# Heating of inhomogeneous electron flow in the hydrodynamic regime

Gu Zhang<EMAIL_ADDRESS>Institute for Quantum Materials and Technologies,
Karlsruhe Institute of Technology, 76021 Karlsruhe, Germany  Valentin
Kachorovskii  Ioffe Institute, 194021 St. Petersburg, Russia  Konstantin
Tikhonov  Skolkovo Institute of Science and Technology, Moscow, 121205, Russia
Condensed-matter Physics Laboratory, National Research University Higher
School of Economics, 101000 Moscow, Russia  Igor Gornyi  Institute for Quantum
Materials and Technologies, Karlsruhe Institute of Technology, 76021
Karlsruhe, Germany  Ioffe Institute, 194021 St. Petersburg, Russia

###### Abstract

We study the electron temperature profiles for an inhomogeneous electron flow
in the hydrodynamic regime. We assume that the inhomogeneity is due to a
weakly non-uniform distribution of the momentum relaxation time within a
spherically constricted area. We show that the temperature profile
dramatically depends on the drive strength and the viscosity of the electron
liquid. In the absence of viscosity, a Landauer-dipole-like temperature
distribution, asymmetrically deformed along the current by the inelastic
electron-phonon scattering, emerges around the inhomogeneity. We find that
both the Landauer-dipole temperature profile and its asymmetry in the
direction of the driving electric field exist in all dimensionalities and are,
therefore, universal features of inhomogeneous hydrodynamic electron flow. We
further demonstrate that the electron viscosity suppresses the thermal
Landauer dipole and leads to the appearance of a “hot spot” exactly at the
center of the constriction. We also calculate the phonon temperature
distribution, which can be directly measured in experiments on thermal
nanoimaging.

##  I Introduction

The study of low-dimensional electronic systems is a key direction in the
condensed matter physics in the last few decades. This is dictated by the
general trend in the reduction of the sizes of the electronic devices, and is
supported by significant advances in modern technology. The semiconducting
heterostructures Ihn (2010) and graphene Novoselov  _et al._ (2005) are among
these technological developments, through which the two-dimensional (2D)
electron gas has been experimentally realized and employed for designing
nanoelectronic devices.

One of the most important properties of an electronic circuit is its ability
to cool efficiently and operate under a sufficiently strong drive, when
transport becomes substantially nonequilibrium. Under the non-equilibrium
conditions, effects related to overheating, dissipation, and thermalization
become decisive for the device functioning. The variety of setups in which
such phenomena define the physical properties of the system is quite wide, and
surprises arise even when studying more conventional structures subject to the
drive. The need of developing a comprehensive theory of heat transfer in
nanosystems has become particularly evident in recent years. Indeed,
miniaturization of electronic devices down to the nanoscale and the use of new
materials with unique properties are expected to affect the thermal properties
of nanostructures in a crucial way.

As the nanoscales are reached, new effects come into play owing to the
increased role of disorder, electron-electron interactions, and their
interplay. Further, it is now possible to change smoothly the dimensionality
of the system. For example, through etched gates, a 2D electron gas can be
divided into multiple areas that are connected by point contacts Goldhaber-
Gordon  _et al._ (1998); Cronenwett  _et al._ (1998). Properly selected
configuration of gates allows one to create a contact in the form of a quasi-
one-dimensional constriction and control the number of channels responsible
for current transfer through such a constriction. By using a selective doping,
one can engineer inhomogeneous low-dimensional gate-controllable structures to
probe charge and heat transport.

At the same time, the development of the SQUID-on-tip (tSOT) Finkler  _et al._
(2010); Vasyukov  _et al._ (2013) and the cryogenic quantum magnetometry Vool
_et al._ (2020) techniques enables precise measurement of the temperature
profiles and electric current distributions in nanostructures. The former
technique has already been successfully applied for the imaging of impurities
Halbertal  _et al._ (2016, 2017) and the quantum Hall edges states of graphene
samples Marguerite  _et al._ (2019); Uri  _et al._ (2020); Aharon-Steinberg
_et al._ (2020). These modern experimental techniques can be applied to the
analysis of the influence of various types of nanoscale inhomogeneities
(intrinsic, geometrical, artificial) on heat balance in nanodevices.

Recently, a new direction in the physics of low-dimensional systems has
emerged—electronic hydrodynamics, which was discussed for many decades Gurzhi
(1963, 1964, 1968); de Jong and Molenkamp (1995), but was scarcely studied
because of the lack of experimental realization at that time. Now this
direction is booming thanks to the technological advances in the production of
ultraclean ballistic systems, and a number of new hydrodynamic regimes were
theoretically predicted (for review, see, e.g., Refs. Narozhny  _et al._
(2017); Lucas and Fong (2018); Narozhny (2019); Polini and Geim (2020)) and
experimentally discovered Bandurin  _et al._ (2016); Crossno  _et al._ (2016);
Moll  _et al._ (2016); Ghahari  _et al._ (2016); Krishna Kumar  _et al._
(2017); Bandurin  _et al._ (2018); Braem  _et al._ (2018); Jaoui  _et al._
(2018); Gooth  _et al._ (2018); Berdyugin  _et al._ (2019); Gallagher  _et
al._ (2019); Sulpizio  _et al._ (2019); Ella  _et al._ (2019); Ku  _et al._
(2020); Raichev  _et al._ (2020); Gusev  _et al._ (2020); Vool  _et al._
(2020); Geurs  _et al._ (2020); Kim  _et al._ (2020); Gupta  _et al._ (2021);
Gusev  _et al._ (2021); Zhang and Shur (2021); Jaoui  _et al._ (2021). In
particular, with the use of modern nanoimaging techniques, it became possible
to visualize hydrodynamic flows in 2D materials Braem  _et al._ (2018); Ella
_et al._ (2019); Sulpizio  _et al._ (2019); Ku  _et al._ (2020); Vool  _et
al._ (2020).

Despite the experimental advances in the thermal detection, the studies of the
hydrodynamic phenomena and transition regimes between hydrodynamics, in which
the electron-electron scattering is the fastest, and the drift-diffusion
regime, where scattering by disorder dominates, mostly focus on the charge
transport. The heat transport features, however, are comparatively less
visited Rokni and Levinson (1995); Gurevich (1997); Gatiyatov  _et al._ (2010)
and overheating of the sufficiently small electron devices—the issue of
crucial importance for possible applications—in fact, remains very poorly
understood. Importantly, as was pointed out more than twenty years ago Rokni
and Levinson (1995), the local Joule heating approximation does not work for
the description of small devices like point contacts. The heat generation
there can be governed by nonlocal dissipation processes. However, nonlocality
is not the only specific feature of the heat dissipation at the nanoscales.

A recent analysis Tikhonov  _et al._ (2019) shows a great variety of different
overheating regimes in a quasi-one-dimensional (quasi-1D) constriction [see
Fig. 1(a)] with an inhomogeneous distribution of transport scattering rate.
One of the distinguishing features of all these regimes is the presence of
strongly asymmetric temperature profiles [see Fig. 1(b)]. The heat transfer in
2D graphene with local defects was addressed theoretically in Refs. Tikhonov
_et al._ (2018); Kong  _et al._ (2018) within the concept of
supercollisions—the impurity-stimulated electron-phonon scattering Song  _et
al._ (2012); Song and Levitov (2015) applied to resonant scatterers, as
suggested by experimental observations Halbertal  _et al._ (2017). The spatial
distribution of dissipation power was linked there to the formation of
Landauer dipoles Landauer (1957) around the local defect [cf. Fig. 1(d)].
However, only the total dissipation power was calculated in Ref. Tikhonov  _et
al._ (2018) (in the linear-response regime), without exploring the spatial
structure of the local temperature profiles.

In this paper, we study in detail the thermal characteristics of an
inhomogeneous 2D sample in the hydrodynamic regime, focusing on the
temperature profiles induced by inhomogeneities. We also discuss the thermal
properties of 3D inhomogeneous systems within a simplified model. The
inhomogeneity is introduced by the presence of a constricted area, where the
impurity scattering rate is different from its uniform value (i.e, its value
far away from the constriction). We consider a circular-shape constriction
[Fig. 1(c)], where the constriction has a higher impurity scattering rate.
Similarly to the quasi-1D case, we predict a strong asymmetry in the
electronic temperature profiles, even for a perfectly symmetric constriction.

We assume that the inhomogeneity is weak, and study the dc-current-induced
variation of temperature, density and drift velocity distributions. One of our
main results is the prediction of a the Landauer-dipole character of the
temperature distribution (similar to the distribution of the electric field in
the original Landauer’s consideration Landauer (1957)). This Landauer-dipole
temperature profile is further shifted asymmetrically with increasing the
driving current, as in the quasi-1D setting Tikhonov  _et al._ (2019).
Remarkably, this Landauer-dipole asymmetric feature universally exists in all
dimensionalities [see Fig. 1]. Our second key result is that the temperature
profile is dramatically modified by the viscosity of the electron liquid. We
demonstrate that the electron viscosity suppresses the Landauer-dipole-like
structure and creates a “hot spot” exactly at the center of the constriction.
We also discuss the corresponding phonon-temperature profiles that can be
experimentally observed using the tSOT technique Finkler  _et al._ (2010);
Halbertal  _et al._ (2016).

Figure 1:  (a) Quasi-1D symmetric constriction with increased scattering rate
and the corresponding asymmetric overheating profile (b). The temperature
asymmetry increases when the ratio between the constriction size $L$ and the
drift inelastic length $l_{\text{in}}$ decreases (see Ref. Tikhonov  _et al._
(2019)). The circular-shape constriction with increased impurity scattering in
2D or 3D electron gas (c) driven by the flow along the $x$ direction, with the
homogeneous drift velocity $v_{\infty}$ at $r\to\infty$. The magnitude of the
momentum relaxation time $\tau$ is shown by the color (the lighter the color,
the larger the value of $\tau$). (d) Constriction-induced local heating in a
2D or 3D sample, calculated with the local-heating approximation, see Appendix
B.

The paper is organized as follows. In Sec. II, we describe our model and
present the basic equations. In Sec. III, we analyze the current-induced
overheating for different dimensionalities, geometries, and interaction models
within the ideal fluid model. In Sec. IV, we discuss the viscous case. Section
V addresses the phonon temperature. Our results are summarized in Sec. VI. The
details of calculations are discussed in Appendices A and B.

##  II Model and basic equations

In this section, we introduce the hydrodynamic equations of ideal electron
fluid (Sec. II.1) and the constriction model (Sec. II.2). The formalism
adopted in the present paper closely follows and generalizes the one of Ref.
Tikhonov  _et al._ (2019), where an ideal-fluid description of a quasi-1D
constriction was developed.

###  II.1 Hydrodynamic approach

We consider a 2D spatially inhomogeneous, weakly disordered electron system,
where the inhomogeneity is induced by the dependence of the electron transport
scattering time $\tau$ on position $\mathbf{r}$. The distribution function
$f(\mathbf{r},\mathbf{V})$ obeys the stationary kinetic equation

$\mathbf{V}\frac{\partial
f}{\partial\mathbf{r}}+\frac{\mathbf{F}}{m}\frac{\partial
f}{\partial\mathbf{V}}=\widehat{\rm St}f.$ (1)

Here $\mathbf{V}$ and $m$ are the electron velocity and mass, respectively,
$\mathbf{F}$ is the total local force. The right-hand side (r.h.s.) of Eq. (1)
is determined by the collision operator

$\widehat{\rm St}=\widehat{\rm St}_{\text{\rm imp}}+\widehat{\rm
St}_{\text{\rm ph}}+\widehat{\rm St}_{\text{\rm ee}},$

which contains three terms describing electron-impurity, electron-phonon, and
electron-electron scattering. The impurity part of the collision integral
reads

$\widehat{\rm St}_{\text{imp}}=\frac{f_{0}-f}{\tau(\mathbf{r})},$ (2)

where $f_{0}=\langle f\rangle_{\theta}$ is the distribution function averaged
over the velocity angle $\theta$. Below, we assume that the transport
relaxation time $\tau(\mathbf{r})$ is constant everywhere except for a certain
constricted area. We model the electron-phonon scattering by the following
Fokker-Planck-type collision integral

$\widehat{\rm St}_{\rm
ph}=\gamma\frac{\partial}{\partial\epsilon}\left\\{\epsilon\left[f_{0}(1-f_{0})+T_{0}\frac{\partial}{\partial\epsilon}f_{0}\right]\right\\},$
(3)

where $\gamma$ is the electron-phonon scattering rate, $\epsilon$ is the
single-particle energy, and $T_{0}$ is the phonon temperature. We assume
$\gamma\tau\ll 1$. The specific form of the electron-electron collision
operator, $\widehat{\rm St}_{\text{ee}}$, is not important to us. We consider
the hydrodynamic limit where the electron-electron collision time,
$\tau_{\text{ee}}$, is much shorter than the momentum relaxation time:
$\tau_{\text{ee}}\ll\tau$. We also assume that $\tau_{\text{ee}}\ll 1/\gamma.$

First, we consider an ideal electron liquid neglecting the electron viscosity
and the heat conductivity, which are both proportional to $\tau_{\rm ee}$. The
viscous liquid will be discussed in Sec. IV. In the limit $\tau_{\rm ee}\to
0$, one can use the hydrodynamic Ansatz for the distribution function

$f(\mathbf{r},\mathbf{V})=\frac{1}{\exp\left[\frac{m[\mathbf{V}-\mathbf{v}(\mathbf{r})]^{2}/2-\mu(\mathbf{r})}{T(\mathbf{r})}\right]+1},$
(4)

where functions $\mu(\mathbf{r})$, $T(\mathbf{r})$, and
$\mathbf{v}(\mathbf{r})$ are the local values of chemical potential, electron
temperature, and the drift velocity, respectively. The equations governing
these collective (hydrodynamic) variables are obtained by averaging Eq. (1)
multiplied by $1$, $\mathbf{V}$, and $mV^{2}$, respectively, over the velocity
$\mathbf{V}$ (see, for example, Appendix A in Ref. Tikhonov  _et al._ (2019)).
After some algebra, we get the following set of equations:

$\displaystyle\\!\\!{\rm div}\left(\mathbf{v}N\right)=0,$ (5)
$\displaystyle\\!\\!(\mathbf{v}\nabla)\mathbf{v}+\frac{\mathbf{v}}{\tau}=\frac{1}{m}\left(\mathbf{F}_{0}+\delta\mathbf{F}-\frac{1}{N}\nabla
W\right),$ (6) $\displaystyle\\!\\!C\leavevmode\nobreak\ {\rm
div}(\mathbf{v}T)=N\left[\frac{mv^{2}}{\tau}-\gamma\leavevmode\nobreak\
(T-T_{0})\right].$ (7)

Here, $N=N(\mathbf{r})$ is the local electron concentration, related to the
local chemical potential as follows ($\nu$ is the 2D density of states):

$\mu=T\ln\left[\exp\left(\frac{N}{\nu T}\right)-1\right].$ (8)

The total force is written as a sum of the external force including the
driving homogeneous electric force $e\mathbf{E}_{0}$ and the Lorenz force in
the external homogeneous magnetic field $\mathbf{B}$,

$\mathbf{F}_{0}=e\left(\mathbf{E}_{0}+\left[\frac{\mathbf{v}}{c}\times\mathbf{B}\right]\right),$

and the inhomogeneity-induced correction $\delta\mathbf{F}$ that depends on
the electrostatics of the problem. For the gated case, when interaction is
screened beyond the gate-to-channel distance $d$, this correction reads

$\delta\mathbf{F}=-\frac{e^{2}\nabla N}{\cal C},$ (9)

where ${\cal C}=\varepsilon/4\pi d$ is the gate-to-channel capacitance per
unit area, and $\varepsilon$ is the dielectric constant. For comparison, in
the case of unscreened long-range Coulomb interaction, the correction reads:

$\delta\mathbf{F}=-e^{2}\nabla\int
d^{2}\mathbf{r}^{\prime}\frac{N(\mathbf{r}^{\prime})}{\varepsilon|\mathbf{r}-\mathbf{r}^{\prime}|}.$

The term $N^{-1}\nabla W$ on the r.h.s. of Eq. (6) represents the
thermoelectric force, where

$W=W(N,T)=\int_{0}^{\infty}d\epsilon\ \epsilon\ f_{F}(\epsilon)$

is the density of energy in the frame moving with flow. Here,
$f_{F}=1/\exp[(\epsilon-\mu)/T+1]$ is the Fermi function and $\mu$ should be
expressed via $N$ by using Eq. (8). In the limiting cases, this function is
given by

$W\approx\left\\{\begin{array}[]{c}\displaystyle\frac{N^{2}}{2\nu}+\frac{\pi^{2}\nu
T^{2}}{6},\quad{\rm for}\quad\nu T\ll N,\nobreak\leavevmode\\\\[11.38092pt]
\displaystyle TN,\quad{\rm for}\quad\nu T\gg N.\end{array}\right.$ (10)

The temperature balance equation (7) contains the heat capacity defined as

$\displaystyle C$ $\displaystyle=$ $\displaystyle C(N,T)=(\partial W/\partial
T)_{N=\text{const}}$ (13) $\displaystyle\approx$
$\displaystyle\left\\{\begin{array}[]{c}\displaystyle\frac{\pi^{2}\nu
T}{3},\quad{\rm for}\quad\nu T\ll N,\nobreak\leavevmode\\\\[11.38092pt]
\displaystyle N,\quad{\rm for}\quad\nu T\gg N.\end{array}\right.$

The term $-\gamma(T-T_{0})$ on the r.h.s. of Eq. (7) governs the heat transfer
from the electron system with temperature $T$ to the phonon bath with
temperature $T_{0}$. Strictly speaking, the phonon temperature $T_{0}$ also
depends on $\mathbf{r}$. Throughout most of the paper we neglect this
dependence assuming that $T_{0}$ is fixed by fast heat exchange with the
substrate. A brief discussion of the dependence $T_{0}(\mathbf{r})$ is
presented in Sec. V.

###  II.2  Model of inhomogeneity

In this paper, we consider a simple model of inhomogeneity, which allows for
an analytical solution. First of all, we assume that the sample inhomogeneity
is completely governed by the spatial dependence of the momentum relaxation
time, $\tau(\mathbf{r})$, within a certain constricted area, while other
coupling parameters and the external fields are position-independent. We limit
ourselves to the weak-inhomogeneity limit, for which the dimensionless
parameter

$\xi(\mathbf{r})=\frac{\tau_{\infty}}{\tau(\mathbf{r})}-1$ (14)

is small, $\xi\ll 1$, and search for solutions of Eqs. (5), (6), and (7)
perturbatively, to the leading order in $\xi$. In Eq. (14), $\tau_{\infty}$ is
the momentum relaxation time far away from the constricted area.

Recently, the quasi-1D case was analyzed in Ref. Tikhonov  _et al._ (2019),
where the temperature distribution of an infinitely long stripe-shape sample
was found. In the quasi-1D geometry, the parameter $\xi$ only depends on the
coordinate $x$ along the strip and is nonzero only in a limited area (see Fig.
1a). The asymmetry of temperature distribution in a variety of different
overheating regimes, both in the hydrodynamic and in the impurity-dominated
(the so-called drift-diffusion) cases, was predicted [see Fig. 1(b)]. This
asymmetry is captured neither by the conventional local Joule heating
approximation, nor a more advanced approach of Ref. Rokni and Levinson
(1995)), and reveals itself in the regime of sufficiently strong non-
equilibrium.

In this paper, we consider the hydrodynamic heat transport in higher
dimensionalities, mostly focusing on 2D cases. We study the heat flow through
a spherical-symmetric constriction characterized by the transport relaxation
time $\tau_{0}$ at the center of the constriction and the radius $L$. For
definiteness, when illustrating our results we use the Gaussian shape of the
inhomogeneity:

$\xi(\mathbf{r})=\left(\frac{\tau_{\infty}}{\tau_{0}}-1\right)\exp\left(-\frac{r^{2}}{L^{2}}\right).$
(15)

##  III Current-induced heating

###  III.1 Homogeneous heating

We start our analysis with the homogeneous case,
$\tau(\mathbf{r})=\tau_{\infty}$, $\xi\equiv 0$. We fix the electric field at
$|\mathbf{r}|\to\infty$ and assume that the magnetic field is parallel to the
$z$ axis: $\mathbf{B}=-B\mathbf{e}_{z}.$ From Eqs. (5), (6), and (7) we find
the homogeneous velocity $\mathbf{v}_{\infty}$ and the homogeneous temperature
$T_{\infty}$

$\mathbf{v}_{\infty}=\chi\frac{\mathbf{E}_{0}-\beta\leavevmode\nobreak\
\mathbf{E}_{0}\times\mathbf{e}_{z}}{1+\beta^{2}},\qquad
T_{\infty}=T_{0}+\frac{mv_{\infty}^{2}}{\tau_{\infty}\gamma}.$ (16)

Here, $\chi=e\tau_{\infty}/m$, $\beta=\omega_{c}\tau_{\infty}=\tan\theta$,
$\omega_{c}=eB/mc$, $\theta$ is the Hall angle,

$v_{\infty}=\frac{\chi E_{0}}{\sqrt{1+\beta^{2}}}=\chi E_{\parallel}$ (17)

is the absolute value of velocity, and $E_{\parallel}=E_{0}\cos\theta$ is the
projection of the electric field on the direction of electric current.
Following Ref. Tikhonov  _et al._ (2019) we define the dimensionless
overheating parameter

$\alpha=1-\frac{T_{0}}{T_{\infty}}=\left(1+\frac{m\gamma
T_{0}}{e^{2}E_{\parallel}^{2}\tau_{\infty}}\right)^{-1},$ (18)

which scales quadratically with a weak electric field and saturates at
$\alpha=1$ in strong fields.

From Eqs. (16) and (17), we realize that the application of the magnetic field
for a fixed direction of external electric field simply introduces a rotation
of all the profiles by the Hall angle $\theta$. In what follows, we thus focus
only on the case of zero magnetic field.

###  III.2 Inhomogeneous heating

In this section, we study the spatial profiles of the hydrodynamic variables
(density, hydrodynamic velocity, and temperature) after the introduction of a
weak inhomogeneity. For $B=0$, $\theta=\beta=0,$
$\mathbf{v}_{\infty}=\mu\mathbf{E}_{0},$ and $E_{\parallel}=E_{0}.$ Below, we
consider the two distinct interacting models.

####  III.2.1 Gated 2D liquid

Let us assume that the electron-electron interaction is screened by a gate, so
that $\delta\mathbf{F}$ is given by Eq. (9) . We introduce small
inhomogeneity-induced corrections $\delta n$, $\delta\mathbf{v}$, and $\delta
T$, which are proportional to $\xi$:

$\displaystyle N$ $\displaystyle=N_{\infty}(1+\delta n),$ (19)
$\displaystyle\mathbf{v}$
$\displaystyle=\mathbf{v}_{\infty}+\delta\mathbf{v},$ (20) $\displaystyle T$
$\displaystyle=T_{\infty}+\delta T.$ (21)

Linearizing Eqs. (5), (6), and (7) with respect to $\xi$ and taking Fourier
transform we get (for the Fourier transformed quantities):

$\displaystyle\ \mathbf{q}\mathbf{v}_{\infty}\delta
n_{\mathbf{q}}+\mathbf{q}\delta\mathbf{v}_{\mathbf{q}}=0,$ (22)
$\displaystyle\left(\\!\frac{1}{\tau_{\infty}}+i\mathbf{q}\mathbf{v}_{\infty}\\!\right)\delta\mathbf{v}_{\mathbf{q}}\\!+\\!i\mathbf{q}\left(\\!s^{2}\delta
n_{\mathbf{q}}\\!+\\!\frac{\delta
T_{\mathbf{q}}}{M}\\!\right)=-\frac{\mathbf{v}_{\infty}}{\tau_{\infty}}\xi_{\mathbf{q}},$
(23)
$\displaystyle\left(\\!\frac{\gamma}{c_{\infty}}\\!+\\!i\mathbf{q}\mathbf{v}_{\infty}\\!\right)\\!\delta
T_{\mathbf{q}}\\!+\\!\left(\\!i\mathbf{q}T_{\infty}\\!-\\!\frac{2M\mathbf{v}_{\infty}}{\tau_{\infty}}\\!\right)\\!\delta\mathbf{v}_{\mathbf{q}}\\!=\\!\frac{Mv_{\infty}^{2}}{\tau_{\infty}}\xi_{\mathbf{q}}.$
(24)

Here

$s^{2}=\frac{e^{2}N_{\infty}}{m\cal C}+\frac{1}{m}\left(\frac{\partial
W}{\partial N}\right)_{T=\text{const}}$ (25)

is the plasma wave velocity, with the second term representing contribution
from the Fermi-liquid velocity. We also define

$M=\frac{m}{c_{\infty}}$ (26)

as the “thermal” mass, and

$c_{\infty}=\frac{C_{\infty}}{N_{\infty}}=\frac{1}{N_{\infty}}\left(\frac{\partial
W}{\partial T}\right)_{N=\text{const}}$ (27)

is the specific heat capacity.

####  III.2.2 Weakly compressible liquid

General solutions of Eqs. (22), (23), and (24) are rather cumbersome. Below we
present solutions for the most interesting case, where the liquid is almost
incompressible. This case is realized for strong electron-electron
interaction, when the plasma wave velocity is large:

$s\gg\sqrt{\frac{T_{\infty}}{M}},\quad s\gg v_{\infty}.$ (28)

We introduce two characteristic lengths: the elastic drift length determined
by the scattering off disorder,

$l=v_{\infty}\tau_{\infty},$ (29)

and the inelastic drift length Tikhonov  _et al._ (2019) characterizing the
electron-phonon scattering,

$l_{\text{in}}=\frac{v_{\infty}c_{\infty}}{\gamma}.$ (30)

We further assume that the elastic drift length is the shortest lengthscale,

$L\gg l,\qquad l_{\rm in}\gg l,$ (31)

while the relation between the constriction size, $L$, and $l_{\rm in}$ can be
arbitrary.

From Eq. (23) one can conclude that, in the limit $s\to\infty$, the correction
to the electron density is small, $\delta n\propto 1/s^{2}$. Then, from Eq.
(22), we find that $\mathbf{q}\delta\mathbf{v}_{\mathbf{q}}\propto 1/s^{2}$,
so that the electron liquid is almost incompressible:
$\text{div}\delta\mathbf{v}\to 0.$ As a consequence, to the zeroth order with
respect to $1/s^{2}$, one can replace Eq. (22) with

$\mathbf{q}\delta\mathbf{v}_{\mathbf{q}}=0.$ (32)

Hence, the velocity correction in the momentum space is perpendicular to
$\mathbf{q}$:

$\delta\mathbf{v}_{\mathbf{q}}\propto\mathbf{t}_{\mathbf{q}},$

where $\mathbf{t}_{\mathbf{q}}=[\mathbf{e}_{z}\times\mathbf{q}/q]$, and
$\mathbf{e}_{z}$ is the unit vector in the $z$ direction. This correction can
be found by taking the scalar product of $\mathbf{t}_{\mathbf{q}}$ and Eq.
(23). Substituting $\delta\mathbf{v}_{\mathbf{q}}$ into Eq. (24) and
neglecting the small term $\mathbf{q}\delta\mathbf{v}_{\mathbf{q}}$, one can
find $\delta T_{\mathbf{q}}$. Having in mind Eq. (31), we finally arrive at
the following expressions for the velocity and temperature corrections:

$\displaystyle\delta\mathbf{v}_{\mathbf{q}}=-\xi_{\mathbf{q}}\mathbf{t}_{\mathbf{q}}(\mathbf{t}_{\mathbf{q}}\mathbf{v}_{\infty}),$
(33) $\displaystyle\delta
T_{\mathbf{q}}=\frac{M\xi_{\mathbf{q}}v_{\infty}}{\tau_{\infty}}\frac{q_{\parallel}^{2}-q_{\perp}^{2}}{q^{2}(iq_{\parallel}+1/l_{\rm
in})}.$ (34)

Here, $q_{\parallel}$ and $q_{\perp}$ are, respectively, the parallel and
perpendicular components of the vector $\mathbf{q}$ with respect to the drift
velocity $\mathbf{v}_{\infty}$.

A correction to the concentration arises only in the first order with respect
to $1/s^{2}$. It can be obtained by taking the scalar product of Eq. (23) and
$\mathbf{q}$:

$\delta
n_{\mathbf{q}}=\frac{\xi_{\mathbf{q}}v_{\infty}}{s^{2}\tau_{\infty}q^{2}}\left(iq_{\parallel}-\frac{q_{\parallel}^{2}-q_{\perp}^{2}}{iq_{\parallel}+1/l_{\rm
in}}\right).$ (35)

Importantly, although $\delta n_{\mathbf{q}}$ approaches zero for large $s$,
one cannot neglect $\delta n_{\mathbf{q}}$ from the very beginning, because
$\delta F_{\mathbf{q}}\propto s^{2}\delta n_{\mathbf{q}}$ remains finite for
$s\to\infty$.

We see that the temperature distribution in the momentum space is described
(up to a constant coefficient) by the product of $\xi_{\mathbf{q}}$ and the
heating kernel

$K(\mathbf{q})=\frac{q_{\parallel}^{2}-q_{\perp}^{2}}{q_{\parallel}^{2}+q_{\perp}^{2}}\leavevmode\nobreak\
\frac{1}{iq_{\parallel}+1/l_{\rm in}}.$ (36)

It is worth stressing that this equation is derived in the incompressible
limit corresponding to very strong interactions and, therefore, does not
contain any specific feature of the 2D system. One can easily show that this
form of the heating kernel universally appears in other dimensions. In
particular, for quasi-1D strips parallel to the $x$-axis, where $\xi=\xi(x),$
the transverse wave vector equals zero, $q_{\parallel}=0$, and we arrive at
equations derived previously in Ref. Tikhonov  _et al._ (2019):

$K(q_{x})=1/(iq_{x}+1/l_{\rm in}),$

see Eq. (27) in that work. By using the same calculations as presented above,
one finds that for a 3D case, the heating kernel is also given by Eq. (36),
with $q_{\perp}^{2}=q_{y}^{2}+q_{z}^{2}$ (for $\mathbf{v}_{\infty}$ parallel
to $x$ axis).

####  III.2.3 Landauer-dipole structure of the heating kernel

Direct calculation of the Fourier transform of (36) yields

$K(\mathbf{r})=\int\limits_{0}^{\infty}dse^{-s/l_{\rm
in}}Q_{D}(x_{\parallel}-s,\mathbf{r}_{\perp}),$ (37)

where $Q_{D}(\mathbf{r})$ is equivalent to the “electric field” of a Landauer
dipole Landauer (1957), which can be written in the universal form for all
dimensions:

$Q_{D}(\mathbf{r})=C_{D}\frac{\partial}{\partial
x_{\parallel}}\left(\frac{x_{\parallel}}{r^{D}}\right),\qquad D=1,2,3,$ (38)

where $C_{1}=1/2,\leavevmode\nobreak\ C_{2}=1/\pi,\leavevmode\nobreak\
C_{3}=1/2\pi$ for 1D, 2D and 3D cases. Choosing $x$ axis along the
$\mathbf{v}_{\infty}$ direction, we get the explicit expressions for Landauer
dipoles in the heating kernel:

$Q_{D}(\mathbf{r})=\left\\{\begin{array}[]{cc}\displaystyle\delta(x),\nobreak\leavevmode&\text{for}\quad
D=1,\\\\[2.84544pt]
\displaystyle\frac{y^{2}-x^{2}}{\pi(x^{2}+y^{2})^{2}},\nobreak\leavevmode&\text{for}\quad
D=2,\\\\[5.69046pt]
\displaystyle\frac{y^{2}+z^{2}-2x^{2}}{2\pi(x^{2}+y^{2}+z^{2})^{5/2}},\nobreak\leavevmode&\text{for}\quad
D=3.\end{array}\right.$ (39)

These functions obey the following property:

$Q_{1}(x)=\int dy\,Q_{2}(x,y)=\int dy\,dz\,Q_{3}(x,y,z).$

In order to illustrate the physics behind the Landauer-dipole-like temperature
distributions, we present in Appendix B the discussion of the Joule heat
distribution in a simple two-component model within the conventional theory of
local Joule dissipation.

As seen from Eq. (37), the heating kernel is given by a Landauer dipole that
is spatially-shifted at the distance $\sim l_{\rm in}$. This shift induces the
asymmetry of the Landauer dipole along the direction of the current.
Physically, the asymmetry stems from “convection” described by the term
$\text{div}(\mathbf{v}T)$ in the heat balance equation which is absent in the
theory of local dissipation and in the weak-drive theory of non-local heat
transport of Ref. Rokni and Levinson (1995). Interestingly, the kernel Eq.
(37) remains finite in the limit $l_{\rm in}\to\infty$, which means that the
temperature distribution in this case is fully determined by convection.

The correction to the electron temperature is given in the coordinate space by

$\delta T(\mathbf{r})=\frac{T_{0}}{l_{\rm in}}\frac{\alpha}{1-\alpha}\int
d^{D}\mathbf{r}^{\prime}K(\mathbf{r}-\mathbf{r}^{\prime})\xi(\mathbf{r}^{\prime}).$
(40)

The asymmetry of temperature distribution manifests itself at distances of the
order of (or smaller than) the inelastic lengths. At larger distances, we get

$K(\mathbf{r})\approx l_{\rm in}Q_{D}(\mathbf{r}),\quad\text{for}\quad r\gg
l_{\rm in}.$ (41)

Hence, away from the inhomgeneity center, for $r\gg\text{max}(L,l_{\rm in})$,
the temperature distribution can be considered as symmetric and given by the
Landauer-dipole profile:

$\delta T(\mathbf{r})\approx\frac{\alpha
T_{0}}{(1-\alpha)}\leavevmode\nobreak\
Q_{D}(\mathbf{r})\int\xi(\mathbf{r}^{\prime})d^{D}\mathbf{r}^{\prime},\quad\text{for}\quad
r\to\infty.$ (42)

For the case of the Gaussian constriction (15), equations for temperature
distribution valid for arbitrary relation between $r,\leavevmode\nobreak\ L$
and $l_{\rm in}$ are derived in Appendix A. They can be written as follows.

$\delta T(\mathbf{r})=A_{D}\int\limits_{0}^{\infty}\frac{ds}{l_{\rm
in}}e^{-s/l_{\rm in}}\tilde{Q}_{D}(x-s,\mathbf{r}_{\perp}),$ (43)

where

$A_{D}=\pi^{D/2}L^{D}\frac{\alpha}{1-\alpha}\left(\frac{\tau_{\infty}}{\tau_{0}}-1\right)T_{0},$
(44)

and

$\displaystyle\tilde{Q}_{1}$
$\displaystyle=\frac{1}{\sqrt{\pi}L}e^{-x^{2}/L^{2}},$ (45)
$\displaystyle\tilde{Q}_{2}$ $\displaystyle=\frac{y^{2}-x^{2}}{\pi
r^{4}}\left[1-e^{-r^{2}/L^{2}}(1+r^{2}/L^{2})\right],$
$\displaystyle\tilde{Q}_{3}$
$\displaystyle=\frac{(y^{2}+z^{2}-2x^{2})\leavevmode\nobreak\
\text{Erf}(r/L)}{2\pi r^{5}}$
$\displaystyle-\frac{e^{-r^{2}/L^{2}}[(y^{2}+z^{2})^{2}-x^{4}+L^{2}(z^{2}+y^{2}-2x^{2})]}{\pi^{3/2}L^{3}r^{4}}.$

Sending $L\to 0$ for fixed $r$, we reproduce Eqs. (39): $\tilde{Q}_{D}\to
Q_{D}$ for $L\to 0$.

Above, we have assumed that the elastic drift length $l$ is much smaller in
comparison to the constriction size: $l\ll L$. The obtained results can be
straightforwardly generalized to the case of arbitrary relation between $l$
and $L$. As follows from Eqs. (22), (23), and (24), the general expression for
the heating kernel, Eq. (36), becomes then

$K(\mathbf{q})=\left(\frac{q_{\parallel}^{2}-q_{\perp}^{2}}{q_{\parallel}^{2}+q_{\perp}^{2}}+iq_{\parallel}l\right)\leavevmode\nobreak\
\frac{1}{iq_{\parallel}+1/l_{\rm in}}\frac{1}{1+iq_{\parallel}l}.$ (46)

In real space, the heating kernel is again expressed in terms of the Landauer-
dipole field:

$\displaystyle K(\mathbf{r})$ $\displaystyle=\frac{l_{\rm in}}{l_{\rm
in}-l}\int\limits_{0}^{\infty}d\rho\left(e^{-\rho/l_{\rm
in}}-e^{-\rho/l}\right)Q_{D}(x_{\parallel}-\rho,\mathbf{r}_{\perp})$ (47)
$\displaystyle+\delta(\mathbf{r}_{\perp})\theta(x_{\parallel})\left(e^{-x_{\parallel}/l}-\frac{l}{l_{\rm
in}}e^{-x_{\parallel}/l_{\rm in}}\right).$

Clearly, this reproduces Eq. (37) in the limit $l\to 0$. As another feature,
similar to Eq. (37), the kernel Eq. (47) remains finite in the limit of
diverging $l_{\rm in}$:

$\displaystyle K(\mathbf{r})$
$\displaystyle\approx\int\limits_{0}^{\infty}d\rho\left(1-e^{-\rho/l}\right)Q_{D}(x_{\parallel}-\rho,\mathbf{r}_{\perp})$
(48)
$\displaystyle+\delta(\mathbf{r}_{\perp})\theta(x_{\parallel})e^{-x_{\parallel}/l},\quad\text{for}\quad
l_{\rm in}\to\infty.$

###  III.3 Spatial profiles of the electron temperature, velocity, and
concentration

The spatial dependences of the temperature, as well as the velocity and
concentration, are plotted in Fig. 2 for a 2D system with a Gaussian
constriction [see Eq. (15)]. We assumed that the liquid is nearly
incompressible ($s\to\infty$) and $l\ll l_{\rm in},\leavevmode\nobreak\ l\ll
L.$ The temperature, velocity and concentration are measured, respectively, in
the following units

$\displaystyle\\!\\!T^{*}=\left(\frac{\tau_{\infty}}{\tau_{0}}-1\right)\frac{\alpha}{1-\alpha}T_{0}=\left(\frac{\tau_{\infty}}{\tau_{0}}-1\right)(T_{\infty}-T_{0}),$
(49)
$\displaystyle\\!\\!v^{*}=\left(\frac{\tau_{\infty}}{\tau_{0}}-1\right)v_{\infty},$
(50)
$\displaystyle\\!\\!n^{*}=\left(\frac{\tau_{\infty}}{\tau_{0}}-1\right)\frac{L}{l}\frac{v_{\infty}^{2}}{s^{2}}.$
(51)

The profiles of $\delta T/T^{*}$, $\delta v/v^{*}$, and $\delta n/n^{*}$
depend only on the relation between characteristic lengths of the problem,
$L$, $l_{\rm in}$, and $l$. Figures 2(a),(b) show the temperature plots
$\delta T/T_{*}$ for $l\ll(L,l_{\rm in})$ for the two different ratios between
$L$ and $l_{\rm in}$: $L=2l_{\rm in}$ for Fig. 2(a) and $L=0.5l_{\rm in}$ for
Fig. 2(b).

Panel (a) of Fig. 2 reproduces an almost symmetric Landauer-dipole structure,
while panel (b) is much more asymmetric because of a larger value of $l_{\rm
in}/L$. In both panels, the electron temperature is reduced along the current
injecting direction (the $x$ direction), and enhanced in the direction
perpendicular to the current (the $y$ direction). The Landauer-dipole
structures are shifted by the inelastic drift length $l_{\text{in}}$, yielding
the temperature asymmetry similar to the one predicted for the quasi-1D
geometry Tikhonov  _et al._ (2019).

Figure 2:  Density plots of $\delta T/T_{*}$ in 2D systems with a Gaussian
constriction, Eq. (15) in the incompressible limit for $L=2l_{\text{in}}$ (a)
and $L=0.5l_{\text{in}}$ (b), respectively; (c) and (d): corresponding density
plots of $\delta n/n^{*}$; (e) and (f): vector density plots of
$\delta\mathbf{v}/v^{*}$.

The density plots of $\delta n/n^{*}$ for the same values of $L$ and $l_{\rm
in}$ are shown in panels (c) and (d) of Fig. 2. We observe that among these
distributions the most asymmetric is the density profile, which emerges in the
limit of nearly incompressible fluid only for finite values of $1/s^{2}$, in
contrast to the temperature and hydrodynamic velocity. The asymmetry of the
temperature profiles corresponds to a similar feature of the density
distributions: particles tend to gather in front of the constricted area,
where a larger impurity scattering rate is present.

The appearance of the Landauer-dipole structure in the temperature
distribution can be understood based on the vector density plots for the
velocity distribution $\delta\mathbf{v}/v^{*}$ that are shown in panels (e)
and (f) of Fig. 2. One sees that particles tend to detour around the
constricted area. Heat current, carried by these detoured particles, thus
leads to the hot spots on the $y$ axis, which are clearly seen in panels (a)
and (b).

##  IV The viscous case

So far, we have considered hydrodynamics of an ideal electron fluid
characterized by zero viscosity (taking the limit $\tau_{\text{ee}}\to 0$). In
the presence of a finite viscosity $\eta$, expressions for the velocity and
temperature corrections are modified as follows:

$\displaystyle\delta\mathbf{v}_{\mathbf{q}}=-\frac{\xi_{\mathbf{q}}\mathbf{t}_{\mathbf{q}}(\mathbf{t}_{\mathbf{q}}\mathbf{v}_{\infty})}{1+iq_{\parallel}l+\kappa
q^{2}}$ (52) $\displaystyle\delta
T_{\mathbf{q}}=\frac{M\xi_{\mathbf{q}}v_{\infty}}{\tau_{\infty}}\frac{1}{iq_{\parallel}+1/l_{\rm
in}}\left(1-\frac{2q_{\perp}^{2}}{q^{2}}\frac{1}{1+iq_{\parallel}l+\kappa
q^{2}}\right).$ (53)

Here,

$\kappa=\eta\tau_{\infty}=l_{\kappa}^{2},$ (54)

where $l_{\kappa}=\sqrt{\eta\tau_{\infty}}$ is a new length scale, arising in
the presence of viscosity. It is worth noting that viscosity by itself gives a
direct contribution to the dissipation Landau and Lifshitz (1959), so that the
term

$P_{\text{vis}}=\eta\iint
dxdy\left[\left(\partial_{x}v_{x}-\partial_{y}v_{y}\right)^{2}+\left(\partial_{x}v_{y}+\partial_{y}v_{x}\right)^{2}\right],$
(55)

appears on the r.h.s. of the heat balance equation, Eq. (7). Here $v_{x}$ and
$v_{y}$ are the spatially dependent velocities along the $x$ and $y$
directions, respectively. As seen from Eq. (55), the direct viscous
dissipation is present only in the second order in the weak inhomogeneity
$\xi$. Hence, in the linear approximation with respect to $\xi$, the viscosity
affects the temperature distribution indirectly by changing velocity
distribution, thus modifying the impurity dissipation $mv^{2}/\tau$. Depending
on the relation between four characteristic lengths: $L$, $l$, $l_{\rm in}$,
and $l_{\kappa}$, various regimes of dissipation are possible. However, the
main effect of viscosity is quite simple: with increasing $l_{\kappa}$, the
Landauer-dipole structure is suppressed. It makes sense, therefore, to start
with the discussion of the high-viscosity limit.

###  IV.1 Strong viscosity, $l_{\kappa}\to\infty$

Let us assume that $l_{\kappa}$ is larger than all other length scales in the
problem. Interestingly, even in the limit $l_{\kappa}\to\infty$, there is a
finite temperature correction, which can be found by neglecting the last term
of Eq. (53). Performing the Fourier transformation, we get

$\delta
T=\frac{Mv_{\infty}^{2}}{l}\int\limits_{-\infty}^{x_{\parallel}}dx_{\parallel}^{\prime}\leavevmode\nobreak\
\xi(\mathbf{r}^{\prime})e^{-(x_{\parallel}-x_{\parallel}^{\prime})/l_{\rm
in}}.$ (56)

Let us comment on the physics behind this expression. For infinitely large
viscosity, all velocity gradients are suppressed, so that
$\mathbf{v}\equiv\mathbf{v}_{\infty}$. In this limit, the heat balance
equation is dramatically simplified

$\frac{d\delta T}{dx_{\parallel}}=\frac{Mv_{\infty}^{2}\leavevmode\nobreak\
\xi(\mathbf{r})}{l}-\frac{\delta T}{l_{\rm in}}.$ (57)

Physically, this equation means that the heating is effectively one
dimensional, regardless of the system dimension. Integrating Eq. (57), we
reproduce Eq. (56). We also notice that Eqs. (56) and (57) do not require
linearization and are valid for an arbitrary relation between $\delta T$ and
$T_{\infty}$.

Figure 3: Evolution of the 2D temperature profiles for fixed $L$ and $l_{\rm
in}$, $L=0.2l_{\text{in}}$, with increasing $l_{\kappa}$: (a) the limit of
negligible viscosity, $l_{\kappa}=0.001l_{\text{in}}$; (b) small viscosity,
$l_{\kappa}=0.05l_{\text{in}}$, which is insufficient to suppress the
Landauer-dipole feature; (c) “critical” viscosity,
$l_{\kappa}=0.1l_{\text{in}}$, at which a hot spot appears in the center of
the constriction, which increases with further increasing viscosity, as shown
in (d) for $l_{\kappa}=0.15l_{\text{in}}$ and (e) for
$l_{\kappa}=0.25l_{\text{in}}$, gradually suppressing the Landauer-dipole
structure; (f) strong viscosity, $l_{\kappa}=10l_{\text{in}}$: the Landauer-
dipole feature is suppressed.

###  IV.2 Evolution of the temperature profile with viscosity

In Figs. 3 and 4, we illustrate the evolution of the temperature distribution
with increasing viscosity. We fix $L$ and $l_{\rm in}$ such that the inelastic
drift length is larger than the size of the constriction,
$L=0.2l_{\text{in}}$, and, thus, asymmetry in the temperature distribution is
sufficiently pronounced. At the same time, the Landauer-dipole structure
remains apparent for zero and very small viscosity, as shown in Figs. 3(a) and
(b). When $l_{\kappa}$ becomes the order of $L$, a hot spot appears in the
center of the constriction, Figs. 3(c), whose amplitude increases with
increasing viscosity, see Figs. 3(d), (e), and (f), while, simultaneously, the
Landauer-dipole structure becomes suppressed. At very large values of
$l_{\kappa}$, Figs. 3(f), the Landauer-dipole feature is fully suppressed, and
the temperature distribution is very well described by Eq. (56).

Figure 4: Evolution of the temperature profiles of the cross-sections at (a)
$x=L$ and (b) $y=0$, corresponding to the solid and dashed lines in Fig. 3(a).
Parameters are the same as in Fig. 3.

We also plot in Figs. 4(a), (b) the cross-sections of the temperature
distribution along the lines $x=L$ and $y=0$, respectively. These plots
clearly demonstrate the suppression of the Landauer dipoles and the formation
of the hot spot, which is symmetric in the $y$ direction but asymmetric in the
$x$ direction. Evolution of the velocity profiles with viscosity is shown in
Fig. 5 for the same values of parameters as in Fig. 3

Figure 5: Evolution of the velocity profiles with increasing viscosity. Panels
(a), (b), (c) and (d) correspond to panels (a), (b), (c) and (d) in Fig. 3,
respectively.  Figure 6: The temperature profile after the removal of the “hot
spot” at the constriction position, see Eq. (56).

We reiterate that, to the leading order in $\xi$, viscosity does not introduce
extra dissipation, but instead redistributes it. To see this point, in Fig. 6
we present the temperature profile ($l_{\kappa}=l_{\text{in}}\gg L$) obtained
when we keep only the second term of Eq. (53) in the parentheses [i.e, after
removing the contribution of Eq. (56)]. Apparently, the Landauer-dipole
structure survives in the strong-viscosity limit, although it is strongly
reduced and obscured by the “hot spot” at the constriction.

##  V Phonon temperature distribution

In previous sections, we have discussed the distribution of the electron
temperature. However, experimentally, the electron temperature profiles are
hard to measure directly. On the other hand, the phonon temperature can be
measured at a very high resolution by means of the tSOT technique Finkler  _et
al._ (2010); Vasyukov  _et al._ (2013); Halbertal  _et al._ (2016, 2017);
Marguerite  _et al._ (2019). Using the results obtained above, we obtain the
phonon temperature, restricting ourselves to the discussion of the 2D case.

Figure 7: The phonon temperature near the constriction that corresponds to the
electron temperature of Fig. 3(d). The phonon temperature profile reproduces
the major features of the electron one when $\varkappa_{\text{ph}}$ is small
(a). On the contrary, the “hot spot” disappears when $\varkappa_{\text{ph}}$
becomes large enough (b).

First, we note that the heat exchange rate between phonons and the substrate,
$\gamma_{0}$, is typically several orders of magnitude larger than the
electron-phonon heat transfer rate $\gamma$ (see experimental measurements
Chen  _et al._ (2009); Persson and Ueba (2010); Li  _et al._ (2017) and
estimates in Ref. Tikhonov  _et al._ (2018)). In the homogeneous case we have

$\gamma(T_{\rm el}-T_{\rm ph})=\gamma_{0}(T_{\rm ph}-T_{0}),$ (58)

where $T_{0}$ is the substrate temperature. For $\gamma_{0}\gg\gamma$, we have
$T_{\rm ph}\approx T_{0}$. It is worth noting that this estimate justifies the
approach used in the previous sections, where, for determining the electron
temperature, we assumed that the lattice temperature is a constant.

Small deviations of the lattice temperature from $T_{0}$ can be found from the
phonon heat balance equation

$-\varkappa_{\text{ph}}\Delta
T_{\text{ph}}=\gamma(T_{\text{e}}-T_{\text{ph}})-\gamma_{0}(T_{\text{ph}}-T_{0}),$
(59)

where $\varkappa_{\text{ph}}$ is the phonon thermal conductivity. Introducing

$\delta T_{\rm ph}=T_{\rm ph}-T_{0},$

and making use of the inequality $\gamma\ll\gamma_{0}$, we find

$\delta T_{\rm
ph}^{\mathbf{q}}=\frac{\gamma}{\gamma_{0}+\varkappa_{\text{ph}}q^{2}}\delta
T_{\mathbf{q}},$ (60)

where $\delta T_{\mathbf{q}}$ is the inhomogeneity-induced correction to the
electronic temperature calculated above. We see that for

$q<q_{0}=\sqrt{\gamma_{0}/\varkappa_{\rm ph}},$

the phonon temperature profile coincides with the electronic one up to a small
factor $\gamma/\gamma_{0}$. Very sharp gradiends of the electronic temperature
with $q>q_{0}$ are suppressed by the phonon heat conductivity. In the
coordinate representation, Eq. (60) becomes

$\delta T_{\rm ph}(\mathbf{r})=\int
d\mathbf{r}^{\prime}K(\mathbf{r}-\mathbf{r}^{\prime})\delta
T(\mathbf{r}^{\prime}),$ (61)

where

$K(\mathbf{r})=\int\frac{d^{2}\mathbf{q}\leavevmode\nobreak\
e^{i\mathbf{q}\mathbf{r}}}{(2\pi)^{2}}\frac{\gamma}{\gamma_{0}+\varkappa_{\rm
ph}q^{2}}=\frac{\gamma}{2\pi\varkappa_{\rm ph}}K_{0}(q_{0}r),$ (62)

and $K_{0}$ is the MacDonald function. Evolution of the phonon temperature
distribution with increasing phonon heat conductivity is illustrated in Fig.
7.

##  VI Summary

To summarize, we have studied the heat balance in a weakly disordered system
with local inhomogeneities. Specifically, we have considered a spherical local
constriction with increased impurity scattering rate as compared to the
scattering in the uniform background. We assumed that electron-electron
interaction is strong in two senses: firstly, fast electron-electron
collisions drive the system into the hydrodynamic regime; secondly, it
guarantees the electrical neutrality, thus making the electron liquid nearly
incommpressible.

In the absence of viscosity, the electron temperature distribution induced by
the inhomogeneity is described by a Landauer-dipole-like structure that is
shifted along the current by the amount of the order of the inelastic drift
length (Fig. 2). The Landauer-dipole distribution stems from the tendency of
particles to travel around the inhomogeneity (in the 2D and 3D cases).
Remarkably, the thermal Landauer dipole and its asymmetry, both induced by the
current flow, exist in systems of arbitrary dimensionality, and of genuine
constriction geometries. Thus, we have found that the heating of inhomogeneous
ideal electron fluid is universally described by a Landauer dipole, which is
deformed by the flow beyond the linear-response regime.

Further, focusing on the 2D case, we have explored the evolution of the
electron temperature and velocity profiles with increasing viscosity of the
electron fluid (Fig. 3). Our main conclusion is that the viscosity
dramatically changes the heat balance in the system. Specifically, we have
found that viscosity suppresses the Landauer-dipole structure and—for
sufficiently high viscosity—essentially reduces the heat-balance problem in
all dimensions to a quasi-1D problem. Most importantly, viscosity leads to the
formation of the “hot spot” exactly in the position of the inhomogeneity. We
have also derived a relation between electron temperature and the temperature
of the phonon system (Fig. 7), which can be directly measured in experiment.

###### Acknowledgements.

We thank A. Dmitriev, A. Mirlin, B. Narozhny, and E. Zeldov for discussions.
The work of GZ and IG is supported by the German-Israeli Foundation (GIF grant
No. I-1505-303.10/2019), the DFG grant (No. GO 1405/5) within the FLAG-ERA
Joint Transnational Call (project GRANSPORT), and European Commission under
the EU Horizon 2020 MSCA RISE-2019 program (project 873028 HYDROTRONICS). The
work of VK and IG was supported by the Russian Science Foundation (Grant No.
20-12-00147) . The work of VK was also supported by the Foundation for the
Advancement of Theoretical Physics and Mathematics “BASIS”.

##  Appendix A Gaussian constriction

In this Appendix, we derive the expressions for Landauer-dipole structures in
$D=1,2,3$ dimensions. To this end, one needs to calculate the Fourier
transform of $K_{\mathbf{q}}\xi_{\mathbf{q}}$, where $K_{\mathbf{q}}$ and
$\xi_{\mathbf{q}}$ are given by Eqs. (36) and (15) of the main text,
respectively. Introducing auxiliary integrals,

$\frac{1}{q^{2}}=\int\limits_{0}^{\infty}dt\,e^{-tq^{2}},\quad\frac{1}{iq_{\parallel}+1/l_{\rm
in}}=\int\limits_{0}^{\infty}dse^{-s(iq_{\parallel}+1/l_{\rm in})},$

we find

$\displaystyle\int\\!\\!\frac{d^{D}\mathbf{q}}{(2\pi)^{D}}\frac{q_{\parallel}^{2}-q_{\perp}^{2}}{q_{\parallel}^{2}+q_{\perp}^{2}}\frac{e^{i\mathbf{q}\mathbf{r}-q^{2}L^{2}/4}}{iq_{\parallel}+1/l_{\rm
in}}$ (63) $\displaystyle=\\!\int\limits_{0}^{\infty}\\!dse^{-s/l_{\rm
in}}\\!\\!\int\limits_{0}^{\infty}\\!dt(\partial_{\perp}^{2}-\partial_{\parallel}^{2})\\!\int\\!\\!\frac{d^{D}\mathbf{q}}{(2\pi)^{D}}e^{-q^{2}(t+L^{2}/4)+i\mathbf{q}\mathbf{r}-iq_{\parallel}s}$
$\displaystyle=\\!\int\limits_{0}^{\infty}dse^{-s/l_{\rm
in}}\tilde{Q}_{D}(x-s,\mathbf{r}_{\perp}),$

where

$\tilde{Q}_{D}(\mathbf{r})=\frac{1}{\pi^{D/2}}(\partial_{\perp}^{2}-\partial_{\parallel}^{2})\int\limits_{0}^{\infty}dt\frac{e^{-r^{2}/(4t+L^{2})}}{4t+L^{2}}.$
(64)

Applying the spatial derivatives
$(\partial_{\perp}^{2}-\partial_{\parallel}^{2})$, we get

$\displaystyle\tilde{Q}_{D}(\mathbf{r})$
$\displaystyle=\int\limits_{L^{2}/4}^{\infty}dte^{-r^{2}/4t}$
$\displaystyle\times\left\\{\begin{array}[]{cc}\dfrac{2t-x_{\parallel}^{2}}{8\sqrt{\pi}t^{5/2}},&\text{for}\quad
D=1,\\\\[8.5359pt] \dfrac{r_{\perp}^{2}-x_{\parallel}^{2}}{16\pi
t^{3}},&\text{for}\quad D=2,\\\\[8.5359pt]
\dfrac{r_{\perp}^{2}-x_{\parallel}^{2}-2t}{32\pi^{3/2}t^{7/2}},&\text{for}\quad
D=3.\end{array}\right.$ (68)

Calculating the remaining integral over $t$ and assuming that electric field
is parallel to the $x$-axis, we arrive at Eq. (45) of the main text.

##  Appendix B Temperature Landauer dipole in 3D and 2D system—a local heating
approximation

In this Appendix, we employ the local heating approximation (Fig. 8), and
study the constriction-induced local heating of 2D and 3D systems. The
calculation here generalizes the one in Ref. Tikhonov  _et al._ (2018).

Figure 8: Phenomenological local-conductivity model adopted for the local
Joule approximation. A constriction of radius $L$, located at the origin, is
characterized by the conductivity $\sigma_{\text{in}}$ that is smaller than
the uniform background conductivity: $\sigma_{\text{in}}<\sigma_{\text{out}}$.
The external field $E_{\infty}$ is directed along the $x$-axis.

###  B.1 Landauer dipole in 3D

Within the local heating approximation, the constriction of radius $L$ has the
uniform conductivity $\sigma_{\text{in}}$ that is smaller than the
conductivity $\sigma_{\text{out}}$ outside the constricted area. The
electrical current is driven by the field $E_{\infty}$ directed along the
$x$-axis. Under the continuity boundary condition, the electric potential
becomes

$\displaystyle\varphi(\mathbf{r})=\left\\{\begin{array}[]{ll}\displaystyle-\frac{3\sigma_{\text{out}}}{\sigma_{\text{in}}+2\sigma_{\text{out}}}\,rE_{\infty}\cos\vartheta,&r<L,\\\\[14.22636pt]
\displaystyle-\left[r+\frac{(\sigma_{\text{out}}-\sigma_{\text{in}})L^{3}}{(2\sigma_{\text{out}}+\sigma_{\text{in}})r^{2}}\right]E_{\infty}\cos\vartheta,&r>L,\end{array}\right.$
(69)

where $\vartheta$ is the angle between the $x$ axis and the direction of the
vector $\mathbf{r}$ in real space. Apparently, $\varphi(\mathbf{r})$ reduces
to the uniform value $-E_{\infty}r\cos\vartheta$ in the
$r\equiv|\mathbf{r}|\gg L$ limit.

With the electric potential from Eq. (69), we find the local electric field
and calculate the local Joule heating $\sigma(\mathbf{r})E(\mathbf{r})^{2}$

$\displaystyle\frac{P(\mathbf{r})}{\sigma_{\text{out}}E_{\infty}^{2}}=\frac{\sigma(\mathbf{r})E(\mathbf{r})^{2}}{\sigma_{\text{out}}E_{\infty}^{2}}$
(70)
$\displaystyle=\left\\{\begin{array}[]{ll}\displaystyle(1+\lambda-2\lambda^{2})\sigma_{\text{out}}E_{\infty}^{2},&r<L,\\\
\\\
\displaystyle\\!\\!\left[1\\!-\\!\frac{L^{3}}{r^{3}}\lambda(1\\!-\\!3\cos\vartheta)\right]^{2}\\!\\!+\\!\frac{9L^{6}}{4r^{6}}\lambda^{2}\sin^{2}(2\vartheta),&r>L,\end{array}\right.$

where

$\lambda\equiv\frac{\sigma_{\text{in}}-\sigma_{\text{out}}}{2\sigma_{\text{out}}+\sigma_{\text{in}}}$
(71)

is the parameter characterizing the inhomogeneity. It vanishes in the
homogeneous situation ($\sigma_{\text{in}}=\sigma_{\text{out}}$), and
approaches $1/2$ when the constriction is insulating ($\sigma_{\text{in}}\to
0$).

To describe the effect of the constriction, we define

$\delta P=P-\sigma_{\text{out}}E_{\infty}^{2},$

as the constriction-induced extra Joule heating and express it in terms of
$\lambda$:

$\displaystyle\frac{\delta P}{\sigma_{\text{out}}E_{\infty}^{2}}$ (72)
$\displaystyle\\!=\\!\left\\{\begin{array}[]{ll}\displaystyle\\!\lambda-2\lambda^{2},&r<L,\\\
\\\ \displaystyle\\!\frac{\lambda
L^{3}[1+3\cos(2\vartheta)]}{r^{3}}\\!+\\!\frac{\lambda^{2}L^{6}[5\\!+\\!3\cos(2\vartheta)]}{2r^{6}}\\!,&r>L.\end{array}\right.$

Clearly, $\delta P$ vanishes when $\lambda=0$. It is also non-monotonous and
approaches the peak value when $\lambda=1/4$. Indeed, all particles detour
around the constriction in the insulating limit, where the Joule heating
vanishes.

For a more intuitive understanding, we provide the profile of the local Joule
heating (when $\lambda=1/3$) in Fig. 1(d) of the main text. Apparently, a
Landauer-dipole heating pattern emerges around the constriction. This heating
pattern naturally induces a shifted Landauer dipole of the electron
temperature profile. Noteworthy, this Landauer-dipole feature is produced by
the residual charges at the boundary of the constriction. These charges also
guarantee the constant electric field in the constricted area.

###  B.2 Landauer dipole in 2D

Let us move to the 2D case with a circular-shape constriction. To begin with,
following the same method as that of Appendix B.1, we arrive at the electric
field of 2D

$\displaystyle\mathbf{E}(\mathbf{r})$
$\displaystyle=\left\\{\begin{array}[]{ll}\displaystyle\mathbf{E}_{\infty}\\!+\delta\mathbf{E}_{\text{in}},&r<L,\\\
\\\
\displaystyle\mathbf{E}_{\infty}\\!+\\!\delta\mathbf{E}_{\text{out}},&r>L,\end{array}\right.$
(73)

where

$(\delta\mathbf{E}_{\text{in}})_{x}=\delta E,\ \ \ \
(\delta\mathbf{E}_{\text{in}})_{y}=0$ (74)

is the electric field in the constricted area, and

$\displaystyle(\delta\mathbf{E}_{\text{out}})_{x}$ $\displaystyle=\delta
E\frac{y^{2}-x^{2}}{(x^{2}+y^{2})^{2}}L^{2},$ (75)
$\displaystyle(\delta\mathbf{E}_{\text{out}})_{y}$ $\displaystyle=\delta
E\frac{-2xy}{(x^{2}+y^{2})^{2}}L^{2},$

refers to that outside of the constriction. In Eqs. (74) and (75), the
inhomogeneity-induced electric field is given by

$\delta
E=E_{\infty}\frac{\sigma_{\text{out}}-\sigma_{\text{in}}}{\sigma_{\text{out}}+\sigma_{\text{in}}}=E_{\infty}\lambda_{2}>0,$
(76)

where the parameter $\lambda_{2}$ characterizes the strength of the
inhomogeneity in the 2D case.

Figure 9: The vector velocity profile of the 2D local heating model.

In the local approximation, the drift velocity is proportional to the electric
field
$\mathbf{v}(\mathbf{r})=\sigma(\mathbf{r})\mathbf{E}(\mathbf{r})/N_{\infty}e$.
Following Eq. (73), we obtain the drift velocity profile

$\displaystyle\mathbf{v}(\mathbf{r})$
$\displaystyle=\left\\{\begin{array}[]{ll}\displaystyle\mathbf{v}_{\infty}+\delta\mathbf{v}_{\text{in}},&\
r<L,\\\ \\\ \displaystyle\mathbf{v}_{\infty}+\delta\mathbf{v}_{\text{out}},&\
r>L,\end{array}\right.$ (77)

where

$(\delta\mathbf{v}_{\text{in}})_{x}=-\delta v,\ \
(\delta\mathbf{v}_{\text{in}})_{y}=0$ (78)

in the constricted area, and

$\displaystyle(\delta\mathbf{v}_{\text{out}})_{x}=\delta
v\frac{y^{2}-x^{2}}{(x^{2}+y^{2})^{2}}L^{2}$ (79)
$\displaystyle(\delta\mathbf{v}_{\text{out}})_{y}=\delta
v\frac{-2xy}{(x^{2}+y^{2})^{2}}L^{2}$

outside of the constriction. In Eqs. (79) and (78), the homogeneous velocity
$\mathbf{v}_{\infty}=\sigma_{\text{out}}\mathbf{E}_{\infty}/Ne$, and

$\delta
v=v_{\infty}\frac{\sigma_{\text{out}}-\sigma_{\text{in}}}{\sigma_{\text{out}}+\sigma_{\text{in}}}\equiv
v_{\infty}\lambda_{2}$ (80)

is the inhomogeneity-induced velocity variation, defined in the same manner as
Eq. (76). In the momentum space, the variation of velocity becomes

$\displaystyle\delta v_{x}(\mathbf{q})$ $\displaystyle=\frac{2\pi
L}{q}[J_{1}(Lq)\cos(2\varphi)-J_{1}(Lq)]\lambda_{2}$ (81) $\displaystyle\delta
v_{y}(\mathbf{q})$ $\displaystyle=\frac{2\pi
L}{q}J_{1}(Lq)\sin(2\varphi)\lambda_{2},$

where $q=|\mathbf{q}|$, the angle $\varphi$ refers to the momentum direction,
and $J_{1}(Lq)$ is the Bessel function.

To better understand Eq. (81), we Fourier transform the circular-shape step
function, $\xi(\mathbf{r})=\xi_{0}\Theta(x^{2}+y^{2}-L^{2})$, into the
expression in the momentum space

$\xi(\mathbf{q})=\xi_{0}\frac{2\pi}{q}LJ_{1}(Lq).$ (82)

The combination of Eqs. (81) and (82) gives us the velocity kernel of the
local-heating model

$\displaystyle K_{u_{x}}$
$\displaystyle=\frac{-2q_{\perp}^{2}}{q_{\parallel}^{2}+q_{\perp}^{2}},$ (83)
$\displaystyle K_{u_{y}}$
$\displaystyle=\frac{2q_{\parallel}q_{\perp}}{q_{\parallel}^{2}+q_{\perp}^{2}},$

where $q_{\parallel}$ and $q_{\perp}$ refer to two momentum components.
Equation (83) coincides with Eq. (33) of the main text, except for an overall
extra factor of two. This extra factor is related to the definition of
$\lambda_{2}$: at the weak inhomogeneity limit,
$\lambda_{2}\approx(\sigma_{\text{out}}-\sigma_{\text{in}})/2\sigma_{\text{out}}$.
Following Eq. (80), we plot the 2D velocity profile in Fig. 9. It captures the
major feature of that in the main text: particles tend to detour around the
constriction.

Figure 10: The temperature profiles of the phenomenological model when (a)
$l_{*}=2L$; (b) $l_{*}=0.5L$. The temperature profiles of the diffusive
phenomenological model are completely symmetric.

With the velocity profile known, we can calculate the temperature profile.
Since this local-heating model is diffusive, $\varkappa_{\text{e}}\gg
v_{\infty}$ (where $\varkappa_{\text{e}}$ is the electronic heat
conductivity), we keep only the diffusive term in the heat diffusion equation

$-\varkappa_{e}\bigtriangledown^{2}\delta
T=ne\mathbf{E}\mathbf{v}-\gamma\delta T.$ (84)

As a simple check, we focus on the weak-inhomogeneity limit $\lambda_{2}\ll
1$. In this limit, the Fourier-transformed temperature profile becomes

$\displaystyle\delta T_{\mathbf{q}}$
$\displaystyle=T_{\infty}\frac{4\pi}{q}\frac{LJ_{1}(Lq)\cos(2\varphi)}{l_{*}^{2}q^{2}+1}\lambda_{2},$
(85)

where $l_{*}=\sqrt{\varkappa_{\text{e}}/\gamma}$. In Eq. (85), the prefactor
$T_{\infty}$ is the homogeneous temperature when $\lambda_{2}=0$. We present
the temperature profile of Eq. (85) in the real space, in Fig. 10. In contrast
to the hydrodynamic model of the main text Eq. (34), the temperature kernel of
Eq. (85) strictly corresponds to a Landauer dipole that sits exactly at the
center of the constricted area: the asymmetry thus completely disappears in
the diffusive situation.

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