## Abstract

The modification of chemical composition to improve desired material parameters is an effective method in materials science and engineering. In this work, Ca_{1-x}Sr_{x}F_{2} solid solution is chosen as the subject. Nd^{3+} and Y^{3+} ions are used as dopants. We have found that spectral properties of Nd^{3+}:Ca_{1-x}Sr_{x}F_{2} and Nd^{3+},Y^{3+}:Ca_{1-x}Sr_{x}F_{2} crystals vary nonlinearly with the ‘*x*’. The X-ray diffraction (XRD) patterns and the density functional theory (DFT) calculations on Ca_{1-x}Sr_{x}F_{2} solid solutions have ruled out the influence of matrix crystals on spectral properties. The rare-earth monomer centers of C_{4v} or C_{3v} symmetry, and the high order clusters are modeled. The calculated results show, that thermodynamic stabilities of the centers vary nonlinearly. Temperature-dependent dielectric losses and the results of projected density of states (pDOS) calculations also show nonlinear dependency. The nonlinearly evolved local structures from cubic to square antiprism sublattice cause the nonlinear variation of spectral properties. The methodology of rare-earth induced nonlinear structural evolutions is then proposed, which is useful for exploring new materials.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Tuning of composition or modification of atomic structure in order to obtain properties of interest is widely used in material science and engineering. For example, solid solutions with alloying elements have been used to improve mechanical properties in alloy fields [1,2]. Atomic sites or interstitials would be partially occupied with the second phase atoms and the inner strains are generated to prevent dislocation movement. Interestingly, many properties vary nonlinearly rather than linearly in the tuning range. Schoen and Mokkath et al. have reported that the magnetic damping and magnetic moments vary sharply in metallic alloys as a consequence of phase transition or magnetic phase transition [3,4]. Even with no phase transition, the electric resistivity varies nonlinearly in Si_{1-x}Ge_{x} miscible alloys [5]. Besides the alloys, the band structure of solid solutions BiOBr_{1-x}I_{x} and NaLa_{1-x}Bi_{x}S_{2} is also tuned nonlinearly. The band edges vary significantly at the point with 20% BiOI or NaBiS_{2} [6,7]. A “magic” composition in solid solution (ZnSnN_{2})_{1-x}(ZnO)_{x} is found and a short-range ordered system with line-compound like properties is discovered [8].

Calcium and strontium fluorides exhibit continuous solubility. The crystal structure does not change regardless of Ca:Sr ratios. The unit cell parameters of Ca_{1-x}Sr_{x}F_{2} solution vary linearly with the ‘*x*’. Other physical parameters like lattice vibrations, thermal conductivity, dielectric constant and microhardness change gradually in the parabolic-like dependency [9–13]. In systems where both components share the same crystal structure (like CaF_{2}-SrF_{2} solution) the atomic structures rarely vary nonlinearly. In order to induce nonlinear atomic structure change we include rare-earth ions into the CaF_{2}-SrF_{2} solution. It is found that spectroscopic properties of Ca_{1-x}Sr_{x}F_{2} solid solution change nonlinearly with the ‘*x*’. The first principles calculation and dielectric property measurements have revealed, that local structures of rare earth ions evolve nonlinearly. Nonlinear evolution in local structures of rare-earth ions contributes to the nonlinear variation of spectroscopic properties.

## 2. Experimental

Two series of crystals 0.5 at.% Nd^{3+}:Ca_{1-x}Sr_{x}F_{2} and 0.5 at.% Nd^{3+},5 at.% Y^{3+}:Ca_{1-x}Sr_{x}F_{2} (*x* = 0, 0.01, 0.05, 0.1, 0.3, 0.5 0.7, 1.0) were grown by the temperature gradient technique method [14]. High purity raw materials (> 99.995%) including NdF_{3}, YF_{3}, CaF_{2} and SrF_{2} were mixed stoichiometrically and 1 wt.% PbF_{2} was added as an oxygen scavenger. The mixtures were loaded to a multi-compartment graphite crucible and then was mounted in the furnace. The pressure in the furnace was kept to be 10^{−3} Pa. The temperature was increased to 1400 °C and was kept for 10 h in order to completely melt the loads. In growth process the temperature gradient was set to 1.5 °C/h.

The grown crystals were cut and polished for spectral measurements. To ensure uniformity of the results, samples were cut from the same parts of crystals. Room temperature absorption spectra were measured with a Jasco V-750 UV/VIS/NIR spectrometer. Photoluminescence properties were measured with a FLS980 time-resolved spectrofluorometer equipped with a 450 W xenon lamp and thermoelectrically cooled InGaAs detector. XRD patterns were collected using Rigaku Ultima-IV diffractometer (Japan) equipped with Cu x-ray lamp. The scans were performed in 20–70° (2θ) with 0.02° per step. For dielectric measurement samples were coated with silver electrodes, and temperature-dependent dielectric losses were measured by Novocontrol Broadband Dielectric Spectrometer (Germany) at frequency of 100 Hz at temperatures from 135 to 330 K.

Calculations were conducted by the plane-wave basis set method in the framework of DFT, as implemented in the VASP code [15,16]. The projector augmented-wave pseudopotential with exchange correlation function in the form of Perdew-Burke-Ernzerhof (PBE) was used to describe the mutual interactions [17,18]. The 3 × 3 × 3 or 2 × 2 × 2 supercells were selectively applied and a 1 × 1 × 1 Gamma *k*-grid was used to ensure that forces on individual atoms reached 0.01 eV·Å^{−1}. The cut-off energy was set to 550 eV with electronic accuracy of 10^{−5} eV. The spin polarization was included in all calculations. The formation energy of a given compound was calculated by Eq. (1) [19]:

_{1-x}Sr

_{x}F

_{2}, CaF

_{2}and SrF

_{2}, respectively. Formation energy of the clusters was given by Eq. (2):

_{tot}and E

_{0}are the relaxed energy of Ca

_{1-x}Sr

_{x}F

_{2}with and without rare-earth ions, respectively; m and n denote the number of trivalent substitutional impurity atoms and interstitial fluorine ions within a cluster; E

_{1}and E

_{2}are the relaxed energy of Ca

_{1-x}Sr

_{x}F

_{2}containing [1RE

^{3+}−0F

_{i}

^{-}] and [0RE

^{3+}−1F

_{i}

^{-}] (RE = Nd, Y); E

_{corr}is the potential alignment and image charge corrections calculated by Eq. (3) [20–22]: where g is scaling factor (for face-centered cubic structure the value of −0.34 was adopted [22]), q is the net charge, α the Madelung constant of 5.038 [23], ɛ and L are the static dielectric constant and supercell dimensions of Ca

_{1-x}Sr

_{x}F

_{2}crystal respectively. For pure CaF

_{2}and SrF

_{2}, ɛ equals to 6.812 and 6.476 at 300 K [24], L equals to 10.9 Å and 11.6 Å respectively. E

_{corr}was approximately the same for both CaF

_{2}and SrF

_{2}(0.323 eV and 0.319 eV respectively). The calculation results show gradual change in the parameters with varied ratios of CaF

_{2}:SrF

_{2}. This agrees well with the experimental results [9,10]. The values of E

_{corr}of Ca

_{1-x}Sr

_{x}F

_{2}were calculated based on the strontium content

*x*linearly. The projected density of states calculations were performed on basis of optimized structure with a 4 × 4 × 4 Gamma

*k*-grid. The Coulomb potential with U = 6 and 1 for neodymium and yttrium, respectively, were considered in the Dudarev’s LSDA + U method [25].

## 3. Results and discussion

Figure 1(a-b) presents absorption spectra of 0.5% Nd^{3+}:Ca_{1-x}Sr_{x}F_{2} and 0.5% Nd^{3+},5% Y^{3+}:Ca_{1-x}Sr_{x}F_{2} crystals. Nd^{3+}-doped alkaline earth fluorides have demonstrated excellent femtosecond laser performances, which is considered as a promising candidate for generation of high repetition rate ultrafast lasers [26,27]. There are two main absorption peaks corresponding to ^{4}I_{9/2} → ^{4}F_{5/2} + ^{2}H_{9/2} transitions of Nd^{3+} ions. These two peaks at 791 and 797 nm have been proved belonging to cubic sublattice and square antiprism structure clusters, respectively, which are denoted as N1 and N2 in the work [28,29]. The absorption intensity of N1 decreases while that of N2 rises with increasing concentration of Sr^{2+}. It is found that absorption intensity of N1 decreases sharply with addition of Sr^{2+} from 0 to 30 at.%, at the same time absorption intensity of N2 increases greatly. When the concentration of Sr^{2+} is higher than 30 at.%, the N1:N2 absorption intensity ratio varies insignificantly. As a comparison, absorption spectra of Nd^{3+},Y^{3+}:Ca_{1-x}Sr_{x}F_{2} samples are shown in Fig. 1(b-c). With addition of Sr^{2+}, absorption of N1 decreases and that of N2 rises. It is similar with that of Nd^{3+}:Ca_{1-x}Sr_{x}F_{2} but varied in a slightly sharper style. The photoluminescence of Nd^{3+},Y^{3+}:Ca_{1-x}Sr_{x}F_{2} under different excitation wavelengths also vary nonlinearly. Figure 1(d) shows emission spectra of samples recorded under 791 or 797 nm excitation. The spectral bandwidth of emission band decreases when Sr^{2+} content increases. Significant change in spectral bandwidth is observed with Sr^{2+} concentration 0–30 at.%, while for 30–100 at.%, the change is small.

XRD patterns of investigated samples are shown in Fig. 2. All diffraction peaks for Nd^{3+}:Ca_{1-x}Sr_{x}F_{2} and Nd^{3+},Y^{3+}:Ca_{1-x}Sr_{x}F_{2} crystals agree well with that of pure CaF_{2} or SrF_{2}. It can be seen that the diffraction positions for mixed crystals gradually shift from CaF_{2} to SrF_{2}. As shown in Fig. 2(c), the lattice parameters shift linearly with increasing concentration of Sr^{2+}. It is quite understandable, since the two crystals share the same crystal structure but with different lattice parameters (CaF_{2}: 5.46295 Å, SrF_{2}: 5.86400 Å). The linear lattice parameter is consistent with Vegard’s relationship, suggesting that complete solid solution is formed. In order to verify the point and then to rule out influence of matrix crystals on nonlinear spectral properties, simulations on Ca_{1-x}Sr_{x}F_{2} are performed. As shown in Fig. 3, total relaxed energy differences of the supercells are less than 0.01 eV with the varied bond distances and angles among strontium ions. Besides, formation energies of Ca_{1-x}Sr_{x}F_{2} vary in a parabolic like behavior when the ‘*x*’ changes from 0 to 1. When ‘*x*’ equals to 0.5, the highest value is obtained in the parabola. The symmetric parabolic character of formation energy with the highest point at *x* = 0.5 means that CaF_{2} and SrF_{2} are miscible in any ratio without phase separation and structure variation. It implies that Ca_{1-x}Sr_{x}F_{2} crystals are complete solid solutions, which is consistent with the XRD patterns and reported results [9].

The linear unit cell expansion cannot explain the nonlinear variation of spectral properties however. In fact, attention needs to be focused on trivalent lanthanides clusters formed in the fluorite crystals due to charge compensation effects [30–34].

In this section, the rare-earth clusters containing Nd^{3+} or Y^{3+} are modeled and relaxed. Since Ca_{1-x}Sr_{x}F_{2} crystals form complete solid solutions and the first coordination sphere of anions and cations influence the local structures of rare-earth ions greatly [35], concentrations of Sr^{2+} in the first cationic coordination shell of rare-earth are therefore considered rather than that of ‘*x*’ in Ca_{1-x}Sr_{x}F_{2} solid solutions. Besides, the Ca_{0.5}Sr_{0.5}F_{2} supercells with fixed strontium concentrations are chosen to study influence of the first cationic coordination sphere on the evolution of rare-earth ions structures. We firstly calculate the monomers of C_{4v} and C_{3v} with interstitials F_{i}^{-} at the nearest and next nearest site, respectively, because the relative stabilities of C_{4v} and C_{3v} determine the rare-earth clustering characteristics [35]. Lattice mismatch occurs in the plane connected two cubes, as shown in Fig. 4(d). Lattice mismatch will compete with the electrostatic interactions of RE^{3+} and F_{i}^{-}. If the electrostatic interactions are stronger, interstitial F_{i}^{-} will locate at the nearest site of the rare-earth creating C_{4v} center. Otherwise, in order to reduce the lattice mismatch, it will locate at the next nearest site creating C_{3v}. As shown in Fig. 4(a-b), the slight variation of formation energies of C_{3v} centers in Nd^{3+} and Y^{3+} suggests that the lattice mismatch in C_{3v} centers is almost independent on Sr^{2+} concentrations. However, for the C_{4v}, lattice mismatch varies significantly with Sr^{2+} concentrations. The bond length of anions surrounding interstitial F_{i}^{-} and RE^{3+} in C_{4v} center is denoted as R(F_{1}-F_{1}) and R(F_{2}-F_{2}) in Fig. 4(d). As presented in Fig. 4(c), R(F_{1}-F_{1}) has a sharp increase when Sr^{2+} content change from 0.333 to 0.667, while R(F_{2}-F_{2}) changes smoothly. The lattice mismatch changes non-linearly instead of linearly, which contributes to the nonlinearly varied stabilities of C_{4v} centers.

The RE^{3+}-F_{i}^{-} pairs could be treated as an electric dipoles in C_{4v} center. Dielectric spectra were measured to study the relationship between dipole’s resonant signals and contents of Sr^{2+}. 0.5% Nd^{3+}:Ca_{1-x}Sr_{x}F_{2} samples were chosen and the temperature-dependent dielectric losses were measured. As presented in Fig. 5, there exists one resonant signal and it is ascribed to C_{4v} center [36]. It is noted that other signals are also observed, but difficult to be distinguished. Stability of C_{4v} centers is very important, so only C_{4v} centers are discussed here. It can be seen that the peak position is at about 210 K when the ‘*x*’ equals to 0 and 0.1. Then, the position jumps to be at about 240 K when the ‘*x*’ equals to 0.3. The sharp increase from 0.1 to 0.3 indicates that activation energy of C_{4v} vary dramatically with increasing concentration of Sr^{2+}. Based on the Arrhenius equation, it is clear that the higher temperature, the larger activation energy. The trend of resonant signals is nearly the same with that of absorption and emission spectra. It is also very close to the varied formation energies of Nd^{3+} C_{4v} centers, except the turning points. The turning points of dielectric and absorption spectra are at about *x* = 0.3. It is a little bit different with the calculated results that the points locate in the area of 0.3 ∼ 0.6. It is because the Sr^{2+} content used in calculations are different with the experimental ones. In order to reduce simulation costs on Ca_{1-x}Sr_{x}F_{2} solid solutions, concentration of Sr^{2+} in the first cationic shell of rare-earth ions rather than the ‘*x*’ in Ca_{1-x}Sr_{x}F_{2} is adopted in calculations. It should be noted here that bandwidths of the resonant signals are much broader in solid solution crystals. The mixing of Ca^{2+} and Sr^{2+} probably contributes to the broad signals. However, contributions from the new emerged centers could not be absolutely ruled out. The varied intensities of signals are the clues that support such statements. The number of C_{4v} centers correlating with resonant intensities are different in the crystals. For solid solutions the resonant intensities are relatively low which suggests that quantities of C_{4v} centers are small. The reduced portion of C_{4v} centers could be transformed into other kinds of clusters. The position of resonant signals of dimer centers is reported to be at shoulders of C_{4v} signals [36], which may contribute to the broad band.

The pDOS of C_{4v} center in Nd^{3+} is also investigated. Two crystals with Sr^{2+} concentrations in the first cationic shells of 0.333 and 0.667 are selected and calculated in Fig. 6. These two concentrations are representative. We also calculated the pDOS of samples with Sr^{2+} concentration of *x* = 0 and 1. The results are very close to that of *x* = 0.333 and 0.667 respectively. For the samples with Sr^{2+} concentration of 0.333, 2p states of F_{i}^{-} energy levels are at −0.287 eV, −0.151 eV and −0.0156 eV, while for the 0.667, they shift to be at −0.231 eV, −0.0858 eV and −0.0130 eV, respectively. Besides, the intensity of lower energy levels at −0.287 eV and −0.151 eV decreases and that of the highest level at −0.0156 eV increases as the concentrations vary from 0.333 to 0.667. The result of Y^{3+} center in Fig. 7 is similar to that of Nd^{3+}. The 2p states of F_{i}^{-} energy levels are at −0.312 eV, −0.146 eV and −0.0138 eV, and they shift to be at −0.215 eV, −0.0495 eV and −0.0165 eV, respectively. The bonding characters indicate that stability of C_{4v} centers is obviously weakened with rising concentration of Sr^{2+}.

Based on the aforementioned discussions, it is known that relative stabilities of C_{4v} and C_{3v} are influenced by the Sr^{2+} content, as well as that of high order clusters. The more stable C_{4v} center, the more stable cubic sublattice clusters; the more stable C_{3v} center, the more stable square antiprism structure clusters [28]. With this in mind, high order clusters of Nd^{3+} and Y^{3+} are modelled and simulated. Formation energy of cubic sublattice and square antiprism structure clusters are presented in Fig. 8(a-b), respectively. As can be seen, when Sr^{2+} content in the first cationic coordination shell of rare-earth increases from 0 to 0.4, formation energy varies slightly, and follows a sharp increase in going from 0.4 to 0.6. Then it gradually approaches to that in pure SrF_{2}. Compared with cubic sublattice clusters, formation energy of square antiprism structure centers firstly decreases and then increases, which means that the rising rate in the range from 0.4 to 0.6 is larger in square antiprism clusters. The square antiprism sublattice clusters are influenced more seriously than the cubic sublattice centers by Sr^{2+} content. Besides, as shown in Fig. 8(a-b), the cubic sublattice centers of Nd^{3+} have lower energy and are more stable than the corresponding Y^{3+}, while the square antiprism sublattice clusters of Y^{3+} have lower energy and are more stable than that of Nd^{3+}, which agrees well with the clustering characters of Nd^{3+} and Y^{3+} doped CaF_{2} and SrF_{2} crystals [28,35]. From Ref. [28], the strong repulsion of F_{i}^{-}-F_{lattice}^{-} around Y^{3+} transforms the cubic sublattice to square antiprism clusters easily. While for Nd^{3+}, the repulsion of F_{i}^{-}-F_{lattice}^{-} is weaker, it is benefit for the stability of cubic sublattice centers. The research indicates that in CaF_{2}, Nd^{3+} ions tend to form cubic lattice centers and the Y^{3+} ions tend to form square antiprism sublattice centers in SrF_{2}. From Nd^{3+} to Y^{3+} and with the matrix from CaF_{2} to SrF_{2}, stability of cubic sublattice clusters is reduced and that of square antiprism clusters enhanced. It is consistent with the results that absorption of N2 is more favored in Nd^{3+},Y^{3+}:Ca_{1-x}Sr_{x}F_{2} than that in Nd^{3+}:Ca_{1-x}Sr_{x}F_{2}. The nonlinearly evolved local structures of rare-earth ions agrees well with the spectral properties.

## 4. Conclusion

In summary, spectral properties of Nd^{3+}:Ca_{1-x}Sr_{x}F_{2} and Nd^{3+},Y^{3+}:Ca_{1-x}Sr_{x}F_{2} crystals are observed to be varied nonlinearly. XRD patterns and the first principles calculations on Ca_{1-x}Sr_{x}F_{2} have confirmed that the crystals are complete solid solutions. Influence of matrix crystals on the spectral properties is then ruled out. Calculations on monomer centers of C_{4v} and C_{3v}, as well as high order clusters are performed and the results show that thermodynamic stabilities of the centers vary nonlinearly with Sr^{2+} concentrations. Temperature-dependent dielectric losses and the projected density of state calculations confirm that local structures of Nd^{3+} evolve nonlinearly. The nonlinear spectral properties are therefore obtained.

## Funding

National Natural Science Foundation of China (61905289, 61925508); Science and Technology Commission of Shanghai Municipality (20511107400, 20520750200); CAS Interdisciplinary Innovation Team (JCTD-2019-12); Instrument Developing Project of CAS (ZDKYYQ20210002).

## Disclosures

The authors declare no conflicts of interests.

## Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

## References

**1. **M. Suzuki, T. Kimura, J. Koike, and K. Maruyama, “Strengthening effect of Zn in heat resistant Mg-Y-Zn solid solution alloys,” Scr. Mater. **48**(8), 997–1002 (2003). [CrossRef]

**2. **M. P. Agustianingrum, S. Yoshida, N. Tsuji, and N. Park, “Effect of aluminum addition on solid solution strengthening in CoCrNi medium-entropy alloy,” J. Alloys Compd. **781**, 866–872 (2019). [CrossRef]

**3. **M. A. W. Schoen, D. Thonig, M. L. Schneider, T. J. Silva, H. T. Nembach, O. Eriksson, O. Karis, and J. M. Shaw, “Ultra-low magnetic damping of a metallic ferromagnet,” Nat. Phys. **12**(9), 839–842 (2016). [CrossRef]

**4. **J. H. Mokkath and U. Schwingenschlögl, “Magnetic phase transition in 2 nm Ni_{x}Cu_{1−x} (0 ≤ x ≤ 1) clusters,” J. Phys. Chem. C **118**(15), 8169–8173 (2014). [CrossRef]

**5. **P. R. Abel, A. M. Chockla, Y. M. Lin, V. C. Holmberg, J. T. Harris, B. A. Korgel, A. Heller, and C. B. Mullins, “Nanostructured Si_{1-x}Ge_{x} for tunable thin film lithium-ion battery anodes,” ACS Nano **7**(3), 2249–2257 (2013). [CrossRef]

**6. **L. Kong, J. Q. Guo, J. W. Makepeace, T. C. Xiao, H. F. Greer, W. Z. Zhou, Z. Jiang, and P. P. Edwards, “Rapid synthesis of BiOBr_{x}I_{1-x} photocatalysts: Insights to the visible-light photocatalytic activity and strong deviation from Vegard’s law,” Catal. Today **335**(SI), 477–484 (2019). [CrossRef]

**7. **A. BaQais, N. Tymińska, T. Le Bahers, and K. Takanabe, “Optoelectronic structure and photocatalytic applications of Na(Bi,La)S_{2} solid solutions with tunable band gaps,” Chem. Mater. **31**(9), 3211–3220 (2019). [CrossRef]

**8. **J. Pan, J. J. Cordell, G. J. Tucker, A. Zakutayev, A. C. Tamboli, and S. Lany, “Perfect short-range ordered alloy with line-compound-like properties in the ZnSnN_{2}:ZnO system,” npj Comput. Mater. **6**(1), 63 (2020). [CrossRef]

**9. **R. E. Youngman and C. M. Smith, “Multinuclear NMR studies of mixed Ca_{1−x}Sr_{x}F_{2} crystals,” Phys. Rev. B **78**(1), 014112 (2008). [CrossRef]

**10. **R. K. Chang, B. Lacina, and P. S. Pershan, “Raman scattering from mixed crystals (Ca_{x}Sr_{1−x})F_{2} and (Sr_{x}Ba_{1-x})F_{2},” Phys. Rev. Lett. **17**(14), 755–758 (1966). [CrossRef]

**11. **P. A. Popov, N. V. Moiseev, D. N. Karimov, N. I. Sorokin, E. A. Sulyanova, B. P. Sobolev, V. A. Konyushkin, and P. P. Fedorov, “Thermophysical characteristics of Ca_{1−x}Sr_{x}F_{2} solid-solution crystals (0 ≤ x ≤ 1),” Crystallogr. Rep. **60**(1), 116–122 (2015). [CrossRef]

**12. **M. V. Subrahmanya Sarma and S. V. Suryanarayana, “Dielectric studies of Ca_{x}Sr_{1−x}F_{2} mixed crystals,” J. Mater. Sci.: Mater. Electron. **1**(4), 182–184 (1990). [CrossRef]

**13. **M. V. Subrahmanya Sarma and S. V. Suryanarayana, “Study of microhardness on Ca_{x}Sr_{1-x}F_{2} mixed crystals,” J. Mater. Sci. Lett. **5**(12), 1277–1278 (1986). [CrossRef]

**14. **Y. Z. Zhou, “Growth of high quality large Nd:YAG crystals by temperature gradient technique (TGT),” J. Cryst. Growth **78**(1), 31–35 (1986). [CrossRef]

**15. **G. Kresse and J. Hafner, “Ab initio molecular-dynamics for liquid-metals,” Phys. Rev. B **47**(1), 558–561 (1993). [CrossRef]

**16. **G. Kresse and J. Furthmuller, “Efficiency of ab-initio total energy calculations for metals and semiconductors using a plane-wave basis set,” Comput. Mater. Sci. **6**(1), 15–50 (1996). [CrossRef]

**17. **P. E. Blöchl, “Projector augmented-wave method,” Phys. Rev. B **50**(24), 17953–17979 (1994). [CrossRef]

**18. **J. P. Perdew, K. Burke, and M. Ernzerhof, “Generalized gradient approximation made simple,” Phys. Rev. Lett. **77**(18), 3865–3868 (1996). [CrossRef]

**19. **S. Benny, R. Grau-Crespo, and N. H. de Leeuw, “A theoretical investigation of α-Fe_{2}O_{3}-Cr_{2}O_{3} solid solutions,” Phys. Chem. Chem. Phys. **11**(5), 808–815 (2009). [CrossRef]

**20. **C. Persson, Y. J. Zhao, S. Lany, and A. Zunger, “N-type doping of CuInSe_{2} and CuGaSe_{2},” Phys. Rev. B **72**(3), 035211 (2005). [CrossRef]

**21. **G. Makov and M. C. Payne, “Periodic boundary-conditions in ab-initio calculations,” Phys. Rev. B **51**(7), 4014–4022 (1995). [CrossRef]

**22. **S. Lany and A. Zunger, “Assessment of correction methods for the band-gap problem and for finite-size effects in supercell defect calculations: Case studies for ZnO and GaAs,” Phys. Rev. B **78**(23), 235104 (2008). [CrossRef]

**23. **A. Jockisch, U. Schröder, F. W. de Wette, and W. Kress, “Relaxation and dynamics of the (111) surfaces of the fluorides CaF_{2} and SrF_{2},” J. Phys.: Condens. Matter **5**(31), 5401–5410 (1993). [CrossRef]

**24. **C. Andeen, J. Fontanella, and D. Schuele, “Pressure and temperature derivatives of the low-frequency dielectric constants of the alkaline-earth fluorides,” Phys. Rev. B **6**(2), 591–595 (1972). [CrossRef]

**25. **S. L. Dudarev, G. A. Botton, S. Y. Savrasov, C. J. Humphreys, and A. P. Sutton, “Electron-energy-loss spectra and the structural stability of nickel oxide: An LSDA + U study,” Phys. Rev. B **57**(3), 1505–1509 (1998). [CrossRef]

**26. **Z. P. Qin, G. Q. Xie, J. Ma, W. Y. Ge, P. Yuan, L. J. Qian, L. B. Su, D. P. Jiang, F. K. Ma, Q. Zhang, Y. X. Cao, and J. Xu, “Generation of 103 fs mode-locked pulses by a gain linewidth-variable Nd,Y:CaF_{2} disordered crystal,” Opt. Lett. **39**(7), 1737–1739 (2014). [CrossRef]

**27. **J. F. Zhu, L. Wei, W. L. Tian, J. X. Liu, Z. H. Wang, L. B. Su, J. Xu, and Z. Y. Wei, “Generation of sub-100 fs pulses from mode-locked Nd,Y:_{SrF2} laser with enhancing SPM,” Laser Phys. Lett. **13**(5), 055804 (2016). [CrossRef]

**28. **F. K. Ma, D. P. Jiang, Z. Zheng, X. Q. Tian, Q. H. Wu, J. Y. Wang, X. B. Qian, Y. Liu, and L. B. Su, “Tailoring the local lattice distortion of Nd^{3+} by codoping of Y^{3+} through first principles calculation for tuning the spectroscopic properties,” Opt. Mater. Express **9**(11), 4256–4272 (2019). [CrossRef]

**29. **F. K. Ma, P. X. Zhang, L. B. Su, H. Yin, Z. Li, Q. T. Lv, and Z. Q. Chen, “The host driven local structures modulation towards broadband photoluminescence in neodymium-doped fluorite crystal,” Opt. Mater. **119**, 111322 (2021). [CrossRef]

**30. **J. Corish, C. R. A. Catlow, P. W. M. Jacobs, and S. H. Ong, “Defect aggregation in anion-excess fluorites. Dopant monomers and dimers,” Phys. Rev. B **25**(10), 6425–6438 (1982). [CrossRef]

**31. **P. J. Bendall, C. R. A. Catlow, J. Corish, and P. W. M. Jacobs, “Defect aggregation in anion-excess fluorites II. Clusters containing more than two impurity atoms,” J. Solid State Chem. **51**(2), 159–169 (1984). [CrossRef]

**32. **Y. K. Voronko, A. A. Kaminskii, and V. V. Osiko, “Analysis of the optical spectra of CaF_{2}:Nd^{3+} (type 1) crystals,” Sov. Phys. JETP **22**(2), 295–300 (1966).

**33. **Y. K. Voronko, V. V. Osiko, and I. A. Shcherbakov, “Investigation of the interaction of Nd^{3+} ions in CaF_{2}, SrF_{2} and BaF_{2} crystals (type 1),” Sov. Phys. JETP **28**(5), 838–844 (1969).

**34. **D. R. Tallant and J. C. Wright, “Selective laser excitation of charge compensated sites in CaF_{2}:Er^{3+},” J. Chem. Phys. **63**(5), 2074–2085 (1975).

**35. **F. K. Ma, F. Su, R. F. Zhou, Y. Y. Ou, L. J. Xie, C. M. Liu, D. P. Jiang, Z. Zhang, Q. H. Wu, L. B. Su, and H. B. Liang, “The defect aggregation of RE^{3+} (RE = Y, La∼Lu) in MF_{2} (M = Ca, Sr, Ba) fluorites,” Mater. Res. Bull. **125**, 110788 (2020). [CrossRef]

**36. **C. G. Andeen, J. J. Fontanella, M. C. Wintersgill, P. J. Welcher, R. J. Kimble, and G. E. Matthews, “Clustering in rare-earth-doped alkaline-earth fluorides,” J. Phys. C: Solid State Phys. **14**(24), 3557–3574 (1981). [CrossRef]