# Phonon hydrodynamics in bulk insulators and semi-metals

Yo Machida  Department of Physics, Gakushuin University, Tokyo 171-8588,
Japan.  Valentina Martelli  Laboratory for Quantum Matter under Extreme
Conditions
Institute of Physics, University of São Paulo, 05508-090, São Paulo, Brazil
Alexandre Jaoui  JEIP, USR 3573 CNRS, Collège de France, PSL Research
University, 11, Place Marcelin Berthelot, 75231 Paris Cedex 05, France.
Laboratoire de Physique et d’Étude des Matériaux
(ESPCI - CNRS - Sorbonne Université), PSL Research University, 75005 Paris,
France.  Benoît Fauqué  JEIP, USR 3573 CNRS, Collège de France, PSL Research
University, 11, Place Marcelin Berthelot, 75231 Paris Cedex 05, France.
Laboratoire de Physique et d’Étude des Matériaux
(ESPCI - CNRS - Sorbonne Université), PSL Research University, 75005 Paris,
France.  Kamran Behnia  Laboratoire de Physique et d’Étude des Matériaux
(ESPCI - CNRS - Sorbonne Université), PSL Research University, 75005 Paris,
France.

###### Abstract

Decades ago, Gurzhi proposed that if momentum-conserving collisions prevail
among heat-carrying phonons in insulators and charge-carrying electrons in
metals, hydrodynamic features will become detectable. In this paper, we will
review the experimental evidence emerging in the last few years supporting
this viewpoint and raising new questions. The focus of the paper will be bulk
crystals without (or with a very dilute concentration) of mobile electrons and
steady-state thermal transport. We will also discuss the possible link between
this field of investigation and other phenomena, such as the hybridization of
phonon modes and the phonon thermal Hall effect.

##  I Introduction

In his seminal 1968 paper, entitled “Hydrodynamic effects in solids at low
temperatures”[1], Gurzhi wrote: “The phenomena of thermal conductivity of
dielectric [i.e. insulators] and the electrical conductivity of metals have
specific properties. In both cases, the total quasi-particle current turns out
to be non-vanishing. It follows that when only normal collisions occur in the
system, there could exist an undamped current in the absence of an external
field which could sustain it.” The adjective ‘normal’ is employed here to
specify those collisions (we will call them ‘Normal’), which are not of the
Umklapp type, i.e. between two quasi-particles with wave-vectors whose sum
exceeds the width of the Brillouin zone.

In the text quoted above, Gurzhi is proposing a thought experiment. For both
electrons in metals and phonons in insulators, one is invited to wonder what
happens if Umklapp collisions are turned off. He proposes that then if there
were no viscosity, the motion would become perpetual. This insight opens a
route towards exploration of the viscosity of these quasi-particles.

When two phonons collide, a third phonon is produced. If the wave-vector of
the latter is larger than the size of the Brillouin zone, then the collision
is called Umklapp (See Fig.1, which was adapted from ref. [2]). Such an event
would degrade the momentum (and the energy) flow of phonons. In contrast,
during a normal collision, the wave-vector of the third phonon remains shorter
than the width of the Brillouin zone and the flow is not degraded. The same is
true for electrons, where because of the conservation of charge, the two
initial electrons are replaced by two and not one.

Now, there is a fundamental difference between electrons and phonons regarding
the Umklapp collisions. The wave-vector of a thermally excited phonon $q_{ph}$
shrinks with temperature. At a given temperature, it is
$q_{ph}=\frac{k_{B}T}{\hbar v_{s}}$, where $v_{s}$ is the sound velocity,
$k_{B}$ is the Boltzmann constant, and $\hbar$ is the reduced Planck constant.
Therefore, in order to perform Gurzhi’s thought experiment for phonons, all
you need [to get rid of Umklapp events] is to cool down your sample well below
a temperature, where $q_{ph}<G/2$, which corresponds to a fraction of the
Debye temperature. The situation is quite different for electrons, whose
wavevector is set by the Fermi radius and does not change with temperature. A
very small Fermi surface, located at the center of the Brillouin zone [3] is
needed to avoid Umklapp collisions.

The role of disorder is another factor which renders the observation of phonon
hydrodynamics easier than its electronic counterpart. Since the wavelength of
heat-carrying acoustic phonons increases with cooling, point defects become
incapable of scattering them. In a single crystalline bulk insulator, the
phonon mean free path becomes equal to the size of the sample at cryogenic
temperatures, and the default transport regime becomes ballistic. Achieving
this for electrons in metals is much more challenging.

Figure 1: The contrast between Normal and Umklapp events during a three phonon
process: In a Normal collision (left), the resultant phonon wave-vector
remains inside the Brillouin zone. This is not the case in an Umklapp event
(right), which leads to the loss of momentum $\overrightarrow{b}$,
corresponding to a unit vector of the reciprocal lattice (adapted from [2]).

These fundamental differences may explain, at least partially, the historical
fact that phonon hydrodynamics was explored before electron hydrodynamics. As
early as 1974, two decades before the first recorded experimental
investigation of electron hydrodynamics [4], Beck and co-workers [5] wrote a
review paper on phonon hydrodynamics. However, after a promising start,
research on phonon hydrodynamics went through hibernation in the last decades
of the previous century.

The renaissance of this field of research began in 2015 with theoretical
studies of heat transport in graphene [6, 7]. The present paper summarizes the
evidence gathered in the last few years by steady-sate transport measurements,
confirming the existence of a hydrodynamic window for thermal transport by
phonons. Our main focus will be a subset of such hydrodynamic phenomena
associated with the Poiseuille regime observed in bulk crystals by steady-
state heat transport studies. For a recent review paper on phonon
hydrodynamics, written from a theoretical perspective, and covering a broader
range of topics, see [8].

##  II Heat transport regimes

Figure 2 presents a generic picture of the evolution of fundamental length
scales and their consequences for thermal transport by phonons in an
insulator. Near the Debye temperature and above, the phonon wave-vector is
comparable to the width of the Brillouin zone, and most phonon-phonon
collisions are Umklapp. Therefore: $l_{N}>l_{U}$; that is, the distance
between two Umklapp collisions ($l_{U}$) is shorter than the one between two
Normal collisions ($l_{N}$). As the temperature decreases and the phonon wave-
vector shrinks, the frequency of Umklapp collisions decreases. Since the
phonon density of states also decreases with cooling, Normal collisions rarefy
too, albeit with a slower rate (Figure 2a). As a consequence of this
inequality, an intermediate temperature window (in red) can pop up between the
high-temperature diffusive (in green) and the low-temperature ballistic
regimes (in blue).

Figure 2: The four regimes of heat transport and the hydrodynamic window: a)
The distance between two Normal collisions $l_{N}$ and two Umklapp collisions
$l_{U}$ increases with cooling. But the increase in $l_{N}$ is slower.
Eventually, both exceed the sample thickness, $d$, shown by a horizontal
dashed line. The sketch distinguishes between three regimes. In the ballistic
regime (in green), the most frequent collisions suffered by a heat-carrying
phonon are boundary collisions. In the diffusive regime (in blue), Umklapp
collisions are more frequent than collisions with boundary. In the
hydrodynamic regime (in red), Boundary collisions are more frequent than
Umklapp collisions, but less frequent than Normal collisions. b) A sketch of
the evolution of the effective phonon mean free path with temperature. In the
hydrodynamic regime, it exceeds the sample thickness. c) The four regimes of
heat transport in a crystalline insulator (adapted from [5]). The Poiseuille
regime is where thermal conductivity evolves faster than T3.

In the ballistic regime, the shortest length scale is the [temperature-
independent] thickness of the sample (d) and transport is dominated by
boundary scattering. In the diffusive regime, transport is dominated by
phonon-phonon scattering, which is mostly Umklapp. Thermal conductivity in the
diffusive regime is intrinsic to the insulating solid, does not depend on the
sample size and can be quantified nowadays by ab initio theory starting from
the computed phonon spectrum. It steadily decreases with rising temperature.
In contrast, thermal conductivity in the ballistic regime depends on the size
of the crystal, the velocity of sound and the heat capacity, and because of
the latter, increases with rising temperature. The hydrodynamic regime, which
may arise between these two, is characterized by a specific hierarchy of the
three length scales: $l_{U}>d>l_{N}$. It implies that compared to boundary
scattering, Umklapp collisions are rare and Normal collisions are abundant.
One remarkable consequence of the emergence of such a hydrodynamic regime is
sketched in Figure 2b. Thanks to Normal collisions, the temperature dependence
of the effective phonon mean-path becomes non-monotonous and its peak value is
expected to be longer than the effective thickness of the sample.

Based on the hierarchy of distinct scattering times and mean free paths, Beck,
Meier and Thellung [5] identified four regimes of heat transport in a generic
insulator (See also [6] and [8]). They are illustrated, together, with the
expected temperature dependence of thermal conductivity in each regime, in
Figure 2c.

In the high-temperature regime, called ‘kinetic’, Umklapp phonon-phonon
scattering dominates and $l_{U}$ is the shortest length scale. The temperature
is high enough for thermal excitation of almost all phonon population. The
specific heat tends to saturate to its universal Dulong-Petit value of 3kB per
atom and the thermal conductivity tends to follow the inverse of temperature.
It has been recently noted that the phonon-phonon scattering time in this
regime approaches but never falls below the so-called Planckian time [9, 10].
Mousatov and Hartnoll [11] have argued that this bound is due to a bound on
the sound velocity. Because of quantum mechanics, it cannot fall below a
velocity set by the melting temperature of the crystal, the interatomic
spacing and $\hbar$.

Cooling the crystal, one enters the so-called ’Ziman’ regime [5]. In this
temperature range, phonons with a large wave-vector are absent, and therefore,
Umklapp collisions become rare. As a consequence, $l_{N}<l_{U}$. Both $l_{N}$
and $l_{U}$ remain shorter than the sample size, $d$. In this regime, thermal
conductivity increases faster than linear with decreasing temperature and is
expected to become eventually exponential, following
$\propto\exp(-\frac{\Theta}{T})$. However, a clear exponential behavior has
only been seen in a handful of solids [12].

Cooling further, leads to a peak in thermal conductivity. At this peak
temperature, the decrease in phonon population and the increase in the phonon
scattering time, both induced by cooling cancel each other. Below this peak,
the temperature dependence of $\kappa$ is mainly set by the temperature
dependence of specific heat, which, asymptotically, is $\propto T^{3}$. The
ballistic regime, also known as the Casimir regime [5], referring to an early
and influential note by Casimir [13], occurs at the lowest temperatures. Here,
$l_{N}$ and $l_{U}$ become longer than the sample size, which sets the
ultimate mean free path of the heat carriers. Thermal conductivity in single
crystals becomes size dependent [14] and follows a $\propto T^{3}$ temperature
dependence.

The Poiseuille regime, which is the focus of the present paper is expected to
emerge between the ballistic regime and the peak. In the Poiseuille regime,
one has $l_{N}<d<l_{U}$. Now, most collision events are Normal and the
frequency of Umklapp collisions is negligible. Therefore phonons can exchange
momentum frequently before colliding with the boundaries. In this case,
phonons closer to the boundary would have a drift velocity lower than those
further from the boundaries. In this temperature range, the thermal
conductivity is expected to have a temperature dependence faster than the
specific heat. This implies that warming the sample leads to an increase in
the effective mean free path, despite an increase in the collision rate.

Figure 3: Early observations: a) The variation of the effective phonon mean
free path with temperature (normalized by the Debye temperature) in different
crystals of 4He grown under different pressures from 60 (sample 1) to 185
(sample 4) bars. Note the presence of a peak [From ref [15]]. b) Temperature
dependence of the effective phonon mean free path in Body Centered Cubic
crystals of 3He grown in pressures ranging from 56 (sample D) to 96 (sample A)
bars. There is a local peak [From ref [16]]. c) Temperature dependence of the
effective phonon mean free path in Bi crystals with different sizes. There is
clearly a local peak and a size dependence. However, the size dependence at
the peak and below the peak are not very different [From ref.[17]].

##  III Thermal conductivity in the ideal Poiseuille regime

With cooling, the collision rate between phonons is reduced and both lU and lN
increase. In the simplest picture of phonon transport, lU rises exponentially
and lN follows a power law. The expected temperature dependence of $\kappa$ in
the Poiseuille regime depends on the exponent of this power law. A simple
argument would lead to l${}_{N}\propto T^{-5}$, reminiscent of the temperature
dependence of the e-ph scattering time for temperatures well below the Debye
temperature [18]. The similarity is not accidental, because the origin is the
same. The volume of Debye sphere, which sets the phonon population is $\propto
T^{3}$. On top of this, the collisions cross section, which is $\propto
q^{2}$, will add another $\propto T^{2}$ factor, leading to a $\propto T^{5}$
phonon-phonon Normal scattering rate.

Consider that thermal conductivity can be expressed as
$\kappa=1/3C_{V}v_{s}\ell_{eff}$. The effective mean free path $\ell_{eff}$ is
expected to be proportional to the cross section of the conductor and
inversely proportional to the distance between Normal collisions:
$\ell_{eff}=\frac{d^{2}}{l_{N}}$. Therefore, in an ideal Poiseuille regime one
expects:

$\kappa=1/3C_{V}v_{s}\frac{d^{2}}{l_{N}}$ (1)

Thus, one expects two scaling relations for $\kappa$. A temperature dependence
as strong as $\propto T^{8}$ and a super-ballistic size dependence $\propto
d^{2}$ [5].

##  IV Experimental detection of the Poiseuille regime

Experiments have found signatures of hydrodynamic heat flow in the Poiseuille
regime. However, save for two experimental studies of 4He crystals by Mezhov-
Deglin [15, 19] and by Hogan et al. [20], they do not approach the spectacular
predictions for size dependence ($\propto d^{2}$) and for temperature
dependence ($\propto T^{8}$). The emerging picture is the following: in a
growing list of solids, one can detect a regime below the $\kappa$ peak where
the temperature dependence is faster for thermal conductivity than for
specific heat. In other words, with $C\propto T^{\gamma}$, $\kappa\propto
T^{\gamma^{\prime}}$, one finds $\gamma<\gamma^{\prime}$ in a narrow
temperature window. The correction to the overall amplitude is modest (often
between 10 to 30 percent). It is, nevertheless, qualitatively important,
because it implies that collisions can help the fluid flow instead of
degrading it, as often assumed, when momentum-conserving collisions are
neglected.

Let us briefly review the results.

Solid 4He- At ambient pressure, both isotopes of helium remain a liquid down
to $T=0$. Under a pressure of 25 bars (i.e. 2.5 MPa), 4He becomes a solid
[21]. Cryogenic crystals can be made under pressures exceeding this pressure.
These are remarkable electrical insulators with virtually no chemical
impurities and a high degree of isotopic purity. The first confirmation of
Gurzhi’s theory was reported by Mezhov-Deglin, working at the Institute for
Physics Problems and the Institute of Solid State Physics in Moscow in 1966.
He measured the thermal conductivity of crystalline 4He in the temperature
range of 0.5K to 2.5K and at various pressures. In the samples grown at 85
bars, the thermal conductivity was found to peak at 0.9 K and below this peak,
it was found to decrease much faster than $T^{3}$ [15]. A clear peak in the
temperature dependence of the effective mean free path could be detected (see
Fig.3a). This study was followed by another work by Mezhov-Deglin quantifying
the size dependence [19] and two other investigations in Duke [20] and in
Urbana [20, 16]. The studies led to the confirmation of Poiseuille flow in
4He, albeit with a nuance.

Crystals grown at low pressures (below 100 bars) displayed strong signatures
of Poiseuille flow, including a temperature dependence as fast as $\propto
T^{6}$ and a quadratic $\propto d^{2}$ size dependence [20, 16]. The
signatures for Poiseuille flow in crystals grown at pressures higher than 185
bars were weaker [22]. Nevertheless, these crystals did show a local peak in
the temperature dependence of the effective mean free path. Interestingly, the
latter crystals did show a prominent Ziman regime and no evidence of any
deficit in purity. This suggests to us that phonon hydrodynamics is linked
with the proximity of a structural instability, as indicated by posterior
research on column V elements. In this line of interpretation, with increasing
pressure, the solid state of helium is further stabilized, and therefore
Normal phonon-phonon collisions become less frequent, gradually closing the
hydrodynamic window.

Solid 3He- 3He becomes a solid when pressure exceed 30 bars [21]. It
crystallizes at low pressures in a Body Centered Cubic (BCC) crystalline phase
and then becomes Face Centered Cubic (FCC). In 1969, Thomlinson [16], almost
simultaneously with those studying 4He crystals, grew single crystals of 3He
under pressure and measured their thermal conductivity in both crystalline
phases. As seen in Fig.3b, his results led to the observation of a local peak
in the temperature dependence of the effective mean free path.

Bismuth- Crystalline bulk bismuth has fascinating electronic properties. It is
a semimetal with an equal density of electrons and holes, which at low
temperatures corresponds to one carrier of each sign per 105 atoms [23].
Because of this low concentration of mobile carriers of charge, the low-
temperature thermal conductivity of bismuth has been measured by several
authors [17, 24, 25, 26]. Like in an insulator, it is almost totally dominated
by phonons and peaks between 3 and 4 K.

In 1973, Kopylov and Mezhov-Deglin [17] studied thermal transport in massive
bismuth crystals of centimetric dimensions between 1.3 and 7 K. They observed
a stronger than cubic temperature-dependent thermal conductivity below the
maximum and attributed it to “a hydrodynamic situation in the phonon system
evoked by frequent normal collisions between phonons”. They quantified the
mean free path of phonons in their crystals, which showed a local peak, as one
can see in Fig. 3c. They could not detect any superlinear size dependence in
$\kappa$ and in $\ell_{eff}$, expected in the ideal Poiseuille flow.

A few years later, Pratt and Uher [24] measured the thermal conductivity of
bismuth crystals below 1 K and found that the phonon mean free path shortens
with decreasing temperature in agreement with what was found by Kopylov and
Mezhov-Deglin [17]. They invoked electron-phonon interactions as a source of
this. We will return to this feature when discussing the case of antimony.

Solid hydrogen- Like helium isotopes, solid hydrogen is a quantum crystal, but
a molecular one, which at ambient pressure solidifies at $\approx$ 13.8 K.
There is a small concentration (0.2%) of orthohydrogen molecules in
equilibrium, which can be decreased.

The first study of phonon hydrodynamics in the present century by steady state
thermal transport was reported by Zholonko [27], who measured the thermal
conductivity of solid parahydrogen in the temperature range of 1.5 to 6.0 K.
He found that below its low-temperature maximum, thermal conductivity displays
a rapid variation with an exponent barely larger than 3 and attributed it to a
Poiseuille flow of phonons. The estimated mean free path of phonons at
$\approx$3 K exceeded the radius of the cylindrical sample and showed a small
local peak, barely larger than the noisy background [27]. The experimental
evidence for Poiseuille flow in this case appears weaker than in other
cryogenic crystals.

Figure 4: Poiseuille flow in black P: a) The extracted phonon mean free path
in black P samples with a heat current applied along the c-axis. The thicker
the sample the longer the mean free path. Note the local peak. b) The same
data plotted after normalizing to the minimum mean free path. Note the shift
towards lower temperature with increasing thickness, which is expected in the
hydrodynamic picture [adapted from ref. [28]].

Black phosphorus- Black phosphorus (BP) is a bulk insulator [29], which
crystallizes in an orthorhombic crystal structure with puckered honeycomb
layers coupled by van der Waals interaction between the layers. It has a band
gap of 2 eV. Like other column V elements and IV-VI binary salts, its crystal
symmetry is the outcome of a close competition between between cubic,
rhombohedric and orthorhombic structures[30, 31, 32]. Such a proximity to a
structural instability may favor Normal phonon-phonon scattering.

In 2018, Machida and co-workers [28] reported on measurements of thermal
conductivity in bulk black phosphorus between 0.1 and 80 K. Several samples
with a variety of thicknesses were studied. The study found a thermal
conductivity showing a faster than cubic temperature dependence between 5 K
and 12 K. The effective phonon mean free path was found to be non-monotonous
in temperature dependence in the Poiseuille regime. A local maximum and a
local minimum are clearly visible in all samples (See Fig.4).

The size dependence of the thermal conductivity, in both ballistic and
Poiseuille regimes, was found to be close to linear, as previously found in
bismuth [17]. This is not what one expects in the ideal Poiseuille regime.
Nevertheless, a subtle size dependence beyond the ballistic picture was also
revealed. It was found that local maximum and the local minimum temperatures
both shift with thickness. In the thicker samples, both occur at lower
temperatures. This is understandable in the hydrodynamic picture, because the
fraction of phonons suffering frequent Normal collisions is larger in thicker
samples (See Fig.4b).

The absence of superlinear size dependence may indicate that the phonon fluid
is non-Newtonian [28]. This is because viscosity is set by the Normal
scattering rate and this rate is not the same for all phonons. At a given
temperature, Normal scattering is much less frequent for phonons close to the
sample boundary. It has been shown that the velocity profile of a non-
Newtonian fluid is significantly flatter than the parabolic one, which is
expected for Newtonian fluids [33]. This would explain a size dependence less
straightforward and drastic than what is expected in the simple and ideal
Poiseuille flow.

Strontium titanate- SrTiO3, known sometimes simply as STO, this large-gap
insulator with a perovskite structure, is a quantum paraelectric, where zero-
point quantum fluctuations play a crucial role in the stabilization of the
ground state [34]. An extremely small concentration of chemical dopants turn
this insulator to a metal with a superconducting instability [35]. Its thermal
conductivity was first measured by Steigmeier [36], who discovered that
electrical field can modify the amplitude of thermal conductivity at cryogenic
temperatures.

In 2018, Martelli et al. [9] reported on a study of thermal conductivity in
SrTi1-xNbxO3. All four regimes of heat transport could be detected in undoped
STO. Moreover, it was found that three different samples showed a $\kappa$
evolving faster than cubic with temperature below its peak and in a narrow
temperature window. The behavior was attributed to a Poiseuille flow of
phonons. It was found to disappear in the presence of Nb dopants.

It is noteworthy that in strontium titanate, at T$\approx$ 2K, the thermal
conductivity is almost cubic, but the phonon mean free path is still orders of
magnitude shorter than the crystal dimensions in the Poiseuille regime. It is
of the order of a few microns, the estimated size of the tetragonal structural
domains [37]. Therefore, the ballistic regime is not attained above 0.1 K.
Nevertheless, as seen in Fig. 5a, the phonon mean free of a typical undoped
STO crystal displays a clearly non-monotonous temperature dependence.

Figure 5: Two cases of non-monotonous phonon mean free path- a) Temperature
dependence of the mean free path of phonons in strontium titanate, estimated
by dividing the thermal conductivity by product of the specific heat and the
sound velocity ($\ell_{ph}=\frac{3\kappa}{v_{s}C}$) [9, 38]. The mean free
path remains well below the thickness of the sample. It shows a local maximum
shown by an arrow. b) Temperature dependence of the in-plane thermal
conductivity and the specific heat in graphite. Thermal conductivity evolves
faster than the specific heat. As seen in the inset, the mean free path of
phonons displays a peak [39].

The peculiarity of SrTiO3 is the presence of two soft phonon modes [40]. The
one at the center of the Brillouin zone is associated with ferroelectric
instability and its frequency does not vanish at zero temperature. It has a
significant dispersion and contributes to heat transport. The other at the
zone boundary drives the structural transition at 105 K. A recent inelastic
neutron scattering study [41] has found evidence for hybridization between a
transverse acoustic mode and the zone center ferroelectric soft mode. This may
play a role in the emergence of phonon hydrodynamic phenomena.

Graphite- Allotropes of carbon are among the best known conductors of heat. At
room temperature, the thermal conductivity of diamond is superior to that of
copper [21]. The other carbon allotrope, graphite, has a layered structure
combining strong interlayer covalent with weak intralayer van der Waals bonds.
The two-hundred-fold difference in the strength of bonding leads to two
distinct in-plane and out-of-plane Debye energy scales. This leads to a
specific heat which in graphite, evolves slower than $T^{3}$ [42]. This has
been known for a long time and discussed as an exception illuminating the
Debye $T^{3}$ rule in a textbook like Ziman’s [18].

In 2020, decades after the previous studies of heat transport in graphite [43,
44, 45], Machida et al. [39] carried out a study of thermal transport in thin
graphite samples and found that the in-plane thermal conductivity varies
faster than the specific heat (See Fig.5 b). This was unnoticed in previous
studies. Moreover, the effect was found to become more pronounced with
thinning of the graphite samples, approaching what was reported in the case of
graphene [46].

The origin of the pronounced size dependence of the thermal conductivity in
thin graphite is still to be properly understood. It is a subject of ongoing
theoretical research [47].

Figure 6: The case of antimony- a) The temperature dependence of the mean free
path $\ell_{ph}$ in single crystals of antimony with different dimensions. The
heat current is applied along the bisectrix axis. There is a local maximum,
which is amplified in larger samples. The arrow highlights the temperature at
which the phonon mean free path becomes size-dependent. Below this
temperature, Umklapp events are rare and ph-ph collisions conserve momentum.
Note that below 1 K, the phonon mean free path becomes sample independent
again. b) The temperature dependence of the exponent, $\gamma$, of electrical
resistivity in different samples (shown in the inset). Despite the difference
in the magnitude of resistivity at low temperature, $\gamma$ (extracted from
$\rho=\rho_{0}+T^{\gamma}$) shows a similar behavior in all of them. With
decreasing temperature, it steadily increases from 1 towards 5, as expected in
the Bloch-Grüneisen picture. But this steady increase in abruptly stopped when
ph-ph collisions cease to be resistive. The arrows in the two panels point to
the same temperature (adapted from ref. [48]).

##  V Phonon-electron hydrodynamics

The phonon and electron fluids can exchange momentum through Normal collisions
between these two types of quasi-particles. As a result, hydrodynamic effects
involving both fluids can emerge. One known example is phonon drag, a non-
diffusive thermoelectric response generated by heat-carrying phonons coupled
to the electron bath, which dominates the low temperature thermoelectric
response of many semiconductors [49].

Evidence for frequent phonon-electron momentum exchange with signatures in
thermal and electrical transport channels was uncovered by a study of
electrical and thermal transport in antimony (Sb) crystals of various sizes
[48]. Jaoui and co-workers found that the phonon thermal conductivity does not
show any size dependence, in contrast to the electronic thermal and electrical
conductivities. How come that the mean free path of electrons increases with
the sample size, but not the mean free path of phonons? In other words, if
disorder is at work, why does it affects phonons but not electrons? It was
also found that phonon thermal conductivity displays prominent quantum
oscillations, implying that the evacuation of each Landau level affects the
phonon mean free path. Taken together, these facts point to a case of frequent
momentum-conserving collisions between electrons and phonons at cryogenic
temperatures. Coupled electron-phonon hydrodynamics has been a subject of
emerging theoretical interest [50, 51, 52].

Fig.6a shows the temperature dependence of the phonon mean free path in
several crystals of antimony of various sizes. It shows a local maximum, like
in other cases of phonon Poiseuille flow. There is a clear size dependence
near this peak, but in addition to a high temperature intrinsic regime, there
is a second one at low temperature with no detectable size dependence in the
phonon mean free path. This is where the scattering of phonons by electrons
dominates the lattice heat transport.

It is interesting to look at the situation from the point of view of
electrons. At low temperatures, the electrical resistivity, as well as the
electronic thermal resistivity of antimony crystals, is dominated by electron-
electron scattering [53]. This leads to quadratic temperature dependence of
both, with a prefactor, which evolves with the sample size [53]. As a result,
the Lorenz number becomes size-dependent, as theoretically expected in the
case of electronic hydrodynamics [54].

The strong coupling between electrons and phonons leads to an abrupt breakdown
of the Bloch-Grüneisen picture [18] below 15 K. This is shown in Fig.6b, which
plots the temperature dependence of the exponent of the electrical resistivity
($\gamma$, with $\rho=\rho_{0}+AT^{\gamma}$). In this picture, $\gamma$ is
expected to smoothly evolve from $\gamma\simeq 1$ above the Debye temperature
to $\gamma\simeq 5$ at temperatures much below. In the high temperature
regime, all phonons are excited and electron-phonon scattering is essentially
elastic. At low temperatures, the phonon population diminishes and the typical
phonon wave-vector shrinks, pulling down the e-phonon cross section. This is
indeed what happens in noble metals, copper, silver or gold. In contrast, as
one can see in Fig.6b, in antimony, the steady increase in $\gamma$ is
suddenly interrupted in antimony when phonons do not suffer Umklapp collisions
and scattering by phonons ceases to decay the flow of electrons. Thus, there
is an asymmetry between the two components of the electron-phonon bifluid.
Momentum can leak from the electron bath, but not from the phonon bath.
Whenever an electron yields momentum to the phonon fluid, the electron fluid
is going to receive it back in another momentum exchange event. The only
source of inelastic dissipation is scattering between electrons.

##  VI Open questions

In the previous century, it was believed that the observation of phonon
Poiseuille flow required single crystals of exceptional purity. The findings
of the past few years indicate that this is not necessarily the case. When
Normal collisions are frequent enough, then even in the presence of disorder,
phonon hydrodynamics becomes detectable. Two remarkable cases are strontium
titanate and graphite. These two solids are far from being isotopically pure.
Moreover, and for different reasons, even in cryogenic temperatures, their
phonon mean free path remains well below the sample size. Therefore, their
ballistic regime is elusive. Nevertheless, not only one can detect signatures
of the Poiseuille flow, as we saw above, but also manifestations of the second
sound have been experimentally observed by optical probes in both materials
[55, 56, 57].

But what sets the amplitude of Normal collisions? Early works on graphene [6,
7] highlighted the quadratic dispersion of acoustic phonons, which amplify the
phonon density of states. More recently, hybridization between acoustic and
optical phonons have been observed in strontium titanium [41] and may play a
role in the amplification of momentum exchange between phonons. In the case of
column V elements, Bi, Sb and black P, one suspects the proximity of a
structural instability [30, 32] to be at work. One can see that the suspicions
go along different directions and one may wonder if a common ground exists
between them.

Another open question is the possible link between phonon hydrodynamics and
the phonon thermal Hall effect. The reason why phonons in some insulators
generate a detectable thermal Hall response is a mystery. Is it a coincidence
that the two non-magnetic insulators, which display a thermal Hall effect are
strontium titanate [58] and black P [59], which were both found to host
Poiseuille flow of phonons? Future studies will tell.

One may also speculate on a link between this thematic and the recent
theoretical progress [60, 61] in unifying the pictures of heat transport in
crystals and glasses, which considering the wave nature of phonons and
diffusion across phonon branches.

##  VII Acknowledgements

This work was supported in France by JEIP-Collège de France, by the Agence
Nationale de la Recherche (ANR-19-CE30-0014-04), a grant by the Ile de France
regional council, and Fundação de Amparo à Pesquisa do Estado de São Paulo
(2018/19420-3).

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