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:

  1. entrance: Si → Ge

  2. exit: Ge → Si

Thus, the total transmission probability is the product of the two steps:

α2Ttot(θ,ω,p)=T1(θi,ω,p)×T2(θt,ω,p)\alpha^{\mathrm{tot}}_{2T}(\theta,\omega,p) = T_{1}(\theta_i,\omega,p)\,\times\,T_{2}(\theta_t,\omega,p)

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:

λm(ω,p)=2πkm(ω,p)\lambda_m(\omega,p) = \frac{2\pi}{k_m(\omega,p)}

The interface specularity PmP_m depends on the roughness η\eta 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]).

Pm(ω,p)=exp ⁣(16π2η2λm(ω,p)2)P_m(\omega,p) = \exp\!\left(-\frac{16\pi^2 \eta^2}{\lambda_m(\omega,p)^2}\right)

In case of specular scattering, the transmitted phonons are diverted, satisfying Snell's law (Eq. 18 in [1]):

vg,i(ω,p)sinθi=vg,j(ω,p)sinθj\frac{v_{g,i}(\omega,p)}{\sin\theta_i} = \frac{v_{g,j}(\omega,p)}{\sin\theta_j}

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:

vg,jvg,isinθi>1\bigl|\tfrac{v_{g,j}}{v_{g,i}}\sin\theta_i\bigr|>1

no real transmitted angle exists: the code sets αijSpec=0\alpha^{\mathrm{Spec}}_{i\to j}=0

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:

  1. Entry Si → Ge

  2. Exit Ge → Si

Both must lie in [1,1][-1, 1], otherwise the code sends a specular reflection.

Specular (AMM-like) transmission.

For the specular specularity we use the classical AMM form with acoustic impedances

Zm=ρmvg,mZ_m=\rho_m v_{g,m}

(which is simplified Eq. 25 in [1]):

αijSpec(θi,ω,p)=4ZiZjcosθicosθj(Zicosθi+Zjcosθj)2\alpha^{\mathrm{Spec}}_{i\to j}(\theta_i,\omega,p) = \frac{4 Z_i Z_j |\cos\theta_i|\,|\cos\theta_j|} {(Z_i |\cos\theta_i| + Z_j |\cos\theta_j|)^2}

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]):

αijDiff(ω,p)=(1Pj)kj(ω,p)2(1Pi)ki(ω,p)2+(1Pj)kj(ω,p)2\alpha^{\mathrm{Diff}}_{i\to j}(\omega,p) = \frac{(1-P_j)\,k_j(\omega,p)^2} {(1-P_i)\,k_i(\omega,p)^2+(1-P_j)\,k_j(\omega,p)^2}

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:

αij(θi,ω,p)=Pi(ω,p)αijSpec(θi,ω,p)+(1Pi(ω,p))αijDiff(ω,p)\begin{aligned} \alpha_{i\to j}(\theta_i,\omega,p) &= P_i(\omega,p)\,\alpha^{\mathrm{Spec}}_{i\to j}(\theta_i,\omega,p) + (1-P_i(\omega,p))\,\alpha^{\mathrm{Diff}}_{i\to j}(\omega,p) \end{aligned}

(The reverse direction jij \to i uses PjP_j and the corresponding angles and specularity.)


Two interfaces in series (Si→Ge then Ge→Si)

For the Ge mini-layer, we evaluate:

T1(θi,ω,p)=PiαijSpec(θi,ω,p)+(1Pi)αijDiff(ω,p)T_{1}(\theta_i,\omega,p) = P_i\,\alpha^{\mathrm{Spec}}_{i\to j}(\theta_i,\omega,p) + (1-P_i)\,\alpha^{\mathrm{Diff}}_{i\to j}(\omega,p)
T2(θt,ω,p)=PjαjiSpec(θt,ω,p)+(1Pj)αjiDiff(ω,p)T_{2}(\theta_t,\omega,p) = P_j\,\alpha^{\mathrm{Spec}}_{j\to i}(\theta_t,\omega,p) + (1-P_j)\,\alpha^{\mathrm{Diff}}_{j\to i}(\omega,p)

and combine them as:

α2Ttot(θ,ω,p)=T1×T2\alpha^{\mathrm{tot}}_{2T}(\theta,\omega,p) = T_1 \times T_2

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 (vg,i/vg,j)sinθi>1(v_{g,i}/v_{g,j})\sin\theta_i > 1, then αijSpec=0\alpha^{\mathrm{Spec}}_{i\to j}=0 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: cosθ|\cos\theta| 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

  1. X. Ran and B. Cao, Roughness dependence of phonon-interface thermal transport: Theoretical model and Monte Carlo simulation, Phys. Rev. B 110, 024302 (2024).

Last updated