# 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/$\tau$ 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 $\tau _{C}$ can be written as:
    ${\frac {1}{\tau _{C}}}={\frac {1}{\tau _{U}}}+{\frac {1}{\tau _{M}}}+{\frac {1}{\tau _{B}}}+{\frac {1}{\tau _{\text{ph-e}}}}$
The parameters $\tau _{U}$, $\tau _{M}$, $\tau _{B}$, $\tau _{\text{ph-e}}$ 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 $\omega$ and umklapp processes vary with $\omega ^{2}$, Umklapp scattering dominates at high frequency. $\tau _{U}$ is given by:
    ${\frac {1}{\tau _{U}}}=2\gamma ^{2}{\frac {k_{B}T}{\mu V_{0}}}{\frac {\omega ^{2}}{\omega _{D}}}$
where $\gamma$ is the Gruneisen anharmonicity parameter, μ is the shear modulus, V$_0$ is the volume per atom and $\omega _{D}$ 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:
    ${\frac {1}{\tau _{M}}}={\frac {V_{0}\Gamma \omega ^{4}}{4\pi v_{g}^{3}}}$
where $\Gamma$ is a measure of the impurity scattering strength. Note that ${v_{g}}$ is dependent of the dispersion curves.

## Boundary scattering

Boundary scattering is particularly important for low-dimensional nanostructures and its relaxation rate is given by:
    ${\frac {1}{\tau _{B}}}={\frac {v_{g}}{L_{0}}}(1-p)$
where $L_{0}$ is the characteristic length of the system and $p$ represents the fraction of specularly scattered phonons. The $p$ parameter is not easily calculated for an arbitrary surface. For a surface characterized by a root-mean-square roughness $\eta$, a wavelength-dependent value for $p$ can be calculated using
    $p(\lambda )=\exp {\Bigg-16{\frac {\pi ^{2}}{\lambda ^{2}}}\eta ^{2}\cos ^{2}\theta {\Bigg )}$
where $\theta$ is the angle of incidence. An extra factor of $\pi$ is sometimes erroneously included in the exponent of the above equation. At normal incidence, $\theta =0$, perfectly specular scattering (i.e. $p(\lambda )=1$) 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 $p=0$ the relaxation rate becomes
    ${\frac {1}{\tau _{B}}}={\frac {v_{g}}{L_{0}}}$
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:
    ${\frac {1}{\tau _{\text{ph-e}}}}={\frac {n_{e}\epsilon ^{2}\omega }{\rho v_{g}^{2}k_{B}T}}{\sqrt {\frac {\pi m^{*}v_{g}^{2}}{2k_{B}T}}}\exp \left(-{\frac {m^{*}v_{g}^{2}}{2k_{B}T}}\right)$
The parameter $n_{e}$ 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.