Phonon scattering

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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.

Contents

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

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References

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