Samiksha Jaiswal (Editor)

Phonon scattering

Updated on
Edit
Like
Comment
Share on FacebookTweet on TwitterShare on LinkedInShare on Reddit

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 which is the inverse of the corresponding relaxation time.

Contents

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

1 τ C = 1 τ U + 1 τ M + 1 τ B + 1 τ p h − e

The parameters τ U , τ M , τ B , τ p h − 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:

1 τ U = 2 γ 2 k B T μ V 0 ω 2 ω D

where γ is Gruneisen anharmonicity parameter, μ is shear modulus, V0 is volume per atom and ω D is Debye frequency.

Mass-difference impurity scattering

Mass-difference impurity scattering is given by:

1 τ M = 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 time is given by:

1 τ B = 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:

1 τ B = V D

This equation is also known as Casimir limit.

Phonon-electron scattering

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

1 τ p h − e = n e ϵ 2 ω ρ V 2 k B T π m ∗ V 2 2 k B T exp ⁡ ( − m ∗ V 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.

References

Phonon scattering Wikipedia
(Text) CC BY-SA