Microscopic mechanism of the ferromagnetic-resonance linewidth and relaxation in europium iron garnet (Eu$_3$Fe$_5$O$_{12}$)

Europium iron garnet (EuIG) shows one of the largest linewidth anomalies of any ferrimagnetic insulator: its ferromagnetic-resonance linewidth is below roughly 1 Oe at low temperature but grows by nearly two orders of magnitude at high temperature, approximately tracking the thermal population of the Eu$^{3+}$ $J=1$ excited levels above the nonmagnetic $J=0$ ground state. LeCraw, Nilsen, Remeika, and Van Vleck (1963) interpreted this via the longitudinal slow-relaxation mechanism; they left the discrimination between longitudinal and transverse mechanisms, the angular dependence, the $J=1$ level constants, impurity broadening, and the uniqueness of the activated fit unresolved. The 1970-1975 quantitative theory of Foglio, Sekerka, and Van Vleck reproduces the temperature, frequency, and directional dependence using the longitudinal mechanism with anisotropic exchange, but requires a short, component-independent phenomenological relaxation time whose microscopic origin is left open (phonon modulation of exchange being excluded), and modern work (2021) still treats the EuIG damping mechanism as an open question. The open problem is to identify the microscopic relaxation channel and rate and to establish quantitatively whether the longitudinal mechanism, with transverse contributions bounded, accounts for the full measured linewidth of bulk EuIG.

Background

Europium iron garnet (EuIG, Eu$_3$Fe$_5$O$_{12}$) is a ferrimagnetic insulator of the garnet family, the same crystal structure as the microwave workhorse yttrium iron garnet: Fe$^{3+}$ ions occupy tetrahedral and octahedral oxygen sites and form two antiparallel sublattices whose difference gives the net magnetization, while the rare-earth ions occupy dodecahedral sites. In ferromagnetic resonance (FMR), a microwave field drives the uniform precession of the net magnetization, and the linewidth $\Delta H$ of the resonance peak measures how fast the precessional energy is dissipated, i.e. the magnetic relaxation. A magnon is a quantized spin wave; the FMR mode is the $k=0$ magnon, and two-magnon scattering, which transfers its energy to degenerate short-wavelength magnons, is a common additional linewidth channel. Gilbert damping, the standard phenomenological damping constant $\alpha$ of the Landau-Lifshitz-Gilbert equation, is the modern parameterization of such losses.

EuIG is exceptional among rare-earth garnets because the Eu$^{3+}$ ion (a $4f^6$ shell) has a nonmagnetic $J=0$ ground level by Hund's rules, with the first excited level $J=1$ only a few hundred kelvin above. That $J=1$ manifold is split into three components by the exchange field that the ordered Fe$^{3+}$ sublattices exert on the Eu$^{3+}$ $4f$ shell and by the low-symmetry (rhombic) crystal field at the dodecahedral site. In the longitudinal (slow-relaxer) mechanism proposed for EuIG, the populations of the thermally occupied $J=1$ components lag behind instantaneous equilibrium with the radio-frequency and exchange fields by a relaxation time $\tau$; the delayed Eu$^{3+}$ susceptibility is then out of phase with the motion of the magnetization and dissipates its energy, so $\Delta H$ grows with the Boltzmann population of $J=1$ and depends on the angular microwave frequency $\omega$ through the product $\omega\tau$. This is the ferrimagnetic analog of the slowly-relaxing-ion linewidth of paramagnetic impurities. Transverse mechanisms, in which relaxation acts on components transverse to the quantization axis, for example anisotropy-enhanced (giant-anisotropy) magnon scattering driven by EuIG's very large magnetocrystalline anisotropy, were the competing proposal (the giant-anisotropy theory itself was formulated after the 1963 measurement).

LeCraw, Nilsen, Remeika, and Van Vleck measured $\Delta H(T)$ and $\Delta H(\omega)$ on flux-grown (crystallized from a molten-salt flux) EuIG single-crystal spheres: a very narrow linewidth (below about 1 Oe) at low temperature, growing by nearly two orders of magnitude at high temperature with an approximately activated (Arrhenius-like, exponential in the inverse temperature) temperature dependence. They interpreted this with the longitudinal mechanism, which had just been formulated for dilute rare-earth-doped iron garnets (Van Vleck and Orbach, 1963). The paper left several items unresolved: the discrimination between the longitudinal and transverse mechanisms, the angular dependence of $\Delta H$, the $J=1$ level constants, residual-impurity influence on the low-temperature linewidth, and the non-unique fit to an activated temperature dependence.

The subsequent theory program shaped the question. Huber developed the spin-magnon relaxation theory of rare-earth ions in iron garnets and applied it to europium. Sparks proposed the giant-anisotropy linewidth theory. Nowik and Ofer measured the Eu-Fe exchange interactions in EuIG. Foglio, Sekerka, and Van Vleck then built the quantitative longitudinal theory, first a conference-level account and then a full treatment combining the crystalline field and anisotropic Eu-Fe exchange, which reproduces the measured temperature, frequency, and directional dependence of $\Delta H$ provided the $J=0\leftrightarrow J=1$ (inter-level) relaxation is fast, of order $10^{-12}$ s, and roughly equal for all $J=1$ components; that requirement excludes phonon modulation of the exchange interaction as the relaxation channel. The microscopic origin of the required relaxation time, and whether the longitudinal mechanism quantitatively exhausts the linewidth against transverse and impurity contributions, were left open. Modern EuIG thin-film work (2021) still states that the bulk linewidth increase raises fundamental questions about the responsible damping mechanism and that the mechanisms of intrinsic damping in EuIG warrant further investigation.

Problem Statement

Determine quantitatively and microscopically the mechanism responsible for the ferromagnetic-resonance linewidth $\Delta H$ of europium iron garnet, the question left open by LeCraw, Nilsen, Remeika, and Van Vleck (1963) and by the later theory program. Concretely: (i) establish the relative importance of the longitudinal relaxation mechanism, in which the populations of the crystal-field- and exchange-split Eu$^{3+}$ $J=1$ levels above the $J=0$ ground state lag behind equilibrium with relaxation time $\tau$, versus transverse mechanisms such as anisotropy-enhanced (giant-anisotropy) magnon scattering; (ii) identify the microscopic relaxation process that produces the required $\tau$, of order $10^{-12}$ s and approximately the same for all three $J=1$ components, and compute its magnitude and temperature dependence; and (iii) demonstrate quantitative agreement with the measured linewidth of bulk EuIG single crystals over its full temperature range, its frequency range from microwave through millimeter-wave frequencies spanning the $\omega\tau \sim 1$ crossover, and its angular dependence. The measured linewidth to be reproduced is that of the flux-grown bulk EuIG single-crystal spheres of LeCraw, Nilsen, Remeika, and Van Vleck [1]. These items are facets of one objective: a single quantitative microscopic account of the EuIG ferromagnetic-resonance linewidth and its temperature, frequency, and angular dependence.

References

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