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/[math]\displaystyle{ \tau }[/math] 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 [math]\displaystyle{ \tau_{C} }[/math] can be written as:

[math]\displaystyle{ \frac{1}{\tau_C} = \frac{1}{\tau_U}+\frac{1}{\tau_M}+\frac{1}{\tau_B}+\frac{1}{\tau_\text{ph-e}} }[/math]

The parameters [math]\displaystyle{ \tau_{U} }[/math], [math]\displaystyle{ \tau_{M} }[/math], [math]\displaystyle{ \tau_{B} }[/math], [math]\displaystyle{ \tau_\text{ph-e} }[/math] 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 [math]\displaystyle{ \omega }[/math] and umklapp processes vary with [math]\displaystyle{ \omega^2 }[/math], Umklapp scattering dominates at high frequency.^[1] [math]\displaystyle{ \tau_U }[/math] is given by:

[math]\displaystyle{ \frac{1}{\tau_U}=2\gamma^2\frac{k_B T}{\mu V_0}\frac{\omega^2}{\omega_D} }[/math]

where [math]\displaystyle{ \gamma }[/math] is the Gruneisen anharmonicity parameter, μ is the shear modulus, V₀ is the volume per atom and [math]\displaystyle{ \omega_{D} }[/math] 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:

[math]\displaystyle{ \frac{1}{\tau_M}=\frac{V_0 \Gamma \omega^4}{4\pi v_g^3} }[/math]

where [math]\displaystyle{ \Gamma }[/math] is a measure of the impurity scattering strength. Note that [math]\displaystyle{ {v_g} }[/math] is dependent of the dispersion curves.

Boundary scattering

Boundary scattering is particularly important for low-dimensional nanostructures and its relaxation rate is given by:

[math]\displaystyle{ \frac{1}{\tau_B}=\frac{v_g}{L_0}(1-p) }[/math]

where [math]\displaystyle{ L_0 }[/math] is the characteristic length of the system and [math]\displaystyle{ p }[/math] represents the fraction of specularly scattered phonons. The [math]\displaystyle{ p }[/math] parameter is not easily calculated for an arbitrary surface. For a surface characterized by a root-mean-square roughness [math]\displaystyle{ \eta }[/math], a wavelength-dependent value for [math]\displaystyle{ p }[/math] can be calculated using

[math]\displaystyle{ p(\lambda) = \exp\Bigg(-16\frac{\pi^2}{\lambda^2}\eta^2\cos^2\theta \Bigg) }[/math]

where [math]\displaystyle{ \theta }[/math] is the angle of incidence.^[6] An extra factor of [math]\displaystyle{ \pi }[/math] is sometimes erroneously included in the exponent of the above equation.^[7] At normal incidence, [math]\displaystyle{ \theta=0 }[/math], perfectly specular scattering (i.e. [math]\displaystyle{ p(\lambda)=1 }[/math]) 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 [math]\displaystyle{ p=0 }[/math] the relaxation rate becomes

[math]\displaystyle{ \frac{1}{\tau_B}=\frac{v_g}{L_0} }[/math]

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:

[math]\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) }[/math]

The parameter [math]\displaystyle{ n_{e} }[/math] 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 ^{[citation needed]}.

See also

• Lattice scattering
• Umklapp scattering
• Electron-longitudinal acoustic phonon interaction

References

0.00 (0 votes) Original source: https://en.wikipedia.org/wiki/Phonon scattering. Read more