The Modified Guyer-Krumhansl Equations Derived From the Linear Boltzmann Transport Equation

Abstract

Phonon hydrodynamics originated from the macroscopic energy and momentum balance equations (called Guyer-Krumhansl equations) proposed by Guyer and Krumhansl by solving the linearized Boltzmann transport equation for studying second sound in the normal-process collision dominated phonon transports in an isotropic nonmetallic crystal with a dispersionless frequency spectrum. However, the low-dimensional dielectric materials and semiconductors are anisotropic, and the different branches in their phonon frequency spectrum usually have different group velocities. For such materials, we derive the macroscopic energy and momentum balance equations from the linear Boltzmann transport equation to describe the phonon hydrodynamic transport, and solve the longstanding debate about whether the energy balance equation contains the second-order spatial derivatives of temperature. Finally, by solving the modified Guyer–Krumhansl equations, we find the minimum and maximum values of the length required by the occurrence of second sound in suspended single-layer graphene with the rectangular shape.