Graphite gets a second sound Between the two extremes of ballistic and diffusive lattice thermal transport is the potential for an exotic wave-like state known as second sound. Huberman et al. used fast, transient thermal grating measurements to show the existence of second sound in graphite between 85 and 125 kelvin (see the Perspective by Shi). Previous observations of second sound have been rare, confined to isotopically pure materials at very low temperatures. The observation of second sound in graphite is likely due to its layered nature, suggesting that this thermal transport mode may be accessible in other two-dimensional materials. Science, this issue p. 375; see also p. 332

Abstract Wavelike thermal transport in solids, referred to as second sound, is an exotic phenomenon previously limited to a handful of materials at low temperatures. The rare occurrence of this effect restricted its scientific and practical importance. We directly observed second sound in graphite at temperatures above 100 kelvins by using time-resolved optical measurements of thermal transport on the micrometer-length scale. Our experimental results are in qualitative agreement with ab initio calculations that predict wavelike phonon hydrodynamics. We believe that these results potentially indicate an important role of second sound in microscale transient heat transport in two-dimensional and layered materials in a wide temperature range.

In nonmetallic solids, heat is carried primarily by lattice vibrations called phonons. In nearly perfect crystals at temperatures below ~10 K, phonons may propagate over macroscopic distances without scattering, yielding ballistic heat transport. At room temperature, the mean free paths of heat-carrying phonons are short because of high rates of phonon-phonon scattering, and heat spreads macroscopically by diffusion. The transition between ballistic and diffusive transport kinetics (transport distance D and time t, related by D ∝ t and D ∝ t , respectively) on micro- and nanoscales has attracted increased interest (1–5). Recent observations of nondiffusive heat transport at room temperature in silicon and other materials (3–5) have important implications for thermal management in microelectronics, as well as for thermoelectric energy conversion, where nanostructured materials help increase efficiency (1, 2).

Second sound, or the wavelike transport of heat in solids (6), has long been considered an exotic phenomenon occurring at very low temperatures in only a few materials within a special regime between ballistic transport and diffusion. In this regime, referred to as phonon hydrodynamics (7, 8), “normal” phonon-phonon scattering processes that conserve the net reduced phonon momentum are much more frequent than “Umklapp” processes capable of reversing the reduced momentum direction. Normal scattering alone cannot dissipate a heat flux and return the lattice to thermal equilibrium; instead, the phonon population relaxes to a “displaced” Bose-Einstein distribution characterized by a nonzero drift velocity (9), akin to a flow of molecules in a gas. This enables temperature waves (i.e., phonon density waves) to propagate at a velocity below the speed of sound. This phenomenon is termed second sound because of the analogy with temperature waves in superfluid He (10).

Observations of second sound were made with heat pulse experiments in 3He (between 0.42 and 0.58 K) (11), in Bi (between 1.2 and 4 K) (12), and in NaF (between 11 and 14.5 K) (13, 14). The reported occurrence of second sound in SrTiO 3 at 30 to 40 K (15, 16) is in dispute, as others argued that the attribution of the low-frequency doublet in the Brillouin spectra to second sound was not supported by the observations (17, 18). Computational work indicated that the temperature window for the phonon hydrodynamics regime is much wider in graphene (9) and other two-dimensional (2D) materials (19) because of strong normal scattering involving the flexural mode. A similar prediction has been made for graphite, which exhibits a phonon dispersion resembling that of graphene because of the weakness of its interlayer van der Waals interactions (20).

Second sound is expected to occur on the time scale that falls in between the characteristic normal and Umklapp relaxation times—i.e., τ N < t < τ U . Theoretical predictions for graphene (9) indicated a nanosecond time scale, making it difficult to use conventional thermal sensors to probe the transport. However, time-resolved laser-based measurements are well suited for measuring thermal responses on this time scale (1, 3–5). We used the transient thermal grating (TTG) technique (Fig. 1A), in which two short (≤60-ps) laser pulses are crossed at the surface of the sample, yielding a spatially sinusoidal heat source whose period L is set by the optical interference pattern. This heat source sets up a spatially sinusoidal temperature field along the surface, ΔT(t,z)cos(qx), where q = 2π/L is the TTG wave vector, T is temperature, and x and z are the in-plane and cross-plane coordinates, respectively. This “thermal grating” subsequently decays via thermal transport. Thermal expansion produces an associated sinusoidal surface displacement modulation, or “ripple” pattern, u(t)cos(qx), that acts as a transient diffraction grating for probe laser light. We monitored the decay of the thermal grating through the time-dependent diffraction of a continuous-wave probe laser beam. We superposed the diffracted beam with a reference beam derived from the same laser source for optical heterodyne detection (21). The signal is proportional to the amplitude of the spatially sinusoidal surface displacement modulation u(t) (22).

Fig. 1 TTG measurements on graphite at room temperature. (A) Schematic illustration of the experiment. A thermal grating is produced by two crossed pump pulses. Thermal expansion gives rise to a surface modulation that is detected via diffraction of the probe beam, and the reference beam is used for optical heterodyne detection. (B) Signal waveforms at 300 K for a range of TTG periods. The dashed lines correspond to the best fits to exponential decays. (C) The apparent thermal diffusivity L2/4π2τ as a function of grating period L at 300 K for grating periods of 3.7 to 37.5 μm, plotted alongside theoretical results from ab initio BTE calculations. In the diffusive regime, this quantity would be a constant, equal to the thermal diffusivity α. The reduction of the apparent diffusivity as L is reduced occurs as the mean free paths of increasing fractions of heat-carrying phonons become comparable to the grating period. The error bars account for the measured systematic laser power effect and the statistical standard error of the mean (22).

Our sample was highly oriented pyrolytic graphite of the highest-quality commercially available grade. The sample was polycrystalline, with a grain size of ~10 μm (fig. S3), but the c axes of all crystal grains were perpendicular to the sample surface. The absorption of 515-nm excitation light produced the initial thermal grating within an optical skin depth of ~30 nm. Thermal transport occurs both along the thermal grating (in plane) and in the direction normal to the surface (cross plane). However, the in-plane thermal conductivity of graphite is ~300 times as high as the cross-plane conductivity; hence, the cross-plane thermal diffusion depth remains much smaller than the TTG period until the thermal grating is washed out by in-plane transport (22). In this case, the normal surface displacement is proportional to the integral of the temperature rise over the cross-plane coordinate z. Consequently, the cross-plane transport does not affect the surface displacement, and the measured signal is sensitive only to 1D in-plane transport in the grating direction x. According to the heat diffusion equation, in the 1D case, a TTG decays exponentially with a time constant τ = L2/4π2α, where L is the grating period and α is the thermal diffusivity (23). At room temperature (300 K), we observed an exponential decay of the TTG (Fig. 1B). At L = 37.5 μm, we obtained a thermal diffusivity of 11 cm2/s, which agrees with the literature value (24). However, as we reduced the TTG period, the apparent thermal diffusivity (L2/4π2τ) did not remain equal to the constant value of α as predicted by the heat diffusion equation. At small TTG periods (Fig. 1C), the decay became slower than predicted by the diffusion model, indicating nondiffusive behavior similar to that observed previously in Si and other materials (3–5). The observed onset of the size effect occurs when the mean free path of heat-carrying phonons becomes comparable to the heat transport distance (25). Still, the decay remained exponential over the whole range of TTG periods we used (3.7 to 37.5 μm).

When we lowered the temperature to 85 K, we observed notably different behavior (Fig. 2A). Unlike the signals at 300 K, which decayed monotonically, signal waveforms at 85 K yielded damped oscillations, with the signal falling below zero (26). In the heterodyne detection scheme, this sign flip of the TTG signal means that the spatial phase of the grating has shifted by π (21). This indicates that the local maxima and minima of the surface displacements (and hence of the temperature) switched places; in other words, the TTG behaves as a thermal standing wave. The sign flip is a hallmark of the wavelike propagation of heat. In the diffusive transport regime, TTG maxima and minima cannot switch places because heat can move only from hotter to colder regions. With an increasing TTG period, the negative dip in the response became shallower and eventually disappeared. The position of the dip shifted to longer times as the period increased, indicating that the frequency of the wavelike dynamics decreased. We determined the frequency from the position of the first minimum of the response (corresponding to half of the oscillation period) as a function of the wave vector q (Fig. 2A, inset). The nearly linear dependence indicated a velocity of about 3200 m/s (determined from the slope of the linear fit multiplied by 2π). TTG signals often contain oscillations due to surface acoustic waves (SAWs), albeit with much lower damping rates (27). However, the oscillation frequency we observed did not match the frequency of SAWs or any other acoustic waves that may propagate in the basal plane of graphite. The SAW velocity is 1480 m/s (28), which is very close to the slow transverse velocity, whereas the fast transverse velocity is 14,700 m/s and the longitudinal velocity is even higher (29). Besides, we know of no reason for acoustic waves to disappear as the background temperature or the grating period is increased.

Fig. 2 Experimental and simulated TTG dynamics for graphite at 85 K. (A) The TTG signal at 85 K for a range of grating periods. In the inset, circles represent the measured second-sound frequency as a function of the wave vector, and the solid line is a linear fit corresponding to a phase velocity of 3200 m/s. (B) Absolute value of complex frequency-domain Green’s functions versus frequency at 80 K for a range of TTG periods. (C) Simulated thermal grating amplitude versus time at 80 K.

To simulate the observed dynamics, we solved the linearized Boltzmann transport equation (BTE) with the full three-phonon scattering matrix in the 1D TTG geometry for the initial temperature profile ΔT 0 cos(qx) (22). Rigorous solutions of the BTE for graphene and graphite were previously used to calculate the thermal conductivities of these materials (30, 31) and to explore the conditions for the phonon hydrodynamics regime (9, 19, 20). However, previous studies dealt with the stationary BTE, which is unable to capture transient phenomena such as second sound. Chiloyan et al. (32) described a technique for calculating frequency-domain Green’s functions for the nonstationary and spatially nonuniform BTE. Such Green’s functions (Fig. 2B) describe a response of the phonon population to a heat source having the form of a harmonic plane wave, exp(iωt-qr) (where i is the imaginary unit, ω is the temporal frequency, and r is the real space coordinate vector). In our case, the heat source was sinusoidal in space, and we modeled it as impulsive in time because the laser pulse duration was short compared with the observed dynamics. Consequently, we can obtain the temporal dynamics of the TTG by performing a Fourier transform of the frequency-domain Green’s function at wave vector q corresponding to the TTG period. We followed the approach developed by Chiloyan et al. to solve the BTE with a collision integral constructed by using inputs from density functional theory calculations, accounting for three-phonon scattering as well as scattering by mass disorder due to the natural isotope content (1.1% of 13C). Our calculations, performed entirely from first principles with no fitting parameters, produced the time dependence of the thermal grating amplitude, which we directly compared with the experimentally measured signal.

Our calculations performed at 300 K over a range of grating periods yielded exponential TTG decays in agreement with the experiment. Calculations of the apparent thermal diffusivity as a function of the TTG period reproduced the trend seen in the experiment (Fig. 1C). By contrast, at 85 K the frequency-domain Green’s functions yielded a resonant peak (Fig. 2B). This peak is a hallmark of second sound and gives rise to damped oscillations in the simulated time-domain waveforms (Fig. 2C). The simulated waveforms agreed qualitatively with the experimental data (fig. S5) and yielded trends with respect to the TTG period that are consistent with our observations. In particular, the simulations reproduced the disappearance of the second-sound signature at large TTG periods. The calculated second-sound velocity determined from the peak position of the frequency resonance in Fig. 2C is 3650 m/s, which is somewhat higher than the measured velocity. The theory correctly predicts that the second-sound velocity falls in between the slow and fast transverse acoustic velocities. By contrast to that in graphite, the velocity of second sound in other materials was found to be lower than the lowest phonon velocity. The peculiarity of graphite is the unusually low velocity of the slow transverse acoustic mode, which is analogous to the flexural (ZA) mode in 2D materials such as graphene. The very large anharmonicity and density of states of this mode lead to intense normal scattering and create conditions for hydrodynamic phonon transport (9).

We collected TTG data with a constant grating period of 10 μm at different temperatures (Fig. 3). Тhe oscillatory behavior was still apparent at 104 K and even at 125 K and eventually disappeared at 150 K. The simulations reproduced this trend. The oscillatory behavior also disappeared when we lowered the temperature to 50 K. At temperatures below 80 K, we saw increasing discrepancy between the experiment and simulations. At 50 K, the simulated response still contained a dip at ~1.5 ns, even though it no longer went negative. One possible origin of the quantitative discrepancy is the assumption of the initial thermal distribution used in our simulations (22). This assumption becomes increasingly inaccurate at low temperatures when transport transitions to the ballistic regime. One would need to consider the details of the electronic excitation by the pump laser pulse and the subsequent electron-phonon interaction to determine the initial phonon distribution in the laser-induced thermal grating. Another pertinent consideration is that the experimental observable is thermal expansion whereas the simulated observable is the temperature (i.e., thermal energy per unit of volume). Whereas in equilibrium, thermal expansion should precisely track the temperature, in a nonequilibrium situation this does not have to be the case because the Grüneisen parameter is different for different phonon modes. The simulated response at 50 K obtained in the ballistic limit, with phonon scattering rates set to zero (dashed curve in Fig. 3), does not have a dip. The disappearance of the second-sound signature we observed at 50 K is generally consistent with the transition to the ballistic regime. Unlike in the heat pulse experiments on NaF and Bi (12–14), where the wavelike behavior persisted in the ballistic regime, in graphite the slow transverse phonon mode, which carries most of the heat, is strongly dispersive. Consequently, in the ballistic limit, simulations do not yield any prominent oscillations that would correspond to a certain propagation velocity. Notably, eliminating scattering altogether in the simulations made the TTG decay slower, as normal phonon-phonon scattering processes facilitate heat transport by redistributing energy to modes with higher group velocity.

Fig. 3 Temperature dependence of TTG dynamics. Measured TTG signals (orange curves) and simulated responses (blue solid curves) at different temperatures. (The label 85 K corresponds to the experimental curve, whereas the simulated curve was obtained for 80 K.) Horizontal dashed lines indicate zero for each pair of curves. The dashed curve for 50 K shows the calculated response in the ballistic limit (i.e., with the scattering matrix set to zero).

We determined the second-sound domain in the temperature–TTG period parameter space. We used the ratio of the maximum at the peak of the magnitude of frequency-domain Green’s functions (Fig. 2B) to the minimum between the peak and zero frequency as a metric for the strength of the second-sound effect. From this, we predicted that second sound occurs between 50 and 250 K (Fig. 4), with higher temperatures corresponding to shorter thermal transport length scales. The temperature window closes at ~150 K for L = 10 μm, in agreement with our observations, whereas at L = 1.5 μm we expect an observable second-sound signature at up to 250 K. However, our experimental setup lacked the temporal resolution needed to probe responses at TTG periods smaller than ~5 μm. At low temperatures and small grating periods, phonon scattering of any kind disappears and transport becomes ballistic. At high temperatures and large TTG periods, transport slowly transitions to the “quasi-diffusive” regime in which the TTG decay is exponential but the decay time τ does not scale as L2 (Fig. 1, B and C). Eventually, with a further increase in L or T, the transport would reach the diffusive limit (τ ∝ L2).

Fig. 4 Second-sound window for graphite with the natural isotope content. The color scale corresponds to the ratio of the maximum at the peak to the minimum between the peak and zero frequency in frequency-domain Green’s functions, such as those shown in Fig. 2B. Second sound occurs in the hydrodynamic window in the temperature-distance parameter space. When the temperature and/or the TTG period is increased, thermal transport transitions to the quasi-diffusive regime, characterized by an exponential TTG decay. In the opposite limit of low temperatures and short TTG periods, transport transitions to the ballistic regime.

The size of the second-sound window depends on the concentration of defects, such as isotopes, which contribute to Umklapp scattering (6, 9, 20). Our simulations showed that in isotopically pure graphite, the second-sound oscillations would be more pronounced and less damped (fig. S6). Whereas the second-sound window (Fig. 4) is shown for the natural carbon isotope content, it should be wider for isotopically pure material (9). Previously, second sound was observed only in isotopically pure solids (with the exception of the SrTiO 3 reports). Our observation of second sound in graphite with the natural isotope content indicates the distinct nature of phonon hydrodynamics in this type of material. This behavior should extend to other layered and 2D materials, including graphene. This has practical implications for the performance of graphite and graphene as heat-spreading materials in microelectronics. Our work also complements recent observations of electronic hydrodynamic transport in graphene and other 2D materials (33–35). We believe that these developments collectively will open up a new understanding of the manipulation of transport phenomena on the micro- and nanoscales.

Supplementary Materials science.sciencemag.org/content/364/6438/375/suppl/DC1 Materials and Methods Figs. S1 to S6 References (36, 37)

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Acknowledgments: We are grateful to L. Paulatto for his help with the construction of the scattering matrix. Funding: This work is supported in part by the Office of Naval Research under MURI grant N00014-16-1-2436 (G.C. for high thermal conductivity materials, including phonon hydrodynamics); in part by the Solid State Solar-Thermal Energy Conversion Center (S3TEC), an Energy Frontier Research Center funded by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences, under award DE-SC0001299 (G.C. for thermoelectric materials); and in part by the NSF EFRI 2-DARE grant EFMA-1542864 (K.A.N. for thermal transport in 2D materials). Author contributions: The project was proposed by S.H. and R.A.D., who led the theoretical and experimental work, respectively, with a substantial experimental contribution by K.C. and B.S., theoretical contributions by V.C. and Z.D., and guidance from A.A.M., G.C., and K.A.N. All the participants contributed to the writing of the paper. Competing interests: None. Data and materials availability: All data are available in the main text or the supplementary materials.