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

Thermal transport in non-metal solids was usually considered to be governed by the three-phonon scattering process,[3] 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 [4] and for certain materials at room temperature.[5] The predicted significance of four-phonon scattering in boron arsenide was confirmed by experiments.[6]

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

${\displaystyle {\frac {1}{\tau _{B}}}={\frac {V}{L_{0}}}(1-p)}$

where ${\displaystyle L_{0}}$ is the characteristic length of the system and ${\displaystyle p}$, which is related to the roughness of the surface, represents the fraction of specularly scattered phonons. The ${\displaystyle p}$ parameter is not easily calculated for an arbitrary surface. For a surface characterized by a root-mean-square roughness ${\displaystyle \eta }$, a wavelength-dependent value for the ${\displaystyle p}$ parameter can be calculated using

${\displaystyle p(\lambda )=\exp {\Bigg (}-{\frac {16\pi ^{3}\eta ^{2}}{\lambda ^{2}}}{\Bigg )}}$

in the case of plane waves at normal incidence.[7] The value ${\displaystyle p=1}$ corresponds to a perfectly smooth surface such that boundary scattering is purely specular. The relaxation time ${\displaystyle \tau _{B}}$ is in this case infinite, implying that boundary scattering does not contribute to the thermal resistance. Conversely, the value ${\displaystyle p=0}$ corresponds to a very rough surface, in which case boundary scattering is purely diffusive and the relaxation rate is given by:

${\displaystyle {\frac {1}{\tau _{B}}}={\frac {V}{L_{0}}}}$

This equation is also known as Casimir limit.[8]

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

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

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.

• Lattice scattering
• Umklapp scattering
• Electron-longitudinal acoustic phonon interaction