  Phonon Boltzmann transport equation (BTE) is an important tool for studying
the nanoscale thermal transport. Because phonons have a large spread in their
properties, the non-gray (i.e. considering different phonon bands) phonon BTE
is needed to accurately capture the nanoscale transport phenomena. However, BTE
solvers generally require large computational cost. Non-gray modeling imposes
significant additional complexity to the numerical simulations, which hinders
the modeling of real nanoscale systems. In this work, we address this issue by
a systematic investigation on the phonon band discretization scheme using real
material properties of four representative materials, including silicon,
gallium arsenide, diamond, and lead telluride. We find that the schemes used in
previous studies require at least a few tens of bands to ensure the accuracy,
which requires large computational costs. We then propose an improved band
discretization scheme, in which we divide the mean free path domain into two
subdomains, one on either side of the inflection point of the mean free path
accumulated thermal conductivity and adopt the Gauss-Legendre quadrature for
each subdomain. With this scheme, the solution of phonon BTE converges (error <
1%) with less than 10 phonon bands for all these materials. The proposed scheme
allows to significantly reduce the time and memory consumption of the numerical
BTE solver, which is an important step towards large-scale phonon BTE
simulation for real materials.
