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Micromagnetic simulation of spin wave propagation in a ferromagnetic film with different thicknesses

in a ferromagnetic film with different thicknesses Hiroshi Hata1, Takahiro Moriyama1, Kenji Tanabe1,2, Kensuke Kobayashi2, Ryo Matsumoto3, Shuichi Murakami3, Jun-ichiro Ohe4, Daichi Chiba5, and Teruo Ono1 1Institute for Chemical Research, Kyoto University, Gokasho, Uji, Kyoto 611-0011, Japan 2Department of Physics, Graduate School of Science, Osaka University, 1-1 Machikaneyama, Toyonaka, Osaka 560-0043, Japan 3Department of Physics, Tokyo Institute of Technology, 2-12-1 Ōokayama, Meguro-ku, Tokyo 152-8551, Japan 4Department of Physics, Toho University, 2-2-1 Miyama, Funabashi, Chiba 274-0072 Japan 5Department of Applied Physics, Faculty of Engineering, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan


Introduction
Spin wave (SW) is a significant phenomenon in the magnetics on the aspects of not only basic physics but also device application. As the SW has the characteristics of waves, it shows various phenomena such as interference [1][2][3][4] and refraction. [5][6][7] SW is also expected for device application, such as SW-based logic circuit [8][9][10][11] or storage [12][13][14][15][16][17] with its advantage of ultralow-power consumption. Consequently, for further understanding and utilizing the SW, it is important to investigate its propagation process.
The refraction law (Snell's law) is one of the most important laws in wave propagation. The refraction occurs when an incident wave is injected into the medium with different wave velocity. The wavelength is also modified in the refraction process. In general, Snell's law is expressed as follows: where θ i(r) , v i(r) and λ i(r) correspond to the incident (refraction) angle, the phase velocity of the wave, and the wavelength, respectively. In ordinal light wave, relationship between the incident and the refraction wave follows Snell's law. However there are some exceptions, such as birefringence in anisotropic materials 18) . In such materials, the light wave does not follow Snell's law. Hence, it is curious whether SW follows Snell's law. There are several reports on the refraction of the SW at the junction between materials with different anisotropies 5) and different materials 6) where f is the frequency of the wave, µ 0 is the vacuum permeability, γ is the gyromagnetic ratio 1.76 × 10 11 s −1 T −1 , H is the external magnetic field, M S is the saturation magnetization, k is the wavevector of the SW, and d is the thickness of the film. If the thickness d is halved, the wavevector k is doubled, since the frequency f is unchanged between the refraction processes. Thus we can modify the wavevector by simply changing the film thickness. Considering Eqs. (1) and (2), Snell's law for MSSW can be written as Eq. (3) clearly shows that the phase velocity, the wavelength, and the wavevector are dependent on the film thickness. Note that the reason why we used MSSW is because the decay length of MSSW is longer than the other mode, and MSSW can be excited by in-plane magnetic field, which makes the experimental setup easier than the other mode.
For the system with different film thickness, Tanabe et al. 7) experimentally showed that the SW wavelength was modified after propagating through the step. Note that they carried out the timedomain measurement using the wave packet of the SW. For further understanding of Snell's law for SWs, it is important to carry out the quantitative evaluation of overall propagation process in the system with different film thickness. In this paper, we report the micromagnetic study of the wavelength modulation during propagating through the step consisting of the films with different thicknesses, and verify MSSW follows Snell's law [Eq.  2 Micromagnetic Simulation

Simulation Setup
The setup of the simulation is shown in Fig. 1(a). The origin of the coordinate system is set at the center of the film. The film has 50 µm-width and 60 µm-length. The film is set to be wide enough to reproduce the plane wave. The thicknesses of the left (x < 0) and right (x > 0) half of the film are d i = 60 nm and d r = 30 nm, respectively, and there is the "step" at the center (x = 0). The simulation cell size is 50 × 50 × 30 nm 3 , which is small enough to reproduce the SW with 5 µm-wavelength. Note that the film consists of 2 cells (1 cell x-direction. This configuration can excite the MSSW mode. We used two types of the SW antenna: "1-terminal antenna" for the calculation of the dispersion relation [ Fig. 1(b)] and 3-terminal antenna for the calculation with the step [ Fig. 1(c)]. In the case of "1-terminal antenna", we apply a radio-frequency (rf) magnetic field h = +h rf sin(2π f t)x along the x direction at "+" signed area (−30 µm ≤ x ≤ −29 µm) in Fig. 1(b). "3-terminal antenna", on the other hand, mimicked the coplanar waveguide, often used in the experiment [ Fig. 1(c)]. "3-terminal antenna" can excite the SW whose wavelength is exactly defined because of precisely Position, x (µm)  positioned three electrodes. In Fig. 1(c), it can excite 5 µmwavelength SW. At "+" signed area (−28 µm ≤ x ≤ −26 µm), rf magnetic field h = +h rf sin(2π f t)x is applied, and at "−" signed area (−30 µm ≤ x ≤ −29 µm and − 25 µm ≤ x ≤ −24 µm), rf magnetic field h = −h rf sin(2π f t)x is applied. The amplitude of the rf field is fixed: µ 0 h rf = 1 mT. In order to prevent SW reflections at the right edge, we set a high damping (α = 1) in x ≥ 25 µm. 20) We utilize mumax3 code [21][22][23] for the micromagnetic simulation.

Dispersion Relation
Since the "3-terminal antenna" is set to excite the SW with 5 µm-wavelength, it is necessary to find the frequency which can excite 5 µm-wavelength SW. We carried out the simulation in the frequency range from 7 to 7.5 GHz. We used "1-terminal antenna" and the thickness was 60 nm throughout the film (namely, there was no step) in this simulation. Dots in Fig. 2(a) show the waveform of the SW at f = 7.206 GHz. This waveform is obtained at the center of the film (y = 0) on the bottom layer, and the time is t = 10.00 ns. In order to evaluate the wavelength of the SW, we fit waveform in Fig. 2(a) with the product of sine curve and exponential decay as follows: where A is the amplitude, B is the decay constant, C and D are the offset for x and g direction, respectively. Fitting result is shown in Fig. 2(a) with solid line. The wavelength of the SW given by the fitting is λ = 5.00 µm at f = 7.206 GHz. Fig. 2(b) shows the frequency dependence of the wavelength. It is found in this calculation that the frequency which can excite 5.00 µm-wavelength SW is 7.206 GHz. Using the relation k = 2π/λ, the wavevector dependence of the frequency (namely, the dispersion relation) is calculated in Fig. 2

Simulation of the film with the step
We carried out the simulation with the step, using "3-terminal antenna". Figure 3 shows the waveform of the SW in the film with the step at f = 7.206 GHz. This waveform is obtained at the center of the film (y = 0) on the bottom layer, and the time is t = 9.06 ns. The wavelength of the SW in the left half (d i = 60 nm) is λ i = 5.00 µm. On the other hand, the wavelength of the SW in the right half (d r = 30 nm) is λ r = 2.59 µm. The wavelength of SW is almost halved after propagating through a half thick layer. Small discrepancy between expected value (2.50 µm) and the simulation result (2.59 µm) is due to roughness of the cell size in x-y direction. Figure 4 shows the snapshot of magnetization at f = 7.206 GHz and at t = 9.06 ns. It clearly shows the modulation of wavelength by passing though the step.
Note that the length of the wave front appears to shrink as the SW propagates. The reason of this phenomenon can be described as follows: T. Schneider et al. 24) pointed out that the SW excited  in Fig. 5 follows a specific crest of the wave. It seems that there is discontinuity in the wave amplitude at the step (x = 0). This can be explained as follows: Figure 5 shows the magnetization at the bottom layer only. Thickness of the left half is 60 nm, and that of the right half is 30 nm. The energy of the SW concentrates on the bottom layer in the right half when the SW propagates into the step. Hence, the amplitude of the SW increases at the right half. Note that the phase does not change in this process, as the wave crest remains unchanged after propagating through the step. In order to quantitatively evaluate the phase shift during propagating through the step, we fit the waveform with Eq.  Fig. 6(a) coincide with that of the SW excitation. The difference of the right and the left value is shown in Fig. 6(b). The difference is oscillating around 0, thus we conclude that there is no phase shift during propagating through the step.
Next, we investigated the phase velocity of the SW from the simulation result.

Summary
We have shown by micromagnetic simulation that the wavelength and phase velocity of the SW is modulated when the SW propagates across a step between two ferromagnetic films with different thicknesses under the condition that the incident angle is zero (θ i = 0). The result well follows the Snell's law. We have also shown that there is no phase shift during propagating across a step. We propose that the SW refraction can be observed experimentally in such an easy system with different film thickness. We investigated the SW propagation in the real space using the continuous wave. The result will serve as a guide for experiments in real space measurements, such as Brillouin Light Scattering Spectroscopy.