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/${\displaystyle \tau }$ 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 ${\displaystyle \tau _{C}}$ can be written as:

${\displaystyle {\frac {1}{\tau _{C}}}={\frac {1}{\tau _{U}}}+{\frac {1}{\tau _{M}}}+{\frac {1}{\tau _{B}}}+{\frac {1}{\tau _{\text{ph-e}}}}}$

The parameters ${\displaystyle \tau _{U}}$, ${\displaystyle \tau _{M}}$, ${\displaystyle \tau _{B}}$, ${\displaystyle \tau _{\text{ph-e}}}$ 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 ${\displaystyle \omega }$ and umklapp processes vary with ${\displaystyle \omega ^{2}}$, Umklapp scattering dominates at high frequency. ^[1] ${\displaystyle \tau _{U}}$ is given by:

${\displaystyle {\frac {1}{\tau _{U}}}=2\gamma ^{2}{\frac {k_{B}T}{\mu V_{0}}}{\frac {\omega ^{2}}{\omega _{D}}}}$

where ${\displaystyle \gamma }$ is the Gruneisen anharmonicity parameter, μ is the shear modulus, V₀ is the volume per atom and ${\displaystyle \omega _{D}}$ 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:

${\displaystyle {\frac {1}{\tau _{M}}}={\frac {V_{0}\Gamma \omega ^{4}}{4\pi v_{g}^{3}}}}$

where ${\displaystyle \Gamma }$ is a measure of the impurity scattering strength. Note that ${\displaystyle {v_{g}}}$ is dependent of the dispersion curves.

Boundary scattering

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

${\displaystyle {\frac {1}{\tau _{B}}}={\frac {V}{L_{0}}}(1-p)}$

where ${\displaystyle L_{0}}$ is the characteristic length of the system and ${\displaystyle p}$ represents the fraction of specularly scattered phonons. The ${\displaystyle p}$ parameter is not easily calculated for an arbitrary surface. For a surface characterized by a root-mean-square roughness ${\displaystyle \eta }$, a wavelength-dependent value for ${\displaystyle p}$ can be calculated using

${\displaystyle p(\lambda )=\exp {\Bigg (}-16{\frac {\pi ^{2}}{\lambda ^{2}}}\eta ^{2}\cos ^{2}\theta {\Bigg )}}$

where ${\displaystyle \theta }$ is the angle of incidence. ^[6] An extra factor of ${\displaystyle \pi }$ is sometimes erroneously included in the exponent of the above equation. ^[7] At normal incidence, ${\displaystyle \theta =0}$, perfectly specular scattering (i.e. ${\displaystyle p(\lambda )=1}$) 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 ${\displaystyle p=0}$ the relaxation rate becomes

${\displaystyle {\frac {1}{\tau _{B}}}={\frac {V}{L_{0}}}}$

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:

${\displaystyle {\frac {1}{\tau _{\text{ph-e}}}}={\frac {n_{e}\epsilon ^{2}\omega }{\rho V^{2}k_{B}T}}{\sqrt {\frac {\pi m^{*}V^{2}}{2k_{B}T}}}\exp \left(-{\frac {m^{*}V^{2}}{2k_{B}T}}\right)}$

The parameter ${\displaystyle n_{e}}$ 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.

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. arXiv: cond-mat/0308587 . Bibcode:2003PhRvB..68k3308M. doi:10.1103/PhysRevB.68.113308.
2. 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. Archived from the original (PDF) on 2010-06-18.
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. arXiv: 1510.00706 . Bibcode:2016PhRvB..96p5202F. doi:10.1103/PhysRevB.93.045202.
5. Feng, Tianli; Lindsay, Lucas; Ruan, Xiulin (2017). "Four-phonon scattering significantly reduces intrinsic thermal conductivity of solids". Physical Review B. 96 (16): 161201. Bibcode:2017PhRvB..96p1201F. doi: 10.1103/PhysRevB.96.161201 .
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: 195308. doi:10.1103/PhysRevB.97.195308.
7. Maznev, A. (2015). "Boundary scattering of phonons: Specularity of a randomly rough surface in the small-perturbation limit". Phys. Rev. B. 91: 134306. doi:10.1103/PhysRevB.91.134306.
8. Casimir, H.B.G (1938). "Note on the Conduction of Heat in Crystals". Physica. 5 (6): 495–500. Bibcode:1938Phy.....5..495C. doi:10.1016/S0031-8914(38)80162-2.