.2$ K. Figure 2: (Color online) The resistance
R(B) as a function of magnetic field at $V_{g}=4$ V for sample sections of
different size (a) and resistance as a function of gate voltage and magnetic
field for the sample section widths $W=3.5\>\text{$\mu$m}$ (b) and
$W=10\>\text{$\mu$m}$ (c), $T=4.2$ K.

The hydrodynamic approach is valid for systems with a linear spectrum, such as
graphene. Moreover, the electron hydrodynamics in graphene has aroused big
interest polini  because the observation of the negative nonlocal resistance
due to current whirpools bandurin1  and the superballistic flow in graphene
point contacts kumar  open up the opportunity to study the so called Dirac
liquids. The shear stress relaxation time calculation in Refs. principi ;
narozhny  highlights the subtle effect of the linear dispersion that
distinguishing graphene from the usual parabolic semiconductor-based 2D
systems.

Contrary to the quasiparticle lifetime in two-dimensional system which one
expects to be expressed as $\tau_{ee}\sim\frac{\hbar
E_{F}}{[(kT)^{2}ln(E_{F}/kT)]}$, it has been predicted that
$\tau_{2,ee}\sim\frac{\hbar E_{F}}{(kT)^{2}}$ for a Fermi gas and
$\tau_{2,ee}\sim\frac{\hbar ln^{2}(E_{F}/kT)}{(kT)^{2}}$ for a strongly
interacting Fermi liquid alekseev ; novikov . Proportionality coefficients
depend on the dispersion relations and are different for graphene and
conventional 2D systems principi ; narozhny ; alekseev .

It has been demonstrated that, similar to single layer graphene, a gapless
phase of single valley Dirac fermions exists in symmetric HgTe quantum wells
with the critical width $d_{c}=6.3$ nm buttner ; kozlov . The equation
describing the electron linear dispersion relation near Dirac point is
$E(k)=\hbar v_{F}k$, where Fermi velocity $v_{F}=7\times
10^{7}\text{\>cm/s}=c/430$ ($c$ is light velocity) is close to the Fermi
velocity in graphene $v_{F}=c/300$. Various methods have been proposed for
observation of viscosity in the hydrodynamic regime in graphene bandurin1 ;
bandurin2 . Although experimental studies of transport in a magnetic field are
mainly focused on observing the Hall viscosity effect berdyugin ; kim ,
subsequent analysis of the results obtained may provide more information on
the mechanism of electron-electron shear stress scattering of Dirac fermions.

Figure 3: (Color online) The resistance R(B) as a function of magnetic field
for different temperatures in a sample segment with a width of 3.5 $\mu$m for
$V_{g}=13$ V. The circles show the resistance R(B) calculated from Eq. (1) for
different temperatures T(K): 7.6, 19.2, 38.5, 46.1, 56.6, 64.

This paper presents the results of an experimental study of the
magnetotransport properties of two-dimensional Dirac fermions in gapless HgTe
quantum wells in a wide temperature range. In samples with a small channel
width at a high electron concentration, a large negative magnetoresistance
with a Lorentzian profile was found, which is in agreement with the
hydrodynamic model. Comparing experiment with theory, we were able to find the
stress relaxation time in a system with a linear spectrum.

Figure 4: (Color online) (a) The relaxations lengths $l_{tr}$ and $l_{2}$ as a
function of the densities, $T=4.2K$. (b)The relaxations length $l_{2,ee}$ as a
function of the temperature for two values of the Fermi energy $E_{F}=91$ meV
( black points) and $E_{F}=114$ meV (red points). The thick red and black
lines represent theory with parameters indicated in the text. Blue line is the
channel width.

Quantum wells $Cd_{0.65}Hg_{0.35}Te/HgTe/Cd_{0.65}Hg_{0.35}Te$ with (013)
surface orientations and a well thickness of 6.3 nm were prepared by molecular
beam epitaxy kozlov . The experimental structures were identical Hall bar
devices consisting of three consecutive sections with different widths W: 3.5,
10 and 50 $\mu$m, respectively. The distance between the voltage probes was
$L=12,30,100\text{\>$\mu$m}$, (fig.1). In addition, a macroscopic Hall bar
device with eight voltage probes was also examined. This Hall bar had the
width $W$ of $50\text{\>$\mu$m}$ and three consecutive segments of different
length $L$ $(100,250,100\text{\>$\mu$m})$ (not shown). A dielectric layer (200
nm of $SiO_{2}$) was deposited on the sample surface and then covered by a
TiAu gate. The density variation with gate voltage was $1.1\times 10^{11}$
cm-2V-1. The resistance in the presence of the perpendicular magnetic field,
R(B), has been measured in the temperature range 4.2 - 70 K using a standard
four point circuit with a 1-13 Hz ac current of 1-10 nA through the sample.

Figure 1 presents the resistivity $\rho=R\frac{W}{L}$ as a function of gate
voltage measured at zero magnetic field for segments of different width. One
can see that the resistivity is weakly size dependent and demonstrates a
typical peak near the charge neutrality point with a value close to
$h/2e^{2}$. Further we focus on the sample behavior in the positive gate
voltage region, corresponding to the Fermi level residing in the conduction
band. In contrast to graphene, the density of states in a zero gap HgTe QW is
not symmetric and it rapidly grows in the valence band when moving away from
the Dirac pointkozlov .

Figure 2(a) shows the resistance as a function of magnetic field for two
narrow segments of the mesoscopic sample with the widths 3.5 and 10 $\mu$m at
$V_{g}=4\text{\>V}$. One can see a large negative magnetoresistance (
$R(B)-R(0)<0)$) with a Lorentzian profile for the 3.5 $\mu$m segment and a
wider triangular-shaped peak for the 10 $\mu$m segment. The magnetoresistance
for the macroscopic sample is shown in the supplementary materials suppl . It
is positive for all gate voltages. The evolution of R(B) with gate voltages in
mesoscoscopic samples is given in Figures 2 b and c. One can see that in the
vicinity of the CNP the magnetoresistance becomes positive. Figure 3 shows the
representative traces illustrating the evolution of R(B) with temperature for
a fixed gate voltage far away from the CNP $V_{g}=13\text{\>V}$. The
resistance curves is almost independent of $T$ at temperatures below 15 K, but
R(B) becomes wider at high temperatures. Note that the resistance at $B=0$
increases with $T$. It was found that the $R(T)$ dependences above 4.2 K can
be well described by a cubic law: $R(T)/R(4.2)=1+\alpha_{mes}T^{3}$, with
$\alpha_{mes}=1.2\times 10^{-6}\>$K-3. In the macroscopic sample the $R(B=0)$
above 4.2 K can be approximated with only a quadratic term:
$R(T)/R(4.2)=1+\alpha_{macr}T^{2}$, with $\alpha_{macr}=8.6\times
10^{-5}\>$K-2 suppl . The temperature independent term is described by the
interface roughness scattering dobretsova , while the temperature dependent
contribution to the scattering is expected to be due to the phonons, similar
to 2D GaAs systems gusev1 . Note, however, that the scattering by acoustic
phonons leads to a linear rather than a $T^{2}$ dependence for $T>T_{BG}\sim
4K$, where $T_{BG}$ is Bloch-Gruneisen temperaturesizov . Further theoretical
study is required for the explanation of this behaviour, which is out of scope
of this experimental paper.

In existing theories, electron transport in mesoscopic samples is considered
within the framework of ballistic, hydrodynamic, or more general models, all
based on a detailed approach assuming solution of the Boltzmann kinetic
equation complemented with the boundary conditions for the electron
distribution function gurzhi ; torre ; alekseev1 ; lucas ; scaffidi ; lucas2 ;
pellegrino2 ; holder ; raichev . The hydrodynamic description of transport in
graphene and other Dirac materials have been mostly focused on the vicinity of
the Dirac point, where several anomalies and collective excitations in the
Dirac fluid have been predicted briskot ; svintsov ; molenkamp2 ; fong . Far
away from the Dirac point, the system is expected to be similar to an ordinary
Fermi liquid fong ; polini .

Below we make use of the model proposed in alekseev ; scaffidi ; alekseev1 ,
because it captures all major magnetohydrodynamic properties, including the
subtle effects related to the relaxation of the second harmonic of the
distribution function by defects and e-e scattering.

The model describes the conductivity as a sum of two independent
contributions: the first one is determined by ballistic effects or static
disorder and the second one is due to viscosity alekseev1 . This approach
assumes the use of the magnetic field dependent viscosity tensor and the
derivation of the resistivity tensor alekseev1 :

$\rho(B)=\rho_{0}\left(1+\frac{\tau}{\tau^{*}}\frac{1}{1+(2\omega_{c}\tau_{2})^{2}}\right),\,\,\,$
(1)

where $\rho_{0}=\frac{m}{e^{2}n\tau}$, $1/\tau$ is the scattering rate due to
static disorder, $m$ and $n$ are the effective mass and the density,
$\tau^{*}=\frac{W(W+6l_{s})}{12\eta}$, where
$\eta=\frac{1}{4}v_{F}^{2}\tau_{2}$ is the viscosity. The shear viscosity
relaxation rate is given by
$\frac{1}{\tau_{2}(T)}=\frac{1}{\tau_{2,ee}}+\frac{1}{\tau_{2,imp}}$. The
relaxation rate $\frac{1}{\tau_{2,imp}(T)}$, determined by process responsible
for relaxation of the second harmonic of the distribution function, such as
scattering by static defects, gives rise to viscosity, while
$\frac{1}{\tau_{2,ee}(T)}$ refers to the shear viscosity relaxation due to e-e
scattering alekseev ; alekseev1 . The momentum relaxation rate is expressed as
$\frac{1}{\tau}=\frac{1}{\tau_{0,ph}}+\frac{1}{\tau_{0,imp}}$, where
$\tau_{0,ph}$ is the term responsible for phonon scattering, and
$\tau_{0,imp}$ is the scattering time due to static disorder (not related to
the second moment relaxation time) alekseev1 . The boundary conditions can be
characterized by a diffusive scattering or by a slip length $l_{s}$ with
extreme cases being no-slip ($l_{s}\rightarrow 0$) and no-stress
($l_{s}\rightarrow\infty$) conditions. It is expected that for
$l_{s}\rightarrow\infty$ no hydrodynamic Poiseuille-like flow should be
observed.

We fit the magnetoresistance curves and the $R(T)$ at zero magnetic field in
Fig. 3 with the 3 fitting parameters : $\tau(T)$, $\tau^{*}(T)$ and
$\tau_{2}(T)$. Note that the 2D resistivity is proportional to the resistance:
$\rho=\frac{W}{L}R$, and below we discuss the resistivity behaviour. Figure 3
shows the examples of the calculated Lorentzian-like magnetoresistance at
different temperatures. Our experimental data are in good agreement with the
theoretical models alekseev ; alekseev1 . We have also compared
magnetoresistance traces for different gate voltages and extracted the
relaxation times versus $N_{s}$ dependences. The corresponding relaxations
lengths $l=v_{F}\tau$ and $l_{2}=v_{F}\tau_{2}$ are shown in figure 4a. Note,
that for massless Dirac fermions the Fermi velocity does not depend on the
carrier density. One can see that hydrodynamic conditions are approximately
met at high densities, where we obtain $l/W\sim 1$ and $l_{2}/W\ll 1$. It is
worth noting, that the model alekseev  is valid for $l/W\sim 1$, because two
channels providing the relaxation of the second harmonic of the electron
distribution function and momentum can be regarded as parallel channels. The
transport features associated with the hydrodynamic properties should be
observed for $l_{2}/W\ll 1$ and $l_{2}/l\ll 1$, which are reasonably satisfied
in our devices at $N_{s}>4.5\times 10^{11}cm^{-2}$ ( Figure 4a). Note,
however, that at densities $N_{s}<4.5\times 10^{12}cm^{-2}$, the hydrodynamic
conditions are no longer well satisfied in which case both $l_{2}$ and its
density dependence become not very meaningful. More stringent conditions
$l/W\gg 1$ and $l_{2,ee}/W\ll 1$ lead to pure hydrodynamic regime, where
second term in the equation 1 becomes dominant and
$\rho\approx\frac{m}{e^{2}n}\eta\frac{12}{W^{2}}$ gurzhi . In this regime the
resistivity is inversely proportional to the square of the temperature,
$\rho\sim T^{-2}$, the so-called “Gurzhi effect”, which has been observed in a
high quality GaAs 2D electron system dejong ; gusev1 ; gusev4  and in a
graphene quantum point contact kim ; polini . Because of a lower mobility, in
our HgTe zero gap quantum well the first term in equation 1 will dominate, and
the Gurzhi effect is suppressed. Another key parameter that determines the
hydrodynamic condition is the slip length $l_{s}$, which should be much less
than the channel width: $l_{s}/W\ll 1$. From comparison with the experiment we
derive $\tau^{*}$, and the product of
$\tau^{*}\tau_{2}=W(W+6l_{s})/3v_{F}^{2}$ yields the slip length $l_{s}$.
$l_{s}$ turns out to be negligibly small, confirming the validity of the
hydrodynamic description. In addition, this makes it possible to exclude one
fitting parameter $\tau^{*}(T)$ from comparison with the theory, which
improves the model readability.

Let us now consider the data on the electron-electron interaction that can be
obtained from the processing of magnetoresistance. The interparticle
scattering time can be found from the equation:

$\frac{1}{\tau_{2,ee}(T)}=\frac{1}{\tau_{2}(T)}-\frac{1}{\tau_{2,imp}}=C\frac{(kT)^{2}}{\hbar
E_{F}},\,\,\ $ (2)

where the numerical factor C would be different for systems with parabolic and
linear spectrum. For a weakly interacting 2D Fermi gas with a parabolic
spectrum $C_{par}$ is defined as alekseev ; novikov :
$C_{par}=8.4r_{s}^{2}ln\left(\frac{1}{kT/E_{F}+r_{s}}\right)$, where the
interparticle interaction parameter $r_{s}=1/(a_{B}\sqrt{\pi n})$ is small in
a Fermi system, $a_{B}$ is the Bohr radius. For a system of massless Dirac
fermions parameter $C_{lin}$ takes the form narozhny : $C_{lin}\approx
16.4\alpha_{ee}^{2}(ln(E_{F}/kT))$, where the fine structure constant
$\alpha_{ee}$ is defined as $\alpha_{ee}=\frac{e^{2}}{\varepsilon\hbar v_{F}}$
and $\varepsilon$ is the dielectric constant. In Dirac systems the
dimensionless parameter and the fine structure constant have the same value.
In HgTe QW we find $\alpha_{ee}=r_{s}=3.2/\varepsilon$.

Comparing the temperature dependence of the relaxation rate $1/\tau_{2,ee}(T)$
with Eq.(2), we can derive a temperature-independent characteristic time
$\tau_{2,imp}=0.65\times 10^{-12}\>\text{s}$. The hydrodynamic approach is
associated with a substantial relaxation of the m-th harmonic of the
distribution function due to disorder scattering with the rates $\tau_{m,imp}$
alekseev . It is usually argued that $\tau=\tau_{1,imp}$ alekseev ; raichev
and $\tau_{2,imp}$ are of the same order of magnitude. In our sample we obtain
$\tau_{1,imp}=4.2\times 10^{-12}\>\text{s}>\tau_{2,imp}$. Note that in high-
mobility GaAs quantum wells it has been found that the time
$\tau=\tau_{1,imp}$ is much longer than $\tau_{2,imp}$ (by 10-100 times
)gusev1 ; gusev4 ; alekseev .

Figure 4b shows the mean free path $l_{2,ee}=v_{F}\tau_{2,ee}$ versus T in a
broad temperature range for two values of the Fermi energy
$E_{F}=91\>\text{meV}$ and $E_{F}=114\>\text{meV}$. Comparing our results with
equation (2) we find parameters $C=1.1$ and $C=0.7$ for lower and higher
energies. Thus, comparison with theory turned out to be possible when the
temperature changes by one order of magnitude. These values are very different
from the values obtained for a 2D electron Fermi system in GaAs well, which
are in order of 5-7 gusev1 ; raichev ; gusev4 ; alekseev . Parameters C
obtained from the experiment coincide in order of magnitude with the results
of calculation, if we assume the dimensionless parameter $r_{s}\sim 0.1-0.15$
which is very different from graphene ($r_{s}\approx 0.7$) due to high
dielectric constant. Although the equations for $C_{par}$ and $C_{lin}$ look
different due to logarithmic term, the differences would not be enough to be
significant in the temperature dependence. Discrimination of the subtle
difference between massive a massless spectrum is still challenging task and
requires more experimental and theoretical work.

In summary, we have performed a detailed study of the magnetotransport in a
single cone massless Dirac fermion system. The negative magnetoresistance was
fitted by a Lorentzian profile in accordance with the hydrodynamic approach
and the shear stress relaxation time was extracted, which determines the
viscosity in the Fermi liquid with a linear dispersion. Compared to graphene,
the most typical Dirac material, our system has a number of advantages. First,
the fact that only one valley is present in the spectrum allows for a more
unambiguous interpretation of the transport measurements, whereas the
intervalley scattering present in graphene can affect boundary scattering.
Secondly, the advantage of a simpler manufacturing method (MBE growth versus
exfoliation) allows for the fabrication of samples of macroscopic size and,
therefore, for discriminating between the transport properties due Poiseuille
flow in narrow channels and those due to scattering in the bulk.

The financial support of this work by Ministry of Science and Higher Education
of the Russian Federation, Grant No. 075-15-2020- 797 (13.1902.21.0024); São
Paulo Research Foundation (FAPESP) Grant No. 2015/16191-5, and the National
Council for Scientific and Technological Development (CNPq) is acknowledged.
We thank O.E. Raichev for the helpful discussions.

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##  I Supplementary material:Magnetohydrodynamics and electro-electron
interaction of massless Dirac fermions

In the main text we have concentrated on the experimental results for samples
with the width of 3.5 $\mu m$. At the same time, measurements were also
carried out on a macroscopic Hall bar sample with 8 voltage probes, the width
$W=50\mu m$ and three consecutive segments of different lengths $L$
$(100,250,100\mu m)$ (not shown).

Figure 5: (Color online) Normilized resistance R(B)/R(O) at different gate
voltages in the macroscopic sample ($W=50\mu m$), T=4.2 K.

Figure 1 shows the typical R(B)/R(O) curves for macroscopic sample at
different gate voltage in the electron part of the spectrum (the Fermi level
lies in the conduction band). One can see that the magnetoresistance (
$R(B)-R(0)>0)$) is positive at low magnetic field $B<0.1T$, but becomes
slightly negative at higher fields. AT $B>0.2T$ we observe Shubnikov- de Haas
(SdH) oscillations. In narrow channel samples the magnetoresistance was always
negative ( $R(B)-R(0)<0)$), had a large $(\sim 50\%)$ magnitude and a
Lorentzian profile, with SdH oscillations appearing at higher fields.
Therefore, one can see that the magnetoresistive is quite sensitive to the
width of the device channel. This observation confirms the assumption about
the hydrodynamic origin of the magnetoresistance in narrow samples.

Figure 6: (Color online) (a) The normalized resistance as a function of the
temperature for the macroscopic and for 3.5 $\mu m$ segment of HgTe QW
mesoscopic sample in zero magnetic field. The circles indicate resistance
calculated from Eq. (1). (b) The resistance difference $\Delta
R(T)=R(T)-R_{4.2K}$ as a function of the temperature for the macroscopic
(black) and for 3.5 $\mu m$ wide samples (red). Solid lines represents
dependencies $\Delta R(T)~{}T^{3}$ (red) and $T^{2}$ (black).

As we indicated in the main text the resistance is almost independent of T at
temperatures below 15 K and increases with T at higher temperatures. Figure 2
shows the data for macroscopic and mesoscopic sample with the width of $3.5\mu
m$. It was found that the R(T) dependences in mesoscopic device above 4.2 K
can be well described by a cubic law: $R(T)/R(4.2)=1+\alpha_{mes}T^{3}$, with
$\alpha_{mes}=1.2\times 10^{-6}K^{-3}$. In the macroscopic sample the R(B=0)
above 4.2 K can be approximated with only a quadratic term:
$R(T)/R(4.2)=1+\alpha_{macr}T^{2}$, with $\alpha_{macr}=8.6\times
10^{-5}K^{-2}$ ( figure 2a). The figure 2b shows the temperature-dependent
difference $\Delta R(T)=R(T)-R_{4.2K}$ for the macroscopic and mesoscopic
samples in log scale.

To further confirm the role of the hydrodynamic factor, we analyze the
magnetoresistance in the sample with the width of 10 $\mu m$.

Figure 7: (Color online) The resistance R(B) as a function of the magnetic
field in the 10 $\mu m$ segment of HgTe QW sample at $V_{g}=6V$. The circles
indicate resistance calculated from Eq. (1) in the main text for different
temperatures T(K): 5, 14.1, 30

.

Figure 8: (Color online) The inverse shear relaxation time for the 10 $\mu m$
sample as a function of temperature for the Fermi energy $E_{F}=80meV$ ( black
points). The thick red line represents the theory with the parameters
indicated in the text.

Figure 3 illustrates the variation of R(B) with temperature at a fixed gate
voltage $V_{g}=6V$ away from the CNP. The R(B) is almost independent of T at
temperatures below 15 K, but grows wider at high temperatures. One can see
that the curves profile is not exactly Lorentzian, but rather triangular. It
could be assumed that the observed profile distortion is a result of the fact
that the hydrodynamic regime is not completely fulfilled in the wider sample.
However, we believe that the model alekseev  is still valid and the
information on the e-e scattering can be extracted from the data.

The R(B) curves shown in Figure 3 were fitted with the equation (1) of the
main text with two adjustable parameters $\tau(T)$, $\tau^{*}(T)$ and
$\tau_{2}(T)$. Note that we have failed in our attempt to improve agreement
with theory by extending the range of the magnetic field. By an appropriate
choice of the fitting parameters $\tau(T)$ and $\tau_{2}(T)$ we obtain
$l_{s}=0$ and $W=9.1\mu m$, which is in excellent agrement with the
geometrical width $W=10\mu m$. The observed scaling of R(B) with the width
strongly supports the validity of the hydrodynamic theory in our samples. The
Poiseuille velocity profile becomes flat with a further increase of the
channel width, and the hydrodynamic model becomes inapplicable.

Figure 9: (Color online) The inverse shear relaxation time for the 3.5 $\mu m$
sample as a function of temperature for the Fermi energy $E_{F}=71meV$ ( black
points). The thick red line represents the theory with the parameters
indicated in the text.

Similar to the $3.5\mu m$ sample, the results for e-e collisions determined
from the magnetoresistance data have been compared with the theory:
$\frac{1}{\tau_{2,ee}(T)}=\frac{1}{\tau_{2}(T)}-\frac{1}{\tau_{2,imp}}=C\frac{\hbar
E_{F}}{(kT)^{2}}$. Figure 4 shows the shear scattering rate $1/\tau_{2,ee}$ in
a temperature range $10K<T<40K$. Fitting the theory to this data yields the
following values: $1/\tau_{2,imp}=2\times 10^{12}s^{-1}$ and C=1. Note,
however that fitting theory to experiment in this sample is considerably less
accurate because of a weaker temperature dependence.

In the main text we focused on the results at high density, where there are
limits imposed by the hydrodynamic conditions. We indeed analysed the data for
lower voltages, these conditions are no longer being met very well. The figure
5 shows the inverse shear relaxation time for the 3.5 $\mu m$ sample for
$E_{F}=71meV$ and comparison with the theory. We fit this data with the
following values: $1/\tau_{2,imp}=1.5\times 10^{12}s^{-1}$ and C=0.7. Note
however, that in this case $l_{tr}/l_{2}=3$, and $l_{2}/W=3$, which does not
completely satisfy all the requirements. We are also limited in the use of
high gate voltages, because of the possibility of electrical breakdown of the
dielectric.

In summary, the dependences R(B) in a wider sample becomes less sensitive to
the temperature and magnetic field in accordance with the prediction of the
model alekseev . However, we have been able to fit the R(B) curves with the
theory which was found to describe the data with a sufficient accuracy.

## References

  * (1) P. S. Alekseev, Phys. Rev. Lett. 117, 166601 (2016).