Electrical spin-wave spectroscopy in nanoscale waveguides with nonuniform magnetization

Spin waves modes in magnetic waveguides with width down to 320 nm have been studied by electrical propagating spin-wave spectroscopy and micromagnetic simulations for both longitudinal and transverse magnetic bias fields. For longitudinal bias fields, a 1.3 GHz wide spin-wave band was observed in agreement with analytical dispersion relations for uniform magnetization. However, transverse bias field led to several distinct bands, corresponding to different quantized width modes, with both negative and positive slopes. Micromagnetic simulations showed that, in this geometry, the magnetization was nonuniform and tilted due to the strong shape anisotropy of the waveguides. Simulations of the quantized spin-wave modes in such nonuniformly magnetized waveguides resulted in spin wave dispersion relations in good agreement with the experiments.

Spin waves are collective excitations of the magnetization in ferromagnetic materials with typical frequencies in the GHz and wavelengths in the nm to µm ranges. Due to their low intrinsic energies, they have recently received increasing interest for ultralow power spintronic applications, 1,2 in particular in Boolean logic circuits 3-6 as well as analog 7 or neuromorphic computing 8 concepts. Recently, several of the proposed device concepts have been realized experimentally. [9][10][11][12][13] To be competitive with conventional electronics, the logic element density in a spintronic circuit must be high and therefore spin-wave devices must be miniaturized to nanoscale dimensions. At the nanoscale, the properties of spin waves are strongly affected by quantization and pinning conditions at the device edges. Spin-wave devices are commonly operated under transverse applied bias fields. However, in this configuration, the magnitude of the shape anisotropy field in nanoscale magnetic devices (in particular in long waveguides) can become comparable to typical bias fields (on the order of hundreds of mT), which leads to a nonuniform magnetization orientation inside the structure. This has a strong impact on the relative excitation efficiency of different quantized spin-wave modes as well as on their propagation. 14 Confined spin waves in uniformly magnetized nanoscale structures have been studied by micromagnetic simulations and imaged by optical and x-ray-based techniques, [15][16][17] revealing the formation of quantized width modes. However, all-electrical experiments [18][19][20][21] at the nanoscale are still lacking due to challenges to detect spin-wave signals in reduced magnetic volumes.
To date, little work has been dedicated to scaled waveguides with nonuniform magnetization. In this letter we report an all-electrical study of spin waves in magnetic waveguides with widths down to 320 nm. We show that the nonuniformity of the anisotropy field strongly affects spin wave transmission spectra for transverse magnetic bias fields whereas the effects are much less pronounced for longitudinal bias fields. The experimental results are in good agreement with micromagnetic simulations.
The studied structures consisted of multiple parallel Ta/Co 40 Fe 40 B 20 /Ta ferromagnetic waveguides of 50 µm length. The number of waveguides was chosen so that the total magnetic width was about 5 µm to ensure sufficient electrical read out signals. 18,20,22  4.3 × 10 −3 and a Landé factor of g = 2.07. 13,23 Spin waves were excited and detected by 500 nm wide Au inductive antennas, electrically connected to coplanar microwave waveguides.
A schematic of the device structure is shown in Fig. 1.
Electrical measurements of the devices were performed using a Keysight E8363B network analyzer (output RF power 0.25 mW). Both the reflected (S 11 ) and transmitted (S 21 ) power was recorded vs. magnetic bias field and frequency. Figure 2 shows the bias-field derivative of S 21 for a device including 20 waveguides (320 nm width, 180 nm gap) and longitudinal bias fields H 1 along the x-direction (cf. Fig. 1). In this geometry, spin waves are of the backward volume type. 24 The data indicate a spin-wave transmission band between 12.5 and 13.8 GHz close to zero bias field, which progressively shifts to higher frequencies with increasing H 1 .
The strong shape anisotropy leads to uniform magnetization in the waveguides even at very low magnetic fields, leading to the observation of the spin waves at magnetic fields close to zero. The spin-wave frequency bandwidth of about 1.3 GHz is in agreement with the spin-wave dispersion relations 24 and is limited by the FMR frequency and the wavenumber spectrum determined by the antenna width. 19 Considering the bias field and the lateral confinement in the waveguide, analytical calculations and micromagnetic simulations (not shown) indicate that the measured signal corresponds to the propagation of the first-order quantized width mode n 1 . The frequency band of the third-order width mode n 3 is expected about 7 GHz above the n 1 band and was not visible in the experiment. Note that the antenna can excite only odd width modes in this configuration. 25 The behavior was much more complex for transverse applied bias fields H 2 in the Damon- Eshbach geometry. Experimental field-frequency maps of the bias-field derivative of S 21 are shown in Fig. 3 for sets of waveguides with different widths between 320 nm and 2.5 µm.
The data reveal several spin-wave branches with positive and negative slopes depending on the bias field magnitude. A feature of all maps is a frequency minimum at a characteristic bias field H m that increases with decreasing waveguide width. The projection of the signal to H 2 = 0 leads to a frequency of ∼ 13.8 GHz for the narrowest waveguides (320 nm width, Fig. 3(a)), in agreement with the FMR frequency found in the backward volume configuration at zero bias field (cf. Fig. 2). This suggests that the strong shape anisotropy  Fig. 2 slightly different magnetization and internal field orientation due to different stray fields.
As a consequence of the internal field distribution, the magnetization was oriented at an average angle around or below θ m = 45 • with respect to the x-axis for fields µ 0 H 2 < 105 mT (see Fig. 4(a)). Only higher bias fields compensate the shape anisotropy and allow for a transveral magnetization orientation of the waveguide. Figures 4(b) and (c)  Furthermore, its intensity was modulated due to the coherent superposition of higher-order width modes. Steady-state snapshot images of the magnetization dynamics for excitation The simulations can explain well the experimental results in Fig. 3. For longitudinal bias fields, the behavior is similar to that confined backward volume spin waves in uniformly magnetized waveguides (cf. Fig. 2). However, for transverse bias fields, the behavior is strongly modified. At small H 2 , the transverse internal field is weak and the magnetization is still oriented along the waveguide. As a result, several width modes of backward-volume-like spin waves are excited. Increasing H 2 leads to a decreasing effective field in the x-direction and decreasing spin-wave frequencies. Above a critical field H m , the magnetization becomes transverse and the internal field further increases with H 2 , leading to upward frequency shift of the spin wave band. In both regimes, higher-order width modes are also visible. Below H m , a series of width modes (including even ones) is observed. Note that the critical fields at minimum frequency are band dependent and shift to higher values with increasing mode number since each mode has different quantization conditions. H m also strongly depends on the waveguide width, decreasing with increasing width (cf. Fig. 3) due to reduced shape anisotropy that allows for the saturation of wider waveguides at lower transverse bias fields.
Above H m , the lowest band corresponds to n 3 for the narrowest waveguide width of 320 nm; for wider waveguides, additional higher-order bands are observed below the firstorder mode n 1 , as shown in Figs. 3(b) to (d). This is qualitatively in agreement with the dispersion relations of spin waves in uniformly magnetized waveguides 24 . Nonetheless, the strongest band corresponds in all cases to the first-order mode n 1 . The frequency bandwidth of n 1 increases for wider waveguides, in keeping with the increasing slope of the dispersion relation. The n 1 band is modulated by the coherent superposition of higher-order modes, whose dispersion relations cross that of n 1 , as shown in Fig. 5(b).
The results indicate that the spin-wave spectra in magnonic devices operated in the Damon-Eshbach geometry are strongly modified when the structure widths reach values of several 100 nm. Since magnonic logic devices have begun to reach such dimensions 13 the modified mode behavior becomes increasingly relevant for the design of next generation of devices and the analysis of their behavior. In addition, the performance of spin-wave transducers is also strongly affected since the excitation efficiency of a spin-wave mode depends on the overlap integral of its magnetization distribution with the exciting (effective) magnetic field. The results thus provide a pathway to understanding the behavior of complex spin-wave devices with dimensions at which the demagnetizing field is comparable to typical bias fields.