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 dynamics of

thermal conduction has been given less attention. The frequency response of thermal conductivity has become

more crucial in recent years, especially in light of the constant rise in the clock frequencies in microprocessors

and terahertz sensing applications. Thermal conductivity in response to a time-varying temperature field starts

decaying when the frequency exceeds a cutoff frequency Omega_c, which is related to the inverse of phonon relaxation time τ, on the order of 2–10 ps in most bulk semiconductors. Phonons in graphene have much longer phonon relaxation times,which we showleads to far lowerc.Our calculations, based on the phonon Boltzmann equation coupled with first-principles dispersion, show that dynamical thermal conductivity of graphene resembles a low-pass filter that decays beyond an Omega_c ranging from 100 MHz to 10 GHz, controlled by temperature and ribbon width. The response parallels the Drude model of electrons, but with far lower cutoff. Moreover, the presence of strong normal processes in graphene results in a complex-valued conductivity and gradual transition around Omega_c, with the resistive contribution to the heat flux having higher cutoff frequency and smaller phase lag than the hydrodynamic part. The dynamical conductivity will impact dissipation in high-frequency applications of graphene. Our findings also provide a platform for future studies of hydrodynamic transport and wavelike, or

second sound, heat transfer by tuning the frequency of the applied temperature field.