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, Template:Mvar is the shear modulus, Template:Mvar 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:

$\frac{1}{{\tau }_M}=\frac{V_0\mathrm{\Gamma }{\omega }^4}{4\pi v_g^3}$

where $\mathrm{\Gamma }$ 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

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

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:

$\frac{1}{{\tau }_{\text{ph-e}}}=\frac{n_e{ϵ}^2\omega }{\rho v_g^2{k}_{B}T}\sqrt{\frac{\pi m^{*}{v}_g^2}{2k_{B}T}}\exp (-\frac{m^{*}{v}_g^2}{2k_{B}T})$

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 Template:Citation needed.

See also

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