Length Divergence or Convergence of the In-Plane Thermal Conductivity of Suspended Monolayer Graphene beyond ~10 Micrometers

Idealized two-dimensional momentum-conserving phonon models predict that the thermal conductivity $k$ grows without bound as $k\sim\log L$ with sample length $L$ . Whether this divergence actually occurs in realistic suspended graphene is undecided: the only systematic measurement (Xu et al., 2014) reaches $\sim 9\ \mu\mathrm{m}$ , and both a convergent ballistic-diffusive model and a logarithmically divergent model fit those data; the two are distinguishable only for $L\gtrsim 10\ \mu\mathrm{m}$ . Recent first-principles calculations including four-phonon scattering find a convergent room-temperature value near $1300\ \mathrm{W\,m^{-1}K^{-1}}$ , while momentum-conserving hydrodynamic transport of flexural phonons, disorder, ripples, and residual coupling keep the realistic asymptotics contested. The problem is whether the length dependence of $k$ of clean suspended monolayer graphene at lengths beyond $\sim 10\ \mu\mathrm{m}$ (toward $100\lambda\approx 60\ \mu\mathrm{m}$ ) continues as a divergence or converges to a finite asymptotic value.

Background

Thermal conductivity $k$ relates heat flux to temperature gradient through Fourier's law; on the microscopic level heat in an electrically insulating crystal is carried by phonons (quantized lattice vibrations). A phonon travels on average one mean free path (MFP) $\lambda$ between scattering events, so a sample shorter than roughly $\lambda$ behaves ballistically (heat flow limited only by the sample boundaries) while longer samples are diffusive; a ballistic-diffusive (BD) model interpolates between the two, giving a size-dependent conductivity $k(L)$ that approaches the bulk value only for lengths several times $\lambda$ . For graphene the relevant phonons include flexural (out-of-plane, ZA) modes, whose vibrations are perpendicular to the sheet. In lattice transport theory (the Peierls-Boltzmann transport equation, BTE, which tracks phonon populations with scattering rates computed from interatomic force constants), scattering splits into normal processes, which conserve crystal momentum, and Umklapp processes, which relax it; when normal processes dominate, heat flow becomes hydrodynamic (collective, fluid-like) rather than diffusive. A long-standing result of nonequilibrium statistical mechanics is that in idealized one- and two-dimensional momentum-conserving nonlinear lattices Fourier's law fails: $k$ diverges with system size, as $L^{\alpha}$ in 1D and logarithmically, $k\sim\log L$ , in 2D (e.g., Wang, Hu and Li, Phys. Rev. E 86, 040101(R) (2012)). Since these ideal models restrict atomic motion to the embedding dimension, while atoms in a real 2D membrane such as graphene move in three dimensions, it was never obvious that the predicted divergence carries over to real materials. Coupling to a substrate suppresses the size dependence entirely for supported graphene (Ong and Pop, Phys. Rev. B 84, 075471 (2011)), so the question is specific to suspended samples. The only systematic experiment is by Xu et al. (Nat. Commun. 5, 3689 (2014)), who measured suspended single-layer graphene at 300 K up to $L\approx 9\ \mu\mathrm{m}$ and interpreted the continuing rise of $k$ with $L$ as the logarithmic divergence. Li, Liu, Lindsay, Xu, Duan and Pop (arXiv:1711.02772 (2017)) reanalyzed those data: a simple one-'color' (single effective MFP) BD model with $k_{\mathrm{diff}}=1790\ \mathrm{W\,m^{-1}K^{-1}}$ and a modified BD model with an added logarithmic term ( $k_{\mathrm{diff}}=1590$ , $k_0=130\ \mathrm{W\,m^{-1}K^{-1}}$ , $L_0=1.1\ \mu\mathrm{m}$ ) both fit within experimental error, and the two classes can be told apart only at $L>10\ \mu\mathrm{m}$ , where no data exist. The same work derives an overall phonon MFP $\lambda\approx 580$ nm for suspended graphene, notes that $k(L)$ converges to 90% of the diffusive value only at $L\approx 16\lambda$ and is not fully diffusive until $L\gtrsim 100\lambda\approx 60\ \mu\mathrm{m}$ , and concludes that divergence tests require samples and simulations beyond $100\lambda$ . It further flags that disorder and higher-order (four-) phonon scattering, already anticipated in earlier work, might eliminate the divergence altogether. This defines the unresolved target: the true large-length behavior of $k$ in realistic suspended graphene.

Problem Statement

Determine the length dependence of the in-plane thermal conductivity $k$ of clean suspended monolayer graphene at room temperature in the unexplored regime $L\gtrsim 10\ \mu\mathrm{m}$ (approaching $100\lambda\approx 60\ \mu\mathrm{m}$ , with $\lambda\approx 580$ nm the overall phonon mean free path): does $k$ continue to grow logarithmically with $L$ , $k\sim\log L$ , as ideal 2D momentum-conserving models and three-phonon hydrodynamic transport of flexural phonons predict, or does it converge to a finite asymptotic value near $1300\ \mathrm{W\,m^{-1}K^{-1}}$ , as first-principles calculations including four-phonon scattering indicate? The two behaviors become distinguishable only for $L\gtrsim 10\ \mu\mathrm{m}$ , at uncertainties small enough to discriminate them; which mechanisms (normal-process hydrodynamic transport versus four-phonon/anharmonic scattering, disorder, ripples, isotope and edge scattering, residual strain and coupling to clamps) set the outcome in realistic samples remains contested and shapes the interpretation of the length dependence.

References

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