Effects of an adjacent metal surface on spin wave propagation

We experimentally investigate and model the effects of a copper surface adjacent to a surface on which spin waves propagate in a thin film of yttrium iron garnet (YIG). Investigation was performed using a phase detection method, which can map out the spin wave velocity as a function wavevector for small wavevector with high resolution. This velocity is in good agreement with a simple model and allows for extraction of the separation between the YIG film and the copper.We experimentally investigate and model the effects of a copper surface adjacent to a surface on which spin waves propagate in a thin film of yttrium iron garnet (YIG). Investigation was performed using a phase detection method, which can map out the spin wave velocity as a function wavevector for small wavevector with high resolution. This velocity is in good agreement with a simple model and allows for extraction of the separation between the YIG film and the copper.


I. INTRODUCTION
There is currently much interest in various effects associated with spin-orbit coupling that arise from metal layers in contact with ferromagnetic films, examples being spin pumping and interfacial Dzyaloshinskii-Moriya interactions (DMI). 1 When considering dynamic responses involving the propagation of the wave, one must also account for the current induced in the adjacent metal layer which, in turn, affects the propagation of the wave itself. 2 In free space, the amplitude of the magnetic field accompanying a spin wave decays exponentially with a characteristic length equal to 1/k, where k in the wavevector of the spin wave.For the case of a perfect conductor in contact with the film, the field must vanish at the interface.
The in-plane dispersion of spin waves in an infinite slab was treated long ago by Damon and Eshbach [3][4][5][6] where two principal modes were identified: a surface wave, now termed the Damon-Eshbach (DE) mode, and a so called backward volume (BV) mode.Here we report measurements of the variation of the velocity with wavevector resulting from an adjacent copper film for the DE mode in an insulating ferromagnet, yttrium iron garnet (YIG).
Figure 1 shows frequency vs. wavevector for a 3.05 micron film in a field of 1.4 kG according to the Damon-Eshbach theory.The velocity in the low k, dipole dominated, regime for the DE mode is given by where γ is the gryomagnetic ratio, s is the sample thickness, ω 0 is the applied microwave frequency, and M is the saturation magnetization of the sample.This equation is derived with the boundary condition that the amplitude goes to zero at infinity.If we introduce a metal film, the equation for the velocity becomes 7,8 V (k) = γ 2 sπM ω 0 4πM(e where N = k/|k|, and t is the separation of the conducting surface from that of the ferromagnet.
The direction of k in the definition of N depends on the field direction such that N = 1 for propagation on the side nearest the metal surface and N = 1 for the side furthest.

II. APPARATUS
Figure 2 shows a block diagram of the experiment.The YIG sample used here was supplied by MDI Corporation: it is 5mm × 10mm with a stated thickness of 3.05µm and was grown epitaxially on a gadolinium galium garnet (GGG) substrate.The copper ground plane was provided by a square section of printed circuit board which forms an integral part of the sample probe and against which FIG. 2. Block diagram of the apparatus used.Microwaves are applied to an antenna on one side of a YIG sample.Depending on the field applied and the frequency of the microwaves, a spin wave with a particular wavevector will be launched across the sample.This spin wave is then picked up by a second antenna on the opposite side of the sample.A portion of the input signal and the output of the second antenna are then combined in a mixer, the output of which depends on the phase difference between the source and the received spin wave.As we vary the field, the wavevector of the spin wave will vary along with the output of the mixer.A small modulation field is applied to allow for lock-in detection of the mixed signal.the sample was pressed.Transmitting and receiving antennas separated by 5mm were positioned on opposing sides of the sample.Each consisted of a wire running from a grounded SMA connector along the 10mm length of the sample which was grounded on the far side.(An alternate antenna design, which will not be described here, was used for metal free propagation experiments).
6 GHz microwaves are applied to one of the antennas.When the conditions defined by the Kittel equation together with the dispersion relation are met for a given magnetic field, spin waves of the corresponding wavevector will be launched perpendicular to the transmitting edge and propagate to the receiving antenna on the other side of the sample.As we vary the field, the wavevector of the spin wave will also vary.If we apply a portion of the input microwave signal and the receiver antenna output to opposite ports of a microwave mixer, the resulting signal will be controlled by the phase difference.This phase difference, ∆φ, is related to the change in wavevector, ∆kd = ∆φ, where ∆k is the change in wavevector for a change in field, ∆H, and d is the width of the sample.
The detected signal for the Damon-Eshbach geometry and in the absence of a metallic boundary is shown in figure 3. The change in wavevector results in an oscillatory signal output from the mixer.Additionally, coupling to the exchange split backwards volume modes where they intersect the Damon-Eshbach mode results in nodes. 9,10The spin wave velocity can be calculated using the oscillation period since a full period involves a change in phase of 2π, which corresponds to a change in wavevector of ∆k = 2π d .When combined with the large path length (5 mm), this allows us to make measurements of the velocity with high resolution in k, which is, in turn, made possible by the low damping in YIG.

III. RESULTS
Figure 4 shows the field dependence of the output of the phase detector for the DE geometry for four different field/wavevector configurations in which a copper layer is in contact with the free side of the YIG film.The upper left panel shows the data for a nominally positive field and positive wavevector.The upper right panel shows the result of reversing the magnetic field direction while the lower left figure is the result of reversing the wavevector direction (simply by reversing which antenna we apply the microwave signal to).The propagation of the spin waves is clearly different between the first case and the second two cases.Instead of fast oscillations with periodic nodes due to the exchange split backwards volume modes, we observe a slow oscillation.Finally, in the lower right of figure 4 we show the data when both the field direction and the wavevector are reversed and we completely recover our original signal.
The large change in oscillation period arises from the differing characteristics of the DE mode when the wave is concentrated at the dielectric and metal interfaces.Given that the coupling to the DE mode is maximally when the magnetic field is perpendicular to the wavevector, there are two This velocity vs. wavevector data can be fit using equation 2 to determine both the separation of the metal to the YIG and the absolute value of the wavevector.The change in velocity as k increases seems to be captured well by equation 2. The distance between the YIG sample and the metal film are determined to be 17.15 ± 2.82 µm and 18.68 ± 3.98 µm for the forward and reversed propagating waves respectively.(a) Velocity versus field for positive field with a fit from equation 2 where the spin wave is nearest the metal film.Corresponds to the lower left plot of figure 4. (b) Velocity versus field for negative field with a fit from equation 2 where the spin wave is nearest the metal film.Corresponds to the upper right plot of figure 4.

FIG. 3 .
FIG. 3. A measurement of propagation in the Damon-Eshbach mode for the sample on a dielectric.

FIG. 4 .
FIG. 4. Examples of data taken for a YIG film pressed against a copper ground plane for various combinations of field and wavevector direction with the upper left being the reference figure.Arrows point to the position of the ferromagnetic resonance (FMR).The figures on the right show the result of reversing the magnetic field direction while the bottom figures show the result of reversing the direction of propagation.Clear nonreciprocity of the spin waves is shown, where the upper right and lower left correspond to a wave concentrated on the YIG/copper interface.The upper left and lower right correspond to a wave concentrated on the YIG/GGG interface.Small asymmetries in the measured field are observed in between the positive and negative field measurements.