Understanding the mechanism of diffuse phonon scattering at disordered surfaces by atomistic wave-packet investigation

Phonon surface scattering is of great importance for understanding thermal transport in nanostructured materials and has been widely utilized to tailor thermal properties. However, the current understanding of phonon surface scattering is largely based on the Ziman's formula which was derived at the continuum limit of the scalar wave equation and, thus, ignoring the atomistic information of the surface. In this work, we applied the atomistic phonon wave-packet simulations to study the impact of surface amorphization and surface roughness on phonon reflection. We found that for both types of surfaces, the energies of the specularly reflected components decrease with increasing wavelength. However, the underlying reflection mechanisms are different. For the amorphous surface, the energy of the incident wave packet will first enter the amorphous layer before being partially and diffusely scattered in this layer. The local density and elastic moduli fluctuations as well as the localized vibrational modes are reasons for the diffuse scattering in the amorphous region, which affects the short wavelength phonons more severely. For the rough surface, the diffuse reflection is the combined effect from surface irregularity, surface Rayleigh waves, and spatially localized modes induced by the surface roughness. The extent of diffuse reflection at the rough surface can be tuned by engineering the surface boundary condition.

Figure 1

(a) Schematic diagram of the simulation domain used in this work. The size of the simulation domain is $200\times 400$ nm${}^2$ in the $xy$ direction. The thickness of the domain in the $z$ direction is $\sim 2$ nm with periodic boundary condition. The incident wave packet hit the right hand side surface with an incident angle of ${45}^{\circ }$. (b) Illustration of amorphous coating surface. The thickness of the amorphous layer varies from 2–10 nm. (c) Illustration of rough surface with random pillars. The separation between adjacent pillars follows a Gaussian distribution with a mean value of 4 nm and standard deviation of 1 nm. The height of the pillar is 2 nm or 8 nm to model surface with different roughness. The outmost layer of the rough surface is set to be fixed or free.

Figure 2

Illustration of wave-packet energy distribution in reciprocal space after reflection. The energy of the incident wave packet is centered at $\mathbf{k}=(0.2,0.2,0)\times 2\pi /a$. After reflection, two major peaks that obey the Snell's law occur, with the $k_y$ value unchanged.

Figure 3

(a) Dependence of specularity on incident wavelength; the thickness of the amorphous layer is 10 nm. (b) Dependence of specularity on the thickness of amorphous coating layer; the wavelength of the incoming phonon is set to be 3.8 nm.

Figure 4

Density variation in amorphous silicon at different length scale.

Figure 5

(a) Dynamics structure factor of transverse and longitudinal modes in amorphous silicon. (b) The linewidths of transverse and longitudinal modes in amorphous silicon.

Figure 6

Transporting of phonon wave packet in amorphous silicon. Color represents the kinetic energy profile of the wave packet (arbitrary units: red, high; blue, low)

Figure 7

Specularity at (a) fixed rough surface with the pillar height of 8 nm, (b) free rough surface with the pillar height of 8 nm, and (c) free surface with the pillar height of 2 nm.

Figure 8

Illustration of energy distribution in real space. The incident wave packet is TA1 polarized with $\mathbf{k}=(0.1,0.1,0)\times 2\pi /a$ and the wavelength is 3.8 nm. The boundary conditions are: (a) fixed rough surface with the pillar height of 8 nm, (b) free rough surface with the pillar height of 8 nm, and (c) free surface with the pillar height of 2 nm. The dashed white rectangle indicates the part of energy that is trapped near the surface region.

Figure 9

Illustration of energy distribution in reciprocal space. The incident wave packet is TA1 polarized with $\mathbf{k}=(0.1,0.1,0)\times 2\pi /a$ and the wavelength is 3.8 nm. The boundary conditions are: (a) fixed rough surface with the pillar height of 8 nm, (b) free rough surface with the pillar height of 8 nm, and (c) free surface with the pillar height of 2 nm.

Figure 10

The participation ratio for bulk silicon and silicon with rough surface.