Evaluation of the diffuse mismatch model for phonon scattering at disordered interfaces

Diffuse phonon scattering strongly affects the phonon transport through a disordered interface. The often-used diffuse mismatch model assumes that phonons lose memory of their origin after being scattered by the interface. Using mode-resolved atomistic Green's function simulation, we demonstrate that diffuse phonon scattering by a single disordered interface cannot make a phonon lose its memory and thus the applicability of the diffusive mismatch model is limited. An analytical expression for diffuse scattering probability based on the continuum approximation is also derived and shown to work reasonably well at low frequencies.

Figure 1

(a) In AGF, the system is partitioned into three parts: two semi-infinite leads and a rough interface as the device region. The lead region in the supercell contains $N_{\mathrm{uc}}$ unitcells and transverse lattice vector ${\mathbf{R}}_{\mathrm{sc}}=N_{\mathrm{uc}}{\mathbf{R}}_{\mathrm{uc}}$. The numbers 0, 1,... denote the index of repeated cells for left and right leads. The period lengths along the direction normal to the interface are $a_{z,L}$ and $a_{z,R}$ for the left and right leads, respectively. (b) The in-plane wave vector for incident, transmitted, and reflected phonons, ${\mathbf{q}}_{\parallel ,n}$, ${\mathbf{q}}_{\parallel ,m}$, and ${\mathbf{q}}_{\parallel ,l}$. $a=0$ corresponds to specular transmission, while $a\ne 0$ corresponds to diffuse transmission. Similarly, $b=0$ corresponds to specular reflection, while $b\ne 0$ corresponds to diffuse reflection. Note the schematic is drawn for two-dimensional system for visual clarity and for three-dimensional system the partitioning and wave-vector conservation laws can be analogously defined.

Figure 2

The angle-resolved (a) diffuse transmittance $T_{\mathrm{d},\mathrm{Si}\to \mathrm{Ge}}\left({\mathrm{\Omega }}_{\mathrm{Si}}\right)$ from the Si side and (b) diffuse reflectance $R_{\mathrm{d},\mathrm{Ge}\to \mathrm{Ge}}\left({\mathrm{\Omega }}_{\mathrm{Ge}}\right)$ from the Ge side at $\omega =5$ THz. The angle-resolved (c) diffuse transmittance $T_{\mathrm{d},\mathrm{Ge}\to \mathrm{Si}}\left({\mathrm{\Omega }}_{\mathrm{Ge}}\right)$ from the Ge side and (d) diffuse reflectance $R_{\mathrm{d},\mathrm{Si}\to \mathrm{Si}}\left({\mathrm{\Omega }}_{\mathrm{Si}}\right)$ from the Si side at $\omega =5$ THz. ${\mathrm{\Omega }}_{\alpha }=({\theta }_{\alpha },{\varphi }_{\alpha })$, $\alpha =\mathrm{Si},\mathrm{Ge}$ is the direction of incident group velocity. The radial coordinate corresponds to the polar angle ${\theta }_{\alpha }$ (angle of incidence) and the polar axis corresponds to the azimuthal angle ${\varphi }_{\alpha }$. (e), (f) The difference between maximum and minimum diffuse scattering probability as a measure of the anisotropy of diffuse scattering probability. The diffuse scattering probability is obtained by taking the ensemble average of calculations for 21 structures of 8 ml disordered configurations with a $20\times 20{\mathbf{q}}_{\mathrm{sc},\parallel }$-point mesh.

Figure 3

(a), (b) The diffuse transmittance $T_{\mathrm{d},\alpha }$ from one side and the diffuse reflectance $R_{\mathrm{d},\beta }$ from the other side as a function of the polar angle of the incident phonon at $\omega =3.3$ THz. The markers are obtained by integrating $T_{\mathrm{d}}\left(\mathrm{\Omega }\right)$ and $R_{\mathrm{d}}\left(\mathrm{\Omega }\right)$ from AGF calculation for 8 ml structures over the azimuthal angle $\varphi$ divided by $2\pi$. The solid lines are predictions from continuum modeling with the number of pairs of swapped atoms per unit area $n=2.78/a^2$ and $a=5.527$ Å. (c), (d) The average diffuse transmittance $T_{\mathrm{d},\alpha }\left(\omega \right)$ from one side and the average diffuse reflectance $R_{\mathrm{d},\beta }\left(\omega \right)$ from the other side as a function of frequency from AGF calculation in solid lines, compared with DMM in dash-dot lines. (e)–(h) are the specular transmittance and specular reflectance corresponding to (a)–(d).

Figure 4

The frequency-resolved (a) diffuse and (b) total transmittance/reflectance from Si and Ge side with different $m_{\mathrm{Ge}}/m_{\mathrm{Si}}$ from AGF calculation for 8-ml structures and DMM. ${\omega }_{\max }$ is the maximum allowed frequency for nonzero transmission function $\mathrm{\Theta }\left(\omega \right)$. ${\omega }_{\max }=17.2$ THz when $m_{\mathrm{Ge}}/m_{\mathrm{Si}}=1.1$, and ${\omega }_{\max }=9.1$ THz when $m_{\mathrm{Ge}}/m_{\mathrm{Si}}=4.0$.

Figure 5

The similarity between the diffuse transmittance from one side and the diffuse reflectance from the other side. The blue lines correspond to the cases when the final states are on the Ge side and the orange lines correspond to the cases where the final states are on the Si side. The diffuse transmittance and reflectance for evaluating the similarity are from the data presented in Fig. 4.

Figure 6

(a) The total transmission function, (b) the specular transmission function, and (c) the diffuse transmission function versus phonon frequency for [001] Si/Ge interfaces from AGF. Inset: The ensemble-averaged mass profile as a function of atom layer number at the interface region obtained by averaging the mass of atoms within the same layer. The right axis is the Ge fraction in each atom layer. The distance between adjacent atom layers is $a/4=1.382$ Å. (d)–(f) The total, specular and diffuse reflection function from the Si side. (g)–(i) The total, specular, and diffuse reflection function from the Ge side.

Figure 7

The transmittance and reflectance for a rough Si/Ge interface predicted from the continuum model. (a), (b) The specular and diffuse transmittance of acoustic phonons at 4 THz in Si and Ge compared with AMM. (c), (d) The specular and diffuse reflectance of acoustic phonons at 4 THz in Si and Ge compared with AMM. (e), (f) The diffuse transmittance from one side and reflectance from the other side of acoustic phonons at 4 THz. $\theta$ is the velocity angle of the incident state.

Figure 8

(a)–(d) The frequency-resolved transmittance and reflectance from Si side and Ge side from continuum modeling. (e) The transmission function $\mathrm{\Theta }\left(\omega \right)$ for Si/Ge interface as a function of frequency. A multiplicity factor of 3 is multiplied in the transmission function as there are three acoustic phonon branches. When the frequency is much higher than 5 Thz, the lowest perturbation theory is no longer valid, as the perturbed part becomes large.