Graded manifold

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

${\frac {1}{\tau _{C}}}={\frac {1}{\tau _{U}}}+{\frac {1}{\tau _{M}}}+{\frac {1}{\tau _{B}}}+{\frac {1}{\tau _{ph-e}}}$

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

${\frac {1}{\tau _{U}}}=2\gamma ^{2}{\frac {k_{B}T}{\mu V_{0}}}{\frac {\omega ^{2}}{\omega _{D}}}$

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

Mass-difference impurity scattering

Mass-difference impurity scattering is given by:

${\frac {1}{\tau _{M}}}={\frac {V_{0}\Gamma \omega ^{4}}{4\pi v_{g}^{3}}}$

where $\Gamma$ 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 time is given by:

${\frac {1}{\tau _{B}}}={\frac {V}{D}}(1-p)$

where D is the dimension of the system and p represents the surface roughness parameter. The value p=1 means a smooth perfect surface that the scattering is purely specular and the relaxation time goes to ∞; hence, boundary scattering does not affect thermal transport. The value p=0 represents a very rough surface that the scattering is then purely diffusive which gives:

${\frac {1}{\tau _{B}}}={\frac {V}{D}}$

This equation is also known as Casimir limit.

Phonon-electron scattering

Template:Mergefrom Phonon-electron scattering can also contribute when the material is lightly doped. The corresponding relaxation time is given as:

${\frac {1}{\tau _{ph-e}}}={\frac {n_{e}\epsilon ^{2}\omega }{\rho V^{2}k_{B}T}}{\sqrt {\frac {\pi m^{*}V^{2}}{2k_{B}T}}}\exp \left(-{\frac {m^{*}V^{2}}{2k_{B}T}}\right)$

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.

References

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