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

1 τ C = 1 τ U + 1 τ M + 1 τ B + 1 τ ph-e {\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 τ U {\displaystyle \tau _{U}}, τ M {\displaystyle \tau _{M}}, τ B {\displaystyle \tau _{B}}, τ ph-e {\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 ω {\displaystyle \omega } and umklapp processes vary with ω 2 {\displaystyle \omega ^{2}}, Umklapp scattering dominates at high frequency. τ U {\displaystyle \tau _{U}} is given by:

1 τ U = 2 γ 2 k B T μ V 0 ω 2 ω D {\displaystyle {\frac {1}{\tau _{U}}}=2\gamma ^{2}{\frac {k_{B}T}{\mu V_{0}}}{\frac {\omega ^{2}}{\omega _{D}}}}

where γ {\displaystyle \gamma } is the Gruneisen anharmonicity parameter, μ is the shear modulus, V0 is the volume per atom and ω D {\displaystyle \omega _{D}} is the Debye frequency.

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:

1 τ M = V 0 Γ ω 4 4 π v g 3 {\displaystyle {\frac {1}{\tau _{M}}}={\frac {V_{0}\Gamma \omega ^{4}}{4\pi v_{g}^{3}}}}

where Γ {\displaystyle \Gamma } is a measure of the impurity scattering strength. Note that v g {\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:

1 τ B = v g L 0 ( 1 − p ) {\displaystyle {\frac {1}{\tau _{B}}}={\frac {v_{g}}{L_{0}}}(1-p)}

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

p ( λ ) = exp ⁡ ( − 16 π 2 λ 2 η 2 cos 2 ⁡ θ ) {\displaystyle p(\lambda )=\exp {\Bigg (}-16{\frac {\pi ^{2}}{\lambda ^{2}}}\eta ^{2}\cos ^{2}\theta {\Bigg )}}

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

1 τ B = v g L 0 {\displaystyle {\frac {1}{\tau _{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:

1 τ ph-e = n e ϵ 2 ω ρ v g 2 k B T π m ∗ v g 2 2 k B T exp ⁡ ( − m ∗ v g 2 2 k B T ) {\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)}