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/ 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 can be written as:
The parameters , , , 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 and umklapp processes vary with , Umklapp scattering dominates at high frequency. is given by:
Mass-difference impurity scattering
Mass-difference impurity scattering is given by:
where is a measure of the impurity scattering strength. Note that is dependent of the dispersion curves.
Boundary scattering is particularly important for low-dimensional nanostructures and its relaxation time is given by:
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:
This equation is also known as Casimir limit.
Phonon-electron scattering can also contribute when the material is lightly doped. The corresponding relaxation time is given as:
The parameter 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.
- 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.
- Zou, Jie; Balandin, Alexander (2001). "Phonon heat conduction in a semiconductor nanowire" (PDF). Journal of Applied Physics 89 (5): 2932. Bibcode:2001JAP....89.2932Z. doi:10.1063/1.1345515.
- 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.