Thermal response functions and second sound in graphene

TL;DR

This paper investigates non-diffusive heat transport and second-sound propagation in graphene using a classical molecular-dynamics framework based on thermal-response functions. By computing $K_{m{q}}(\tau)$ from equilibrium fluctuations and analyzing its spectral content, the authors demonstrate a clear second-sound signal at 300 K for length scales up to $L=68.1$ nm, and extract a $q$-dependent lifetime $\tau_{ss}$ and velocity $v_{ss}$ that show strong decoherence from phonon dispersion rather than anharmonic scattering. The results deviate significantly from Boltzmann transport equation predictions, which yield much longer lifetimes and $q$-independent $\tau_{ss}$, and align qualitatively with graphite TTG experiments where $\tau_{ss}$ depends on the excitation scale. The work also develops a framework to predict the response to time-dependent heating sources, linking the observed temperature oscillations to the magnitude and phase of $\tilde{K}_{\bm{q}}(\omega)$, and proposes time-resolved experiments as a direct probe of second-sound spectra. Overall, the study highlights the importance of phonon dispersion and coherence effects at nanoscale lengths and offers a path toward first-principles reductions via DFT-based phonon calculations and time-dependent probes for validating second-sound physics in graphene and related materials.

Abstract

The propagation of second sound, and more broadly the ballistic transport of heat, is of central importance in heat dissipation from electronic devices at very short length and time scales. Specifically, there is an interest in the practical implications of violations of Fourier's law. Recently, we have developed a simulation approach based on thermal-response functions that is appropriate for elucidating physics beyond the diffusive regime, including time-dependent sources and second-sound propagation. The methods are applied to free-standing graphene simulated using molecular-dynamics (MD) with empirical potentials. The simulations predict a strong second-sound signal at T=300K for length scales of at least L=68.1nm. It is demonstrated that the second-sound dissipation time is determined primarily by decoherence that emerges from the details of the phonon band structure. It is also shown that the decay time for second sound depends sensitively on the length scale that characterizes the thermal excitation. This is in contrast with theories based on the Boltzmann transport equation (BTE), where second-sound dissipation is determined primarily by the resistive anharmonic phonon scattering rate. Calculations using the linearized BTE are also presented, along with analysis of second sound based on the BTE. This approach results in significantly longer lifetimes for second sound in comparison to our MD simulation results. Predictions for the response due to time-dependent sources are also presented, including insight into how time-dependent sources could be tuned to result in weak or strong temperature oscillations, and how time-dependent experiments might probe the spectra associated with second sound. Results are discussed in the context of second sound in graphite in the temperature range from 100-200K.

Figures (12)

• Figure 1: Thermal conductivity integral in Eq. \ref{['GK']} plotted as a function of the upper integration limit $\tau$ for different equilibration times $\tau_{eq}$.
• Figure 2: Response function $K_{\bm{q}}(\tau)$ plotted as a function of time for three different wave vectors with magnitudes $q_{1}={2 \pi \over L}$, $q_{2}={4 \pi \over L}$, and $q_{3}={6 \pi \over L}$. As described in the text, the length scale is $L=68.1$nm.
• Figure 3: Imaginary part $K^{\prime \prime}_{\bm{q}}(\omega)$ of the Fourier transformed response function plotted as a function of frequency ${\omega \over 2 \pi}$ for $q_{1}={2 \pi \over L}$. The green curve shows the predicted curve based on Fourier's law using the classical heat capacity and computed thermal conductivity values $\kappa$.
• Figure 4: Real part $K^{\prime}_{\bm{q}}(\omega)$ of the Fourier transformed response function plotted as a function of frequency ${\omega \over 2 \pi}$ for $q_{1}={2 \pi \over L}$. The green curve shows the predicted curve based on Fourier's law using the classical heat capacity and computed thermal conductivity values $\kappa$.
• Figure 5: Imaginary component $K^{\prime \prime}_{\bm{q}}(\omega)$ plotted as a function of time for three different wave vectors with magnitudes $q_{1}={2 \pi \over L}$, $q_{2}={4 \pi \over L}$, $q_{3}={6 \pi \over L}$ with $L=68.1$nm.