#
Electronic and Optical Properties of Alkaline Earth Metal Fluoride Crystals with the Inclusion of Many-Body Effects: A Comparative Study on Rutile MgF_{2} and Cubic SrF_{2}

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Computational Methods and Resulting Ground-State Properties

_{2}, the experimental values are from Reference [51], while for c-SrF

_{2}, experimental values were obtained from References [52,53].

_{2}, calculated with PBEsol, shows a deviation in the order of 0.1%, while in the case of c-SrF${}_{2}$, the deviation from the experiment for the same observable is of the order of 0.4%. The AM05 results are approximately of the same quality. For the PBE XC scheme (with the typical under-binding effect due to gradient corrections), the results show more significant deviations from the experiment; for LDA calculations (strong over-binding due to the local approximation), the comparison with experiment is worse. For these reasons, structures obtained with PBEsol were used for the present study for the calculation of energy bands and optical properties.

## 3. Electronic Excitations in r-M$\mathrm{g}$F${}_{\mathbf{2}}$ and c-S$\mathrm{r}$F${}_{\mathbf{2}}$

#### 3.1. Energy Gaps for r-MgF${}_{2}$ and c-SrF${}_{2}$

#### 3.2. Quasiparticle Energy Bands for r-MgF${}_{2}$ and c-SrF${}_{2}$

## 4. Dielectric Function and Optical Absorption Spectrum of M$\mathrm{g}$F${}_{\mathbf{2}}$ and S$\mathrm{r}$F${}_{\mathbf{2}}$

## 5. Summary and Conclusions

## 6. Additional Material: Bulk Systems versus Clusters

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**QP energy bands for r-MgF${}_{2}$ in the GW${}_{0}$ scheme (

**top**) and within the G${}_{0}$W${}_{0}$ method (

**bottom**). See text for details.

**Figure 2.**QP energy bands for c-SrF${}_{2}$ in the GW${}_{0}$ scheme (

**top**) and the scQP-GW (

**bottom**) scheme. See text for details.

**Figure 3.**Dielectric function components for r-MgF${}_{2}$ (

**top**) and c-SrF${}_{2}$ (

**bottom**) in the BSE scheme on top of GW${}_{0}$ bands (see text). Red lines refer to the real part and black lines to the imaginary parts of the dielectric functions. In the case of c-MgF${}_{2}$, full lines refer to the $zz$ component and broken lines to the $xx$ and $yy$ components.

**Figure 4.**Dielectric functions of r-MgF${}_{2}$ (

**top**) resulting in the BSE scheme (for the two components of light polarization) and of c-SrF${}_{2}$ (

**bottom**) on top of G${}_{0}$W${}_{0}$ and scQP-GW respectively self-energy calculations (see text). Red lines refer to the real part and black lines to the imaginary parts of the dielectric functions. In the case of r-MgF${}_{2}$, full lines refer to the $zz$ component and broken lines to the $xx$ and $yy$ components.

**Figure 5.**Dielectric functions components (

**top**: imaginary part,

**bottom**: real part) of r-MgF${}_{2}$ in the BSE scheme on top of GW${}_{0}$ energies in comparison with the experiment reported in Reference [37] (see text). Red lines refer to our calculated spectra and black lines to the experimental data. Isotropic averages only are displayed.

**Figure 6.**Dielectric function of c-SrF${}_{2}$ calculated in the BSE scheme on top of GW${}_{0}$ energies (

**top**: imaginary part,

**bottom**: real part) in comparison with the experiment from Reference [29]. Red lines refer to our calculated spectra and black lines to the experimental data.

**Table 1.**Ground-state properties of the difluoride crystals r-MgF${}_{2}$ and c-SrF${}_{2}$. The lattice parameter a of both structures and parameter c for the rutile are reported together with the bulk modulus and its pressure derivative.

PBEsol | r-MgF${}_{2}$ | c-SrF${}_{2}$ |
---|---|---|

a [Å] | 4.6313 | 5.7744 |

c [Å] | 3.0558 | — |

$c/a$ | 0.6598 | — |

B${}_{0}$ [MPa] | 97.1 | 72.8 |

dB${}_{0}$/dp | 4.69 | 4.71 |

**Table 2.**Cutoff parameters and ground-state energy of the difluoride crystals under study. The total energy for the simulation cell ${E}_{0}$, the cutoff energy ${E}_{cut}$ for non-norm-conserving (partial) wave functions, and the cutoff energy ${E}_{aug}$ for the plane-wave expanded “intermediate” PAW augmentation charges are given. On the last line, the k-point mesh used for the BZ integration is listed.

PBEsol | r-MgF${}_{2}$ | c-SrF${}_{2}$ |
---|---|---|

E${}_{cut}$ [eV] | 1020 | 640 |

E${}_{aug}$ [eV] | 1700 | 1640 |

E${}_{0}$ [eV] | −30.1122 | −16.3389 |

k-point set | 12 × 12 × 18 | 12 × 12 × 12 |

**Table 3.**Ground-state properties of r-MgF${}_{2}$ and c-SrF${}_{2}$. For r-MgF${}_{2}$ in the upper part of the table, the lattice parameters a, c, their ratio $c/a$, and the x parameter are reported as functions of the XC potential used for the calculations in the first four rows. In the following three rows, the bulk modulus, its pressure derivative, and the total energy of the unit cell are given. In the last column, the experimental values from Reference [51] are listed. In the lower part of the table, we report the ground-state properties of c-SrF${}_{2}$. The lattice parameter a was reported as in the first row, i.e., as a function of the different XC potentials used. In the following three rows, we provide the bulk modulus, its pressure derivative, and the total energy of the unit cell. In the last column, the experimental data after References [52,53] are given.

${\mathbf{r}-\mathbf{MgF}}_{2}$ | $\mathbf{PBEsol}$ | $\mathbf{PBE}$ | $\mathbf{AM}05$ | $\mathbf{LDA}$ | $\mathbf{EXP}$ |
---|---|---|---|---|---|

a[Å] | 4.6313 | 4.6928 | 4.6649 | 4.5638 | 4.6249 |

c[Å] | 3.0558 | 3.0875 | 3.0741 | 3.0194 | 3.0520 |

$c/a$ | 0.6598 | 0.6579 | 0.6590 | 0.6616 | 0.6599 |

x | 0.3033 | 0.3035 | 0.3037 | 0.3030 | 0.3027 |

B${}_{0}$[GPa] | 97.1 | 90.1 | 91.6 | 111.2 | 101 ± 3 |

dB${}_{0}$/dP | 4.69 | 4.74 | 4.73 | 4.64 | 4.2 ± 1.1 |

E${}_{0}$[eV] | −30.1122 | −28.7552 | −29.7466 | −33.0805 | |

${\mathbf{c}-\mathbf{SrF}}_{\mathbf{2}}$ | $\mathbf{PBEsol}$ | $\mathbf{PBE}$ | $\mathbf{AM}\mathbf{05}$ | $\mathbf{LDA}$ | $\mathbf{EXP}$ |

a[Å] | 5.7744 | 5.8712 | 5.8094 | 5.6813 | 5.7994 |

B${}_{0}$[GPa] | 72.8 | 64.5 | 67.5 | 84.9 | 67.1 − 74.6 |

dB${}_{0}$/dp | 4.71 | 4.73 | 4.74 | 4.61 | 4.2 ± 1.1 |

E${}_{0}$[eV] | −16.3389 | −15.6630 | −16.0187 | −17.8951 |

**Table 4.**Quasi-particle energies for the fundamental energy gaps of r-MgF${}_{2}$ and c-SrF${}_{2}$ calculated with different approximations are reported and compared with available experimental results (for r-MgF${}_{2}$ from Reference [37] and for c-SrF${}_{2}$ from Reference [1]). B3PW refers to hybrid exchange–correlation potential calculations: the value for r-MgF${}_{2}$ from Reference [68], the values for c-SrF${}_{2}$ after References [69,70]. The row “Other” refers to the theoretical data for r-MgF${}_{2}$ from Reference [37] and for c-SrF${}_{2}$ from References [32,71] in parentheses.

${\mathbf{r}-\mathbf{MgF}}_{2}$ | Direct Gap | ${\mathbf{\Delta}}_{\mathit{cf}}$ |
---|---|---|

[eV] | [eV] | |

PBEsol | 6.921 | −0.320 |

HSE06 | 9.433 | −0.289 |

G${}_{0}$W${}_{0}$ | 12.800 | −0.291 |

GW${}_{0}$ | 13.243 | −0.285 |

scQP-GW | 13.945 | −0.277 |

B3PW | 9.48 | - |

Other | 12.17 | - |

Exp. | 12.4 | −0.2 |

${\mathbf{c}-\mathbf{SrF}}_{\mathbf{2}}$ | Direct Gap | Indirect Gap |

[eV] | [eV] | |

PBEsol | 6.932 | 6.827 |

HSE06 | 9.172 | 9.072 |

G${}_{0}$W${}_{0}$ | 11.437 | 11.316 |

GW${}_{0}$ | 11.820 | 11.700 |

scQP-GW | 12.490 | 12.375 |

B3PW | 11.306/10.96 | - |

Other | 11.24 | 11.20(7.55) |

Exp. | 11.25 | — |

**Table 5.**Relevant optical absorption observables and dielectric constants of the difluorides r-MgF${}_{2}$ and c-SrF${}_{2}$. The first peak energy position at the onset, the binding energy of the exciton and the ${\epsilon}_{\infty}$ values are reported as obtained in different BSE and GW schemes as discussed in the text and compared with corresponding experimental values from Reference [37] and Reference [29] (fourth column). In the case of r-MgF${}_{2}$, both the values for the two principal directions of the crystal were reported.

${\mathbf{r}-\mathbf{MgF}}_{2}$ | BSE (G${}_{0}$W${}_{0}$) | BSE (GW${}_{0}$) | BSE (scQP-GW) | $\mathbf{EXP}$ |
---|---|---|---|---|

E${}_{Peak,\left|\right|}$[eV] | 11.37 | 11.81 | 12.23 | 11.6 |

E${}_{Peak,\perp}$[eV] | 11.76 | 12.21 | 12.62 | 12.1 |

E${}_{Bind,\left|\right|}$[eV] | 1.14 | 1.14 | 1.43 | 0.8 |

E${}_{Bind,\perp}$[eV] | 1.04 | 1.03 | 1.32 | 0.5 |

${\epsilon}_{\infty ,\left|\right|}$ | 1.88 | 1.85 | 1.84 | 1.67 |

${\epsilon}_{\infty ,\perp}$ | 1.91 | 1.890 | 1.87 | 1.68 |

${\mathbf{c}-\mathbf{SrF}}_{\mathbf{2}}$ | BSE (G${}_{\mathbf{0}}$W${}_{\mathbf{0}}$) | BSE (GW${}_{\mathbf{0}}$) | BSE (scQP-GW) | $\mathbf{EXP}$ |

E${}_{Peak}$[eV] | 10.01 | 10.40 | 10.83 | 10.6 |

E${}_{Bind}$[eV] | 1.43 | 1.42 | 1.66 | 0.65 |

${\epsilon}_{\infty}$ | 2.18 | 2.15 | 2.13 | 2.08 |

**Table 6.**Excited and optical properties of the clusters (MgF${}_{2}$)${}_{n}$ and (SrF${}_{2}$)${}_{n}$, $n=1,2,3$ and the crystalline solids (r-MgF${}_{2}$ and c-SrF${}_{2}$). The quasiparticle gap ${\mathrm{E}}_{gap}$, the optical onset ${\mathrm{E}}_{opt}$, and the binding energy of the exciton ${\mathrm{E}}_{\mathrm{b}}$ are displayed. In the first row, we report the data on the (MgF${}_{2}$)${}_{n}$ clusters, in the second, we present the data on the solid r-MgF${}_{2}$ from Reference [37], in the third, we present the outcomes of the present work for solid r-MgF${}_{2}$ with experimental data in parentheses. In the last two rows, the data for (SrF${}_{2}$)${}_{n}$ clusters and solid c-SrF${}_{2}$ are given. The outcomes for the clusters are from Reference [38].

${\mathbf{E}}_{\mathbf{gap}}$ | ${\mathbf{E}}_{\mathbf{Peak}}$ | ${\mathbf{E}}_{\mathbf{Bind}}$ | |
---|---|---|---|

[eV] | [eV] | [eV] | |

Clusters (MgF${}_{2}$)${}_{n}$ | 11.45–12.49 | 6.56–6.78 | 4.49–5.71 |

Solid r-MgF${}_{2}$ (Present) | 13.24 (12.4) | 11.8 (11.6) | 1.4 (0.8) |

Solid r-MgF${}_{2}$ (Other) | 12.17 | 10.90 | 1.13 |

Clusters (SrF${}_{2}$)${}_{n}$ | 9.33–10.16 | 5.10–5.26 | 4.23–4.9 |

Solid c-SrF${}_{2}$ (Present) | 11.82 (11.25) | 10.4 (10.6) | 1.4(0.65) |

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## Share and Cite

**MDPI and ACS Style**

Cappellini, G.; Furthmüller, J.; Bechstedt, F.; Botti, S.
Electronic and Optical Properties of Alkaline Earth Metal Fluoride Crystals with the Inclusion of Many-Body Effects: A Comparative Study on Rutile MgF_{2} and Cubic SrF_{2}. *Symmetry* **2023**, *15*, 539.
https://doi.org/10.3390/sym15020539

**AMA Style**

Cappellini G, Furthmüller J, Bechstedt F, Botti S.
Electronic and Optical Properties of Alkaline Earth Metal Fluoride Crystals with the Inclusion of Many-Body Effects: A Comparative Study on Rutile MgF_{2} and Cubic SrF_{2}. *Symmetry*. 2023; 15(2):539.
https://doi.org/10.3390/sym15020539

**Chicago/Turabian Style**

Cappellini, Giancarlo, Jürgen Furthmüller, Friedhelm Bechstedt, and Silvana Botti.
2023. "Electronic and Optical Properties of Alkaline Earth Metal Fluoride Crystals with the Inclusion of Many-Body Effects: A Comparative Study on Rutile MgF_{2} and Cubic SrF_{2}" *Symmetry* 15, no. 2: 539.
https://doi.org/10.3390/sym15020539