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-xGe_x miscible alloys [5]. Besides the alloys, the band structure of solid solutions BiOBr_1-xI_x and NaLa_1-xBi_xS₂ is also tuned nonlinearly. The band edges vary significantly at the point with 20% BiOI or NaBiS₂ [6,7]. A “magic” composition in solid solution (ZnSnN₂)_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-xSr_xF₂ 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 CaF₂-SrF₂ solution) the atomic structures rarely vary nonlinearly. In order to induce nonlinear atomic structure change we include rare-earth ions into the CaF₂-SrF₂ solution. It is found that spectroscopic properties of Ca_1-xSr_xF₂ 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.% Nd³⁺:Ca_1-xSr_xF₂ and 0.5 at.% Nd³⁺,5 at.% Y³⁺:Ca_1-xSr_xF₂ (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 NdF₃, YF₃, CaF₂ and SrF₂ were mixed stoichiometrically and 1 wt.% PbF₂ 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]:

3. Results and discussion

Figure 1(a-b) presents absorption spectra of 0.5% Nd³⁺:Ca_1-xSr_xF₂ and 0.5% Nd³⁺,5% Y³⁺:Ca_1-xSr_xF₂ crystals. Nd³⁺-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 ⁴I_9/2 → ⁴F_5/2 + ²H_9/2 transitions of Nd³⁺ 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 Sr²⁺. It is found that absorption intensity of N1 decreases sharply with addition of Sr²⁺ from 0 to 30 at.%, at the same time absorption intensity of N2 increases greatly. When the concentration of Sr²⁺ is higher than 30 at.%, the N1:N2 absorption intensity ratio varies insignificantly. As a comparison, absorption spectra of Nd³⁺,Y³⁺:Ca_1-xSr_xF₂ samples are shown in Fig. 1(b-c). With addition of Sr²⁺, absorption of N1 decreases and that of N2 rises. It is similar with that of Nd³⁺:Ca_1-xSr_xF₂ but varied in a slightly sharper style. The photoluminescence of Nd³⁺,Y³⁺:Ca_1-xSr_xF₂ 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 Sr²⁺ content increases. Significant change in spectral bandwidth is observed with Sr²⁺ concentration 0–30 at.%, while for 30–100 at.%, the change is small.

 

Fig. 1. Area normalized absorption spectra of (a) Nd³⁺:Ca_1-xSr_xF₂ and (b) Nd³⁺,Y³⁺:Ca_1-xSr_xF₂ solid solution crystals. (c) Sr²⁺ concentrations dependent normalized absorption intensity at 791 and 797 nm. (d) Photoluminescence spectra of Nd³⁺,Y³⁺:Ca_1-xSr_xF₂ excited at 791 and 797 nm.

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XRD patterns of investigated samples are shown in Fig. 2. All diffraction peaks for Nd³⁺:Ca_1-xSr_xF₂ and Nd³⁺,Y³⁺:Ca_1-xSr_xF₂ crystals agree well with that of pure CaF₂ or SrF₂. It can be seen that the diffraction positions for mixed crystals gradually shift from CaF₂ to SrF₂. As shown in Fig. 2(c), the lattice parameters shift linearly with increasing concentration of Sr²⁺. It is quite understandable, since the two crystals share the same crystal structure but with different lattice parameters (CaF₂: 5.46295 Å, SrF₂: 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-xSr_xF₂ 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-xSr_xF₂ 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 CaF₂ and SrF₂ are miscible in any ratio without phase separation and structure variation. It implies that Ca_1-xSr_xF₂ crystals are complete solid solutions, which is consistent with the XRD patterns and reported results [9].

 

Fig. 2. Powder XRD patterns of (a) Nd³⁺:Ca_1-xSr_xF₂ and (b) Nd³⁺,Y³⁺:Ca_1-xSr_xF₂ crystals. (c) Strontium concentrations dependent lattice parameter of Nd³⁺:Ca_1-xSr_xF₂ crystals.

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Fig. 3. (a) Bond distances and (b) angles dependent relaxed total energy of Ca_1-xSr_xF₂. (c) The nominal strontium concentrations dependent formation energy of Ca_1-xSr_xF₂ crystal.

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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 Nd³⁺ or Y³⁺ are modeled and relaxed. Since Ca_1-xSr_xF₂ 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 Sr²⁺ in the first cationic coordination shell of rare-earth are therefore considered rather than that of ‘x’ in Ca_1-xSr_xF₂ solid solutions. Besides, the Ca_0.5Sr_0.5F₂ 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 RE³⁺ 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 Nd³⁺ and Y³⁺ suggests that the lattice mismatch in C_3v centers is almost independent on Sr²⁺ concentrations. However, for the C_4v, lattice mismatch varies significantly with Sr²⁺ concentrations. The bond length of anions surrounding interstitial F_i^- and RE³⁺ in C_4v center is denoted as R(F₁-F₁) and R(F₂-F₂) in Fig. 4(d). As presented in Fig. 4(c), R(F₁-F₁) has a sharp increase when Sr²⁺ content change from 0.333 to 0.667, while R(F₂-F₂) changes smoothly. The lattice mismatch changes non-linearly instead of linearly, which contributes to the nonlinearly varied stabilities of C_4v centers.

 

Fig. 4. Formation energy of C_4v and C_3v center in the (a) Nd³⁺ and (b) Y³⁺. (c) Concentrations dependent bond lengths in C_4v center of the Nd³⁺ or Y³⁺ doped Ca_1-xSr_xF₂. (d) Structure of C_3v center with interstitial F_i^- at the next nearest site and C_4v center with interstitial F_i- at the nearest site. F₁^- and F₂^- are the normal fluorine and F_i^- the interstitial fluorine. The R(F₁-F₁) and R(F₂-F₂) represent the corresponding bond length. Grey balls are F^- anions and orange balls the RE³⁺ ions.

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The RE³⁺-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 Sr²⁺. 0.5% Nd³⁺:Ca_1-xSr_xF₂ 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 Sr²⁺. 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 Nd³⁺ 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 Sr²⁺ content used in calculations are different with the experimental ones. In order to reduce simulation costs on Ca_1-xSr_xF₂ solid solutions, concentration of Sr²⁺ in the first cationic shell of rare-earth ions rather than the ‘x’ in Ca_1-xSr_xF₂ 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 Ca²⁺ and Sr²⁺ 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.

 

Fig. 5. Temperature dependent dielectric losses of Nd³⁺:Ca_1-xSr_xF₂ crystals at the frequency of 100 Hz.

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The pDOS of C_4v center in Nd³⁺ is also investigated. Two crystals with Sr²⁺ 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 Sr²⁺ 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 Sr²⁺ 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 Y³⁺ center in Fig. 7 is similar to that of Nd³⁺. 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 Sr²⁺.

 

Fig. 6. pDOS of Nd³⁺ and interstitial F_i^- in C_4v center. Concentrations of Sr²⁺ in the first cationic coordination shell of Nd³⁺ are 0.333 and 0.667 in (a) and (b), respectively. (c) and (d) are the corresponding enlarged images.

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Fig. 7. pDOS of Y³⁺ and interstitial F_i^- in C_4v center. Concentrations of Sr²⁺ in the first cationic coordination shell of Y³⁺ are 0.333 and 0.667 in (a) and (b), respectively. (c) and (d) are the corresponding enlarged images.

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Based on the aforementioned discussions, it is known that relative stabilities of C_4v and C_3v are influenced by the Sr²⁺ 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 Nd³⁺ and Y³⁺ 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 Sr²⁺ 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 SrF₂. 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 Sr²⁺ content. Besides, as shown in Fig. 8(a-b), the cubic sublattice centers of Nd³⁺ have lower energy and are more stable than the corresponding Y³⁺, while the square antiprism sublattice clusters of Y³⁺ have lower energy and are more stable than that of Nd³⁺, which agrees well with the clustering characters of Nd³⁺ and Y³⁺ doped CaF₂ and SrF₂ crystals [28,35]. From Ref. [28], the strong repulsion of F_i^--F_lattice^- around Y³⁺ transforms the cubic sublattice to square antiprism clusters easily. While for Nd³⁺, the repulsion of F_i^--F_lattice^- is weaker, it is benefit for the stability of cubic sublattice centers. The research indicates that in CaF₂, Nd³⁺ ions tend to form cubic lattice centers and the Y³⁺ ions tend to form square antiprism sublattice centers in SrF₂. From Nd³⁺ to Y³⁺ and with the matrix from CaF₂ to SrF₂, 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 Nd³⁺,Y³⁺:Ca_1-xSr_xF₂ than that in Nd³⁺:Ca_1-xSr_xF₂. The nonlinearly evolved local structures of rare-earth ions agrees well with the spectral properties.

 

Fig. 8. Formation energy of Nd³⁺ or Y³⁺ clusters versus concentration of Sr²⁺ in the first cationic shell of RE³⁺ ions. (a) and (b) represent the cubic sublattice and square antiprism structure clusters, respectively.

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4. Conclusion

In summary, spectral properties of Nd³⁺:Ca_1-xSr_xF₂ and Nd³⁺,Y³⁺:Ca_1-xSr_xF₂ crystals are observed to be varied nonlinearly. XRD patterns and the first principles calculations on Ca_1-xSr_xF₂ 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 Sr²⁺ concentrations. Temperature-dependent dielectric losses and the projected density of state calculations confirm that local structures of Nd³⁺ evolve nonlinearly. The nonlinear spectral properties are therefore obtained.