EPR parameters of $E^{\prime }$ centers in $v-{\mathrm{SiO}}_2$ from first-principles calculations

A first-principles investigation of $E^{\prime }$  centers in vitreous silica $(v-{\mathrm{SiO}}_2)$ based on calculations of the electron paramagnetic resonance (EPR) parameters is presented. The EPR parameters are obtained by exploiting the gauge including projector augmented wave method as implemented in the quantum-espresso package. First, we analyze the EPR parameters of a large number of ${\mathrm{Si}}_2$ dimers. The $g$ tensor of the ${\mathrm{Si}}_2$ dimers is shown to possess an average rhombic symmetry and larger $g$ principal values with respect to those observed, e.g., for the $E^{\prime }$${}_{\gamma }$ center in silica. Furthermore, the $g$ principal values clearly show a linear trend with the Si-Si dimer length. Our results suggest that the ${\mathrm{Si}}_2$ dimers could correspond to an unidentified paramagnetic center, though occasionally the calculated $g$ principal values of the ${\mathrm{Si}}_2$ dimer might be compatible with those found experimentally for the $E^{\prime }$${}_{\delta }$ center. Next, we generate nondimer configurations by a procedure involving structural relaxations in the subsequent positively charged states. In particular, puckered, unpuckered, doubly puckered, and forward-oriented configurations are generated. The distributions of the calculated EPR parameters of the puckered and unpuckered configurations further support the assignment of the $E^{\prime }$${}_{\gamma }$ center to an unpaired spin localized at a threefold coordinated silicon dangling bond. Moreover, by analyzing Fermi contacts and $g$ tensors of the puckered and forward-oriented configurations, we suggest the assignment of the $E^{\prime }$${}_{\alpha }$ center to the latter type of configurations. This work also suggests that the differences in the EPR parameters of $E^{\prime }$${}_{\alpha }$ and $E^{\prime }$${}_{\gamma }$ centers mainly arise from the strained geometry of the silicon dangling bond. In the forward-oriented configurations, one Si-O bond is about 0.2 Å longer than the remaining two, whereas in the silicon dangling bond of the puckered and unpuckered configurations, all three bonds have a length of $\sim$1.6 Å  each.

Figure 1

Summary of the configuration types of positively charged oxygen vacancies generated by following the procedure outlined in Sec. 2: (a) ${\mathrm{Si}}_2$ dimer, (b) back-projected unpuckered, (c) forward-projected unpuckered, (d) back-projected puckered $4\times$, (e) forward-projected puckered $4\times$, (f) back-projected puckered $5\times$, (g) forward-projected puckered 5$\times$, (h) forward-oriented, and (i) doubly puckered configurations. Oxygen (filled disk) and silicon (empty disk) atoms and the unpaired electron (dot) are shown. The shaded area indicates the relevant local environment of the paramagnetic center. Si-Si distances discussed in the text $\left(d_{\mathrm{SiSi}}\right)$ are taken between silicon atoms labeled with ${\mathrm{Si}}_{\mathrm{DB}}$ and ${\mathrm{Si}}_{\mathrm{P}}$ in (b)–(g) and between ${\mathrm{Si}}_{\mathrm{A}}$ and ${\mathrm{Si}}_{\mathrm{B}}$ in (a) and (h).

Figure 2

Ball and stick models and spin densities of (a) ${\mathrm{Si}}_2$ dimer, (b) puckered, (c) forward-oriented (FO), and (d) doubly puckered (DP) configurations in $v-{\mathrm{SiO}}_2$ [40]. Red spheres correspond to oxygen atoms, light brown spheres correspond to silicon atoms, and spin density is shown with a transparent gray color. The same axis scales are used in (a)–(c), while (d) is plotted by using a scale factor of 0.5 with respect to (a).

Figure 3

Calculated (a) $g$-tensor principal values $g_1$ (circles), $g_2$ (squares), $g_3$ (triangles) and (b) Fermi contact of ${}^{29}$Si atoms in positively charged ${\mathrm{Si}}_2$ dimers vs Si-Si dimer length. The shaded area shows the range of experimental $g$ values [24].

Figure 4

Calculated (a) $g$ principal values: $g_1$ (circles), $g_2$ (squares), $g_3$ (triangles), and (b) Fermi contacts of FO (red discs), back-projected puckered (green squares), forward-projected puckered (blue circles), and unpuckered (black triangles) configurations vs Si-Si distance.

Figure 5

Fermi contacts of back-projected (green squares), forward-projected (blue circles), and puckered and unpuckered (black triangles) configurations plotted vs average dangling bond angle $\langle$O-Si${}^{\mathrm{DB}}$-O$\rangle$.

Figure 6

Fermi contacts of forward-projected puckered $5\times$ configurations plotted vs Si${}^{\mathrm{DB}}-{\mathrm{O}}_{\mathrm{R}}$ distance, where ${\mathrm{O}}_{\mathrm{R}}$ is the nearest oxygen of the two-membered ring.

Figure 7

Calculated (a) relative energy and (b) $g_3$ principal values plotted vs bond length $\left(d_{{\mathrm{Si}}^{\mathrm{DB}}{\mathrm{O}}^{\left[3\right]}}\right)$ between the threefold silicon and the threefold oxygen atom in forward-oriented (FO) configurations [Fig. 1]. Shaded area shows the experimental range of $g_3$ values of the $E^{\prime }$${}_{\alpha }$ [25]. The dotted line and empty symbols show the decreasing of $g_3$ vs $d_{{\mathrm{Si}}^{\mathrm{DB}}{\mathrm{O}}^{\left[3\right]}}$ calculated as explained in the text.

Figure 8

(a) Relative energy, (b) $g_2$, and (c) $g_3$ principal values of forward-projected (blue circles) and back-projected (green squares), puckered, unpuckered (black filled triangles), and FO (red discs) configurations plotted versus the corresponding Fermi contacts. Shaded area shows the experimental range of $g_2$ and $g_3$ values of the $E^{\prime }$${}_{\alpha }$ [25].

Figure 9

Ball and stick models and spin densities of (a) a back-projected configuration in which the silicon dangling bond points towards a background oxygen placed at 3.05 Å  distance, and (b) a forward-projected configuration in which the silicon dangling bond points towards an oxygen atom belonging to a twofold ring and placed at 2.78 Å  from the silicon atom. This spin density in (a) corresponds to the $-45$ mT point (square) of Fig. 8, while the spin density in (b) corresponds to the $-49$ mT point (disc) of Fig. 8.

Figure 10

Relative energy $\left(\Delta E\right)$ with respect to the lowest-energy dimer configuration. (a) Relative energy $\left(\Delta E\right)$ with respect to the lowest-energy configuration of positively charged ${\mathrm{Si}}_2$ dimers vs Si-Si dimer length $\left(d_{\mathrm{SiSi}}\right)$. (b) Puckered $4\times$ (green filled rhombs), $5\times$ (blue rhombs), forward-oriented FO (red discs), unpuckered (black filled triangles), and doubly puckered DP (triangle) configurations. (c) Relative energy of forward-projected (blue circles) and back-projected (green squares) puckered configurations. For each configuration type, vertical and horizontal lines are used, respectively, to indicate the average $d_{\mathrm{SiSi}}$ and the average relative energy.