# 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/${\displaystyle \tau }$ 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 ${\displaystyle \tau _{C}}$ can be written as:

${\displaystyle {\frac {1}{\tau _{C}}}={\frac {1}{\tau _{U}}}+{\frac {1}{\tau _{M}}}+{\frac {1}{\tau _{B}}}+{\frac {1}{\tau _{ph-e}}}}$

The parameters ${\displaystyle \tau _{U}}$, ${\displaystyle \tau _{M}}$, ${\displaystyle \tau _{B}}$, ${\displaystyle \tau _{ph-e}}$ 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 ${\displaystyle \omega }$ and umklapp processes vary with ${\displaystyle \omega ^{2}}$, Umklapp scattering dominates at high frequency.[1] ${\displaystyle \tau _{U}}$ is given by:

${\displaystyle {\frac {1}{\tau _{U}}}=2\gamma ^{2}{\frac {k_{B}T}{\mu V_{0}}}{\frac {\omega ^{2}}{\omega _{D}}}}$

where ${\displaystyle \gamma }$ is Gruneisen anharmonicity parameter, μ is shear modulus, V0 is volume per atom and ${\displaystyle \omega _{D}}$ is Debye frequency.[2]

## Mass-difference impurity scattering

Mass-difference impurity scattering is given by:

${\displaystyle {\frac {1}{\tau _{M}}}={\frac {V_{0}\Gamma \omega ^{4}}{4\pi v_{g}^{3}}}}$

where ${\displaystyle \Gamma }$ is a measure of the impurity scattering strength. Note that ${\displaystyle {v_{g}}}$ is dependent of the dispersion curves.

## Boundary scattering

Boundary scattering is particularly important for low-dimensional nanostructures and its relaxation time is given by:

${\displaystyle {\frac {1}{\tau _{B}}}={\frac {V}{D}}(1-p)}$

where D is the dimension of the system and p represents the surface roughness parameter. The value p=1 means a smooth perfect surface that the scattering is purely specular and the relaxation time goes to ∞; hence, boundary scattering does not affect thermal transport. The value p=0 represents a very rough surface that the scattering is then purely diffusive which gives:

${\displaystyle {\frac {1}{\tau _{B}}}={\frac {V}{D}}}$

This equation is also known as Casimir limit.[3]

## Phonon-electron scattering

Phonon-electron scattering can also contribute when the material is heavily doped. The corresponding relaxation time is given as:

${\displaystyle {\frac {1}{\tau _{ph-e}}}={\frac {n_{e}\epsilon ^{2}\omega }{\rho V^{2}k_{B}T}}{\sqrt {\frac {\pi m^{*}V^{2}}{2k_{B}T}}}\exp \left(-{\frac {m^{*}V^{2}}{2k_{B}T}}\right)}$

The parameter ${\displaystyle n_{e}}$ is conduction electrons concentration, ε is deformation potential, ρ is mass density and m* is effective electron mass.[2] It is usually assumed that contribution to thermal conductivity by phonon-electron scattering is negligible.