We investigate the coupled magnetization dynamics in heterostructures of a single crystal of the chiral magnet (CSO) and a polycrystalline ferromagnet (Py) thin film using broadband ferromagnetic resonance (FMR) at cryogenic temperatures. We observe the excitation of a hybrid mode (HM) below the helimagnetic transition temperature of CSO. This HM is attributed to the spin dynamics at the CSO/Py interface. We study the HM by measuring its resonance frequencies for in plane rotations of the external magnetic field. We find that the HM exhibits dominantly fourfold anisotropy in contrast to the FMR of CSO and Py.
Chiral magnets exhibit non-collinear spin structures, such as spin helices and magnetic skyrmions, below their critical temperature Tc and critical field .1 Skyrmions are topologically protected non-coplanar magnetization configurations that can behave as particle-like objects. Furthermore, they are small yet stable, making them suitable to become the carriers of information in future devices.2–12 The non-collinear spin structure of chiral magnets gives rise to intriguing magnetization dynamics, in particular in their skyrmion lattice phase.13 The recently discovered low-temperature skyrmion phase14 leads to additional striking spin dynamical signatures15 in the low-damping16 chiral magnet (CSO). The periodicity of the magnetic lattice leads to naturally formed magnonic crystals17 and the skyrmion eigenmodes can be coupled to photonic resonators with high cooperativity.18 The chiral properties of skyrmions give rise to non-reciprocal spin-wave dynamics.19 Thus, the emerging field of spin dynamics of chiral magnets has already revealed important fundamental insight with perspectives for practical applications.
In topologically trivial magnets, the now well studied coupling between multiple magnetic layers20–26 resulted in the discovery of some of the most technologically relevant effects, such as tunneling magnetoresistance27 or giant magnetoresistance.28,29 Heterostructures of topologically trivial magnets can exhibit coupled spin dynamics that can lead to, e.g., excitation of nanoscale spin waves.30 Much less is known about spin dynamics in heterostructures of collinear and chiral magnets. The coupling between distinct order parameters across interfaces has explained important phenomena, such as proximity effects, exchange bias, or exchange spring-induced hard magnets.31 However, the studies of excitations in chiral magnets are so far limited to a single magnetically ordered constituent.32 Even though the formation of novel topological order at the chiral magnet/ferromagnet interface was predicted by theory, it has not yet been observed in experiment.33
In this work, we investigate the hybrid magnetization dynamics of heterosturctures of thin film metallic ferromagnets and bulk chiral magnets, in this case the ferrimagnetic insulator (CSO). To study the magnetization dynamics of the chiral magnet/ferromagnet heterostructures in the GHz frequency regime, we use broadband ferromagnetic resonance (FMR) spectroscopy. We experimentally determine and phenomenologically model the resonance frequencies in such heterostructures. Thereby, we find that a hybrid mode of the chiral magnet/ferromagnet heterostructure is excited, which we attribute to the spin dynamics at the interface of the two magnetic layers. We investigate a (CSO/Py) sample, where the Py thin film has a thickness of 40 nm. The CSO crystal is (111)-oriented and cut to a cuboid shape with dimensions mm, mm, and mm. It was grown by a chemical vapor transport method35 (for details on the crystal orientation of the CSO and the sample preparation, see supplementary material S1 and S2). We place the CSO/Py hybrid on top of a coplanar waveguide (CPW) with a center conductor width of as shown in Fig. 1(a). The CPW is connected to two ports P1 and P2 of a vector network analyzer (VNA), which measures the change of transmission from P1 to P2 defined as the complex transmission parameter S21 as a function of frequency f and external magnetic field at a fixed microwave power of 1 mW (0 dBm). We then place the CPW/CSO/Py assembly into the variable temperature insert of a superconducting 3D-vector magnet. By applying a static external magnetic field in the plane of the Py thin film and setting the temperature to 5 K, we can access the helical (H), conical (C), and ferrimagnetic (F) phases of the CSO17 as schematically depicted in Fig. 1(b). See the supplementary material S3 for an illustration of the phases of CSO in dependence of the frequency and field.
As a reference measurement, we first place the CSO/Py hybrid sample on the CPW with the Py facing away from the CPW. Due to the large thickness of the CSO layer, the oscillating magnetic field generated by the CPW does not reach the Py layer. In this way, we only excite the magnetization in CSO itself with no influence of the Py layer. We apply a fixed external magnetic field mT along the x-axis () at 5 K, as illustrated schematically in the top panel of Fig. 2(a). For this temperature and external magnetic field strength, the CSO magnetization is in the field polarized phase. To correct for the microwave background of the complex transmission parameter S21, we use the derivative divide method36 to obtain the field derivative of the complex transmission parameter divided by S21. On the bottom panel of Fig. 2(a), for the CPW/CSO/Py assembly at 5 K is shown as a function of the frequency f. In the gray marked frequency range, several resonances appear. These are attributed to the excitation of magnetostatic modes of the cuboid-shaped CSO crystal. In the frequency range 7 GHz < f < 12 GHz, no additional modes are observed (inset). After determining the response of the isolated CSO magnetization dynamics, we place the CSO/Py hybrid on the CPW with the Py facing the CPW and again apply a fixed magnetic field mT along the x-axis as schematically shown in the top panel of Fig. 2(b). Now, the field generated by the CPW interacts with the Py layer as well as the CSO as the Py layer is a thin film. In the bottom panel of Fig. 2(b), at 5 K is shown for the CPW/Py/CSO assembly as a function of the frequency f. In addition to the resonance lines of CSO (gray marked frequency range), the Py FMR line also appears close to 10 GHz as expected. The change in the CSO mode spectrum is attributed to the presence of the metallic film and concomitant shielding of the microwave field in the bulk of CSO. Furthermore, we observe an additional medium frequency mode, which is shifted by about 1 GHz to lower frequencies than the Py FMR mode. This hybrid mode (HM) is also observed in the CSO/Py hybrid in the conical phase of CSO (see Fig. S2 in the supplementary material). For the dependence of the HM on the magnitude of the external magnetic field, see supplementary material S3. We attribute the appearance of this additional mode to the spin dynamics at the CSO/Py interface. Thus, in a simplified macrospin picture, we may treat our bilayer as a trilayer with a new interfacial layer inheriting properties from both sides. The HM is then modeled as a result of macrospin dynamics of the interlayer as discussed in the following.
To investigate the dependence of the HM on the direction of the external magnetic field, we apply a field with a fixed magnitude of mT and rotate the field direction by 360° in the Py film plane. For a quantitative analysis of the HM, we simultaneously fit the Py FMR peak and the HM peak in the frequency domain of the transmission parameter S21 for each fixed external field direction (for detailed information on the fit model, see supplementary material S4). In Fig. 3, three exemplary fits of the frequency spectrum are shown for a fixed magnitude of the field mT and for different directions under which it was applied. In Fig. 3(a), the external field is applied in the Py thin film plane along the x-axis as schematically depicted at the top of Fig. 3(a). We find the peak of the hybrid mode at 9.2 GHz, which has small amplitude compared to the Py FMR peak-dip at 10.4 GHz. In Fig. 3(b), we show the fit result for the field applied under an angle with respect to the x-axis as depicted at the top of Fig. 3(b). Now, the two resonance frequencies of the Py mode and the HM are less separated than in Fig. 3(a), as the hybrid mode moved to higher frequencies and the Py mode to lower frequencies. In Fig. 3(c), the field is applied under an angle . The two resonance frequencies of the Py mode and the HM are not separable anymore as the HM resonance frequency is presumably superimposed on the Py resonance frequency. Furthermore, the amplitude of the FMR signal decreases in Figs. 3(b) and 3(c). As the applied field shifts away from the x-axis, so does the equilibrium magnetization. As a result, the CPW oscillating field component transverse to the static magnetization also diminishes. This results in a reduction of the recorded FMR signal amplitude.
To better understand the behavior of the spin dynamics in the sample in dependence of the external magnetic field and temperature, we fit the resonance frequencies of each layer for a full rotation of the external field applied in the Py thin film plane, and we evaluate the dependence of the HM resonance frequency on temperature by fitting its resonance frequency for a fixed field direction at different temperatures. In Fig. 4(a), the fitted resonance frequencies at 5 K of the Py FMR mode (blue dots), the CSO modes (green shading indicates the frequency range over which the CSO magnetostatic modes are observed), and the hybrid mode (orange dots) are plotted against the angle . In the fit model, we only consider the CSO mode with the strongest amplitude (green dots). Furthermore, for we are not able to fit the HM, as it either vanishes or merges with the Py FMR mode. Due to the demagnetization field in the CSO crystal, the CSO resonance frequencies show a cosine like dependence on the angle under which the external field is applied. Furthermore, we observe an uniaxial anisotropy in the Py resonance frequencies. This uniaxial anisotropy is strongest for 5 K and weakens for increasing temperatures [see Fig. 4(b) and supplementary material S6]. Additionally, a broadening of the Py resonance minima and a narrowing of the respective peaks is observed. The HM shows a striking feature with its angle dependence being inverted with respect to the CSO and Py modes. Its presence and frequency difference with respect to the Py mode at vanishes when surpassing the CSO ordering temperature as shown in Fig. 4(c).
The angle and temperature dependence of the observed modes reveals a complex interaction within the heterostructure. In the following, we critically analyze various mechanisms that could potentially underlie the observed HM.
First, we start by noting the effect of CSO on the system. By increasing the temperature, a reduced uniaxial anisotropy not only of the CSO but also of the Py and HM mode becomes evident [see Figs. 4(a) and 4(b) and supplementary material S6]. This reduction agrees with a decreased spin ordering in CSO at higher temperatures, such that the effect of magnetic stray field of the bulk CSO crystal on the Py layer can be assumed to be the dominant effect seen by the uniaxial anisotropy in the Py mode (see calculation of the anisotropy field according to Ref. 37 in the supplementary material S6).
It also supports our inference that CSO plays the main role in determining the frequency of the HM. In the following, we argue that the HM is indeed the result of a coupling of the multilayer system. Figure 4(c) visualizes the dependence of the Py and the HM resonance frequencies on temperature where the external field has a magnitude of 120 mT and is applied under an angle . With increasing temperature the HM mode approaches the Py mode until it vanishes above 58 K. Thus, above the critical temperature of CSO, the coupling of the CSO and Py dynamics vanishes and the HM cannot be excited any more. This demonstrates that the existence of the HM is a direct consequence of the magnetic ordering of CSO. We may, thus, exclude non-uniformity of the Py layer as its origin.
Assuming exchange interaction and spin torques at the interface similar to in Ref. 30 and including such terms in the Landau–Lifshitz–Gilbert (LLG) equation would result in four solutions for resonance frequencies. Namely, the uncoupled CSO and Py layer as well as two coupled modes. It might be possible that the second coupled mode is hidden in the series of magnetostatic modes of the CSO. However, in this case, the solutions for the hybrid modes would only lead to a constant shift of the HM frequencies compared to the Py resonance frequencies without an angle dependent gap between the Py and the HM resonance frequencies. Nonetheless, if the exchange interaction was to be anisotropic, a reason for which is not evident to us, the additional degree of freedom could result in the observed symmetry of the HM.
CSO is known for its intrinsic inversion symmetry breaking and the existence of Dzyaloshinskii–Moryia interaction (DMI). As a result, the FMR mode at k = 0 is no longer the lowest frequency mode. The DMI gives rise to a linear-in-k contribution that causes further lowering of frequency for a finite-k spin wave mode.38 Such a finite-k mode does not experience the same shape anisotropy, parameterized by the demagnetization tensor components, as the FMR modes.39 Hence, it makes sense for the HM mode to bear a different angular dependence than the Py and CSO FMR modes dominated by the shape anisotropy. Furthermore, a lowering of the frequency by the DMI term38 is consistent with the HM mode appearing below the Py FMR modes. It is, however, not clear to us why the finite-k mode at the lowest frequency should be excited in this bilayer and act as an additional mode, although it cannot be ruled out. The observed angle-dependence of the HM is still not obvious in that case and may require considering the pinning caused by the DMI at the edges.38
Another observation that makes a clear interpretation difficult is the broadening/narrowing of the Py resonance minima/maxima. Such a behavior is usually seen in material systems with a cubic, i.e., fourfold symmetry superimposed with the previously described uniaxial anisotropy.40 This contradicts the threefold symmetry that is expected from the (111)-orientation of the CSO crystal. Naively, a cubic magnetocrystalline anisotropy in the Py layer could explain the observation.40 Yet, it is not clear why such a crystalline structure should have developed when considering either the (111)-oriented substrate or the continuous rotation of the sample during the deposition process.
The chiral magnet/ferromagnet heterostructure proves to be a highly complex material system raising further questions about anisotropic interaction parameters and fundamental symmetry manifestations that go beyond the scope of this work. To phenomenologically model the data in Figs. 4(a) and 4(b), we use a Landau–Lifshitz–Gilbert (LLG) approach, where we treat the magnetization dynamics in the sample as a three-layer system consisting of the uncoupled Py and CSO layer as well as an interfacial layer as shown at the top of Fig. 2(b). Here, introducing an interfacial layer that inherits properties from Py and CSO layers effectively captures the coupling between the two and phenomenologically models a new mode that becomes active in this bilayer. Thereby, the magnetization dynamics of each of these three layers are treated as macrospins. In our model, we treat CSO as a ferromagnet, as it has been demonstrated that ferrimagnets can be modeled as ferromagnets when considering dynamics in the GHz regime.41,42 The equation of motion for magnetization then reads
Here, γi is the gyromagnetic ratio and αi is the Gilbert damping parameter of layer i. For all three magnetic layers, the effective field accounts for the external field, the driving field, and the demagnetization field. The effective magnetic field of the CSO is, thus, given by
The angle dependent demagnetization factors , and Nz as well as the ansatz for the dynamic magnetization are given in supplementary material S7. In the case of the Py layer, additionally includes a phenomenological uniaxial magnetic anisotropy field , which is expected due to the demagnetization fields of the CSO and can be regarded effectively as a shape anisotropy of the CSO. However, a finite contribution due to the trapped flux in the superconducting coils is also to be expected (see supplementary material S6). Furthermore, it includes a phenomenological cubic anisotropy field that we assume to exist at the interface to explain the data. We note, again, that the fourfold symmetry does not correspond to the cubic anisotropy of CSO, because the projection of the cubic anisotropy to the (111) plane would result in a threefold symmetry
The effective magnetic field of the HM as well additionally accounts for a phenomenological cubic anisotropy field
In conclusion, we investigated the coupled magnetization dynamics in a CSO/Py heterostructure by broadband ferromagnetic resonance experiments at cryogenic temperatures. We found that a hybrid mode at the CSO/Py interface is excited. While a microscopic picture for the formation of the HM is so far missing, our experimental findings pave the way for future experiments on coupled spin dynamics in topologically non-trivial magnetic bilayer systems, which have recently attracted great interest from the viewpoint of possible applications in high-performance memory devices.7,32,43
See the supplementary material for additional measurements of the CSO/Py heterostructure and detailed information on the fitting and simulation methods.
We gratefully acknowledge the financial support by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) via Nos. WE 5386/5-1 and BA 2181/19-1 and financial support from the Spanish Ministry for Science and Innovation—AEI Grant No. CEX2018-000805-M (through the “Maria de Maeztu” Programme for Units of Excellence in R&D). We would also like to thank M. Müller for support in carrying out some of the measurements.
Conflict of Interest
The authors have no conflicts to disclose.
Carolina Lüthi and Luis Flacke contributed equally to this work.
Carolina Lüthi: Data curation (equal); Formal analysis (equal); Writing – original draft (equal). Luis Flacke: Data curation (equal); Formal analysis (equal); Writing – original draft (equal). Aisha Aqeel: Resources (lead). Akashdeep Kamra: Formal analysis (equal); Writing – review & editing (equal). Rudolf Gross: Writing – review & editing (equal). Christian Back: Formal analysis (equal); Writing – review & editing (equal). Mathias Weiler: Conceptualization (equal); Formal analysis (equal); Supervision (equal); Writing – review & editing (equal).
The data that support the findings of this study are available from the corresponding author upon reasonable request.