Propagation of spin waves through a Néel domain wall

Magnonics offers a promising concept for devices with the potential to go beyond CMOS technology in cases when low-power computing is desired as Joule losses are effectively circumvented by the nature of the operation of magnonic devices. It was shown that it is possible to fabricate an all magnon transistor¹ and spin-wave logic gates based on a Mach–Zehnder interferometer.^2–5 These gates rely on a controlled phase shift of spin waves, which might be achieved, e.g., by magnetic domain walls (DWs).^6–9 

For this reason, the propagation of spin waves through domain walls (DWs) is of great interest. Recently, it has been shown that domain walls can serve as a spin-wave valve depending on their magnetic state (head-to-head and tail-to-tail spin configurations).¹⁰ In samples with perpendicular magnetic anisotropy, it was demonstrated that DWs can affect both the spin-wave amplitude and phase and that spin waves can move the DW by spin torque phenomena.^11–14 Moreover, DWs may also serve as a source and probe of spin waves with short wavelengths¹⁵ or can be used as magnonic conduits by themselves.^16,17 However, there was no experimental study on the spatial evolution of spin waves transmitted through a DW in a magnonic waveguide. A thorough understanding of the behavior of spin waves in magnonic waveguides hosting DWs is necessary for designing future integrated magnonic circuits as they present the most elementary building block of magnonic circuitry.

In this study, we focused on spatial imaging of spin-wave propagation through a Néel domain wall confined in an in-plane magnetized magnonic waveguide. To prepare a sample suitable for this study, we used a metastable fcc $Fe78Ni22$ thin film with a nominal thickness of $16 nm$⁠. The films were grown on a Cu(001) single crystal substrate under UHV conditions by evaporation from an $Fe78Ni22$ rod (⁠$2 mm$ thick, purity 99.99%, heated by electron bombardment) at a residual CO pressure of $5×10−10 mbar$⁠.¹⁸ After deposition, the sample was transferred to a FIB-SEM microscope for FIB irradiation. We oriented the long axis of the waveguides along the fcc [010] direction of the Cu substrate in order to obtain the highest possible magnetocrystalline anisotropy.¹⁹ To imprint the anisotropy direction perpendicular to the long axis of the waveguide, we wrote the structures with a single pass of a 30 keV Ga⁺ ion beam (30 nm spot, beam current of 150 pA, and 5 μs dwell time) with the fast scanning direction rotated by $80°$ from the waveguide's long axis. The resulting ion dose of 4 × 10¹⁵ cm^–2 has given reliable growth conditions for all the waveguides.

The produced waveguides had an effective anisotropy field of 40 mT with an in-plane easy axis perpendicular to the waveguide's long axis [see Fig. 1(a)]. The effective anisotropy field is the result of the two competing contributions—the uniaxial magnetocrystalline anisotropy coming from the FIB writing process and the shape anisotropy having its origin in the waveguide's demagnetizing field. This configuration allows for spin wave propagation in Damon–Eshbach (DE) geometry in zero external magnetic field and for stabilization of a domain wall in the vicinity of a spin-wave source. Spin waves were excited by a strip line antenna placed on top of the waveguide [see Fig. 1(a)]. The 1 μm wide antenna, made of a multilayer structure consisting of a $20 nm SiO2$ insulating layer, $5 nm$ Ti, $85 nm$ Cu, and a $10 nm$ Au capping layer, was patterned by electron beam lithography using PMMA resist, followed by e-beam evaporation of the layers and the subsequent lift-off process. The desired two-domain state of the waveguide was obtained by applying an external magnetic field in the $+y$ direction while observing the magnetization using a Kerr microscope (evico magnetics GmbH). The DW was positioned at a distance of approximately 3 μm from the antenna in order to obtain a sufficient Brillouin light scattering microscopy (BLS) signal on both sides of the domain wall. The position of the domain wall was stable in zero external magnetic field with a depinning field of $1.0±0.1$ mT (obtained by Kerr microscopy experiments).

To further reveal the internal structure of the DW, we performed magnetic force microscopy measurements (Bruker Dimension Icon, ASYMFMLM probe tip), supplemented by micromagnetic simulations using MuMax3 software²⁰ and the following parameters: a simulation area of $3×16 μm2$⁠, a cell size of $4×4×16 nm3,Ms=1409 kA/m, Ku=28.9 kJ/m3, Aex=11 pJ/m$⁠. These material parameters were obtained from fitting the spin-wave dispersion measured by phase-resolved BLS using the model of Kalinikos and Slavin²¹ as described in Ref. 22. We also confirmed that a simulation with a twice smaller cell size gives identical results. From the results shown in Figs. 1(b) and 1(c), it is apparent that the domain wall is a symmetric Néel wall with a circular Bloch line (vortex) in the middle.^23,24 The Bloch line is enforced by the surrounding topology (antiparallel magnetization along the two edges). The occurrence of the Bloch line results in a better dipolar energy minimization between the charges of the wall and those on the sides of the strip.

Spin-wave propagation was studied using micro-focused Brillouin light scattering (μBLS).²⁵ In order to get better insight into the internal magnetic field structure, we measured thermally excited spin waves while scanning the μBLS spot (⁠$300 nm$ in diameter) in the center of the waveguide, across the DW [Fig. 1(d)]. The total length of the line scan was 2 μm, with a spatial step of 50 nm, and the acquisition time for a single spectrum was $240 s$⁠. We observe an increase in spin-wave frequencies by approximately $2 GHz$ in the DW region (the DW is located at x = 0 μm) in comparison to the uniformly magnetized area. This increase can be attributed to an approximately $15 mT$ increase in the local effective magnetic field. The effective magnetic field obtained from the micromagnetic simulation is plotted together with the thermal spectra [Fig. 1(d), red curve]. Note that at the exact position of the DW, the effective field exceeds $600 mT$⁠, but this peak is spatially restricted only to a region of approximately $10 nm$ (an order of magnitude below the μBLS spatial resolution), and its contribution is not present in the acquired spectra.

In the second experiment, we excited spin waves by passing a RF signal (⁠$5 dBm$⁠, 7–10 GHz, frequency step $50 MHz$⁠, and the acquisition time for a single spectrum was $150 s$⁠) through the microwave antenna. The spin-wave intensity for each excitation frequency was acquired at two positions: the first position was $x=−0.5$ μm (before the DW) and the second position was x = 0.5 μm (after the DW). The resulting spectra are shown in Fig. 1(e). Based on these measurements, we selected two frequencies (⁠$7.15 GHz$ and $9.00 GHz$⁠, indicated by red arrows), where we observed an increased spin-wave intensity at both positions, before and after the DW.

For the selected frequencies, we performed intensity- and phase-resolved μBLS experiments. The measurements were done for the magnonic waveguide in a single domain state (SD) and in the DW state (here, the DW state means the two-domain state separated by the Néel domain wall). For the SD state, we observe only the central waveguide mode [Fig. 2(a)]. The corresponding spin-wave phase (⁠$Φ$⁠), obtained by the technique described in Refs. 26 and 27, shows a linear increase in the region −1 μm < x < 1 μm [Fig. 2(d), blue region] as expected for a propagating wave. Outside of this region, the phase could not be reconstructed due to either nonlinear spin-wave behavior (closer to the antenna) or insufficient spin-wave signals (farther from the antenna). We determine the wavelength of the excited spin waves from the slope of the fitted line [Fig. 2(d), red line], $λ=4.5±0.2$ μm, which is in agreement with the calculated dispersion relation.²¹ The propagation of the spin waves excited at 7.15 GHz through the DW is shown in Fig. 2(b). The spin-wave propagation pattern before the DW resembles closely the one seen in Fig. 2(a). After passing the DW, the spin waves are split into two beams. A similar effect of splitting into two beams was observed in a different system by Demidov et al.^28,29 Note that the spin-wave signal is suppressed more strongly at the position of the circular Bloch line, compared to the sides of the domain wall. The phase evolution in the lower spin-wave beam [Fig. 2(e)] exhibits a clear discontinuity; a phase shift of approximately $0.6π$ is observed at the DW position. For the second measured frequency of 9.00 GHz, the 2D spin-wave intensity map appears remarkably different. In this case, the transmitted spin wave is confined to the middle of the waveguide. Also note that the region of the suppressed spin wave signal at the DW position is wider for this frequency than in the case of excitation at $7.15 GHz$⁠. The corresponding spin-wave phase, shown in Fig. 2(f), exhibits a phase shift of approximately $0.5π$ at the position of the DW (x = 0 μm).

To further understand this frequency-dependent behavior and phase evolution, we conducted dynamic micromagnetic simulations, using the same material parameters as those for the calculation of the static magnetization configuration. Spin waves were excited by a spatially varying field (the field from the rectangular antenna was calculated by Finite Element Method Magnetics - FEMM software³⁰). To prevent reflections at the ends of the simulated waveguide, we implemented regions with increasing damping at both ends of the waveguide. A continuous RF excitation was applied during the initial 20 periods. Then, the spatial distribution of the magnetization was sampled for two periods with a time step of $2 ps$⁠. The magnetization was transformed to in-plane and out-of-plane components, and the spin-wave phase was calculated using the procedure described in Ref. 31. The results of these simulations are shown in Figs. 3(a)–3(c) in the form of the time-averaged squared out-of-plane component of magnetization. Panel (a) shows the spin-wave propagation in the SD state. Panel (b) shows the spin wave-propagation through the DW at the excitation frequency of 7.15 GHz and panel (c) at the frequency of 9.00 GHz. We can observe qualitatively the same behavior of the spin wave propagation before and after DW as in the experiment. We do not see the variations of the spin wave suppression within the DW, contrary to what we observed in the experiments.

Simulated phase maps corresponding to the previously described intensity maps are shown in Figs. 3(d)–3(f). For the waveguide in the SD state, we can observe that the phase is almost constant across the y coordinate of the μBLS spot [see green dots in Fig. 3(d) and the black line in Fig. 3(g)]. On the other hand, the phase after passing the DW is much more distorted across a relatively small distance [for $7.15 GHz$ see green dots in Fig. 3(e) and the red line in Fig. 3(g)] and the same applies to the $9.00 GHz$ spin wave [see Fig. 3(f), green dot and Fig. 3(g), blue line]. This distortion explains why we were not able to see the linear phase evolution in our experiments. We were probing the spin waves with a $300 nm$ μBLS spot, and the phase is changing by more than $0.5 rad$ within the spot size.

To check if a change in the internal configuration of the domain wall can be used to modify the spin-wave propagation, we performed a simulation where we applied a small external magnetic field of 5 mT along the x direction. This moves the circular Bloch line in the y-direction toward the edge of the waveguide [see Fig. 4(a)]. The field dependence of the Bloch line position along the y axis is shown in Fig. 4(b).

This displacement of the circular Bloch line changes the transmitted spin-wave profile significantly. In Fig. 4(c), we can observe that for 7.15 $GHz$⁠, the Bloch line still casts a shadow in the spin-wave intensity; only little space and intensity are left for the lower beam (y < 0.7 μm). At $f=9 GHz$⁠, as shown in Fig. 3(c), a maximum is seen downstream of the Bloch line, but now displaced sideways, and the wave pattern appears to be bent. The corresponding phases for the frequencies of 7.15 $GHz$ and 9 $GHz$ are shown in Figs. 4(e) and 4(f).

In summary, we stabilized a symmetric Néel DW with a topologically enforced circular Bloch line confined in a magnetic waveguide with imprinted uniaxial anisotropy in the vicinity of a microwave antenna. This configuration allowed us to experimentally observe zero-magnetic-field propagation of DE spin waves through a DW by phase-resolved μBLS. We observed two different regimes of spin-wave propagation appearing at frequencies of 7.15 GHz and 9.00 GHz. In the first regime, which was observed at the frequency of 7.15 GHz, spin-wave propagation in the vicinity of the circular Bloch line is suppressed and two spin-wave beams are created. In contrast, at 9.00 GHz, spin waves propagate through the circular Bloch line and create a single spin-wave beam. Phase-resolved measurements reveal that spin waves exhibit a phase shift of approximately $0.6π$ upon transmission through the DW. We observed that the DW spatially distorts the phase of the spin wave. This effect needs to be taken into account when designing DW-based phase shifters or other devices relying on spin-wave phase manipulation. The other point that needs to be taken into consideration is the occurrence of topologically enforced spin structures, which are unavoidable in certain geometries. They can limit the performance or functionality of the DW-based device. On the other hand, they can also be an advantage: We propose an interesting technique for spin-wave guidance by manipulating the circular Bloch line position in the DW by external magnetic fields. This technique can be used for dynamical turning or blocking spin waves in future magnonic devices.

The authors thank R. Schäfer and O. Fruchart for the discussions on the DW classification.

This research was supported by the CEITEC Nano+ project (No. CZ.02.1.01/0.0/0.0/16013/0001728) and Austrian Science Fund (FWF) project I1937. M. Staňo acknowledges support by the ESF under the project CZ.02.2.69/0.0/0.0/19_074/0016239. CzechNanoLab project LM2018110 funded by MEYS CR is gratefully acknowledged for the financial support of the measurement and sample fabrication at the CEITEC Nano Research Infrastructure.