Temperature in a Peierls-Boltzmann treatment of nonlocal phonon heat transport

摘要

In nonmagnetic insulators, phonons are the carriers of heat. If heat enters in a region and temperature is measured at a point within phonon mean free paths of the heated region, ballistic propagation causes a nonlocal relation between local temperature and heat insertion. This paper focuses on the solution of the exact Peierls-Boltzmann equation (PBE), the relaxation time approximation (RTA), and the delinition of local temperature needed in both cases. The concept of a nonlocal "thermal susceptibility" (analogous to charge susceptibility) is defined. A formal solution is obtained for heating with a single Fourier component P(r, t) = P_0exp(ik · r - iωt), where P is the local rate of heating). The results are illustrated by Debye model calculations in RTA for a three-dimensional periodic system where heat is added and removed with P(r, t) = P(x) from isolated evenly spaced segments with period L in x. The ratio L/ℓ_(min) is varied from 6 to ∞, where ℓ_(min) is the minimum mean free path. The Debye phonons are assumed to scatter anharmonically with mean free paths varying as ℓ_(min)(q_D/q)~2 where q_D is the Debye wave vector. The results illustrate the expected local (diffusive) response for ℓ_(min) ≪ L, and a diffusive to ballistic crossover as ℓ_(min) increases toward the scale L. The results also illustrate the confusing problem of temperature definition. This confusion is not present in the exact treatment but occurs in RTA.