Magnetism of Yttrium Iron Garnet

Yttrium iron garnet (YIG) is an insulating ferrimagnet at room temperature with extremely low spin wave damping, making it one of the most intensively studied magnetic materials. The field of spintronics aims to replace conventional charge based operations as the building block of computing technology with devices using spin waves, which are smaller in energy and therefore cause less heat dissipation. Because spin waves in YIG can propagate over macroscopic distances with negligible attenuation, this material is particularly important for fundamental experiments and proof-of-principle demonstrations.

First principles calculations of YIG, and the family of iron garnets (where different rare earth elements replace yttrium) has largely been limited to DFT+U. LDA and GGA underestimate the iron magnetic moments and the fundamental gap is much smaller than experiment, so these methods only give a qualitative description. Because the gap lies between occupied and unoccupied iron d states, both errors in the iron moment and $E_g$ can be ameliorated to some extent using a local Hubbard correction; meanwhile more reliable, parameter free electronic structure methods have until now been too demanding numerically for the large, 80-atom structure of the garnets.

The choice of U is naturally delicate; often it is guided by matching one experimental characteristic with the hope that the chosen U then reproduces other properties equally well. For the case of YIG, the gap and the ferrimagnetic critical temperature are well known properties that can be matched by +U, but seemingly they are not simultaneously reproduced with a single +U parameter. Quasiparticle self-consistent GW (QSGW) is a method that is able to describe the physics of LDA+U–most importantly local exchange interactions–very accurately and with no free parameters; this feature is a consequence of the improvement of the single-particle potential under iteration to QSGW self-consistency (and is not present, for example, in single shot GW). Because QSGW treats all bands on an equal footing, there is no need for the specification of a correlated subspace, double counting corrections or interaction parameters: the effective U is calculated explicitly from the QSGW self energy $\Sigma = GW$. In fact, this is similar to the most successful methods for calculating U, the constrained random phase approximation.

Figure 1. Density of states for YIG: site projected $n(E)$ for tetrahedral Fe is coloured red, octahedral Fe grey.

In a recent paper invited to the IOP journal “Electronic Structure”, we present QSGW band structures for YIG. These calculations demonstrate Questaal’s capability to tackle complex functional materials using the QSGW method: in this case an 80 atom ferrimagnetic oxide. These calculations were made possible by the hybrid parallel strategy in Questaal’s GW code involving MPI, OpenMP and CUDA that was developed during 2020. For YIG, 16 GPU enabled nodes (totalling 64 Nvidia V100 cards) of the Marconi M100 machine at CINECA were used; from an LDA starting point, the converged QSGW self-energy on a 4x4x4 q-mesh (which was found to be more than sufficient) could be obtained in under 8 hours, with individual self-energy calculations taking roughly 10 minutes. Clearly this performance shows that complex systems with large unit cells–eg, as is often the case in high temperature superconductors–are now well within the capability of QSGW.

The magnetism of YIG is due to iron which occupies two distinct sets of sites (with 12 tetrahedral sites and 8 octahedral sites) in the garnet lattice; these align antiferromagnetically such that YIG overall is ferrimagnetic with 20 distinct modes in the spin wave spectrum. Using the QSGW band structure, the transverse (spin) susceptibility $\chi^{+-}(\omega,q)$ can be calculated; the static limit of this quantity can be related to the magnetic interaction parameters of the Heisenberg model. Heisenberg models can be defined either in terms of quantum spins or classical spins; these two models show very different behaviour and match only in the limit of very large |S|. Because the quantum Heisenberg model is difficult to solve, it has become customary to treat the moments entering the spin model as classical objects and to solve them using Monte Carlo or dynamics (via the Landau-Lifshitz or Landau-Lifshitz-Gilbert equations) at finite temperatures, yielding rapidly an prediction of the critical temperature from $M(T)$ or $C_v(T)$.

Although the use of classical statistics for modelling magnetic systems has been questioned for a long time, more or less good agreement with experimental critical temperatures have been found for various systems, and it has not been clear whether this agreement is a success of methods such as the LDA or GGA (or their +U derivatives) or is accidental. In the YIG study, this question was addressed by showing the effect of a more thorough treatment of the spin model by using a quantised heat bath: this means that the occupation of spin waves during a finite temperature simulation obeys the correct statistics, without requiring the solution of the full quantum problem. We have shown that the critical temperatures calculated using classical statistics is roughly 1/2 of that using quantum statistics. When the Heisenberg model is parameterised from the LDA, this effect roughly cancels the LDA’s overestimate of the coupling parameters J. On the other hand, when using the Heisenberg parameters calculated using the more accurate QSGW band structure, we find that this cancellation is unnecessary: the QSGW Js coupled with the semi-quantum statistics gives a realistic estimate (535 K calculated v. 560 K experiment). Strong support for this approach comes from the spin wave spectra: the QSGW derived interactions give a very close agreement to the spin waves of YIG (Fig. 2), particularly for the first (acoustic) branch, while the LDA description is significantly at error.

Figure 2. Calculated spin wave spectra at 10 K. Red and blue indicate magnon polarisations. Dots are experimental data, insensitive to polarisation¹.

Close agreement in such a sensitive property as the spin wave spectrum emphasises the usefulness of QSGW as a general, parameter free method for materials studies: even when significantly more expensive than DFT+U, the accuracy – in particular its reliability – makes it well suited for diverse problems, even for high-throughput materials studies.

¹ “Spinwave dispersion curves for yttrium iron garnet” J S Plant, Journal of Physics C: Solid State Physics, Volume 10, Number 23