Phonon scattering
Phonons 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/formula_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 formula_2 can be written as: The parameters formula_4, formula_5, formula_6, formula_7 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 (processes which conserve the phonon wave vector - N processes) are ignored in favor of Umklapp processes (U processes). Since normal processes vary linearly with formula_8 and umklapp processes vary with formula_9, Umklapp scattering dominates at high frequency. formula_10 is given by: where formula_12 is the Gruneisen anharmonicity parameter, is the shear modulus, is the volume per atom and formula_13 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 formula_15 is a measure of the impurity scattering strength. Note that formula_16 is dependent of the dispersion curves. Boundary scattering. Boundary scattering is particularly important for low-dimensional nanostructures and its relaxation rate is given by: where  formula_18 is the characteristic length of the system and formula_19 represents the fraction of specularly scattered phonons. The formula_19 parameter is not easily calculated for an arbitrary surface. For a surface characterized by a root-mean-square roughness formula_21, a wavelength-dependent value for formula_19 can be calculated using where formula_24 is the angle of incidence. An extra factor of formula_25 is sometimes erroneously included in the exponent of the above equation. At normal incidence, formula_26, perfectly specular scattering (i.e. formula_27) would require an arbitrarily large wavelength, or conversely an arbitrarily small roughness. Purely specular scattering does not introduce a  boundary-associated increase in the thermal resistance. In the diffusive limit, however, at formula_28 the relaxation rate becomes This equation is also known as Casimir limit. These phenomenological equations can in many cases accurately model the thermal conductivity of isotropic nano-structures with characteristic sizes on the order of the phonon mean free path. More detailed calculations are in general required to fully capture the phonon-boundary interaction across all relevant vibrational modes in an arbitrary structure. Phonon-electron scattering. Phonon-electron scattering can also contribute when the material is heavily doped. The corresponding relaxation time is given as: The  parameter formula_31 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.