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Phonon hydrodynamics and ultrahigh–room-temperature thermal conductivity in thin graphite

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Science  17 Jan 2020:
Vol. 367, Issue 6475, pp. 309-312
DOI: 10.1126/science.aaz8043

Thin graphite gets cool fast

In nonmetallic solids, heat is transported primarily through crystal vibrations called phonons. These phonons can have wavelike properties under certain conditions, which increases the thermal conductivity of the material. Machida et al. found that making graphite samples thin expands the hydrodynamic regime from cryogenic to room temperatures. The researchers measured an extremely high thermal conductivity in the very thin graphite samples, which may be important for a variety of electronics applications.

Science, this issue p. 309

Abstract

Allotropes of carbon, such as diamond and graphene, are among the best conductors of heat. We monitored the evolution of thermal conductivity in thin graphite as a function of temperature and thickness and found an intimate link between high conductivity, thickness, and phonon hydrodynamics. The room-temperature in-plane thermal conductivity of 8.5-micrometer-thick graphite was 4300 watts per meter-kelvin—a value well above that for diamond and slightly larger than in isotopically purified graphene. Warming enhances thermal diffusivity across a wide temperature range, supporting partially hydrodynamic phonon flow. The enhancement of thermal conductivity that we observed with decreasing thickness points to a correlation between the out-of-plane momentum of phonons and the fraction of momentum-relaxing collisions. We argue that this is due to the extreme phonon dispersion anisotropy in graphite.

Heat travels in insulators because of the propagation of collective vibrational states of the crystal lattice called phonons. The standard description of this transport phenomenon invokes quasiparticles losing their momentum to the underlying lattice because of collisions along their trajectory (1). Gurzhi proposed decades ago that phonons in insulators and electrons in metals can flow hydrodynamically if momentum-conserving collisions among carriers become abundant (2). Recently, hydrodynamic regimes for electrons (35) and for phonons (610) have become a subject of renewed attention, partially driven by the aim of quantifying the quasiparticle viscosity.

Unlike particles in an ideal gas of molecules, the phonon momentum is not conserved in all collisions. When scattering between two phonons produces a wave vector exceeding the unit vector of the reciprocal lattice, the excess of momentum is lost to the underlying lattice. These are called Umklapp (U) scattering events, and they require sufficiently large wave vectors. Because cooling reduces the typical wavelength of thermally excited phonons, U scattering rarefies with decreasing temperature, and most collisions among phonons conserve momentum, becoming normal (N) scattering events. In this context, a regime of phonon hydrodynamics emerges that is sandwiched between diffusive and ballistic regimes (2). Observations of the hydrodynamic regime include several solids (8, 9, 1114). In this narrow temperature window, warming multiplies normal collisions, and this enhances the ratio of thermal conductivity to specific heat—called the thermal diffusivity. Observations of this behavior tend to be at cryogenic temperatures.

The domination of N events over U events across a very broad temperature range in graphene led two groups to propose that phonon hydrodynamics might be observed at temperatures outside the cryogenic range (6, 7). However, heat transport measurements in graphene (15) are challenging to study by using the standard four-probe steady-state technique. Evidence for second sound, a manifestation of phonon hydrodynamics, was recently found at temperatures exceeding 100 K in graphite (10). These observations were in agreement with theoretical expectations (16).

The two-dimensional lattice of graphite (Fig. 1A, inset) consists of strong interlayer sp2 covalent bonds combined with weak intralayer van der Waals bonds. The strength of the in-plane and the out-of-plane couplings differs by two orders of magnitude. This dichotomy makes graphite easily cleavable down to the single-layer graphene form (17). The bonding of graphite also creates two distinct Debye temperatures, one for the in-plane and the other for the out-of-plane atomic vibrations (18). This induces a large difference between in-plane and out-of-plane thermal conductivities (19). The experimentally measured thermal conductivity (1923) shows a roughly similar temperature dependence. However, there is a large variety in the reported magnitude of in-plane thermal conductivity, which at room temperature can vary between 72 and 2100 W/m·K (19), a feature attributed to the unavoidable presence of the stacking faults and contamination of the in-plane data by a contribution from c-axis flow. As we will see below, new insight is provided by a thickness-dependent study on the same sample.

Fig. 1 Thermal conductivity and experimental setup.

(A) Temperature dependence of in-plane thermal conductivity of graphite with thicknesses ranging from 580 to 8.5 μm on a logarithmic scale. Inset shows side view of the crystal structure of graphite. A schematic illustration (B) and a photo (C) of the measurement setup for the thermal conductivity. Heat current jq generated by a heater on one end of the sample passes through the sample toward the thermal bath. Temperature difference developed in the sample is determined by two pairs of thermocouples.

We measured the in-plane thermal conductivity (κ) of commercially available highly oriented pyrolytic graphite (HOPG) samples, all peeled from a thick mother sample, with a standard steady-state one-heater–two-thermometers technique in high vacuum (Fig. 1). We tested the reliability by measuring the thermal conductivity of a long, thin silver foil with a thermal resistance comparable to that of our most thermally resistive sample and quantifying the small deviation from the Wiedemann-Franz law (24). For samples with thicknesses ranging from 8.5 to 580 μm, we found identical κ behavior below 20 K and a steady thickness evolution for κ with increasing temperature above 20 K.

We compared the temperature dependence of κ in the thickest sample (580 μm) with the measured specific heat (Fig. 2A). We found that κ peaks around 100 K, similar to other measurements (20, 21, 23). Below this maximum, κ quickly decreases and roughly follows a T2.5 dependence, close to the specific heat trend below 10 K (25). The specific heat (C) temperature behavior deviated from the T3 expected from the Debye approximation. However, this behavior is not strictly observed in most real solids, owing to unequal distribution of phonon weights. The 2.5 exponent has been attributed in graphite to an admixture of T3 and T2 contributions by out-of-plane and in-plane phonons, respectively (26). This unusual exponent may have obscured the Poiseuille regime, which is usually associated with faster-than-cubic thermal conductivity (2).

Fig. 2 Hydrodynamic heat transport.

(A) Temperature dependence of in-plane thermal conductivity κ (left axis) and specific heat C (right axis) of the 580-μm-thick graphite sample. (B) κ divided by T2.5 (left axis) and C divided by T2.5 (right axis) as a function of temperature. A pronounced maximum is seen only in κ/T2.5 above 10 K. This yields a maximum in temperature dependence of thermal diffusivity Dth (C). Dominant phonon contribution in κ is indicated by a large Lorenz ratio L/L0 shown in (D).

Closer examination of the parallel evolution of thermal conductivity and specific heat can help unveil the Poiseuille regime as κ evolves faster than C above 10 K and slower below 10 K (Fig. 2A). Plotting κ/T2.5 and C/T2.5 makes this difference easier to recognize (Fig. 2B). Upon warming, κ/T2.5 shows a pronounced maximum above 10 K, whereas C/T2.5 gradually decreases. The thermal diffusivity, Dth, is the ratio of thermal conductivity to specific heat (expressed in proper units of J/K mol). We found that Dth has a nonmonotonic temperature dependence between 10 and 20 K (Fig. 2C). The phonon hydrodynamic picture provides a straightforward interpretation of this feature. Warming leads to enhanced momentum exchange among phonons, because the fraction of collisions that conserve momentum increases. As a consequence, heat diffusivity rises. If all phonons had the same mean free path irrespective of their branch and wave vector, this would also imply a rise in the effective mean free path. The Poiseuille maximum around 40 K and the Knudsen minimum around 10 K, where diffuse boundary scattering rate is effectively increased because of N scattering, define the boundaries of this hydrodynamic window.

We found that the electron contribution is negligibly small in the temperature range of interest by determining the Lorenz ratio (L/L0). We measured electrical conductivity, σ, to quantify L=κσT and compare it with L0 = 2.44 × 10−8 W-ohm/K2. This results in a ratio between 100 and 1000 above the Knudsen minimum (Fig. 2D).

The behavior that we observed for κ and C is not due to outstanding sample quality. Comparable features can be found in published data (20, 21, 24) but appear to have gone unnoticed. Our mother sample was an average HOPG containing both stable isotopes of carbon, (~99% 12C, ~1% 13C). Our results support the conjecture that phonon hydrodynamics can occur without isotopic purity (8).

We measured an increased κ as we decreased sample thickness (Fig. 3A). We performed successive measurements after changing the thickness (t) of the sample along the c axis, maintaining the sample width (w = 350 μm) and the distance (l) between contacts for the thermal gradient to be long enough compared to the thickness (l/t > 10) (24). The trend is the opposite of observations for black phosphorus (8). With respect to the hydrodynamic regime, thinning leads to an amplification of the nonmonotonic behavior of thermal diffusivity. This drives the Poiseuille peak to become sharper and toward higher temperatures. Eventually, Dth of the thinnest sample shows a sharp maximum at 100 K. Second sound in graphite was observed near this temperature (10). The thickness dependence vanishes below 10 K, presumably because the phonon mean free path in this range is set by the average crystallite size (19), which does not depend on thickness. Another possible origin of the thickness-independent low-temperature thermal conductivity is an intrinsic scattering of phonons by mobile electrons.

Fig. 3 Thickness dependence of thermal conductivity.

(A) Temperature dependence of in-plane thermal conductivity κ for various sample thicknesses. In the thinnest sample, κ attains the largest value (~4300 W/m·K) known in any bulk system near room temperature. (B) Temperature dependence of thermal diffusivity Dth for various sample thicknesses. The maximum in Dth forms a sharp, single peak with decreasing thickness. (C) Our data are compared with those of ultrahigh–thermal conductivity materials (22, 2729). The inset shows thickness dependence of thermal conductivity at 250 K. κ of the thinnest sample is comparable with the high values reported in single-layer graphene (27, 32).

The thermal conductivity in our 240-μm-thick sample is in reasonable agreement with previous observations on a similar thickness graphite (22). The in-plane κ that we measured for the 8.5-μm-thick sample was ~4300 W/m·K. This exceeds the value for an isotopically pure graphene sample (27) and is higher than that of other bulk solids. The value is twice that of natural abundance diamond (28) and about three times larger than high-purity crystalline BAs (2931). At room temperature, reducing the thickness by two orders of magnitude leads to a fivefold increase in κ (Fig. 3C). Although the κ that we measured is already comparable with the highest values reported in single-layer graphene (κ ≈ 3000 to 5000 W/m·K) (27, 32), our data do not saturate in the low-thickness limit. In contrast to suspended graphene over a trench of 3 μm (32), our samples are millimetric in length. Given the quasi-ballistic trajectory of phonons, we make the reasonable assumption that in-plane dimensions matter in setting the amplitude of thermal conductivity. This would imply that the ceiling is higher than previously believed, and thinner samples with larger aspect ratio should display even larger conductivity. Although several theoretical works have predicted a robust hydrodynamic regime in graphene (6, 7) and its persistence in graphite (16), none examined the issue of thickness dependence.

To try to understand the origin of our observation, we scrutinized the occurrence of U and N collisions, given the phonon dispersion of graphite (15, 33) (Fig. 4). We show the calculation of Nihira and Iwata (33) from a semicontinuum model for the in-plane and out-of-plane dispersion of longitudinal (LA), in-plane transverse (TA), and out-of-plane transverse (ZA) acoustic phonons along the ΓM and ΓA directions (Fig. 4B). The model parameters (velocities and elastic constants) were determined by using the best account of experimental specific heat data from 0.5 to 500 K (33). The two orientations show a marked contrast regarding the typical wavelength of thermally excited phonons and requirements for U scattering. At 300 K (or 200 cm−1), the typical in-plane wave vector of the LA mode is only 0.1 of the Brillouin zone (BZ) width. This makes U collisions extremely rare (Fig. 4C), because to create a phonon with a wave vector larger than half of the BZ width, the average wave vector of each colliding phonon needs to be 0.25 of the BZ width. The fundamental reason behind the scarcity of U collisions and the emergence of hydrodynamics resides within this simple feature. The situation is radically different for out-of-plane wave vectors. Even at 50 cm−1, a thermally excited phonon can have an out-of-plane wave vector that is one-fourth of the BZ height. Above 90 cm−1 (corresponding to 130 K), out-of-plane phonons are all thermally excited (33), and their peak wavelength is half of the BZ height. Any additional momentum along this orientation can “kick” them out of the BZ. A small c-axis component in the momentum exchanged by colliding phonons suffices for the collision to become a U event (Fig. 4D) and the heat flow to degrade.

Fig. 4 Phonon dispersions.

(A) First Brillouin zone (BZ) of graphite. (B) Calculated dispersions of acoustic phonon blanches along the ΓA and ΓM directions of BZ (33), together with the experimental data obtained by neutron (34) and Raman scattering (35). BZ in the ΓKM plane (C) and ΓMA plane (D). Collision between the in-plane component of an incident phonon (green arrow) and a thermally excited phonon (blue arrow) remains N, because the in-plane wave vector of the thermal phonon is only a small fraction of the BZ width even at 300 K (or 200 cm−1). Hence, the wave vector of the outcome phonon (red arrow) does not exceed one-half of the BZ width. By contrast, the out-of-plane wave vector of a thermal phonon is one-fourth of the BZ height for frequencies as low as 50 cm−1. Therefore, the collision becomes U, if the in-plane traveling phonon happens to possess a small out-of-plane component.

Our observation implies a reduction in the relative weight of U collisions as the sample is thinned, because attenuating the relative rate of U collisions would extend the hydrodynamic window and enhance thermal conductivity. We note that the spacing between discrete available states in the reciprocal space depends on thickness. Therefore, the total number of states with out-of-plane momentum is inversely proportional to the thickness. It is true that only a small fraction of the BZ is wiped out by the finite size. However, different collision mechanisms are competing for phase space, and reducing the thickness not only reduces the population of the out-of-plane phonons but also amplifies boundary scattering. Heat-carrying phonons can suffer either a U collision with an out-of-plane phonon or a (more or less) specular collision at the boundary. Thus, reducing the thickness, by substituting a fraction of U collisions with specular boundary reflection, would limit the degradation of the heat flow.

A satisfactory account of thickness dependence of thermal conductivity in both HOPG and black phosphorus (8) is lacking. Scattering at the boundaries and imperfect transmission across interfaces between partially twisted graphene layers require further scrutiny. Serious theoretical calculations are needed to explain our findings.

Supplementary Materials

science.sciencemag.org/content/367/6475/309/suppl/DC1

Materials and Methods

Supplementary Text

Figs. S1 to S7

Table S1

References (3641)

References and Notes

  1. Materials and Methods are available as Supplementary Materials on Science Online.
Acknowledgments: We thank S. Kurose for the contribution in the early stage of this study and A. Subedi for discussions. We also thank M. Tsubota and M. Watanabe for technical support. Funding: This work was supported by the Japan Society for the Promotion of Science Grant-in-Aids KAKENHI 16K05435, 17KK0088, and 19H01840 and by the Agence Nationale de la Recherche (ANR-18-CE92-0020-01). Author contributions: Y.M. and K.B. conceived of and designed the study. Y.M., N.M., and T.I. performed the transport and specific heat measurements. Y.M. and K.B. wrote the manuscript with assistance from all the authors. Competing interests: The authors declare no competing interests. Data and materials availability: All data are available in the manuscript or the supplementary materials.
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