Vibrations and thermal transport in nanocrystalline silicon

We use a combination of vibrational-mode analysis and molecular-dynamics simulations to study the effect of grain size on the nature of thermal vibrations, their localization, and their ability to carry heat in nanocrystalline silicon. Vibrational-mode analysis demonstrates that the vibrations that carry most of the heat in small-grain $(\lesssim 3\mathrm{nm})$ structurally heterogeneous nanocrystalline silicon are almost identical in nature to those in homogeneous amorphous silicon, where the majority of the vibrations are delocalized and unpolarized. Consequently, the principal thermal conductivity mechanism in such a nanocrystalline material is the same as in the amorphous material. With increasing grain size, the vibrational modes become progressively more like that of a crystalline material; this is reflected in a crossover in the mechanism of thermal transport to that of a crystalline material.

Figure 1

Atomic positions and bonding in a high-angle (001) $\Sigma 29$ twist grain boundary in silicon with GB separation $d_z=2.7\mathrm{nm}$. Two grains are separated by two GB’s, one at the center and one at the edges of the simulation box. Periodic boundary conditions are used in all three directions.

Figure 2

Bottom panel: schematic of three-dimensional periodic simulation cell for estimation of ${\kappa }_{\mathit{MD}}$ using MD simulations. Both the source and sink are slabs of equal thickness $\delta$ wherein, at each time step, energy $\Delta ϵ$ is added and removed, respectively. Top panels show atomic structures corresponding to 1D GB superlattices and 2D textured columnar microstructures.

Figure 3

Local GB and grain interior vibrational DOS as a function of frequency for GB structure with $d_z=2.7\mathrm{nm}$.

Figure 4

Participation ratio vs frequency for amorphous silicon, (111) GB structure with $d_z=2.7\mathrm{nm}$, (001) GB structures with $d_z=2.7$, $5.4\mathrm{nm}$, and perfect crystal structure of silicon.

Figure 5

Fraction of each mode contained in the grain-boundary region of the GB structure. Mode weight factor as a function of frequency for (001) GB model structure with (top panel): $d_z=2.7\mathrm{nm}$ and (bottom panel): $d_z=1.6\mathrm{nm}$. The dashed lines at 0.66 (bottom panel) and 0.4 (top panel) mark the volume fraction of GB region in the structure with $d_z=1.6\mathrm{nm}$ and $2.7\mathrm{nm}$, respectively.

Figure 6

Spatial variation of amplitude of a mode projected along the $z$ direction. The region between the vertical lines as shown is the region of grain boundary in (001) GB structure with $d_z=2.7\mathrm{nm}$. Top panel: $\nu \sim 11\mathrm{THz}$, localized in grain interiors. Bottom panel: $\nu \sim 12\mathrm{THz}$, localized on grain boundaries.

Figure 7

Projection of normalized unit polarization vector components $z$ vs $x$ (left-hand column) and $z$ vs $y$ (right-hand column) for (A) crystalline silicon, (B) (001) GB with $d_z=5.4\mathrm{nm}$, (C) (001) GB with $d_z=2.7\mathrm{nm}$, and (D) amorphous silicon. Each of the polarization vectors plotted here is characteristic for a mode with a participation ratios $p_{\lambda }\sim 0.5$ and frequency ${\nu }_{\lambda }\sim 3\mathrm{THz}$ in the respective structures.

Figure 8

Mode phase quotients as a function of frequency for crystalline, (001) GB with $d_z=2.7\mathrm{nm}$, $1.6\mathrm{nm}$, and amorphous silicon.

Figure 9

Mode diffusivities as a function of frequency for (001) GB with $d_z=2.7\mathrm{nm}$, $1.6\mathrm{nm}$, and amorphous silicon. An increase in grain size leads to an increase in mode diffusivity across the spectrum.

Figure 10

Comparison of temperature profiles for (A) amorphous silicon, (001) GB structure with (B) $d_z=2.7\mathrm{nm}$, (C) $d_z=10.8\mathrm{nm}$, and (D) individual (001) GB’s. The incipience of the temperature discontinuities at GB’s is evident in (001) GN structure with $d_z=10.8\mathrm{nm}$.

Figure 11

Thermal conductivity of the (001) GB structures in the direction normal to the GB plane divided by the GB separation, which is the nominal GB conductance, as a function of $1∕d_z$. Two data sets are for thermal conductivity obtained in MD simulations and from AF theory. Individual GB conductance is shown at $1∕d_z=0$.

Figure 12

Thermal conductivity obtained from MD simulations for normal to the GB plane direction for GB superlattices and for in-plane direction for textured 2D microstructures, normalized by the GB separation $d_g$ and plotted as a function of $1∕d_g$.