Theoretical studies of the EPR parameters for Ni2+ and Co+ in MgO

Theoretical studies of the EPR parameters for Ni²⁺ and Co⁺ in MgO

Zhi-Hong Zhang^I; Shao-Yi WuI, II, ^*; Pei Xu^I; Li-Li Li^I

^IDepartment of Applied Physics, University of Electronic Science and Technology of China, Chengdu 610054, P.R. China

^IIInternational Centre for Materials Physics, Chinese Academy of Sciences, Shenyang 110016, P.R. China

ABSTRACT

The electron paramagnetic resonance (EPR) parameters (g factors and the hyperfine structure constants) for Ni²⁺ and Co⁺ in MgO are theoretically studied from the perturbation formulas of these parameters for a 3d⁸ ion in octahedral crystal-fields. In the computations, the ligand orbital and spin-orbit coupling contributions are taken into account using the cluster approach. The calculated EPR parameters are in good agreement with the experimental data. The larger g factor and the smaller magnitude of the hyperfine structure constant for Ni²⁺ as compared with those for Co⁺ can be attributed to the higher spin-orbit coupling coefficient and the lower dipolar hyperfine structure parameter of the former, respectively.

Keywords: Impurity and defects; Electron paramagnetic resonance; Ni²⁺; Co⁺; MgO.

1. INTRODUCTION

Magnesium oxide (MgO) is usually regarded as a model system to investigate magnetic [1,2], adsorption [3,4], structure [5,6] and optical properties [7] of doped transition-metal impurities. Particularly, MgO containing Ni²⁺ and Co⁺ can exhibit unique catalytic [8-10] and tunable laser properties [11,12]. Normally, these properties are closely correlated with the electronic states of the transition-metal ions in the hosts, which can be investigated by means of electron paramagnetic resonance (EPR) technique. For example, EPR experiments were carried out for Ni²⁺ and Co²⁺ doped MgO, and the EPR parameters (i.e., the isotropic g factors and the hyperfine structure constants) were also measured for the cubic Ni²⁺ and Co⁺ centers [13,14].

Up to now, however, the above EPR experimental results have not been satisfactorily explained. On the other hand, the EPR spectra and magnetic properties have been extensively investigated for Ni²⁺ in various chlorides by considering only the central ion orbital and spin-orbit coupling contributions [15]. Nevertheless, the contributions to the EPR parameters from the ligand orbital and spin-orbit coupling interactions were not taken into account in the previous studies. In fact, for the 3d⁸ ions in oxides, the systems may still show some covalency and impurity-ligand orbital admixtures. In addition, the EPR spectra for MgO:Co⁺ have not been interpreted until now. Considering that (i) investigations on the EPR parameters for Ni²⁺ and Co⁺ in MgO can reveal useful information about electronic structures which would be helpful to understand the properties of these systems and that (ii) 3d⁸ ions can be treated as model systems containing only two unpaired holes, further quantitative studies on the EPR spectra for the Ni²⁺ and Co⁺ centers are of scientific and practical significance. In this work, the improved perturbation formulas of the EPR parameters based on the cluster approach are applied to the theoretical analysis of the Ni²⁺ and Co⁺ centers in MgO. In the calculations, the ligand orbital and spin-orbit coupling contributions are considered in a uniform way. The results are discussed.

2. CALCULATIONS

Judging from the observed isotropic g factors and the hyperfine structure constants, the experimental EPR signals [13,14] can be assigned to the substitutional cubic Ni²⁺ and Co⁺ centers in MgO. When a 3d⁸ ion locates on an octahedral (O_h) site, the free-ion configuration ³F would be separated into two orbital doublets ³T_1g and ³T_2g and one singlet ³A_2g, with the latter lying lowest corresponding to the isotropic g and A signals [16]. As for the g factor of a 3d⁸ ion in octahedra, the perturbation formula was established using the conventional crystal-field model [17,18], by including only the contributions from the central ion orbital and spin-orbit coupling interactions. In order to study the EPR spectra of the 3d⁸ centers more exactly, the ligand orbital and spin-orbit contributions may be taken into account. Thus, the improved g formula based on the cluster approach is applied here. Meanwhile, the perturbation formula of the hyperfine structure constant for a 3d⁸ ion in regular octahedra can be similarly derived. Thus, we have [19] :

Here g_s ( ≈ 2.0023) is the spin-only value. ζ and ζ´ are the spin-orbit coupling coefficients, k and k´ are the orbital reduction factors, and P and P´ are the dipolar hyperfine structure parameters for a 3d⁸ ion in crystals. κ is the core polarization constant. The energy denominators E_i(i = 1 ~ 3) denote the energy separations between the excited ³T₂, ¹T_2a and ¹T_2b and the ground ³A₂ states [17-19]. They can be described in terms of the cubic field parameter Dq and the Racah parameters B and C for the 3d⁸ ion in crystals: E₁≈ 10 Dq, E₂≈ 10 Dq + 12 B and E₃≈ 8 B + 2C + 10 Dq [17-19]. From the cluster approach containing the ligand p- and s-orbital contributions [20], the spin-orbit coupling coefficients ζ and ζ´, the orbital reduction factors k and k' and the dipolar hyperfine structure parameters P and P' can be expressed as

Here and are the spin-orbit coupling coefficients of the free 3d⁸ and the ligand ions, respectively. A denotes the integral , where R is the impurity-ligand distance of the studied systems. N_γ and λ_γ (here the subscripts γ = t and e denote the irreducible representations T_2g and E_g of O_h group, respectively) are the normalization factors and the orbital admixture coefficients. They are determined from the approximation relationships [20]

and the normalization conditions [20]

Here N is the average covalency factor, characteristic of the covalency of the studied systems. S_dpγ (and S_ds) are the group overlap integrals. In general, the orbital admixture coefficients increase with increasing the group overlap integrals, and one can approximately adopt the proportional relationship λ_s/λ_e ≈ S_ds/S_dpe between the orbital admixture coefficients and the related group overlap integrals within the same irreducible representation Eg. Obviously, omission of the ligand contributions (i.e., ζ´ = ζ = N, k´ = k = N, P´ = P = P₀N), the above g formula returns to that of the previous work based on the conventional crystal-field model [17,18].

Usually, the impurity-ligand distance R is different from the host cation-anion distance R_H in a pure crystal due to the difference between the ionic radius r_i of impurity and the radius r_h of host ions. Fortunately, studies based on experimental superhyperfine parameter and extended X-ray absorption fine structure (EXAFS) measurements have verified that the empirical formula R ≈ R_H + (r_i – r_h)/2 is approximately valid for impurity ions in crystals [21]. From R_H ( ≈ 2.105 Å [22] ) for MgO, r_i ( ≈ 0.69 and 0.82 Å [22] ) for Ni²⁺ and Co⁺ as well as r_h ( ≈ 0.66 Å [22] ) for Mg²⁺, the distances R are obtained and listed in Table 1. From the distances R and the Slater-type self-consistent field (SCF) wave functions [23,24], the group overlap integrals are calculated and shown in Table 1. According to the optical spectra for Ni²⁺ in MgO [25,26], the cubic field parameter Dq ≈ 860 cm^-1 and the covalency factor N ≈ 0.90 can be obtained. Since the isoelectronic monovalent Co⁺ suffers weaker crystal-fields (i.e., lower Dq) and covalency effect (i.e., higher N) [27] when coordinated to the same oxygen ligands, the spectral parameters Dq ≈ 780 cm^-1 and N ≈ 0.91 may be estimated for Co⁺ in MgO. Then the Racah parameters are determined from the relationships B ≈ N²B₀ and C ≈ N²C₀ [28], with the corresponding free-ion values B₀≈ 1208 and 878 cm^-1 and C₀≈ 4459 and 3828 cm^-1 [27] for Ni²⁺ and Co⁺, respectively. Using Eqs. (3) and (4), the molecular orbital coefficients N_γ and λ_γ (and λ_s) can be calculated. From the free-ion values ≈ 649 and 456 cm^-1 [27] for Ni²⁺ and Co⁺ and ≈ 151 cm^-1 [29] for O^2-, the parameters in Eq. (2) are obtained and shown in Table 1. In the formula of the hyperfine structure constant, the dipolar hyperfine structure parameters are P₀≈ 112 × 10^-4 cm^-1 and 228 × 10^-4cm^-1 [30] for Ni²⁺ and Co⁺, respectively. The core polarization constant can be determined from the empirical relationship κ ≈ -2χ/(3 < r^-3 > ), where χ is characteristic of the density of unpaired spins at the nucleus of the central ion and < r^-3 > the expectation value of the inverse cube of the radial wave function of the 3d⁸ orbital [30]. Applying < r^-3 > ≈ 7.094 and 5.388 a.u. [16] and χ ≈ -3.15 and -3.31 a.u. [30] for Ni²⁺ and Co⁺ in oxides, one can obtain κ ≈ 0.3 and 0.41 for MgO:Ni²⁺ and MgO:Co⁺, respectively. Substituting the above values into the formulas of the EPR parameters, the corresponding theoretical results (Cal. b) are calculated and shown in Table 2. To clarify the importance and the tendency of the covalency and the ligand contributions for Ni²⁺ and Co⁺ in MgO, the results (Cal. a) based on omission of the ligand contributions are also collected in Table 2 for comparison.

Table 2

3. DISCUSSION

1) The experimental isotropic g and A factors are attributed to the cubic Ni²⁺ and Co⁺ centers on the substitutional Mg²⁺ site in MgO. It is noted that there are some low symmetrical 3d⁸ centers in other oxides, e.g., the trigonal Ni²⁺ and Cu³⁺ centers in α – Al₂O₃ [31]. The anisotropic g factors g_// and g_⊥ [31] for the trigonal centers may be similarly analyzed from the perturbation formulas [32] for a trigonally distorted octahedral 3d⁸ cluster, and the contributions from the trigonal distortion can be quantitatively considered in the calculations of the trigonal crystal-field parameters from the superposition model [33] and the local geometrical relationship of the impurity centers. Interestingly, the larger g factors [31] for Cu³⁺ than those for Ni²⁺ are attributable to the higher spin-orbit coupling coefficient ( ≈ 876 cm^-1 [16] ) of the former than that ( ≈ 649 cm^-1 [27] ) of the latter. In addition, Co⁺ replacing the host Mg²⁺ in MgO may induce some means of charge compensation due to the fewer charge of the impurity. On the other hand, larger size of Co⁺ than Mg²⁺ can lead to enhancement of the local metal-ligand interactions around the impurity and thus make Co⁺ stable on Mg²⁺ site. Further, local charge compensation (e.g., oxygen vacancy nearby) would break the original cubic symmetry of the ideal Mg²⁺ site and yield anisotropic EPR parameters. In view of the observed isotropic EPR signals [13,14], the charge compensation may occur in the outer ligand spheres far away from the impurity Co⁺, and the possible disturbance of the local structure of this center can be regarded as very small and negligible for simplicity. Of course, further experimental investigations of possible charge compensation for Co⁺ in MgO seem necessary and meaningful.

2) From Eqs. (1) and (2), the g factor largely depends upon the spin-orbit coupling coefficient ζ' (related to the central ion spin-orbit coupling coefficient). Thus, the larger g factor for MgO:Ni²⁺ than that for MgO:Co⁺ can be illustrated by the higher spin-orbit coupling coefficient ( ≈ 649 cm^-1) of the former than that ( ≈ 456 cm^-1) of the latter. On the other hand, the hyperfine structure constant is sensitively related to the dipolar hyperfine structure parameter P₀ and the dominant contribution proportional to the core polarization constant κ. So, the larger magnitude of the hyperfine structure constant for MgO:Co⁺ than that for MgO:Ni²⁺ can be understood in view of the higher values of P₀ and κ in the former.

3) The average covalency factors N ( ≈ 0.90 and 0.91 for Ni²⁺ and Co⁺) in this work still show some influences of the covalency on the EPR parameters, although the spin-orbit coupling coefficient ( ≈ 151 cm^-1 [28] ) of the oxygen ligand is much smaller than that ( ≈ 649 or 456 cm^-1 [26] ) of the impurity Ni²⁺ or Co⁺. Thus, omission of the ligand contributions yields larger g factors and slightly lower hyperfine structure constants in magnitude (Cal. ª). It seems that the improved formulas of the EPR parameters adopted in this work are superior to the previous ones [17,18] based on the conventional crystal-field model. Further, the covalency and the strength of the crystal-fields exhibit the decreasing tendency from Ni²⁺ to Co⁺ in the same MgO host, i.e., N (Ni²⁺) < N (Co⁺) and Dq (Ni²⁺) < Dq (Co⁺). This point is consistent with the lower valence state and hence weaker covalency and impurity-ligand interactions of the latter.

4. CONCLUSION

The EPR parameters for Ni²⁺ and Co⁺ in MgO are satisfactorily explained from the perturbation formulas based on the cluster approach. Inclusion of the ligand contributions yields better theoretical results as compared with those in the absence of these contributions. The larger g factor and the smaller magnitude of the hyperfine structure constant for Ni²⁺ than those for Co⁺ can be attributed to the higher spin-orbit coupling coefficient and the lower dipolar hyperfine structure parameter of the former.

(Received on 21 June, 2010)

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Publication Dates

• Publication in this collection
  27 Sept 2010
• Date of issue
  Sept 2010

History

• Received
  21 June 2010