Phonon hydrodynamics in two-dimensional materials

Abstract

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.

ARTICLE

Received 3 Oct 2014 | Accepted 26 Jan 2015 | Published 6 Mar 2015

The conduction of heat in two dimensions displays a wealth of fascinating phenomena of key relevance to the scientic 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 rst-principles phonon transport and heat conductivity in graphene, boron nitride, molybdenum disulphide and the functionalized derivatives graphane and uorographene. 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 conned 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.

DOI: 10.1038/ncomms7400

Phonon hydrodynamics in two-dimensional materials

Andrea Cepellotti1,2,*, Giorgia Fugallo1,3,*, Lorenzo Paulatto3, Michele Lazzeri3, Francesco Mauri3

& Nicola Marzari1,2

1 Theory and Simulations of Materials (THEOS),cole Polytechnique Fdrale de Lausanne, Station 12, 1015 Lausanne, Switzerland. 2 National Center for Computational Design and Discovery of Novel Materials (MARVEL),cole Polytechnique Fdrale de Lausanne, Station 12, 1015 Lausanne, Switzerland.

3 Institut de Minralogie, de Physique des Matriaux, et de Cosmochimie (IMPMC), Sorbonne Universits, UPMC University Paris 06, UMR CNRS 7590, Musum National dHistoire 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: mailto:[email protected]

Web End [email protected] ) or to N.M. (email: mailto:[email protected]

Web End [email protected] ).

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The 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 scientic

questions and technological opportunities that are arising in the eld 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 insufcient, 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 value1), and unexpected phenomena have been observed experimentally, with, for example, a size dependence in the thermal conductivity for samples reaching the micron scale3. This is consistent with the theoretical prediction that heat carriers in graphene can propagate for distances up to the millimetre scale4. The origin of such behaviour is under active investigation and much debated, with arguments centring around dimensionality effects3,5,6 and the reduction of the scattering phase space7.

It has been noted that for graphene and boron nitride4,79 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 behaviour4,10 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-ux 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 specic 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 distance11,12. 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 helium13, sodium uoride11,14, bismuth12, strontium titanate15,16 (theoretically it is also been hypothesized for diamond17).

In this work, we study the lattice thermal conductivity of graphene, graphane, boron nitride, uorographene and molybdenum disulphide by solving the linearized BTE using a variational approach devised by some of us18,19 that is particularly robust in converging to the exact solution of the BTE under all possible conditions (iterative solutions have limited domains of convergence2022 and this is particularly challenging in 2D materials4,7, because of the collective nature of the excitations). In addition, all calculations are parameter free, as scattering rates and lifetimes for phononphonon interactions and for isotopic elastic scattering are computed using phonon frequencies and phononphonon interactions from density-functional pertubation theory2328, in a reciprocal-space formulation able to deal with any arbitrary wavevector29. Isotopic concentrations of the ve materials use the natural abundances (see the Methods section for details).

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

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 propagation30. 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.

ResultsBoltzmann transport equation. The thermal conductivity k is the quantity, which relates the heat ux Q with a gradient of temperature rT, so that Q 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 q, s labels collectively the Brillouin zone

wavevectors and phonon branches). For small perturbations, nv is

linearized around the BoseEinstein thermal equilibrium distribution

nn

1

exp on=kBT 1 so that nn

nn

nn

nn 1rTFn,

and all deviations from equilibrium are expressed by the set of functions Fv. Using the microscopic expression of the heat ux31,32, k can be expressed in terms of microscopic quantities as k

1 NV

Pn nn nn 1 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

nn; un; on, which can

be computed routinely and inexpensively even from rst-principles23,24, and the challenge is to nd the Fv deviations from equilibrium.

To obtain nv, we solve the linearized BTE18

vnrT

@

nn@T X

n0

On;n0 nn0 ; 1

where the scattering operator is represented by a matrix of scattering rates On;n0 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, nite 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 event18. 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

Xn0On;n0 nn0 nn nntn ; 2

and the SMA thermal conductivity depends directly on the individual phonon lifetimes: k

1 NV

Pn nn nn 1u2no2ntn.

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 ve 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 rst-principles scattering rates for graphene4,7 or empirical potentials8,9. As expected, we nd that such failure occurs also in graphane and molybdenum disulphide, and, by a smaller amount, in uorographene, making it common in 2D materials. Interestingly, errors affect also qualitative predictions: for

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a

101

100

104

k (W m1 K1 )

GrapheneGraphaneBoron nitride Fluorographene Molybdenum disulphide

Exact

SMA

[afii9796] (THz)

101

103

102

Graphene

103

400

800

102

101

200

400

200 600Temperature (K)

800

600 Temperature (K)

100

b

GrapheneGraphaneBoron nitride Fluorographene Molybdenum disulphide

Exact

Callaway

[afii9796] (THz)

101

104

k (W m1 K1 )

102

Graphane

103

400

200 600

Temperature (K)

800

103

101

102

100

400

800

[afii9796] (THz)

101

102

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

200 600Temperature (K)

Boron nitride

103

400

200 600

Temperature (K)

800

101

100

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 rst 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 known18 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, dened as

Gi

2p ti

[afii9796] (THz)

101

102

Fluorographene

103

400

200 600

Temperature (K)

800

101

100

[afii9796] (THz)

101

102

Molybdenum disulphide

103

400

P

200 600Temperature (K)

800

n Cn2p=tin

Pn Cn; 3

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

nn

nn 1 on

2

Figure 2 | Average phonon linewidths. Average linewidths of normal (N ),

Umklapp (U) and isotopic (I) scattering processes in innite suspended

sheets of graphene (a), graphane (b), boron nitride (c), uorographene (d) and molybdenum disulphide (e); I processes are calculated at natural

isotopic abundances. N scattering in all these two-dimensionalmaterials is

clearly dominant.

kBT2 is the specic 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

out18, and even then only provided that I rates or other extrinsic

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

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20,000

Table 1 | Classication of thermal transport regimes.

Ballistic E N and E R

Poiseuille N E R

Ziman N R E

Kinetic R N and R E

Classication of different regimes of thermal conductivity as a function of the linewidths of different scattering events: normal (N ), resistive (Rcombining both Umklapp and isotopic)

and extrinsic (E). Poiseuille and Ziman hydrodynamics are characterized by dominant N

scattering against all other mechanisms.

k kL

kLballistic

Representative 3D solid

k

Ballistic

Poiseuille

Ziman

Kinetic

10 20

30

Temperature (K)

Graphene

15,000

k (W m1K1)

10,000

5,000

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 identies the regime(s) under which the phonon gas is dened 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-of-equilibrium 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 ux is dissipated at every scattering event that becomes invalid. Instead, given that most processes are normal, heat ux 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 Guyer33. In the ballistic regime, the dominanant scattering events are extrinsic (E), and are

typically due to the nite size of the sample (line defects, surfaces, grain boundaries); in the Poiseuille regime, normal (N ) processes

dominate, and the heat ux is dissipated by E events (this is the

rst 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 ux 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 innite sample (kN) or a nite ribbon (here of width L 100 mm, kL), with an

extrinsic scattering rate given by

nn

nn 1 vnL (ref. 22),

considering this also in the ballistic limit kballisticL obtained by removing all the internal sources of scattering N , U and J , but

preserving the extrinsic scattering E. The comparison of kballisticL

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 ux, and extrinsic sources of scattering dissipate it (the term Poiseuille comes from an analogy33 between the ow of phonons in a material with the ow of a uid in a pipe). The thermal conductivity though does not grow indenitely with temperature, because of the limit imposed by the intrinsic thermal conductivity of an innite sheet kN. Thus, there is a thermal conductivity peak, and for higher temperatures, the heat ux is mostly dissipated by intrinsic sources (U J ). As

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 U J 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.

Callaways model. Callaways 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 Klemens35 that the most probable boson distribution in a system where momentum is conserved is given by the drifting distribution ndriftn

1

exp on q V=kBT

1, where V is a Lagrange

multiplier enforcing momentum conservation. Therefore, N

processes will tend to relax the phonon population nv towards ndriftn. Instead, resistive processes R U J tend to relax the

phonon populations towards local thermal equilibrium (that is, the BoseEinstein distribution

nn), just as described by the SMA.

Therefore, Callaway approximated the scattering operator as

00

200 400 600

800

Temperature (K)

Figure 3 | Thermal transport regimes in two- and three-dimensional (3D) materials. Thermal conductivity of a graphene ribbon of width L 100 mm

(kL, solid line) together with its ballistic limit (see text; kballisticL, dashed), and of an innite 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 conned around those temperatures.

Xn0On;n0 nn0 nn ndriftntNn nn nntRn; 4

where the Matthiessen rule (that is, arithmetic sum of the scattering rates for independent events) is used 1tR

n

1tUn 1tIn : As

N scattering rates are very large, an out-of-equilibrium

distribution will mostly decay rst into the drifting distribution, and from this state it will relax towards static equilibrium. The results for the thermal conductivity in Callaways 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 Callaways thermal conductivity can be found in the

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100

102

GrapheneGraphaneBoron nitride Fluorographene Molybdenum disulphide

80

101

GrapheneGraphaneBoron nitride Fluorographene Molybdenum disulphide

% Drifting component

60

[afii9838] ss (m)

100

40

101

20

0

102

400

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.

200 600Temperature (K)

400

800

800

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

200 600Temperature (K)

Supplementary Note 1 and Supplementary Fig. 1). It can be seen that the model broadly reproduces the exact solutions, even if, for example, in uorographene 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 Klemens34,35.

Second sound. Materials in hydrodynamic regimes can host second sound1116, and is thus interesting to discuss its existence or characteristics in two dimensions. We dene 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 sound36; the common requirement is that a mechanism exists, which is causing a slow decay of the heat ux. This can indeed happen in case of a large presence of N

events. As mentioned before, the conservation of momentum drives nv towards ndriftn, and once this drifting distribution ndriftn 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 BoseEinstein 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-of-equilibrium distribution closely resembles the drifting one, we rst 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 dene the drifting component as:

Pn CnFnq==

P

n

nn

nn 1 onvinqi=tRn

Pn nn nn 1 onvinqi: 7

The second-sound relaxation timesreported in the Supplementary Fig. 2indicate that the energy ux 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 computedthey are reported in the Supplementary Fig. 3 as a function of temperature, and compared with the average velocity of acoustic phonons. We nd 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.

; 5

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

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 uorographene and 10% molybdenum disulphide. Therefore, in graphene, graphane and boron nitride, a large fraction of the heat ux is carried by the drifting distribution (which is a good approximation of nv for the study of the heat ux propagation), whereas in uorographene 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 dene a second-sound length lss

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 prole, characterized by the following quantities:

vss2 P

n Cn vn vn2

Pn Cn; 6

1 tss

Pn CnF2n

r

Pn Cnq2==

r

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The propagation lengths lss 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 second-sound lengths of a fraction of a micron (the values of lss for

molybdenum disulphide and uorographene 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 veried experimentally in the past1115 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.

DiscussionWe have investigated heat transport in some key 2D materials graphene, boron nitride, molybdenum disulphide and the functionalized derivatives graphane and uorographene. 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 conrming 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 rst into a momentum-conserving drifting distribution before relaxing into the nal BoseEinstein 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 theory23,27,28,37 as implemented in the Quantum ESPRESSO distribution38, using the local-density approximation, norm-conserving pseudopotentials from the PSlibrary39 and a plane-wave cutoff of90 Ry. The electronic-structure calculations for graphene (G), boron nitride (BN), uorographene (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 for all materials except MoS2, where it is taken to be 18 . 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, nding 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 ones40: hydrogen of99.9885% 1H, 0.0115% 2H; carbon of 98.93% 12C, 1.07% 14C; boron of 19.9% 10B,80.1% 11B, nitrogen of 99.632% 14N, 0.368% 15N; uorine of 100% 19F; sulphur of94.93% 32S, 0.76 33S, 4.29 34S, 0.02% 36S; molybdenum of 14.84% 92Mo, 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 uorographene and1.945 for MoS2.

All calculations have been managed using the AiiDA materials informatics platform41.

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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

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

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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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