Consistent microscopic theory of europium iron garnet: the factor-of-two Eu quadrupolar-splitting discrepancy and the undetermined fourth-degree anisotropic-exchange coefficients

The 1974 theory of Foglio and Van Vleck explained the magnetic anisotropy and the Eu nuclear magnetic resonance (n.m.r.) of europium iron garnet (EuIG, Eu3Fe5O12) through the virtual mixing of the Eu3+ J=0 ground state by the crystalline field and by anisotropic exchange with the ordered Fe sublattices, extracting the second-degree anisotropic-exchange coefficients from the n.m.r. hyperfine anisotropy. Two issues were explicitly left open: the predicted Eu nuclear quadrupolar splitting averaged over the three principal local axes is about twice the single-crystal n.m.r. value of Streever and Caplan, with 'unexpectedly large biquadratic superexchange' suggested but not justified; and the fourth-degree anisotropic-exchange coefficients $G_4^m$, to which the first anisotropy constant $K_1$ is very sensitive, cannot be obtained from the available experiments and were only fitted. Neither the subsequent EuIG anisotropy measurements of Yamaguchi and Sakuraba (1980, which re-fitted the fourth-degree coefficients from the anisotropy constants they were meant to predict) nor the modern EuIG thin-film literature resolves the quadrupolar discrepancy, quantifies biquadratic (two-ion rank-2) superexchange in EuIG, or independently determines $G_4^m$. The open problem is to complete the microscopic description of Eu3+ in EuIG with a single parameter set simultaneously consistent with the Eu hyperfine fields, the quadrupolar splittings, the anisotropy constants $K_1$ and $K_2$, and the ferromagnetic-resonance data, thereby identifying the origin of the factor-of-two quadrupolar discrepancy and removing the $G_4^m$ parameter ambiguity.

Background

Europium iron garnet (EuIG, Eu3Fe5O12) is a ferrimagnetic insulator of the garnet family: its Fe3+ sublattices order well above room temperature, while the Eu3+ ions sit on dodecahedral (eightfold oxygen-coordinated) sites whose local symmetry is only orthorhombic, so the cubic unit cell contains six magnetically inequivalent Eu sites with three mutually perpendicular local principal axes. The Eu3+ ion (4f6) has a 7F0 ground multiplet with total angular momentum $J=0$: it carries no moment and produces no hyperfine or anisotropic effect to first order. All of its low-temperature magnetic behavior is 'induced': it arises from virtual mixing of the $J=0$ state with the low-lying excited multiplets $J=1$ (mean energy about 350 cm^-1) and $J=2$ (about 1000 cm^-1) caused by (i) the local crystalline electric field, with parameters $V_n^m$, and (ii) the exchange coupling of the Eu 4f shell with the ordered Fe sublattices. Following Wolf and Van Vleck, this exchange is expanded into an isotropic part, represented by an effective exchange field $H^{ex}$, plus anisotropic higher-degree harmonics with dimensionless coefficients $G_n^m$ ($n=2,4,\dots$), the 'anisotropic exchange', whose reality in rare-earth iron garnets was demonstrated by optical and Moessbauer work (Atzmony et al. 1969). Superexchange means exchange interaction mediated by the intervening O2- ligand ions rather than by direct overlap; Moessbauer spectroscopy (the recoilless resonant absorption of nuclear gamma rays, which measures hyperfine fields and electric-field gradients at the nucleus) supplies several of the experimental inputs used below.

Two classes of observables probe this induced structure. First, the macroscopic magnetocrystalline anisotropy constants $K_1$ and $K_2$, the coefficients of the standard expansion of the anisotropy energy in even powers of the direction cosines of the magnetization, receive a Eu contribution only through the virtual mixing. Second, the nuclear magnetic resonance (n.m.r.) of the isotopes 151Eu and 153Eu in the magnetically ordered solid: at each inequivalent Eu site the nucleus experiences a large internal hyperfine field (hundreds of kG) plus a much smaller electric quadrupole interaction that splits each resonance line. The quadrupole splitting is set by the electric-field gradient at the Eu nucleus, produced by the induced 4f quadrupole moment (with the Sternheimer antishielding factor $1-R$ describing how polarization of the closed electron shells amplifies the field gradient acting on the nucleus) and by the lattice.

The problem chain is as follows. Wolf and Van Vleck (1960) created the theory of induced Eu magnetism in garnets. Levy (1966) computed the Eu anisotropy from the crystalline field alone and obtained the wrong sign, showing that anisotropic exchange dominates. Moessbauer studies (Gilat and Nowik 1963; Nowik and Ofer 1967; Stachel et al. 1969) measured exchange fields and hyperfine interactions. Streever (1969, powders) and Streever and Caplan (1971, single crystals) measured the 151/153Eu hyperfine fields along the three local axes and the quadrupole couplings. Foglio and Van Vleck (1974) then constructed the quantitative operator-respecting theory: from the anisotropy of the measured hyperfine fields they extracted the second-degree anisotropic-exchange coefficients ($G_2^0=-0.020$, $G_2^2=-0.034$) and the isotropic exchange field ($\beta H^{ex}=15.2$ cm^-1, where $\beta$ is the Bohr magneton in cm^-1-per-gauss units so that $\beta H^{ex}$ is an energy); from the field sum they obtained the radial expectation value $\langle r^{-3}\rangle$ (the expectation value of the inverse cube of the 4f-electron distance from the nucleus, which sets the magnetic hyperfine coupling; found as $10^{-3}\,\mathrm{nm}^3\,\langle r^{-3}\rangle=52.5$); and by diagonalizing the $J=0,1,2$ manifold for the six sites they reproduced the sign and order of magnitude of the Eu contribution to $K_1+\frac{1}{3}K_2$ with the fourth-degree exchange coefficients set to zero. The paper showed that the fourth-degree coefficients $G_4^m$ (of order 10^-3), which no then-available experiment determines, strongly control $K_1$, which is very sensitive to small changes in them, and that small nonzero values of the $G_4^m$ can be chosen to satisfy simultaneously the measured anisotropy and the ferromagnetic-resonance data (ferromagnetic resonance, f.m.r.: the microwave resonance of the uniform magnetization precession, including here its linewidth); the companion paper of Foglio, Sekerka and Van Vleck (1975) carried the same parameter set into the resonance linewidth. Microscopic calculations of the rare-earth-iron superexchange in garnets existed (Pearson and Herrmann 1967; Copland 1970), but the 1974 analysis treated the fourth-degree coefficients as fitted or zero parameters rather than microscopically computed quantities. The second-degree crystal-field parameters $V_2^0$, $V_2^2$ were transferred from optical spectroscopy of Eu3+ in yttrium gallium garnet (Koningstein 1965), for which several distinct fits exist. Yamaguchi and Sakuraba (1980) measured the temperature dependence of $K_1$ and $K_2$ of EuIG and of Ga- and Al-substituted EuIG single crystals and analyzed them within the same formalism, but still had to fix the $G_4^m$ by trial values because the two anisotropy constants cannot determine the three fourth-degree coefficients, and they identified missing two-ion correlation effects as a likely cause of the remaining deviations.

One failure remained, stated in the paper's own abstract as 'the only apparent inconsistency in the theory': the quadrupolar substructure of the n.m.r. observed by Streever and Caplan, averaged over the three principal local axes, is half as large as predicted theoretically, and the discrepancy 'cannot be blamed on higher order harmonics or uncertainty in the parameters'; a possible explanation 'could be unexpectedly large biquadratic superexchange', by which is meant exchange terms beyond the bilinear Hamiltonian that couple the quadrupolar (rank-2) operator moments of the two interacting ions, the two-ion analogue of biquadratic exchange for spin ions.

After 1974 the microscopic superexchange-operator formalism for rare-earth-iron garnets was developed by Veltrusky and Nekvasil (1980; 1990), the framework in which such rank-2 two-ion (biquadratic) exchange terms would be derived microscopically, and the anisotropic-exchange program was carried forward experimentally by Streever for other rare-earth ions (YbIG, ErIG, 2002-2004). Neither line revisited the EuIG quadrupolar discrepancy, and the modern EuIG literature (thin films for spintronics) treats anisotropy at the continuum level. The question asked below is the completion of the 1974 program.

Problem Statement

Complete the microscopic theory of the Eu3+ ion in europium iron garnet so that a single parameter set, comprising the crystalline-field parameters $V_n^m$ (second and fourth degree: $V_2^0$, $V_2^2$ and $V_4^0$, $V_4^2$, $V_4^4$), the isotropic exchange field $\beta H^{ex}$, the second-degree anisotropic-exchange coefficients $G_2^m$, and the fourth-degree coefficients $G_4^0$, $G_4^2$, $G_4^4$, simultaneously reproduces, within experimental uncertainties: (a) the measured 151/153Eu hyperfine fields along the three local principal axes (single-crystal n.m.r. data of Streever and Caplan, 1971); (b) the measured Eu nuclear quadrupole splittings, in particular their value averaged over the three principal local axes; (c) the Eu-sublattice contribution to the macroscopic anisotropy constants, in particular the combination $K_1+\frac{1}{3}K_2$ to which the 1974 calculation is directly compared, and $K_2$; and (d) the available ferromagnetic-resonance and linewidth data. Two obstacles left open by Foglio and Van Vleck (1974) must be removed.

1. The quadrupolar-splitting discrepancy. The 1974 prediction of the exchange-induced quadrupolar splitting averaged over the three principal local axes is about twice the single-crystal n.m.r. value of Streever and Caplan (1971); equivalently, the observed quadrupolar substructure averaged over the three principal local axes is half as large as predicted theoretically. Identify the physical origin of this factor-of-two gap: in particular, determine whether biquadratic (two-ion, rank-2) superexchange terms of the magnitude, sign, and microscopic origin needed to remove the discrepancy actually exist in EuIG, or whether another mechanism accounts for it. The 1974 analysis stated that the discrepancy 'cannot be blamed on higher order harmonics or uncertainty in the parameters'; any candidate explanation - biquadratic superexchange, revised nuclear factors (the Sternheimer antishielding factor $1-R$, the radial expectation value $\langle r^{-3}\rangle$), inclusion of non-diagonal nuclear terms, contributions from $J\ge 3$ states, or a re-determination of the experimental value itself - must be quantified, not merely asserted.

2. The fourth-degree anisotropic-exchange coefficients. Determine $G_4^0$, $G_4^2$, $G_4^4$ independently, that is, from experiments that do not use the anisotropy constants these coefficients are meant to predict, or from microscopic (superexchange or electronic-structure) calculation, so that the anisotropy calculation is no longer circular. The later fitting by Yamaguchi and Sakuraba (1980) of $K_1(T)$ and $K_2(T)$ within the same formalism, which still had to fix the $G_4^m$ by trial values because two anisotropy observables cannot determine three fourth-degree coefficients, shows that this circularity was not removed by the subsequent EuIG literature.

These parts are linked by one objective, that of a fully self-consistent microscopic description of Eu3+ in EuIG. The perturbative treatment of the virtual mixing of the $J=0$ ground state with the low-lying excited multiplets, the low-order expansion in $G_n^m$ and $V_n^m$ (especially where cancellations make $K_1$ small), and the ambiguity of the second-degree crystal-field parameters $V_2^0$, $V_2^2$ (several distinct fits of the same optical data, parameters transferred from Eu3+ in yttrium gallium garnet) limit the precision with which the observables (a)-(d) can be reproduced simultaneously. No particular method (new n.m.r. or Moessbauer experiment, first-principles calculation, or analytic theory) is prescribed.

References

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