Phase-resolved optical characterization of nanoscale spin waves Open Access

Spin waves and their quantum counterpart magnons are promising candidates for beyond CMOS technology.¹ In the framework of spin waves, a lot of ideas for wave-based computing were proposed, such as directional coupler² or inverse-design devices.^3,4 The biggest drawback of these devices is a large group delay caused by an unfavorably low ratio of spin-wave group velocity and wavelength. In recent years, magnonic-research community strives to overcome this drawback by moving toward nanoscale, where the spin waves propagate faster and do not need to travel long distances, and, thus, the group delay is minimized.⁵ 

When the spin wave wavelengths approach the exchange length of a magnetic material or when dimensions of the magnonic system are reduced, new phenomena such as spin unpinning condition,⁶ generation of spin waves using parametric pumping,^7,8 or collective dynamics⁹ can arise. Also, the interaction between spin waves and spin textures arises at exchange length scales.^10–14 However, currently, the only possibility to image nanoscale spin waves is x-ray microscopy, requiring synchrotron radiation and making investigation of nanoscale-related phenomena very time- and resource-demanding.^15–18 The development of an optical method for spin wave measurement capable to go beyond diffraction limit is one of the major challenges in the field of magneto-optics.¹⁹ 

In this Letter, we demonstrate that the phase of spin waves can be imaged optically with a subdiffraction resolution and nanometer precision by using Mie resonance-enhanced Brillouin light scattering (BLS).²⁰ This is achieved by concentrating the light into subdiffractional regions (hotspots) in the magnetic material using high-refractive-index dielectric nanoresonators. The induced spatial restriction allows the light to interact with spin waves that have much shorter wavelengths than the wavelength of free-space light. When we arrange the individual nanoresonators into a weakly interacting array, it is possible to extract the information about the spin wave at predefined positions. By using standard electron beam lithography (EBL) techniques, the nanoresonators can be placed with nanometer precision.

To probe the nanoscale spin waves, we used a standard phase-resolved micro-BLS setup,²¹ see Fig. 1(a). The light from a single-mode laser (COBOLT Samba) with a wavelength of 532 nm passes through an electro-optical modulator (EOM, QUBIG TW-15M1-VIS), where the frequency of a tiny fraction of the light is blue- and red-shifted by the pumping frequency. The light is focused on the sample by objective lens with numerical aperture NA = 0.75. The incident power of the laser is 3 mW. The backscattered light is analyzed by the Tandem Fabry–Pérot interferometer (TFPi, Tablestable, TFP2HC) with a single photon detector.²² 

A single radio frequency (RF) generator (Rohde & Schwarz SMB100B) is used to excite the spin waves by an excitation antenna and to drive the EOM at the same time. To ensure a temporal coherence of the EOM and the excited spin waves, the signal is split by a 3 dB power divider. Two RF switches are used to individually control the sample and EOM excitation. The power of both branches can be independently set by varying the output power of the signal generator and by a variable attenuator, which is placed in the EOM branch. A motorized phase shifter, also placed in the EOM branch, allows to precisely set the phase difference between the excited spin waves and the reference signal from the EOM. The excitation antenna is contacted by RF picoprobes (GGB industries, Model 40A).

The sample consisted of a continuous Permalloy layer (30 nm-thick) deposited on a silicon substrate. The spin waves were excited by a strip line nanoantenna (180 nm wide, 105 nm thick multilayer stack: 10 nm SiO₂/85 nm Cu/10 nm Au), fabricated by EBL and the liftoff process. The antenna enabled excitation of spin waves with wavelengths down to $≈ 200 nm ( k ≈ 30 rad / μ m )$⁠.^23,24 The square array consisting of 200 nm-wide and 60 nm-thick sputter-deposited silicon disks was fabricated by EBL and the liftoff process in the vicinity of the antenna. The array had a lattice constant of 500 nm and a tilt of 8° with respect to the antenna. A scanning electron microscope image of the studied structure is shown in Fig. 1(b).

Here, ω denotes the frequency, k denotes the wavenumber, $ℏ$ denotes the reduced Planck constant, and indices i, s, and m stand for incident light, scattered light, and magnons, respectively.

To confirm these findings, we performed a 2D scan of the BLS intensity in the vicinity of the excitation antenna, which was connected to the RF generator with a frequency set to 14.5 GHz, in an external magnetic field of 50 mT. The antenna, thus, excited coherent spin waves with wavevector $k = 27 rad / μ m$ (λ = 232 nm).^33,34 Such short-wavelength spin waves are beyond the detection limit of conventional μ-BLS (the detection limit of our μ-BLS is $k ≈ 11 rad / μ m )$⁠,²⁰ and, thus, without any further enhancement, we would see no signal. However, due to the subdiffraction localization of the E-field by the silicon disks, we can observe a propagating spin wave with even such a short wavelength (see Fig. 2). The BLS signal naturally decays as the wave propagates further from the excitation antenna, and it is strongly enhanced at the positions of the silicon disks. The low signal in between the positions of the silicon disks is caused by the fact that the waist of the Gaussian beam (440 nm)²⁰ is comparable to the lattice constant (500 nm) of the silicon disk array. These observations agree well with expectations based on Eq. (5).

In the second experiment, we focused on the measurement of the spatial evolution of the spin wave phase. During the BLS process, the scattered photon acquires the phase of the spin wave, and this phase can be reconstructed with the help of reference phase signal produced by the EOM. In this measurement, both RF switches were switched on [see Fig. 1(a)], and we observed the interference of the light inelastically scattered on spin-waves with the light modulated by EOM.³⁵ In order to acquire the phase with spatial resolution sufficient to measure spin waves with very short wavelengths, we exploited the symmetry along y-axis of the spin wave propagating perpendicularly from the excitation antenna and performed a linescan over one row of the silicon disk array [see Fig. 1(b)]. As the array is tilted by 8, the distance of the disks from the antenna increases by 70 nm for each disk in the row.

When we look more carefully at the linescan data, we can observe a periodic pattern [either areas with an increased signal nicely visible in Fig. 3(b) or stair-like pattern in Fig. 4(d)]. This pattern has a periodicity of 70 nm, which is the same as periodicity of the positions of the individual silicon disks under the scanning tilt. The linescans were acquired with oversampling, where the spatial step in the spin-waves propagation direction (x axis) was approximately 13 nm, while the distance between the silicon disks was approximately 70 nm. This periodic pattern in extracted phase [Fig. 4(e)] suggests that the detected phase is defined by the positions of the disks on the thin film and is not overly sensitive to the exact positioning of the laser spot relative to the position of the nanoresonator. Also note that we were able to reliably measure the spin-wave wavelength λ = 4.2 μm by scanning over a distance of 450 nm only. This would not be possible without precisely positioned detection points and without the full-phase reconstruction technique.

To clarify the phase stability when scanning over the array of silicon disks, we performed finite-differences-time-domain (FDTD) simulations of the sample with the same parameters as in the experiments. The simulations were carried out in the Lumerical software with the mesh cell set to 5 nm in all directions. More details can be found in Ref. 20. Figure 5(a) shows the localization of electromagnetic field into hotspots that allow the detection of high-k spin waves. In this case, the Gaussian beam is precisely positioned on the silicon disk. On the other hand, if we position the beam spot directly between the silicon disks [Figs. 5(b) and 5(c)], this localization is suppressed, and the overall light intensity is lower. This results in a decrease in the detection sensitivity to high-k spin waves and overall decrease in the BLS signal. We also extracted the phase of the electric field from the simulated data [Figs. 5(d)–5(f)]. The phase in the axis of the linear polarization of the incident light (x) is homogeneous across the high intensity regions (hotspots), while the inner area of the disk has the opposite phase, and this is the same for all simulated positions of the beam. This independence of the electric field phase on exact position of the beam suggests the robustness of the phase reconstruction to, e.g., mechanical vibrations. Note that even though the two hotspots are localized on the disk edges and separated by 200 nm, this does not influence the capability of the system to detect even nanometer changes in the spin-wave wavelength. This is analogous to the case of inductive detection of spin waves by coplanar waveguides or meander antennas in propagating spin-wave spectroscopy experiments. There, the electromagnetic field distribution under relatively large detection antennas with complex geometries is also non-trivial, and still, the spin wave phase can be measured.^23,41

In conclusion, we demonstrated an efficient and simple technique for phase-resolved characterization of nanoscale spin waves based on conventional phase-resolved BLS microscopy. Presented technique uses weakly interacting arrays of Mie resonators, which localize and enhance the electric field involved in the magneto-optical coupling and the connected BLS process. Short-wavelength spin waves can be detected, and their phase can be characterized with nanometer precision. This precision depends on the exact positioning of the nanoresonator and the design of the array. With modern electron beam lithography techniques, sub-10 nm precision of placement of individual array elements is easily achievable. The presented experiments were limited only by the excitation efficiency of the used strip line nanoantenna. With the use of a different type of the spin-wave source, it should be possible to detect spin waves with sub-100 nm wavelengths. The presented technique is not material specific and can be transferred to any magnetic material system, including ferrimagnetic insulators, such as YIG. Also, with the presented technique, other types of BLS experiments, such as time-resolved measurements,⁴² can be easily performed on nanoscale spin waves. These results show that Mie-enhanced BLS microscopy is relevant and highly versatile tool for nanoscale magnonics research.

We gratefully acknowledge CzechNanoLab Project No. LM2018110 for the financial support of the measurements and sample fabrication at CEITEC Nano Research Infrastructure. O.W. was supported by Brno Ph.D. talent scholarship and acknowledges BUT specific research project. F.L. acknowledges the support by the Grant Agency of the Czech Republic (21-29468S). J.H. acknowledges a support from the project Quality Internal Grants of BUT (KInG BUT), Reg. No. CZ.02.2.69/0.0/0.0/19_073/0016948, which is financed from the OP RDE.

The authors have no conflicts to disclose.

Ondřej Wojewoda: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Investigation (lead); Methodology (lead); Visualization (equal); Writing – original draft (lead). Michal Urbánek: Conceptualization (equal); Funding acquisition (lead); Project administration (lead); Supervision (lead); Writing – review & editing (lead). Martin Hrtoň: Investigation (equal); Methodology (equal); Software (lead). Meena Dhankhar: Resources (equal). Jakub Krčma: Investigation (equal); Software (supporting); Writing – review & editing (equal). Kristýna Davídková: Resources (equal). Jan Klíma: Resources (equal); Writing – review & editing (equal). Jakub Holobrádek: Resources (supporting). Filip Ligmajer: Conceptualization (supporting); Writing – review & editing (equal). Tomáš Šikola: Project administration (supporting); Writing – review & editing (equal).

The data that support the findings of this study are available from the corresponding authors upon reasonable request.