Linearly polarized GHz magnetization dynamics of spin helix modes in the ferrimagnetic insulator Cu$_{2}$OSeO$_{3}$

TL;DR

This paper demonstrates that the ferrimagnetic insulator Cu$_{2}$OSeO$_{3}$ exhibits linearly polarized magnetization dynamics at around 2 GHz, with polarization properties tunable by sample geometry, offering new insights for microwave technology applications.

Contribution

It reveals that Cu$_{2}$OSeO$_{3}$ shows bulk linear dichroism in magnetization dynamics at GHz frequencies, a phenomenon not previously observed in such materials.

Findings

Linearly polarized GHz magnetization dynamics observed in Cu$_{2}$OSeO$_{3}$

Polarization plane switchable by slight changes in sample width

Broad shape range supports linear polarization in bulk material

Abstract

Linear dichroism -- the polarization dependent absorption of electromagnetic waves -- is routinely exploited in applications as diverse as structure determination of DNA or polarization filters in optical technologies. Here filamentary absorbers with a large length-to-width ratio are a prerequisite. For magnetization dynamics in the few GHz frequency regime strictly linear dichroism was not observed for more than eight decades. Here, we show that the bulk chiral magnet Cu$_{2}$OSeO$_{3}$ exhibits linearly polarized magnetization dynamics at an unexpectedly small frequency of about 2 GHz. Unlike optical filters that are assembled from filamentary absorbers, the magnet provides linear polarization as a bulk material for an extremely wide range of length-to-width ratios. In addition, the polarization plane of a given mode can be switched by 90$^\circ$ via a tiny variation in width. Our findings shed a new light on magnetization dynamics in that ferrimagnetic ordering combined with anisotropic exchange interaction offers strictly linear polarization and cross-polarized modes for a broad spectrum of sample shapes. The discovery allows for novel design rules and optimization of microwave-to-magnon transduction in emerging microwave technologies.

Equations

$$\varGamma_{\sigma}(N_{x},N_{y})\big{|}_{H=0}\propto\frac{\varTheta[\sigma\cdot(N_{x}-N_{y})]}{1+(2+N_{x})\chi_{\mathrm{con}}^{\mathrm{int}}/6}$$

Full text

Skyrmions; dichroism; spin dynamics; magnonics; coplanar waveguides; chiral magnets

Microwave components such as power limiters, oscillators and tunable bandpass filters exploit the precession of magnetic moments (spins). The magnetization dynamics allows one to process microwave signals in the few GHz frequency regime. In microwave filters bulk spheres prepared from the ferrimagnetic insulator yttrium iron garnet (YIG) provide for instance an excellent quality factor and a wide tunability of frequencies. In microwave circulators and isolators the nonreciprocal characteristics of magnetic materials are functionalized. For an efficient operation, the magnetization dynamics and the microwaves require a common polarization. However, this is usually not the case as the equation of motion for spin precession (Landau-Lifshitz equation) provides a circular polarizationLanLif1935 ; Gurevich . It may be elliptically deformed by demagnetization fieldsLanLif1935 ; Gurevich ; Gruenberg::NobelLecture : For very thin films Zivieri2002 and filamentary magnetic rods with extremely large length-to-diameter ratios the ellipticity $\epsilon$ can take values close to one, but strictly linear polarization ($\epsilon=1$) is not possible (Supplementary Eqs. (1)-(3)). The sphere-like and bulk samples used so far in microwave components are neither filamentary nor in thin-film form. Hence their magnetization dynamics exhibit a polarization close to circular ($\epsilon=0$). In contrast, microwaves provided by standard coaxial cables, coplanar waveguides (CPWs) and microwave cavities are linearly polarized ($\epsilon=1$). As a consequence, 50 % of the microwave energy would be wasted in case of a YIG sphere and nonreciprocal devices do not function due to the mismatch of polarization. This is a drawback for future on-chip resonators exploiting coupled magnons and photons for quantum information processing Huebl2013 ; Tabuchi2014 ; Zhang2014 ; Goryachev2014 . On the one hand, involved waveguides and cavity designs were invented to produce microwaves with a high degree of circular polarization Wen1969 ; Duncan1957 ; Poole67 ; Yasukawa2016 . On the other hand magnetocrystalline anisotropy (e.g. easy-plane anisotropy) and antiferromagnetism Gurevich might help to overcome the bad coupling with linearly polarized electromagnetic waves as these two qualities enhance the ellipticity in bulk materials. But at the same time, the frequency increases to several 10 GHz or even near-infrared frequencies (terahertz) Sievers::PhysRev::1963 ; Klausen2004 ; Kampfrath08 where efficient waveguides do not exist. The integration of magnets with microwave electronics could be significantly improved by a ferrimagnetic insulator similar to YIG that possesses linearly polarized magnetization dynamics. However, such a material and the underlying blueprint have not yet been identified.

Here, we report the discovery of linearly polarized magnetization dynamics in the chiral ferrimagnet Cu2OSeO3. This magnet supports a helical spin order stabilized by anisotropic exchange interaction, i.e., Dzyaloshinskii-Moriya interaction (DMI) Roessler2006 ; 2009:Muhlbauer:Science ; 2010:Yu:Nature ; 2012:Seki:Science . Its phase diagram includes helical (H), conical (C) and skyrmion lattice (SkL) phases. Each phase is known to support characteristic spin excitations, introduced as modes +Q and $-$Q in phases H and C, as well as clockwise (CW), counterclockwise (CCW) and breathing (BR) modes in the SkL phase 2012:Mochizuki:PhysRevLett ; 2012:Onose:PhysRevLett ; Schwarze2015 . We observe the linearly polarized magnetization dynamics for the modes +Q and $-$Q of the spin-helix phase. Their frequencies are near 2 GHz which are three orders of magnitude smaller compared to easy-plane antiferromagnets Sievers::PhysRev::1963 . Surprisingly, the linear polarization is present also in almost circularly shaped samples. Furthermore, the polarization characteristics are modified and controlled by fields on the order of 10*-2* T. This is not possible with antiferromagnets. A ferrimagnet with DMI thus opens an unexpected perspective for the polarization control of microwaves in frequency regimes that are key for modern telecommunication networks. Our findings are paradigm-shifting and a breakthrough for the efficient integration of magnets into future microwave circuitry. Magnetic insulators such as Cu2OSeO3 are particularly interesting as they provide negligible eddy current loss and small spin-wave damping, which adds to their fascinating topological properties 2009:Neubauer:PhysRevLett ; 2010:Jonietz:Science ; 2012:Schulz:NaturePhys ; 2013:Milde:Science ; 2013:Fert:NatureNano ; 2013:Nagaosa:NatureNano .

Results

Broadband spectroscopy across magnetic phase diagram

For our experiments we mounted a bar-shaped single crystal of Cu2OSeO3 with lateral dimensions $2.3\times 0.4\times 0.3\,\mathrm{mm}^{3}$ on a CPW with a 1 mm-wide signal line [Fig. 1]. The assembly was cooled down to a temperature $T<T_{\rm c}$, i.e., $T$ was smaller than the critical temperature $T_{\rm c}$ below which the spin-helix phase is stabilized. A static magnetic field $\mathbf{H}$ was applied perpendicular to the substrate, i.e. along the $z$ axis, being collinear with a $\langle 100\rangle$ axis of the crystal. Considering a placement as shown in Fig. 1 and assuming $N_{x}+N_{y}+N_{z}=1$ for the approximated ellipsoid with $\mathbf{H}$ parallel to a semi-principal axis, we estimated the components of the sample’s demagnetization tensor to have components $N_{x}=0.07$, $N_{y}=0.40$, and $N_{z}=0.53$ Aharoni1998 . Note that $N_{x}$ and $N_{y}$ interchanged their values if the sample was rotated by $90^{\circ}$ in the $xy$ plane Gurevich . Radiofrequency (rf) signals were applied to the CPW using a vector-network analyzer. The dynamic field $\mathbf{h}$ created by the linearly-polarized electromagnetic wave in the CPW is depicted in Fig. 1b. We find a dominant in-plane component $h_{x}$ above the signal line and a pronounced out-of-plane component $h_{z}$ above the gaps between the signal (S) and ground (G) lines (for additional experimental details see Methods, Supplementary Notes I and Supplementary Figure 1).

To compare with previous reports on the GHz magnetization dynamics of Cu2OSeO3 2012:Onose:PhysRevLett ; Schwarze2015 we show different sets of spectra taken throughout the magnetic phase diagram at $T=57~{}$K$<T_{\rm c}$. In particular we present spectra taken for different sample orientations (Fig. 2). In Fig. 2a we place the 0.4 mm wide sample on the central axis of the 1 mm wide signal line of the CPW similar to earlier setups 2012:Onose:PhysRevLett . Here, the component $h_{x}$ of $\mathbf{h}$ provides the relevant torque for spin excitation Gurevich . At $H=0$, i.e. in the helical phase, we resolve a single resonance at $f\approx 1.8$ GHz. For increased $H$, the sample enters the conical phase and this resonance shifts to a smaller frequency (detailed spectra are found in the Supplementary Figure 2). At intermediate fields in Fig. 2a, two weak resonances are observed (red lines at 18 and 24 mT). We attribute the corresponding field regime to the SkL phase 2012:Onose:PhysRevLett . Following Refs. 2012:Mochizuki:PhysRevLett ; 2012:Onose:PhysRevLett , $h_{x}$ excites the modes CCW and CW. The allocation of magnetic phases is consistent with both Ref. [2012:Onose:PhysRevLett, ] and measurements on thermodynamic properties performed on Cu2OSeO3 (such as specific heat or magnetic susceptibility; not shown). For the sample being collinear with the CPW and within the gap between ground and signal line [Fig. 2b], we observe a similar sequence of spectra as a function of $H$. A detailed analysis [Fig. 2e] of eigenfrequencies $f$ in the SkL phase reveals, however, a different field dependence of the prominent SkL mode residing at a higher frequency compared to the setup in Fig. 2a. We relate this difference to the out-of-plane component $h_{z}$ of $\mathbf{h}$ [Fig. 1b] favoring the excitation of a prominent breathing mode BR 2012:Mochizuki:PhysRevLett ; 2012:Onose:PhysRevLett .

When rotating the sample by 90∘ [Fig. 2c], spectra at finite field significantly change. At 22 mT, in the conical state, we resolve two modes that are of similar signal strength. With increasing $H$, the low-frequency mode successively vanishes (compare spectra taken at 38 and 46 mT). In the SkL phase, all three modes CCW, BR and CW are seen in one-and-the-same spectrum (compare 28 mT and Supplementary Figure 2), due to simultaneous excitation via both $h_{x}$ (signal line) and $h_{z}$ (gap). Still, only one resonance is seen at $H=0$ (helical phase) in Fig. 2c. Strikingly, for a rotation of 45∘ (Fig. 2d) we identify two modes at $H=0$. The diagonal positioning was not reported before. In the following, we explain the observation of the two modes by an extraordinary linear polarization of magnetization dynamics in the helical phase.

Comparison of experiment with theory

To compare to theory, it is instructive to summarize both the different number of modes and measured eigenfrequencies $f$ in normalized units following Ref. Schwarze2015 in Fig. 2e-g. The magnetic phases H, C, SkL and FP (field-polarized phase) are indicated by different background colors. From Fig. 2e-g we find that, qualitatively, in all the different magnetic phases, the measured eigenfrequencies (symbols) follow well the field dependencies (lines) predicted by the theory of Ref. [Schwarze2015, ] if we assume a finite wave vector $k$ transferred by the CPW. A nonzero wave vector is provided by the inhomogeneity of $\mathbf{h}(x,z)$ [Fig. 1b]. The wave-vector dependence will be discussed elsewhere.

In the following, we focus on the different numbers of modes that we resolved experimentally at small $H$ for phases H and C. Figure 2g displays that for the sample placed at 45∘ with respect to the CPW the two resonances detected at $H=0$ are consistent with the two helical modes +Q and $-$Q. This diagonal placement used in Fig. 2g is usually not executed in the literature. In Figs. 2e and 2f, where we used the conventional placement, only a single resonance is detected at $H=0$. In Fig. 2e, theory attributes the single resonance in phases H and C to mode +Q. We do not detect mode $-$Q when the long axis of the sample and the CPW are collinear. Changing the sample orientation by 90∘ on the CPW (Fig. 2f), the single resonance detected near $H=0$ is the complementary mode $-$Q. Mode +Q is no longer resolved. Our experiments suggest that at small $H$ a cross-polarization exists between the linearly polarized excitation field $\mathbf{h}$ and either mode $-$Q or $+$Q in Fig. 2e and 2f, respectively. The corresponding linear polarization of spin helix modes +Q and $-$Q occurring near 2 GHz has not yet been considered. Note that the detection of the ferromagnetic resonance in the FP phase does not depend on the sample placement [Fig. 2e-g]. The DMI-induced spin texture at low $H$ is thus decisive for the extraordinary polarization characteristics. The occurrence or absence of a double peak at $H=0$ was not explained before 2012:Onose:PhysRevLett .

Discussion

Polarization and ellipticity

We now explain the observed polarization dependencies by considering the ellipticity of magnetization precession in the spin helix phases (Fig. 3). For this, we make use of the theoretical approach outlined in detail in the Supplementary Information of Ref. [Schwarze2015, ]. There, spectral weights $\Gamma$ of different modes were discussed, but ellipticities and polarization were not evaluated. An rf field with $\mathbf{h}$ in a plane perpendicular to ${\bf Q}$ excites the two modes +Q and -Q with a weight that depends on its polarization within this plane, as we show in the following. It is instructive to introduce the corresponding homogeneous dynamic magnetization $\mathbf{m}_{\sigma}$ averaged along a helix period ($\sigma=+1$ and $-1$ for the +Q and $-$Q mode, respectively) [Fig. 1c,d]. For a helix with $\mathbf{Q}$ in $z$ direction, $\mathbf{m}_{\sigma}=(m^{x}_{\sigma},m^{y}_{\sigma})$ oscillates in the $xy$ plane, characterized by its handedness and ellipticity $|\varepsilon_{\sigma}|=\left|{m^{x}_{\sigma}}^{2}-{m^{y}_{\sigma}}^{2}\right|^{1/2}/\,\mathrm{max}[m^{x}_{\sigma},m^{y}_{\sigma}]$. The handedness of the ${\bf m}_{\sigma}$ oscillations is always counterclockwise (left-circular) and clockwise (right-circular) for the +Q and -Q mode, respectively. Here we consider $\mathbf{Q}$ pointing towards the observer.

For pedagogical reasons, we first discuss the polarization of $\mathbf{m}_{\sigma}$ in a disk-shaped sample ($N_{x}=N_{y})$ at $H$=0 (Fig. 3a). In this case, the oscillation is circular with zero ellipticity for symmetry reasons. As a linearly polarized excitation field $\mathbf{h}$ in the $xy$ plane is equivalent to the superposition of left- and right-circularly polarized components, it couples equally well to the two modes +Q and -Q and leads to the same nonzero spectral weight $\varGamma_{\sigma}$ for both modes. Note that for a disk with $N_{x}=N_{y}$, the screw symmetry of the helix ensures that the two excitation modes are degenerate in frequency at $H=0$. By appropriately combining the two circularly polarized modes, any polarization can be achieved. For $H\neq 0$ applied along $\mathbf{Q}$, the precessional motion of each individual mode stays circularly polarized. The handedness of the +Q mode (left-circular) coincides with the one of the local spin precession and thus with the Kittel mode in the FP phase. Mode +Q smoothly connects to the Kittel mode at $H_{c2}$. Contrastingly, the weight of the -Q mode reduces to zero close to the critical field $H_{c2}$. This is because mode $-$Q is right-circular and does not comply to the handedness imposed by the Landau-Lifshitz equation of motion Gurevich . The resulting imbalance in precession amplitude compared to +Q is indicated by bold and broken lines in Fig. 3c (see $\varGamma_{\sigma}$ at $N_{x}=0.235$). The imbalance increases with increasing $H$. For a disk-shaped sample with $N_{x}=N_{y}$ (= 0.235 in our case) the modes $\pm Q$ are thus circularly polarized for all fields $H<H_{\rm c2}$ [Fig. 3a and Supplementary Videos 5-8 (at $H=0$) and 9-12 (at $H=0.1\,H_{\rm c2}$)] and exhibit $\varepsilon_{\sigma}=0$.

For a sample with ellipsoidal shape within the $xy$ plane, i.e., $N_{x}\neq N_{y}$, the polarization of the $\pm$ Q modes is no longer circular and $\varepsilon_{\sigma}$ becomes nonzero. This is illustrated in the main panel of Fig. 3b, where we show ellipticities calculated for $0\leq N_{x}\leq 1-N_{z}$ at different field values. We set $\varepsilon_{\sigma}$ positive and negative for the long principal axis being along $\hat{x}$ and $\hat{y}$, respectively [sketches on the right side of Fig. 3b].

We first focus on $\varepsilon_{\sigma}$ in zero field (black lines). As soon as $N_{x}\neq N_{y}$ the frequency degeneracy of modes $+$Q and $-$Q is lifted, and the modes are found to be strictly linearly polarized. In zero field, the spins of the chiral magnet align perpendicular to the helical propagation vector Q, so that the helix possesses a 180*∘*-rotation symmetry around each of its spins. By virtue of this symmetry, there exist always pairs of spins within the helix whose precessional motion conspire to yield a linear polarization along one of the principal axes, i.e. $x$ or $y$. This scenario is similar to the easy-plane AFM Sievers::PhysRev::1963 . As a consequence, the ellipticities of modes +Q (solid line) and $-$Q (dotted line) follow step functions $\varepsilon_{\sigma}|_{H=0}=\mathrm{sgn}[\sigma\cdot(N_{x}-N_{y})]$ exhibiting $\sigma=\pm 1$ with an unexpectedly pronounced sensitivity regarding the sample shape. This striking sensitivity is known neither from electronic excitations nor from ferrimagnets or easy-plane antiferromagnets.

Spectral weights

The characteristic shape dependence of $\varepsilon_{\sigma}$ of a DMI-containing magnet has an impact on the spectral weights $\varGamma_{\sigma}$ [Fig. 3c]. For the spectral weight of the $\pm Q$ modes at $H=0$ we find

[TABLE]

with the Heaviside step function $\varTheta(s)=1$ for $s>0$ and zero otherwise ($\chi_{\mathrm{con}}^{\mathrm{int}}$ is the internal conical susceptibility Schwarze2015 ). Here we consider an rf field $\bf h$ along the $x$ direction. Hence, selective excitation of either mode +Q or $-$Q can be realized. For instance, mode +Q is polarized along $\hat{y}$ ($\epsilon_{+}=-1$) for $N_{x}<N_{y}$ [Fig. 1c]. An rf field $\mathbf{h}$ along $\hat{x}$ does not couple to its precessional motion and, as a consequence, the spectral weight $\varGamma_{+}$ is zero in the helical phase at $H=0$ [full black line in Fig. 3c for $N_{x}<0.235$]. For mode $-$Q, $\epsilon_{-}=+1$ [polarized along $\hat{x}\,||\,\mathbf{h}$ in Fig. 1d] and $\varGamma_{-}$ is large [dotted black line in Fig. 3c for $N_{x}<0.235$].

For $H>0$, spins within the helix cant towards the field direction forming the conical state. Note that considering the ellipticities and spectral weights of both modes in Fig. 3b and c, respectively, their field dependencies are not symmetric with respect to $N_{x}=0.235$ (disk-like shape). For $N_{x}\ll N_{y}$, $\varepsilon_{-}$ of mode $-$Q stays close to $+1$ [colored dotted lines in Fig. 3b for $N_{x}<0.235$ and Supplementary Videos 13-16] and its linear polarization persists over a remarkably large field range. We extract $\varepsilon_{-}=0.999$ at $H=0.5\,H_{c2}$ for $N_{x}=0.07$ (our sample) so that an rf field ${\bf h}\parallel\hat{x}$ can couple to the -Q mode. Nevertheless, the spectral weight $\Gamma_{-}$ decreases significantly with increasing $H$ towards $H_{c2}$ (colored dotted lines in Fig. 3c) because of the mismatch of handedness with the Kittel mode within the FP phase as discussed above. Conversely, $\varGamma_{+}$ grows much with $H$ as the ellipticity $|\varepsilon_{+}|$ of the mode +Q first decreases reaching circular polarization at $H_{\rm circ}\approx 0.76\,H_{\rm c2}$ and then increases again staying below $\varepsilon_{+}=1$ [colored full lines in Fig. 3b, Supplementary Equations (5)-(8) and Supplementary Notes II]. These features explain our experimental observations in Fig. 2c and 2f in which signal strengths of mode $-$Q and +Q interchange completely with increasing $H$. The diagonal placement of Fig. 2d allows us to detect both modes +Q and $-$Q at $H=0$ as $\mathbf{h}\,||\,\hat{x}$ can be decomposed into equal components along and transverse to the bar-shaped Cu2OSeO3. Still, the signal strengths of mode +Q and $-$Q are found to be different at $H=0$. We have applied the same formalism to the three eigenmodes of the SkL phase. As the SkL phase only exists at a finite magnetic field $H$, the clockwise and counterclockwise modes are always elliptically polarized with a finite $\varepsilon$, whereas the breathing mode is linearly polarized by symmetry (Supplementary Notes II).

In Fig. 4 we compare the real part of the complex effective microwave susceptibility $\chi_{\rm eff}(f)$ Kalarickal::JAPL::2006 recorded at $57.3$ K in different phases of Cu2OSeO3. The resonance positions (inflection points) of the three curves coincide but the signal strength (peak-to-peak amplitude) varies. The largest signal strength is obtained for the field-polarized phase when the magnetization $M$ of the ferrimagnet is maximum. Note that Cu2OSeO3 and the technologically relevant ferrimagnet YIG exhibit a similar magnetization $\mu_{0}M$ of $0.13$ T 2012:Adams:PhysRevLett ; Schwarze2015 and $0.18$ T 2013:AllivyKelly:APL , respectively. At $H=0$ where the overall magnetization of the helical phase amounts to zero, we do not find the susceptibility of Cu2OSeO3 to be reduced significantly compared to its field-polarized phase (Fig. 4). We hence expect an optimized chiral insulator with low damping at room temperature to become technologically relevant. The linear polarization of the helical phase and the related linear dichroism in the few GHz frequency regime open new avenues for microwave components. For example, a conventional linear polarizer for free-space GHz radiation consists of a large grid prepared from macroscopic metallic rods mounted on a large frame. Contrastingly, our results offer a field-tunable magnetic polarizer based on a compact material. The linear polarization in chiral magnets reported in our manuscript is very robust and arises for any generic sample shape deviating from a circular shape, as it is controlled by the symmetries of the helical spin structure that is stabilized by the Dzyaloshinskii-Moriya-interaction. Magnon-photon cavities is one of many future technologies where DMI-induced linear dichroism offers novel, exciting perspectives for microwave control with integrated magnetic components by boosting the coupling between linearly polarized electromagnetic waves and magnons.

In conclusion, we studied collective spin excitations of an insulating ferrimagnet coupled to a GHz magnetic field and discovered the shape-controlled linear polarization of cross-polarized spin-helix modes. In our experiments we selected the complementary spin-helix modes by rotation of the sample, i.e., by interchanging demagnetization factors. To explain this, we explored the ellipticity of magnetization dynamics in the presence of Dzyaloshinskii-Moriya interaction. On the one hand, the extraordinary linear polarization of magnetization precession is of fundamental interest as it is not known from the extensive studies on other magnetic materials at low GHz frequencies. On the other hand, it allows for the efficient exploitation of magnetization dynamics by standard microwave equipment such as cavities or coplanar waveguides offering a high degree of on-chip integration.

Methods

The crystals were grown by means of chemical vapor transport using HCl as transport agent Gnezdilov2010 and crystallized as cubic chiral magnets in space group $P2_{1}3$. Cu2OSeO3 is considered as a local-moment ferrimagnetic insulator 1977:Kohn:JPhysSocJpn ; 2010:Belesi:PhysRevB ; Janson2014 . The combination of exchange and Dzyaloshinskii-Moriya interaction (DMI) results in long-wavelength helimagnetism (wavelength $\lambda_{\mathrm{h}}=620\,\mathrm{\AA}$) below a critical field $H_{c2}$ and below $T_{c}$ 2012:Adams:PhysRevLett ; 2012:Seki:PhysRevB . We connected a vector network analyzer to both ends of the CPW to excite and detect the spin dynamics by measuring the scattering parameter $S_{12}$ in transmission configuration Schwarze2015 . A reference spectrum was subtracted to obtain $\Delta|S_{12}|$ and enhance the signal-to-noise ratio. We do not address the nonreciprocal directional dichroism attributed to magnetoelectric coupling in Cu2OSeO3  2012:Seki:PhysRevB2 ; 2012:White:JPhysCondensMatter ; 2013:Okamura:NatCommun ; WhitePRL2014 ; 2015::Okamura ; Mochizuki2015PRL and therefore limit our discussion to the magnitude of the parameter $|S_{12}|$. Note that for a substrate with a relative dielectric constant $\epsilon_{\rm r}=1$, electrical and magnetic components of the electromagnetic wave are linearly polarized in a 50 $\Omega$ matched CPW Wen1969 . We assumed our Rogers substrate to approximate this condition as $\epsilon_{\rm r}=3.5$ was small. The spatial distribution of $\mathbf{h}$ does not vary much in the few GHz frequency regime addressed here. The field dependency of eigenfrequencies $f$, ellipticity $\epsilon$ and spectral weight $w$ of modes were modelled using the theoretical approach of Ref. [Schwarze2015, ] that included, both, DMI and dipolar effect due to the sample shape. Note that the propagation vector $\mathbf{Q}$ for helical and conical modes +Q and $-$Q [Fig. 1c,d] was oriented along the $z$ axis ($\hat{z}$) in our experiments as we measured spectra after saturating the sample along the corresponding $\langle 100\rangle$ axis being parallel to $\hat{z}$. For the analysis in Fig. 2e-g, the frequencies and field values are normalized as introduced in Ref. [Schwarze2015, ]. The normalization allows us to present consistently the different datasets taking into account the slight variation of temperature $T$ when remounting the sample in a different orientation (Supplementary Notes I). The lines indicate eigenfrequencies for different modes as predicted by the theoretical approach outlined in Ref. [Schwarze2015, ]. When extracting the susceptibility from scattering parameters $S_{\rm 12}$ we corrected for the slight impedance mismatch between the sample and CPW by adjusting the delay time of the electromagnetic wave. The real part of the complex effective microwave susceptibility $\chi_{\rm eff}(f)$ Kalarickal::JAPL::2006 is calculated via \chi_{\rm eff}(f)=i\,\Bigg{(}\frac{\text{ln}\big{[}S_{\rm 12}^{\text{meas}}(f)\big{]}}{\text{ln}\big{[}S_{\rm 12}^{\text{ref}}(f)\big{]}}-1\Bigg{)}, where $f$ is the frequency, $S_{\rm 12}^{\text{meas}}$ is the scattering parameter measured in the corresponding magnetic phase and $S_{\rm 12}^{\text{ref}}$ represents a reference spectrum containing the field-independent background signal.

Acknowledgements

We thank S. Mayr for assistance with sample preparation, as well as A. N. Slavin, M. Bailleul, N. Kanazawa, F. Lisiecki, and S. Watanabe for fruitful discussions and support on the design of antennas for broadband spectroscopy. Financial support through DFG TRR80 (From Electronic Correlations to Functionality), DFG FOR960 (Quantum Phase Transitions), and ERC AdG (291079, TOPFIT) is gratefully acknowledged. A.B. acknowledges financial support through the TUM graduate school.

Author contributions

I.S., J.W. and D.G. planned the experiment. I.S. and S.W. conducted the experiments. I.S., S.W., A.B., D.G. analysed the experimental data. H.B. fabricated the samples. J.W. and M.G. performed the theoretical calculations. D.G. and C.P. proposed this study. I.S, A.B., M.G. and D.G. wrote the manuscript. All authors discussed the data and commented on the manuscript.

Additional information

Supplementary information is available in the online version of the paper. Reprints and permissions information is available online at www.nature.com/reprints. Correspondence and requests for materials should be addressed D.G.

Competing financial interests

The authors declare no competing financial interests.