[1]Mingran Xu These authors contributed equally to this work. These authors
contributed equally to this work. [1,2]Dirk Grundler

[1]Laboratory of Nanoscale Magnetic Materials and Magnonics, Institute of
Materials (IMX), School of Engineering, École Polytechnique Fédérale de
Lausanne (EPFL), Lausanne, 1015, Vaud, Switzerland

2]Institute of Electrical and Micro Engineering, School of Engineering, École
Polytechnique Fédérale de Lausanne (EPFL), Lausanne, 1015, Vaud, Switzerland

# Room temperature realization of artificial chiral magnets with
reprogrammable magnon nonreciprocity at zero field

<EMAIL_ADDRESS>Axel J. M. Deenen  Huixin Guo  *<EMAIL_ADDRESS>*
[

###### Abstract

Chiral magnets are materials which possess unique helical arrangements of
magnetic moments, which give rise to nonreciprocal transport and fascinating
physics phenomena. On the one hand, their exploration is guided by the
prospects of unconventional signal processing, computation schemes and
magnetic memory. On the other hand, progress in applications is hindered by
the challenging materials synthesis, limited scalability and typically low
critical temperature. Here, we report the creation and exploration of
artificial chiral magnets (ACMs) at room temperature. By employing a mass
production compatible deposition technology, we synthesize ACMs, which consist
of helical Ni surfaces on central cylinders. Using optical microscopy, we
reveal nonreciprocal magnon transport at GHz frequencies. It is controlled by
programmable toroidal moments which result from the ACM’s geometrical
handedness and field-dependent spin chirality. We present materials-by-design
rules which optimize the helically curved ferromagnets for 3D nonreciprocal
transport at room temperature and zero magnetic field.

###### keywords:

Nonreciprocity, Artifical Chiral Magnet, Brillouin Light Scattering, 3D
Magnonics

##  1 Introduction

Chirality is a characteristic quality of objects which are non-superimposable
mirror images of each other. It plays a crucial role in nature, various
scientific disciplines and technologies [1, 2, 3, 4]. Manifested by
nonreciprocal transmission of signals [5, 6], chiral objects manipulate the
polarization of electromagnetic waves. In communication technologies,
chirality is being widely integrated for signal isolation, noise reduction and
efficient energy transfer. In condensed matter physics and materials science,
chiral crystals form a fascinating class of magnetic materials, which give
rise to rich physical phenomena. Among them, there are the nonreciprocal
propagation of phonons [7] and magnons [8], skyrmions as topologically
protected spin structures, the magnon Hall effect [9], bulk electronic diode
effect [10] and emergent inductor [11]. Magnetic chirality arises from
breaking spatial-inversion and time-reversal symmetries simultaneously [12,
13], which occurs in specific noncentrosymmetric crystals [14]. Typically, the
crystals require complex synthesis routes and possess the relevant magnetic
ordering at low temperatures. These aspects delay the development of room-
temperature applications.

In recent years, the advancements in nanotechnologies [15] have opened the
possibility of endowing conventional materials with complex topologies in all
three spatial directions [16]. They allow one to artificially design optical
[17], mechanical, and electrical properties. In magnetism, the interplay
between 3D geometry and magnetic properties is intensively investigated [18,
19, 20, 21, 22]. It has been found that curved surfaces and geometric topology
lead to the emergence of an extrinsic Dzyaloshinskii-Moriya interaction (DMI)
[23, 24], chiral symmetry breaking and unconventional magnetic anisotropies.
For spin waves (magnons), nonreciprocal band structures are expected.
Nonreciprocity means that the energy flow is different in opposite directions.
In a nonreciprocal magnonic device, either the signal intensity, the
frequency, or both differ for magnons propagating with opposite wave vectors
$-k$ and $+k$. For bulk crystals and thin films nonreciprocity was examined by
means of broadband electrical spectroscopy with coplanar waveguides [8],
magnetoelastic spectroscopy with surface acoustic waves [25, 26], and
Brillouin light scattering (BLS) spectroscopy with wave vector resolution [27,
28, 29], confocal microscopy [30] or time-resolved scanning transmission X-ray
microscopy (TR-STXM) [31]. The symmetry breaking and magnon nonreciprocity
arose from interfacial DMI provided through an additional heavy metal layer,
magnetic surface anisotropy [32, 33], dipolar coupling and asymmetric coupling
of hybridized modes [34]. In a more general picture, the spatial-symmetry
breaking of a magnetic system via e.g. a heavy-metal interface,
antiferromagnetically coupled bilayers, a Bloch domain wall, or curved shapes
generates a non-zero toroidal moment vector
$\boldsymbol{\tau}=\sum_{i}\boldsymbol{r}_{i}\times\boldsymbol{m}_{i}$ with
$\boldsymbol{r}_{i}$ ($\boldsymbol{m}_{i}$) the position vector (magnetic
moment) at site $i$ (Fig. 1a). It is used to estimate the helicity $\chi$ of a
magnetic configuration via integration across its cross-sectional area $A$
using [35]

$\centering\boldsymbol{\tau}\left(\boldsymbol{m}_{0}\right)\stackrel{{\scriptstyle\text{
def
}}}{{=}}\frac{1}{A}\iint_{A}\mathrm{~{}d}x\mathrm{d}y\boldsymbol{~{}r}\times\boldsymbol{m}_{0}(\boldsymbol{r}),\@add@centering$
(1)

where $\boldsymbol{m_{0}}$ represents the static magnetization. If the
relation

$\centering\boldsymbol{k}\cdot\boldsymbol{\tau(\boldsymbol{m}_{0})}\neq
0\@add@centering$ (2)

is fulfilled, the spin-wave dispersion relation exhibits nonreciprocity which
is induced by dipolar interaction between the spins [35].

Figure 1: Helical spin texture of artificial chiral magnets, toroidal moments
and reprogrammable nonreciprocity of magnons at zero magnetic field. a, Sketch
of ferromagnetic tubes with helical magnetic states of left- and right-
handedness in a small magnetic field $H$ (top) and corresponding toroidal
moment vectors $\boldsymbol{\tau}$ motivating nonreciprocal (NR) propagation
of magnons with wave vectors $+k$ and $-k$ along $z$-direction (bottom). b,
Ferromagnetic tubes incorporating screw-like surfaces with left-handedness and
right-handedness prepared by additive manufacturing. The surface topology of
the Ni shell gives rise to an artificial chiral magnet (ACM). c, Magnon
spectra obtained on a RH-ACM (pink symbols) and LH-ACM (green symbols) in a
field $\mu_{0}H=+250$ mT at position $z=-0.5$
$\text{\,}\mathrm{\SIUnitSymbolMicro m}$. Curves reflect Lorentzian fits. d,
Nonreciprocity parameter $I_{\rm NR}$ extracted from spectra taken at
$\mu_{0}H=0$ mT after applying a magnetic field of 30 mT in opposite axial
directions of a LH-ACM. $I_{\rm NR}$ was measured at $f=10~{}$GHz five times
(cycles) before realizing a reversed magnetic field history at position
$z=-0.7$ $\text{\,}\mathrm{\SIUnitSymbolMicro m}$. The data reflect
reprogrammable nonreciprocity at zero field. e, Micromagnetic simulation
revealing nonreciprocal magnon transport into negative $z$-direction when
exciting an LH-ACM in its central region at zero field (here, at 3 GHz) after
it has been exposed to a field pointing in positive $z$-direction beforehand.

Via non-zero $\tau$ and the dipole-induced symmetry breaking a helical spin
structure, one can introduce chirality in a conventional but curved magnet.
Since the beginning of this decade, considerable efforts have been dedicated
to investigating the spin dynamics of curved ferromagnetic shells forming
nanotubes [36, 37, 38] and lattices of interconnected ferromagnetic curved
segments [20, 22]. However, a conclusive experimental evidence of NR magnon
propagation has remained elusive as noted in Ref. [37]. Helicity and chirality
are two closely related physical concepts but refer to different
characteristics of an object. A helical spin texture stands for spins rotating
relative to an axis in space, while a chiral spin texture describes a non-
superimposable arrangement of spins. In a bulk chiral magnet, spins relax into
the helical texture at zero magnetic field. Its toroidal moment vanishes [39],
and the chiral magnet does not exhibit an asymmetric dispersion relation in
zero field [29].

Beyond the intriguing physics emerging from curved surfaces, 3D nanomagnetic
structures captivate vast attention from a technological perspective. Compared
with planar thin-film devices, 3D magnetic architectures innately come with a
greater data storage capacity and higher dimensional connectivity. These
aspects are key for sustainable nanoelectronics with low energy consumption
and high computational power. Towards magnonics-based signal processing,
nonreciprocal waveguides are of utmost importance for an efficient routing of
signals. Furthermore, artificially created 3D magnetic objects naturally have
a higher degree of design flexibility. If prepared from magnetically ordered
materials with high critical temperature the materials-by-design approach is
expected to enable chiral properties ready for room-temperature applications.

Here, we present a novel route to chirality in magnetic materials at room
temperature. Using the materials-by-design approach and additive manufacturing
of nickel, we create artificial chiral magnets (ACMs) and explore their
unprecedented magnetic properties by means of inelastic light scattering and
simulations. By combining two-photon lithography (TPL) and mass production
compatible atomic layer deposition (ALD), we prepare tubular ferromagnetic
structures with screw-like surface topology of left-handedness (LH) and right-
handedness (RH). The ACMs consist of a photo-resist template (illustrated in
gray color in Fig. 1a,) which is covered by a uniformly thick nickel shell
(green). A scanning electron microscopy image of LH-ACMs is shown in Fig. 1b.
The suspended magnetic tubes (bright) are investigated in the present work.
The helical relief (helix region) is integrated into an otherwise cylindrical
Ni tube (tube region). In a magnetic field $H$ applied parallel to the ACM,
the magnetization in the tube region aligns with $H$ while in the helix region
it spirals around the tube forming locally a screw-like spin texture. The
handedness of the texture follows the corresponding structural chirality,
generating oppositely polarized toroidal moments $\mp\boldsymbol{\tau}$ in LH-
and RH-ACMs and inverted nonreciprocities
$\boldsymbol{\tau}\cdot\boldsymbol{k}$ (Fig. 1a). By means of micro-focused
Brillouin light scattering microscopy ($\mu$-BLS) (Fig. 1a and c), we detect
nonreciprocal spin dynamics with high spatial resolution. By studying Stokes
and anti-Stokes scattering processes of the focused laser light with magnons,
we identify differences in both resonance frequencies $\Delta f$ (Fig. 1c) and
intensities of BLS peaks (Fig. 1d) which we describe by the asymmetry
parameter $I_{\rm NR}$. We anticipate that the observed nonreciprocities
originate from asymmetric dispersion relations induced by chiral spin textures
following the helical reliefs. By scanning the laser spot across the curved
surfaces of the helices, we resolve a gradual evolution of Stokes and anti-
Stokes peaks. We attribute the variations to momentum conservation dependent
on the light scattering geometry. The nonreciprocities are inverted by either
changing the polarity of the applied magnetic field or exploring an ACM of
opposite handedness. We demonstrate also a programmable magnetic chirality at
zero field through the magnetic field history applied to the helical Ni tubes.
Exploring the remanent state of the same tube after applying either +30 or
$-$30 mT, we detect opposite nonreciprocities at zero magnetic field (labelled
$\pm 0$ mT) by means of $\mu$-BLS (Fig. 1d). Micromagnetic simulations show
that the surface curvature of the helical relief added to the Ni tubes
stabilizes a chirally arranged magnetic texture. The observed signatures of
nonreciprocity are qualitatively reproduced by numerical simulations (Fig.
1e). Hence, our research unveils the possibility of empowering a conventional
single-layered ferromagnet such as Ni with chiral properties in the remanent
state through engineered helical surface curvature. Using an additive
manufacturing strategy to create the observed nonreciprocity of spin dynamics,
we facilitate a mass production technology for chirality-based nanoelements
which offer unidirectional signal transmission in 3D nanomagnetic
architectures. We expect them to enhance connectivity in hardware-implemented
3D neural networks.

##  2 Observation of asymmetric magnon spectra in artificial chiral magnets

Figure 2: Nonreciprocal magnon spectra detected along right- and left-handed
ACMs via Brillouin light scattering spectroscopy. a, Sketch of micro-focus BLS
spectroscopy performed along the long axis of an RH-ACM. The black arrow on
the ACM indicates the scanning direction in a field $H$ applied the long axis.
Light scattering geometry at positions b
$z<0~{}$\text{\,}\mathrm{\SIUnitSymbolMicro m}$$ and c
$z>0~{}$\text{\,}\mathrm{\SIUnitSymbolMicro m}$$ leading to transferred magnon
wave vectors with oppositely directed propagation directions as sketched for
the Stokes signal. Color-coded spectra taken along an RH-ACM in a field of d
+250 mT and e -250 mT. Color-coded spectra taken along a LH-ACM in a field of
f +250 mT and g -250 mT. h Quantitative analysis of the nonreciprocity in
terms of frequency shift $\Delta f$ for the LH-ACM (top) and the RH-ACM
(center) extracted from spectra shown on the left. The bottom panel shows the
magnitude of $f_{\rm NR}(\mathbf{H})$ for the RH-ACM (pink) and LH-ACM (green)
as a function of position $z$.

We investigated the spin dynamics of ACMs by $\mu$-BLS at room temperature
(Fig. 2a). The scattering process between photons and magnons in an opaque
thin film obeys the momentum conservation law for the in-plane momenta. In the
BLS spectrum, the Stokes (anti-Stokes) peak corresponds to the creation
(annihilation) of magnons. In the $\mu$-BLS setup, the focused laser light
offers a cone of incidence angles around the optical axis of the lens. The
backscattered light hence contains photons which interact with magnons
exhibiting a specific distribution of in-plane wave vectors leading to the
inhomogeneous broadening of BLS peaks. Assuming isotropic scattering
processes, a frequency asymmetry (nonreciprocity) of the magnon band structure
with respect to wave vectors $k$ parallel to the film plane is not resolved.
This is no longer true if the laser light is focused on a curved surface whose
normal direction does not coincide with the optical axis of the lens. In Fig.
2b and c we depict the scattering configurations near the helix of an ACM. At
$z<0~{}$\text{\,}\mathrm{\SIUnitSymbolMicro m}$$ and
$z>0~{}$\text{\,}\mathrm{\SIUnitSymbolMicro m}$$ the Stokes (anti-Stokes)
signal mainly comes from magnons propagating in parallel with (anti-parallel
to) the momentum of the incident light projected into the film, i.e., the
resulting magnon wave vectors $k$ are in $-z$\- and $+z$-direction,
respectively ($+z$\- and $-z$-direction, respectively), depending on the laser
spot position. The resulting imbalance in BLS spectra allows one to
qualitatively analyze the frequency nonreciprocity of magnon band structures
and propagation directions $-k$ versus $+k$, respectively, along the $z$-axis.

In Fig. 1c, we show spectra taken on an RH-ACM (pink) and LH-ACM (green) with
a field of +250 mT applied in $z$-direction. Strikingly, the height difference
between Stokes and anti-Stokes peaks is reversed for the two samples.
Consequently, the sign of intensity asymmetry parameter $I_{\rm NR}$ depends
on the geometrical handedness of the ACMs only as their Ni shells were
deposited in the same ALD process. Also the asymmetry in frequency $\Delta f$
(Fig. 1c) is inverted when changing the handedness. We attribute the
handedness-dependent signal asymmetries to magnetochiral properties and
nonreciprocal magnon characteristics in the ferromagnetic shell. The
nonreciprocal characteristics introduced by geometry are further verified by
spatially resolved measurements presented in Fig. 2d to g. The magnon spectra
were taken at different positions along the long axis of the tubular shell.
The position $z=0$ is put near the topmost point on the surface helix. The
$z$-position-dependent spectra show clear asymmetries with respect to both
$z=0$ and $f=0$ in signal strength and frequencies of signal maxima. As a
function of position, both the frequency of the signal maximum and its
intensity vary characteristically for the Stokes (left) and anti-Stokes
(right) spectra. The observed asymmetries are reversed by either an opposite
field direction (Fig. 2d,f versus Fig. 2e,g) or opposite handedness (Fig. 2d,e
versus Fig. 2f,g). In the central position $z=0$ the spectra show BLS peaks
with local minima of the resonance frequency. At this position, the asymmetric
intensity distribution and frequency difference between Stokes and anti-Stokes
signals change sign.

To quantitatively analyse the asymmetries in frequency as a function of spot
position (Fig. 2h), we apply a Lorentzian fit to the peak of each Stokes and
anti-Stokes spectrum according to

${I_{\rm total}=I_{0}+\frac{I}{(f-f_{\rm c})^{2}/\Gamma_{f}^{2}+1}},$ (3)

where $\Gamma_{f}$ is the half width at half maximum (HWHM), $f_{\rm c}$ is
the central frequency, $I$ is the signal strength at the maximum, and $I_{0}$
is the background signal in the spectrum. We then calculate $\Delta
f=\left|f_{\rm c,\text{Stokes }}\right|-\left|f_{c,\text{ anti-Stokes
}}\right|$ and $\Delta I=I_{\text{Stokes }}-I_{\text{anti-Stokes}}$. However,
a systematic frequency shift $f^{\prime}$, independent on $z$, is inevitably
present due to a slight misalignment of the laser scanning path with respect
to the center of the magnetic tubes. In order to compensate for this
systematic offset, we introduce $\Delta f^{\prime}=\Delta f-f^{\prime}$ and
$\Delta I^{\prime}=\Delta I-I^{\prime}$, representing the position-dependent
component of $\Delta f$ and $\Delta I$. Using the extracted parameters $\Delta
f^{\prime}$ and $\Delta I^{\prime}$, we calculate the asymmetry parameters as
a function of $z$-position. The parameters $\Delta I^{\prime}$ for RH- and LH-
ACMs at $\pm$ 250 mT (red and blue symbols) are summarized in Fig. S5, while
values $\Delta f$ are summarized in the two top panels of Fig. 2h. The
parameters $\Delta f$ and $\Delta I$ for RH- and LH-ACMs at $\pm$ 250 mT are
summarized in Fig. S8. The maximum asymmetry amounts to 0.67 GHz (0.8 GHz) in
case of the LH-ACM (RH-ACM). Considering $\Delta f^{\prime}=\left(2\gamma/\pi
M_{S}\right)Dk$ [27, 28], and taking the maximum possible wave vector value
$k=4\pi/\lambda_{laser}$ offered by the focussed laser light, we estimate an
extrinsic DMI constant $D^{\prime}>0.112~{}\rm
mJ/m^{2}$($D^{\prime}<-0.134~{}\rm mJ/m^{2}$) for LH-ACM and RH-ACM,
respectively. The extracted $D^{\prime}$ values are comparable to interfacial
DMI constants reported for magnetic multilayers incorporating a heavy metal
[40]. The latter component introduces relativistic spin-orbit interaction and
enlarges magnetic damping in ferromagnetic metals. The curvature-induced ACM
reported here avoids the detrimental effect of the heavy metal.

To analyze the relative strength of the magnetochiral effect in ACMs and
compare them to natural materials, we evaluate the relative magnitude of
frequency nonreciprocity according to

${\beta_{\rm NR}(\mathbf{H})=\frac{\Delta f^{\prime}(\mathbf{+H})}{f_{\rm
S}(\mathbf{+H})+f_{\rm AS}(\mathbf{+H})}-\frac{\Delta
f^{\prime}(\mathbf{-H})}{f_{\rm S}(\mathbf{-H})+f_{\rm AS}(\mathbf{-H})}}$ (4)

where $f_{\rm S}$ refers to $\left|f_{c},{\text{Stokes}}\right|$ and $f_{\rm
AS}$ refers to $\left|f_{c},{\text{anti-Stokes }}\right|$. We note that
$\beta_{\rm NR}$ is nominally equivalent to $g_{\mathrm{MCh}}$ as defined in
Refs. [7, 41]. In the bottom panel of Fig. 2h we display $\beta_{\rm NR}$ as a
function of position for the RH-ACM (pink) and the LH-ACM (green). Each
dataset is anti-symmetric with respect to $z=0$. The consistent anti-phase
variations observed for all three different asymmetry parameters as a function
of BLS spot position are a clear signature of chirality effects controlled by
geometrical handedness and magnetic field direction. The maximum peak-to-peak
difference amounts to $5.4\times 10^{-2}$ at
$z=-0.5~{}$\text{\,}\mathrm{\SIUnitSymbolMicro m}$$ for RH-ACM. This value is
more than three orders of magnitude larger than $g_{\mathrm{MCh}}$ reported
for a natural chiral material ($g_{\mathrm{MCh}}\approx 40\times 10^{-6}$) in
Ref. [7, 41].

##  3 Helical magnon channel around an ACM

In the following we interpret the observed frequency variation in Fig. 2d to
g. For a polycrystalline thin film of Ni one expects a magnetic hard axis
along the surface normal. Along the scanned path, the hard axis of the ACM
shells gradually rotates from the $-z$-direction via the $-y$-direction toward
the $+z$-direction (Fig. 2a). Considering a magnetic field of 250 mT applied
collinear with the $z$-axis, the LLG equation assumes the largest effective
field in the straight part of the tube (Fig. 2a) and a smaller effective field
in the topmost point of the surface helix. At both of these positions, the
applied field is tangential to the Ni film and the local hard axis is
perpendicular to the field, Consistently, the BLS spectra shown in Fig. 2d to
g show local maxima (minima) of the eigenfrequencies detected within the tube
(helix) regions. To further explore the microscopic nature of the low-
frequency BLS signals near $z=0$ we present further experimental BLS spectra
and micromagnetic simulations in the following.

We scanned the BLS laser focus over three neighboring devices as sketched in
Fig. 3a. The spatially resolved BLS signal (magnon distribution) was obtained
by moving the sample in the $xz$-plane with 100 nm steps along $x-$ and
$z-$directions while keeping a field of +250 mT along the $z$-direction and a
fixed BLS detection frequency of -15 GHz (Fig. 3b). Further data taken at $\pm
13$ and $\pm 17~{}$GHz are displayed in Supplementary Fig. S6. The BLS mapping
of Fig. 3b shows a spatial intensity distribution which exhibits a weak signal
strength in the tube region and a large one in the helix region. The largest
signals are found in the elevated helical reliefs indicated by the dashed
ovals. Within an oval, the signal distribution at -15 GHz is inhomogeneous
suggesting a nonreciprocal character of magnons along the helical surface
relief. The signals at $\pm 13~{}$GHz are strong in the ovals as well.
However, at $\pm 17~{}$GHz the large signals occur in the tube region (compare
Supplementary Fig. S6).

Using the GPU-accelerated software MuMax3 [42] we simulated a Ni ACM
consisting of a tube with inner radius of 220 nm and thickness of 30 nm which
intersects a hollow helix with an ellipsoidal cross-section. The helix had a
pitch of 2000 nm, diameter of 740 nm, cross-sectional inner major and minor
radii of 120 nm and 70 nm, respectively, and a thickness of 30 nm, as detailed
in the Methods section. The diameter is smaller compared to the real sample as
a consequence of the limited computational power. An external field of $+250$
mT was applied along the $z$-axis. To investigate the dynamic response a small
dynamic field $h=h_{0}\operatorname{sinc}\left(2\pi f_{\rm
c}\left(t_{-}t_{0}\right)\right)\hat{x}$ with pulse amplitude
$\mu_{0}h_{0}=3~{}{\rm mT}$, cut-off frequency $f_{\rm c}=15~{}\rm GHz$, time
$t$ and time-offset $t_{0}=26.7~{}{\rm ns}$ was applied to a 20 nm wide strip
forming a ring-like excitation region around the center. Six repetitions of
periodic boundaries (PBCs) were applied along the $z$-axis to prevent finite-
size induced vortex states at the ends. Following the procedures outlined in
the Methods, we obtain a spectrum as presented in Fig. 3c when considering the
spin configuration of an RH-ACM shown in Fig. 3d (left).

Figure 3: Helical magnon channel around an ACM. a, Sketch of the measurement
configuration. b, Spatial map of BLS signal taken at -15 GHz (Stokes signal)
while +250 mT is applied to the RH-ACMs. The dashed red ovals indicate the
most elevated regions due to the helix on the surface. c, Simulated spectrum
at +250 mT for an RH-ACM with the spin structure shown in d. d, Simulated spin
precession occurring along the helix at 11.12 GHz (helix mode).

Multiple peaks are resolved in Fig. 3c which appear mainly in two groups.
Coming from low frequencies, one group exists up to about 12.3 GHz. The
maximum signal strength shows up at 12.56 GHz followed by a second group of
peaks extending to larger frequencies. We are particularly interested in the
low-frequency regime and display the spatial distribution of the magnitude of
the complex dynamic amplitude at 11.14 GHz in Fig. 3d (right). The spin-
precessional motion of this low-frequency excitation occurs along the helical
relief on the surface. The spin-precessional amplitude is distributed
asymmetrically along the surface helix. The tube region is not excited. The
simulations are qualitatively consistent with the experimental observation
presented above and in Supplementary Fig. S6. The remaining discrepancy
concerning eigenfrequencies is attributed mainly to the different diameters.

##  4 Nonreciprocity at remanence induced by field-reprogrammable toroidal
moment vectors

The data of Fig. 2d to g showed that for a given handedness the nonreciprocity
of BLS signals depended on the field orientation. Considering Eq. 2, we expect
a sign change of the toroidal moment vector with positive and negative $H$.
Figure 4a to c summarizes the simulated magnetic field dependence of
$\boldsymbol{\tau}$ in an LH-ACM. In Fig. 4a, we show computed toroidal moment
values according to Eq. 1 when sweeping the magnetic field from negative
(dashed curves) to positive direction and back (solid curves). We note that
the value of the toroidal moment vector depends on the choice of origin. As
long as this choice is made consistently, the change in toroidal moment is
independent of the choice of origin [39]. We have taken the center of the tube
as the origin.

Figure 4: Programmable toroidal moments and nonreciprocal transport at
remanence.  a, Toroidal moment $\tau$ of a LH-ACM (blue curve) simulated as a
function of axial field $H$ and its sweep direction indicated by yellow
arrows. The magenta and orange curves decompose the toroidal moment of the LH-
ACM in the contributions from the tube and helix regions, respectively. The
field was swept from negative to positive (dashed lines) and back (solid
lines). The hysteretic behavior enables a reprogrammable $\boldsymbol{\tau}$
at $H=0$. b, Simulated density distribution of $\tau$ of a LH-ACM at remanence
state ($+0$ mT) following saturation at $+250$ mT. c, Simulated cross-section
of $\tau$ distribution on a LH-ACM when varying the magnetic field from +250
mT, to +0 mT, $-$250 mT, and $-$0 mT. The cross-section was taken at the
central plane of the ACM. d, Intensity nonreciprocity $I_{\rm NR}$ as a
function of magnetic field and sweep direction, showing a hysteretic behavior
at position $z=-0.7$ $\text{\,}\mathrm{\SIUnitSymbolMicro m}$. e, Color-coded
magnon spectra $+0$ mT and $-0$ mT obtained along the central axis along the
top surface of a LH-ACM at $\pm$0 mT after applying $+50$ mT and $-50$ mT,
respectively. The white (blue) circles indicate the same positions on the
$z$-axis and highlight clearly asymmetric Stokes (anti-Stokes) signals which
substantiate their dependence on the magnetic history.

In the supplementary material, we show that simulations performed for the RH-
ACM result in a mirror image of Fig. 4a, which reflects the opposite
handedness of the ACMs.

In Fig. 4a we decompose $\tau_{\rm full}$ into a contribution from the tube
region $\tau_{\rm tube}$ (purple curves) and a contribution from the helix
region $\tau_{\rm helix}$ (red curves). As shown in Fig. 4a, $\tau_{\rm tube}$
vanishes when the magnetic field increases (decreases) to +10 mT (-10 mT),
whereas $\tau_{\rm helix}$ remains at a finite value in the presence of a
strong magnetic field ($\pm$ 250 mT). Besides, at the remanent state ($\pm$ 0
mT), the LH-ACM sustains a substantial amplitude of $\tau_{\rm full}$, which
is further visualized in Fig. 4b where the full toroidal moment density
$\boldsymbol{T}$ is shown for +0 mT. Note that in Fig. 4a, $\tau_{\rm tube}$
appears to have a larger contribution to $\tau_{\rm full}$, which is
attributed to the larger volume of the tube region. Figure 4c shows cross-
sections of the toroidal moment density $\boldsymbol{T}$ of the LH-ACM for a
sweep from -250 mT to $-$0 mT and a sweep from 250 mT to 0 mT. In both cases,
the toroidal moment is large in remanence, with an appreciable contribution
originating from the helix region.

In Fig. 4d, we report BLS spectra taken on one and the same LH-ACM after
saturating it with a magnetic field along the $+z$-($-z$-)direction expecting
$-\tau(+\tau)$ from Fig. 4c. The BLS spectra were obtained in zero magnetic
field after saturating at +50 mT (labeled +0 mT) and $-$50 mT (labeled $-$0
mT). Comparing the BLS signals near the dashed circles in Fig. 4d, the Stokes
peak of +0 mT is located at higher frequency with stronger intensity than in
the $-$0 mT spectra, while anti-Stokes peaks show the opposite behavior. The
spectra exhibit position-dependent nonreciprocity consistent with the +250 mT
spectra of Fig. 2d to g. The corresponding spectra on RH-ACM are given in Fig.
S3, which shows opposite position dependence with respect to Fig. 4d. In Fig.
4e we display the full hysteresis curve of the evaluated intensity asymmetry
$I_{\rm NR}$. Here, we swept the magnetic field from +30 mT to $-$30 mT and
then back to +30 mT while acquiring BLS intensities at frequencies $+10$ GHz
and $-10$ GHz at the $z$ coordinate highlighted by the dashed circles. $I_{\rm
NR}$ was extracted via

${I_{\mathrm{NR}}(f_{0},\mathbf{H})=\frac{I_{\text{S}}(f_{0},\mathbf{H})-I_{\text{AS
}}(f_{0},\mathbf{H})}{(I_{\text{S }}(f,\mathbf{H})+I_{\text{AS
}}(f_{0},\mathbf{H}))/2}}.$ (5)

These independent data substantiate the non-vanishing chirality of an ACM at
zero magnetic field and the field-programmable nonreciprocity. The unique
surface topography of ACMs favors a spirally aligned spin texture which
consequently preserves the chirality with $\tau\neq 0$ at zero magnetic field.
To explore the reproducibility of the chirality memory effect in an ACM, we
sequentially applied $\pm$30 mT, and extracted $I_{\mathrm{NR}}$ at $\pm 0$ mT
for five times before applying the opposite magnetic field history. The data
are displayed in Fig. 1d and Fig. S4c,d. Within the noise level, $I_{\rm NR}$
exhibits only two levels reflecting two different magnon reciprocities
reproducibly induced at zero field. The hysteretic curves of the LH- and RH-
ACM are acquired at position $z=-0.7$\text{\,}\mathrm{\SIUnitSymbolMicro m}$$,
showing opposite nonreciprocity in Fig. S4a and S4b.

##  5 Optimization of magnon nonreciprocity in ACMs

In the following we discuss how to engineer and optimize nonreciprocity in
ACMs. To this end, first, the dispersion relation of magnons $f(k_{\rm z})$ on
the tubular segment is obtained via dynamic micromagnetic simulations. Both
right-handed and left-handed ACMs are considered. Figure 5a shows the
numerical results as color-coded graphs obtained for four configurations
provided by differently oriented spin textures and the different handedness
(shown on the right of each graph). Thereby, we realize four combinations of
helicity of spins $\chi=\pm 1$ and polarization $p=\pm 1$. $p$ is evaluated
from $\boldsymbol{m}_{0}$ projected on the $z$-axis (for details, see
Supplymentary materials.

Figure 5: Optimization of reprogrammable nonreciprocity in ACMs. a, Numerical
simulation results of spin wave dispersion relations $f(k_{\rm z})$ on RH and
LH Ni tubes at $\pm 0$ mT. The 3D illustrations provide the remanent magnetic
textures leading to $f(k_{\rm z})$ shown on their left sides. b, Contributions
of $\pm k$ magnons to the spectra of a RH-ACM at +0 mT. c, Phase distribution
of magnon modes at 1.49 GHz on an ACM. d, Optimization of nonreciprocity
($\Delta f$) according to Eq. 6 by varying geometrical and materials
parameters as well as the angle the helical spins take with respect to the
longitudinal axis.

The top row of Fig. 5a shows the dispersion obtained for an RH- and LH-ACM at
zero field, where the system was first saturated along the positive
$z$-direction. Due to their opposite handedness, the spin textures of the two
tubes have identical polarity but opposite helicity (handedness). The color
intensity map shows the resulting eigenfrequencies of the tube segments. The
noisy appearance is due to the finite frequency and field resolution of the
performance-hungry simulations on ACMs. The numerical results demonstrate a
clear frequency nonreciprocity between $\pm k_{z}$, whose sign is opposite for
RH- and LH-ACMs. We compare the numerical results obtained on ACMs with
dispersion relations calculated analytically by an expression provided for
nanotubes with conical spin texture [43] (see Methods). The analytical results
(solid lines) agree almost quantitatively with the branches resolved by the
full-fledged numerical simulations applied to ACMs, when we consider an
effective radius $r_{\rm st}$ of 300 nm for the straight (st) nanotube. The
two minima shown in the numerical dispersion can be identified with the two
azimuthally propagating modes $m=\pm 1$, with $m$ being the azimuthal index.
The frequency nonreciprocity is opposite for RH- and LH-ACMs highlighting the
geometrical programmability of magnon transport via the surface helix.
Stimulated by [44] and considering their peculiar geometry and the
unidirectional magnon flow along the $z$-axis we call the created ACMs
magnonic Archimedean screws.
Furthermore, we show that the nonreciprocity at remanence is field tunable.
After first saturating the system with a field applied along the
$-z$-direction and then approaching 0 mT, we reverse the nonreciprocal
dispersions for the RH- and LH-ACMs (Fig. 5a, bottom row). The different
contributions of $+k_{z}$ and $-k_{z}$ magnons can be clearly shown by
integrating the intensity of the dynamic magnetization (Fig. 5a) over the
positive and negative wavevectors, respectively. The resulting spectrum of the
RH-ACM is shown in Fig. 5b and clearly demonstrates that $+k_{z}$ magnons
dominate the absolute frequency minimum in the dispersion. In Fig. 5c we
display the phase profile of the mode with the frequency of the main peak at
1.49 GHz. The magnons are seen to propagate in a helical manner along the
tube. Unwrapping the ACM onto a plane reveals a phase jump induced by the
surface helix (Fig. 5c).
There are two crucial parameters responsible for the nonreciprocity that are
related to the geometry of the ACM: one is the angle $\eta$ of the
magnetization $\boldsymbol{m}_{0}$ with respect to the longitudinal tube axis,
which is related to the surface helix angle, and the other one is the radius
$r$ of the tube region. Using the theoretical formalism introduced by Salazar-
Cardona et al.[43], we studied the parameter-dependent nonreciprocity of Ni
ACMs in greater detail. For this, we assumed that the opening angle $\alpha$
of the conical spin structure relates to $\eta$ and the straight tube radius
$r_{\rm st}$ to $r$. In Fig. 5d, the computed nonreciprocity $\Delta f$ for
long tubes is shown as a function of angle $\alpha$ and radius $r_{\rm st}$.
The nonreciprocity increases for both smaller radius and smaller angles. We
show line cuts for three radii: $r=168,~{}294,~{}{\rm and}~{}654$ nm. Here,
294 nm is the effective radius of the tube, whereas the radius of 654 nm is
comparable to the experimentally realized ACM presented in this work. We do
not display the parameter regime of very small values in which the hierarchy
of azimuthal modes $m$ is reversed as TPL sets a lower limit to $r_{\rm st}$.
The minimum radius reached by state-of-the-art TPL amounts to 50 nm. The non-
reciprocity can be further enhanced by the choice of polycrystalline
ferromagnetic material. In Fig. 5d the close-up shows a comparison of the
nonreciprocity for three ferromagnets: Ni, Py, Co (Methods). Both Py and Co
have a larger nonreciprocity than Ni. In particular, Py shells are also
accessible via ALD [45, 38].

##  6 Conclusion

We reported the successful creation of artificial chiral magnets which support
nonreciprocal magnon transport at zero magnetic field. Our work unveils the
interplay between geometrical handedness and nonreciprocity. It demonstrates
that chirality can be imprinted at room temperature by surface engineering of
a conventional ferromagnet. Magnons on LH- and RH-ACMs show opposite
nonreciprocities in both magnon frequencies and intensities. The
nonreciprocity can be reversed by reversing the magnetic field direction. The
artificial magnetic chirality manifests itself in a pronounced magnon
nonreciprocity chiral parameter $\beta\rm_{NR}\rm=5.4\times 10^{-2}$, which
are three orders of magnitude higher than the reported value in bulk chiral
magnets [41].

The presented ACMs are created by combining TPL and ALD, which is compatible
with mass production of components in microelectronics. In particular, the
ACMs have been realized based on a polycrystalline magnetic material. Such a
material is preferred in technologies over single crystals which were explored
in the case of natural chiral magnets. The chirality of ACMs originates from
the geometrical handedness and a helical Ni surface on a central cylinder.
This structure gives rise to a magnonic Archimedean screw. The significant
magnon nonreciprocity of our 3D nanomagnetic elements in the GHz frequency
regime fuels the prospects of 3D interconnected magnonic circuits with uni-
directional signal transmission in vertical directions. Such elements enhance
the integration density and energy efficiency of magnonics.

## References

  *   * Barron [1972] Barron, L.D.: Parity and optical activity. Nature 238(5358), 17–19 (1972) https://doi.org/10.1038/238017a0
  * Rikken and Raupach [1997] Rikken, G.L.J.A., Raupach, E.: Observation of magneto-chiral dichroism. Nature 390(6659), 493–494 (1997) https://doi.org/10.1038/37323
  * He et al. [2023] He, J.J., Tanaka, Y., Nagaosa, N.: The supercurrent diode effect and nonreciprocal paraconductivity due to the chiral structure of nanotubes. Nature Communications 14(1), 3330 (2023) https://doi.org/10.1038/s41467-023-39083-3 arXiv:2212.02183
  * Eslami et al. [2014] Eslami, S., Gibbs, J.G., Rechkemmer, Y., Slageren, J., Alarcón-Correa, M., Lee, T.-C., Mark, A.G., Rikken, G.L.J.A., Fischer, P.: Chiral Nanomagnets. ACS Photonics 1(11), 1231–1236 (2014) https://doi.org/10.1021/ph500305z
  * Rikken et al. [2002] Rikken, G.L.J.A., Strohm, C., Wyder, P.: Observation of Magnetoelectric Directional Anisotropy. Physical Review Letters 89(13), 133005 (2002) https://doi.org/10.1103/PhysRevLett.89.133005
  * Cheong et al. [2018] Cheong, S.W., Talbayev, D., Kiryukhin, V., Saxena, A.: Broken symmetries, non-reciprocity, and multiferroicity. npj Quantum Materials 3(1), 19 (2018) https://doi.org/10.1038/s41535-018-0092-5
  * Nomura et al. [2019] Nomura, T., Zhang, X.X., Zherlitsyn, S., Wosnitza, J., Tokura, Y., Nagaosa, N., Seki, S.: Phonon magnetochiral effect. Physical Review Letters 122(14), 145901 (2019)
  * Seki et al. [2016] Seki, S., Okamura, Y., Kondou, K., Shibata, K., Kubota, M., Takagi, R., Kagawa, F., Kawasaki, M., Tatara, G., Otani, Y., Tokura, Y.: Magnetochiral nonreciprocity of volume spin wave propagation in chiral-lattice ferromagnets. Physical Review B 93(23), 235131 (2016) https://doi.org/10.1103/PhysRevB.93.235131
  * Onose et al. [2010] Onose, Y., Ideue, T., Katsura, H., Shiomi, Y., Nagaosa, N., Tokura, Y.: Observation of the magnon Hall effect. Science 329(5989), 297–299 (2010) https://doi.org/10.1126/science.1188260
  * Yokouchi et al. [2017] Yokouchi, T., Kanazawa, N., Kikkawa, A., Morikawa, D., Shibata, K., Arima, T., Taguchi, Y., Kagawa, F., Tokura, Y.: Electrical magnetochiral effect induced by chiral spin fluctuations. Nature Communications 8(1), 866 (2017) https://doi.org/10.1038/s41467-017-01094-2 arXiv:1703.04897
  * Yokouchi et al. [2020] Yokouchi, T., Kagawa, F., Hirschberger, M., Otani, Y., Nagaosa, N., Tokura, Y.: Emergent electromagnetic induction in a helical-spin magnet. Nature 586(7828), 232–236 (2020) https://doi.org/10.1038/s41586-020-2775-x
  * Barron [2000] Barron, L.D.: Chirality, magnetism and light 405(6789), 895–896 (2000) https://doi.org/10.1038/35016183
  * Cheong and Xu [2022] Cheong, S.-W., Xu, X.: Magnetic chirality. npj Quantum Materials 7(1), 40 (2022) https://doi.org/10.1038/s41535-022-00447-5
  * Tokura and Nagaosa [2018] Tokura, Y., Nagaosa, N.: Nonreciprocal responses from non-centrosymmetric quantum materials. Nature Communications 9(1), 374 (2018) https://doi.org/10.1038/s41467-018-05759-4
  * Saha et al. [2019] Saha, S.K., Wang, D., Nguyen, V.H., Chang, Y., Oakdale, J.S., Chen, S.C.: Scalable submicrometer additive manufacturing. Science 366(6461), 105–109 (2019) https://doi.org/10.1126/science.aax8760
  * Donnelly et al. [2017] Donnelly, C., Guizar-Sicairos, M., Scagnoli, V., Gliga, S., Holler, M., Raabe, J., Heyderman, L.J.: Three-dimensional magnetization structures revealed with X-ray vector nanotomography. Nature 547(7663), 328–331 (2017) https://doi.org/10.1038/nature23006
  * Wang et al. [2023] Wang, J., Valligatla, S., Yin, Y., Schwarz, L., Medina-Sánchez, M., Baunack, S., Lee, C.H., Thomale, R., Li, S., Fomin, V.M., Ma, L., Schmidt, O.G.: Experimental observation of Berry phases in optical Möbius-strip microcavities. Nature Photonics 17(1), 120–125 (2023) https://doi.org/10.1038/s41566-022-01107-7
  * Fernández-Pacheco et al. [2017] Fernández-Pacheco, A., Streubel, R., Fruchart, O., Hertel, R., Fischer, P., Cowburn, R.P.: Three-dimensional nanomagnetism. Nature Communications 8(1), 15756 (2017) https://doi.org/10.1038/ncomms15756
  * Makarov et al. [2022] Makarov, D., Volkov, O.M., Kákay, A., Pylypovskyi, O.V., Budinská, B., Dobrovolskiy, O.V.: New Dimension in Magnetism and Superconductivity: 3D and Curvilinear Nanoarchitectures. Advanced Materials 34(3), 2101758 (2022) https://doi.org/10.1002/adma.202101758
  * Guo et al. [2023] Guo, H., Deenen, A.J.M., Xu, M., Hamdi, M., Grundler, D.: Realization and Control of Bulk and Surface Modes in 3D Nanomagnonic Networks by Additive Manufacturing of Ferromagnets. Advanced Materials 35(39), 2303292 (2023) https://doi.org/10.1002/ADMA.202303292
  * Volkov et al. [2024] Volkov, O.M., Pylypovskyi, O.V., Porrati, F., Kronast, F., Fernandez-Roldan, J.A., Kákay, A., Kuprava, A., Barth, S., Rybakov, F.N., Eriksson, O., Lamb-Camarena, S., Makushko, P., Mawass, M.A., Shakeel, S., Dobrovolskiy, O.V., Huth, M., Makarov, D.: Three-dimensional magnetic nanotextures with high-order vorticity in soft magnetic wireframes. Nature Communications 15(1), 2193 (2024) https://doi.org/10.1038/s41467-024-46403-8
  * Gołȩbiewski et al. [2024] Gołȩbiewski, M., Hertel, R., D’Aquino, M., Vasyuchka, V., Weiler, M., Pirro, P., Krawczyk, M., Fukami, S., Ohno, H., Llandro, J.: Collective Spin-Wave Dynamics in Gyroid Ferromagnetic Nanostructures. ACS Applied Materials & Interfaces (2024) https://doi.org/10.1021/acsami.4c02366
  * Volkov et al. [2018] Volkov, O.M., Sheka, D.D., Gaididei, Y., Kravchuk, V.P., Rößler, U.K., Fassbender, J., Makarov, D.: Mesoscale Dzyaloshinskii-Moriya interaction: Geometrical tailoring of the magnetochirality. Scientific Reports 8(1), 866 (2018) https://doi.org/10.1038/s41598-017-18835-4
  * Makarov and Sheka [2022] Makarov, D., Sheka, D.D.: Curvilinear Micromagnetism: from Fundamentals to Applications vol. 146. Springer, Springer Nature Switzerland AG 2022 (2022). https://doi.org/10.1007/978-3-031-09086-8
  * Sasaki et al. [2017] Sasaki, R., Nii, Y., Iguchi, Y., Onose, Y.: Nonreciprocal propagation of surface acoustic wave in Ni/LiNbO3. Physical Review B 95(2), 020407 (2017) arXiv:1605.04058
  * Xu et al. [2020] Xu, M., Yamamoto, K., Puebla, J., Baumgaertl, K., Rana, B., Miura, K., Takahashi, H., Grundler, D., Maekawa, S., Otani, Y.: Nonreciprocal surface acoustic wave propagation via magneto-rotation coupling. Science Advances 6(32), 1724 (2020) https://doi.org/10.1126/sciadv.abb1724 arXiv:2001.05135
  * Di et al. [2015] Di, K., Zhang, V.L., Lim, H.S., Ng, S.C., Kuok, M.H., Yu, J., Yoon, J., Qiu, X., Yang, H.: Direct observation of the Dzyaloshinskii-Moriya interaction in a Pt/Co/Ni film. Physical Review Letters 114(4), 047201 (2015) arXiv:1409.8570
  * Nembach et al. [2015] Nembach, H.T., Shaw, J.M., Weiler, M., Jué, E., Silva, T.J.: Linear relation between Heisenberg exchange and interfacial Dzyaloshinskii-Moriya interaction in metal films. Nature Physics 11(10), 825–829 (2015)
  * Ogawa et al. [2021] Ogawa, N., Koehler, L., Garst, M., Toyoda, S., Seki, S., Tokura, Y.: Nonreciprocity of spin waves in the conical helix state. PNAS 118(8), 2022927118 (2021) https://doi.org/10.1073/pnas.202292711
  * Madami et al. [2011] Madami, M., Bonetti, S., Consolo, G., Tacchi, S., Carlotti, G., Gubbiotti, G., Mancoff, F.B., Yar, M.A., Åkerman, J.: Direct observation of a propagating spin wave induced by spin-transfer torque. Nature Nanotechnology 6(10), 635–638 (2011) https://doi.org/10.1038/nnano.2011.140
  * Gallardo et al. [2023] Gallardo, R., Weigand, M., Schultheiss, K., Kakay, A., Mattheis, R., Raabe, J., Schütz, G., Deac, A., Lindner, J., Wintz, S.: Coherent Magnons with Giant Nonreciprocity at Nanoscale Wavelengths. ACS Nano 18, 5249–5257 (2023) https://doi.org/10.1021/acsnano.3c08390
  * Hillebrands [1990] Hillebrands, B.: Spin-wave calculations for multilayered structures. Physical Review B 41(1), 530–540 (1990) https://doi.org/10.1103/PhysRevB.41.530
  * Cochran et al. [1990] Cochran, J.F., Rudd, J., Muir, W.B., Heinrich, B., Celinski, Z.: Brillouin light-scattering experiments on exchange-coupled ultrathin bilayers of iron separated by epitaxial copper (001). Physical Review B 42(1), 508–521 (1990) https://doi.org/10.1103/PhysRevB.42.508
  * Yamamoto et al. [2020] Yamamoto, K., Yu, W., Yu, T., Puebla, J., Xu, M., Maekawa, S., Bauer, G.: Non-reciprocal pumping of surface acoustic waves by spin wave resonance. Journal of the Physical Society of Japan 89(11), 113702 (2020) https://doi.org/10.7566/JPSJ.89.113702
  * Körber et al. [2022] Körber, L., Verba, R., Otálora, J.A., Kravchuk, V., Lindner, J., Fassbender, J., Kákay, A.: Curvilinear spin-wave dynamics beyond the thin-shell approximation: Magnetic nanotubes as a case study. Physical Review B 106(1), 014405 (2022) https://doi.org/10.1103/PhysRevB.106.014405 arXiv:2204.05065
  * Giordano et al. [2020] Giordano, M.C., Baumgaertl, K., Escobar Steinvall, S., Gay, J., Vuichard, M., Fontcuberta I Morral, A., Grundler, D.: Plasma-Enhanced Atomic Layer Deposition of Nickel Nanotubes with Low Resistivity and Coherent Magnetization Dynamics for 3D Spintronics. ACS Applied Materials and Interfaces 12(36), 40443–40452 (2020) https://doi.org/10.1021/acsami.0c06879 arXiv:2004.01592
  * Körber et al. [2021] Körber, L., Zimmermann, M., Wintz, S., Finizio, S., Kronseder, M., Bougeard, D., Dirnberger, F., Weigand, M., Raabe, J., Otálora, J.A., Schultheiss, H., Josten, E., Lindner, J., Kézsmárki, I., Back, C.H., Kákay, A.: Symmetry and curvature effects on spin waves in vortex-state hexagonal nanotubes. Physical Review B 104(18), 184429 (2021) https://doi.org/10.1103/PhysRevB.104.184429 arXiv:2108.07103
  * Giordano et al. [2023] Giordano, M.C., Hamdi, M., Mucchietto, A., Grundler, D.: Confined spin waves in magnetochiral nanotubes with axial and circumferential magnetization. Physical Review Materials 7(2), 1–6 (2023) https://doi.org/10.1103/PhysRevMaterials.7.024405
  * Spaldin et al. [2008] Spaldin, N.A., Fiebig, M., Mostovoy, M.: The toroidal moment in condensed-matter physics and its relation to the magnetoelectric effect. J. Phys. Cond. Mat. 20, 434203 (2008) https://doi.org/10.1088/0953-8984/20/43/434203
  * Kuepferling et al. [2023] Kuepferling, M., Casiraghi, A., Soares, G., Durin, G., Garcia-Sanchez, F., Chen, L., Back, C.H., Marrows, C.H., Tacchi, S., Carlotti, G.: Measuring interfacial Dzyaloshinskii-Moriya interaction in ultrathin magnetic films. Reviews of Modern Physics 95(1), 15003 (2023) https://doi.org/10.1103/RevModPhys.95.015003 arXiv:2009.11830
  * Nomura et al. [2023] Nomura, T., Zhang, X.X., Takagi, R., Karube, K., Kikkawa, A., Taguchi, Y., Tokura, Y., Zherlitsyn, S., Kohama, Y., Seki, S.: Nonreciprocal Phonon Propagation in a Metallic Chiral Magnet. Physical Review Letters 130(17), 176301 (2023) https://doi.org/10.1103/PhysRevLett.130.176301 arXiv:2212.00225
  * Vansteenkiste et al. [2014] Vansteenkiste, A., Leliaert, J., Dvornik, M., Helsen, M., Garcia-Sanchez, F., Van Waeyenberge, B.: The design and verification of MuMax3. AIP Advances 4(10), 107133 (2014) https://doi.org/10.1063/1.4899186 . Accessed 2024-02-09
  * Salazar-Cardona et al. [2021] Salazar-Cardona, M.M., Körber, L., Schultheiss, H., Lenz, K., Thomas, A., Nielsch, K., Kákay, A., Otálora, J.A.: Nonreciprocity of spin waves in magnetic nanotubes with helical equilibrium magnetization. Applied Physics Letters 118(26), 262411 (2021) https://doi.org/10.1063/5.0048692 . Accessed 2024-02-08
  * del Ser et al. [2021] Ser, N., Heinen, L., Rosch, A.: Archimedean screw in driven chiral magnets. SciPost Physics 11(1), 009 (2021) https://doi.org/10.21468/SCIPOSTPHYS.11.1.009 arXiv:2012.11548
  * Giordano et al. [2021] Giordano, M.C., Escobar Steinvall, S., Watanabe, S., Fontcuberta I Morral, A., Grundler, D.: Ni80Fe20 nanotubes with optimized spintronic functionalities prepared by atomic layer deposition. Nanoscale 13(31), 13451–13462 (2021) https://doi.org/10.1039/d1nr02291a arXiv:2105.01969
  * Exl et al. [2014] Exl, L., Bance, S., Reichel, F., Schrefl, T., Peter Stimming, H., Mauser, N.J.: LaBonte’s method revisited: An effective steepest descent method for micromagnetic energy minimization. Journal of Applied Physics 115(17), 17–118 (2014) https://doi.org/10.1063/1.4862839 . Accessed 2024-02-09
  * Körber [2023] Körber, L.: Spin waves in curved magnetic shells. PhD thesis, Technische Universität Dresden (2023). https://doi.org/10.25368/2023.131 . https://nbn-resolving.org/urn:nbn:de:bsz:14-qucosa2-869431 Accessed 2024-02-08

##  7 Methods

###  7.1 Sample Preparation

The magnetic chiral tubes were fabricated by combining two-photon lithography
(TPL) and atomic layer deposition (ALD). We applied the additive manufacturing
methodology described in [20] to 3D polymer wires which contained helical
reliefs. They were prepared by TPL using a Photonic Professional GT+ system
(Nanoscribe Inc., Germany) in three steps. Firstly, negative photoresist IP-
Dip was dropped onto a fused-silica substrate (25 x 25 $\mathrm{mm^{2}}$ with
a thickness of 0.7 mm). Secondly, an infrared femtosecond laser (wavelength:
780 nm, power: 20 mW) was focused inside the resist exploiting the dip-in
laser lithography (DILL) configuration for the exposure. Thirdly, the whole
substrate was immersed in propylene glycol monomethyl ether acetate (PGMEA)
for 20 min and isopropyl alcohol (IPA) for another 5 min. After the polymer
was dried in ambient conditions, the sample was put into a hot wall Beneq
TFS200 ALD system. We conformally coated the polymer with a 30-nm-thick Ni
shell after depositing 5-nm-thick Al2O3 using the plasma-enhanced ALD process
presented in [36]. The detailed preparation process is presented in (Fig. S1).

###  7.2 Brillouin light scattering microscopy

The spin dynamics were investigated by $\mu$-BLS at room temperature (Fig.
S2). The samples were mounted on a piezo stage by which the sample was moved
in steps of 50 nm underneath the laser focus. Positive and negative external
magnetic fields were applied by permanent magnets mounted in different
orientations along the $x$-axis. The ACMs were positioned parallel to the
$x$-axis. The green laser (wavelength: 532 nm) with a power of 3 mW was
focused on the surface of the helical magnet using a 100X objective lens with
a numerical aperture of 0.75. The s-polarized component of the scattered light
was passed through a Glan-Taylor polarizer and sent to a six-pass tandem
Fabry-Pérot interferometer. In the $\mu$-BLS setup, the focussed laser light
offers a cone of incidence angles around the optical axis of the lens. The
backscattered light contains photons which interact with magnons with
different in-plane wave vectors $+k$ and $-k$. The magnitudes of $k$ vectors
ranges from 0 to about $17.7~{}\mathrm{rad}/\mu$m.

###  7.3 Simulation

Micromagnetic simulations were conducted using the MuMax3 software [42] which
solves the Landau-Lifshitz-Gilbert (LLG) equation on a finite difference grid.
We considered a Ni ACM consisting of a tube with inner radius 220 nm and
thickness 30 nm which intersects a hollow helix with ellipsoidal cross-
section. The helix had a pitch of 2000 nm, diameter of 740 nm, cross-sectional
inner major and minor radii of 120 nm and 70 nm respectively, and thickness 30
nm. The saturation magnetization was set to $M_{\rm s}=490~{}\mathrm{kA/m}$
and the exchange stiffness to $A_{\rm ex}=8~{}\rm{pJ/m}$. The system was
discretized into 160×160×384 cells of dimension $5\times 5\times
5.2~{}\rm{nm^{3}}$. Six repetitions of periodic boundary conditions along the
$z$-direction were used.

Hysteresis diagrams of the structures were computed by sweeping an applied
field parallel to the tube axis with a $2^{\circ}$ misalignment between
$+1~{}\rm{T}$ and $–1~{}\rm{T}$ and back to $+1~{}\rm{T}$. Additionally, a
constant background field of 0.7 mT along the $x,y$-diagonal was applied. The
magnetic ground state was computed in between specified field increments by
first using the steepest conjugate gradient method [46] to minimize the energy
and then solving the LLG without precessional term. The resulting ground
states provided the initial state for the computation of the toroidal moment
and the dynamic behavior.

The toroidal moment was computed per layer using Eq. 1 using the tube axis as
the origin. The plots in Fig. 4c-e show the product
$\mathbf{r}\times\mathbf{m}\left(\mathbf{r}\right)$ projected on the $z$-axis.

The dynamic simulations were conducted following the methodology in the main
text. The dynamic field was confined to a strip of width 20 nm. The
simulations were run for a total time of 53.3 ns and the magnetization was
sampled on the surface of the tube along the tube axis every 33.3 ps. The
damping was set to $\alpha=10^{=-3}$ and increased quadratically to 1 near the
ends of the structure. For the numerical results shown in Fig. 3d, care had to
be taken that no asymmetries due to field misalignment were present.
Therefore, the ground state from the hysteresis was used as an initial state,
then the background signal was removed and the applied field was fully aligned
with the tube axis. The ground state was subsequently recalculated. The
gilbert damping was set to $\alpha=10^{-3}$. A fast Fourier transform (FFT)
was performed over the dynamic magnetization components $m_{\rm x},m_{\rm y}$.
Then, the intensity of the complex dynamic magnetization $m_{\rm d}=m_{\rm
x}+im_{\rm y}$ was computed. Its spatial distribution is shown in Fig. 5 d,e
and the integrated intensity is shown in Fig. 5 c. Finally, the dispersion
shown in Fig. 5 was obtained by performing a 2D FFT over the magnetization
sampled on the tube along the $z$-axis.

To clearly visualize the magnon propagation in real space, simulations were
performed on longer tubes (4000 nm). No periodic boundaries were applied. To
minimize stray field effects at the edges, the tube radius was reduced to 125
nm and the helix end-to-end diameter was set to 200 nm. The dispersion of this
structure is shown in Supplementary.

###  7.4 Analytical dispersion

The dispersion in Fig. 5 is plotted together with data obtained from the
analytical model proposed by Salazar et al. [43] for nanotubes with helical
equilibrium magnetization. The dispersion is given by

$\omega_{m}\left(k\right)=\omega_{M}\left[\mathcal{A}_{m}\left(k\right)+\sqrt{\mathcal{B}_{m}\left(k\right)\mathcal{C}_{m}\left(k\right)}\right],$
(6)

with $\omega_{M}=\gamma\mu_{\rm 0}M_{\rm s}$, $\gamma$ is the gyromagnetic
ratio and $k$ the wavevector. The index $m$ denotes the azimuthal mode.
$\mathcal{A}_{m}\left(k\right),\mathcal{B}_{m}\left(k\right),\mathcal{C}_{m}\left(k\right)$
are the dynamic stiffness fields. The frequency non-reciprocity is determined
by the magnetochiral stiffness field
$\mathcal{A}_{k}^{m}=-\chi\mathcal{K}\quantity(m,k)\sin\quantity(\theta)+p\quantity(\mathcal{N}\quantity(m,k)-\frac{2m\lambda_{exc}^{2}}{b^{2}})\cos\quantity(\theta)$.
Here, $\theta$ is the angle of the magnetization with respect to the tube
axis, $p=\pm 1$ is the polarity of the magnetization and $\chi=\pm 1$ is the
helicity. We note that there is a minus sign difference for the expression of
$\mathcal{A}$ [47]. The functions
$\mathcal{K}\quantity(m,k),\mathcal{N}\quantity(m,k_{z})$ are demagnetizing
factors. The data shown in Fig. 5 is obtained from Eq. 6 in the ultra-thin
shell approximation where $t\approx\lambda_{\rm exc}\ll R$, with $t$ the
thickness, $\lambda_{\rm exc}$ the exchange length and $R$ the mean radius of
the tube. Complete expressions are given in the supplementary information.

## Declarations

Acknowledgments This research was supported by the SNSF via grant number
197360. The authors thank PD Dr. Oleksandr Dobrovolskiy, Dr. Jun He, Dr. Yang
Liu for the helpful discussion. The simulations and post-processing have been
performed using the facilities of the ScientificIT and Application Support
Center of EPFL.

Availability of data and materials The data presented in the main text and the
supplementary information are available from the corresponding authors upon
reasonable request. Code availability The code used in this study is available
from the corresponding author upon reasonable request. Competing interests The
authors declare no competing interests. Author contributions M.X. and D.G.
conceived the study and the design of the devices. M.X., A.D., and D.G.
prepared figures. M.X. wrote the main manuscript text, with support from D.G.
and A.D. M.X. performed the measurements. A.D. conducted micromagnetic
simulations. H.G. fabricated the devices. M.X., A.D., and D.G. processed and
analyzed the measurements and simulations. All authors reviewed the manuscript
and provided critical feedback. D.G. supervised this project.