Phonon hydrodynamics in two-dimensional materials

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The conduction of heat in two dimensions displays a wealth of fascinating phenomena of key relevance to the scientific understanding and technological applications of graphene and related materials. Here, we use density-functional perturbation theory and an exact, variational solution of the Boltzmann transport equation to study fully from first-principles phonon transport and heat conductivity in graphene, boron nitride, molybdenum disulphide and the functionalized derivatives graphane and fluorographene. In all these materials, and at variance with typical three-dimensional solids, normal processes keep dominating over Umklapp scattering well-above cryogenic conditions, extending to room temperature and more. As a result, novel regimes emerge, with Poiseuille and Ziman hydrodynamics, hitherto typically confined to ultra-low temperatures, characterizing transport at ordinary conditions. Most remarkably, several of these two-dimensional materials admit wave-like heat diffusion, with second sound present at room temperature and above in graphene, boron nitride and graphane.

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### ARTICLE

Received 3 Oct 2014 | Accepted 26 Jan 2015 | Published 6 Mar 2015 DOI: 10.1038/ncomms7400
## Phonon hydrodynamics in two-dimensional materials

Andrea Cepellotti 1,2, *, Giorgia Fugallo 1,3, *, Lorenzo Paulatto 3, Michele Lazzeri 3, Francesco Mauri 3

& Nicola Marzari 1,2

The conduction of heat in two dimensions displays a wealth of fascinating phenomena of

key relevance to the scientific understanding and technological applications of graphene

and related materials. Here, we use density-functional perturbation theory and an exact,

variational solution of the Boltzmann transport equation to study fully from first-principles

phonon transport and heat conductivity in graphene, boron nitride, molybdenum disulphide

and the functionalized derivatives graphane and fluorographene. In all these materials, and at

variance with typical three-dimensional solids, normal processes keep dominating over

Umklapp scattering well-above cryogenic conditions, extending to room temperature and

more. As a result, novel regimes emerge, with Poiseuille and Ziman hydrodynamics, hitherto

typically confined to ultra-low temperatures, characterizing transport at ordinary conditions.

Most remarkably, several of these two-dimensional materials admit wave-like heat diffusion,

with second sound present at room temperature and above in graphene, boron nitride and

graphane.

1 Theory and Simulations of Materials (THEOS), E´cole Polytechnique Fe´de´rale de Lausanne, Station 12, 1015 Lausanne, Switzerland. 2 National Center for
Computational Design and Discovery of Novel Materials (MARVEL), E ´ cole Polytechnique Fe´de´rale de Lausanne, Station 12, 1015 Lausanne, Switzerland.
3 Institut de Mine´ralogie, de Physique des Mate´riaux, et de Cosmochimie (IMPMC), Sorbonne Universite´s, UPMC University Paris 06, UMR CNRS 7590,
Muse´um National d’Histoire Naturelle, IRD UMR 206, 4 Place Jussieu, F-75005 Paris, France. - These authors contributed equally to this work.
[Correspondence and requests for materials should be addressed to F.M. (email: francesco.mauri@impmc.jussieu.fr) or to N.M. (email: nicola.marzari@epfl.ch).](mailto:francesco.mauri@impmc.jussieu.fr)

he thermal conductivity of materials is a key property that
is attracting much interest both for its role and relevance to
thermoelectric waste heat recovery, and for the scientific
# T
questions and technological opportunities that are arising in the
field of low-dimensional materials (see, for example, refs 1,2 for a
review). When dimensionality is reduced or the relevant sizes
reach microscopic scales (from tens of nanometers to tens of
microns), the knowledge developed in the past on conventional
bulk crystals becomes insufficient, and novel properties emerge.
The prototypical two-dimensional (2D) material, graphene, has
the highest known thermal conductivity (even if there is to-date
no complete consensus on its actual value 1 ), and unexpected
phenomena have been observed experimentally, with, for
example, a size dependence in the thermal conductivity for
samples reaching the micron scale 3 . This is consistent with the
theoretical prediction that heat carriers in graphene can
propagate for distances up to the millimetre scale 4 . The origin
of such behaviour is under active investigation and much debated,
with arguments centring around dimensionality effects 3,5,6 and
the reduction of the scattering phase space 7 .

It has been noted that for graphene and boron nitride 4,7–9 it is
essential to consider the exact solution of the Boltzmann
transport equation (BTE) when studying thermal conductivity,
and that the widely used single-mode relaxation time
approximation (SMA; where individual phonons thermalize
independently, without collisions repopulating them) fails in
describing correctly thermal transport. This failure arises because
the simultaneous interaction of all phonon populations is crucial,
and such collective behaviour 4,10 can lead to the emergence of
composite excitations as the leading heat carriers.

Such collective behaviour is driven by the dominance of
normal (that is, heat-flux conserving) phonon scattering events,
which allow the phonon gas to conserve to a large extent its
momentum before other resistive scattering mechanisms can
dissipate it away. A phonon gas in such state is said to be in the
hydrodynamic regime, because of the similarity with the case of
an ideal gas where particles scatter without dissipating momentum. Phonons in the hydrodynamic regime can, under specific
conditions, form packets that alter the typical diffusive behaviour
of heat and make it propagate as a damped wave, giving origin to
the phenomenon of second sound, where a localized heating
perturbation generates two sound wavefronts when probed at a
certain distance 11,12 . This phenomenon has not been extensively
studied, as to-date it was possible to observe it only in
few materials at cryogenic temperatures (10 K or less): solid
helium 13, sodium fluoride 11,14, bismuth 12, strontium titanate 15,16
(theoretically it is also been hypothesized for diamond 17 ).

In this work, we study the lattice thermal conductivity
of graphene, graphane, boron nitride, fluorographene and
molybdenum disulphide by solving the linearized BTE using a
variational approach devised by some of us 18,19 that is
particularly robust in converging to the exact solution of the
BTE under all possible conditions (iterative solutions have limited
domains of convergence 20–22 and this is particularly challenging
in 2D materials 4,7, because of the collective nature of the
excitations). In addition, all calculations are parameter free, as
scattering rates and lifetimes for phonon–phonon interactions
and for isotopic elastic scattering are computed using
phonon frequencies and phonon–phonon interactions from
density-functional pertubation theory 23–28, in a reciprocal-space
formulation able to deal with any arbitrary wavevector 29 . Isotopic
concentrations of the five materials use the natural abundances
(see the Methods section for details).

Remarkably, we show that all five materials considered fall into
the hydrodynamic regime even at room temperature or above it,
and we highlight the persistence of second sound in graphene,

individual phonon lifetimes: k ¼ NV1 n n�nðn�n þ 1Þu n 2o2 n tn.

The comparison between the SMA thermal conductivity as a
function of temperature and the exact solution of the BTE is
reported in Fig. 1a for all five materials considered here, and the
qualitative failure of the SMA at all temperatures becomes
immediately apparent. This failure has already been highlighted,
for example, for the case of graphene and boron nitride, using
either first-principles scattering rates for graphene 4,7 or empirical
potentials 8,9 . As expected, we find that such failure occurs also
in graphane and molybdenum disulphide, and, by a smaller
amount, in fluorographene, making it common in 2D materials.
Interestingly, errors affect also qualitative predictions: for

graphane and boron nitride at temperatures well-above cryogenic.
In graphene at room temperature, second sound propagates over
distances of the order of 1 mm, hinting at the possibility of
devising thermal devices based on coherent propagation 30 . Last,
we also provide validation for the Callaway model of thermal
conductivity, developed for heat transport at very low
temperatures, and shown here to be valid for these 2D
materials up to room temperature and above.

Results
Boltzmann transport equation. The thermal conductivity k is
the quantity, which relates the heat flux Q with a gradient of
temperature rT, so that Q ¼ - krT; for simplicity, the thermal
conductivity, which in general is a tensor, is written as a scalar
property, as we are considering only the in-plane conductivity,
that is, isotropic. To express k as a function of microscopic
quantities, one introduces the out-of-equilibrium phonon
distribution nv (n ¼ q, s labels collectively the Brillouin zone
wavevectors and phonon branches). For small perturbations, nv is
linearized around the Bose–Einstein thermal equilibrium distribution n�n ¼ expð‘on=1kBTÞ � 1 so that nn ¼ �nn þ �nnðn�n þ 1ÞrTFn,
and all deviations from equilibrium are expressed by the set of
functions Fv. Using the microscopic expression of the heat
flux 31,32, k can be expressed in terms of microscopic quantities as
1 P
k ¼ NV n n�nðn�n þ 1Þ‘onunFn, where ‘on is the energy of the

k ¼ NV1 n n�nðn�n þ 1Þ‘onunFn, where ‘on is the energy of the

phonon n, un is the projection of the phonon group velocity on the
direction parallel to rT and NV is the volume of the crystal.
Thus, k depends on harmonic quantities ðn�n; un; onÞ, which can
be computed routinely and inexpensively even from firstprinciples 23,24, and the challenge is to find the Fv deviations
from equilibrium.

To obtain nv, we solve the linearized BTE 18

vnrT @n�n

@T ¼

n 0

On;n 0 nn 0 ; ð1Þ

where the scattering operator is represented by a matrix of
scattering rates On;n 0 acting on the phonon populations nv. We
focus here mostly on intrinsic processes and isotopic disorder, so
the scattering matrix is built using three-phonon processes and
isotopic elastic rates as detailed in refs 19,26; extrinsic sources of
scattering (typically, finite sizes) are discussed in connection with
the Poiseuille thermal conductivity regime.

For simplicity, the BTE is often solved in the SMA, which relies
on the assumption that heat-current is dissipated every time a
phonon undergoes a scattering event 18 . Therefore, the phonon
distribution relaxes towards thermal equilibrium at a rate which is
given by the phonon lifetime tv, the average time between
phonon scattering events at equilibrium. The scattering operator
is thus approximated as

n 0

On;n 0 nn 0 �� nn � n�n

; ð2Þ

and the SMA thermal conductivity depends directly on the
1 P
individual phonon lifetimes: k ¼ NV n n�nðn�n þ 1Þu n 2o2 n tn.

n 2o2 n

n n�nðn�n þ 1Þu n 2

10 4

10 3

10 2

10 4

10 3

10 2

Graphene

200 400 600 800
Temperature (K)

Graphane

200 400 600 800

Temperature (K)

Boron nitride

200 400 600 800

Temperature (K)

Fluorographene

200 400 600 800

Temperature (K)

Molybdenum disulphide

200 400 600 800
Temperature (K)

Graphene
Graphane
Boron nitride
Fluorographene
Molybdenum disulphide

200 400 600 800

Temperature (K)

200

Graphene
Graphane
Boron nitride
Fluorographene
Molybdenum disulphide

400 600 800

Temperature (K)

Figure 1 | Thermal conductivity as a function of temperature. Ab-initio
estimates of the thermal conductivity k as a function of temperature for
infinite suspended sheets, at natural isotopic abundances. Two
approximations are tested against the exact solution of the first-principles
phonon Boltzmann transport equation: (a, top panel) the single-mode
relaxation time approximation (SMA); (b, bottom panel) Callaway’s model.

example, the SMA solution predicts graphane to have a
conductivity larger than graphene at room temperature.

Poiseuille and Ziman hydrodynamics. To make further progress, it is very instructive to inspect the scattering rates for the
different processes involved, that is, three-phonon processes
(divided into normal (N ) and Umklapp (U) events) and isotopic
scattering (I ). For N processes, total momentum is conserved
and all three phonon wavevectors belong to the first Brillouin
Zone, whereas in U processes momentum is not conserved, and
the sum of the three wavevectors corresponds to a non-zero
reciprocal lattice vector. Such distinction is key for heat transport,
as it has been long known 18 that N processes do not actively
dissipate heat-current, as a consequence of energy and
momentum conservation, and U processes act as the source of
intrinsic dissipation.

We show in Fig. 2 the average linewidths for all these processes,
defined as

10 1

10 0

10 –1

10 –2

10 –3

10 1

10 0

10 –1

10 –2

10 –3

10 1

10 0

10 –1

10 –2

10 –3

10 1

10 0

10 –1

10 –2

10 –3

10 1

10 0

10 –1

10 –2

10 –3

Cn2p=t i n

~~P~~

�
- - 2p
G i ¼ ¼
t i

n

; ð3Þ

Figure 2 | Average phonon linewidths. Average linewidths of normal (N ),
Umklapp (U) and isotopic (I) scattering processes in infinite suspended
sheets of graphene (a), graphane (b), boron nitride (c), fluorographene
(d) and molybdenum disulphide (e); I processes are calculated at natural
isotopic abundances. N scattering in all these two-dimensionalmaterials is
clearly dominant.

sources of scattering are negligible. Quite surprisingly, it is seen
here that not only N processes are very relevant, as argued in
refs 8,29, but also that they represent the dominant scattering

where i labels either N, U or I processes and Cn ¼

n�nðn�n þ 1Þ ð‘ kB o T n2Þ2 is the specific heat of the phonon mode n (this

weight is the same appearing in the second-sound velocity shown
later). Conventionally, it is expected that N events become
dominant only at very low temperatures, when U processes freeze
out 18, and even then only provided that I rates or other extrinsic

200 400 600 800

Temperature (K)

Table 1 | Classification of thermal transport regimes.

Ballistic E �N and E �R
Poiseuille N �E �R
Ziman N �R �E
Kinetic R �N and R �E

Classification of different regimes of thermal conductivity as a function of the linewidths of
different scattering events: normal (N ), resistive (R—combining both Umklapp and isotopic)
and extrinsic (E). Poiseuille and Ziman hydrodynamics are characterized by dominant N
scattering against all other mechanisms.

mechanisms in these 2D materials at any temperature (with the
exception of natural-abundance molybdenum disulphide and
boron nitride, where I is comparable to N because of the large
isotopic disorder of molybdenum and boron).

The predominance of normal scattering processes identifies the
regime(s) under which the phonon gas is defined to be in the
hydrodynamic conditions mentioned above. We nevertheless
underscore that, even if the momentum-conserving nature of N
processes prevents them from dissipating heat-current, they still
affect the total thermal conductivity by altering the out-ofequilibrium phonon distribution. Last, we highlight that the
failure of the SMA in these conditions is a rather conceptual one,
as it is the assumption that heat flux is dissipated at every
scattering event that becomes invalid. Instead, given that most
processes are normal, heat flux is just being shuttled between
phonon modes.

To compare these 2D materials with conventional solids, we
summarize in Table 1 the different regimes of phonon transport
according to the nomenclature of Guyer 33 . In the ballistic regime,
the dominanant scattering events are extrinsic (E), and are
typically due to the finite size of the sample (line defects, surfaces,
grain boundaries); in the Poiseuille regime, normal (N ) processes
dominate, and the heat flux is dissipated by E events (this is the
first hydrodynamic regime where the gas of phonons is ‘normal’
but it feels the ‘walls’ of the container). In the Ziman regime, N
events still dominate, but the heat flux is now dissipated by
resistive R scattering (either Umklapp (U) or isotopic (J ))—in
this regime, the ‘walls’ have become irrelevant for the normal gas.
Finally, in the kinetic regime intrinsic resistive processes R
(again, typically U or J ) have the highest linewidth, the sample
size is much larger than all typical mean free paths and N events
have become negligible.

It becomes instructive then to compare graphene, taken as
reference, with a typical three-dimensional solid, like silicon or
germanium. We show in Fig. 3 the thermal conductivity of
graphene both in the case in which we have an infinite sample
(kN) or a finite ribbon (here of width L ¼ 100 mm, kL), with an
extrinsic scattering rate given by n�nðn�n þ 1Þ v L n (ref. 22),
considering this also in the ballistic limit k ballistic L obtained by

removing all the internal sources of scattering N, U and J, but
preserving the extrinsic scattering E. The comparison of k ballistic L

with the exact thermal conductivity kL in a ribbon of width L
shows that the ballistic regime is not a good approximation to kL.
In fact, the introduction of N scattering events, that is,
hydrodynamic conditions, enhances the thermal conductivity
well above the ballistic limit. As the thermal conductivity grows
with temperature, graphene is in the Poiseuille regime, where N
events facilitate the heat flux, and extrinsic sources of scattering
dissipate it (the term Poiseuille comes from an analogy 33 between
the flow of phonons in a material with the flow of a fluid in a
pipe). The thermal conductivity though does not grow
indefinitely with temperature, because of the limit imposed by
the intrinsic thermal conductivity of an infinite sheet kN. Thus,
there is a thermal conductivity peak, and for higher temperatures,
the heat flux is mostly dissipated by intrinsic sources (U þ J ). As

where the Matthiessen rule (that is, arithmetic sum of the
scattering rates for independent events) is used t 1R n ¼ t 1U n þ t 1I n : As

N scattering rates are very large, an out-of-equilibrium
distribution will mostly decay first into the drifting distribution,
and from this state it will relax towards static equilibrium. The
results for the thermal conductivity in Callaway’s approximation
are shown in Fig. 1b, and compared again against the results of
the exact solution of the BTE (more details on the expression
of Callaway’s thermal conductivity can be found in the

20,000

15,000

10,000

5,000

0
0

_k_ ∞
_kL_

_kL_ ballistic

Graphene

Representative

3D solid

10 20 30

Temperature (K)

Figure 3 | Thermal transport regimes in two- and three-dimensional (3D)
materials. Thermal conductivity of a graphene ribbon of width L ¼ 100 mm
(kL, solid line) together with its ballistic limit (see text; kL ballistic, dashed), and

of an infinite graphene sheet (kN, dashed line). Inset: thermal conductivity
regimes in a standard 3D solid (for example, silicon or germanium), where
peak conductivity is obtained at cryogenic temperatures and the
hydrodynamic conditions, if present, are confined around those
temperatures.

the N processes have the largest linewidth at any temperature,
the regime at temperatures higher than the peak is still
hydrodynamic, but it shifts to Ziman, where the intrinsic
resistive processes R set the typical length scales, and the
boundaries become irrelevant. Last, if the intrinsic events
R ðU þ J Þ had the largest linewidth, the thermal conductivity
would be in the kinetic regime: this is the regime of conventional
materials at ordinary temperatures, but it is not ever reached in
these 2D cases.

In conclusion, the two major differences between these 2D
materials and a conventional solid are (i) in the temperature
scales involved (the non-kinetic regimes in a conventional solid
are all condensed in narrow energy window around cryogenic
conditions, and hydrodynamic regimes may not be present at all),
and (ii) in the fact that even at the highest temperatures normal
processes remain dominant, restricting conductivities to just the
hydrodynamic regimes of Poiseuille and Ziman.

Callaway’s model. Callaway’s model is an approximation to the
scattering matrix that was originally developed in order to
describe the thermal conductivity of germanium at very low
temperatures (20 K or so) 34 . It had previously been shown by
Klemens 35 that the most probable boson distribution in a system
where momentum is conserved is given by the drifting
distribution n drift n ¼ expðð‘on � q1�VÞ=kBTÞ � 1 , where V is a Lagrange

multiplier enforcing momentum conservation. Therefore, N
processes will tend to relax the phonon population nv towards
n drift n . Instead, resistive processes ðR ¼ U þ J Þ tend to relax the

phonon populations towards local thermal equilibrium (that is,
the Bose–Einstein distribution n�n), just as described by the SMA.
Therefore, Callaway approximated the scattering operator as

n 0

n

On;n 0 nn 0 �� nn � ndrift

- nn � n�n
t R n

n

n

t N n

; ð4Þ

t 1R n ¼ t 1U n

t 1U n þ t 1I n

100

Graphene
Graphane
Boron nitride
Fluorographene
Molybdenum disulphide

200 400 600 800
Temperature (K)

Graphene
Graphane
Boron nitride
Fluorographene
Molybdenum disulphide

200 400 600 800
Temperature (K)

10 2

10 1

10 0

10 –1

10 –2

Figure 4 | Second-sound phonon component. Projection of the
out-of-equilibrium phonon distribution onto the drifting distribution
(percentage values).

Supplementary Note 1 and Supplementary Fig. 1). It can be seen
that the model broadly reproduces the exact solutions, even if, for
example, in fluorographene it overestimates the exact results, or
in molybdenum disulphide underestimates them. So, even more
than providing simple but fairly accurate estimates of the phonon
distribution without the need of solving the BTE, the value of the
model is in the clear physical insight it offers on the microscopic
nature of heat transport, allowing us to conclude
not only that the dominant presence of N processes at all
temperatures characterizes heat transport in these materials, but
also that the distribution function can be broadly described as a
drifting distribution according to the prescriptions of Callaway
and Klemens 34,35 .

Second sound. Materials in hydrodynamic regimes can host
second sound 11–16, and is thus interesting to discuss its existence
or characteristics in two dimensions. We define second sound,
following ref. 36, as the propagation of heat in the form of a
damped wave. In a conventional material, one expects only
diffusive propagation; it is the ‘second’ heat-wave transport that
gives the name to this phenomenon.

In principle, more than one microscopic mechanism could lead
to the formation of second sound 36 ; the common requirement is
that a mechanism exists, which is causing a slow decay of the heat
flux. This can indeed happen in case of a large presence of N
events. As mentioned before, the conservation of momentum
drives nv towards nn drift, and once this drifting distribution n drift n

has accumulated, heat propagates as a wave that is eventually
damped on longer timescales by the resistive processes that relax
the phonons to the equilibrium Bose–Einstein distribution, and
determine diffusive heat transport at longer lengths.

To observe second sound, the out-of-equilibrium distribution
needs to be in the form and well described by the drifting
distribution. We have already shown in the previous section that
its introduction in the scattering operator provides a good
estimate of the thermal conductivity. To see whether the out-ofequilibrium distribution closely resembles the drifting one, we
first expand the drifting distribution in a Taylor series, so that it is
simply proportional to the projection of the wavevector parallel to
the temperature gradient (see Supplementary Note 2). Then, we
define the drifting component as:

n drift, and once this drifting distribution n drift n

Figure 5 | Second-sound scattering lengths. Characteristic propagation
length of second sound, describing the wave-like propagation of heat.
Dotted lines are used for the materials where second sound has a low
probability of being observed. We note that for graphene and graphane at
room temperature the typical decay length of second sound is of the order
of microns.

materials considered in Fig. 4 (as a percentage), and we observe
that more than 40% of the phonon distribution of graphene at
room temperature is determined by the drifting distribution, and
that this ratio grows at lower temperatures; this ratio is slightly
smaller in all other materials, falling below 30% for fluorographene and 10% molybdenum disulphide. Therefore, in graphene,
graphane and boron nitride, a large fraction of the heat flux is
carried by the drifting distribution (which is a good approximation of nv for the study of the heat flux propagation), whereas in
fluorographene and molybdenum disulphide, the drifting distribution is a worse descriptor of the out-of-equilibrium
distribution, as also hinted by the reduced effectiveness of the
Callaway model.

When the drifting distribution is a good approximation to the
out-of-equilibrium distribution, the material will display second
sound. The second-sound heat wave is characterized by a velocity
vss and a relaxation time tss that define a second-sound length
lss ¼ vss tss, that is, the characteristic distance that the heat wave
propagates before decaying. It is possible to show (details are
reported in the Supplementary Note 3) that, approximating the
out-of-equilibrium distribution with the drifting distribution and
using the Callaway approximation for the scattering operator, the
BTE can be rewritten as a wave-like equation for the temperature
profile, characterized by the following quantities:

Cn vn 2 �vn

ðvssÞ 2 ¼

n

2
~~P~~

; ð6Þ

n�nðn�n þ 1Þ‘onvn i qi=tR n

1
tss

n
~~P~~

n�nðn�n þ 1Þ‘onvn i qi

n

: ð7Þ
n�nðn�n þ 1Þ‘onvn i qi

CnFnq==
~~f~~ f i ~~f~~ f i ~~f~~ f i ~~f~~ f i ~~f~~ n f i ~~f~~ f i ~~f~~ f i ~~f~~ ~~rf~~ f i ~~f~~ f i ~~f~~ f i f i f i f i f i f i
~~P~~ ~~P~~

CnF 2

f i f i f i f in f i f i f i

~~rf~~ f i ~~f~~ f i ~~f~~ f i ~~f~~ f i ~~f~~ n f i ~~f~~ f i ~~f~~ f i ~~f~~ ~~rf~~ f i ~~f~~ f i ~~f~~ f i ~~f~~ f i ~~f~~ f i ~~f~~ f i ~~f~~ f i ~~f~~ f i ~~f~~

~~P~~ ~~P~~

CnFn 2 Cnq 2 ==

The second-sound relaxation times—reported in the
Supplementary Fig. 2—indicate that the energy flux dissipation
decays on average on a time scale of the order of hundred
picoseconds at room temperature for the three materials of higher
conductivity. The second-sound velocities depend only on
harmonic properties, and can be easily computed—they are
reported in the Supplementary Fig. 3 as a function of
temperature, and compared with the average velocity of acoustic
phonons. We find that the different vss typically lay in between
the larger average velocities of the longitudinal and tranverse
acoustic branches, and the much lower average velocity of the
out-of-plane acoustic mode.

f i f i f i f i f i f i f i

CnFn 2

~~rf~~ f i ~~f~~ f i ~~f~~ f i ~~f~~ f i ~~f~~ f i ~~f~~ f i ~~f~~ f i ~~f~~ f i ~~f~~ ; ð5Þ

~~P~~

Cnq 2 ==

f i f i f i f i f i f i f i f i

Cnq 2

which is equal to one if the out-of-equilibrium distribution is
equal to the drifting distribution. We plot this quantity for all

The propagation lengths lss ¼ tss vss for the second-sound wave
for graphene and graphane reach the micron scale, even at room
temperature, whereas boron nitride is characterized by secondsound lengths of a fraction of a micron (the values of lss for
molybdenum disulphide and fluorographene are shown in Fig. 5
for completeness, even if those materials do not host second
sound). At lower temperatures, where U processes tend to freeze
and only I are present, damping of the second sound becomes
less effective and the heat wave propagates over longer distances.

As mentioned, the presence of second sound has been verified
experimentally in the past 11–15 in 3D materials at cryogenic
temperatures, and it is conveniently studied as the response of a
material to a temperature pulse. One should then be able to
observe at a distance standard pulses due to the diffusive
propagation by the longitudinal, tranverse and out-of-plane
acoustic modes; in addition, the second-sound signature will
appear as a further peak due to the formation of the drifting
distribution. One should also always consider the size of the
experimental setup, as this will affect phonons, which travel
ballistically, that is, without scattering between the heat source
and the detector, and that would diminish the component of heat
travelling in the other modes. Their effect on second sound
should be negligible as long as the distance between pump and
probe is larger than lss. Finally, let us note that the estimate of lss
is obtained here as a statistical average, and thus we cannot
exclude the existence of tails of the second-sound mode
propagating at much longer distances.

Discussion
We have investigated heat transport in some key 2D materials—
graphene, boron nitride, molybdenum disulphide and the
functionalized derivatives graphane and fluorographene. We have
found that momentum-conserving normal processes are the
dominant scattering mechanisms at all temperatures, requiring an
exact solution of the linearized BTE for qualitative and
quantitative accuracy, and confirming that graphene provides
the highest thermal conductivity among all materials considered.
Heat transport in all cases falls into the conditions of Poiseuille
and Ziman hydrodynamics (below and above the conductivity
peak, respectively), never reaching the ‘ordinary’ conditions of
diffusive transport even at very high temperatures. The Callaway
model, developed originally for the study of heat transport in
conventional three-dimensional solids in the hydrodynamic
regimes that could arise at cryogenic temperatures, provides a
broadly accurate description of heat transport in two dimensions
at all temperatures. Moreover, its microscopic picture of two
relaxation time scales, where the out-of-equilibrium distribution
accumulates first into a momentum-conserving drifting distribution before relaxing into the final Bose–Einstein equilibrium,
provides both microscopic insight and understanding. Graphene,
graphane and boron nitride all display second sound, that is, a
component in heat transport that behaves as a damped wave and
is not described by diffusive propagation, with length scales that
reach 1 mm for graphene at room temperature.

Methods
First-principles simulations. Second- and third-order force constants have been
calculated using density-functional perturbation theory 23,27,28,37 as implemented in
the Quantum ESPRESSO distribution 38, using the local-density approximation,
norm-conserving pseudopotentials from the PSlibrary 39 and a plane-wave cutoff of
90 Ry. The electronic-structure calculations for graphene (G), boron nitride (BN),
fluorographene (CF) and graphane (CH) use a Gamma-centred Monkhorst-Pack
Brillouin zone sampling of 24 � 24 � 1 k-points, and 16 � 16 � 1 k-points for
MoS2. All systems are simulated in a supercell geometry, relaxing the in-plane
lattice parameter a to equilibrium, and with interlayer distances of 7 Å for all
materials except MoS2, where it is taken to be 18 Å. For graphene, that is a
semimetal, we use a Methfessel-Paxton cold smearing of 0.02 Ry. For all materials,

but not for MoS2, the harmonic and anharmonic force constants have been
computed on meshes of 16 � 16 � 1 and 4 � 4 � 1 q-points in the Brillouin zone,
respectively, whereas for MoS2 they have been computed on 16 � 16 � 1 and
6 � 6 � 1, respectively.

Thermal conductivity simulations. The BTE is solved using the variational
method of ref. 19, finding well-converged results for a mesh of 128 � 128 � 1
q-points and a Gaussian broadening of 10 cm � 1, except for the case of MoS2,
where a mesh of 140 � 140 � 1 q-points and a broadening of 1 cm � 1 has been
used. The isotopic concentrations are chosen to be the natural ones 40 : hydrogen of
99.9885% 1 H, 0.0115% 2 H; carbon of 98.93% 12 C, 1.07% 14 C; boron of 19.9% 10 B,
80.1% 11 B, nitrogen of 99.632% 14 N, 0.368% 15 N; fluorine of 100% 19 F; sulphur of
94.93% 32 S, 0.76 33 S, 4.29 34 S, 0.02% 36 S; molybdenum of 14.84% 92 Mo, 9.25%
94Mo, 15.92% 95Mo, 16.68% 96Mo, 9.55% 97Mo, 24.13% 98Mo, 9.63% 100Mo.
The numerical results presented in the text are normalized by the volume of the
unit cell of the crystal; the thickness in the perpendicular direction is chosen as the
experimental one of the corresponding 3D material, using c/a ratios of 1.367 for
graphene and graphane, 1.317 for boron nitride, 1.206 for fluorographene and
1.945 for MoS2.
All calculations have been managed using the AiiDA materials’ informatics
platform 41 .

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Acknowledgements
We gratefully acknowledge support from the Swiss National Science Foundation (AC),
the Swiss National Supercomputing Center CSCS (A.C., N.M.), the ANR project
Accattone (G.F., L.P., M.L. and F.M.) and the EU Graphene Flagship (F.M., N.M.).

Author contributions
A.C. and G.F. carried out all the simulations, using software written by A.C., G.F., L.P.
and M.L.; A.C., G.F., F.M. and N.M. conceived the paper, and A.C. and N.M. wrote it.

Additional information
Supplementary Information accompanies this paper at [http://www.nature.com/](http://www.nature.com/naturecommunications)
[naturecommunications](http://www.nature.com/naturecommunications)

Competing financial interests: The authors declare no competing financial interests.

Reprints and permission information is available online at [http://npg.nature.com/](http://npg.nature.com/reprintsandpermissions/)
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How to cite this article: Cepellotti, A. et al. Phonon hydrodynamics in two-dimensional
materials. Nat. Commun. 6:6400 doi: 10.1038/ncomms7400 (2015).

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