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 → SiGe

2. exit: SiGe → Si

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

$$\alpha^{\mathrm{tot}}_{2T}(\theta,\omega,p) = T_{1}(\theta_i,\omega,p)\,\times\,T_{2}(\theta_t,\omega,p)$$

where $\theta_i$ is the incident angle in Si, $\theta_t$ is the transmitted angle inside SiGe determined by Snell’s law (4), $\omega$ is the angular frequency, and $p$ the branch/polarization. The SMMM model is used in the simulation and is summarized here. A detailed description is provided in [1].

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Equation used in the code

For each material m and branch p, the code interpolates the dispersion relation to obtain k_m and thus the wavelength:

$$\lambda_m(\omega,p) = \frac{2\pi}{k_m(\omega,p)}$$

The interface specularity $P_m$ depends on the roughness $\eta$ 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]).

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

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

$$\frac{v_{g,i}(\omega,p)}{\sin\theta_i} = \frac{v_{g,j}(\omega,p)}{\sin\theta_j}$$

with $\theta_i$ the incident angle between the projection to the x-y plane and y-axis and $\theta_j$ the exit angle, $v_i$ and $v_j$ are the group velocities of the incident and transmitted material, respectively. In case of:

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

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

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 → SiGe

2. Exit SiGe → Si

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

Specular (AMM-like) transmission.

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

$Z_m=\rho_m v_{g,m}$

(Eq. 20 in [1], which that paper writes as $4AB/(A+B)^2$ with $A=Z_j/Z_i$ and $B=\cos\theta_j/\cos\theta_i$):

$$\alpha^{\mathrm{Spec}}_{i\to j}(\theta_i,\omega,p) = \frac{4 Z_i Z_j |\cos\theta_i|\,|\cos\theta_j|} {(Z_i |\cos\theta_j| + Z_j |\cos\theta_i|)^2}$$

where $\theta_j$ is obtained from Snell’s law.

Note that the cosines in the denominator are crossed: $\theta_j$ pairs with $Z_i$ and $\theta_i$ with $Z_j$. This is the standard oblique-incidence acoustic result $4W_iW_j/(W_i+W_j)^2$ written with the angle-corrected impedances $W_m = Z_m/\cos\theta_m$. The uncrossed form agrees at normal incidence but over-estimates transmission at oblique incidence (for Si/SiGe LA at 1 THz, 0.990 instead of 0.928 at 60°). FreePATHS used the uncrossed denominator until 2026-09-25; results produced with VerticalPlane before that date are slightly under-reflective at oblique incidence.

The MTM detailed-balance prefactor of Eq. 25 in [1],

$$\min\!\left\{1,\;\frac{P_j C_j v_{g,j}^3}{P_i C_i v_{g,i}^3}\right\},$$

is not applied — a deliberate simplification. For Si/SiGe it equals 0.98, 0.91 and 0.70 at 0.5, 1 and 2 THz, so omitting it makes the layer slightly more transmissive at low frequency (of order 3 percentage points in $R$ at 1 THz).

Diffuse (DMM-like) transmission.

For the diffuse scattering, we use the equation below (Eq. 30 in [1]):

$$\alpha^{\mathrm{Diff}}_{i\to j}(\omega,p) = \frac{(1-P_j)\,k_j(\omega,p)^2} {(1-P_i)\,k_i(\omega,p)^2+(1-P_j)\,k_j(\omega,p)^2}$$

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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:

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

(The reverse direction $j \to i$ uses $P_j$ and the corresponding angles and specularity.)

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

For the SiGe mini-layer, we evaluate:

$$T_{1}(\theta_i,\omega,p) = P_i\,\alpha^{\mathrm{Spec}}_{i\to j}(\theta_i,\omega,p) + (1-P_i)\,\alpha^{\mathrm{Diff}}_{i\to j}(\omega,p)$$

$$T_{2}(\theta_t,\omega,p) = P_j\,\alpha^{\mathrm{Spec}}_{j\to i}(\theta_t,\omega,p) + (1-P_j)\,\alpha^{\mathrm{Diff}}_{j\to i}(\omega,p)$$

and combine them as:

$$\alpha^{\mathrm{tot}}_{2T}(\theta,\omega,p) = T_1 \times T_2$$

In practice, we compute wave vectors from the tabulated dispersions, then wavelengths, then $P_i$, $P_j$; we obtain $\theta_t$ from Snell’s law, and finally evaluate the equations above.

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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 $(v_{g,i}/v_{g,j})\sin\theta_i > 1$, then $\alpha^{\mathrm{Spec}}_{i\to j}=0$ 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: $|\cos\theta|$ is used in the AMM formula to avoid sign issues for grazing angles in numerical implementation.

• MTM prefactor omitted: the specular channel implements Eq. 20 of [1], not the MTM-corrected Eqs. 25–26; see the note under Specular (AMM-like) transmission above.

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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→SiGe) and an exit (SiGe→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/SiGe 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.

Not currently implemented. The 1T code path (alpha_total_1T and the RectangularBulk scatterer that was its only caller) was removed in October 2026 as unfinished work — it had never been functional. The single-interface equations above remain correct and are the building blocks alpha_total_2T uses for each of its two crossings, so a 1T scatterer can be recoded from this page if a single-boundary inclusion is needed.

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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).