Theoretical Studies of the Local Structures and EPR Parameters for the Rhombic Cu²⁺ Center in Cu_0.5Zr₂(PO₄)₃ Phosphate

1 Introduction

Cu_0.5Zr₂(PO₄)₃ phosphate belongs to the Nasicon-type family with a three-dimensional network built of PO₄ tetrahedra sharing corners with ZrO₆ octahedra [1, 2]. This material has been extensively studied as solid electrolytes, ionic conductors, low thermal expansion ceramics, sensors, etc. [3–6]. In order to improve the efficiency of the aforementioned properties of this material, knowledge of the local structures of Cu²⁺ is useful. In Cu_0.5Zr₂(PO₄)₃ phosphate, there are two sites labeled M(1) and M(2), can be occupied by copper ions [7, 8]. The M(1) site is coordinated to a trigonally (C_3v) distorted octahedron, formed by the common faces of two neighboring ZrO₆ octahedra perpendicular to the c-axis of the hexagonal cell. The M(2) site with a coordination number varying from eight to ten, is located in a large and distorted cavity between these layers. Generally, the symmetry of the available sites should decrease, because Cu²⁺ ion has a tendency to show its own environment in many oxide compounds. As is known, electron paramagnetic resonance (EPR) technique is a powerful tool to study the structural model for the paramagnetic defects in crystals [9–11]. For example, EPR studies [12] were carried out for Cu_0.5Zr₂(PO₄)₃, and the parameters (anisotropic g-factors g_i (i=x, y, z) and hyperfine structure constants A_i) were measured. From the obtained EPR parameters, Taoufik et al. assumed that Cu²⁺ on M(1) site presents a rhombically distorted octahedron (D_2h) [12]. Since, trigonal distortion cannot remove the degeneracy of the ground doublet ²E state of d⁹ ions [13], Cu²⁺ (3d⁹) ion on an original trigonal octahedral site may suffer the transformation from trigonal to tetragonal (or rhombic) owing to the Jahn–Teller effect, which can remove the degeneracy of doublet ²E. To explain the EPR results, simple second-order perturbation formulas of the EPR parameters for a 3d⁹ ion under elongated octahedra were adopted in the previous work [14]. However, there are some imperfections in their treatments. First, the rough point-charge model was applied to determine the crystal-field parameters, which may induce obvious errors in the final results. Second, the hyperfine structure constants were not analysed in the previous work.

Considering that (i) information about local structures and electronic states for Cu²⁺ site in the Cu_0.5Zr₂(PO₄)₃ phosphate would be helpful to understand the microscopic mechanisms of EPR behaviours of Nasicon-type materials containing Cu²⁺ and (ii) the anisotropic g factors for a d⁹ ion in crystals are sensitive to its immediate environment, further investigations on EPR parameters and defect structures of the Cu²⁺ center in Cu_0.5Zr₂(PO₄)₃ phosphates are of fundamental and practical significance. In this work, the high-order perturbation formulas of EPR parameters for a 3d⁹ ion under rhombically elongated octahedra are adopted in the EPR analysis of Cu²⁺ center in Cu_0.5Zr₂(PO₄)₃ phosphate. In calculations, the rhombic crystal-field parameters are determined from the superposition model; the covalency and the admixture of d-orbitals in the ground state to EPR parameters are also taking into account. The theoretical results are in good agreement with the experimental values.

2 Calculations

For a Cu²⁺ (3d⁹) ion with low spin (S=1/2) under octahedra, its ground state is the orbital doublet ²E_g [15], which is Jahn–Teller unstable in a trigonal symmetry. The Jahn–Teller effect may relax the two Cu²⁺-O^2- bonds along z axis and contracting the other four bonds in the perpendicular plane, which transforms the local point symmetry from the original trigonal into tetragonal (D_4h) or rhombic (D_2h). Then, the original trigonal distortion of the host M(1) site can be entirely depressed. These low symmetrical distortions can be mainly characterized by the axial distortion angle Δα=α – α₀ with α ≈ tan^–1(R_∞/R_‖) and α₀ ≈ 45° of the cubic symmetry, where R_‖ and R_⊥ denote the metal-ligand distances parallel and perpendicular to the z axis (see Fig. 1). In addition, the bond angle θ between the x and y axes lies in the plane perpendicular to the z-axis. Thus, the impurity-ligand bonding lengths can be written in terms of the local distortion angle Δα and the average distance R̅ as

Figure 1:

Projective view of the local lattice structure for Cu²⁺ center in Cu_0.5Zr₂(PO₄)₃ phosphate.

(1)R|| ≈ 3R¯1 + 2tan(45° − Δα),  R⊥ ≈ 3R¯2 + cot(45° − Δα). (1)

Then, from superposition model [16] and local structures of Cu²⁺ in Cu_0.5Zr₂(PO₄)₃, the rhombic field parameters Ds, Dt, Dξ and Dη can be expressed as follows:

(2)Ds = 27A¯2(R0)[(R0R||)t2 − (R0R⊥)t2],Dt = 47A¯4(R0)[(7cos2θ + 3)(R0R⊥)t4 + 4(R0R||)t4],Dξ = 421A¯2(R0)(R0R⊥)t2cosθ,Dη = 2021A¯4(R0)(R0R⊥)t4cosθ. (2)

Here, t₂ ≈ 3 and t₄ ≈ 5 are the power-law exponents due to the dominant ionic nature of the bonds [16–19]. A̅₂(R₀) and A̅₄(R₀) are the intrinsic parameters with the reference distance R₀ (taken as the average Cu-O distance, i.e., R₀=R̅= 2.24 Å [3]). The ratio A̅₂(R₀)/A̅₄(R₀) was found to be in the range 9 ∼ 12 as in several studies of optical and EPR spectra using superposition model for 3dⁿ ions in crystals [19–23], we take the average value, i.e., A̅₂(R)/A̅₄(R) ≈ 10.5, here. It is noted that the shortcomings of the previous calculations [14], where the crystal-field parameters are acquired from point-charge model, are thus overcome in this work by adopting the more powerful superposition model.

For a 3d⁹(Cu²⁺) ion in rhombically elongated octahedra, its lower orbital doublet ²E_g would be separated into two singlets ²A_1g(θ) and ²A_1g’(ε), with the latter lying lowest. Meanwhile, the higher cubic orbital triplet ²T_2g would be split into three singlets ²B_1g(ζ), ²B_2g(η), and ²B_3g(ξ) [16]. For the studied [CuO₆]^10- octahedral cluster, the spin-orbit coupling parameter ζ_p⁰ (≈ 150 cm^–1 [24]) of ligand O^2– is much smaller than that (≈ 829 cm^–1 [25]) of the central ion Cu²⁺. Thus, the ligand orbital and spin-orbit coupling contributions to the EPR parameters are expected to be very small and reasonably ignored for simplicity here. Then, from perturbation theory, the high-order perturbation formulae of EPR parameters (g factors g_x, g_y, g_z and hyperfine structure constants A_x, A_y, A_z) for 3d⁹ ions in rhombic (D_2h) symmetry can be expressed as [11]:

(3)gx = gs + 2kζ/E2 + kζ2[(2/E1 – 1/E3)/E2 – 4/(E1E3)] + gsζ2[2/E12 – (1/E22 – 1/E32)/2] – kζ3×{(1/E2 – 1/E3) (1/E3 + 1/E2)/(2E1) + (2/E1 – 1/E2)(2/E1 + 1/E2)/2E3 – (1/E2 – 1/E3)/(2E2E4)}+ (gsζ3/4) [(1/E3 – 2/E1)/E22 + (2/E3 – 1/E2)/E32 + 2(1/E2 – 1/E3)/E12 + 2(1/E22 – 1/E32)/E1],gy = gs + 2kζ/E3 + kζ2[(2/E1 – 1/E2)/E3 – 4/(E1E2)] + gsζ2[2/E12–(1/E32 – 1/E22)/2] + kζ3×{(1/E2 – 1/E3)(1/E3 + 1/E2)/(2E1) + (2/E1 – 1/E3)(2/E1 + 1/E3)/2E2 – (1/E3 – 1/E2)/(2E3E4)}+ (gsζ3/4) [(1/E2 – 2/E1)/E32 + (2/E2 – 1/E3)/E22 + 2(1/E3 – 1/E2)/E12 + 2(1/E32 – 1/E22)/E1],gz = gs + 8kζ/E1+kζ2[1/(E3E2) + 2(1/E1E2 + 1/E1E3)] – gsζ2[1/E12–(1/E22 + 1/E32)/4]+ kζ3[8/E1–(1/E2 + 1/E3)]/(2E2E3) – 2kζ3[1/(E1E2) + 1/(E1E3)–1/(E2E3)]/E1+ (gsζ3/4)[2(1/E22 + 1/E32)/E1–(1/E2 + 1/E3)/(E2E3)],Ax = P[−κ–k′ + 2N2/7 + (gx − gs)– 3(gy − gs)/14]Ay = P[− κ + k′ + 2N2/7 + (gy − gs) – 3(gx − gs)/14]Az=P[− κ – 4N2/7 + (gz − gs) + 3(gx − gs)/14 + 3(gy − gs)/14] (3)

Here, g_s (≈ 2.0023) is the spin-only value. k (≈ N) is the orbital reduction factor; ζ and P are, respectively, the spin-orbit coupling coefficient and the dipolar hyperfine structure parameter for 3d⁹ ion in crystals; κ and κ′ are the isotropic and anisotropic core polarization constants, respectively; N (≈ k) is the average covalency factor, characteristic of the covalency effect of the studied system. Taking into account the covalency effect (characterized by the factor N), the spin-orbit coupling parameter ζ and dipolar hyperfine constant P can be given as [26]

(4)ζ = Nζd0 and P = NP0 (4)

here, ζ_d⁰ and P₀ (≈ 388×10^–4 cm^–1 [27]) are the corresponding parameters of free dⁿ ion. The denominators E_i (i=1–4) can be obtained from the energy matrices for a 3d⁹ ion under rhombic symmetry in terms of the cubic field parameter Dq and the rhombic field parameters Ds, Dt, Dξ and Dη:

(5)E1 ≈ 4Ds + 5Dt,E2 ≈ 10Dq,E3 ≈ 10Dq + 3Ds – 5Dt – 3Dξ + 4Dη,E4 ≈ 10Dq + 3Ds – 5Dt + 3Dξ – 4Dη. (5)

Thus, EPR parameters are correlated to the rhombic field parameters and, hence, to the local structure of studied [CuO₆]^10- cluster in Cu_0.5Zr₂(PO₄)₃ phosphate. From optical spectral studies for Cu²⁺ in oxides with similar [CuO₆]^10- clusters [28, 29], Dq≈ 1200 cm^–1 and N ≈ 0.82 can be obtained and used for the studied system here.

Therefore, in the formulas of EPR-g factors, only the structure parameters Δα and θ are unknown. Substituting the aforementioned values into (3) and fitting the calculated EPR-g factors to the experimental data, one can obtain

(6)Δα ≈ 5.1° and θ ≈ 83.8°. (6)

As for the core polarization constant κ, it satisfies the relationship κ ≈ –2Nχ/(3<r^-3>) [30]. Here, χ is the characteristic of density of unpaired spins at the nucleus of central ion, and <r^-3> is the expectation value of inverse cube of the radial wave function of 3d orbital in crystals. From the data χ (≈ –3.12 a.u.) [30] and <r^-3> ≈ 8.252 a.u. [31] for Cu²⁺ in similar oxides, one can obtain the value of κ (≈ 0.2) for the studied system here.

Substituting these data into (3) and fitting the calculated hyperfine structure constants A_i to the observed values, the anisotropic core polarization constant is obtained:

(7)k′ ≈ 0.0061. (7)

The corresponding EPR parameters (Cal. ^b) are shown in Table 1. For comparisons, the previous theoretical results (Cal. ^a) based on point-charge model [14] are also collected in Table 1.

3 Discussion

Table 1 reveals that the calculated EPR parameters for Cu²⁺ center in Cu_0.5Zr₂(PO₄)₃ phosphate based on the local axial distortion angle Δα (≈ 5.1°) and bond angle θ (≈ 83.8°) between x and y axes show better agreement with the observed values. As compared with the previous results [14], some improvements are achieved in this work. Thus, the EPR parameters of rhombic Cu²⁺ center in Cu_0.5Zr₂(PO₄)₃ phosphate are satisfactorily interpreted, and information about local structure around Cu²⁺ in Cu_0.5Zr₂(PO₄)₃ phosphate is also obtained.

1. EPR spectroscopy is an effective technique for the study of transition (or rare earth) metal ions in crystals [32–34], the information of local environment and the nature of interactions between central metal ions and ligands can be obtained by analysis EPR parameters. The angular distortions (particularly Δα≈ 5.1°) for Cu²⁺ center obtained in this work are mainly originated from relaxation and contraction of R_|| and R_⊥ due to Jahn–Teller effect. From (1) and (2), one can find that the contributions from the axial crystal fields (Ds and Dt) are mainly affected by the axial distortion angle Δα, whereas the contributions from perpendicular (rhombic) ones (Dξ and Dη) are influenced by the planar bond angle θ. Of course, the tentative local structures acquired by analyzing the EPR parameters of Cu²⁺ center in this work remain to be further checked with more powerful theoretical (e.g., ab initio) and experimental (e.g., EXAFS) investigations. Interestingly, similar local lattice modifications (i.e., elongation of the ligand octahedron) around impurity ions in crystals were also found for LiNbO₃:Cu²⁺, where the host Nb⁵⁺ site belongs to trigonal symmetry, but the substitutional Cu²⁺ actually locates in a rhombically elongated octahedron due to Jahn–Teller effect [35].

2. From (1), the hyperfine structure constants A_i (i=x, y, z) originate mainly from the isotropic contributions proportional to core polarization constant κ, characteristic of the Fermi contact between the ground 3s²3d⁹ configuration and excited s-orbitals (e.g., 3s¹3d⁹4s¹) for the central ion in crystal. The anisotropy of A factors ΔA (= (A_x + A_y)/2 –A_z) are mainly related to the covalency factor N and g shifts (= g_i – g_s), which are somewhat relevant to the local structure of the [CuO₆]^10- cluster. The smaller optimal anisotropic core polarization constants κ′ (≈0.0061), which is much smaller than the isotropic κ (≈0.2), and g anisotropy δg (= g_x – g_y) also contributes some anisotropic contributions δA (= A_x – A_y) to A factors. The signs of hyperfine structure constants A_i of Cu²⁺ ion in crystals are very difficult to determine solely from EPR experiment [30]. So the constants A_i obtained by EPR experiment are actually the absolute values. Based on the aforementioned calculations, the negative sign of A_z and positive signs of A_x and A_y can be determined here. This point is in agreement with the opinion for Cu²⁺ in many crystals [11, 13, 26, 30, 35, 36].

4 Conclusions

In conclusion, local structure of the rhombic Cu²⁺ center in Cu_0.5Zr₂(PO₄)₃ phosphate were theoretically studied from the perturbation formulas of the EPR parameters for the 3d⁹ (Cu²⁺) ion in rhombic symmetry. Based on the studies, the local axial distortion angle Δα (≈ 5.1°) and the planar bond angle θ (≈ 83.8°) of the Cu²⁺ center in Cu_0.5Zr₂(PO₄)₃ phosphate was obtained. By applying the uniform theoretical formulas and fewer adjustable parameters, the calculated EPR parameters in this work show some improvements compared with those in the previous studies.