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:

$\frac{1}{{\tau }_M}=\frac{V_0\mathrm{\Gamma }{\omega }^4}{4\pi v_g^3}$

where $\mathrm{\Gamma }$ 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:

$\frac{1}{{\tau }_{ph-e}}=\frac{n_e{ϵ}^2\omega }{\rho V^2{k}_{B}T}\sqrt{\frac{\pi m^{*}{V}^2}{2k_{B}T}}\exp (-\frac{m^{*}{V}^2}{2k_{B}T})$

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.

References

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