Hydrogen/silicon complexes in silicon from computational searches

Defects in crystalline silicon consisting of a silicon self-interstitial atom and one, two, three, or four hydrogen atoms are studied within density-functional theory (DFT). We search for low-energy defects by starting from an ensemble of structures in which the atomic positions in the defect region have been randomized. We then relax each structure to a minimum in the energy. We find a new defect consisting of a self-interstitial and one hydrogen atom (denoted by $\left\{I,\text{H}\right\}$) which has a higher symmetry and a lower energy than previously reported structures. We recover the $\left\{I,{\text{H}}_2\right\}$ defect found in previous studies and confirm that it is the most stable such defect. Our best $\left\{I,{\text{H}}_3\right\}$ defect has a slightly different structure and lower energy than the one previously reported, and our lowest-energy $\left\{I,{\text{H}}_4\right\}$ defect is different to those of previous studies.

Figure 1

(Color online) Magnitudes of the energy differences between the converged energy (taken from a multi-B calculation with $n=20$) and energies obtained with different $k$-point grids for a two-atom cell of diamond-structure silicon. The energies are in eV per cell and the number of points in each grid is $n^3$. The standard Monkhorst-Pack grid, labeled MP, is shown in red (dashed line), multi-$\Gamma$ denotes grids centered on $\mathbf{k}=0$ shown in green (dotted line), and multi-B denotes the multi-$k$-point generalization of the Baldereschi scheme is shown in blue (solid line).

Figure 2

Formation energies per hydrogen atom using Eq. (7) for various hydrogen defects in silicon calculated with respect to silicon containing a self-interstitial defect and silicon containing hydrogen molecules.

Figure 3

(Color online) Two projections of the new $\left\{I,\text{H}\right\}$ defect of $C_{3v}$ symmetry found in this work. Silicon atoms are shown in yellow (gray) and the hydrogen atom is white. This defect is based on a hexagonal self-interstitial rather than the split-$⟨110⟩$ self-interstitial of previous work.

Figure 4

(Color online) The most stable $\left\{I,{\text{H}}_2\right\}$ defect which has $C_2$ symmetry and was also found in previous work (Refs. 2, 9, 10, 11, 12, 13). Silicon atoms are shown in yellow (gray) and the hydrogen atoms are white. The defect is based on a split-$⟨110⟩$ silicon interstitial defect, with hydrogen atoms saturating the two dangling bonds.

Figure 5

(Color online) The second most stable ${\left\{I,{\text{H}}_2\right\}}^{\ast }$ defect which has $C_2$ symmetry and was also found in previous work (Ref. 10). Silicon atoms are shown in yellow (gray) and the hydrogen atoms are white. The defect is based on a buckled bond-centered silicon self-interstitial.

Figure 6

(Color online) The new $\left\{I,{\text{H}}_3\right\}$ defect of $C_1$ symmetry found in this work. Silicon atoms are shown in yellow (gray) and the hydrogen atoms are white. The defect is based on a deformed split-$⟨110⟩$ self-interstitial.

Figure 7

(Color online) The new $\left\{I,{\text{H}}_4\right\}$ defect of $C_1$ symmetry found in this work. Silicon atoms are shown in yellow (gray) and the hydrogen atoms are white. It is based on the new $\left\{I,{\text{H}}_3\right\}$ (see Fig. 6) with a defect similar to the ${\text{H}}_1^{\ast }$ defect of Chadi (Ref. 35) adjacent to it on the antibonding site.

Figure 8

(Color online) Relative abundances of the defects at (a) zero temperature and (b) 1200 K. $n_I/n_{\text{H}}$ is the ratio of the concentration of interstitial silicon atoms to hydrogen atoms. At low $n_I/n_{\text{H}}$ there is a large relative abundance of ${\text{H}}_2$ molecules. As $n_I/n_{\text{H}}$ increases, ${\text{H}}_2$ molecules bind strongly to the Si self-interstitials, forming $\left\{I,{\text{H}}_4\right\}$ defects. As $n_I/n_{\text{H}}$ increases further, formation of $\left\{I,{\text{H}}_2\right\}$ defects is favored. However, an increase in $n_I/n_{\text{H}}$ above 0.5 does not lead to formation of $\left\{I,\text{H}\right\}$ because the mixed state of $\left\{I\right\}$ and $\left\{I,{\text{H}}_2\right\}$ is more favorable. The most significant differences at 1200 K are that the abundance of $\left\{I,{\text{H}}_4\right\}$ is somewhat reduced and that ${\text{H}}_2$ and $\left\{I,{\text{H}}_2\right\}$ defects are favored instead, and $\left\{I,{\text{H}}_3\right\}$ has a small but finite abundance in a region which peaks at around $n_I/n_{\text{H}}=0.4$. The abundance of $\left\{I,\text{H}\right\}$ at 1200 K is negligible.