Computational generation of voids in $a$-Si and $a$-Si:H by cavitation at low density

Use of amorphous silicon ($a$-Si) and hydrogenated amorphous silicon ($a$-Si:H) in photovoltaics has been limited by light-induced degradation (the Staebler-Wronski effect) and low hole mobilities, and voids have been implicated in both problems. Accurately modeling the void microstructure is critical to theoretically understanding the cause of these issues. Previous methods of modeling voids have involved removing atoms according to an a priori idea of void structure and/or using computationally expensive molecular dynamics. We propose a new fast and unbiased approach based on the established and efficient Wooten-Winer-Weaire Monte Carlo method, by using a range of fixed densities to generate equilibrium structures of $a$-Si and $a$-Si:H that maintain 4-coordination. We find a smooth evolution in bond lengths, bond angles, and bond angle deviations $\mathrm{\Delta }\theta$ as the density is changed around the equilibrium value of $4.9\times {10}^{22}$ $\mathrm{atoms}/{\mathrm{cm}}^3$. However, a significant change occurs at densities below $4.3\times {10}^{22}$ $\mathrm{atoms}/{\mathrm{cm}}^3$, where voids begin to form to relieve tensile stress, akin to a cavitation process in liquids. We find both small voids (radius $\sim 3$ Å) and larger ones (up to 7 Å), which compare well with available experimental data. The voids have an influence on atomic structure up to 4 Å beyond the void surface and are associated with decreasing structural order, measured by $\mathrm{\Delta }\theta$. We also observe an increasing medium-range dihedral order with increasing density. Our method allows fast generation of statistical ensembles, resembles a physical process during experimental deposition, and provides a set of void structures for further studies of their effects on degradation, hole mobility, two-level systems, thermal transport, and elastic properties. The basic concept of generating voids at low density is applicable to other amorphous materials.

Figure 1

Keating energy throughout a chassm calculation at a density of $4.5\times {10}^{22}$ $\mathrm{atoms}/{\mathrm{cm}}^3$. The first eight to ten steps are high-$T$ randomization. If the structure fails to obtain enough energy to escape the barrier to the amorphous phase, it recrystallizes to a strained $c$-Si (blue). If the structure randomizes at too high a $T$, it does not relax to a reasonable energy (red) or bonding network. A run producing a desired realistic amorphous structure has an intermediate behavior (black).

Figure 2

Top: An example low-density ($4.3\times {10}^{22}$ $\mathrm{atoms}/{\mathrm{cm}}^3$) $a$-Si structure with a large void. The green void points fill in the largest 10% of the void size distribution. Bottom: The pore size histogram of a low-density ($4.05\times {10}^{22}$ $\mathrm{at}/{\mathrm{cm}}^3$) post-DFT structure. Large voids (4.9 Å) and interstitial voids (2.5 Å, dashed line) appear as strong signals in this histogram. The area underneath the solid region constitutes the void volume, excluding interstitials. Only void points belonging to the largest 10% of voids (green) are considered for the void proximity ($r_v$) analysis.

Figure 3

Averaged partial pair distribution functions, $g\left(r\right)$, for $a$-Si:H at all densities. The Si-Si $g\left(r\right)$ for $a$-Si is indistinguishable from that of $a$-Si:H. Decreasing density increases the height of the H-H 2.2 Å peak, but has little effect on the other curves.

Figure 4

chassm (dashed lines) and chassm $+$ DFT (solid lines) calculated pressures vs densities. As density is decreased in $a$-Si and $a$-Si:H, negative pressure is induced, but then relieved near the void onset density of $(4.3–4.5)\times {10}^{22}$ $\mathrm{atoms}/{\mathrm{cm}}^3$, similar to the cavitation process of bubble formation. chassm pressures are systematically too low compared to DFT, but have the correct trend. $a$-Si has a more abrupt transition than $a$-Si:H.

Figure 5

Evolution of structural parameters with density. Bond lengths and angles change trends around $4.3\times {10}^{22}$ $\mathrm{atoms}/{\mathrm{cm}}^3$, the density of void onset shown in Fig. 6. Relaxed $c$-Si has a chassm energy of 0 eV and $\mathrm{\Delta }\theta =0^{\circ }$. $a$-Si DFT energies are relative to $c$-Si, and $a$-Si:H energies are relative to the lowest $a$-Si:H energy in our data set.

Figure 6

Top: Renormalized densities indicate the density of the nonvoid regions, showing that void formation allows the rest of the sample to retain a constant density. Bottom: Voids start forming at $(4.3–4.5)\times {10}^{22}$ $\mathrm{atoms}/{\mathrm{cm}}^3$. Above the critical density, the largest void radii are about the size of the interstitial, and the total void volume is essentially zero.

Figure 7

Top: The relationship between density and the dihedral oscillation amplitude, $A$. $A$ measures dihedral order, increasing as the density increases. Bottom: Dihedral distributions for a pair of low-density ($4.1\times {10}^{22}$ $\mathrm{atoms}/{\mathrm{cm}}^3$) and relaxed ($4.9\times {10}^{22}$ $\mathrm{atoms}/{\mathrm{cm}}^3$) $a$-Si structures. Dihedral order vanishes at the lowest densities.

Figure 8

Locally resolved structure of low-density structures as a function of void proximity: average bond length $\langle r\rangle$ with respect to global average bond length for the given density; average Si-Si coordination number $\langle C\rangle$; average bond angle deviation $\langle \mathrm{\Delta }\theta \rangle$; and average bond angle $\langle \theta \rangle$. $a$-Si and $a$-Si:H lines are averaged over all structures with densities $4.3\times {10}^{22}$ $\mathrm{atoms}/{\mathrm{cm}}^3$ and below. Higher-density structures (green) are plotted for comparison; since the largest voids in these structures are not distinguishable from interstitials, there is little correlation between structure and void proximity. The results are consistent with a local rearrangement of bonds to accommodate a void as shown in the bottom-right sketch.