#
Difference in Structure and Electronic Properties of Oxygen Vacancies in α-Quartz and α-Cristobalite Phases of SiO_{2}

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Background

#### 2.1. O Vacancy Configurations in SiO${}_{2}$

_{2}, the general structure of which can be seen in Figure 2c. The ${E}^{\prime}$ center naming convention uses Greek letters for a-SiO

_{2}and Arabic numerals for $\alpha $-Q. Recent electron paramagnetic resonance (EPR) studies distinguished at least 15 different EPR signals attributed to ${E}^{\prime}$ centers in different types of $\alpha $-Q perturbed by different defects and impurities [23,24,25].

_{S}-O and Si

_{L}-O bonds, where Si

_{S}is the Si atom associated with the shorter bond and Si

_{L}—with the slightly longer one. The calculations show that the Si

_{L}atom can back-bond to O to form the puckered configuration of the ${E}_{1}^{\prime}$ center, as discussed in more detail below. The most stable configuration of this ${E}^{\prime}$ center has a forward projection of electrons on the Si${}_{S}$ towards the vacant O site. The less stable ${E}_{di}^{\prime}$ configuration has not yet been observed experimentally in either $\alpha $-Q or a-SiO${}_{2}$ [15].

#### 2.2. Properties of $\alpha $-Cristobalite

_{2}. This makes $\alpha $-C a good model for the a-SiO

_{2}surface structure without the need for computationally demanding melt-quench methods to produce theoretical models of a-SiO

_{2}surface structures. Therefore, interfaces of $\alpha $-C with water [52,53] and with solids have been extensively calculated [48,54,55].

_{S}and Si

_{L}cannot be made. However, as the E′ center is investigated in both, the Si atom involved in the back bonding must be labeled. To distinguish $\alpha $-C as separate from the $\alpha $-Q Si

_{S}notation, a new labeling was introduced, where Si${}_{A}$ is the Si involved in the back bonding (Si

_{L}equivalent) and Si${}_{B}$ is the other Si at the oxygen vacancy (Si

_{S}equivalent). For the sake of comparison, $\alpha $-Q is also labeled in the same way, see Figure 3.

## 3. Materials and Methods

## 4. Results of Calculations

#### 4.1. Pristine Crystals

^{3}) and a-SiO

_{2}(2.20 g/cm

^{3}) [82] showed a difference of 0.15 g/cm

^{3}, and, thus, the band gaps were expected to differ with a lower density indicating a lower band gap value. However, El-Sayed et al. [83] calculated a band gap value of 8.9 eV for a-SiO

_{2}using the hybrid HSE06 XC functional, while experimental band gap values of a-SiO

_{2}ranged from 8.9–9.7 eV [75,84,85,86].

**Table 1.**The structural properties of $\alpha $-cristobalite and $\alpha $-quartz compared to the previous experimental and theoretical literature. The geometric structure is also shown, with the band gap separated at the bottom of the table. All bond lengths and lattice vectors are in Å, all angles are in degrees (°), the density is in g/cm

^{3}, and the band gap is in eV.

Parameter | $\mathit{\alpha}$-Cristobalite | $\mathit{\alpha}$-Quartz | ||
---|---|---|---|---|

This Work | Literature Data | This Work | Literature Data | |

Lattice Vectors | ||||

a = b | 5.05 | 4.97 [87] | 4.93 | 4.91 [88] |

c | 7.08 | 6.93 [87] | 5.43 | 5.40 [88] |

Bond length | ||||

Si${}_{A}$-O | 1.604–1.609 | 1.600–1.607 [87] | 1.608 | 1.604 [88] |

Si${}_{B}$-O | 1.604–1.609 | 1.600–1.607 [87] | 1.612 | 1.613 [88] |

Bond Angles | ||||

O-Si-O | 108–111.8 | 108.1–111.3 [89] | 109.2–110.5 | 109.0–109.5 [88] |

Si-O-Si | 150–153 | 146.6 [87] | 144.8–145.1 | 143.7 [88] |

Density | 2.35 | 2.18–2.37 [44,45] | 2.41 | 2.47–2.70 [44,45,90] |

Band gap | 8.57 | 8.54 [44] | 8.5 | 9.65 [75] |

#### 4.2. Oxygen Vacancies

#### 4.2.1. Geometric Structure and Stability

#### Neutral Vacancies

#### Positively Charged Vacancies

#### Negatively Charged Vacancies

#### 4.2.2. Charge Transition Levels

#### 4.2.3. Optical Absorption

#### 4.2.4. EPR Parameters

_{$\delta $}center [101], as well as in calculations of dimer vacancy configurations [11]. However, the analysis of experimental EPR spectra of the E′

_{$\delta $}center in silica glass concluded [15,27] that this center had a more complex structure than the ${E}_{di}^{\prime}$ configuration described here and in previous calculations. Since ${E}_{di}^{\prime}$ was the only stable configuration of positively charged vacancy in $\alpha $-C, EPR measurements for the positively charged O vacancy in $\alpha $-C offered a unique opportunity to identify this configuration and to verify the theoretical predictions.

## 5. Discussion and Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

DFT | Density Functional Theory |

$\alpha $-C | $\alpha $-cristobalite |

$\alpha $-Q | $\alpha $-quartz |

a-SiO${}_{2}$ | Amorphous silicon dioxide |

CT | Charge transition |

ODC | Oxygen deficient center |

EPR | Electron paramagnetic resonance |

Opal-CT | Opal cristobalite tridymite |

LDA | Local density approximation |

GTH | Goedecker-Teter-Hutter |

BFGS | Broyden–Fletcher–Goldfarb–Shanno |

XC | Exchange-correlation |

PBE | Perdew-Burke-Ernzerhof |

ADMM | Auxiliary density matrix method |

TD-DFT | Time-dependent density functional theory |

TC-LRC | Truncated coulomb long-range correction |

NEB | Nudged elastic band |

CI-NEB | Climbing image nudged elastic band |

GGA | Generalized gradient approximation |

HSE | Heyd-Scuseria-Ernzerhof |

VB | Valence band |

CB | Conduction band |

CBM | Conduction band minimum |

CTL | Charge transition level |

TS | Transition state |

OA | Optical absorption |

AE | all-electron |

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**Figure 1.**Pristine structures of SiO${}_{2}$ in different phases, viewed along the

**a**axis. Yellow atoms correspond to Si, red atoms correspond to oxygen atoms. (

**a**) $\alpha $-cristobalite structure; (

**b**) $\alpha $-quartz structure. Both phases are on the same scale as $\alpha $-quartz has a smaller unit cell than $\alpha $-cristobalite.

**Figure 2.**Schematic representation of oxygen vacancy defects in SiO${}_{2}$[8]. Yellow atoms are silicon and red atoms are oxygen, the blue oval represents electron localization. (

**a**) Neutral dimer configuration, such as ODC(I). (

**b**) Positive dimer configuration, such as ${E}_{di}^{\prime}$ with elongated Si-Si bonds. (

**c**) Puckered ${E}_{1}^{\prime}$ center, showing the back bond between Si${}_{L}$ and the back oxygen in the surrounding ring, O${}_{B}$. The labeling is consistent with $\alpha $-quartz and is described in the text.

**Figure 3.**Local pristine structures of (

**a**) $\alpha $-cristobalite with two connected tetrahedrons and (

**b**) $\alpha $-quartz with two connected tetrahedrons with a ring containing back bonding O${}_{B}$. Yellow atoms are silicon and red atoms are oxygen. In $\alpha $-quartz, Si${}_{A}$ and Si${}_{B}$ are not equivalent, but in $\alpha $-cristobalite they are equivalent. The Si${}_{A}$ and Si${}_{B}$ notations are, therefore, used to distinguish the Si atoms in defects which are later introduced to the pristine structure. More detail is provided in the text.

**Figure 4.**Square modulus of the wavefunction of the highest occupied state in the studied dimer oxygen defects in $\alpha $-C shown in purple, the isosurface shown is $\left|0.1\right|$. Yellow atoms are silicon and red atoms are oxygen. (

**a**) ${V}_{O}^{0}$ neutral dimer configuration. (

**b**) ${E}_{di}^{\prime}$, positive dimer. (

**c**) ${V}_{O}^{-}$, negative dimer. (

**a**–

**c**) are general HOMOs applicable to either $\alpha $-cristobalite or $\alpha $-quartz.

**Figure 5.**Square modulus of the wavefunction of the highest occupied state in the studied oxygen ${E}^{\prime}$ defects shown in purple, the isosurface shown is $\left|0.1\right|$. Yellow atoms are silicon and red atoms are oxygen. (

**a**) ${E}_{puck}^{\prime}$, $\alpha $-quartz puckered, forward projected; (

**b**) ${E}_{bp}^{\prime}$, $\alpha $-cristobalite, back projected; (

**c**) ${E}_{bp}^{\prime}$, $\alpha $-quartz, puckered, back projected.

**Figure 6.**Charge transition level diagram of oxygen vacancy defects in (

**a**) $\alpha $-cristobalite and (

**b**) $\alpha $-quartz. 0 eV is the valence band maximum and the graph cuts off at the conduction band minimum.

**Table 2.**Bond length (Å) between the atoms surrounding the oxygen vacancy: Si${}_{A}$, Si${}_{B}$, O${}_{B}$ and O surrounding the Si${}_{A/B}$ atoms in $\alpha $-C and $\alpha $-Q. For some Si${}_{A/B}$-O bond lengths a range is reported as the bond lengths slightly differ due to asymmetries in the relaxation around the defect. The atoms and labeling conventions are described in Figure 3, Figure 4 and Figure 5. $\alpha $-cristobalite and $\alpha $-quartz are compared to understand how they are perturbed between each defect.

Bond | $\mathit{\alpha}$-Cristobalite | $\mathit{\alpha}$-Quartz | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Pristine | ${\mathit{V}}_{\mathit{O}}^{0}$ | ${\mathit{E}}_{\mathit{di}}^{\prime}$ | ${\mathit{E}}_{\mathit{bp}}^{\prime}$ | ${\mathit{V}}_{\mathit{O}}^{-}$ | Pristine | ${\mathit{V}}_{\mathit{O}}^{0}$ | ${\mathit{E}}_{\mathit{di}}^{\prime}$ | ${\mathit{E}}_{\mathit{puck}}^{\prime}$ | ${\mathit{E}}_{\mathit{bp}}^{\prime}$ | ${\mathit{V}}_{\mathit{O}}^{-}$ | |

Si${}_{A}$-Si${}_{B}$ | 3.11 | 2.38 | 2.88 | 4.59 | 2.39 | 3.09 | 2.41 | 3.00 | 4.45 | 5.38 | 2.47 |

Si${}_{A}$-O${}_{B}$ | 5.31 | 5.54 | 5.22 | 4.83 | 5.63 | 3.66 | 4.10 | 3.71 | 1.83 | 1.84 | 4.07 |

Si${}_{A}$-O | 1.61 | 1.61–1.63 | 1.57–1.58 | 1.62–1.63 | 1.67–1.70 | 1.61 | 1.63 | 1.58 | 1.62 | 1.61–1.64 | 1.69–1.73 |

Si${}_{B}$-O | 1.61 | 1.62–1.63 | 1.57–1.58 | 1.54–1.55 | 1.67–1.71 | 1.61 | 1.62 | 1.57 | 1.58 | 1.57–1.59 | 1.66–1.68 |

**Table 3.**Optical absorption of oxygen vacancies calculated using different exact exchange cutoff radii. 2 Å is the cutoff used in geometry optimization. The values 6.4 Å and 7.5 Å were the highest cutoff points for the phases of $\alpha $-quartz and $\alpha $-cristobalite, respectively. Transition types were determined by the state that had the largest contribution to the excitation. In the spin-polarized calculations, i.e., positively charged defects, the $\alpha $ spin channel contributed to a large extent to the transitions. Transition types are notated by the symmetry of the states, the Si. → Si. indicates a promotion into the singly occupied dangling bond on the same Si atom.

Cutoff Radius (Å) | Phase | Defect | Peak Energy (eV) | Oscillator Strength | Transition Type |
---|---|---|---|---|---|

7.5 | $\alpha $-C | ${V}_{O}^{0}$ | 7.62 | 0.22 | $\sigma \to \pi $ |

7.72 | 0.20 | ||||

8.13 | 0.16 | ||||

8.4 | 0.13 | ||||

${E}_{di}^{\prime}$ | 7.56 | 0.12 | $\sigma \to \pi $ | ||

7.72 | 0.11 | $\sigma \to \pi $* | |||

6.4 | $\alpha $-Q | ${E}_{puck}^{\prime}$ | 6.18 | 0.12 | Si. → deloc. ring |

2 | $\alpha $-C | ${V}_{O}^{0}$ | 7.86 | 0.27 | $\sigma \to \pi $ |

7.94 | 0.19 | ||||

8.20 | 0.13 | ||||

8.48 | 0.12 | ||||

${E}_{di}^{\prime}$ | 6.27 | 0.15 | $\sigma \to \sigma $* | ||

7.86 | 0.12 | $\sigma \to \pi $ | |||

$\alpha $-Q | ${E}_{puck}^{\prime}$ | 6.38 | 0.14 | Si${}_{B}$ → Si${}_{A}$ | |

6.43 | 0.12 | Si. → deloc. ring |

**Table 4.**Calculated isotropic hyperfine coupling constants employing the pseudopotential structures and the all-electron wavefunction with the PBE0 functional, as the quality of the pcJ basis sets systematically improved.

AE Basis | $\mathit{\alpha}$-Quartz (mT) | $\mathit{\alpha}$-Cristobalite (mT) | |||
---|---|---|---|---|---|

${\mathbf{E}}_{\mathbf{di}}^{\prime}$ | ${\mathbf{E}}_{\mathbf{puck}}^{\prime}$ | ${\mathbf{E}}_{\mathbf{di}}^{\prime}$ | |||

Silica | Si${}_{\mathrm{A}}$ | Si${}_{\mathrm{B}}$ | Si${}_{\mathrm{A}}$ | Si${}_{\mathrm{A}}$ | Si${}_{\mathrm{B}}$ |

pcJ-0 | 12.29 | 10.11 | 42.25 | 11.16 | 10.78 |

pcJ-1 | 11.14 | 8.70 | 38.96 | 10.44 | 10.01 |

pcJ-2 | 10.26 | 8.06 | 37.46 | 9.55 | 9.17 |

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## Share and Cite

**MDPI and ACS Style**

Milton, K.L.; Durrant, T.R.; Cobos Freire, T.; Shluger, A.L.
Difference in Structure and Electronic Properties of Oxygen Vacancies in *α*-Quartz and *α*-Cristobalite Phases of SiO_{2}. *Materials* **2023**, *16*, 1382.
https://doi.org/10.3390/ma16041382

**AMA Style**

Milton KL, Durrant TR, Cobos Freire T, Shluger AL.
Difference in Structure and Electronic Properties of Oxygen Vacancies in *α*-Quartz and *α*-Cristobalite Phases of SiO_{2}. *Materials*. 2023; 16(4):1382.
https://doi.org/10.3390/ma16041382

**Chicago/Turabian Style**

Milton, Katherine L., Thomas R. Durrant, Teofilo Cobos Freire, and Alexander L. Shluger.
2023. "Difference in Structure and Electronic Properties of Oxygen Vacancies in *α*-Quartz and *α*-Cristobalite Phases of SiO_{2}" *Materials* 16, no. 4: 1382.
https://doi.org/10.3390/ma16041382