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:

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

The parameters ${\displaystyle \tau _{U}}$, ${\displaystyle \tau _{M}}$, ${\displaystyle \tau _{B}}$, ${\displaystyle \tau _{\text{ph-e}}}$ are due to Umklapp scattering, mass-difference impurity scattering, boundary scattering and phonon-electron scattering, respectively.

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.

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 rate is given by:

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

where ${\displaystyle L_{0}}$ is the characteristic length of the system and ${\displaystyle p}$ 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 ${\displaystyle p}$ can be calculated using

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

where ${\displaystyle \theta }$ is the angle of incidence.^[6] An extra factor of ${\displaystyle \pi }$ is sometimes erroneously included in the exponent of the above equation.^[7] At normal incidence, ${\displaystyle \theta =0}$, perfectly specular scattering (i.e. ${\displaystyle p(\lambda )=1}$) 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 ${\displaystyle p=0}$ the relaxation rate becomes

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

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

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

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

The parameter ${\displaystyle n_{e}}$ 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 ^{[citation needed]}.

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