Field-controlled rotation of spin-wave nanochannels in bi-component magnonic crystals

Bi-component magnonic crystals (BMCs) consisting of two different periodically arranged magnetic materials have attracted great interest in recent years [1–10]. Two-dimensional (2D) arrays of chessboard-like arranged micromagnets or ferromagnetic nanodiscs in a ferromagnetic thin film have been explored [5, 8] For magnetic fields H applied in the sample plane, 2D arrays have been found to exhibit an artificially tailored band structure for spin waves [8, 9]. Here, forbidden frequency gaps reflect the difference between material parameters such as the saturation magnetization (M_S) and the exchange stiffness constant (A). So far, however, investigations were focused on a single magnetic field orientation collinear with a high-symmetry crystal direction. In previously studied magnonic crystals consisting of magnetic antidot lattices (ADLs), i.e. 2D arrays of air holes in a ferromagnetic film, the rotation of H varied the nature of spin-wave modes significantly [11]. Due to the inhomogeneous internal field H_int and geometrical boundaries [12], spin-wave channels were found to extend through the lattice only for selected orientations of H, otherwise localization and quantization occurred [13–15]. A BMC where one avoids air holes to create an artificially tailored band structure, might offer different characteristics but has not yet been explored in tilted in-plane fields. In addition, as experimentally demonstrated by Demidov and his co-workers in individual transversely magnetized waveguides [16, 17], it is possible to channel, manipulate and split spin-wave beams for possible application in microwave-frequency signal processing and spin-wave based nano-optics devices.

In this paper, we report a combined study of all-electrical spin-wave spectroscopy (AESWS), Brillouin light scattering (BLS), and dynamic micromagnetic simulations. We address BMCs which are formed by Permalloy (Ni₈₀Fe₂₀) thin films incorporating a periodic square lattice of Co nanodiscs. Lattice constants range from 200 to 1000 nm. For all devices, we find the formation of spin-wave channels for a field orientation for which spin-wave localization was decisive in ADLs. Bi-component devices thus help us to avoid the reported mode conversion from propagating to fully-quantized spin-wave modes in 2D magnonic crystals [15]. BMCs offer new perspectives for the optimization of spin-wave based signal transmission in magnonic applications and magnetic logic [18–20].

The BMCs were prepared on semi-insulating GaAs substrates. First, a 300 µm by 120 µm large Permalloy (Py) mesa was deposited extending in the xy-plane. Using electron beam lithography, a square lattice of circular holes was exposed in a resist on top of the mesa and then developed (see figure 1(a)). The resist mask was used for Ar milling of nanotroughs into the Py in z-direction followed by in situ deposition of Co. After lift-off processing Co nanodiscs were embedded in Py as sketched in figure 1(b). Here, we report experimental data obtained on two BMCs. Considering figure 1(b), the relevant parameters were for sample #A, t_mesa = 26 nm, t_e = 7 nm, t_d = 15 nm, d = 435 nm and p = 1000 nm; for sample #B, t_mesa = 24 nm, t_e = 8 nm, and t_d = 15 nm, d = 310 nm and p = 600 nm (compare figure 1 of [8]). On top of the BMCs, we deposited an insulating layer of either 4 nm thick SiO₂ or up to 8 nm thick Al₂O₃ using magnetron sputtering and atomic layer deposition, respectively.

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Finally, coplanar waveguides (CPWs) were prepared using thin films of Cr, Ag and Au. The width of the signal lines was 2 or 4 µm³. Using markers, we aligned the inner conductor of the CPW with one side of the cubic unit cell of the BMCs. A vector network analyser was connected to the CPWs to measure the real part of the scattering parameter S₁₁, i.e. in reflection configuration. Using a 2D vector magnet, we varied the field orientation in the xy-plane to explore the spin-wave eigenfrequencies f as a function of η (figure 1(c)). From each S₁₁ spectrum, we subtracted a reference spectrum acquired at η = 90° and μ₀H = 100 mT, where we expected vanishing spin excitation. This procedure provided us with the AESWS difference signal ℜ(a₁₁) discussed here. We used a field value of μ₀H = 20 mT being beyond switching fields of the Py mesa and Co nanodiscs⁴. The magnetization M is thus expected to follow H. The CPWs explored a narrow stripe of the BMC. To address f(η) on a larger sample area, we performed inelastic light scattering experiments by BLS using laser light at a wavelength of 532 nm with a spot diameter of about 30 µm [21]. In this work BLS probes the thermally excited spin waves. We applied the backscattering configuration for laser light focused at normal incidence upon the sample surface, i.e. we addressed spin-wave excitations with zero wave vector. Spectra were detected using a Sandercock type (3+3)-pass tandem Fabry–Pérot interferometer. Micromagnetic simulations based on the MicroMagus software package were performed to model the angular dependence f(η) [22]. We considered the experimentally determined geometrical parameters, and for Py M_S;Py;#A = 806 kA m⁻¹ (M_S;Py;#B = 780 kA m⁻¹), and exchange constant A_Py = 13×10⁻¹² J m⁻¹. For Co, we took M_S;Co;#A = M_S;Co;#B = 1000 kA m⁻¹ and A_Co = 20 × 10⁻¹² J m⁻¹. We attribute the remaining discrepancy in M_S;Py for samples #A and #B to the different thickness modifying the contribution of the surface anisotropy. We considered a damping constant α = 0.01 and no further magnetic anisotropies reflecting the polycrystalline nature of the films. Spin precession was induced by a uniform 4 mT strong field pulse (with a Gaussian temporal profile) applied in η = 90° direction in-plane and 45° out-of-plane in order to reproduce out-of-plane components of the CPW excitation. In each simulation, one unit cell with 2D periodic boundary conditions was simulated. To explore a BMC patterned on a deep-submicrometre length scale we simulated sample #C with p = 200 nm and d = 87 nm. The lateral side length of a simulations cell varied from 15.625 nm for sample #A to 3.125 nm for sample #C. In vertical direction the samples were subdivided in three layers following [8].

In figures 2(a) and (b) we show experimental data obtained on the p = 1000 nm and p = 600 nm BMC, respectively. The grey-scale plots reflect AESWS data where dark colour stands for absorption due to spin-wave excitation. White circles reflect BLS data. Overall, we find a good agreement between both experimental methods. BLS is found to be more sensitive to the highest frequency mode where the signal detected by AESWS is almost vanishing. The remaining discrepancies near the local minima of the low-frequency branches are attributed to the misalignment of the BMC unit cell and the field direction in the BLS experiment. In the following, we concentrate on the two prominent branches (labelled in figure 2(a)), which were often discussed for ADLs. The most pronounced angular dependence is found for the low-f branch. The variation of f(η) for the p = 600 nm BMC is larger compared to the p = 1000 nm BMC. In contrast to data reported for ADLs, the signal strength of the low-f branch does not depend very much on η. For the high-f branch the intensity is found to vary markedly with η. In figures 2(c) and (d) we show results from the corresponding micromagnetic simulations. The angular dependencies f(η) are quantitatively and qualitatively well reproduced. Simulated and experimental spectra show local minima for the eigenfrequencies if H is along high symmetry directions at η = 0° and 45°. In the micromagnetic simulations, the variation in f(η) is found to be larger for the p = 600 nm BMC (#B) compared to p = 1000 nm (#A), consistent with the experimental observation. At the same time, the intensity of the low-f branch does not vary much with η. The almost constant intensity of the low-f branch is also found for the p = 200 nm BMC (#C) shown in figure 2(e). Here, the smallest eigenfrequencies at η = 0° and 45° are almost degenerate.

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A further local minimum exists in the low-f branch at about η = 22°. A closer inspection in figure 2(f) shows that, surprisingly, the low-f-branch frequency at η = 22° does not depend much on p for p = 1000 and 600 nm while it is smaller for p = 200 nm. Also the high-f-branch frequency does not vary much between p = 1000 and 600 nm at this intermediate angle. At η = 0° and 45°, the miniaturization of the lattice constant p is found to change eigenfrequencies much and enhance considerably the frequency splitting between the low-f and high-f branches.

In figure 3 we show spatial profiles of simulated spin-precession amplitudes for resonance frequencies which are marked by open squares in figures 2(c)–(e). Consistent with [4] for η = 0° and p = 1000 nm, regions of largest spin-precession amplitude are channel-like and extend throughout the lattice. The channels are in particular perpendicular to the field H.

The channels go through the Co nanodiscs for the low-f excitations. For the high-f branch, the channels are complementary and shifted by half the lattice constant in x direction. These channels extend almost exclusively through the Py; Co nanodiscs are avoided. The channels reflect the inhomogeneity of H_int as elaborated in [4]. These systematics hold true for p = 600 nm and p = 200 nm (left column of figure 3). In figure 3(p), the width of the spin-wave channel of sample #C is slightly smaller than 100 nm. At η = 45° (right column), the channels in all three samples rotate by the same angle and thus remain perpendicular to H. Spatial profiles of spin-precession amplitudes are again complementary when the low-f and high-f modes are compared for one-and-the-same BMC. The exact transverse widths of the channels depend on p and η. The interesting case of η = 22° is shown in the central column of figure 3. At this angle, H does not point in a high symmetry direction of the square lattice. For this angle, low-frequency modes have been found to be fully localized in ADLs consisting of a periodic lattice of air holes [23, 24]. This was due to both the inhomogeneous H_int and the geometrical boundary conditions (confinement) imposed by the holes. For the BMC, we find a completely different behaviour in that relatively wide channels of large spin-precession amplitudes extend through the whole lattice. Interestingly, the eigenfrequencies of precession profiles in figures 3(b), (h) and (n) are found to be nearly degenerate though p varies by a factor of 5.

Low-f and high-f modes show somewhat complementary profiles for p = 1000 and 600 nm. For p = 200 nm only the low-f mode is extracted from the simulations at η = 22°. For all three BMCs, the intensity of the low-f branch does not vary much for 10° < η < 40° in figures 2(a)–(e), suggesting extended spin-wave channels to exist also for the further angles. Using the BMC, one thus circumvents the strong localization of spin-wave modes known from ADLs. Still, the singly-connected ferromagnetic material of the BMC shows a different mode configuration compared to a plain film in that the spin-wave channels are formed. In the channels, the propagating spin waves represent relatively fast waves⁵. Their group velocity might be higher compared to Damon–Eshbach like modes (k perpendicular to M) in wider nanochannel [17] or etched nanochannels of the same width [24] considering the different boundary condition as suggested in [28]. Depending on the exact composition and filling fraction of the second ferromagnetic material in the unit cell, we expect BMCs to provoke characteristic caustic effects different from the plane film [25–27] when a spin-wave beam is injected into the BMC for a specific η.

In conclusion, we have reported spin-wave excitations in BMCs for different field orientations. For the BMCs, we embedded a periodic lattice of Co nanodiscs into a Permalloy thin film. The BMCs allowed us to rotate spin-wave channels over a much larger angular regime compared to results known from previously investigated magnetic antidot lattices. Such rotatable channels support relatively fast Damon–Eshbach-type spin waves. The findings allow one to optimize signal transmission in different spatial directions making use of bi-component magnonic crystals in rotated fields. Depending on the excitation frequency, complementary channels can be functionalized.

The research leading to these results has received funding from the European Community's Seventh Framework Programme (FP7/2007-2013) under Grant Agreement No 228673 MAGNONICS, from the German Excellence Cluster Nanosystems Initiative Munich (NIM) and from the Ministero Italiano dell'Università e della Ricerca (MIUR) under PRIN Project No 2010ECA8P3 'DyNanoMag'.