  While using first-principles-based Boltzmann transport equation approach to
predict the thermal conductivity of crystalline semiconductor materials has
been a routine, the validity of the approach is seldom tested for
high-temperature conditions. Most previous studies only focused on the phononic
contribution, and neglected the electronic part. Meanwhile, the treatment on
phonon transport is not rigorous as a few ingredients, such as four-phonon
scatterings, phonon renormalization and thermal expansion, are ignored. In this
paper, we present a Boltzmann transport equation study on high-temperature
thermal conduction in bulk silicon by considering the effects of both phonons
and electrons, and explore the role of the missing parts in the previous
studies on the thermal conductivity at elevated temperature. For the phonon
transport, four-phonon scattering is found to considerably reduce the thermal
conductivity when the temperature is larger than 700 K, while the effects of
phonon renormalization and thermal expansion on phononic thermal conductivity
are negligible. Bipolar contribution to the electronic thermal conductivity
calculated from first-principles is implemented for the first time. More than
25% of heat is shown to be conducted by electrons at 1500 K. The computed total
thermal conductivity of silicon faithfully reproduces the measured data. The
approach presented in this paper is expected to be applied to other
high-temperature functional materials, and the results could serve as
benchmarks and help to explain the high-temperature phonon and electron
transport phenomena.
