Length divergence of the lattice thermal conductivity in suspended graphene nanoribbons

Abstract

Thermal properties of graphene have attracted much attention, culminating in a recent measurement of its length dependence in ribbons up to 9 $\ensuremath{\mu}\mathrm{m}$ long. In this paper, we use the improved Callaway model to solve the phonon Boltzmann transport equation while capturing both the resistive (umklapp, isotope, and edge roughness) and nonresistive (normal) contributions. We show that for lengths smaller than 100 $\ensuremath{\mu}\mathrm{m}$, scaling the ribbon length while keeping the width constant leads to a logarithmic divergence of thermal conductivity. The length dependence is driven primarily by a ballistic-to-diffusive transition in the in-plane (LA and TA) branches, while in the hydrodynamic regime when $10\phantom{\rule{0.28em}{0ex}}\ensuremath{\mu}\mathrm{m}lLl100$ $\ensuremath{\mu}\mathrm{m}$, the contribution from the in-plane branches saturates and the out-of-plane (ZA) branch shows a clear logarithmic trend, driven by the nonresistive normal contribution. We find that thermal conductivity converges beyond $Lg100$ $\ensuremath{\mu}\mathrm{m}$ due to the coupling between in-plane and flexural modes. This coupling leads to renormalization of ZA phonon dispersion in the long-wavelength range, preventing further divergence of thermal conductivity. We also uncover a strong dependence on sample width, which we attribute to the interplay between nonresistive normal and diffusive edge scattering in the Poisseuille flow regime. We conclude that normal processes play a crucial role in the length and width dependence of thermal transport in graphene in the hydrodynamic regime and dictate the relative in-plane (LA+TA) to out-of-plane (ZA) contribution to transport.