Heat transport in semiconductor crystals: Beyond the local-linear approximation

Abstract

We extend the application of the nonlocal theory of Mahan and Claro [Phys. Rev. B 38, 1963 (1988)] to solve the steady-state Boltzmann–Peierls transport equation within the framework of the single mode relaxation time approximation using the modified Debye–Callaway model. We consider the case of a semi-infinite semiconductor (SC) crystal with a boundary condition at its top surface that can be considered reasonably representative of time domain thermoreflectance (TDTR) and frequency domain thermoreflectance (FDTR) techniques. The approach allows us to obtain three different contributions to the heat flux density current that shed further light on the fundamental role of nonlocality and nonlinearity in heat transport by phonons in SC crystals. Through their intrinsic and implicit shuffling effect of the crystal momentum, phonon–phonon Normal scattering processes play a key role in the onset of thermal conduction as they introduce the temperature Laplacian as a second driving potential force for the heat flux density current in addition to the conventional Fourier's temperature gradient. The developed model suits quite fairly to interpret the frequency behavior of the reduced effective thermal conductivity of SC crystals that is observed in TDTR and FDTR experiments. We obtain an expression of the effective thermal conductivity of the SC crystal that is characterized with a universal spectral suppression function that captures and describes the role, the weight, and the contribution of quasi-ballistic and non-diffusive phonons. The spectral suppression function only depends on the ratio between the phonon mean free path and the thermal penetration depth as defined based on the diffusive Fourier's law.