Linearly polarized GHz magnetization dynamics of spin helix modes in the ferrimagnetic insulator Cu2OSeO3

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 Cu2OSeO3 exhibits linearly polarized magnetization dynamics at an unexpectedly small frequency of about 2 GHz at zero magnetic field. Unlike optical filters that are assembled from filamentary absorbers, the magnet is shown to provide 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° via a small variation in width. Our findings shed a new light on magnetization dynamics in that ferrimagnetic ordering combined with antisymmetric exchange interaction offers strictly linear polarization and cross-polarized modes for a broad spectrum of sample shapes at zero field. The discovery allows for novel design rules and optimization of microwave-to-magnon transduction in emerging microwave technologies.


I. EXPERIMENTAL ASPECTS
A. Crystals of Cu 2 OSeO 3 Single crystals were grown by vapor transport [1]. Samples that crystallized in the space group P 2 1 3 were oriented by Laue x-ray diffraction, cut with a wire saw, and carefully polished to size. The samples had a size of about 0.4 × 2.3 × 0.3 mm 3 . Their corresponding edges were collinear to crystallographic orientations along as follows: 110 , 110 and 100 (sample #1), 100 , 100 and 100 (sample #2), 110 , 211 and 111 (sample #3), respectively. In the main text we focus on sample #1. Samples #2 and #3 were used for additional measurements with a coplanar waveguide of small signal-line width.

B. Experimental setup
In Fig. 1a we show a photograph of sample #1 mounted in the cryostat. The coplanar waveguide (CPW) was contacted via non-magnetic microwave probe tips (not shown; sup-plier GGB Industries). Via holes provided electrical and thermal contact between ground lines and the copper sample stage, where the heater is mounted. The superconducting magnet of the LakeShore probe station CPX-VF provides a perpendicular field H of up to 2.5 T.
Near-and far-field in-and out-of-plane amplitude profiles of the dynamic magnetic fields h have been calculated for different CPWs [2]. For the simulations we considered the layout of coplanar waveguides with different geometrical parameters such as widths of signal and ground lines, gaps and layer thicknesses (see Table I). This was done to optimize for an impedance of 50 Ω. Via holes in the commercial CPW contacted an additional metallic ground plane on the backside of the CPW substrate [ Fig. 1a]. They redistributed the back-flowing radiofrequency currents in the ground lines, but were not expected to alter considerably the strong magnetic field components close to the signal line [3]. Hence, they were not considered in the simulations. Field profiles of the broad CPW are summarized in in the same way on the narrow CPW as sample #2. Here, the high symmetry direction 111 was collinear with the applied field H. The spectra resembled the ones observed for H parallel 100 . We encountered a small shift of eigenfrequencies only, consistent with the cubic anisotropy being the weakest energy scale in the chiral magnet [4].

D. Sample temperature
In our experiment the sample temperature was measured at the sample stage, which is thermally anchored to a cold finger. For each probing configuration described in this work, there was a slightly different thermal contact between the sample, the CPW and the sample stage. At each cool down, we chose a temperature T in the vicinity of 57 K, where the SkL was stabilized. The exact values for T and the H c2 values used to normalize the data in

A. Ferromagnets
According to the orthodox understanding of magnetic resonance [5], the ellipticity ε of a general ellipsoid of a field-polarized ferromagnetic material is given by where directionsx,ŷ,ẑ and magnetic field H are oriented as sketched in Fig. 1a of the main text. M is the magnetization and N i (i = x, y, z) are the components of the demagnetization tensor. For Eq.
(1), one assumes N y ≥ N x and N x + N y + N z = 1 with the external field H applied along the ellipsoidalẑ axis [5].
For a cylindrical sample shape N y = N x the oscillating magnetization is circularly polarized with ellipticity ε = 0. For N y > N x the polarization is elliptically deformed so that ε > 0. The maximal value of ε is obtained in the limit of zero internal field for which the Kittel resonance frequency reduces to In order to obtain → 1, we get N x /N y → 0. At the same time, however, according to

B. Chiral magnets
The field dependence of the eigenfrequencies in Cu 2 OSeO 3 as well as ellipticities were modelled in the framework of the theory outlined in Ref. [6] by T. Schwarze et al.. The result depends on the internal conical susceptibility χ int con , i.e., the magnetic susceptibility within the conical phase with χ int con = 1.76 for Cu 2 OSeO 3 , and the temperature dependent critical field H c2 (T ) (see Table II) where the transition into the field-polarized phase occurs.
The sample shape is modelled by an ellipsoid with demagnetization factors N x , N y , and N z where the field is applied along the principalẑ axis.

Ellipticity for the helix at zero and finite magnetic fields
The mean magnetization for the +Q and −Q modes of the helix, σ = 1 and σ = −1, respectively, oscillates, and its dynamic part has the form where we assumed that the pitch is aligned with the principalẑ axis, and m x σ and m y σ are the amplitudes along the principalx andŷ axes. An explicit expression for the eigenfrequencies ω σ was given in Ref. [6]. The mean magnetization oscillates counterclockwise for the +Q mode and clockwise for the −Q mode within the plane perpendicular to the pitch, i.e., thê z axis. We neglected the influence of cubic anisotropies which will be discussed elsewhere.
The result depends on the constant internal susceptibility of the conical phase χ int con = χ con /(1−χ con N z ), the demagnetization factors N x , N y , N z , and the magnetic field h = H/H c2 where H c2 is the critical field. The ellipticity ε σ > 0 is given by In the main text, we also introduced a negative ellipticity in case m y σ > m x σ and positive otherwise.
In the limit h → 0, the ellipticity reduces to a step function as discussed in the main text.
In the other limit h → 1 − , the ellipticity of the +Q mode, ε + , reduces to the expression of Eq. (1) for a field-polarized magnet. The +Q mode has the same handedness as the Kittel mode so that they are smoothly transformed into each other at the second-order transition at the critical field H c2 .
Interestingly, the +Q mode changes the polarization axes at some intermediate field h circ + as can be seen in Fig. 3b of the main text. The +Q mode is circular polarized, ε + = 0, at this field h circ + irrespective of demagnetization factors N x and N y . We obtain for this field the explicit expression h circ + = 2 + χ int con (2 + χ int con )(2 + (2 − N z )χ int con ) + (1 − N z )χ int con (2 + χ int con ) 2 (4 + (1 − N z )χ int con ) .
For our Cu 2 OSeO 3 sample with χ int con = 1.76 and N z = 0.53 we get h circ + ≈ 0.76.

Ellipticity in the Skyrmion lattice phase
In Fig. 3 we show the results of our calculation for the ellipticity and spectral weight distribution of the gyrational Skyrmion lattice (SkL) modes, CCW and CW. We concentrate on field values around 0.5 H c2 , where the SkL phase is stable. We find that linear polarization, i.e. |ε| = 1 is not achieved for any sample shape. The spectral weight Γ of the counterclockwise (CCW) mode is larger compared to the clockwise (CW) mode for every sample shape. The mean magnetization oscillates counter-clockwise and clockwise for the CCW and CW mode, respectively, within the plane perpendicular to the applied magnetic field.
The spectral weight of the CCW mode increases with increasing field H. For the CW mode one finds the opposite trend. Note that the ellipticity of the CCW mode is practically independent of the field and already well described by the Kittel expression of Eq. (1) indicating that it is basically dominated by the polarized background. As the CCW and CW modes are elliptically polarized, either sample placement that covers CPW signal line and gaps allows one to monitor them in the same spectrum. The breathing mode is linearly polarized by symmetry and can be excited only with the help of a longitudinal dynamical field h z . [