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/τ 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 τC can be written as:

1τC=1τU+1τM+1τB+1τph-e

The parameters τU, τM, τB, τph-e are due to Umklapp scattering, mass-difference impurity scattering, boundary scattering and phonon-electron scattering, respectively.

## Contents

* 1 Phonon-phonon scattering
* 2 Three-phonon and four-phonon process
* 3 Mass-difference impurity scattering
* 4 Boundary scattering
* 5 Phonon-electron scattering
* 6 See also
* 7 References

## 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 ω and umklapp processes vary with ω2, Umklapp scattering dominates at high frequency.[1] τU is given by:

1τU=2γ2kBTμV0ω2ωD

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

## Three-phonon and four-phonon process

Thermal transport in non-metal solids was usually considered to be governed by the three-phonon scattering process,[3] 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 [4] and for certain materials at room temperature.[5] 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:

1τM=V0Γω44πvg3

where Γ is a measure of the impurity scattering strength. Note that vg is dependent of the dispersion curves.

## Boundary scattering

Boundary scattering is particularly important for low-dimensional nanostructures and its relaxation rate is given by:

1τB=vgL0(1−p)

where L0 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 η, a wavelength-dependent value for p can be calculated using

p(λ)=exp⁡(−16π2λ2η2cos2θ)

where θ is the angle of incidence.[6] An extra factor of π is sometimes erroneously included in the exponent of the above equation.[7] At normal incidence, θ=0, perfectly specular scattering (i.e. p(λ)=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

1τB=vgL0

This equation is also known as Casimir limit.[8]

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:

1τph-e=neϵ2ωρvg2kBTπm*vg22kBTexp⁡(−m*vg22kBT)

The parameter ne 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 [citation needed].

## See also

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

## References

1. ↑ Mingo, N (2003). "Calculation of nanowire thermal conductivity using complete phonon dispersion relations". Physical Review B 68 (11): 113308. doi:10.1103/PhysRevB.68.113308. Bibcode: 2003PhRvB..68k3308M. https://zenodo.org/record/1233747.
2. ↑ 2.0 2.1 Zou, Jie; Balandin, Alexander (2001). "Phonon heat conduction in a semiconductor nanowire". Journal of Applied Physics 89 (5): 2932. doi:10.1063/1.1345515. Bibcode: 2001JAP....89.2932Z. Archived from the original on 2010-06-18. https://web.archive.org/web/20100618011126/http://ndl.ee.ucr.edu/jap-zou-1.pdf.
3. ↑ Ziman, J.M. (1960). Electrons and Phonons: The Theory of transport phenomena in solids. Oxford Classic Texts in the Physical Sciences. Oxford University Press.
4. ↑ Feng, Tianli; Ruan, Xiulin (2016). "Quantum mechanical prediction of four-phonon scattering rates and reduced thermal conductivity of solids". Physical Review B 93 (4): 045202. doi:10.1103/PhysRevB.93.045202. Bibcode: 2016PhRvB..96p5202F.
5. ↑ Feng, Tianli; Lindsay, Lucas; Ruan, Xiulin (2017). "Four-phonon scattering significantly reduces intrinsic thermal conductivity of solids". Physical Review B 96 (16): 161201. doi:10.1103/PhysRevB.96.161201. Bibcode: 2017PhRvB..96p1201F.
6. ↑ Jiang, Puqing; Lindsay, Lucas (2018). "Interfacial phonon scattering and transmission loss in > 1 um thick silicon-on-insulator thin films". Phys. Rev. B 97 (19): 195308. doi:10.1103/PhysRevB.97.195308. Bibcode: 2018PhRvB..97s5308J.
7. ↑ Maznev, A. (2015). "Boundary scattering of phonons: Specularity of a randomly rough surface in the small-perturbation limit". Phys. Rev. B 91 (13): 134306. doi:10.1103/PhysRevB.91.134306. Bibcode: 2015PhRvB..91m4306M.
8. ↑ Casimir, H.B.G (1938). "Note on the Conduction of Heat in Crystals". Physica 5 (6): 495–500. doi:10.1016/S0031-8914(38)80162-2. Bibcode: 1938Phy.....5..495C.

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