Understanding spin-wave dynamics in chiral magnets is a key step for the development of high-speed, spin-wave based spintronic devices that take advantage of chiral and topological spin textures for their operation. Here, we present an experimental and theoretical study of spin-wave dynamics in a cubic B20 FeGe single crystal. Using the combination of waveguide microwave absorption spectroscopy (MAS), micromagnetic simulations, and analytical theory, we identify the resonance dynamics in all magnetic phases (field polarized, conical, helical, and skyrmion phases). Because the resonance frequencies of specific chiral spin textures are unique, a quantitative agreement between our theoretical predictions and experimental findings for all resonance frequencies and spin wave modes enables us to unambiguously identify chiral magnetic phases and to demonstrate that MAS is a powerful tool to efficiently extract a magnetic phase diagram.

## I. INTRODUCTION

Identifying and understanding of spin-wave excitations in chiral and topological magnetic materials is a fundamental step towards their integration into spintronic devices. Moreover, some of these materials host magnetic skyrmions, in which spins form a topologically non-trivial excitation with an integer winding number.^{1,2} Magnetic skyrmions can be driven to move by spin torques with an ultralow current threshold because there is weak or no pinning,^{3,4} which makes them appealing for low-power memory,^{5} magnetic logic,^{6} and auto-oscillator^{7} devices. In support of these potential applications, magnetic skyrmions have been investigated using several methods including Lorentz transmission electron microscopy (LTEM),^{8–10} inelastic neutron scattering,^{1,11} magnetic susceptibility,^{12,13} and recently by microwave absorption spectroscopy.^{14,15}

Microwave absorption spectroscopy (MAS), in particular, is a powerful tool for identifying and studying dynamical magnetic excitations in materials. In materials with several complex magnetic phases, such as chiral magnetic materials with a volume Dzyaloshinskii–Moriya interaction, it may be possible to use MAS to quickly establish a magnetic phase diagram because the spin texture in each magnetic phase has a unique resonance response. This specificity gives MAS an advantage as compared to electrical measurements such as the topological Hall effect,^{16–21} which is more difficult to interpret because the signals are not unique to skyrmion phases or helical phases.^{18–23}

In this letter, we present a study of spin-wave dynamics in a single crystal of B20 FeGe using MAS. Specifically, we measure the gigahertz frequency dynamic susceptibility of FeGe using a field-referenced lock-in technique, which significantly increases our sensitivity to magnetic phase boundaries as compared to conventional MAS.^{14} We observe the resonance response in all the magnetic phases (field polarized, conical, helical, and skyrmion phases) of FeGe, including a secondary skyrmion phase pocket previously observed by neutron scattering and magnetic susceptibility experiments in FeGe.^{11,12} In addition, we adapted a recently developed analytical formalism for chiral spin dynamics and micromagnetic simulations to make definitive identification of the resonance frequencies for each phase. This combination of MAS and theoretical modeling enables us to uniquely identify the magnetic phase diagram of FeGe sensitively and efficiently.

We focus on FeGe because it has the highest critical temperature (280 K) among the noncentrosymmetric B20 compounds,^{2,9,12,24} making the skyrmion phase accessible by cooling with a Peltier element. FeGe also has the closest lattice match to the Si [111] substrate and it can be grown by magnetron sputtering for a scalable fabrication of relevant applications.^{21,23,25} While FeGe has been the object of great interest in recent imaging, transport, and neutron scattering experiments,^{9,10,24,26,27} a complete understanding of the spin dynamics of its magnetic phases is lacking, even though the earliest studies date back to 1978 by Haraldson *et al.*^{28}

## II. EXPERIMENTS

The single crystal of B20 FeGe was grown by chemical vapor transport as described by Richardson.^{29} We determined the structure and crystallographic orientation of a millimeter-sized, pyramid-shaped B20 FeGe crystal using Laue X-ray diffraction prior to MAS experiments (see Fig. S1, supplementary material). To perform MAS, we placed the crystal inside an environmentally controlled sample box in which it sits directly on a broadband coplanar waveguide [CPW, Fig. 1(a)]. We apply both a DC magnetic field (*H _{DC}*) and a weak a.c. magnetic field with amplitude

*ΔH*= 6 Oe at frequency ω

_{ac}_{s}/2π = 503 Hz. The sample temperature was controlled using a thermoelectric cooling element. To excite magnetic dynamics, we applied radio-frequency (RF) power to the CPW, which generated an RF magnetic field (

*H*) just above the coplanar waveguide. Our set-up enables measurement frequencies in the range of 0.1–18 GHz; additional details can be found in Ref. 30. We aligned the [110] crystal axis along the DC magnetic field. We also performed the experiment when the [110] axis and the DC field were perpendicular, but obtained the same results.

_{RF}To obtain the crystal's gigahertz dynamic susceptibility, we measured the transmitted microwave power using an RF diode. We amplified its output and passed the resulting signal to a lock-in amplifier that demodulates with respect to *ΔH _{ac}* [Fig. 1(b)]. This corresponds to the field derivative of the transmitted RF power, (dP/dH), in the in-phase (X) and the out-of-phase (Y) components of the lock-in amplifier, that rejected non-magnetic contributions. As illustrated in Fig. 1(c), we scanned the DC magnetic field, the frequency of

*H*, and the temperature of the sample to map the resonances that exist in the magnetic phases of bulk FeGe, including field polarized, helical, conical, skyrmion, and paramagnetic phases [Fig. 1(d)].

_{RF}^{9,11,12,30}

## III. RESULTS AND DISCUSSION

We first study the field-polarized phase of the FeGe single crystal, which appears at large magnetic fields both above and below the critical temperature *T _{c}* = 280 K, which is defined by the formation of the helical phase under zero-field. To isolate the field-polarized phase, we acquired MAS data at magnetic fields above 1500 Oe and frequencies between 6 and 11 GHz, thus rejecting the conical phase resonances. Because we measure dP/dH, we fit the results to a derivative of Lorentzian lineshape [Fig. 2(a) at 264 K] to extract the resonance fields (

*H*) and the linewidths (

_{0}*ΔH*) (see the supplementary material). Next, we use Kittel's formula to show the relationship between the resonance frequencies and fields in the field polarized phase [Fig. 2(b)], which signifies the uniform (Kittel) mode. Then, to find intrinsic and extrinsic linewidths, we use $\Delta H=4\pi \alpha f0\gamma \mu 0+\Delta H0$ and fit

*ΔH*to the RF frequency ($f0$) linearly [Fig. 2(c)], where $\gamma $ is the electron's gyromagnetic ratio, $\mu 0$ is the vacuum magnetic susceptibility, $\alpha $ is the Gilbert damping factor, and $\Delta H0$ is inhomogeneous broadening.

^{31}Determining the intrinsic damping is relevant for applications, because it influences power dissipation, spin torque critical currents, and the velocity of chiral domain walls.

^{32}Therefore, we also study the temperature dependence of $\alpha $ and $\Delta H0$ at the 256–286 K temperature range [Fig. 2(d)]. While $\alpha $ increases monotonically up to T

_{c}, it drops substantially above T

_{c}. In contrast, $\Delta H0$ has the opposite behavior.

We note that $\alpha $ has complex behavior above the *T _{c}*, which may complicate the interpretation of helimagnetic behavior near the

*T*. For example, Zhang

_{c}*et al.*

^{33}recently reported an ultrasmall damping (α = 0.0036 ± 0.0003) in FeGe thin films on an MgO substrate at 310 K, which may significantly differ at lower temperatures. Moreover, our measurement of $\alpha $ in a FeGe single crystal is 0.066 ± 0.003 at 264 K, which is three times larger than the value we recently reported in sputtered FeGe films on Si [111] (α = 0.021 ± 0.005) at 263 K,

^{25}and which could be influenced by the lower

*T*in thin films with substrate induced strain and anisotropy. Another factor in these comparisons is that, our experiment is performed within the 6 to 11 GHz RF frequency range, while Zhang

_{c}*et al.*used higher frequencies at 17–24 GHz.

^{33}Analysis of linewidths with wider and more complete frequency-field measurements would be interesting for a more comprehensive study of damping in chiral FeGe.

Next, we investigate the spin dynamics in the helical and conical phases. First, we model the spin dynamics with micromagnetic simulations using Mumax3.^{34} We use the material parameters for T = 274 K to rigorously replicate the experiment (see the supplementary material). After initializing the magnetic state, we relax the system to its ground state. In Fig. 3(a), we show that the corresponding equilibrium magnitude of the *z*-component (both the magnetic field direction and the direction of the helical vector in *z*) of the magnetization (*M _{z}*) increases linearly with the magnetic field and reaches saturation at 850 Oe. This value is consistent with the saturation magnetic field,

*H*= 840 Oe, that we measured in the experiment [Fig. 3(d)]. Next, we computationally apply a 5 Oe magnetic pulse, which excites spin precession, and monitor the spatially and temporally resolved magnetization dynamics (the ring-down method).

_{d}^{35}By taking the discrete Fourier transform at each pixel, we find the resonance frequencies and corresponding mode profiles as a function of magnetic field. From these data, we construct a power spectral density plot as shown in Fig. 3(b). We observe a uniform mode and two chiral modes, i.e.,

*Q*and

_{–}*Q*, which are the clockwise (CW) and counter-clockwise (CCW) precession of the average magnetization around its equilibrium orientation.

_{+}^{14}We note that the uniform mode in the micromagnetic simulation is an artifact of the ring-down method, and it is not present in the analytical or the experimental results. Moreover, in Fig. 3(c), we plot the result of a separate calculation of the resonance frequencies for these two chiral modes as a function of the ratio of magnetic field with respect to the saturation field ($Hc2\u2261Hd$) using the analytical model derived in Ref. 15 for an isotropic sample. The analytical calculation is in close agreement with the micromagnetic calculation.

Next, we perform MAS in the range of 0.1–6.3 GHz and –1200–1200 Oe and at 274 K. We note that the *Q _{–}* mode has a strong response at the low field, but it loses weight as the field approaches the saturation field. The

*Q*mode behaves oppositely. This agrees with the analytical calculation of the mode weights in Ref. 37. Interestingly, while we observe conical and field polarized phase resonances strongly in the in-phase component (

_{+}*X*) of the signal [Fig. 3(d)], the out-of-phase component (

*Y*) of the signal [Fig. 3(e)] most strongly reflects the helical to conical phase boundary.

We explain the difference between the X and Y component response by analyzing our detection scheme. For the absorbed microwave power *P(H)* as a function of *H*, the *X* and *Y* of the *ΔH _{ac}* referenced lock-in components are proportional to $X=\u222b02\pi \omega sPH\u2009cos\u2009\omega st\u2009dt$ and $Y=\u222b02\pi \omega sPH\u2009sin\u2009\omega st\u2009dt$. We expand

*P(H)*as the Taylor series

where $H\u2212H0=\Delta Hac\u2009cos\u2009\omega st$. If we assume *P(H)* is a well-defined and continuous function for all *H*, the *X* integral will be non-zero for odd orders of $H\u2212H0$, while the *Y* integral vanishes for all the orders. This is true for the field-polarized phases and their boundaries when the transition is gradual. It is also consistent with their strong response in only the *X* component of our signal. On the other hand, for the helical phase with the multiple *q* vector (and also for the skyrmion phase), microwave absorption *P(H)* is not a continuous function at magnetic phase boundaries because the boundaries are abrupt, likely indicating a first-order phase transition. Thus, neither the X or the *Y* components of the lock-in vanish at such a “kink” in *P(H)*. Although the extra contribution at the phase boundary is present in X, it is small compared to the direct resonant absorption signal. The Y component, however, is nearly zero except at abrupt phase boundaries, which makes abrupt magnetic phase boundaries stand out. Teasing apart phase boundaries from direct magnetic resonance is more challenging using the vector network analyzer (VNA) technique that has been used previously,^{14} because it probes *P(H)* directly; thus, the two methods are complimentary. For a direct comparison, see the supporting material in which we present VNA measurements of *P(H)* side-by-side with lock-in measurements of *dP/dH*.

Finally, we studied spin dynamics in the magnetic skyrmion phase with micromagnetic simulations (Fig. 4) and MAS experiments (Fig. 5). In our simulations, we used FeGe material parameters at 277 K and employed the ringdown method again. We find the equilibrium magnetic configuration, a hexagonal skyrmion lattice, by relaxing the system to its ground state under a 300 Oe out-of-plane magnetic field [Fig. 4(a)]. The skyrmion size is 70 nm, which matches the value measured using LTEM,^{9} which verifies the D and J terms in our model. We also added a minuscule easy axis uniaxial anisotropy of *K _{u}* = 200 J/m

^{3}, which helps to numerically stabilize the skyrmion phase against the conical phase that is energetically favored with zero

*K*.

_{u}^{12,30,36}In our Fourier analysis, to exclude boundary effects, we focus on the central skyrmion, whose 3-dimensional spin configuration is also shown to confirm its topological state [Fig. 4(a)].

To gain insight into the skyrmion modes, we calculated the spatially averaged spectra of the dynamics [Fig. 4(b)]. Because of the thickness profile of the sample, we observe very sharp resonances corresponding to thickness modes. These are an artifact of limited simulation volume, and they are marked with pink dots in Fig. 4(b). Additionally, we find three resonances at 1.925, 2.55, and 3.55 GHz frequencies, whose spin-wave modes are in the plane and uniform through the thickness. The spatial profile of the 1.925 GHz resonance is shown in Figs. 4(c1)–4(c3) by taking the Fourier transform of individual *m _{x}*,

*m*, and

_{y}*m*magnetizations. Using these mode images directly, we identify the spin-wave mode as either CW or CCW, i.e., the center of skyrmion is large in

_{z}*m*and

_{x}*m*, while

_{y}*m*is large in the intermediate regions. To find the exact nature of this spin-wave mode, we drive the system with an {H

_{z}_{x}, H

_{y}, H

_{z}} = {0, $5\u2009sin\u2009(2\pi \xd7t\xd71.925\u2009GHz)$, 0} Oe magnetic field with a realistic damping coefficient $\alpha =0.066$ and record the magnetization eight times per period for a total of 36 periods. In doing so, we eliminate all other resonances but the 1.925 GHz one, and more closely model the experiment. Then, we plot the deviation of magnetization from its equilibrium state at each time step for a full period in Fig. 4(d). We observe that the resonance at 1.925 GHz is the CCW mode. This conclusion is consistent with analytical calculations in Refs. 15 and 37, in which the CCW mode has a significantly larger spectral weight.

We also tried to computationally drive the system in a similar way at 2.55 and 3.55 GHz, but the resulting dynamics always lead to a CCW mode due to the weak spectral weight of these modes. However, using a previously reported analytical model^{14,37} and numerical simulations,^{38} the CW mode is expected to have resonances around 3.56 GHz (∼1.85 times of the CCW mode frequency), which agrees well with the third resonance at 3.55 GHz. We could not identify the 2.55 GHz resonance, which may be a micromagnetic artifact or a mix of CCW and CW modes.

Next, we perform MAS in the temperature range of skyrmion formation, 279–276 K, and plot the results as a function of frequency and magnetic field (Fig. 5). As before, the *Y* component is sensitive to abrupt magnetic phase boundaries with the helical or skyrmion phases [Figs. 5(a)–5(d)], whereas the *X* component is most sensitive to the field polarized and conical phases [Figs. 5(e)–5(h)]. In addition, at 276 and 277 K, the *X* component has some conical resonances between the helical and skyrmion phases and there are multiple competing phases at narrow field ranges as previously found in Cu_{2}OSeO_{3}.^{15}

We note that there are CCW and CW resonances within the skyrmion phases between 279 K and 276 K. Because we drive the system with an in-plane RF field, we do not expect to observe breathing modes, which can be excited only by an out-of-plane H_{RF} field.^{14,15,37} In addition, we observe two skyrmion pockets between 278 K and 276 K and only one at 279 K, an observation that is different from previous spin dynamics studies in other bulk B20 materials.^{14,15} For example, the out-of-phase MAS at 279 K reveals only two skyrmion resonance features at positive fields in Fig. 5(a) (the CW mode marked by the blue feature between 2 and 4 GHz and the CCW mode marked with the red feature between 1 and 2 GHz), whereas at 278 K in Fig. 5(b), a total of four skyrmion features appear (see the SM for individual plots and details on feature tracking). Moreover, at 276 K, the A_{1} skyrmion resonance becomes weak, but the high field A_{2} skyrmion resonance is more evident. These multiple skyrmion phases were previously found in susceptibility, specific heat, and neutron scattering experiments in bulk FeGe and named A_{1} (low field) and A_{2} (high field) phases.^{11,12,30} However, a more recent re-interpretation of the susceptibility measurements in FeGe reported only one skyrmion pocket using analysis based upon the universality of susceptibilities among B20 compounds.^{39} The origin of the A_{1} and A_{2} phases has been attributed to skyrmion stabilization by thermodynamic fluctuations in the vicinity of *T _{c}*. It was suggested that the interaction between skyrmions alters from an attractive to a repulsive at higher magnetic fields and this results in the different susceptibilities, specific heats, and neutron scattering signatures.

^{11,30,36,40,41}Here, we find that both phases are also accessible by MAS. However, we note that to match MAS experimental data for each skyrmion phase to the micromagnetic theory, further calculations including mean field theory calculations, Monte Carlo simulations, and micromagnetic simulations will be required, and which remains as a future challenge.

Next, we plot the magnetic phase diagram of the FeGe single crystal determined by MAS in Fig. 6 as a function of magnetic field and temperature. We track the sharp phase boundaries for the helical, conical, field polarized, and skyrmion phases as shown in Figs. 3 and 5. On the other hand, the transition from the paramagnetic to the field polarized phase above T_{c} happens at a relatively wide field range, where both phases coexist by showing field polarized/ferromagnetic resonance dynamics at paramagnetic phases until 284 K (see the supplementary material). The previous low-frequency susceptibility studies also draw a wide boundary between the two phases.^{12,39} We note that our phase diagram is identical to the one determined by neutron scattering,^{11} however, using magnetic susceptibility Wilhelm *et al.* observed four skyrmion phase pockets; the extra two were labeled A_{0} and A_{3} and they were located just below (in field) the A_{1} phase and to the left of (in temperature) the A_{1} phase.^{12} In a separate specific heat study,^{30} only the A_{1} and A_{2} phases were found, consistent with our study.

We note that our MAS method is complementary to the previously used methods, including neutron scattering^{11} and Lorentz-TEM^{9} for establishing a complicated phase diagram. In particular, MAS supported by micromagnetic modeling offers a practical and cost-efficient approach for unambiguous phase mapping. In addition, MAS provides useful information about spin dynamics that may be relevant for device applications. On the other hand, MAS has shortcomings when micromagnetic simulations are unable to find unique solutions or it finds solutions that disagree with the experimental measurements. For example, our choice of FeGe is well-suited to this approach because of its relatively large skyrmion diameter (∼70 nm) that can be accurately simulated using standard micromagnetic codes. A counter-example would be MnGe, which has a 3 nm skyrmion diameter^{17,42,43}—too small for standard micromagnetic codes and thus requiring the development of a different theoretical method to quantify resonance frequencies in each phase.

## IV. SUMMARY

In summary, we studied spin dynamics in a non-centrosymmetric B20 FeGe single crystal using microwave absorption spectroscopy. We found that phase-sensitive measurement of differential microwave absorption is an efficient method to establish chiral magnetic phase boundaries, including helical and skyrmion phases. Additionally, the quantitative correspondence between experimentally measured resonance frequencies and theoretical calculations enables unambiguous phase identification. Our findings are relevant to the development of skyrmionic logic and storage devices, spin-torque oscillators, and nonreciprocal magnetic devices, because a quantitative understanding of spin-waves in chiral magnets will be crucial to creating active spintronic devices with topological phases.

## V. SUPPLEMENTARY MATERIAL

See supplementary material for X-ray characterization of the FeGe single crystal sample, calculations of the temperature dependent material parameters, how some of these parameters were used in the micromagnetic simulations, additional VNA MAS measurements, and spin-wave modes for 2.55 and 3.55 GHz frequencies in the skyrmion phase.

## ACKNOWLEDGMENTS

The microwave absorption spectroscopy experiment, micromagnetic simulation, and data analysis were supported by the DOE Office of Science (Grant No. # DE-SC0012245) and the single crystal FeGe growth was supported by the NSF (Grant No. # ECCS-1609585). We also acknowledge use of facilities of the Cornell Center for Materials Research (CCMR), an NSF MRSEC (Grant No. # DMR-1120296). We further acknowledge facility use at the Cornell Nanoscale Science and Technology Facility (Grant No. # ECCS-1542081), a node of the NSF-supported National Nanotechnology Coordinated Infrastructure. M.J.S. also acknowledges support from the NSF Graduate Research Fellowship Program (Grant No. # DGE-1256259).