Phonon scattering

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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.

Phonon-phonon scattering[edit]

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.[1] is given by:

where is Gruneisen anharmonicity parameter, μ is shear modulus, V0 is volume per atom and is Debye frequency.[2]

Mass-difference impurity scattering[edit]

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[edit]

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.[3]

Phonon-electron scattering[edit]

Phonon-electron scattering can also contribute when the material is heavily 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.[2] It is usually assumed that contribution to thermal conductivity by phonon-electron scattering is negligible.

See also[edit]


  1. ^ Mingo, N (2003). "Calculation of nanowire thermal conductivity using complete phonon dispersion relations". Physical Review B. 68 (11): 113308. Bibcode:2003PhRvB..68k3308M. arXiv:cond-mat/0308587Freely accessible. doi:10.1103/PhysRevB.68.113308. 
  2. ^ a b 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. 
  3. ^ Casimir, H.B.G (1938). "Note on the Conduction of Heat in Crystals". Physica. 5 (6): 495. Bibcode:1938Phy.....5..495C. doi:10.1016/S0031-8914(38)80162-2.