Structure and diffusion of excess Si atoms in $\mathrm{Si}{\mathrm{O}}_2$

Using gradient corrected density functional theory calculations, we have investigated the structure and diffusion of excess Si atoms in amorphous $\mathrm{Si}{\mathrm{O}}_2$, with comparisons to their behavior in $\alpha$-quartz. From the first principles calculations of their configuration, bonding, and energetics, we find that excess Si atoms can be fully incorporated into the amorphous oxide network while yielding oxygen vacancies. The incorporation turns out to gain energy as high as about $1.8\mathrm{eV}$, relative to the bond center state where the excess Si atom is located at a $\mathrm{Si}\text{−}\mathrm{O}$ bond center. Based on the results, we propose a novel mechanism for Si diffusion in amorphous $\mathrm{Si}{\mathrm{O}}_2$ in the presence of excess Si atoms, which involves the fourfold-coordinate ${\mathrm{Si}}^{2+}$ state creation via oxygen vacancy diffusion and pairing and its reconfiguration to the bond center state. The overall diffusion barrier is approximated to be $4.5–5.0\mathrm{eV}$, in good agreement with recent measurements. Our calculation results also predict that excess Si atoms, if they exist, may undergo diffusion with a moderate barrier of $<3.0\mathrm{eV}$ in $\alpha$-quartz.

Figure 1

(Color online) Configurations of excess Si in $\alpha$-quartz: (a) ${\mathrm{FC}}_{\mathrm{I}}$; (b) ${\mathrm{FC}}_{\mathrm{II}}$; (b) ${\mathrm{BC}}_{\mathrm{I}}$; (d) ${\mathrm{BC}}_{\mathrm{II}}$; (e) RC. The inset shows the structure of $\alpha$-quartz. [The fourfold coordinated (FC) and bond centered (BC) structures are projected on a plane perpendicular to the $c$-axis, while the ring centered (RC) and $\alpha$-quartz structures are on a plane perpendicular to the $a$-axis.] An excess Si atom is indicated I, and its neighboring Si [in yellow (light)] and O [in red (dark)] lattice atoms are also indicated accordingly.

Figure 2

(Color online) Isosurfaces of electron localization functions (at the value of 0.82) (left panels) and decomposed charge densities (right panels) of (a) the ${\mathrm{FC}}_{\mathrm{I}}$, (b) ${\mathrm{BC}}_{\mathrm{I}}$, and (c) RC state of excess Si in $\alpha$-quartz. The decomposed charge densities represent the energy ranges where two shaded sharp peaks appear in the plot of densities of states (Fig. 3) for corresponding excess Si states.

Figure 3

(Color online) Local density of states of excess Si in the ${\mathrm{FC}}_{\mathrm{I}}$, ${\mathrm{BC}}_{\mathrm{I}}$, and RC states, together with that of lattice $\mathrm{Si}\left({\mathrm{Si}}^{4+}\right)$ in defect free $\alpha$-quartz. The Fermi level for ${\mathrm{Si}}^{4+}$ is set at $0\mathrm{eV}$, and excess Si DOS plots are aligned at the O $2s$ band of ${\mathrm{Si}}^{4+}$.

Figure 4

(Color online) Cluster model used to examine the relative stability of BC and FC structures of excess Si in amorphous $\mathrm{Si}{\mathrm{O}}_2$ where the structures are assumed to be fully relaxed. Big yellow (light), small red (dark), and small white balls represent Si, O, and H atoms, respectively.

Figure 5

(Color online) Proposed diffusion mechanism for an excess Si atom in amorphous $\mathrm{Si}{\mathrm{O}}_2$: (a) O hopping to a $\mathrm{Si}\text{−}\mathrm{Si}$ BC site (or O vacancy diffusion); (b) Si hopping from a FC site to a BC site, (i.e., FC-to-BC reconfiguration of excess Si); (c) Si hopping from a BC site to a BC site; (d) Si hopping from a BC site to a FC site, (i.e., BC-to-FC reconfiguration of excess Si); (e) O hopping to a $\mathrm{Si}\text{−}\mathrm{Si}$ BC site (to create two separate ${\mathrm{Si}}^{3+}\text{−}{\mathrm{Si}}^{3+}$ charge configurations, i.e., two separate oxygen vacancies). Big yellow (light) and blue (dark) balls represent lattice Si and excess Si atoms, respectively, and small red (dark) balls indicate lattice O atoms. Solid arrow indicates single step events and dashed arrows represent multiple step events (or a series of single step events).