Using a multielement antenna, we have performed measurements of the angular dependence of propagating magnetostatic spin waves in a thin yttrium iron garnet film, both in-plane and out-of-plane, and compared the measurements with existing theoretical models. For most magnetic field directions, theory and experiments agree reasonably well. However, there is a range of magnetic field directions where differences between theory and experiment become large.

## I. INTRODUCTION

The dispersion of long wavelength spin waves in the absence of exchange is governed by magnetostatic effects arising from the requirement that the Maxwell boundary conditions are satisfied at the free surfaces when solving the Landau-Lifshitz equation for the spin precession. Building on an earlier work by Walker,^{1} a closed form solution for an infinite slab with an in-plane magnetic field was given long ago by Damon and Eshbach^{2} and for the perpendicular field case by Damon and Van De Vaart.^{3} Later a full solution for an arbitrarily directed field was given by Bajpai and Srivastava^{4} (BS). At shorter wavelengths, the effects of exchange enter, which were treated by Kalinikos and Slavin^{5} (KS) using perturbation theory. A microscopic theory utilizing a model Hamiltonian has also been given.^{6}

The propagation of magnetostatic spin waves in yttrium iron garnet (YIG) films has received much attention over the years.^{7,8} However, past studies have largely been limited to the cases where the external magnetic field, **H**, lies (i) parallel to the in-plane wave vector, **k**; (ii) along the film plane normal, **n**; or (iii) in the direction **n **×** k**. The associated modes are designated as backward volume (BV), forward volume (FV), and Damon-Eshbach (DE), respectively; for the first two, the amplitude is distributed throughout the volume of the film, and hence their names, while for the last, it is largest at one of the surfaces and exponentially decays into the interior. The BV mode has the unusual property that the group velocity is the direct *opposite* to the wave vector, and hence its designation. Here, we present a systematic study in which the magnetic field lies in the three principal planes that are perpendicular to the vectors **n**, **k**, and **n** × **k**. We also present a comparison between experimental data and theoretical predictions. A comparison with an approximate dispersion relation given by KS^{5} shows a reasonable agreement over a moderate range of field directions. However, for certain field directions, there is a coexistence of two distinct modes, in contrast to this theoretical model, which predicts only a single mode. The BS theory^{4} has the virtue of being an exactly solved model and, in addition, shows the best agreement with our data over the widest range of angles in the magnetostatic regime. Furthermore, it predicts two branches in the regime where these are observed experimentally.

Historically, spin waves have been generated using a wire, a single strip (or three-element strip line), or a meander line.^{9} In the experiments to be described here, we monitor the steady state absorption from a multielement “ladder” antenna in which a microwave current is passed through the “rungs.” This type of antenna has been modeled by Kalinikos^{10} who showed that it will couple to the propagating magnetostatic modes as well as the exchange split modes having a vertical standing wave component.

The yttrium iron garnet (YIG) film used in these experiments was grown by liquid phase epitaxy on a lattice matched (111) gadolinium gallium garnet (GGG) substrate (5 × 10 mm^{2}) and was supplied by the MTI Corp. The surface was scanned using an AFM (Bruker, Dimension Fastscan™ AFM). The tapping mode was employed, and the linear background was removed from the measured data to eliminate sample tilting. The resulting RMS surface roughness was 3.51 nm.

Our antenna was patterned on a Si substrate [see Fig. 1(a)] and is in direct contact with an epitaxial YIG film deposited on a (111) gadolinium gallium garnet (GGG) substrate;^{11} the film thickness, as measured by ellipsometry, was 2.8 *μ*m. A copper ground plane (not shown) can be positioned against the opposing GGG surface. The presence of an adjacent metal can alter the dispersion relation of various modes, particularly the Damon-Eshbach mode; for an extended discussion of this effect, see Ref. 12. For the data reported here, the rung spacing, which fixes the fundamental spin wavelength, λ, is 50 *μ*m; measurements at other wavelengths will be reported elsewhere. Subwavelength responses, λ/*p* with odd *p*, can also be detected (the linewidth of our antenna is equal to the spacing, so the spatial Fourier series of the magnetic field generated by the antenna has only odd terms). In addition to producing a spatially oscillating microwave field, there is also a quasiuniform component that couples to the uniform ferromagnetic resonance (FMR) mode, thereby providing a reference point from which the spin wave absorption resonances are measured.

## II. MEASUREMENT TECHNIQUE

The experimental setup is shown in Fig. 1(b). Microwave power from a Hewlett Packard HP 8360 Series Synthesized Sweeper, operating at a power level of 25 dBm, is applied to one side of the ladder; the other side is connected to a microwave diode whose the output is fed to a Princeton 124 lock-in amplifier. A reference oscillator operating near 150 Hz is applied to both the lock-in reference input and a power amplifier where the latter drives a pair of field modulation coils mounted on the pole faces of a Varian electromagnet. The current for this magnet is supplied by a pair of Kepco ±20 A operational amplifier power supplies operating in a constant current mode and in parallel. The input for the power supplies is a ramp voltage generated by a personal computer; this same computer records the lock-in output. A National Instruments card performs all D/A operations. The system operates under LabView and is programmed to do multisweep signal averaging.

## III. DISCUSSION

The ladder rungs run perpendicular to the long axis of a 5 × 10 mm^{2} YIG sample, and the signals are typically largest when the field lies along either **k** or **n** × **k**. Figure 2(a) shows a field sweep at 3 GHz for the magnetic field parallel to **k**, the BV geometry. The large full-scale feature corresponds to FMR. Figures 2(b) and 2(c) show the data with an expanded vertical scale where two additional features (indicated by arrows) are resolved. Taken together, the three features arise from spin wave resonances for λ = 50 *μ*m, 16.7 *μ*m, and 10 *μ*m, corresponding to the fundamental and the *p* = 3 and *p* = 5 subfundamental wavelength ladder resonances. A comparison of these data with the theoretical predictions is given in Table I.

In what follows, we will compare the field positions of our fundamental resonances for the magnetic field in all three principal planes with the dispersion equations for magnetostatic modes obtained in the magnetostatic theory based on the formalism of BS. The BS analysis used a well-known form for the susceptibility tensor,^{13} which incorporates the Maxwell boundary condition but does not include effective fields arising from exchange and anisotropy energies. This approach is equivalent to the magnetostatic assumption adopted by Damon and Eshbach.^{2} Then BS analysis introduced a rotation matrix^{14} to orient the film in an arbitrary magnetic field direction. They also treat the case where there is a conducting layer that is arbitrarily spaced from the magnetic layer, which is not required in the present work. The closed form dispersion relation of magnetostatic modes is then given by

where

and

Here, *γ*, $Hi$, *M*, and *t* are the gyromagnetic ratio, *internal* magnetic field, magnetization, and film thickness, respectively; *θ _{i}* and

*φ*are the polar coordinates of the internal magnetic field measured respectively from

_{i}**n**and

**n**×

**k**(and along which the equilibrium magnetization lies); and

**k**= 2

*π*/λ = 2

*pπ*/

*d*, where

*d*is the ladder period and

*p*is an odd integer. The internal angle

*θ*follows from the external angle

_{i}*θ*by applying the condition that the torque vanishes; knowing

*θ*, we can relate

_{i}**H**to the external field,

_{i}**H**; here, we neglected the effects arising from uniaxial anisotropy energy. Also, the internal angle

*φ*is equal to the external angle

_{i}*φ*in the absence of in-plane anisotropy.

Figure 3 shows the angular dependence of the spin wave resonances observed in the x-y plane when **H** ⊥ **n** (i.e., *θ _{i}* = 90°) for frequencies of 3 GHz (a) and 5 GHz (b). As noted above, at some angles, and depending on how long we signal average, we are able to resolve the subwavelength resonances associated with

*p*= 3 and, in some cases,

*p*= 5. Note that in this plane, the angular dependence of the FMR line is minimal, being nominally given by the usual in-plane Kittel formula; in practice, we observe a small anisotropy with twofold symmetry which likely arises from shape and field-dependent demagnetization effects associated with our rectangular sample dimensions. The curves show the expected angular dependence based on Eq. (1); the parameters used are

*γ*= 2.81 GHz/kOe, 4

*π*M = 1.88 kOe for Fig. 3 and 1.80 kOe for Figs. 4 and 5 (obtained from the FMR position), and

*t*= 2.8

*μ*m. For comparison, we also show the predictions of the magnetostatic theory of Damon and Eshbach; we note that this theory predicts an angular range where the velocity is double valued (consisting of one negative and one positive group velocity), although this is not resolved in these experiments. Note that we do not directly obtain the group velocity from these experiments, but we are able to determine its sign.

Figure 4 shows the angular dependence in a second principal plane, the x-z plane, defined by **H** ⊥ **k** (i.e., *φ _{i}* = 0 and 180°) for frequencies of 3 GHz (a) and 4 GHz (b). Note that in this plane, the FMR frequency varies strongly with the angle, being given for

**H**∥

**n**by the perpendicular field Kittel expression, ω =

*γ*(Η − 4

*π*M). For intermediate angles,

^{15,16}ω =

*γ*[H

_{i}(H

_{i}+ 4

*π*M sin

^{2}

*θ*)]

_{i}^{1/2}where H

_{i}and

*θ*are the magnitude and inclination of the internal magnetic field, both of which are evaluated numerically. Note, the external angle

_{i}*θ*is generally smaller (measured relative to

**n**) than

*θ*. Both of these angles are shown in Fig. 4.

_{i}In this plane, both the FV and DE resonances occur at fields below the FMR field, making their character (surface vs volume) difficult to distinguish; in particular, no discontinuity, in the angular dependence, which might otherwise distinguish them, is discernable. We were unable to determine the character of the modes, surface vs bulk, with the angle at λ = 50 *μ*m. The amplitude of the time-varying magnetic field generated by the multielement antenna exponentially decays along the film thickness direction^{17} so that these fields couple more easily with the surface modes than the bulk modes when the film thickness is comparable to or larger than the antenna period. This enhancement of the amplitude when the microwave signal coupled with surface modes has been measured by Lim *et al.*^{17} for an antenna with a 1 *μ*m period and the same YIG film used in this study. No such enhancement was observed in the present measurements since the characteristic decay length of the time-varying field is 50 *μ*m, which is more than ten times larger than thickness of the YIG film, i.e., there is no effective difference between bulk modes (which have a nearly constant amplitude along thickness) and surface modes (which then have the characteristic decay length of 50 *μ*m). Whether there is some simple way to distinguish the surface and bulk magnetostatic modes is an open question. Note that the BS theory predicts the surface and bulk mode distributions in this plane, which are included in Fig. 4 as a black solid line (bulk) and a blue dashed line (surface). Regardless of the surface and bulk character, the measured data are in fair agreement with the theoretical prediction given by the BS theory, but the differences between the BS theory and the experiment reach a maximum around at *θ _{i}* = 45°.

Figure 5 shows the behavior for the observed resonances in the y-z plane with **H** ⊥ **n** × **k** (i.e., φ_{i} = ±90°) for the frequencies of 3 GHz (a) and 4 GHz (b). The FMR frequency again varies strongly with the angle, having the same behavior as discussed in the previous paragraph, so Fig. 5 shows both *θ* and *θ _{i}*. A distinguishing feature of this plane, which includes the BV and FV geometries, is the existence of two different modes (BV and FV) at the same angle. This feature was initially unexpected since the approximate dispersion of KS predicts only a single mode;

^{5}however, this discrepancy had earlier been reported by Slavin and Fetisov.

^{18}They also used the BS theory to compare with KS and the measured data with the result that the BS theory and measurements generally agreed in this plane (

*φ*= ±90°). Here, we present a more complete comparison for the two branches with a higher density of points covering a wider range of angles. The BS theory and the experimental data agree well for near

_{i}*θ*= 0° and 90°. The discrepancy on both the BV and FV branches increases as

_{i}*θ*approaches 45°. The difference occurring for the FV branch may possibly be explained by introducing mode repulsion between the lowest FV mode and other exchange split modes having a nonzero wavevector along the z direction. Mode repulsion happens when two modes crossover, and the crossover only occurs when we consider both dipolar and exchange interactions. However, differences occurring for the BV branch cannot be explained with such an approach since no crossover is expected, even if we introduce the exchange interaction, since the mode frequency is lower than FMR frequency.

_{i}An alternate possibility is that the wavevector of the lowest magnetostatic mode along the thickness direction is nonzero. This flexibility is not part of the BS theory (in which the lowest mode has zero wavevector along the thickness direction in the magnetostatic regime) but can be incorporated into the KS theory, which allows this if spins at both surfaces are not totally free, the wavevector along thickness direction can have values comparable to *π*/*t*. However, according to the BS theory, any higher BV modes that have a nonzero wavevector along the thickness direction have higher resonance frequency than the lowest mode, and this remains true if an exchange interaction is introduced. Therefore, this hypothesis cannot explain the discrepancy because the measured resonance frequency at λ = 50 *μ*m is lower than the BS theoretical prediction. This discrepancy in the BV branch appears to be rather unusual and deserves further study, in particular, its nonmonotonic behavior with *θ _{i}*.

Also, Kalinikos *et al.*^{19} state that cubic anisotropy in a film that is normal to the (111) direction produces no angular dependence of the magnetic field and is equivalent to using a modified magnetic field that uniformly shifts the whole spin wave dispersion. Therefore, cubic anisotropy in a YIG crystal does not change with H.

Table I shows a comparison of the measured and predicted positions of the observed resonances along the principal directions for the frequencies studied.

## IV. CONCLUSION

In summary, we report detailed measurements of the angular dependence of 50 *μ*m magnetostatic spin waves in a YIG film in the three principal planes. The resulting data present a number of challenges. The differences (or relative errors) between the BS theory and the experimental data for both **H** ⊥ **k** and **H** ⊥ **n** × **k** planes were not uniform with the angle *θ _{i}* and were largest near

*θ*= 45°. Specifically, the discrepancy occurring in the BV branch cannot be explained by introducing the exchange interaction. It is possible that the full version of the KS theory might correct this discrepancy. Perhaps less serious is that theory and experiments agree best for the backward volume case (

_{i}**H**∥

**k**), whereas there are larger differences for the field along

**n**and

**n**×

**k**. Some of the discrepancy may result from the effects of uniaxial anisotropy that has yet to be incorporated in any fully analytic (as opposed to perturbation) theory for dipole-exchange spin waves. Finally, perturbations introduced by the metallic antenna elements themselves may need to be included, which presents a formidable boundary value problem.

## ACKNOWLEDGMENTS

The resonance measurements were performed at Northwestern under support from the U.S. Department of Energy (DOE) through Grant No. DE-SC0014424. Device fabrication was carried out at Argonne and supported by the U.S. Department of Energy, Office of Science, Materials Science and Engineering Division. Lithography was carried out at the Center for Nanoscale Materials, an Office of Science user facility, which is supported by DOE, Office of Science, Basic Energy Science under Contract No. DE-AC02-06CH11357.