Phonon scattering explained

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:(1)/(\tau _{C}) = (1)/(+ \tau _{U}) (1)/(+ \tau _{M}) (1)/(+ \tau _{B}) (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:(1)/(\tau _{U}) =2 \gamma ^{2 (k _{B} T)/(\mu V _{0})} (\omega ^{2})/(\omega _{D})

where

\gamma is the Gruneisen anharmonicity parameter, is the shear modulus, is the volume per atom and \omega _{D} 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:

(1)/(= \tau _{M}) (V _{0} \Gamma \omega ^{4})/(4 \pi v ^{3} _{g})

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:

(1)/(= \tau _{B}) (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 η , a wavelength-dependent value for p can be calculated usingp( λ ) = \exp ( -16 (\pi ^{2})/(λ ^{2}) η ^{2\cos} ^{2\theta} )

where

\theta is the angle of incidence.^[6] An extra factor of \pi is sometimes erroneously included in the exponent of the above equation.^[7] At normal incidence, \theta =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(1)/(= \tau _{B}) (v _{g})/(L _{0})

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:

(1)/(= \tau _{ph-e}) (n _{e} \epsilon ^{2} \omega)/(\sqrt{ \rho v ^{2} _{g} k _{B} T) (\pi m ^{*} v ^{2} _{g})/(2 k _{B} T) } \exp \left(-\frac\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 .

See also

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

Notes and References

1. Mingo . N. 2003. Calculation of nanowire thermal conductivity using complete phonon dispersion relations. Physical Review B. 68. 11. 113308. cond-mat/0308587. 2003PhRvB..68k3308M . 10.1103/PhysRevB.68.113308 . 118984828.
2. Zou . Jie . Balandin, Alexander . 2001 . Phonon heat conduction in a semiconductor nanowire . Journal of Applied Physics . 89 . 5 . 2932 . 10.1063/1.1345515 . 2001JAP....89.2932Z . dead . https://web.archive.org/web/20100618011126/http://ndl.ee.ucr.edu/jap-zou-1.pdf . 2010-06-18.
3. Book: Electrons and Phonons: The Theory of transport phenomena in solids . Oxford Classic Texts in the Physical Sciences . Oxford University Press. 1960. Ziman. J.M..
4. 10.1103/PhysRevB.93.045202. Quantum mechanical prediction of four-phonon scattering rates and reduced thermal conductivity of solids. Physical Review B. 93. 4. 045202. 2016. Feng. Tianli. Ruan. Xiulin. 2016PhRvB..96p5202F . 1510.00706. 16015465 .
5. 10.1103/PhysRevB.96.161201. Four-phonon scattering significantly reduces intrinsic thermal conductivity of solids. Physical Review B. 96. 16. 161201. 2017. Feng. Tianli. Lindsay. Lucas. Ruan. Xiulin. 2017PhRvB..96p1201F . free.
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 . 10.1103/PhysRevB.97.195308. 1712.05756 . 2018PhRvB..97s5308J . 118956593 .
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 . 10.1103/PhysRevB.91.134306. 1411.1721 . 2015PhRvB..91m4306M . 54583870 .
8. Casimir . H.B.G. 1938. Note on the Conduction of Heat in Crystals. Physica. 5. 1938Phy.....5..495C. 6. 10.1016/S0031-8914(38)80162-2. 495–500.