Elasticity reshapes heat flow in graphene

Abstract

Classical thermal transport theories that preserve rotational symmetry, predict strong anharmonic scattering of out-of-plane lattice vibrational modes called flexural phonons in flat suspended graphene sheets. Such strong scattering processes cause a breakdown of the phonon quasiparticle picture, which remains valid only when several cycles of lattice vibrations occur before the mode decays. Here we show that the renormalization of elastic bending rigidity ($D$), caused by the coupling between the in-plane and the out-of-plane thermal lattice fluctuations, restores phonon quasiparticles in suspended graphene. Importantly, this $D$-renormalization weakens the momentum-dissipating Umklapp phonon scattering processes, resulting in improved thermal conductivity and amplified phonon hydrodynamics in suspended graphene. Our results unveil a previously-unrecognized connection between the macroscopic elasticity and the microscopic flexural phonon scattering in two-dimensional (2D) materials that does not occur in three-dimensional bulk crystals, thereby motivating a re-examination of the classical theories and opening up new avenues to engineer the thermal as well as the phonon-limited electronic transport and relaxation in two- and lower-dimensional materials.

Figures (3)

• Figure 1: SCSA renormalization of ZA phonons, and its effect on $\kappa$ and phonon quasiparticle picture in graphene.a. Ratio of phonon frequency to square of the magnitude of the wave vector $\mathbf{q}$ ($\nu/q^2$) for the ZA phonons of graphene in the long wavelength limit. In the x-axis, $q$ is in units of $2\pi/a$, where $a$ is the lattice constant of graphene. When rotational invariance and stress-free equilibrium conditions are enforced on the harmonic IFCs in SCAP, the ZA phonons exhibit a quadratic dispersion relation. To stabilize the 2D flat phase of the suspended graphene sheet, the SCSA further renormalizes the quadratic ZA dispersion to a sub-quadratic form for $q \lesssim 0.2 (2\pi/a)$, even reaching a universal scaling of $\nu\sim q^{1.6}$ beyond a critical wavelength of $\sim$ 4 nm ($\sim$ 5 nm) at 300 K (150 K) ravichandran_low_2026. b. Calculated temperature-dependent $\kappa$ of naturally-occurring graphene with the lowest-order three-phonon scattering, $\kappa^{\left(3\right)}$, and with the additional inclusion of higher-order four-phonon scattering, $\kappa^{\left(3+4\right)}$, calculated with the bare and SCSA ZA phonons. The $\kappa^{\left(3+4\right)}$ is strongly affected by SCSA renormalization, while the $\kappa^{\left(3\right)}$ remains nearly unchanged. Comparisons with available experimental data are presented in the Supplementary Fig. S1a-b. c. Total phonon scattering rates ($\Gamma$) vs. frequency ($\nu$) for naturally-occurring graphene sheets at 150 K, 300 K and 600 K. The two dashed grey lines indicate $\Gamma = \omega (= 2\pi \nu)$ and $\Gamma = \omega/10$. The scattering rates for the bare ZA phonons are significantly stronger than those of the SCSA phonons over the entire Brillouin zone. In fact, $\Gamma$ exceeds $\omega$ at room temperature and beyond for the low-frequency bare ZA phonons, resulting in the breakdown of the phonon quasiparticle picture. Introduction of the SCSA renormalization restores the quasiparticle nature of these low-frequency ZA phonons with $\Gamma \lesssim \omega/10$ even up to 600 K.
• Figure 2: Microscopic origin of weak four-phonon scattering of the SCSA ZA phonons in graphene.a. Four-phonon scattering phase space, b. Four-phonon scattering matrix element-weighted phase space and, c. Four-phonon scattering rates of the ZA phonons, with and without the SCSA renormalization, at 300 K. Since the SCSA renormalization affects only the low-frequency ZA phonons, their four-phonon scattering phase space remains unaffected. However, the scattering matrix element-weighted phase space and the scattering rates for the four-phonon processes are strongly suppressed for the ZA phonons due to the SCSA renormalization, as discussed in the main text.
• Figure 3: Origin of the amplified phonon hydrodynamics upon SCSA renormalization of ZA phonons in graphene. a. Momentum-conserving Normal (N) and momentum-dissipating Umklapp (U) scattering rates vs. $\nu$ for isotopically pure graphene at 300 K. N scattering is much stronger than U scattering for the calculations with and without SCSA renormalization of the ZA phonons, even at 300 K. The difference between N and U scattering rates is greater for the calculations with the bare ZA phonons than for those with the SCSA ZA phonons. Yet, $\kappa^{\left(3+4\right)}_{\text{SCSA}} > \kappa^{\left(3+4\right)}_{\text{bare}}$, contrary to conventional understanding. b. Cumulative $\kappa$ contributions from the eigenmodes of the phonon collision operator ($\mathbf{\Omega}$) in isotopically pure graphene at 150 K and 300 K for the calculations with the bare and the SCSA ZA phonons. The $\kappa$-accumulation computed with the SCSA ZA phonons shows large contribution to $\kappa^{\left(3+4\right)}$ from individual eigenmodes with smaller eigenvalues (called the drift eigenmodes, $\sigma^D$, in the main text) compared to that computed with the bare ZA phonons at both temperatures, indicating amplified hydrodynamic heat flow due to SCSA renormalization of the ZA phonons. c. Contributions of phonons to $\sigma^D$ at 150 K and 300 K in graphene, with and without the SCSA renormalization of the ZA phonons. $\sigma_D \approx \mel{\mathfrak{e}^D}{\mathbf{\Omega}^{\left(U\right)}}{\mathfrak{e}^D}/\braket{\mathfrak{e}^D}{\mathfrak{e}^D}$ contains contributions of individual phonons to the matrix element of the collision operator for U processes only, which are smaller for the calculations using the SCSA ZA phonons compared to those using the bare ZA phonons, thus lowering $\sigma^D$ and amplifying phonon hydrodynamics in the former.