can scatter through several mechanisms as they travel through the material. These scattering mechanisms are: Umklapp phonon-phonon scattering, phonon-impurity scattering, phonon-electron scattering, and phonon-boundary scattering. Each scattering mechanism can be characterised by a relaxation rate 1/ which is the inverse of the corresponding relaxation time. All scattering processes can be taken into account using Matthiessen's rule. Then the combined relaxation time can be written as: The parameters,,, are due to Umklapp scattering, mass-difference impurity scattering, boundary scattering and phonon-electron scattering, respectively.
Phonon-phonon scattering
For phonon-phonon scattering, effects by normal processes are ignored in favor of Umklapp processes. Since normal processes vary linearly with and umklapp processes vary with, Umklapp scattering dominates at high frequency. is given by: where is the Gruneisen anharmonicity parameter, is the shear modulus, is the volume per atom and is the Debye frequency.
Three-phonon and four-phonon process
Thermal transport in non-metal solids was usually considered to be governed by the three-phonon scattering process, and the role of four-phonon and higher-order scattering processes was believed to be negligible. Recent studies have shown that the four-phonon scattering can be important for nearly all materials at high temperature and for certain materials at room temperature. The predicted significance of four-phonon scattering in boron arsenide was confirmed by experiments.
Mass-difference impurity scattering
Mass-difference impurity scattering is given by: where is a measure of the impurity scattering strength. Note that is dependent of the dispersion curves.
Boundary scattering
Boundary scattering is particularly important for low-dimensional nanostructures and its relaxation time is given by: where is the characteristic length of the system and, which is related to the roughness of the surface, represents the fraction of specularly scattered phonons. The parameter is not easily calculated for an arbitrary surface. For a surface characterized by a root-mean-square roughness, a wavelength-dependent value for the parameter can be calculated using in the case of plane waves at normal incidence. The value corresponds to a perfectly smooth surface such that boundary scattering is purely specular. The relaxation time is in this case infinite, implying that boundary scattering does not contribute to the thermal resistance. Conversely, the value corresponds to a very rough surface, in which case boundary scattering is purely diffusive and the relaxation rate is given by: This equation is also known as Casimir limit.
Phonon-electron scattering
Phonon-electron scattering can also contribute when the material is heavily doped. The corresponding relaxation time is given as: The parameter is conduction electrons concentration, ε is deformation potential, ρ is mass density and m* is effective electron mass. It is usually assumed that contribution to thermal conductivity by phonon-electron scattering is negligible.
