The steady-state behavior of thermal transport in bulk and nanostructured semiconductors has been widely studied, both theoretically and experimentally. On the other hand, fast transients and frequency response of thermal conduction has been given less attention. The frequency response of thermal conductivity has become more crucial in recent years, especially because of the constant increase in the clock frequencies in microprocessors and other terahertz applications. It has been theoretically predicted in 3-d materials that thermal conductivity in response to a time-varying temperature gradient, starts decaying when the frequency of the temperature gradient (f) exceeds a cut-off (fc), where fc has been found to be in the order of phonon relaxation time. The phonon relaxation time in semiconductors like silicon is short, on the order of 2-10 ps, leading to thermal conductivity that is independent of frequency up to very high values exceeding 10 GHz. In contrast, 2-d materials like graphene have much longer phonon relaxation times. Therefore, in suspended graphene ribbons, fc can be expected to be much lower than that of silicon.
We calculate frequency-dependent thermal conductivity of graphene ribbons from the solution of phonon Boltzmann transport equation using improved Callaway model. The phonon dispersion is calculated from the first principles using Quantum Espresso and the scattering rates are taken from our previous work. We observe that thermal conductivity remains constant at low frequencies of temperature gradient and exhibits a decaying behavior at higher frequencies, therefore behaving like low-pass thermal filters. We define cut-off frequency to be the frequency of the temperature gradient at which thermal conductivity becomes less than 70% of its dc value. The cut-off frequency in graphene ribbons is found to be relatively low as compared to that of silicon, ranging from few μHz to 100 μHz at 20 K and in the order of GHz at room temperature, thereby showing dependence on both temperature and size of the ribbons. However, the rate of decay of thermal conductivity beyond cut-off frequency is, surprisingly, found to be independent of temperature and size of ribbons. At low temperatures (20 K), dynamical thermal conductivity as well as the cut-off frequency is dominated by the resistive contribution, and at room temperature, the contribution from non-resistive (normal) processes governs the behavior of thermal conductivity with frequency. We also found that out-of-plane (ZA) modes contribute the most to dynamical thermal conductivity as compared to in-plane modes (LA and TA) at low temperatures, with narrow spectral distribution of phonon mean free path, whereas at room temperature the contribution from in-plane modes dominates the dynamical thermal conductivity, with spectral phonon mean free path spread over a much wider range as compared to that of low temperature.
 A.K. Majee and Z. Aksamija, Phys. Rev. B 93, 235423, 2016.