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/[math]\displaystyle{ \tau }[/math] 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 [math]\displaystyle{ \tau_{C} }[/math] can be written as:

[math]\displaystyle{ \frac{1}{\tau_C} = \frac{1}{\tau_U}+\frac{1}{\tau_M}+\frac{1}{\tau_B}+\frac{1}{\tau_\text{ph-e}} }[/math]

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

[math]\displaystyle{ \frac{1}{\tau_U}=2\gamma^2\frac{k_B T}{\mu V_0}\frac{\omega^2}{\omega_D} }[/math]

where [math]\displaystyle{ \gamma }[/math] is the Gruneisen anharmonicity parameter, μ is the shear modulus, V0 is the volume per atom and [math]\displaystyle{ \omega_{D} }[/math] 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:

[math]\displaystyle{ \frac{1}{\tau_M}=\frac{V_0 \Gamma \omega^4}{4\pi v_g^3} }[/math]

where [math]\displaystyle{ \Gamma }[/math] is a measure of the impurity scattering strength. Note that [math]\displaystyle{ {v_g} }[/math] is dependent of the dispersion curves.

## Boundary scattering

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

[math]\displaystyle{ \frac{1}{\tau_B}=\frac{v_g}{L_0}(1-p) }[/math]

where [math]\displaystyle{ L_0 }[/math] is the characteristic length of the system and [math]\displaystyle{ p }[/math] represents the fraction of specularly scattered phonons. The [math]\displaystyle{ p }[/math] parameter is not easily calculated for an arbitrary surface. For a surface characterized by a root-mean-square roughness [math]\displaystyle{ \eta }[/math], a wavelength-dependent value for [math]\displaystyle{ p }[/math] can be calculated using

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

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

[math]\displaystyle{ \frac{1}{\tau_B}=\frac{v_g}{L_0} }[/math]

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:

[math]\displaystyle{ \frac{1}{\tau_\text{ph-e}}=\frac{n_e \epsilon^2 \omega}{\rho v_g^2 k_B T}\sqrt{\frac{\pi m^* v_g^2}{2k_B T}} \exp \left(-\frac{m^*v_g^2}{2k_B T}\right) }[/math]

The parameter [math]\displaystyle{ n_{e} }[/math] 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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