# 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:

$\frac{1}{\tau_C} = \frac{1}{\tau_U}+\frac{1}{\tau_M}+\frac{1}{\tau_B}+\frac{1}{\tau_{ph-e}}$

The parameters $\tau_{U}$, $\tau_{M}$, $\tau_{B}$, $\tau_{ph-e}$ 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 $\omega$ and umklapp processes vary with $\omega^2$, Umklapp scattering dominates at high frequency.[1] $\tau_U$ 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 Gruneisen anharmonicity parameter, μ is shear modulus, V0 is volume per atom and $\omega_{D}$ is Debye frequency.[2]

## 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 time is given by:

$\frac{1}{\tau_B}=\frac{V}{D}(1-p)$

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:

$\frac{1}{\tau_B}=\frac{V}{D}$

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

## Phonon-electron scattering

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

$\frac{1}{\tau_{ph-e}}=\frac{n_e \epsilon^2 \omega}{\rho V^2 k_B T}\sqrt{\frac{\pi m^* V^2}{2k_B T}} \exp \left(-\frac{m^*V^2}{2k_B T}\right)$

The parameter $n_{e}$ 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.

## References

1. ^ Mingo, N (2003). "Calculation of nanowire thermal conductivity using complete phonon dispersion relations". Journal reference: Phys. Rev. B Phys Rev B 68: 113308. arXiv:cond-mat/0308587. Bibcode:2003PhRvB..68k3308M. doi:10.1103/PhysRevB.68.113308.
2. ^ a b Zou, Jie; Balandin, Alexander (2001). "Phonon heat conduction in a semiconductor nanowire". Journal of Applied Physics 89 (5): 2932. Bibcode:2001JAP....89.2932Z. doi:10.1063/1.1345515.
3. ^ Casimir, H.B.G (1938). "Note on the Conduction of Heat in Crystals". Physica, 5 6 (6): 495. Bibcode:1938Phy.....5..495C. doi:10.1016/S0031-8914(38)80162-2.