Transmission equations
Formalism used to model transmission through thin layers or another material
Spectral hybrid transmission model
Scope and geometry
For example, we want to model phonon transport through a SiGe layer embedded in Si crystal. A phonon crossing this inclusion experiences two sequential interface events:
entrance: Si → Ge
exit: Ge → Si
Thus, the total transmission probability is the product of the two steps:
where θi is the incident angle in Si, θt is the transmitted angle inside Ge determined by Snell’s law (4), ω is the angular frequency, and p the branch/polarization. The SMMM model used in the simulation and is summarized here. A detailed description is provided in [1].
Equation used in the code
For each material m and branch p, the code interpolates the dispersion relation to obtain km and thus the wavelength:
The interface specularity depends on the roughness and follows Ziman’s exponential law. It is different from the angle-dependent specularity equation used in FreePATHS. For the SMMM model, this equation is angle-independent. (Eq. 32 in [1]).
In case of specular scattering, the transmitted phonons are diverted, satisfying Snell's law (Eq. 18 in [1]):
with θi the incident angle between the projection to the x-y plane and y-axis and θj is the exit angle, vi and vj are the group velocities of the incident and transmitted material, respectively. In case of:
no real transmitted angle exists: the code sets
and returns a specular reflection. In that case, only the diffuse channel can still transmit phonons across the layer.
In scattering primitives with “2T” signature, two tests are made:
Entry Si → Ge
Exit Ge → Si
Both must lie in , otherwise the code sends a specular reflection.
Specular (AMM-like) transmission.
For the specular specularity we use the classical AMM form with acoustic impedances
(which is simplified Eq. 25 in [1]):
where θj is obtained from Snell’s law.
Diffuse (DMM-like) transmission.
For the diffuse scattering, we use the equation below (Eq. 30 in [1]):
One interface: spectral convex combination (SMMM form)
Following the spectral mixed-mismatch idea, the total transmissivity of one interface is the convex combination of the specular and diffuse parts weighted by the incident-side specularity:
(The reverse direction uses and the corresponding angles and specularity.)
Two interfaces in series (Si→Ge then Ge→Si)
For the Ge mini-layer, we evaluate:
and combine them as:
In practice, we compute wave vectors from the tabulated dispersions, then wavelengths, then Pi, Pj; we obtain θt from Snell’s law, and finally evaluate the equations above.
Modeling notes (as coded)
Elastic, branch-conserving scattering: no inelasticity or polarization conversion; transmissions are branch-wise (LA/TA) and elastic.
Total internal reflection (TIR): if , then and only the diffuse formula contributes.
Independent sequential events: the two interfaces (entry/exit) are treated as independent; coherent interference and internal multiple reflections are neglected for simplicity; this matches the intended use for thin, rough, incoherent mini-layers.
Angle handling: is used in the AMM formula to avoid sign issues for grazing angles in numerical implementation.
Justification for the 2T construction
Representing the SiGe layer as a finite mini-rectangle naturally yields two heterogeneous boundaries. The 2T product allows the inclusion to act as a virtual slab, forcing an entry (Si→Ge) and an exit (Ge→Si) event, each mixed by roughness-dependent specularity via the SMMM equations.
A simple per-frequency, per-branch model that captures roughness at both interfaces while respecting detailed balance between AMM and DMM.
By contrast, if the geometry involves only a single Si/Ge boundary (for example, a rectangle of SiGe inside the Si sample), the proper choice is the 1T case: drop the 2T formula and use the single-interface SMMM instead.
References
X. Ran and B. Cao, Roughness dependence of phonon-interface thermal transport: Theoretical model and Monte Carlo simulation, Phys. Rev. B 110, 024302 (2024).
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