Dynamic unidirectional anisotropy in cubic FeGe with antisymmetric spin-spin-coupling

Strong unidirectional anisotropy in bulk polycrystalline B20 FeGe has been measured by ferromagnetic resonance spectroscopy. Such anisotropy is not present in static magnetometry measurements. B20 FeGe exhibits inherent Dzyaloshinskii-Moriya interaction, resulting in a nonreciprocal spin-wave dispersion. Bulk and micron sized samples were produced and characterized. By X-band ferromagnetic resonance spectroscopy at 276 K ± 1 K, near the Curie temperature, a distribution of resonance modes was observed in accordance with the cubic anisotropy of FeGe. This distribution exhibits a unidirectional anisotropy, i.e. shift of the resonance field under field inversion, of KUD = 960 J/m3 ± 10 J/m3, previously unknown in bulk ferromagnets. Additionally, more than 25 small amplitude standing spin wave modes were observed inside a micron sized FeGe wedge, measured at 293 K ± 2 K. These modes also exhibit unidirectional anisotropy. This effect, only dynamically measurable and not detectable in static magnetometry measurements, may open new possibilities for directed spin transport in chiral magnetic systems.


Sample preparation
Stoichiometric FeGe was melted, using induction heating and, to guarantee homogeneity, re-melted twice and annealed for 130 h at 1000 K. Cylinders were formed and a high pressure high temperature synthesis inside a Kawai-type 30 multianvil apparatus with Walker-type 31 module was applied. This resulted in 95% polycrystalline B20 FeGe, confirmed by X-ray diffraction. A maximum of 5% of the sample material could consist of secondary phase Iron Germanium. Energy-dispersive X-ray spectroscopy (EDX) measurements also reveal local composition variations with accumulation of iron (Fe:Ge 55:45). Further investigations with Lorentz microscopy show the formation of helices and skyrmions ( Fig. 1(a)) in accordance to 32 . Micron sized samples ( Fig. 1(b)) with wedge shaped geometries were cut using a Focused Ion Beam (FIB -FEI Helios nanolab 600) and placed inside an R-Type microresonator [33][34][35] using standard lift-off FIB (Omniprobe manipulator with Pt gas insertion system) technique. During the lift-off process a carbon coating with up to 100 nm thickness and up to 15% platinum contamination 36 could not be avoided. Furthermore, the lift-off process used Gallium as cutting ions and resulted in a localized deposition of a maximum of 2.6% of Ga (as verified by EDX). experimental FMR spectra of a bulk polycrystalline, nearly disc shaped piece of FeGe with a diameter of 3.78 mm and a thickness of 0.78 mm (2(b) inset) was acquired in a range of 800 mT to 0 mT at a frequency of 9.517 GHz ± 0.006 GHz. The field was applied at angles of -8° to 172° (corresponding to the directions "up" and "down" in Fig. 2(a)) in steps of 0.5° from out-of-plane to in-plane and to the opposite out-of-plane orientation. The measurement of the uniform FMR mode can be seen in Fig. 2(a) shown as an amplitude contour plot. The temperature is 276 K ± 1 K, which is below the Curie Temperature of T C = 280 K 19 , where the sample is ferromagnetic 32 . The angular precision of our experimental setup is better than 0.05° and the precision of the magnetic field is better than 0.5 mT with a relative precision of 0.005 mT.
Resonance lines in the FMR spectra are identified by a successive local maximum and minimum amplitude. We observe a distribution of resonances, which is in agreement to previous FMR investigations 22 of single crystalline FeGe. Each crystallite in the sample is contributing to this resonance distribution. They are all influenced by the applied external field and the demagnetization field in the sample, due to its general shape. However, their resonance fields vary with respect to the applied magnetic field due to the different symmetry axis of the cubic anisotropy in each crystallite. We simulated the resonance distribution using the known magnetocrystalline anisotropy of FeGe 22 and a random orientation of crystallites and compared it to the measurement. This can be found in the Supplementary Sec. S1. Figure 2(a) shows the differentiated angular dependent FMR spectra as a grey scale contour plot. The out-of-plane orientations are depicted in detail in Fig. 2(b). The measured resonance line exhibits a unidirectional anisotropy, indicated by a difference in the positon of maximum microwave absorption comparing opposite magnetic field directions. A similar anisotropy is observed in systems with exchange-bias 37 . Hence we performed additional magnetometry measurements, to exclude the presence of exchange bias in our system (Fig. 3). No such anisotropic behaviour is observed in static magnetometry using vibrating sample magnetometry (VSM). We therefore conclude, that this anisotropy must be dynamically induced under resonant excitation. Note that it cannot be equated with the linear contribution to the spin wave dispersion, as this changes directionality in accordance with the magnetic field direction. Due to the skin depth of approximately 10 -3 mm 22 one must, to fully reproduce the FMR lineshape, solve the non-uniform LLG 24 taking the shape of the sample into account. However, we show exemplary in the Supplementary Sec. S1 that a Dysonian lineshape 38,39 and the known magnetocrystalline anisotropy of FeGe 18 are able to reproduce the measured FMR lineshape satisfyingly, which is sufficient for our needs. The position of resonance was obtained by subtracting the background and locating the zero crossing of the resonance line. We analyzed the angular dependent spectra using Eq. 1 as a model for the free energy density F. To account for the observed unidirectional symmetry in the angular dependent resonance field position, an additional unidiretional field contribution needs to be introduced. In this model an additional anisotropy field B U = K UD /M is used. This unidirectional contribution is merely a descriptive model to account for the observed phenomenon. It cannot be seen as an additional magnetocristalline anisotropy but rather as an emergent symmetry contribution which arises under dynamic excitation. In the Supplementary Sec. S2 the shape of such a unidirectional free energy density is shown. Additionally, a demagnetization and Zeeman term are considered.
The demagnetisation tensor N ( = . = . N N 0 676, 0162 zz xx yy , ) was deduced, using the demagnetisation tensor of a cylinder as described in 40 . The known g-factor of FeGe (g = 2.07) 23 was used. θ is the out-of-plane angle of the magnetisation M, and B is the external magnetic field. Additionally, the magnetisation M is considered as a fit parameter. The obtained parameters are = ± K 960 J/m 10 J/m . The magnetization matches the magnetization measured by VSM at 281 K, 5 K above the temperature measured by a sensor below the sample. This offset is likely due to microwave heating. Figure 4 shows the angular dependent FMR spectra (293 K ± 2 K, f Microwave = 9.134 GHz ± 0.006 GHz) of the wedge shaped FeGe sample ( Fig. 1(b)) measured inside a microresonator as a grey scale contour plot. Multiple resonances are visible in the spectra, which exhibit anisotropic behavior. The anisotropy is directed such that the resonance field increases when the static field is applied parallel to the long (dipolar-easy) axis of the sample. This suggests that these modes are spinwaves with energies below that of the gamma point (FMR mode), which may be induced by strong dipolar coupling 41 or DMI. Around ±90°, the number of superimposing resonances and the complex mode intesity distribution 15 make it difficult to separate individual lines. We assume that these resonances arise due to geometrical confinement of the modes in our specimen (Fig. 1). Consequently, the inclined surface of our wedge results in different geometrical boundary conditions at the same time. Bidirectional measurements along the ±81° direction as shown in Fig. 5, however, reveal a clear unidirectional shift of the resonances under field reversal. Figure 5 shows the spectra of the same FMR measurement at the positions "up" and "down" marked in (a) . The resonance spectrum for the field applied along the "down" direction consists of two resonance lines. This is due to edge resonances inside the sample 24,42 superimposing with the uniform FMR mode 23 . Additionally a schematic representation of the sample can be seen with the most important field positions marked.

Scientific RepoRtS |
(2020) 10:2861 | https://doi.org/10.1038/s41598-020-59208-8 www.nature.com/scientificreports www.nature.com/scientificreports/ down, whereas Fig. 5(b) illustrates that under field reversal, the resonance position of the spinwaves has shifted. Hence, we find a unidirectional anisotropy.   www.nature.com/scientificreports www.nature.com/scientificreports/ conclusion From angular dependent ferromagnetic resonance spectroscopy we find an unexpected dynamic unidirectional anisotropy (Fig. 2) in the magnetic excitation of FeGe just below the Curie temperature. This anisotropy is of a dynamic character, since it is not detectable in static hysteresis measurements. The magnitude of the unidirectional anisotropy of the bulk resonance line is = ± K 960 J/m 10 J/m UD 3 3 . Spin waves, detected at 293 K ± 2 K for sample sizes with micrometer dimensions, also exhibit unidirectional anisotropy (Fig. 5).

Methods
A conventional Bruker X-band FMR spectrometer was used for FMR measurements on the millimetre sized FeGe sample (see Fig. 1) inside a cylindrical TE 011 cavity. FMR measurements on the micron sized FeGe sample (Figs. 1(b), 4 and 5) were performed inside an R-Type microresonator [33][34][35] . The resonator was connected to a Varian E102 microwave bridge. The modulated microwave reflection was recovered using a SRS SR830DSP lock-in amplifier.

Figure 5.
Analysis of the bidirectional (differentiated) FMR measurements (~293 K) at 81° and 3.587 GHz ± 0.006 GHz, of the specimen shown in Fig. 1(b). (a,ii) and (b,ii) comparison between two different (differentiated) FMR measurements, the former with the same magnetic field direction, the latter at opposite field directions. (a,i) and (b,i) show a schematic representation of the sample and the respective magnetic field directions of the compared measurements. (a,iii) and (b,iii) depict the highlighted areas in (a,ii) and (b,ii) in detail. (a,iv) and (b,iv) are the plotted differences between the compared measurements. (a,v) noise floor of the measurement in a magnetic field region without resonances.