Heat transport in semiconductor crystals under large temperature gradients

We analyze and discuss the fundamental behavior of the nonlocal/nonlinear contributions to heat transport by phonons in bulk cubic semiconductor (SC) crystals subject to large temperature gradients. The calculation approach is based on solving steady-state Boltzmann-Peierls Transport Equation (BPTE) by expanding the phonon distribution function in a series of temperature gradients. We work out the modeling within the framework of the single relaxation time approximation and using modified Debye-Callaway model in which both longitudinal and transverse phonon modes are included explicitly. The SC system is treated as a continuum, elastic, isotropic and dispersionless medium. The frequency and temperature dependences of three-phonon anharmonic Normal and Umklapp scattering processes are kept the same for all SC crystals. Our model allows us to obtain compact expressions for the first three nonlocal/nonlinear thermal coefficients to which we limit our calculations. We assume these three coefficients to be the leading ones over the whole temperature range considered in our study. Their fundamental behaviors are studied by changing ambient temperature, longitudinal and transverse Grüneisen parameters as well as the mass-fluctuation parameter. In the simplest case of the grey spectrum approximation, we shed light on a very interesting result regarding the expression of the effective thermal conductivity κeff of the bulk SC crystal when the latter is subject to a space-periodic temperature profile that is typically encountered in Transient Thermal Grating (TTG) experiments. We find an expression that undoubtedly proves both nonlocality and nonlinearity to be a sound and robust explanation of the reduced measured thermal conductivity that was reported in TTG experiments.