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

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 the Gruneisen anharmonicity parameter, μ is the shear modulus, V0 is the volume per atom and 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:

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

Boundary scattering

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

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

where is the angle of incidence.[6] An extra factor of is sometimes erroneously included in the exponent of the above equation.[7] At normal incidence, , perfectly specular scattering (i.e. ) 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 the relaxation rate becomes

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:

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

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. S2CID 118984828.
  2. 1 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. S2CID 16015465.
  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 (19): 195308. arXiv:1712.05756. Bibcode:2018PhRvB..97s5308J. doi:10.1103/PhysRevB.97.195308. S2CID 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. arXiv:1411.1721. Bibcode:2015PhRvB..91m4306M. doi:10.1103/PhysRevB.91.134306. S2CID 54583870.
  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.
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