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.