In recent years much attention has been paid to the investigation of rare-earth (RE) doped GaN semiconductors, due to their potential in micro- and optoelectronics devices applications.^[1–11] Of particular interest has been the electrical and optical properties of trivalent Er³⁺ in GaN host,^[5–11] since the 1.54 μm luminescence center arising from the ⁴I_13/2 → ⁴I_15/2 transition is in the region of minimal loss in optical fibers.

It is well known that the open 4f shell of the trivalent 4f¹¹ ion Er³⁺ is spatially protected from the 5s and 5p shells due to a greater radial extent. Accordingly, the central RE ions are less affected by the surrounding ligand ions when doped in a certain host, thus resulting in a larger spin–orbit interaction than the crystal field if compared with 3d^N transition-metal complexes.^[12–16] However, the case may be different for the RE doping from one host to another because of the distinct features between the central ion and the ligand ion, e.g., ionic radius, charge, mass, etc.

To gain a better insight into the local structures and optical properties of the RE materials, electronic paramagnetic resonance (EPR) is regarded as a suitable tool to investigate the interrelation between the electronic structure and molecular structure. For instance, Palczewska et al. has performed the EPR measurement on Er³⁺-doped single GaN crystal, and the corresponding EPR parameters g_∥ = 2.861 ± 0.003, g_⊥ = 7.645 ± 0.003, A_∥ = (110 ± 5) × 10⁻⁴ cm⁻¹, and A_⊥ = (290 ± 5) × 10⁻⁴ cm⁻¹ were obtained.^[17] It should be noted that a rule of average g-factor g_av = 1/3(g_∥ + 2g_⊥), obtained from experiment, may be applied to predict the assignment of the lowest Kramers doublet by comparing with the theoretical one.^[18–20] Namely, if the average g-factor g_av ≈ 6.0, where the contributions from all high excited states are ignored and only the pure ⁴I_15/2 manifold is taken into account, the ground resonance states may be assigned to Kramers doublet Γ₇ in T_d symmetry, while the lowest states may be attributed to Γ₆ if g_av ≈ 6.8 for the same site symmetry.^[21] The rule is appropriate for an axial C_3v or C_4v symmetry.^[22,23] Consequently, Palczewska et al. assigned the ground states to Γ₇ for the tetrahedral Er³⁺ center in GaN epilayers based on the fact that the experimental average g-factor (g_av ≈ 6.05) is approximately equal to the expected g_av of the doublet Γ₇.^[17] In order to account for these EPR parameters, Maâlej et al. recently carried out a theoretical study on the EPR parameters of the Er³⁺ ions in GaN crystal by means of the first-order perturbation method,^[24] where only the ground manifold ⁴I_15/2 is included. They also claimed that the lowest Kramers doublet belongs to the Γ₇ based on the calculated average g-factor g_av ≈ 6.14. However, to the best of our knowledge, the experimental g_av of the most Er³⁺-doped complexes is generally less than the expected value 6.0 or 6.8 for the ground Kramers doublet Γ₇ or Γ₆, respectively.^[19] Hence, the average g-factor g_av for the hexagonal Er³⁺ center in GaN seems to be somewhat abnormal if the ground doublet is Γ₇. For that matter, it is necessary to perform a detailed theoretical analysis on the EPR parameters of GaN:Er³⁺ system to examine the actual assignment associated with the ground Kramers doublet.

In this work, we will perform a quantitative theoretical study on the J–J mixing effects, arising from the high-lying excited states (such as the terms ⁴I_13/2, ²K_15/2, ²L_15/2, etc.), on the EPR parameters based on a complete energy matrix method, where all excited states are included. Furthermore, the relation between the EPR g-factors and the orbit reduction factor k reflecting the covalent bonding and nephelauxetic effects is studied systematically. The plausible abnormal g-factors particularly for the average g-tensor g_av of the Er³⁺ ion in GaN are reasonably expounded.

In general, the perturbation Hamiltonian for a 4f¹¹ configuration ion Er³⁺ in an axial crystal field can be expressed as^[25–29]

To study the EPR parameters without spoiling the overall coupling of the various interactions, the actual Zeeman operator has been included in Eq. (1) instead of the approximate operator-equivalent technique, when it is subjected to an external magnetic field H. The general matrix elements of the Zeeman operator Ĥ_ZEE in the intermediate coupling scheme |γSLJM_J⟩ may be expressed as^[30]

Likewise, the magnetic hyperfine interaction Ĥ_HF, which arises from the interaction between the unfilled electrons and nucleus, may be written as

Fig. 1.

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	Fig. 1. Local structure of hexagonal (GaN4)9− cluster in the GaN crystal. The central cation Ga3+ is replaced by the Er3+ ion with approximate C3v symmetry when Er3+ is embedded in GaN host.

In this work, a refined least-squares fitting method, rather than the regular fitting routine, has been developed by simulating the EPR parameters and optical spectra simultaneously, which may reflect the macro- and microscopic optical properties of the GaN:Er³⁺ system, respectively. In order to reduce the number of adjustable parameters in our calculations, all free-ion parameters except the spin–orbit coupling parameter ζ are taken as the same values as the free-ion parameters of Er³⁺ obtained by Carnall et al,^[34] in view of the fact that the small change of free-ion parameters only affects the displacement of the centre of gravity of various manifolds, and hardly affects the Stark splitting and the EPR parameters. The obtained crystal–field parameters together with the free-ion parameters are tabulated in Table 1. The comparison between the theoretical values and experimental values of the Stark energy levels of the manifolds ⁴I_15/2 and ⁴I_11/2 is listed in Table 2 by diagonalizing the 364×364 complete energy matrices. The calculated and experimental values of the EPR parameters including the g-factors g_∥, g_⊥, and hyperfine structure constants A_∥ and A_⊥ are projected in the Table 3. From Table 2 it can be found that the calculated Stark splittings are in good agreement with the observed values, though the third level (No. Z3) of ⁴I_15/2 and the second level (No. A2) of ⁴I_11/2 show a slightly large deviation (but not more than 5.0 cm⁻¹). The errors can be expected because the spin-correlated crystal–field interaction of perturbation Hamiltonian has been ignored considering the absence of the complete optical spectra.^[35,36] However, there are moderate deviations for the EPR g-factors as shown in Table 3, if the covalent bonding and nephelauxetic effects characterized by an average orbital reduction factor k are not taken into account, viz., k = 1. Just as pointed out by Maâlej et al.,^[24] the J–J mixing effects between the ground states and excited states (e.g., ⁴I_13/2, ²K_15/2, ²L_15/2) by the coupling of the crystal–field, spin–orbit, and the orbital angular momentum interactions should be evaluated in order to get some insight into the actual effects from the high-lying manifolds. For this purpose, the eigenfunctions of the lowest Kramers doublet comprised of the ground manifold ⁴I_15/2 and the predominant excited manifolds ⁴I_13/2, ²K_15/2, and ²L_15/2 are obtained and tabulated in Table 4. Note that the coefficients for each state satisfy the following relation: if one state of the Kramers doublet is expressed as

Table 1.

Table 1.

Table 1. The free-ion parameters and crystal–field parameters for Er3+ in GaN crystal. The values of all parameters are expressed in units of cm−1. . Parameteraa Parameter E1 6636.0 T6 –160 E2 32.91 T7 358 E3 652.06 T8 0.0 ζ4f 2363.1 α 22.549 B20 –16.1 β –814.58 B40 720.7 γ 1553.9 B43 –25.0 T2 0.0 B60 –185.5 T3 75.0 B63 –20.7 T4 155 B66 –289.9 aThe free-ion parameters were obtained by Carnall et al.[34]

Table 1. The free-ion parameters and crystal–field parameters for Er3+ in GaN crystal. The values of all parameters are expressed in units of cm−1. .

Table 2.

Table 2.

Table 2. The calculated and experimental Stark levels of the hexagonal Er3+ centres in GaN. . Stark level No. ECalc/cm−1 EExpt/cm−1[35,36] 4I15/2 Z1 0.0 0.0 Z2 4.4 4.5 Z3 38.4 33.9 Z4 103.5 – Z5 132.9 – Z6 134.8 – Z7 167.6 – Z8 193.2 – 4I11/2 A1 10114.8 10115.6 A2 10128.2 10132.9 A3 10137.4 10138.7 A4 10152.6 10150.5 A5 10156.3 10153.0 A6 10165.6 10168.1

Table 2. The calculated and experimental Stark levels of the hexagonal Er3+ centres in GaN. .

Table 3.

Table 3.

Table 3. The theoretical and experimental EPR parameters for the Er3+ centers in single GaN crystal. The hyperfine structure parameters A∥ and A⊥ are in units of 10−4 cm−1. . g-factors g∥ g⊥ A∥ A⊥ g(4I13/2 |4I13/2)a 0.000315 0.001088 g(4I15/2 |4I15/2) –3.464816 8.135542 g(2K15/2 |2K15/2) –0.091034 0.212923 g(2L15/2 |2L15/2) –0.000817 0.001880 g(4I13/2 |4I15/2) –0.003299 0.016300 Cal.b –3.55801 8.37189 6.76726 130.09 305.34 Cal.c –3.17645 7.47650 6.04315 109.27 276.49 Expt.[17] 2.861(3) 7.645(3) 6.050(3) 110(2) 290(2) aThe lines 1–5 represent the g-factors from the different terms based on the obtained eigenfunctions. bCalculated using the orbital reduction factor k = 1 by diagonalizing the complete energy matrices. cCalculated using the orbital reduction factor k ≈ 0.84 by diagonalizing the complete energy matrices. dThe value of average g-tensor gav = (|g∥| + 2| g⊥|)/3.

Table 3. The theoretical and experimental EPR parameters for the Er3+ centers in single GaN crystal. The hyperfine structure parameters A∥ and A⊥ are in units of 10−4 cm−1. .

Table 4.

Table 4.

Table 4. The eigenfunctions of the lowest Kramers doublet for Er3+ ions in GaN crystal. . |2S +1LJ,MJ⟩i |2S +1LJ, ∓ 1/2⟩ |2S +1LJ,± 5/2⟩ |2S +1LJ, ∓ 7/2⟩ |2S +1LJ, ± 11/2⟩ |2S +1LJ, ∓ 13/2⟩ –0.00039 ∓ 0.01115 ± 0.00695 –0.00042 –0.00031 ± 0.00885 0.56840 0.80361 ∓ 0.04225 ± 0.01168 ∓ 0.00153 –0.09744 –0.13802 ± 0.00730 ∓ 0.00204 ∓ 0.00016 –0.00971 –0.01386 ± 0.00075 ∓ 0.00022 aThe represent the preceding coefficients of the states |2S +1LJ,MJ⟩i.

Table 4. The eigenfunctions of the lowest Kramers doublet for Er3+ ions in GaN crystal. .

With the aid of the operator-equivalent technique developed by Stevens et al.,^[37] the J–J mixing effects on the EPR parameters from the manifolds ⁴I_15/2, ⁴I_13/2, ²K_15/2, and ²L_15/2 have been evaluated and collected in Table 3 (lines 1–5). It is shown from Table 3 that the leading J–J contribution to the g-factors, in addition to the ground ⁴I_15/2, arises from the excited manifold ²K_15/2, which accounts for about 2.5%. However, the second important contribution comes from the crystal–field mixing effect between the ground ⁴I_15/2 and the first excited state ⁴I_13/2, where the ratio is less than 0.2% and the J–J contribution to g_⊥ is nearly twice larger than that to g_∥. The next J–J mixing contribution is from the state ²L_15/2 which merely accounts for about 0.023% for g_∥ and 0.022% for g_⊥, respectively. Moreover, the contribution from the first excited state ⁴I_13/2 to the EPR parameters is least compared with the above terms, which may be ascribed to the weaker crystal–field strength than the spin–orbit interaction for the RE complexes. The J–J contribution to the g-factors from the rest states, such as the terms ⁴I_11/2, ⁴I_9/2, ⁴F_9/2, and ²K_13/2, may be almost negligible. The case for the hyperfine structure constants A_∥ and A_⊥ is similar to the EPR g-factors. It follows that the J–J mixing effects from the high manifolds indeed cannot contribute more to the final EPR parameters. Furthermore, we consider that the lowest resonance states may originate from the Kramers doublet Γ₆ based on the evaluated average g-tensor g_av ≈ 6.76 when the orbit reduction factor k is taken as 1.0. However, the noticeable deviation of the calculated g-factors from the measured values needs to be expounded if the above conjecture on the ground state assignment is reasonable.

Early studies have indicated that trivalent erbium ion Er³⁺ will occupy the position of Ga³⁺ site with local C_3v symmetry when implanted in GaN host.^[5] However, there is considerable difference of the ionic radii for the impurity ion Er³⁺ and the replaced cation Ga³⁺ of host (r_Er³⁺ ≈ 0.89 Å, r_Ga³⁺ ≈ 0.62 Å). More overlap of electronic cloud between the central Er³⁺ ion and the ligand ions N^{3 −} in the GaN:Er³⁺ system can be expected as a result of the larger ionic radius of impurity Er³⁺, and thus produce a stronger covalent bonding effect to some extent apart from the dominant covalency of host GaN. In the paper, in order to evaluate the covalent and nephelauxetic effects characterized by an average orbital reduction factor k as a whole,^[38,39] the calculated results of the EPR g-factors (g_∥, g_⊥, g_av) as a function of the orbital reduction factor k are obtained and collected in Table 5, and the dependences of EPR g-factors are also plotted in Fig. 2. As shown by Fig. 2, the EPR g-factors are very sensitive to the orbital reduction factor k. The g-factors will increase linearly along with the increase of the factor k. Furthermore, we find that the agreement of the calculated g-factors may be further improved if the orbital reduction factor k is taken as 0.84, which reflect the stronger covalent bonding effect as expected. In particular, as for the average g-tensor g_av, the theoretical value is in accord with the experimental value. At the same time, it is of interest to note that the calculated hyperfine structure parameter is in excellent agreement with the experimental value if the same reduction factor k = 0.84 is used, whereas the calculated A_∥ = 130.09 × 10⁻⁴ cm⁻¹ when the orbital reduction effect is not taken into account. It is concluded that the stronger covalent bonding and nephelauxetic effects may be mainly responsible for the large deviation of the g_av from the predicted value because of the mismatch of ionic radii. Our recent studies on the Er³⁺ ion implanted in ScVO₄ crystal also show that there is more covalent effect on the EPR parameters if the ionic radius of central ion Er³⁺ is bigger than the substituted ion of host, where a comparable orbit reduction factor k ≈ 0.826 has been obtained based on a more detailed theoretical analysis.^[27] Of course, to evaluate the effect of the hybridization of 4f states with the states of N^{3 −} ions on the EPR parameters with care, more theoretical studies based on the first-principle calculations are needed. However, we do not mean to undertake it in the present study, since the mechanism controlling the atomic rearrangement has not been established, and the overall optical spectra data have not been reported yet, when the Er³⁺ is doped in single GaN crystal.

Table 5.

Table 5.

Table 5. The EPR g-factors (g∥, g⊥, gav) as a function of the orbital reduction factor k for Er3+ ions doped in GaN. . k g∥ g⊥ gav 0.60 2.60412 6.13334 4.95693 0.64 2.69951 6.35721 5.13798 0.68 2.79490 6.58107 5.31901 0.72 2.89029 6.80493 5.50005 0.76 2.98568 7.02879 5.68109 0.80 3.08106 7.25265 5.86212 0.84 3.17645 7.47650 6.04315 0.88 3.27184 7.70036 6.22419 0.92 3.36723 7.92420 6.40521 0.96 3.46262 8.14805 6.58624 1.00 3.55801 8.37189 6.76726

Table 5. The EPR g-factors (g∥, g⊥, gav) as a function of the orbital reduction factor k for Er3+ ions doped in GaN. .

Fig. 2.

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	Fig. 2. Dependence of the EPR g-factors (g∥, g⊥, gav) on the orbital reduction factor k for the GaN: Er3+ system.

A complete theoretical method has been established by diagonalizing the 364×364 energy matrix to study the EPR parameters of the trivalent Er³⁺ ions doped in hexagonal GaN crystal. The results show that the ground states may be derived from the Kramers doublet Γ₆ for the GaN: Er³⁺ system, and the strong covalent bonding and nephelauxetic effects, as a result of the mismatch of ionic radii of the impurity Er³⁺ and replaced Ga³⁺ ion, may be mainly responsible for the larger deviation of the average EPR g-tensor g_av from the theoretical value (g_av ≈ 6.8) as expected. Moreover, the J–J mixing contribution to the EPR parameters from the high-lying excited states has been evaluated. Differing from the previous studies, it is found that the predominant J–J mixing contribution is the state ²K_15/2, which accounts for about 2.5%, and the second major J–J contribution arises from the crystal–field mixture between the ground state ⁴I_15/2 and the first excited state ⁴I_13/2, and is usually less than 0.2%. The next J–J mixing contribution is from the state ²L_15/2 which merely accounts for about 0.023% for g_∥ and 0.022% for g_⊥, respectively. The contributions from the rest states such as the ⁴I_13/2, ⁴I_11/2, ⁴I_9/2, ⁴F_9/2, and ²K_13/2 may be ignored. It follows that the J–J mixing effects from the high manifolds could not really add more contribution to the final EPR parameters, which may be due to the larger spin–orbit coupling interaction than the crystal–field interaction for the RE-doped complexes.