Phonon scattering · Lablypedia

All scattering processes can be taken into account using Matthiessen's rule. Then the combined relaxation time $τ_{C}$ can be written as:

\frac{1}{τ_{C}} = \frac{1}{τ_{U}} + \frac{1}{τ_{M}} + \frac{1}{τ_{B}} + \frac{1}{τ_{ph-e}}

The parameters $τ_{U}$ , $τ_{M}$ , $τ_{B}$ , $τ_{ph-e}$ 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 $ω$ and umklapp processes vary with $ω^{2}$ , Umklapp scattering dominates at high frequency. $τ_{U}$ is given by:

\frac{1}{τ_{U}} = 2 γ^{2} \frac{k_{B} T}{μ V_{0}} \frac{ω^{2}}{ω_{D}}

where $γ$ is the Gruneisen anharmonicity parameter, μ is the shear modulus, V₀ is the volume per atom and $ω_{D}$ is the Debye frequency.

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Three-phonon and four-phonon process

Thermal transport in non-metal solids was usually considered to be governed by the three-phonon scattering process, 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 and for certain materials at room temperature. 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:

\frac{1}{τ_{M}} = \frac{V_{0} Γ ω^{4}}{4 π v_{g}^{3}}

where $Γ$ 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:

\frac{1}{τ_{B}} = \frac{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 using

p (λ) = \exp (- 16 \frac{π^{2}}{λ^{2}} η^{2} \cos^{2} θ)

where $θ$ is the angle of incidence. An extra factor of $π$ is sometimes erroneously included in the exponent of the above equation. At normal incidence, $θ = 0$ , 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

\frac{1}{τ_{B}} = \frac{v_{g}}{L_{0}}

This equation is also known as Casimir limit.

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:

\frac{1}{τ_{ph-e}} = \frac{n_{e} ϵ^{2} ω}{ρ v_{g}^{2} k_{B} T} \sqrt{\frac{π m^{*} v_{g}^{2}}{2 k_{B} T}} \exp (- \frac{m^{*} v_{g}^{2}}{2 k_{B} T})

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 .