The Final Chapter In The Saga Of YIG

The magnetic insulator Yttrium Iron Garnet can be grown with exceptional quality, has a ferrimagnetic transition temperature of nearly 600 K, and is used in microwave and spintronic devices that can operate at room temperature. The most accurate prior measurements of the magnon spectrum date back nearly 40 years, but cover only 3 of the lowest energy modes out of 20 distinct magnon branches. Here we have used time-of-flight inelastic neutron scattering to measure the full magnon spectrum throughout the Brillouin zone. We find that the existing model of the excitation spectrum, well known from an earlier work titled"The Saga of YIG", fails to describe the optical magnon modes. Using a very general spin Hamiltonian, we show that the magnetic interactions are both longer-ranged and more complex than was previously understood. The results provide the basis for accurate microscopic models of the finite temperature magnetic properties of Yttrium Iron Garnet, necessary for next-generation electronic devices.

caloritronics has also recently emerged as a potential application of YIG, utilising the spin Seebeck effect (SSE) and the spin Peltier effect (SPE) to interconvert between magnon and thermal currents, either for efficient large-scale energy harvesting, or the generation of spin currents using thermal gradients 13 .
If the research into classical and quantum aspects of spin wave propagation in YIG is to achieve its potential, it is absolutely clear that the community requires the deep understanding of its mode structure, which only neutron scattering measurements can offer. In many theories and experiments, YIG is treated as a ferromagnet with a single, parabolic spin wave mode 14,15 , simply because the influence of YIG's complex electronic and magnetic structure on spin transport is not known in sufficient detail. Such approaches must break down at high temperature when the optical modes are appreciably populated and a detailed knowledge of the structure of the optical modes is a necessary first step in any realistic model of the magnetic properties of YIG in this operational regime. Despite this, surprisingly little data exists relating to the detail of its magnon mode structure. The key previous work in this area is due to Plant et. al. 2 , and dates back to the 70s. Using a triple-axis spectrometer, these early measurements were able to record 3 of the spin wave modes up to approximately 55 meV, but crucially there are 20 such modes and they are predicted to extend up to approximately 90meV (22 THz) 3 .
Data were collected (see methods section) as a large, 4-dimensional hypervolume in frequency and momentum space, covering the complete magnon dispersion over a large number of Brillouin zones. Fig. 2 shows two-dimensional energy-momentum slices from this hypervolume with the wave-vector along three high-symmetry directions, normalised to a measurement on vanadium (see methods section). A large number of modes can be seen up to an energy of 80meV, whilst data in other slices show modes extending up to nearly 100 meV. The spectrum is dominated by a strongly dispersing and well-isolated acoustic mode at low energies (the so-called 'ferromagnetic' mode), and a strongly dispersing optical mode separated from this by a gap of approximately 30meV at the zone centre. Intersecting this upper mode is a large number of more weakly dispersing optical modes in the region of 30-50meV.
We model the data using a Heisenberg effective spin Hamiltonian, appropriate to YIG as it is both a good insulator and the Fe 3+ ions (S = 5 /2, g = 2) possess a negligible magnetic anisotropy due to the quenched orbital moment.
= ∑ , + ∑ We nevertheless include a magnetic anisotropy Ai in our analysis to take into account crystal field effects, but find this term to be vanishingly small, consistent with previous results. The exchange matrix Jij is a general 3x3 matrix, whose elements are restricted by the symmetry of the bond connecting the spins Si and Sj. Following common practice, Jij is restricted to having only identical diagonal components (i.e. isotropic exchange) since anisotropic and off-diagonal contributions are likely to be small due to the lack of significant orbital angular momentum.
The spin Hamiltonian was diagonalized using the SpinW software package 16 and the calculated magnon dispersion was fitted by a constrained nonlinear least squares method to 1D cuts taken through the 2D intensity slices. We do not include any scaling factors for the magnon intensity, so the agreement between the model and the data in terms of absolute units is indicative of the quality of the model. Our final/best-fit model includes isotropic exchange interactions up to the 6 th nearest neighbour, labelled J1-J6 in Fig 1. The exchanges in this work can be mapped to the exchanges commonly considered for YIG as follows: Jad=J1, Jdd=J2, and Jaa ={J3a,J3b}, where the subscript refers to the majority tetrahedral (d) and minority octahedral (a) sites. Due to the extremely large number of magnetic atoms (20) within the primitive cell, and the consideration of so many exchange pathways, this analysis would be impossible without the use of sophisticated software such as SpinW as the construction of an analytic model would be prohibitively time consuming. During the fitting process, features in the spectrum were weighted so that weak but meaningful features in the data were considered as significant as strong features. The SpinW model output is then convoluted with the calculated experimental resolution of the MAPS spectrometer, including all features of the neutron flight path and associated focussing/defocussing effects, as well as the detector coverage and effects from symmetrisation (see Supplementary Information for details). The final fitted values of the exchanges are listed in Table 1.
An important difference between our results and those of previous authors 2,3 is that there are two symmetry-distinct 3 rd -nearest-neighbour bonds (the so-called Jaa in the literature, which we label J3a and J3b) which have identical length but differ in symmetry. The J3b exchange lies precisely along the body-diagonal of the crystal, and thus has severely limited symmetry-allowed components owing to the high symmetry of the bond (point group D3). The J3a exchange connects the same atoms with the same radial separation, but represents a different Fe-O-Fe exchange pathway as a result of the different point group symmetry (point group C2), so it is distinguished from J3b by the environment around the Fe atoms. Models including anisotropic exchange or Dzyaloshinskii-Moriya interactions on the 1 st -4 th neighbour bonds were tested, but such interactions were found to destabilise the magnetic structure for arbitrarily small perturbations. We also find that J2 (Jdd) is much smaller than previously supposed -the main effect of this exchange is to increase the bandwidth and split the optic modes clustered around 40 meV in a way contradicted by the data.
It has been pointed out 17 that the magnetic structure of YIG is incompatible with the cubic crystal symmetry, although to date no measurements have found any evidence for departures from the ideal cubic structure. Nevertheless, it is necessary to refine the magnetic structure in a trigonal space group (a symmetry that is experimentally observed in terbium rare earth garnets where the magnetoelastic coupling is much stronger 18 ), in order to obtain a satisfactory goodness of fit, and a magnetic moment which agrees with bulk magnetometry 19 . Treating the unit cell of YIG in this fashion for the purposes of the SpinW simulation would be feasible, but would introduce a large number of free parameters which (given the very small size of the departure from cubic symmetry) would nevertheless be expected to change very little from a cubic model. This expectation is borne out by the excellent agreement between the data and the cubic spinwave model (see Fig. 2 and the supplementary materials for more details).
Features absent from the data which are relevant to technological applications (such as the conversion of microwave photons into magnons) include any strong indications of magnon-phonon or magnon-magnon coupling. The data are well described by a linear spinwave model, although the size of the 5 th -neighbour exchange is perhaps indicative of some small deviations not easily captured without such couplings. A strong magnon-phonon coupling would be expected to cause both broadening and anomalies in the dispersion of the magnon modes 20 . We do not observe any such effects, although our measurements would not be sensitive to any magnon-phonon coupling that shifts or broadens the spin wave signal by less than 3meV (the instrumental resolution).
Our results require a substantial revision of the impact of the optical modes on the room-temperature magnetic properties. The differences compared with the existing model are illustrated in Fig. 3, in which we plot the antisymmetric combination of transverse scattering functions Sxy(Q,ω) − Syx(Q,ω), which is proportional to the sign and magnitude of the measured spin-Seebeck effect arising from the associated magnon mode 21 . Most strikingly the absence of spectral weight in the flat mode at ~35meV, as well as a compression and shift of the 'positively' polarised (red) optical modes i.e. those modes which would precess counterclockwise with respect to an applied field. As has recently been shown, the thermal population, broadening, and softening of these modes at elevated temperatures substantially modifies the magnitude of the measured spin-Seebeck effect, which places limitations on device performance and determines the optimum operating temperature. Our results show that the distribution of optical modes is very different from what had previously been assumed, which has consequences for the temperature-  21 . Our new measurements can therefore be used as the basis for a precise microscopic model of the temperature dependent dynamical magnetic properties in YIG.
We have also estimated the parabolicity of the lowest lying 'ferromagnetic' magnon mode i.e. the point where the error of a quadratic fit becomes greater than 5%. We find this region to extend 14.8% of the way towards the Brillouin zone boundary along the H direction (the (0,0,1) direction in the centred unit cell), representing about 0.3% of the entire Brillouin zone. The departure from a purely parabolic acoustic magnon dispersion, as well as the population of optic magnon modes directly generates the temperature dependence of the spin Seebeck effect and our model can be used to fully understand such effects even at elevated temperatures through extension to a multimagnon picture following the procedure in Refs. 21, and 22.
We have presented the most detailed and complete measurement of the magnon dispersion of YIG in a pristine, high quality crystal. Using linear spin-wave theory analysis we are able to reproduce the entire magnon spectrum across a large number of brillouin zones including a reproduction of the absolute intensities of the modes. We confirm the importance of the nearest-neighbour exchange, but are forced to radically reinterpret the nature and hierarchy of longer-ranged interactions. Our work has uncovered substantial discrepancies between previous models and the measured dispersion of the optical magnon modes in the 30-50 meV region, as well as the total magnon bandwidth, and the detailed nature of the magnetic exchanges. Through a detailed consideration of the symmetries of the exchange pathways and long-ranged interactions, we are able to fully reproduce the entire measured spectrum. Technological applications of YIG, particularly those utilising the spin Seebeck effect, are very sensitive to the optical magnons in the region of 30-40meV. This work overturns 40 years of established work on magnons in YIG and will be an essential tool for accurate modelling of the optical magnon modes in the room temperature regime.

Methods
Crystal Growth. YIG crystal growth was carried out in high-temperature solutions applying the slow cooling method 23 . Starting compounds of yttrium oxide (99.999%) and iron oxide (99.8%) as solute and a boron oxide -lead oxide solvent were placed in a platinum crucible and melted in a tubular furnace to obtain a high-temperature solution 24 . Using an appropriate temperature gradient only a few single crystals nucleate spontaneously at the cooler crucible bottom and forced convection, obtained by accelerated crucible rotation technique (ACRT), allows a stable growth which results in nearly defect-free large YIG crystals 25 . The YIG crystal used in this study exhibits a size of 25 mm x 20 mm x 11 mm and a weight of 12 g. It was confirmed by neutron and X-ray diffraction that the YIG crystal was a single grain with a crystalline mosaic of approximately 0.07 degrees FWHM. Preliminary measurements were made on a crystal that was grown by the optical floating-zone method, starting from a pure powder of YIG.

Neutron Scattering Data Collection and Reduction.
Data were collected on the MAPS time-of-flight neutron spectrometer at the ISIS spallation neutron source at the STFC Rutherford Appleton Laboratory, UK. On direct geometry spectrometers such as MAPS, monochromatic pulses of neutrons are selected using a Fermi chopper with a suitably chosen phase. In our experiment neutrons with an incident energy (Ei) of 120 meV were used with the chopper spun at 350 Hz, giving energy resolution of 5.4 meV at the elastic line, 3.8 meV at an energy transfer of 50 meV, and 3.1 meV at an energy transfer of 90 meV. The spectra were normalized to the incoherent scattering from a standard vanadium sample measured with the same incident energy, enabling us to present the data in absolute units of mb sr -1 meV -1 f.u. -1 (where f.u. refers to one formula unit of Y3Fe5O12). Neutrons are scattered by the sample onto a large area detector on which their time of flight, and hence final energy, and position are recorded. The two spherical polar angles of each detector element, time of flight, and sample orientation allow the scattering function S(Q, ω) to be mapped in a four dimensional space (Qx,Qy,Qz,E). In our experiment the sample was oriented with the (HHL)-plane horizontal, while the angle of the (00L) direction with respect to the incident beam direction was varied over a 120 degree range in 0.25 degree steps. This resulted in coverage of a large number of Brillouin zones, which was essential in order to disentangle the 20 different magnon modes. Due to the complex structure factor resulting from the number of Fe atoms in the unit cell, the mode intensity varies considerably throughout reciprocal space. The large datasets recording the 4D space of S(Q,ω), ~100 GB in this case, were reduced using the Mantid framework 26 , and both visualized and analyzed using the Horace software package 27 . Taking advantage of the cubic symmetry of YIG, the data were folded into a single octant of reciprocal space (H>0, K>0, L>0), adding together data points that are equivalent in order to produce a better signal-tonoise. 2D slices were taken from the 6 reciprocal space directions depicted in Figure 2 and in Supplementary Information.