#  Time-resolved measurement of spin excitations in Cu2OSeO3

D. M. Burn  Diamond Light Source, Harwell Science and Innovation Campus,
Didcot, OX11 0DE, United Kingdom  S. L. Zhang  School of Physical Science and
Technology, ShanghaiTech University, Shanghai 200031, China  ShanghaiTech
Laboratory for Topological Physics, ShanghaiTech University, Shanghai 200031,
China  G. van der Laan  Diamond Light Source, Harwell Science and Innovation
Campus, Didcot, OX11 0DE, United Kingdom  T. Hesjedal  Clarendon Laboratory,
Department of Physics, University of Oxford, Parks Road, Oxford, OX1 3PU,
United Kingdom

###### Abstract

Magnetic diffraction in combination with x-ray detected ferromagnetic
resonance (DFMR) is a powerful technique for performing time-resolved
measurements on individual spin textures. Here, we study the ferromagnetic
resonance (FMR) modes of both the conical and field-polarized phases in the
chiral magnet Cu2OSeO3. Following the identification of the FMR modes at
different temperatures using broadband vector network analyzer FMR, we use
DFMR on the crystalline (001) Bragg peak to reveal the time-dependent spin
configurations of the selected FMR modes. By being able to measure both the
amplitude and phase response of the spin system across the resonance, a
continuous phase advance (of $180^{\circ}$) in the conical mode, and a phase
lag (of $180^{\circ}$) in the field-polarized mode is found. By performing
dynamic measurements in the conical phase as a function of the linear
polarization angle of the x-rays, i.e., successively probing the dynamics of
the moments, we find an inversion of the dynamics along the conical axis upon
inverting the applied field direction. By allowing for time-resolved
measurements of the phase and amplitude of individual magnetic phases, DFMR
opens up new opportunities for obtaining a deeper understanding of the complex
dynamics of chiral magnets.

##  I Introduction

Topological spin textures in chiral magnets such as Cu2OSeO3 are of great
interest for potential advances in spintronics [1, 2]. Key to the success of
future technologies are their dynamic properties, enabling fast and energy-
efficient transitions of the magnetic states. One of the most commonly used
techniques for the study of the magnetization dynamics is ferromagnetic
resonance (FMR) [3, 4, 5, 6]. In broadband coplanar waveguide (CPW) FMR, the
resonances of a sample are probed by placing it on the CPW and subjecting it
to a variable magnetic field. By sweeping an RF signal fed to the CPW in the
range from typically 0 to 20 GHz, and by measuring the transmitted RF power,
the resonances reveal themselves as absorption maxima in frequency vs field
maps.

Previous vector network analyzer (VNA) FMR studies have probed the
magnetization dynamics of the resonance modes in Cu2OSeO3, i.e., the helical,
conical, and skyrmion phase, as well as the field-polarized phase, revealing
their evolution in field-frequency space [1, 7, 8, 9, 10]. Supported by
micromagnetic simulations, three internal breathing and rotational modes have
been identified in the skyrmion phase [1], two each in the helical and conical
states, and one in the field-polarized state [7].

Within the conical phase, the two resonance modes that have been resolved can
be attributed to vibrational modes in the conical structure and are labeled
$+Q$ and $-Q$ [1]. While their magnetization oscillates uniformly, the two
modes have opposite helicity and oscillate (counter)clockwise in the ($+Q$)
$-Q$ mode [10]. For their excitation, the AC field has to be perpendicular to
the bias field direction, and hence the conical propagation axis [1]. As well
as the geometry of the field direction, the excitation also depends on the
sample geometry due to the anisotropy [7, 8]. The Gilbert damping parameter is
small and was found to be $<$0.003 in Cu2OSeO3 at 5 K (from the FMR linewidth)
[9]. Note that the counterclockwise oscillating $+Q$ mode seamlessly connects
to the also counterclockwise oscillating Kittel mode present in the field-
polarized phase above a critical field, while the spectral weight of the $-Q$
mode is vanishing. The counterclockwise rotation sense in the field-polarized
state is a result of the fact that single electron moments, representative of
a uniform magnetic state, simply precess counterclockwise around a magnetic
field vector.

Resonant elastic x-ray scattering (REXS) provides a unique, element-specific
view of ordered magnetic structures, and has been extensively used for the
study of Cu2OSeO3 [11, 12, 13, 14, 15] and other chiral magnetic systems [16,
17, 18, 19, 20, 21]. The use of soft x-rays (via x-ray magnetic circular
dichroism) to measure magnetization dynamics through x-ray detected
ferromagnetic resonance (XFMR) is a valuable technique [22, 23, 24, 25],
giving access to the dynamics of individual layers via the element-selectivity
of x-ray spectroscopy in few-layer stacks such as spin valves [26] and
magnetic tunnel junctions [27]. The combination of the two, REXS and XFMR,
allows for dynamic studies of diffraction-selected specific magnetic modes,
and is particularly useful for systems in which the eigenmodes are complex and
their interpretation crucially relying on micromagnetic simulations. The
capabilities of this diffractive FMR (DFMR) technique was demonstrated on the
rich magnetic structure of a Y-type hexaferrite [28], and it was further
expanded into the probing of superlattices via x-ray reflectometry [29]. A
similar technique, REXS-FMR, was successfully applied for the study of, most
notably, the three skyrmion eigenmodes in Cu2OSeO3 via the crystalline (001)
Bragg peak and its magnetic satellites [30]. While REXS-FMR is a homodyne
detection technique which relies on the difference of probed magnetic peak
intensities between RF on and off, heterodyne detection DFMR is a truly time-
resolving technique, giving access the phase of the dynamic response [28].
Also note that both REXS-FMR and DFMR can be carried out on the magnetic
satellites as well, however, due to the complexity of the analysis, the
present study will focus on dynamic Bragg peak studies.

Here, we report a time-resolved dynamic study of the conical and field-
polarized phase in Cu2OSeO3 using the diffractive ferromagnetic resonance
technique, employing both circularly and linearly polarized x-rays. First, the
dynamic phase diagram is mapped out using VNA-FMR with the field applied in-
plane, identifying the conical and field-polarized modes and characterizing
their temperature and field dependence. The characteristics of the resonance
modes in the conical and field-polarized phase, as well as the dependence of
the critical field on temperature, is well-captured using the established
models [10]. Using DFMR, we investigate the dynamic behavior of the conical
$+Q$ and the field-polarized mode, as well as the transition between them, in
greater detail. As the field-polarized phase does not result in magnetic
scattering peaks, we restrict our study to the analysis of the (001) Bragg
peak, whose magnetic intensity is susceptible to both phases. Note that, in
principle, any reflection peak could be probed which has a magnetic
contribution. We find a 180∘ phase shift across the resonances of the conical
and field-polarized modes, which is negative and positive, respectively.
Further, while the conical resonance peak position is temperature-dependent,
the field-polarized resonance is temperature-independent. Finally, by
employing linear polarized x-rays of variable polarization angle, we are able
to access the dynamics of the projected moments along the conical texture at
resonance. We find that upon reversing the applied field, the dynamic response
is inverted.

##  II Experimental

The dynamic behavior of the conical and field-polarized phase in Cu2OSeO3 was
measured using broadband VNA-FMR [31] and synchrotron-based DFMR [28]. Both
techniques were performed in-situ in the RASOR diffractometer on beamline I10
at the Diamond Light Source (Oxfordshire, UK). A high-quality, (001)-cut
Cu2OSeO3 sample, measuring 2 $\times$ 2 $\times$ 0.1 mm3, was polished and
placed on a coplanar waveguide (CPW). The CPW was attached to a cold-finger
cryostat at the center of the diffractometer, and cooled to temperatures in
the range from 25 to 60 K in ultrahigh vacuum. A magnetic bias field was
applied in the plane of the sample by way of two permanent magnets, whereby
the field strength can be controlled by varying the separation between the
magnets. This bias field was oriented in the scattering plane, and
perpendicular to the radio frequency (RF) field produced by the CPW. The
schematic of the experimental setup is illustrated in Fig. 1.

First, the static magnetic scattering from the sample was characterized.
Cu2OSeO3 [cubic space group $P$213 (198)] has a lattice constant of 8.925 Å.
At the Cu $L_{3}$ edge (931.25 eV), this results in the (001) structural Bragg
peak at $2\theta=96.5^{\circ}$ [13]. Note that this (001) Bragg peak is
‘forbidden’ and only observed in resonant x-ray scattering [11, 12, 32, 33].
In resonant scattering, the magnetic state of Cu2OSeO3 can be accessed via the
structural Bragg peak [30], i.e., effectively probes the $k=0$ dynamics, or,
alternatively the magnetic satellites ($k\neq 0$ modes, e.g., the skyrmion
breathing mode) [11, 12].

Figure 1:  Schematic of the ferromagnetic resonance setup, allowing for both
VNA-FMR and DFMR measurements in the RASOR diffractometer at the Diamond Light
Source. The Cu2OSeO3 sample is placed on the CPW, which is mounted on the cold
finger inside the diffractometer. The incident x-rays, which are tuned to the
Cu $L_{3}$ edge, are scattered off the sample and detected via a photodiode in
a $\theta$-$2\theta$ geometry. Circularly as well as linearly polarized x-rays
were used, whereby the linear polarization $\epsilon_{i}$ was continuously
variable, covering $\epsilon$ being in the scattering plane ($\pi$) and
perpendicular to it ($\sigma$). A linear polarization angle of $0^{\circ}$
corresponds to linear horizontal ($\sigma$) polarization. The variable
magnetic field is applied in the scattering plane via pairs of permanent
magnets whose distance can be controlled. An RF signal is fed to the CPW to
drive the ferromagnetic resonance in the magnetic sample, with its field
effectively being perpendicular to the scattering plane at the probed sample
position. As the synchrotron conveniently produces x-ray pulses at a frequency
of $\sim$500 MHz, a comb generator is used to produce higher harmonics, which
are selected and fed to the CPW. To probe the time-dependence of the scattered
x-ray intensity, a tunable delay line is used, which shifts the phase between
the pump (the RF signal) and the probe (the pulsed x-rays).

To cover an extended field range, the measurements were performed in two
parts, whereby the smaller permanent magnets were replaced by larger ones for
the second part of the measurements. As their field ranges were not perfectly
overlapping, fields around 30 mT could not be covered (white areas in the
resonance plots in Fig. 2).

As a first step, the dynamic phase space of the system was mapped out using
VNA-FMR. The RF power transmitted through the CPW, $S_{21}$, is measured,
whereby its loss ($\Delta S_{21}$) as a function of frequency represents the
FMR spectrum. In VNA-FMR measurements, this absorption is typically mapped out
as a function of RF frequency and magnetic bias field. However, in this
investigation, the absorbed RF power also depends on the magnetic phase of the
sample, and thus the excitable magnetic spin configurations, resulting in a
temperature and extra field dependence of the spectra. Therefore, in this
investigation, we map out the RF absorption of the sample as a function of all
three quantities, i.e., frequency, field, and temperature. In practice, this
is best achieved by performing frequency sweeps while slowly decreasing the
temperature. The difference in sweep rates of both quantities means that the
temperature deviation over the time taken to measure a frequency sweep is
negligible. These measurements are then repeated at various magnetic fields.
This effectively measures the dynamic properties of the system as prepared
using a field-cooling protocol.

As the temperature was varied continuously while taking frequency sweeps, the
temperature steps need to be interpolated to enable further processing and the
analysis of two-dimensional slices of frequency-field at fixed temperature or
field-temperature at fixed frequency. In VNA-FMR measurements, the background
resulting from the electrical characteristics of the measurement setup is
subtracted. This background is typically obtained from a high-field
measurement where the moments are saturated and any dynamic features are
driven to sufficiently high frequencies to be out of the measurement range.
Due to the limitations in the field range in the RASOR diffractometer, we
adopt an alternative approach, by subtracting a background measured in the
field polarized state at 60 K, i.e., at a temperature above $T_{\mathrm{C}}$.

After the detection of the dynamic modes using VNA-FMR, the modes were further
investigated using polarized soft x-rays. Both circularly and linearly
polarized x-rays were used, with an energy tuned to the Cu $L_{3}$ edge at
931.25 eV. The beam spot size at the sample position is typically 100 $\times$
100 $\mu$m2, and the scattered beam was measured using a photodiode in a
$\theta$-$2\theta$ configuration with slits providing an angular resolution of
$\sim$3 mrad.

Initially measurements of the scattered intensity as a function of reciprocal
lattice space coordinates were used to map out the static magnetic phase
diagram of the sample and to optimize the alignment of the diffractometer. The
detector was then positioned at the (001) Bragg peak of Cu2OSeO3.

The polarized soft x-rays provide a probe of the magnetization state in the
sample due to circular and linear magnetic dichroism in resonant elastic x-ray
scattering [34]. As noted above, in resonant scattering on Cu2OSeO3, both the
(001) Bragg peak [30] as well as its surrounding magnetic satellites [11, 12]
can be used to probe the magnetic states of Cu2OSeO3. Here, due to the
complexity of depth-dependent information a resonant soft x-ray study of the
magnetic satellites provides, we focus on the analysis of the (001) Bragg peak
data.

For the dynamic measurements, the scattered intensity at the (001) Bragg peak
was measured stroboscopically, taking advantage of the pulsed nature of the
synchrotron radiation at a frequency of $\sim$500 MHz. Here, the RF magnetic
field generated from the CPW beneath the sample was phase-locked to an integer
multiple of the synchrotron master clock. Due to the reduced thickness of the
sample ($\sim$100 $\mu$m), the RF power delivered from the CPW was sufficient
to excite the magnetization precession dynamics on the Cu2OSeO3 top surface,
probed by the x-rays. The intensity of the scattered x-rays provides a measure
of the magnetic state of the system for a particular point in time during the
precession of the magnetization. Taking such snapshots for increasing delay
time between the RF pump and the x-ray probe are then giving the time-
dependent magnetization behavior. The signal-to-noise ratio in the dynamic
component of the scattered intensity is further enhanced through lock-in
detection, using a 180∘ modulation of the phase of the RF pump at a frequency
of 700 Hz. For our measurements, the RF frequency was set to 4 GHz where a
clear FMR mode crossing is found for both the conical and field polarized
phases. Both of these mode crossings are within the field range accessible in
the diffractometer. Measurements of the dynamic signal as a function of pump-
probe delay and magnetic field were performed at various temperatures with
both left and right circularly polarized x-rays, and also with linearly
polarized x-rays as a function of the incident polarization angle.

##  III Results

The ferromagnetic resonance modes of the Cu2OSeO3 sample were first studied in
the RASOR diffractometer using VNA-FMR. The frequency spectra were measured
during a series of field-cooling sweeps, i.e., as a function of temperature.
Slices of this data as a function of frequency and field at different
temperatures, and also as a function of field and temperature at different
frequencies are shown in Fig. 2(a-d) and Fig. 2(e-h), respectively. The color
scale represents the magnitude of transmitted RF power, $S_{21}$, revealing
strong absorption features corresponding to magnetic resonance modes in the
sample.

Figure 2: Vector network analyzer ferromagnetic resonance measurements showing
plots of the FMR spectra ($\Delta S_{21}$, linear scale in arbitrary units)
for different measurement conditions. The data was obtained from a series of
field-cooling temperature sweeps at different fields, with the field applied
perpendicular to the RF excitation direction. Slices of the data show the
absorption as a function of (a-d) frequency and field, and (e-h) field and
temperature. The strong absorption features represent magnetic resonance modes
in the sample. The vertical dashed line (a-d) and curved lines (e-h) represent
the magnetic phase transition field, $H_{c2}$, between the conical and field
polarized phase. Model plots of the resonance curves for both phases (black
dashed lines) are overlaid on top of the data in panels (a-d), and the dotted
white diagonal line represents the behavior of the Kittel mode in a perfect
sphere. The horizontal dashed line at 4 GHz (a-d), and the vertical dashed
lines at 25, 30, 35, and 40 K in (h) indicate the parameters for which the
DFMR measurements were carried out.

At the critical field $H_{c2}$, an abrupt change in the magnetic resonance
behavior is observed, which is indicated in Fig. 2(a-d) by the vertical, and
in Fig. 2(e-h) by the curved dashed lines. This phase boundary separates the
conical phase at lower fields from the field-polarized phase which exists
above the transition. Both of these phases show different magnetic resonance
behavior and there is a continuous transition between them crossing the phase
boundary.

The resonance behavior of the field-polarized mode above $H_{c2}$ is
adequately described by the Kittel equation, i.e., the description for a
uniform ferromagnet (see Eq. (14) in Ref. [10]). Note that while the sample in
the present case is a flat plate with its surface normal along the
$z$-direction (i.e. with demagnetization factors of $N_{x}\approx N_{y}\approx
0$ and $N_{z}\approx 1$), the static magnetic field is applied along the
$x$-direction, while the RF field, $H_{\mathrm{RF}}$, is applied along the
$y$-direction (see Fig. 1). The measured resonance curves [Fig. 2(a-d)] lie
above the straight (white dotted) frequency-field line representing the
behavior of a perfect sphere, i.e., when demagnetization is irrelevant
(demagnetization factors $N_{x}=N_{y}=N_{z}=1/3$) and
$\hbar\omega=g\mu_{\mathrm{B}}\mu_{0}H$, which is the resonance frequency in
the paramagnetic limit. Note that the field-polarized phase branch can either
lie above (as here) or below the paramagnetic resonance (dotted white lines),
depending on whether the shape anisotropy effectively enhanced or reduces the
applied static field [35].

From fitting the Kittel equation to the data [dashed black lines above
$H_{c2}$ in Fig. 2(a-d)], the following parameters were obtained: for the
demagnetization factors, $N_{x}=N_{y}=0.0012$ and $N_{z}=0.9976$, and for the
internal susceptibility, $\chi_{\mathrm{con}}^{\mathrm{int}}=1.7$. This value
for $\chi_{\mathrm{con}}^{\mathrm{int}}=\mu_{0}M_{s}^{2}/(2AQ^{2})$, where
$M_{s}$ is the saturation magnetization, $Q$ is the pitch vector, and $A$ is
the stiffness, is consistent with the value of 1.76 reported in Ref. [10].

The conical phase, which exists below $H_{c2}$, has a more complex dynamic
behavior with two modes, $+Q$ and $-Q$, corresponding to the two vibrational
modes within the chiral spin texture [1, 7]. The conical modulation period in
Cu2OSeO3 is between 50 and $\sim$60 nm [36, 37]. Both modes show a decrease in
frequency with increasing field, and are separated by a difference in
frequency. At low field, the lower frequency $-Q$ mode contributes to a large
RF absorption. With increasing field, the $-Q$ mode decreases in intensity and
the $+Q$ mode becomes the dominant mode, in agreement with the theoretically
calculated behavior [10]. At $H_{c2}$, the $+Q$ mode continuously transforms
into the field-polarized Kittel mode [8, 7, 10]. These dashed black lines in
Fig. 2(a-d) represent fits to the data using the expression given by Garst _et
al._ in Eq. (30) of Ref. [10].

Further, the resonance modes have a subtle temperature dependence when
traversing the field-temperature phase diagram [Figure 2(e-h)]. In the field-
polarized phase, the resonance field experiences a slight increase up to
$T_{\mathrm{C}}$ = 57 K, above which the intensity of the mode vanishes. In
the conical phase, the resonant field of the $+Q$ and $-Q$ modes both decrease
with increasing temperature at fixed frequencies.

The VNA-FMR measurements shown in Fig. 2 reveal the behavior of the resonance
modes within the frequency-temperature-field phase space in Cu2OSeO3. However,
while the presence of power-absorbing resonances is clear from these
measurements, nothing can be learned about their nature. To gain further
insight into their time-dependent spin structure, the dynamic contribution to
the scattering at the (001) Bragg peak was measured while exciting specific
ferromagnetic resonance modes.

During resonance, the precession of the magnetic moments about the equilibrium
spin configuration results in a time-dependent variation of the projection of
the moments along the direction of the x-ray polarization vector. This time-
dependent projection in turn results in a time-dependent contribution to the
scattered intensity at the (001) Bragg peak. Figure 3(a) shows a colormap plot
of sequences of such time-dependent scattering intensities for an excitation
frequency of 4 GHz. The dynamic contributions to the scattering were recorded
as a function of the time delay between the RF pump and x-ray probe pulse, and
these so-called delay scans were further obtained as a function of external
bias field. The yellow regions indicate that the dynamic scattering is equal
to the scattered intensity from the static state, while red (blue) regions
indicate an increase (decrease) in the scattered intensity, respectively. The
delay scans show sinusoidal oscillations with a period of 250 ps, and
demonstrate that the dynamics in the spin configuration is coupled to the 4
GHz excitation frequency.

Figure 3: (a) Colormap showing the time dependent contribution to the (001)
Bragg intensity, plotted as a function of the magnetic field and the delay
between the RF pump and x-ray probe. The measurements were done with an
excitation frequency of 4 GHz, at a temperature of 25 K with the field applied
in-plane and with right circularly polarized x-rays at the Cu $L_{3}$ edge.
The color scale represents the value of the time-dependent contribution to the
(001) Bragg peak signal (in arbitrary units). The amplitude and phase data
extracted from sinusoidal fits to the delay scans as a function of field are
shown in panels (b) and (c), respectively. The vertical lines indicate
$H_{c2}$ for the respective temperature, separating the conical and field
polarized magnetic phases.

From the contrast in Fig. 3(a) it can be seen that the amplitude and the phase
of the oscillations of the (001) Bragg peak intensity change as a function of
the applied magnetic field. A strong enhancement of the contrast is observed
at 44 mT, and a comparably weaker enhancement between 80 and 100 mT,
corresponding to the two resonance modes observed in the 4 GHz VNA-FMR
measurements at 25 K, shown in Fig. 2(a). From the shifts of the patterns
across Fig. 3(a), especially around the fields at which the contrast is
strong, it can also be concluded that the phase of the sinusoidal dynamic
signal strongly depends on the applied magnetic field.

The dynamic contribution to the (001) Bragg intensity shown in Fig. 3(a) can
be quantified via sinusoidal fits to the delay scan data at each field point.
Figures 3(b) and 3(c) show the corresponding amplitude and phase,
respectively, as a function of the magnetic field at 25, 30, 35, and 40 K. For
all four temperatures, a peak is visible in the amplitude at $\sim$40 mT,
corresponding to the resonance mode in the conical phase. These peaks are
accompanied by a $-180$∘ phase change across the resonance. Both the magnitude
and the position of the amplitude peaks are a function of temperature, with
the amplitude and resonant field being largest at low temperature, and
decreasing with increasing temperature. Both of these features are consistent
with the VNA-FMR measurement data presented in Fig. 2(a-d).

The amplitude [Fig. 3(b)] also shows features in the field range from 80-100
mT, consisting of a more complex peak structure, which is also accompanied by
a (positive) phase change of 180∘ [Fig. 3(c)]. These features correspond to
the resonance of the field-polarized phase and they show, in contrast to the
conical mode, no apparent dependence on temperature in either peak amplitude
or resonance field. This behavior is consistent with the flatness of the
resonance over this temperature range, which can be seen in Fig. 2(h).

Note that the +180∘ phase shift over the field-polarized resonance with field
is opposite to the $-180$∘ phase shift over the conical resonance. This
leading or lagging behavior of the phase, is due to the different ways in
which the resonance is crossed, i.e., with a negative frequency-field slope in
case of the conical and a positive slope in case of the field-polarized
resonance [see Fig. 2(a) for comparison]. Note that the results shown in Fig.
3 are measured with right-circularly polarized light with the field applied
in-plane in the same orientation. When the field direction is reversed, the
dynamic signal still shows a peak in amplitude for the conical resonance and
the associated 180∘ phase change still occurs in the same direction.

The field-polarized resonance is rather broad and shows a detailed fine
structure in both amplitude and phase (Fig. 3). This behavior is due to the
existence of multiple magnetostatic modes in the sample, which are enhanced
due to the thin thickness of the sample [38]. These multiple overlapping
resonance modes combine, resulting in the complex feature in which the higher
frequency-field component is the strongest contribution. In the conical phase,
these magnetostatic modes are also present, but occur over a narrower
frequency range and are not resolved. Note that these modes are also present
in VNA-FMR measurements in both the field-polarized and conical phase,
however, hard to discern in the usual representation of VNA-FMR absorption
maps (they become visible upon differentiation of the signal). Further, the
low-field tail of the field-polarized resonance mode appears to ‘leak’ into a
field range below the $H_{c2}$ transition, i.e., into the conical phase range.
The temperature-dependent $H_{c2}$ values are indicated by the vertical dashed
lines in Fig. 3(b,c). This behavior points towards the existence of metastable
states, resulting from the field-cooling protocol employed in our
measurements.

Figure 4: Dynamic contribution to the (001) Bragg peak of the conical phase of
Cu2OSeO3 measured at the Cu $L_{3}$ edge with linearly polarized light as a
function of the linear polarization angle and the delay between RF excitation
and x-ray probe pulse. The applied field was oriented in-plane, and (a)
parallel ($H_{p}$) and (b) antiparallel ($H_{n}$) to the projection of the
incident beam onto the sample plane. The sample was excited with RF at 4 GHz
in an applied field of 43 mT at 25 K, i.e., exciting the conical $+Q$
resonance. (c) Phase of the dynamic contribution to the (001) Bragg peak
intensity as a function of the linear polarization angle.

The VNA-FMR and XFMR measurements (with circularly polarized x-rays) allow us
to identify the resonance conditions, and also obtain insight into the phase
changes taking place over those resonances. However, further insight into the
time evolution of the magnetic structure can be obtained by measuring the
(001) Bragg diffraction of linearly polarized x-rays as a function of the
linear polarization angle. The linearly polarized light introduces a mechanism
to provide directional contrast into the measurements of the magnetic
configuration of the system.

We have shown earlier that the $+Q$ resonance mode in the conical phase in our
Cu2OSeO3 sample occurs at 4 GHz in an in-plane bias field of 43 mT at 25 K.
With the system excited at this resonance, Fig. 4(a,b) shows the how the
magnetic contrast during this dynamic process varies as a function of both the
linear polarization angle and the time delay.

Similar to the measurements as a function of field in Fig. 3, the dynamic
signal has a sinusoidal dependence on the delay with a period of 250 ps,
originating from the 4 GHz RF excitation frequency. This dynamic signal shifts
in phase as a function of the incident linear polarization angle. The
particular phase of $\sim$$30^{\circ}$ that is measured for a linear
polarization angle of $0^{\circ}$ is given by the probed position within the
resonance curve at 43 mT [see Fig. 3(b,c) for comparison]. Further, the
reversal of the applied field direction reverses the direction of the phase
shift as a function of the incident linear polarization angle, as shown in
Fig. 4(a,b) for the projection of the field along the beam direction applied
parallel and antiparallel, respectively. Sinusoidal fits to the delay scan
data were used to extract the phase of the dynamic signal for both positive
($H_{\mathrm{p}}$) and negative field ($H_{\mathrm{n}}$), shown in Fig. 4(c).
The phase of the dynamic signals in Fig. 4(c) varies sinusoidally with the
linear polarization angle. Most importantly, the phase is inverted when the
direction of the magnetic field is reversed. This phase inversion can be
interpreted as an inversion of the polarization angle-dependent contrast,
which in resonant scattering probes the contribution of the spins along and
perpendicular to the polarization direction.

Figure 5: Illustration of equilibrium conical spin structures for applied
fields along the helical propagation axis (a,b), and opposite to it (c,d),
respectively. The sign of the Dzyaloshinskii–Moriya interaction (DMI), which
is specific to the two enantiomers, results in (a,c) left-handed (LH) and
(b,d) right-handed (RH) conical screws, respectively. See Supplemental
Material for a closer look at the LH conical screw in a positive applied field
shown in (a) [39].

In general, when the field direction is reversed and assuming that the spins
simply follow the applied field, the conical spins change their equilibrium
orientation, as illustrated in Fig. 5. Note that while the spins follow the
magnetic field direction, the conical helix will remain either right- and
left-handed, independent of the field direction. As the rotation sense of the
dynamical precession is determined by the direction of the applied field,
which is counterclockwise for a free electron spin, the rotation sense
reverses (in the reference frame of the sample) if the field direction is
reversed. As stated above, the conical dynamic mode which has the same
(counterclockwise) rotation sense as a free spin, and therefore the field-
polarized mode, is the $+Q$ modes. Both modes are seamlessly connected at the
transition field $H_{c2}$ (see upper row in Fig. 2). Note that this behavior,
i.e., the helicity of the modes, is unaffected by the chirality of the crystal
[10].

In our experiments, we selected the $+Q$ mode via the RF frequency at a given
applied field. Figure 6 shows an illustration of the dynamic behavior of the
$+Q$ mode upon field reversal. The panels show subsequent snapshots of the
precession of the spins about the local effective field. For a positive
applied field (along the helical propagation axis), a compression wave
propagates to the right, in the direction of the applied field. For a negative
applied field, the propagation direction inverts as well. This model is
consistent with our experimental results (Fig. 4).

Figure 6: Illustration of the dynamic conical spin structures (for $+Q$) in
applied fields along the helical propagation axis (a-c), and opposite to it
(d-f), respectively. The panels show sequential snapshots at $t=0$, $\Delta
t$, and $2\Delta t$. The spins precess about the local effective field,
describing a cone. The axes of the cones coincide with the directions of the
spins in the equilibrium conical structures (green lines) shown in Fig. 5. A
color scale has been superimposed as the background to indicate a compression
wave where spins point closer (red) and further apart (blue) from one another.
This compression wave propagates as a function of time. The propagation of the
compression wave is in opposite directions upon inversion of the applied
field. See Supplemental Material for videos of both dynamical scenarios [39].

##  IV Conclusions

In summary, we explored the dynamic phase space of the resonance modes in the
conical and field-polarized states in Cu2OSeO3 as a function of field,
temperature, and RF excitation frequency. First, we identified the resonances
and measured their field and temperature dependence using VNA-FMR. Taking into
account the shape (and demagnetization factors) of the Cu2OSeO3 sample [7],
the observed modes could be reproduced using the established models [10].
Employing the same setup, synchrotron-based DFMR allows for a deeper insight
into the dynamics of the identified resonance modes. We observed an inversion
of the phase of the dynamic signal at 4 GHz across the resonances in the
field-polarized and conical state, owing to the way the resonances are
traversed. Furthermore, linearly polarized incident x-rays provide additional
contrast, projecting out the specific orientation of the magnetization along
the incident beam propagation direction, giving access to the time-dependent
orientation of the magnetic spins during precession. We find that upon
reversing the applied magnetic field direction, the direction of the
propagating compression wave is inverted. In general, being able to measure
the time-dependent orientation of the magnetization allows for the exploration
of the dynamics of chiral systems, which has so far only been accessible via
RF absorption measurements in VNA-FMR. The technique of DFMR therefore
provides a unique view into the magnetization dynamics of complex
topologically structured spin systems.

##  V Acknowledgments

We thank Prof. Markus Garst from KIT in Karlsruhe for fruitful discussion
about the dynamical behavior of magnetization patterns in Cu2OSeO3. We
acknowledge the Diamond Light Source (Oxfordshire, UK) for the provision of
synchrotron radiation beamtime at beamline I10 under proposals SI16043 and
SI17612. This research was funded in part by the Engineering and Physical
Sciences Research Council (UK) through grant EP/N032128/1. For the purpose of
Open Access, the author has applied a CC BY public copyright license to any
Author Accepted Manuscript version arising from this submission.

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