Hybridization‐Induced Spin‐Wave Transmission Stop Band within a 1D Diffraction Grating

Spin wave propagation is studied through a diffraction grating in a 200 nm thick YIG film by using scanning time resolved magneto‐optic Kerr microscopy (TR‐MOKE) and supported by micromagnetic simulations. Caustic‐like spin wave emission and the hybridization of Damon Eshbach (DE) type spin wave modes within the grating region, depending on the magnetic field and the dimensions of the grating, are observed. Hybridization leads to an increased attenuation length for propagating spin waves and consequently to a transmission stop‐band for spin waves at the grating for a certain magnetic field range.


Introduction
The manipulation and control of spin waves is at the heart of the field of magnonics which has become a topic of increasing interest in recent years. [1][2][3][4][5][6][7][8][9][10][11][12] Understanding the manipulation of propagating spin waves is necessary for the fabrication of potential magnonic devices. Magnonic waveguides, [13][14][15][16] periodic structures [17,18] as well as externally controlled magnetic anisotropy [15] were used to control the direction of propagation as well as band gaps for spin waves in ferro-or ferrimagnetic DOI: 10.1002/apxr.202200104 materials. To achieve these kinds of effects, one takes advantage of the wave nature of spin waves, which not only allows one to use wave variables such as wavelength, amplitude, and propagation direction as control knobs, but also to enter a caustic realm where it is possible to control the propagation direction of spin wave beams. [13,14,19] The causticlike regime is easily accessible for inplane magnetized magnetic thin films due to the anisotropic behavior of the dispersion relation for dipolar-dominated spin waves. [13] This anisotropy leads to Damon Eshbach (DE) (k⊥M) spin wave modes and Magneto-Static Backward Volume (BV) spin wave modes (k∥M) or a mixture if the in-plane magnetisation has an arbitrary angle with respect to these two symmetry directions. [20] It was already demonstrated that the direction of propagation of spin wave beams from a magnetic-stripe waveguide into an extended film can be controlled by varying the external magnetic field. [14,15] Furthermore, it was shown experimentally and theoretically that manipulating the propagation of spin waves can be achieved by patterning magnetic films. [17,[21][22][23] In this report, we demonstrate the conversion of a plane spin wave into caustic-like beams in a 200 nm thick patterned Yttrium Iron Garnet (YIG) film, where the pattern consists of square shaped antidots forming a spin wave diffraction grating. [21] This grating diffracts spin waves if the spin wave wavelength is smaller than the antidot size. On the other hand, if the spin wave wavelength is larger than the antidot size, the diffraction grating itself acts as a point-like spin wave source leading to emission of caustic-like spin wave beams. A spin wave propagation stop band through the grating can be achieved when locally changing the effective internal magnetic field between the square shaped antidots. We experimentally observe the propagating spin waves using time resolved magneto-optic Kerr microscopy (TR-MOKE). [24][25][26][27][28] The influence of the diffraction grating on the effective internal magnetic field is investigated by micromagnetic simulations. [29] Furthermore, the dispersion relation for DE and BV spin wave modes was calculated by the zeroth-order perturbation theory of Kalinikos and Slavin (KS-theory). [30] In addition to the KS-theory, we used the open-source finite-element micromagnetic code TetraX [31,32] to calculate the DE dispersion while considering the hybridization of DE-type spin wave modes. We first demonstrate that TetraX gives a suitable solution for the DE-dispersion for a 200 nm thick YIG film by comparing the analytic calculations with the experimentally observed dispersion. Moreover, we show that the transmission stop band occurs due to the hybridization of two DE-type spin wave modes leading to a reduced group velocity and subsequently to a reduced attenuation length. where the gray-scale corresponds to the z-component of the dynamic magnetization measured by TR-MOKE. To record an image of the dynamic magnetization at a fixed rf-phase, femtosecond laser pulses with a center-wavelength of 800nm and a repetition rate of 80MHz where phase locked to the frequency of the rfcurrent sent through a coplanar wave guide (CPW) which in turn generates the rf magnetic field locally exciting the magnetization. Plane dipole-dominated spin waves are excited within a 200nm thick YIG film through the CPW which is deposited directly on top of the YIG film, and schematically represented on the lefthand side of Figure 1a. The CPW consisting of 5 nm Cr and 100 nm Au was patterned by photo-lithography. The ground and the signal lines have a width of 5 μm and 10 μm, respectively, and are separated by 5 μm. Three different diffraction gratings were produced by photo-lithography and Ar etching through the entire 200 nm YIG film, corresponding to lattice constants of d = 10 μ , 20 μ m and 30 μ m. In each case, the antidot width was fixed to d/2. An example for spin waves propagating through the diffraction grating with d = 10 μ m is shown in Figure 1a. Here, a static external magnetic bias field of 0 H = 41mT was applied along the y-direction (Damon-Eshbach geometry) and the frequency of the rf-current was fixed to 2.8GHz. The plane spin wave propagates through the lattice of square shaped antidots, indicated by the black squares in Figure 1a.

Experimental Section
For this condition, we observe emission of spin wave beams from the diffraction grating not only in the positive x-direction, but also in the negative x-direction, back toward the CPW (in Figure 1a). This can be seen by the diamond-shaped spin wave pattern which interferes with the plane spin wave on both sides of the diffraction grating. Consequently, the intensity of the diamond-shaped patterns is modulated by the wavelength of the plane spin waves. We conclude that the regions between any two neighboring antidots acts as a point-like spin wave source as the spin wave wavelength > d/2. This area is a source of spherical wavelets due to Huygens principle. [33] However, due to the anisotropic dispersion relation for magnetostatic spin waves in in-plane magnetized thin films, we do not observe a spherical wavelet but a rather beam-like shaped spin wave emission from the area between the antidots. The k-vector distribution in the x-(k x ) and y-direction (k y ) is extracted by a two dimensional fast Fourier transformation (2D-FFT) from the dynamic component of the magnetization on the right-hand side of the diffraction grating (x > 0), see Figure 1b. The iso-frequency curve calculated using the KS-theory (red dashed line) is shown overlaid on the 2D-FFT data. Every vector which connects the origin of the coordinate system with any point on the iso-frequency curve, represents a specific allowed wave vector k. The direction of the group velocity v g for a specific wave vector k is given by the normal to the isofrequency curve at this point. A caustic beam appears at points where the curvature of the isofrequency curve formed in the wave-vector space is zero, resulting in a divergence in the power flow. [34] A so-called caustic point occurs when dk 2 x ∕dk 2 y = 0, which is out of the k-range which we present in Figure 1(b). Nevertheless, the slope of the iso-frequency curve in the k-range which we observe is close to linear which means that there are enough wave vectors k with a similar direction of the group velocity v g to form spin wave beams. As a result, we will refer to these spin wave beams as caustic-like spin waves.
The absolute value of k and in-turn the wavelength can be tuned by varying the strength of the external magnetic field at a given rf-frequency. Figure 2a-d illustrates the effect of the diffraction grating with d = 10 μm on plane spin waves for different external magnetic field values. The spin wave transmission through the diffraction grating is highly suppressed for a certain magnetic field range. This can be seen in Figure 2b,c in which the spin wave amplitude on the right-hand side of the grating (x > 0) is close to or even zero. Furthermore, it can be seen that the formation of a caustic-like spin wave beam becomes less prominent for higher magnetic field values, see Figure 2a, as there are fewer wave vectors with the same group velocity direction.
To gain further insight into the propagation of the spin waves between the antidots, we additionally measured the propagation through a diffraction grating with d = 130 μm. For several external magnetic field values, a waist-like shape of the spin wave pattern is observed between neighboring antidots. At μ 0 H = 33.0mT strong attenuation of the spin wave occurs between the antidots resulting in almost complete suppression of the spin wave transmission, Figure 2g.
The effective magnetic field for the d = 10 μm and d = 30 μm gratings were simulated for an external magnetic field of 0 H = 33.0mT by micromagnetic simulations, [29] see Figure 2e,j. We observe that the effective magnetic field between the antidots is clearly reduced due to magnetic surface charges at the edges of the antidots. Comparing the iso-field lines in Figure 2e,j (black solid lines) between the antidots, we can see that they have a similar waist-like shape as the measured spin waves in In magnetic films it is well-known that perpendicular standing spin wave (PSSW) modes can be strongly coupled by dynamic dipolar fields. PSSWs are spin waves which are confined within the thickness of the magnetic film as indicated in the inset of Figure 3. This coupling may lead to a hybridization and consequently to an anti-crossing between different PSSW branches of the spin wave dispersion. As a result, the slope of the dispersion in the vicinity of the anti-crossing approaches zero and in turn the group velocity as well. [30,35,36] Note that since the spin wave attenuation length is directly proportional to the group velocity, we expect the excited spin waves to be rapidly attenuated. The strength of the anti-crossing defines the coupling strength between the spin wave modes. Figure 3 presents the calculated dispersion relation of DE and BV spin waves by using the zerothorder perturbation theory by Kalinikos-Slavin (KS-theory). [30] The dispersion relations were calculated for μ 0 H = 31 mT static magnetic field of the first two modes, namely the homogeneous and first PSSWs in the t = 200 nm thick YIG film. Note that a potential mode crossing of the DE-type n = 0 and n = 1 spin wave modes at about 1.4 rad μm -1 and 2.8 GHz for the dispersion calculated by the KS-theory. However, this mode crossing of the DE-type modes can lead to a hybridization and in turn to a reduced attenuation length, as described above. Considering the coupling between these modes requires taking higher order terms within the KS-theory into account requiring a  [30] A crossing of the n = 0 and n = 1 DE-type modes is observed at a frequency of 2.8 GHz and a k-value of about 1.4 rad μm −1 whereas there is no crossing for the BV spin wave modes. b) Calculated spin wave dispersion relation for DE spin waves, using TetraX. [31,32] The coupling between the n = 0 and n = 1 DE-type modes leads to a hybridization and consequently to the avoided mode crossing. The single PSSW modes can hardly be distinguished in the vicinity of the avoided level crossing as a consequence of the hybridization.
considerable computational effort. The open-source finiteelement micromagnetic code TetraX, [31,32] on the other hand, provides a comfortable method for calculating the dispersion relation including the coupling between the n = 0 and n = 1 PSSW modes, resulting in the anti-crossing behavior of those spin wave modes, shown in Figure 3b. Recently, the propagating-wave dynamic-matrix approach was extended for mono-and multilayers and thus TetraX can be used to calculate the dispersion relation for films with a finite thickness, or layers of films with varying thickness and spacing. To obtain the spin wave frequencies and spatial mode profiles for a given wave vector, the linearized equation of motion of magnetization is discretized on a line-trace mesh along the thickness of the film, and subsequently solved numerically as an eigenvalue problem. This approach yields the exact hybridized normal modes of the magnetic system.
To observe a potential hybridization of the two DE-type spin wave modes, we measured spin wave propagation in the xdirection as a function of the external magnetic field in the plain YIG film, far away from any diffraction grating, see Figure 4a. We observe an external magnetic field range around 31 mT in which the spin wave propagation is highly attenuated. We extracted the absolute value of the wave vector k from these measurements by fitting a sinusoidal function to the propagating spin waves in the x-direction for every applied external magnetic field value. With this, we extracted the spin wave wavelength for the respective magnetic field values and calculated the absolute values of the wave vector by using the relation |k| = 2 . These k-values are indicated by the black spheres shown in Figure 4b. Comparing this experimental data with the calculated dispersion relation by using TetraX in Figure 4b shows that the avoided crossing measured by TR-MOKE is in good agreement with the calculations (red spheres). This proves that the high attenuation observed around 31 mT in Figure 4a originates from the coupling and hybridization of the two DE-type spin wave modes. It furthermore confirms that the spin wave transmission stop band at the diffraction grating occurs as a result of the locally reduced www.advancedsciencenews.com www.advphysicsres.com effective field between the antidots. The reduced effective field locally shifts the dispersion relation into the hybridization area for a corresponding external magnetic field range and with this to the high attenuation regime.

Conclusion
In conclusion, we demonstrated the behavior for plane dipolardominated spin waves propagating through a spin wave diffraction grating. The regions between the antidots act like point-like spin wave sources if the wavelength of the spin waves becomes equal to or larger than half the lattice constant d. This can be seen by the emission of caustic-like spin wave beams in both the positive and negative x-directions. Diffraction effects are only observable on the right-hand side of the diffraction grating for spin waves with a wavelength smaller than half the lattice constant d. In addition to the observed diffraction phenomena, we observed a spin wave transmission stop band through the grating for a certain external magnetic field range. This was demonstrated for two different lattice constants (d = 10 μm and d = 30 μm) and the magnetic field range for the stop bands was slightly different for the two gratings. It was found that the hybridization of two DE-type spin wave modes, in combination with the reduced effective internal magnetic field between the antidots, caused this effect. We demonstrated that the hybridization of two spin wave modes can lead to an enhanced attenuation, and by locally changing the effective magnetic field even to a stop-band condition for propagating spin waves. Furthermore, we compared finally the calculated dispersion relation by using the open-source TetraX package, as well as using Mumax for simulations of the effective magnetic field, with our experimental data and found that they are in qualitative agreement. No transmission stop band was observable for BV spin waves in our experiments (see supplementary material) consistent with the theory as we do not expect a mode crossing in the dispersion relation for the n = 0 and n = 1 BV modes. Furthermore, the effective magnetic field between the antidots is barely reduced as the external magnetic field points perpendicular to the grating directing for the BV geometry.

Supporting Information
Supporting Information is available from the Wiley Online Library or from the author.