# Transmission equations

## 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:

$$
\alpha^{\mathrm{tot}}*{2T}(\theta,\omega,p)
\= T*{1}(\theta\_i,\omega,p),\times,T\_{2}(\theta\_t,\omega,p)
$$

where θ<sub>i</sub> is the incident angle in Si, θ<sub>t</sub> 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 *k*<sub>*m*</sub> 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 *θ*<sub>*i*</sub> the incident angle between the projection to the x-y plane and y-axis and *θ*<sub>*j*</sub> is the exit angle, *v*<sub>*i*</sub> and *v*<sub>j</sub> 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 → Ge
2. Exit Ge → Si

Both must lie in $$\[-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&#x20;

$$Z\_m=\rho\_m v\_{g,m}$$&#x20;

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

$$
\alpha^{\mathrm{Spec}}\_{i\to j}(\theta\_i,\omega,p) =
\frac{4 Z\_i Z\_j |\cos\theta\_i|,|\cos\theta\_j|}
{(Z\_i |\cos\theta\_i| + Z\_j |\cos\theta\_j|)^2}
$$

where θ<sub>j</sub> is obtained from Snell’s law.

**Diffuse (DMM-like) transmission.**

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

$$
\alpha^{\mathrm{Diff}}\_{i\to j}(\omega,p) =
\frac{(1-P\_j),k\_j(\omega,p)^2}
{(1-P\_i),k\_i(\omega,p)^2+(1-P\_j),k\_j(\omega,p)^2}
$$

***

#### 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.)

***

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

For the Ge 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<sub>i</sub>, P<sub>j</sub>; we obtain *θ*<sub>*t*</sub> 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 $$(v\_{g,i}/v\_{g,j})\sin\theta\_i > 1$$, 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.

***

#### 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).