Determination of the rank-4 and rank-6 anisotropic Eu3+–Fe3+ exchange parameters and the origin of the anomalously large zero-field-splitting shift of Fe3+ in europium gallium garnet

In europium gallium garnet (EuGaG, Eu3Ga5O12) doped with Fe3+, electron paramagnetic resonance (EPR) shows anisotropic g-shifts together with an anomalously large shift of the axial zero-field-splitting parameter D, of magnitude 0.161 ± 0.014 K relative to diamagnetic gallium garnets. Huang and Van Vleck (1968) analyzed the exchange interaction between the S-state Fe3+ probe ion and the Van Vleck Eu3+ host ion, expanded in tesseral harmonics $Y_{\mu\nu}(\mathbf{L}_{\mathrm{Eu}})$ of the Eu3+ orbital angular momentum with even ranks μ = 0, 2, 4, 6: from the g-shifts they extracted the isotropic parameter (summed over the six nearest Eu neighbors ≈ 0.06 K) and a rank-2 parameter (≈ −0.12 K), and showed that the entire rank ≤ 2 exchange plus dipolar contribution to the D shift is of order 10^-4–10^-3 K, roughly two orders of magnitude too small, while Moriya antisymmetric exchange would require an implausible strength. They concluded that anisotropic exchange harmonics of rank μ = 4 and μ = 6 must be of overriding importance — but the strengths of these higher-rank parameters have never been determined, and the complete calculation including all harmonics up to μ = 6 that fits the observed D shift has never been performed. The problem is to determine the rank-4 and rank-6 Eu3+–Fe3+ exchange parameters $a_{4\nu}$ and $a_{6\nu}$ and thereby account quantitatively for the observed D shift while remaining consistent with the g-shift constraints on the rank ≤ 2 sector.

Background

Europium gallium garnet (EuGaG, Eu3Ga5O12) is a cubic oxide with the garnet structure (space group Ia-3d): Eu3+ ions occupy the eightfold-coordinated dodecahedral sites and Ga3+ ions occupy octahedral and tetrahedral sites. When a small amount of Fe3+ is substituted for Ga3+ it enters mainly the octahedral ('a') sites, whose local symmetry is trigonal $C_{3i}$. Electron paramagnetic resonance (EPR) of such Fe3+ impurities is described by a spin Hamiltonian

$H = g\,\mu_B\,\mathbf{B}\cdot\mathbf{S} + D\left[S_z^2 - \tfrac{1}{3}S(S+1)\right] + E\,(S_x^2 - S_y^2), \qquad S = 5/2,$

where $\mu_B$ is the Bohr magneton and $D$ and $E$ are the axial and rhombic zero-field-splitting parameters, which split the six $M_S$ sublevels of the Fe3+ ground state in zero applied field. Fe3+ (configuration 3d5) is an 'S-state' ion: its $^{6}S$ ground term has zero orbital angular momentum, so the usual crystal-field mechanism for zero-field splitting operates only through higher-order spin–orbit admixture and is small. In diamagnetic gallium garnets $D$ is small and well characterized, so an extrapolated diamagnetic baseline exists for comparison.

Eu3+ (configuration 4f6) in EuGaG has the nonmagnetic singlet ground level $^{7}F_0$ ($J = 0$): the crystal is a 'Van Vleck paramagnet', whose magnetization arises from field-induced mixing of the excited $^{7}F_J$ multiplets ($J = 1,\dots,6$). In the analysis of Huang and Van Vleck the first excited level lies $h\nu_1 \approx 480$ K above the ground level and the higher levels were approximated by the Landé interval rule $E_J \propto J(J+1)$, giving $h\nu_2 = 3h\nu_1$ and $h\nu_3 = 6h\nu_1$; crystal-field ('Stark') splittings of the excited $J$ multiplets were ignored. Projected onto the ground Russell–Saunders terms of both ions, the exchange interaction between the S-state Fe3+ and the Eu3+ ion is expanded in tesseral harmonics $Y_{\mu\nu}$ (real combinations of spherical harmonics, used here as operator equivalents built from the Eu3+ orbital angular momentum $\mathbf{L}$, $L = 3$; for example $Y_{20}(\mathbf{L}) = 3L_z^2 - L(L+1)$):

$V_{\mathrm{ex}} = -2\left[a_{00} - \tfrac{1}{6}\sum_{\mu,\nu} a_{\mu\nu}\,Y_{\mu\nu}(\mathbf{L}_{\mathrm{Eu}})\right]\mathbf{S}_{\mathrm{Eu}}\cdot\mathbf{S}_{\mathrm{Fe}}.$

Time-reversal symmetry restricts the rank $\mu$ to even values, and for 4f electrons $\mu = 0, 2, 4, 6$. Here $a_{00}$ is the isotropic exchange constant (denoted differently, rendered $a_\infty$, in the 1968 paper) and $a_{2\nu}$, $a_{4\nu}$, $a_{6\nu}$ are anisotropic exchange parameters, each a fixed linear combination of fundamental 4f–3d two-electron exchange integrals. Anisotropic exchange of this type was first demonstrated experimentally in ytterbium iron garnet (Wickersheim and White, 1962), is the accepted mechanism for rare-earth contributions to the anisotropy of the magnetic iron garnets, and its general high-rank theory was given by Birgeneau, Hutchings, Baker and Riley (1969).

The anomaly is the following. Hutchings, Windsor and Wolf (1966) measured EPR of Fe3+ in EuGaG and found, relative to the diamagnetic baseline, $g$-shifts $\Delta g_{\parallel} = 0.0095 \pm 0.002$ and $\Delta g_{\perp} = -0.0015 \pm 0.002$ (of which magnetic dipole–dipole coupling accounts for only about a quarter) and a shift of the zero-field-splitting parameter of magnitude $0.161 \pm 0.014$ K (Huang and Van Vleck describe an increase by this amount; Yang and Huang Liu, with a different sign convention for $\Delta D$, write $\Delta D = -0.161 \pm 0.014$ K). Tachiki and Šroubek (1967) computed the exchange contribution to the $D$-parameters of Fe3+ and Cr3+ in EuGaG using a single second-degree anisotropic term. Huang and Van Vleck (1968) then showed from the $g$-shifts — which by angular-momentum selection rules constrain only the isotropic and rank-2 harmonics, since the Zeeman operator couples $J = 0$ only to $J = 1$ — that the rank-2 anisotropic exchange is comparable in strength to the isotropic term (summed over the six nearest Eu3+ neighbors of the octahedral Fe3+ site, whose bonds make an angle $\theta \approx 43^{\circ}51'$ with the trigonal axis: $\sum_k a_{00}^k \approx 0.06$ K and $\sum_k a_{20}^k \approx -0.12$ K), yet the entire rank $\le 2$ exchange contribution to the $D$ shift is only about $-4\times10^{-4}$ K, roughly 400 times smaller than observed; the Moriya (Dzyaloshinskii–Moriya) antisymmetric exchange — the term $\mathbf{d}\cdot(\mathbf{S}_{\mathrm{Fe}}\times\mathbf{S}_{\mathrm{Eu}})$ of the pair Hamiltonian — is excluded because it would require a strength of about 2 K, some twenty times the isotropic exchange. Their conclusion was that exchange harmonics of rank $\mu = 4$ and $\mu = 6$ must be of overriding importance in the Eu3+–Fe3+ exchange: these cannot contribute to the $g$ shift but do contribute to the zero-field splitting. Within the $^{7}F$ term of Eu3+, whose levels run from $J = 0$ to $J = 6$, the product of a rank-$\mu$ orbital harmonic with the Eu3+ spin operator is a tensor of total angular-momentum rank $\mu-1$, $\mu$, or $\mu+1$, so that rank-4 and rank-6 harmonics can couple the $J = 0$ ground level to excited levels with $J$ up to 5 and 6 respectively. Yang and Huang Liu (1975) completed the rank-2 sector with the full tensor treatment (all of $a_{20}$, $a_{21}$, $a_{22}$ with real and imaginary parts, plus the dipolar term of strength $J_d \approx -0.087$ K): the trace of the $\Delta g$ tensor fixes $a_{00} \approx 0.01$ K per Eu–Fe pair, the anisotropy fixes one linear combination of $a_{20}$ and $\mathrm{Re}\,a_{22}$, and the rank $\le 2$ contribution to $\Delta D$ is about $-10^{-3}$ K, still two orders of magnitude too small; independent evidence from europium iron garnet (nuclear magnetic resonance (NMR) hyperfine data of Streever and Caplan, 1971; the anisotropy analysis of Foglio and Van Vleck, 1974) indicates that rank-2 parameters are only a few percent of $a_{00}$, so enlarging the rank-2 sector is not a viable escape. In the closely related, magnetically ordered compound europium iron garnet, Foglio and Van Vleck (1974) found explicitly that the fourth- and higher-degree exchange parameters 'cannot be obtained' from the available experimental data and must be treated as adjustable quantities.

What the source analyses left undetermined is the quantitative content of the higher-rank anisotropic exchange: the values of $a_{4\nu}$ and $a_{6\nu}$ for the Eu3+–Fe3+ pair in EuGaG, and a complete calculation, including all harmonics up to $\mu = 6$, that reproduces the observed $D$ shift while remaining consistent with the $g$-shift constraints. Both source analyses used the Landé interval rule for the Eu3+ level energies, ignored the crystal-field (Stark) splittings of the excited $J$ multiplets, and retained only the six nearest Eu neighbors.

Problem Statement

Determine the quantitative strengths of the higher-rank anisotropic exchange parameters — the rank-4 parameters $a_{4\nu}$ ($\nu = 0,\dots,4$) and the rank-6 parameters $a_{6\nu}$ ($\nu = 0,\dots,6$) — real for $\nu = 0$, complex with independent real and imaginary parts for $\nu \ge 1$ — of the Eu3+–Fe3+ pair in europium gallium garnet, in the exchange Hamiltonian

$V_{\mathrm{ex}} = -2\left[a_{00} - \tfrac{1}{6}\sum_{\mu,\nu} a_{\mu\nu}\,Y_{\mu\nu}(\mathbf{L}_{\mathrm{Eu}})\right]\mathbf{S}_{\mathrm{Eu}}\cdot\mathbf{S}_{\mathrm{Fe}},$

and thereby give a complete microscopic account of the anomalously large zero-field-splitting shift of Fe3+ at the octahedral site of EuGaG (magnitude $0.161 \pm 0.014$ K relative to diamagnetic gallium garnets).

The second-order perturbation calculation of the Fe3+ spin Hamiltonian must be carried to all allowed exchange harmonics up to $\mu = 6$, with the correct intermediate Eu3+ levels ($J = 1$ through $J = 6$; the $^{7}F$ term of Eu3+ comprises $J = 0,\dots,6$) and their energy denominators; it must include the correct geometry of the Eu3+ neighbors (at least the six nearest neighbors of the octahedral Fe3+ site); and it must reproduce the observed $D$ shift within its experimental uncertainty while remaining consistent with the measured $g$-shifts (total values relative to diamagnetic garnets $\Delta g_{\parallel} = 0.0095 \pm 0.002$ and $\Delta g_{\perp} = -0.0015 \pm 0.002$, of which the magnetic dipole–dipole contributions are only $\Delta g_{\parallel}^{\mathrm{dip}} = 0.002$ and $\Delta g_{\perp}^{\mathrm{dip}} = -0.001$) and with the established bounds on the rank $\le 2$ sector. It must also establish which combinations of the higher-rank parameters are actually determined by the data and which require microscopic theory or additional experiment.

References

1. N. L. Huang and J. H. Van Vleck, 'Strong orbital anisotropy in the exchange interaction in Fe3+ : EuGaG', Solid State Communications 6(8), 557–559 (1968). DOI: 10.1016/0038-1098(68)90511-5, https://doi.org/10.1016/0038-1098(68)90511-5.
2. M. T. Hutchings, C. G. Windsor and W. P. Wolf, 'Determination of Eu3+–Fe3+ and Eu3+–Gd3+ Exchange Interactions from Electron Paramagnetic Resonance Shifts in Europium Gallium Garnet', Physical Review 148, 444–458 (1966). DOI: 10.1103/PhysRev.148.444, https://doi.org/10.1103/PhysRev.148.444.
3. M. Tachiki and Z. Sroubek, 'Exchange contribution to the D-parameters of Fe3+ and Cr3+ in europium gallium garnet', Solid State Communications 5, 361–363 (1967). DOI: 10.1016/0038-1098(67)90775-2, https://doi.org/10.1016/0038-1098(67)90775-2.
4. Y. Yang and N. L. Huang Liu, 'Effects of Eu3+–Fe3+ exchange interaction on the spin Hamiltonian parameters of Fe3+ in EuGaG', Journal of Magnetism and Magnetic Materials 1, 161–166 (1975). DOI: 10.1016/0304-8853(75)90163-8, https://doi.org/10.1016/0304-8853(75)90163-8.
5. M. E. Foglio and J. H. Van Vleck, 'Theory of the magnetic anisotropy and nuclear magnetic resonance of europium iron garnet', Proceedings of the Royal Society of London A 336, 115–140 (1974). DOI: 10.1098/rspa.1974.0010, https://doi.org/10.1098/rspa.1974.0010.
6. N. L. Huang Liu, 'Magnon dispersion relation in europium iron garnet', International Journal of Quantum Chemistry 5, 661–665 (1971). DOI: 10.1002/qua.560050875, https://doi.org/10.1002/qua.560050875.
7. R. J. Birgeneau, M. T. Hutchings, J. M. Baker and J. D. Riley, 'High-Degree Electrostatic and Exchange Interactions in Rare-Earth Compounds', Journal of Applied Physics 40, 1070–1079 (1969). DOI: 10.1063/1.1657531, https://doi.org/10.1063/1.1657531.
8. R. L. Streever and P. J. Caplan, 'Nuclear-Magnetic-Resonance Studies of Eu151 and Eu153 in Europium Iron Garnet Single Crystals', Physical Review B 3, 2910–2917 (1971). DOI: 10.1103/PhysRevB.3.2910, https://doi.org/10.1103/PhysRevB.3.2910.
9. Y. Yang, N. L. Huang Liu and R. Orbach, 'Theoretical interpretations of the angular dependence of the ESR resonance field of Dy in Pd', Solid State Communications 18, 1443–1446 (1976). DOI: 10.1016/0038-1098(76)90365-3, https://doi.org/10.1016/0038-1098(76)90365-3.
10. K. A. Wickersheim and R. L. White, 'Anisotropy of Exchange in Ytterbium Iron Garnet', Physical Review Letters 8, 483–485 (1962). DOI: 10.1103/PhysRevLett.8.483, https://doi.org/10.1103/PhysRevLett.8.483.
11. J. M. Baker, 'Interactions between ions with orbital angular momentum in insulators', Reports on Progress in Physics 34, 109–173 (1971). DOI: 10.1088/0034-4885/34/1/303, https://doi.org/10.1088/0034-4885/34/1/303.
12. R. L. Cone and R. S. Meltzer, 'Ion–Ion Interactions and Exciton Effects in Rare Earth Insulators', in Spectroscopy of Solids Containing Rare Earth Ions, Modern Problems in Condensed Matter Sciences (Elsevier, 1987), pp. 481–556. DOI: 10.1016/B978-0-444-87051-3.50014-6, https://doi.org/10.1016/B978-0-444-87051-3.50014-6.
13. I. Veltrusky and V. Nekvasil, 'Superexchange operator and crystal-field-like terms for rare-earth-iron interaction in garnets', Journal of Physics C: Solid State Physics 13, 1685–1696 (1980). DOI: 10.1088/0022-3719/13/9/015, https://doi.org/10.1088/0022-3719/13/9/015.
14. A. I. Popov and B. Yu. Sokolov, 'Magnetic linear birefringence of europium-gallium garnet', Russian Physics Journal 40, 387–390 (1997). DOI: 10.1007/BF02508863, https://doi.org/10.1007/BF02508863.
15. R. L. Streever, 'Anisotropic exchange in YbIG', Journal of Magnetism and Magnetic Materials 263, 219–227 (2003). DOI: 10.1016/S0304-8853(02)01568-8, https://doi.org/10.1016/S0304-8853(02)01568-8.
16. R. L. Streever, 'Anisotropic exchange in ErIG', Journal of Magnetism and Magnetic Materials 278, 223–230 (2004). DOI: 10.1016/J.JMMM.2003.12.1231, https://doi.org/10.1016/J.JMMM.2003.12.1231.