# Low Temperature Multi-mode Microwave Spectroscopy of Single Crystal Calcium
Tungstate

Elrina Hartman  Quantum Technologies and Dark Matter Labs, Department of
Physics, University of Western Australia, 35 Stirling Highway, Crawley, WA
6009, Australia<EMAIL_ADDRESS>
<EMAIL_ADDRESS>Michael E Tobar  Quantum Technologies and Dark Matter
Labs, Department of Physics, University of Western Australia, 35 Stirling
Highway, Crawley, WA 6009, Australia.  Ben T McAllister  Quantum Technologies
and Dark Matter Labs, Department of Physics, University of Western Australia,
35 Stirling Highway, Crawley, WA 6009, Australia.  Jeremy Bourhill  Quantum
Technologies and Dark Matter Labs, Department of Physics, University of
Western Australia, 35 Stirling Highway, Crawley, WA 6009, Australia.  Maxim
Goryachev  Quantum Technologies and Dark Matter Labs, Department of Physics,
University of Western Australia, 35 Stirling Highway, Crawley, WA 6009,
Australia.

###### Abstract

Single crystal calcium tungstate (CaWO4) is an interesting material for
investigating spins in solids using microwave quantum electrodynamics (QED).
We construct a dielectrically loaded microwave cavity resonator from a
cylindrical single crystal of CaWO4 and perform whispering gallery multi-mode
spectroscopy at 30 mK. This study found many high-$Q$ modes, indicating a low
dielectric loss tangent ($\tan\delta<10^{-7}$) which enables high sensitivity
analysis of photon-spin interactions. Measurements of spin $g_{L}$ factors and
zero field splittings (ZFS) allow for identification and characterization of
spins that arise from the presence of paramagnetic impurities. We found Gd3+
concentrations of $\mathcal{O}(10^{13})$ cm-3 and lower bounds on the spin
concentration could be placed on the order of ppb. Additionally, we observe
the presence of unknown rare-earth metal impurities at much lower
concentrations.

††preprint: APS/123-QED

##  I INTRODUCTION

Scheelite, or calcium tungstate (CaWO4) is a crystalline dielectric material
whose crystal structure has been well characterized [1] and is an important
material that is implemented as a scintillation detector for rare events, such
as neutrinoless double $\beta$-decay [2], radioactive decay of very long-
living isotopes [3] and searches for weakly interacting massive particles
(WIMPs), a candidate for Dark Matter (DM) [4, 5]. The low nuclear-spin density
of CaWO4 makes it a promising platform for hosting spin qubits, with
particular interest in utilizing rare-earth ions for quantum information
storage and processing. Spin populations can be introduced via doping
mechanisms but commonly arise from contamination during the crystal growth
process [6]. Samples may be subject to contamination from radioactive isotopes
(leading to the creation of colour centres), and rare-earth impurities [7, 8].
Previous experimental investigations have been performed on CaWO4, studying
Er3+[9, 10, 11, 12, 13, 14, 15, 16], Yb3+ [17], Nd3+ [18] and Pr3+[19, 20, 21]
dopants.

This work explores the spectroscopic response of CaWO4 manufactured by
SurfaceNet GmbH in the microwave regime and at low temperatures, which is
shown in Fig. 1. We identified some residual paramagnetic impurities that
exist within the crystal lattice, in particular, the large spin system of the
rare earth ion, Gd3+ [22]. Knowledge of the properties of these impurities in
CaWO4 at cryogenic temperatures represents a gap in the scientific literature
at present. The prospect for employing this crystal in rare-event searches
either as a cryogenic bolometer or qubit host motivates the use of non-
destructive spectroscopy which will be outlined in the following section.

##  II Experiment Setup

Figure 1: The CaWO4 crystal purchased from SurfaceNet was cut along the
$\langle 100\rangle$ axis so that the c-axis of a unit cell aligns with the
radial-axis of the cylinder. A displays a schematic of the dielectric sample
(dim: $r_{d}=14.68\,$mm, $h_{d}=20.00\,$mm), positioned inside a copper cavity
(dim: $r_{c}=25.00\,$mm, $h_{c}=40.00\,$mm), and mounted on a sapphire post
(dim: $r_{p}=1.75\,$mm, $h_{p}=14.50\,$mm) that slots into a central hole at
the base of the crystal. The sample was photographed from the top view (B) and
the side view (C). The sample exhibits slight discoloration due to the
presence of the paramagnetic impurities. The borehole that appears white in
the photograph is in a region of low field density for a WGM so frequency
perturbations due to the presence of the sapphire post are negligible.

Figure 2: The crystal was placed in an oxygen-free copper cavity and mounted
within the magnet bore of a dilution refrigerator. A microwave signal was
input to the cavity using a Vector Network Analyser (VNA). The VNA and
cryogenic dilution refrigerator were controlled by a computer that sets all
parameters such as scan rate, power input, magnetic field strength, and
frequency span.

Figure 3: A large number of high-$Q$ resonances are present in this sample
(blue). A spread of resonances were selected (red) in the present study with
additional lower-$Q$ modes at lower frequencies to resolve zero field
splittings (ZFS).

The CaWO4 crystal lattice is made of diamagnetic atoms, thus, any paramagnetic
impurities and lattice distortions can be detected using electron spin
resonance spectroscopy (ESR). Coupling photonic resonances to spin resonances
allows us to determine zero-field splittings (ZFS) and Landé g factors
($g_{L}$) via Zeeman splitting [23]. The energies of such interactions are
characteristic markers for ion impurities that allow for their identification.
Even at low concentrations of paramagnetic impurity ($<0.0001\%$) [24], the
spin magnetic moment can be sensed. In order to discern weakly coupled spin
systems, we excite spin transitions using high-$Q$ photonic modes, namely,
transverse magnetic whispering gallery modes (TM-WGM) from $7$ GHz to $22$
GHz. When the energy of a spin transition matches the energy of an injected
WGM, the photon resonance is coupled to the spin resonance to induce the spin
transition. The spin transition can be considered a hybridisation of the spin
and photon resonances. This hybridisation results in a frequency shift of the
photonic mode as well as a sharing of loss mechanisms between the two systems,
typically resulting in a transmission amplitude shift and change in microwave
$Q$-factor.

To determine the paramagnetic impurities present in the sample, the
dielectrically loaded cavity was cooled to 30 mK in a dilution refrigerator as
shown in Fig. 2. A VNA was swept to determine the frequencies of the high-$Q$
WGMs. An external DC magnetic field was then ramped, tuning the spin
transition energies across the high-$Q$ WGMs. The transmission spectra of each
of these modes were recorded at all magnetic field values so the specific
interaction field could be captured as well as the impacts on the photon
mode’s frequency, $Q$-factor, and transmission. By compiling all of the ESR
interactions across the 23 recorded WGMs, a complete picture of paramagnetic
impurity interactions could be constructed. Characteristic Hamiltonians for
spin impurities in CaWO4 could then be used to fit the ZFS and magnetic field
dependence of the observed interactions, allowing a determination of the
impurity species. Further, shifts in $Q$-factors of the WGMs and fitted
coupling rates between the WGM and ESR allowed the determination of limits on
spin line-width $\Gamma_{s}$ and spin concentration, respectively.

Evanescent fields exist beyond the dielectric surface which decay
exponentially in the axial direction above and below the crystal and as a
modified Bessel function of the second kind in the radial direction [25]. WGMs
are measured by coupling to these radiative fields using transmission line
probes. Co-axial probes were inserted parallel to the z-axis at the base of
the cavity at a depth adjusted for under-coupling ($\beta\ll 1$, for
$\beta=\kappa_{in}/\kappa_{0}$, where $\kappa_{in}$ and $\kappa_{0}$ are the
loss rates of the input probes and the microwave resonance, respectively). The
crystal was mounted on a sapphire post which was clamped to the base of the
cavity to prevent loss from thermal contact with the conducting surfaces of
the cavity. The associated readouts are specified by scattering parameters or
s-parameters ($S_{11}$ is the input port power reflection coefficient and
$S_{21}$ is the forward power gain). Vector network analysis extracted the
$S_{21}$ parameter and the transfer function was modelled as a Breit-Wigner
distribution or asymmetric Fano pulse [26]:

$|S_{21}|=1-\frac{(q\Gamma_{p}/2+\Delta)^{2}}{(\Gamma_{p}/2)^{2}+\Delta^{2}}\,,$
(1)

where q is the Fano parameter measuring the ratio of resonant scattering to
background scattering, $\Gamma_{p}$ is the photonic resonance line-width, and
$\Delta$ is the frequency detuning ($f-f_{0}$). The Q-factors of the modes can
be calculated from the line-widths and are shown for zero DC magnetic field in
Fig. 3. The best Q-factors in the crystal are of order of $3\times 10^{7}$
close to 14 GHz, indicating a loss tangent of order $tan\delta\approx 3\times
10^{-8}$, which could be determined more precisely using the Whispering
Gallery technique [27, 28, 29, 30], which for our purposes here this was not
necessary to implement..

##  III Multimode Spectroscopy

We undertook multi-mode spectroscopy by varying the DC magnetic field and
continually measuring the high-Q WGMs of Fig. 3. This allowed us to
experimentally measure multiple paramagnetic spins interacting with the
multiple microwave modes. During the interactions the photons and the spins
hybridize, and may be modelled as a coupled harmonic oscillator with
$\omega_{+}$ and $\omega_{-}$ as the eigensolutions to the characteristic
equation [31];

$\omega_{\pm}=\frac{1}{\sqrt{2}}\sqrt{\omega_{s}^{2}+\omega_{p}^{2}\pm\sqrt{\omega_{s}^{4}-2\omega_{s}^{2}\omega_{p}^{2}+4\Delta_{ps}\omega_{s}^{2}\omega_{p}^{2}-\omega_{p}^{4}}}\,,$
(2)

where $\omega_{p}$ is a particular photonic resonance frequency and
$\omega_{s}$ is the spin resonance frequency, and $\Delta_{ps}$ is the
inductive coupling between these two resonant systems. $\Delta_{ps}$
parameterises the rate of coupling $g$ ($g\coloneqq\Delta_{ps}\omega_{p}/2$).
In the limit where $\omega_{s}$ is detuned far from $\omega_{p}$,
$\omega_{s}=\gamma B$ exhibits linear dependence on the magnetic field ($B$)
where $\gamma$ is the gyromagnetic ratio. As
$\omega_{s}\xrightarrow[]{}\omega_{p}$, $\omega_{+}$ and $\omega_{-}$ are
detuned asymptotically around the intersection point representing a forbidden
region of energy for the coupled system, called an avoided level crossing
(ALC). When $\omega_{s}=\omega_{p}$, the two resonant systems are maximally
hybridised, and $\omega_{+}$ and $\omega_{-}$ represent the normal mode
solutions of the coupled system. The frequency separation of these two modes
($\omega_{+}-\omega_{-}=2\Delta_{ps}\omega_{p}=2g$) can be extracted by
fitting equation 2 to the ALC. With a single photonic resonance, multiple spin
interactions can be observed as the magnetic field tunes the latter over the
former. If the photonic mode’s frequency increases (decreases) as
$\omega_{s}\xrightarrow[]{}\omega_{p}$, then the Zeeman energy levels of the
spin resonance transition are diverging (converging) as magnetic field is
increased. In the present study, all ALCs were categorized accordingly to
better track the spin resonances over multiple TM-WGMs and denoted as ‘+ve
ALC’ (‘-ve ALC’).

Figure 4: A shows the frequency detuning ($\Delta$) for a selected high-order
TM-WGM where $f_{0}=14.934048$ GHz. The vertical black lines mark the magnetic
field positions of ALCs. As an example, the first dashed line from the left is
a -ve ALC, the second is +ve ALC. A large proportion of the ALC sites
identified spectroscopically occurred close together and/or exhibited very
weak coupling (such as the three dot-dashed lines on the right). The fourth
dashed line demonstrates a stronger +ve ALC that was fitted to by $\omega_{+}$
and $\omega_{-}$. The frequency splitting of the ALC in A at the magnetic
field value where $\omega_{s}=\omega_{p}$ is equal to the magnitude of
$2g/2\pi$ and secondary information such as lower bounds on $Q_{s}$ and can be
extracted by fitting B as per equation 3, with the results presented in Table
2. Each frequency detuning in A has a corresponding $Q$ detuning in B.
Reduction in the amplitude of the transfer function due to absorption of the
spin is also shown (C). D is the transfer function of the photonic resonance
that relates to A, B, and C.

Figure 5: A The ALC map summarises the results of the spectroscopy. Shown are
the locations of ALCs in the frequency to magnetic field parameter space. Spin
transitions in CaWO4:Gd3+ are derived from the spin Hamiltonian (see equation
5) and the corresponding transitions are shown in the CaWO4:Gd3+ energy-level
diagram in B. The dashed red line marks the resonant frequency of the WGM
shown in Fig. 4. The grey lines, A and B, track the spin transitions of
unidentified impurities as a function of magnetic field and allow the
determination of the ZFS. The gradient gives $g_{L}$
($\frac{df}{dB}=g_{L}\mu_{B}h^{-1}$ where $h$ is Plank’s constant).

From the ALCs, information about the spin systems could be extracted. The $Q$
factor of the coupled photon-spin system can be modelled with the following
equation [32]:

$\begin{pmatrix}\frac{\omega_{+}}{Q_{+}}\\\
\frac{\omega_{-}}{Q_{-}}\end{pmatrix}=\begin{pmatrix}1&1\\\
\omega_{-}^{2}&\omega_{+}^{2}\end{pmatrix}^{-1}\begin{pmatrix}1&1\\\
\omega_{s}^{2}&\omega_{p}^{2}\end{pmatrix}\begin{pmatrix}\frac{\omega_{p}}{Q_{p}}\\\
\frac{\omega_{s}}{Q_{s}}\end{pmatrix}\,.$ (3)

This expression allows for the determination of upper bounds on spin resonance
line widths ($\Gamma_{s}=\frac{\omega_{s}}{Q_{s}}$) and lower bound on spin
quality factor ($Q_{s}$). Fig. 4B shows how the $Q$ of the hybrid system in
the coupling regime is limited by the highest loss system, which in this case
is the spin resonance, leading to the degradation of the photonic resonance
quality factor ($Q_{p}$). There is a corresponding dip in transmission,
demonstrated by Fig. 4C.

The concentration of impurity spins ($n$) was derived from the rate of
coupling:

$g=g_{L}\mu_{B}\sqrt{\frac{\mu_{0}\omega{{}_{p}}n\xi_{\perp}}{4\hbar}}\,,$ (4)

where $\mu_{B}$ is the Bohr Magneton, $\hbar$ is the reduced Planck’s
constant, $\xi_{\perp}$ is the perpendicular magnetic filling factor [31]
($\xi_{\perp}\approx 1$ for TM-WGMs), and $\mu_{0}$ is the permeability of
free space. With sufficient coupling rate, spin interactions such as those
shown in Fig. 4 help derive spin quality factor ($Q_{s}$), $g_{L}$, and
impurity concentrations. Zooming out along the frequency axis allows us to
follow ALC locations across an ensemble of WGMs and allows us to track the
$\omega_{s}$ as magnetic field strength increases. This gives us the $g_{L}$
and ZFS (see Tab. I), which are typical for a paramagnetic impurity in a given
crystal field environment, allowing for the identification of the impurity.

Species |  $\Delta$M |  ZFS (GHz) |  Transition
---|---|---|---
CaWO4:Gd3+ |  $1$ |  $4.61$ |  $|-3/2\rangle\rightarrow|-1/2\rangle$
$g_{L}=1.99$ |  |  $10.42$ |  $|+5/2\rangle\rightarrow|+3/2\rangle$
|  |  |  $|-5/2\rangle\rightarrow|-3/2\rangle$
|  |  $17.90$ |  $|+7/2\rangle\rightarrow|+5/2\rangle$
|  |  |  $|-7/2\rangle\rightarrow|-5/2\rangle$
|  $2$ |  $15.03$ |  $|-5/2\rangle\rightarrow|-1/2\rangle$
|  |  |  $|+5/2\rangle\rightarrow|+1/2\rangle$
|  $3$ |  $15.03$ |  $|-5/2\rangle\rightarrow|+1/2\rangle$
|  |  |  $|+5/2\rangle\rightarrow|-1/2\rangle$
|  $4$ |  $10.42$ |  $|-5/2\rangle\rightarrow|+3/2\rangle$
|  |  |  $|+5/2\rangle\rightarrow|-3/2\rangle$
|  $5$ |  $2.22$ |  $|+7/2\rangle\rightarrow|-3/2\rangle$
|  |  $28.33$ |  $|-5/2\rangle\rightarrow|+5/2\rangle$
A |  |  |
$g_{L}=7$ |  - |  $6.10$ |  -
B |  |  |
$g_{L}=4.3$ |  - |  $2.20$ |  -
Table 1: Determined properties of some spin transitions calculated form the
multi-mode spectroscopy. Here $\Delta$M is the change in spin quantum number

From the results of the spectroscopy, an ‘interaction map’ can be plotted as
shown in Fig. 5A from which the spin transition frequencies can be fitted as a
function of magnetic field, using +ve or -ve ALC designation as a helpful
guide. The energies of these spin transitions can be described using a spin
Hamiltonian that takes into consideration the following contributions; the
Zeeman splitting term, the crystal field, and hyperfine splitting due to spin-
orbit coupling. On the scale of GHz, spin-orbit coupling is not elucidated by
the data, as the detuning resolution and the uncertainty of the detuning from
the fitting is on the order of magnitude as the hyper-fine structure
constants. Thus, their contribution is neglected for the present study and we
turn our focus to the crystal field parameters and the Zeeman splitting term.
The Zeeman term gives us the $g_{L}$ and we extract this value from the slopes
of the Zeeman levels. The crystal field parameters are constructed by well-
defined tesseral harmonics that describe the spin energy levels. We choose the
Stevens representation which builds operators from spin operators as a basis,
and determine the Stevens coefficients experimentally. We rely here on the
work of previous investigations that analysed doped CaWO4 samples to check
which Hamiltonians correspond to the data in the ALC map. Pitfalls arise as
not all the energy levels are occupied or coupled to with the present setup.
Thus, not every energy level theoretically predicted is reflected in the
experiment. Nevertheless, with ZFS values and $g_{L}$ values, we can check
through various elements to match this experiment to previously derived
results. It is through this process of matching, checking, and eliminating
that we find the results indicate that the CaWO4 sample is contaminated with
trace amounts ($n\approx\mathcal{O}(10^{13})$) of Gd3+ ions, and other rare-
earth metals yet to be identified.

$Q_{p}$ $\times 10^{6}$ |  $Q_{s}$ $\times 10^{3}$ lower bound |  $\Gamma_{p}/2$ (kHz) |  $\Gamma_{s}/2$ (MHz) upper bound |  n $\times 10^{13}$ (cm-3)
---|---|---|---|---
6.53 |  5.5 |  1.14 |  1.35 |  8.28
Table 2: Gd3+ concentration ($n$) was estimated from frequency and $Q$
detuning models as shown in Fig. 4.

The crystal field of CaWO4:Gd3+ was modelled in this work by the following
Hamiltonian;

$\begin{split}\mathcal{H}=\,&g_{L}\mu_{B}H_{z}S_{z}+B^{0}_{2}O^{0}_{2}+B^{0}_{4}O^{0}_{4}\\\
&+B^{4}_{4}O^{4}_{4}+B^{0}_{6}O^{0}_{6}+B^{4}_{6}O^{4}_{6}\,,\end{split}$ (5)

where $O^{j}_{k}$ are Stevens operators, $S=\frac{7}{2}$ and $g_{L}=1.99$. The
crystal field parameters are; $B^{0}_{2}=-9.215\cdot 10^{-1}$,
$B^{0}_{4}=-1.139\cdot 10^{-3}$, $B^{4}_{4}=-7.015\cdot 10^{-3}$,
$B^{0}_{6}=5.935\cdot 10^{-7}$ , and $B^{4}_{6}=4.747\cdot 10^{-7}$ in units
of GHz. This Hamiltonian allows the determination of the Zeeman energy levels
of Gd3+ in CaWO4, which are shown in Fig. 5B. The energy difference (vertical
dashed lines) between the allowed transitions of these levels (coloured lines)
allows one to identify the spin transitions in Fig. 5A. This result is in good
agreement with the literature [33] with the only discrepancy being a $3.14\%$
increase in the value of $B^{0}_{2}$ which dictates the ZFS values. This is
due to the difference in orientation of this sample’s c-axis to that of
Baibekov et.al. who use a $\langle 001\rangle$ cut single crystal.

##  IV Conclusion

This study found the presence of rare-earth impurities in a scheelite sample
as indicated by the high $g_{L}$ factors and low ZFS which is characteristic
of f-shell elements. The limitation in magnetic field resolution ($0.001$ T)
made sound model fitting unfeasible in many instances, which would require
more refined magnetic fields steps.

This work adds to the literature of paramagnetic spin impurities in
crystalline CaWO4, useful for designs which require coherent spin ensembles to
perform quantum information processes and as well as applications, which would
utilise such a crystal for novel sensing methods, such as bolometers for dark
matter detection.

##  V Acknowledgements

This work was funded by the ARC Centre of Excellence for Engineered Quantum
Systems, CE170100009, and Dark Matter Particle Physics, CE200100008. BM was
also funded by the Forrest Research Foundation.

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# Topological magnons driven by the Dzyaloshinskii-Moriya interaction in the
centrosymmetric ferromagnet Mn5Ge3

M. dos Santos Dias<EMAIL_ADDRESS>Peter Grünberg Institut
and Institute for Advanced Simulation, Forschungszentrum Jülich $\&$ JARA,
D-52425 Jülich, Germany  Faculty of Physics, University of Duisburg-Essen and
CENIDE, D-47053 Duisburg, Germany  Scientific Computing Department, STFC
Daresbury Laboratory, Warrington WA4 4AD, United Kingdom  N. Biniskos
<EMAIL_ADDRESS>Forschungszentrum Jülich GmbH, Jülich Centre
for Neutron Science at MLZ, Lichtenbergstr. 1, D-85748 Garching, Germany
Current address: Charles University, Faculty of Mathematics and Physics,
Department of Condensed Matter Physics, Ke Karlovu 5, 121 16, Praha, Czech
Republic  F. J. dos Santos<EMAIL_ADDRESS>Laboratory for Materials
Simulations, Paul Scherrer Institut, 5232 Villigen PSI, Switzerland  Theory
and Simulation of Materials (THEOS), and National Centre for Computational
Design and Discovery of Novel Materials (MARVEL), École Polytechnique Fédérale
de Lausanne, 1015 Lausanne, Switzerland  K. Schmalzl  Forschungszentrum Jülich
GmbH, Jülich Centre for Neutron Science at ILL, 71 Avenue des Martyrs, F-38000
Grenoble, France  J. Persson  Forschungszentrum Jülich GmbH, Jülich Centre for
Neutron Science (JCNS-2) and Peter Grünberg Institut (PGI-4), JARA-FIT,
D-52425 Jülich, Germany  F. Bourdarot  Université Grenoble Alpes, CEA, IRIG,
MEM, MDN, F-38000 Grenoble, France  N. Marzari  Theory and Simulation of
Materials (THEOS), and National Centre for Computational Design and Discovery
of Novel Materials (MARVEL), École Polytechnique Fédérale de Lausanne, 1015
Lausanne, Switzerland  Laboratory for Materials Simulations, Paul Scherrer
Institut, 5232 Villigen PSI, Switzerland  S. Blügel  Peter Grünberg Institut
and Institute for Advanced Simulation, Forschungszentrum Jülich $\&$ JARA,
D-52425 Jülich, Germany  T. Brückel  Forschungszentrum Jülich GmbH, Jülich
Centre for Neutron Science (JCNS-2) and Peter Grünberg Institut (PGI-4), JARA-
FIT, D-52425 Jülich, Germany  S. Lounis  Peter Grünberg Institut and Institute
for Advanced Simulation, Forschungszentrum Jülich $\&$ JARA, D-52425 Jülich,
Germany  Faculty of Physics, University of Duisburg-Essen and CENIDE, D-47053
Duisburg, Germany

###### Abstract

The phase of the quantum-mechanical wave function can encode a topological
structure with wide-ranging physical consequences, such as anomalous transport
effects and the existence of edge states robust against perturbations. While
this has been exhaustively demonstrated for electrons, properties associated
with the elementary quasiparticles in magnetic materials are still
underexplored. Here, we show theoretically and via inelastic neutron
scattering experiments that the bulk ferromagnet Mn5Ge3 hosts gapped
topological Dirac magnons. Although inversion symmetry prohibits a net
Dzyaloshinskii-Moriya interaction in the unit cell, it is locally allowed and
is responsible for the gap opening in the magnon spectrum. This gap is
predicted and experimentally verified to close by rotating the magnetization
away from the $c$-axis with an applied magnetic field. Hence, Mn5Ge3 realizes
a gapped Dirac magnon material in three dimensions. Its tunability by chemical
doping or by thin film nanostructuring defines an exciting new platform to
explore and design topological magnons. More generally, our experimental route
to verify and control the topological character of the magnons is applicable
to bulk centrosymmetric hexagonal materials, which calls for systematic
investigation.

##  I Introduction

Recent breakthroughs in the physics of electrons in solids resulted from the
application of topological concepts to the quantum-mechanical wave function,
highlighting the role of the Berry phase Berry (1984). For instance, the
modern understanding of the integer quantum Hall effect in a two-dimensional
(2D) system is that of a gapped bulk with non-zero Chern numbers which imply
the existence of chiral edge states responsible for the quantized conduction
Thouless  _et al._ (1982). Three-dimensional (3D) topological insulators are
gapped by the spin-orbit interaction, leading to Dirac-like surface states
with a linear dispersion and spin-momentum locking that underpin the quantum
spin Hall effect Hasan and Kane (2010). There is now a systematic
classification of the possible topological phases in electronic systems,
encompassing also gapless systems such as Weyl semimetals, where the
dimensionality but also spatial and magnetic symmetries are prominent Bradlyn
_et al._ (2017); Po  _et al._ (2017); Kruthoff  _et al._ (2017).

Topology is agnostic to whether the (quasi)particles are fermions or bosons,
so that magnons can also be responsible for novel physical effects McClarty
(2022). Perhaps the first example is the Hall effect experienced by a
thermally-induced magnon current in ferromagnetic (FM) Lu2V2O7, with the
Dzyaloshinskii-Moriya interaction (DMI) playing a similar role to the one of
spin-orbit coupling (SOC) for electronic systems Onose  _et al._ (2010);
Matsumoto and Murakami (2011); Mena  _et al._ (2014). Magnetic materials with
an hexagonal lattice generally exhibit a Dirac-like magnon dispersion at the
K-point in the Brillouin zone, which if gapped signals the existence of non-
trivial topology Zhang  _et al._ (2013); Mook  _et al._ (2014); Chisnell  _et
al._ (2015); Fransson  _et al._ (2016); Pershoguba  _et al._ (2018). Such
topological magnon insulators were experimentally identified in 2D FM
materials Chen  _et al._ (2018, 2021); Zhu  _et al._ (2021), while gapless
Dirac magnons were characterized in 3D magnetic materials such as
antiferromagnetic (AFM) Cu3TeO6 Yao  _et al._ (2018) and CoTiO3 Yuan  _et al._
(2020); Elliot  _et al._ (2021), and the elemental FM Gd Scheie  _et al._
(2022). While a symmetry-based approach has been proposed to predict materials
hosting topological magnons Karaki  _et al._ (2023), experimentally confirming
the topological character is challenging, as the magnon Hall effect is
difficult to measure and other signatures such as a characteristic winding of
the scattering intensity have only recently been detected Elliot  _et al._
(2021).

In this article, we study the 3D centrosymmetric ferromagnet Mn5Ge3 Forsyth
and Brown (1990); Maraytta  _et al._ (2020). This material exhibits
significant anomalous Hall Zeng  _et al._ (2006) and Nernst Kraft  _et al._
(2020) effects which are signatures of large electronic Berry phases. Its
Curie temperature ($T_{\text{C}}$) can be enhanced by carbon doping Gajdzik
_et al._ (2000) as successfully described by a previous theoretical work
Slipukhina  _et al._ (2009), but its magnonic properties remain unexplored.
Here, we theoretically predict and experimentally confirm the existence of
gapped Dirac magnons at the K-point due to the DMI. This gap can be closed by
rotating the magnetization direction with an external magnetic field, thus
validating the proposed gap mechanism and confirming the topological character
of the magnons at the K-point. Our experimental route to verify and control
the topological character of the magnons is not limited to Mn5Ge3 and should
also be applicable to other bulk centrosymmetric hexagonal materials.

##  II Results

###  II.1 Basic properties of Mn5Ge3

A high-quality Mn5Ge3 single crystal of about $10\text{\,}\mathrm{g}$ has been
grown using the Czochralski method. The space group is $P6_{3}/mcm$ and the
unit cell contains 10 Mn atoms (and 6 Ge atoms), with Mn1 and Mn2 occupying
the Wyckoff positions 4$d$ and 6$g$, respectively Forsyth and Brown (1990). In
the $ab$-plane, Mn1 adopts a honeycomb lattice while Mn2 adopts an hexagonal
arrangement, Fig. 1a. Along the $c$-axis, Mn1 forms chains and Mn2 columns of
face-sharing octahedra, Fig. 1b. The Curie temperature ($T_{\mathrm{C}}$) is
around $300\text{\,}\mathrm{K}$ and the magnetic moments of the Mnl and Mn2
atoms are 1.96(3) $\mu_{\mathrm{B}}$ and 3.23(2) $\mu_{\mathrm{B}}$,
respectively, aligned along the $c$-axis Maraytta  _et al._ (2020); Forsyth
and Brown (1990).

###  II.2 Magnetic properties from first principles

Density functional theory (DFT) calculations were performed prior to the
inelastic neutron scattering measurements in order to provide an initial
picture of the expected magnon excitations and to identify interesting regions
in ($\mathbf{Q}$,$E$) space. The theoretical magnetic moments from juKKR (2.11
$\mu_{\mathrm{B}}$ and 3.14 $\mu_{\mathrm{B}}$ for Mn1 and Mn2, respectively)
and the computed Heisenberg exchange interactions are comparable to the ones
previously reported Slipukhina  _et al._ (2009), as seen in Table 1. Spin-
orbit coupling leads to significant DMI (c.f. Table 1), much weaker symmetric
anisotropic exchange (not included in Eq. (1)), and a small uniaxial magnetic
anisotropy energy ($K\approx$-0.1\text{\,}\mathrm{meV}$$), so that the
relevant spin Hamiltonian (with $|\mathbf{S}_{i}|=1$) reads:

$\mathcal{H}=K\sum_{i}(S_{i}^{z})^{2}-\sum_{i,j}J_{ij}\,\mathbf{S}_{i}\cdot\mathbf{S}_{j}-\sum_{i,j}\mathbf{D}_{ij}\cdot(\mathbf{S}_{i}\times\mathbf{S}_{j})\;.$
(1)

Here $J_{ij}$ are the Heisenberg exchange interactions and $\mathbf{D}_{ij}$
are the DMI vectors which mostly align along the $c$-axis, with the strongest
shown in Fig. 1b. We discovered that some of the magnetic interactions, namely
the AFM ones, are quite sensitive to small changes in the unit cell parameter
and the atomic positions. The impact of this can be seen in the theoretical
magnon dispersions shown in Fig. 1c, where we compare the results obtained
with the experimental crystal structure parameters and with the theoretically
optimized ones. In both cases there are two magnon bands in the energy range
of experimental interest, with an energy gap at the K-point where otherwise a
Dirac-like crossing of the bands would be expected by symmetry. This is
straightforwardly verified to arise from the DMI, as omitting it from the
magnon calculation leads to the closing of the gap.

Figure 1:  Crystal structure and theoretical magnon bands of Mn5Ge3. a Top view of the crystal structure, showing bonds with length up to $4.3\text{\,}\mathrm{\SIUnitSymbolAngstrom}$. The unit cell is indicated by the thin black lines. b Perspective view of the crystal structure, omitting the Ge atoms. The first few magnetic exchange interactions are marked with the respective non-vanishing Dzyaloshinskii-Moriya interaction (DMI) vectors located in the bond centers. The values are collected in Table 1. c Computed magnon bands using the magnetic exchange interactions extracted from DFT calculations performed for the DFT optimized structure (blue) and for the experimental structure reported in Ref. Forsyth and Brown (1990) (red). The solid and dashed lines show the magnon bands obtained with and without the DMI, respectively. With DMI, a gap opens at the K-point. The magnetization is set along the $c$-axis, which is the easy-axis of the material. The inset indicates the location of the high-symmetry points in the hexagonal Brillouin zone.  |  | Ref. Slipukhina  _et al._ (2009) | Experimental structure Forsyth and Brown (1990) | DFT structure
---|---|---|---|---
| Type | Distance (Å) | Value (meV) | Distance (Å) | Value (meV) | Distance (Å) | Value (meV)
$J_{1}$ | Mn1-Mn1 | 2.526 | 29.1 | 2.527 | 30.87 | 2.485 | 31.59
$J_{2}$ | Mn1-Mn2 | 3.068 | 8.0 | 3.065 | 8.65 | 3.021 | 7.82
$J_{3}$ | Mn2-Mn2 | 2.974 | -2.0 | 2.983 | -1.27 | 3.013 | -0.21
$J_{4}$ | Mn2-Mn2 | 3.055 | 6.9 | 3.058 | 6.84 | 3.033 | 6.10
$J_{5}$ | Mn1-Mn1 | 4.148 | -1.4 | 4.148 | -3.86 | 4.112 | -2.52
$J_{6}$ | Mn2-Mn2 | 4.263 | 9.4 | 4.271 | 9.97 | 4.276 | 9.86
$|\mathbf{D}_{2}|$ | Mn1-Mn2 | — | — | 3.065 | 0.57 | 3.021 | 0.59
$|\mathbf{D}_{3}|$ | Mn2-Mn2 | — | — | 2.983 | 0.50 | 3.013 | 0.45
Table 1:  Magnetic exchange interactions in Mn5Ge3 from DFT. We compare our
results using the experimental crystal structure Forsyth and Brown (1990) with
those of Ref. Slipukhina  _et al._ (2009), and to our results using the
theoretically optimized crystal structure. Positive (negative) values
characterize FM (AFM) coupling. The corresponding pairs are indicated in Fig.
1b. The calculated magnetic interactions are long-ranged and only the first
few values are given in the table. The listed values are significant to the
displayed decimal precision.

###  II.3 Dzyaloshinskii-Moriya interaction in centrosymmetric systems

The DMI is the key magnetic interaction for the subsequent interpretation of
our experimental findings, so before we continue we wish to clarify how it can
be present and have an effect in a centrosymmetric material. In his seminal
paper Moriya (1960), Moriya established the symmetry rules that the
interaction named after Dzyaloshinskii and himself must obey. The most famous
of these rules is that if two spins are connected by an inversion center then
the respective DMI must identically vanish. This pointedly explains why we
find finite DMI in our calculations for Mn5Ge3: it is finite for those spin
pairs that do not contain an inversion center in the midpoint of the
corresponding bond, such as those illustrated in Fig. 1b.

Centrosymmetry does ensure that the net DMI of the unit cell is zero, which is
also verified in our simulations. This means that the ferromagnetic domain
walls are not chiral and that magnetic skyrmions cannot form, in agreement
with Neumann’s principle. In contrast, magnons can still be influenced by the
local DMI. In a semiclassical picture, the spins at different sites precess
with different phases and/or amplitudes, so that certain pairs of spins are
noncollinear and can be affected by the torque arising from the DMI. This is a
strong effect at the K-point, where two magnon modes with opposite chirality
cross and the degeneracy is lifted in a non-perturbative way by the DMI. We
now report the experimental observation of this effect and its implications.

###  II.4 Magnons from inelastic neutron scattering

Inspired by these theoretical predictions, the experimental magnon spectrum of
Mn5Ge3 was investigated by INS. Several constant-$\mathbf{Q}$ and constant-$E$
scans have been performed at $T=10$ K along the reciprocal space directions
$(h,0,0)$, $(h,0,2)$, $(h,h,0)$, and $(0,0,l)$, with representative examples
shown in Fig. 2a. The measured scattering intensity (circles) was fitted with
Gaussian line shapes on top of a sloping background (lines).

Figure 2:  Magnon bands of Mn5Ge3 determined by inelastic neutron scattering.
a Representative measurements obtained at IN22 with constant-$\mathbf{Q}$
scans (symbols) at $T=10$ K with $|\mathbf{k}_{f}|=2.662$ Å-1 and fits used to
determine the excitation energies (lines). The purple arrows and rectangles
indicate the peak positions in the corresponding ($\mathbf{q}$,$E$) region of
the dispersion. The dashed lines show the individual contributions of the
various Gaussian peaks and of the linear background.  The error bars indicate
one standard deviation (square root of the neutron counts). b Experimentally
determined magnon bands (symbols) and fit to a simplified model described in
the text (lines). The solid and the dashed lines show the magnon bands
obtained with and without the Dzyaloshinskii-Moriya interaction (DMI),
respectively.  The error bars indicate the uncertainty in the peak location
from least-squares fitting.

The magnetic nature of the excitations has been confirmed through their
temperature dependence (see Supplementary Figs. 4-8 com  and Fig. 3a below),
and the obtained $\mathbf{q}$ and $E$ position is given as red symbols in Fig.
2b. For the in-plane directions ($\Gamma-\text{M}-\text{K}-\Gamma$) one can
distinguish the presence of three modes: two acoustic-like modes dispersing
upwards in energy away from $\Gamma$ and additionally a third higher-energy
mode with a steep dispersion along $\Gamma-\text{M}$ and weakly dispersive
along $\text{K}-\text{M}$. In contrast, in the same investigated $E$ range a
single stiff spin-wave mode is observed for the out-of-plane direction
($\Gamma-\text{A}$).

We now turn to the theoretical interpretation of these measurements. Comparing
the experimental results of Fig. 2b with Fig. 1c, we find that the spin model
derived from the DFT calculations qualitatively reproduces several features.
The absence of a clearly visible gap in the spin-wave spectrum near zero
energy transfer ($\Gamma$-point) agrees with the expected weakness of the
uniaxial magnetic anisotropy Maraytta  _et al._ (2020), as also computed from
DFT. The theoretical dispersion along $\Gamma-\text{A}$ is slightly stiffer
than the experimental one, and a simulation of the INS intensity reveals that
the second mode which is higher in energy should be invisible (see
Supplementary Fig. 14 com ). We find rather poor quantitative agreement
between theory and experiment in the $\Gamma-\text{K}-\text{M}$ plane, which
is probably related to the already-identified strong dependence of the
magnetic interactions computed from DFT on small changes of the structural
parameters. We verified that this dependence is systematic by considering
various deformations of the unit cell (see Supplementary Figs. 11 and 12 com
). However, the most crucial feature is observed both in theory and in
experiment, that is the existence of a gap at the K-point where two magnon
bands should otherwise cross.

In order to provide a more quantitative description of the experimentally
obtained magnon bands for the in-plane directions, we constructed a simplified
effective spin model (additional details are given in the Supplementary
Information, Section II.E com ). We replace each Mn2 triangle at a constant
height in the unit cell with a single effective spin $S=9/2$ ($S=1$ for Mn1),
so that the column of Mn2 octahedra is replaced by a spin chain. Seen from the
$c$-axis, the unit cell for this model thus contains two Mn1 chains and one
effective Mn2 chain, and we determine the model parameters using the measured
magnon energies at the high-symmetry points. The lines in Fig. 2b show that
the results of this model approach indeed provide a realistic band dispersion,
and confirm once more that the gap at the K-point is a consequence of a finite
DMI. The model results also highlight a peculiarity of the measured magnon
energies in the vicitiny of $\Gamma$, to which we shall briefly return in the
Discussion.

###  II.5 Closing the topological magnon gap

Although there are strong theoretical arguments in favor of the topological
character of the magnon gap at the K-point, a convincing experimental
demonstration is in order. The gap is expected to arise from the DMI, but such
a microscopic interaction cannot easy be manipulated experimentally. However,
it is known that the impact of the DMI on the FM magnon spectrum depends on
the relative alignment between the vectors that characterize this interaction
and the FM magnetization Udvardi and Szunyogh (2009); Zakeri  _et al._ (2010);
dos Santos  _et al._ (2020). Adapting these arguments to the hexagonal
symmetry of Mn5Ge3 leads to the prediction that the magnitude of the gap
should depend on the relative alignment of the FM magnetization and the
crystalline $c$-axis, and in particular should vanish if the two are
perpendicular. We have verified that the magnon gap closes both in the DFT-
based spin model and in the one fitted to the experimental measurements when
the magnetisation is perpendicular to the $c$-axis (as it does when
disregarding the DMI). The magnon dispersions computed with DMI and setting
the magnetization along the $a^{*}$-axis are identical to the dashed lines
shown in Fig. 1c and Fig. 2b, which were obtained by excluding the DMI from
the calculations.

Figure 3:  Temperature and magnetic field dependence of the magnon peaks at
the K-point determined by inelastic neutron scattering. a Temperature
dependence. At $T=$10\text{\,}\mathrm{K}$$ three peaks are seen between
$8\text{\,}\mathrm{meV}$ and $22\text{\,}\mathrm{meV}$. At
$T=$398\text{\,}\mathrm{K}$$ (above $T_{\mathrm{C}}$) only a broad
quasielastic signal is observed. Neutron intensity for the data at 10 K and
398 K is given on the right and left axis, respectively. b Dependence on
magnetic field applied along the $c$-axis at $T=$10\text{\,}\mathrm{K}$$. The
field has almost no effect on the location of the two peaks. c Dependence on
magnetic field applied along the $a^{*}$-axis at $T=$10\text{\,}\mathrm{K}$$.
In zero field the magnetization is along the $c$-axis and two peaks are
visible. The application of a transverse magnetic field saturates the
magnetization along the $a$-axis and merges the two peaks into one,
demonstrating the closure of the magnon energy gap. The data shown in a were
obtained at IN22
($|\mathbf{k}_{f}|=$2.662\text{\,}{\mathrm{\SIUnitSymbolAngstrom}}^{-1}$$) and
the data in b-c at IN12
($|\mathbf{k}_{f}|=$1.971\text{\,}{\mathrm{\SIUnitSymbolAngstrom}}^{-1}$$).
The error bars indicate one standard deviation (square root of the neutron
counts).

This hypothesis can be experimentally tested by applying an external magnetic
field. First we rule out the possibility of a phonon contribution to the
inelastic peaks at the K-point, by heating the sample above its
$T_{\mathrm{C}}$, as seen in Fig. 3a, and verifying that the peaks disappear
conforming their magnetic origin. The measurements reported so far in this
work were performed in zero field, for which the magnetization is parallel to
the $c$-axis due to the uniaxial magnetic anisotropy. Applying a magnetic
field along the $c$-axis should lead to a very small Zeeman shift of the
magnon energies, and this is indeed what we observed, as seen in Fig. 3b. If a
magnetic field of similar magnitude is applied along the $a^{*}$-axis it
overcomes the magnetic anisotropy energy and saturates the magnetization
Maraytta  _et al._ (2020). The results obtained in this way are shown in Fig.
3c. Now the effect of the field cannot be explained by a simple Zeeman shift,
and instead we find the anticipated closing of the magnon gap. The two peaks
observed in zero field coalesce into a single one with an integrated intensity
approximately matching the sum of intensities for the two peaks in zero field
(see also Supplementary Fig. 10 com ), although shifted to slightly higher
energy than anticipated. The distinct response of the magnon excitation to a
magnetic field applied to orthogonal crystal directions is consistent with the
DMI mechanism, and so the gapped Dirac magnons at the K-point should
consequently have a topological nature.

###  II.6 Ruling out alternative explanations for a gap at the K-point

Next we rule out potential alternative mechanisms to the DMI that could lead
to a magnon gap opening at the K-point, such as dipolar, Kitaev and magnon-
phonon interactions:

(i) Dipolar interactions are long-ranged but much weaker than the magnetic
exchange interactions, so their effect is usually seen for rather small wave
vectors in the vicinity of the $\Gamma$-point. Even if they did lift the
magnon degeneracy at the K-point, their intrinsic weakness could not account
for the observed magnitude of the gap.

(ii) Kitaev interactions were proposed for instance in Ref. Lee  _et al._
(2020) to explain measurements on CrI3 but are ruled out for Mn5Ge3 both by
our simulations and by general considerations. The interactions extracted from
our DFT calculations include both the Heisenberg exchange, the DMI and the
symmetric anisotropic exchange (SAE), which includes the Kitaev interaction.
The SAE was found to be rather weak and unable to open the observed magnon gap
at the K-point. The weakness of the SAE (and so of potential Kitaev
interactions) could be anticipated from the weak magnetic anisotropy measured
for this system. This reflects the lack of heavy elements in the material that
could supply a strong spin-orbit interaction, which is a key ingredient for
obtaining a sizeable Kitaev interaction. We are also not aware of any Kitaev
material candidates containing only elements from the first four rows of the
periodic table (i.e. $Z<36$), likely due to the preceding reason. To make this
argument more quantitative, we employ the theory of magnetic exchange
interactions for systems with weak SOC presented by Moriya Moriya (1960), Eqs.
2.3 and 2.4. The DMI is first-order in the weak SOC, while the Kitaev
interaction is part of the SAE which is second-order in SOC and so is much
weaker than the DMI. The Kitaev interaction, if present, would contribute to
the magnetic anisotropy energy, which is about 1 meV/f.u. for Mn5Ge3 (the DMI
does not contribute to the magnetic anisotropy energy of the ferromagnetic
state). To give an estimate of the potential magnitude of the Kitaev
interaction using only experimental input, we distribute the magnetic
anisotropy energy on one of the set of bonds for which we identified the DMI,
bond #2 indicated in Fig. 1b. This set of bonds occurs four times in the unit
cell, as it connects each Mn1 site to its six Mn2 neighbours, and so could
have 1 meV/24 = 0.04 meV maximum Kitaev strength. The SAE obtained directly
from the DFT calculations is about 0.02 meV in magnitude, which is in line
with this estimate, and is one order of magnitude smaller than the values
found for the DMI (0.57 meV for the set of bonds #2), as expected from the
theory of the magnetic exchange interactions for systems with weak SOC.

(iii) Magnon-phonon interactions can result in gaps at the crossing points
between the magnon and phonon branches. However, our INS data ruled out this
possibility. The measured excitations at different $\mathbf{Q}$ vectors around
and at the K-point are solely of magnetic origin. This has been verified
through their temperature dependence. All the peaks observed in the energy
range from 10 to 20 meV at $T=10$ K are replaced by a broad quasi-elastic
signal (centered at 0 meV) above the ordering temperature as shown in Fig. 3a.
Hence no phonon modes were detected in the vicinity of the K-point with an
energy compatible with that of the magnons, which is a requirement for the gap
opening mechanism through magnon-phonon interactions.

Therefore, we can assert that the only reasonable mechanism for the gap
opening at the K-point is the DMI.

###  II.7 Simulation of magnon surface states

Figure 4:  Magnon surface states in a $(01\bar{1}0)$-oriented slab of Mn5Ge3.
a Relation between the primitive cell and the rectangular cell used to define
the slab, viewed along the $c$-axis, using the simplified model that fits the
experimental data in Fig. 2b. The Mn1 chains are indicated by the two
different shades of red, while the effective spin taken to represent the Mn2
sites is shown in blue. b Path in reciprocal space used for the magnon
calculation. c Magnon bands of a 20-unit-cell-wide slab computed with periodic
boundary conditions (PBC). d Magnon bands of a 20-unit-cell-wide slab computed
by changing periodic to open boundary conditions (OBC) along the
$(01\bar{1}0)$ orientation. Results obtained excluding/including the
Dzyaloshinskii-Moriya interaction (DMI) are shown by dashed/solid lines. The
green arrow in (d) indicates the DMI-enabled magnon surface state.

Here we explore the expectation that if the bulk magnon band structure has
some topologically non-trivial character it should be accompanied by magnon
surface states. To do so, we compute the magnon band structure of slabs which
are finite in one direction and periodic (i.e., infinite) in the other two
directions. Comparing the simulations performed for the same slab with
periodic and open boundary conditions along the chosen surface normal enables
us to identify the energy range corresponding to the surface projection of the
bulk magnon bands. Surface magnons are then expected to appear in the regions
of $(E,\mathbf{q})$ where bulk magnon bands are absent.

To illustrate this point, we performed simulations using the simplified spin
model depicted in Fig. 4a with parameters fitted to the experimental
measurements in Fig. 2b. We created a rectangular supercell and extended it by
20 unit cells along the $(01\bar{1}0)$ direction of the original hexagonal
lattice. The chosen path in reciprocal space is perpendicular to the surface
normal and shown in Fig. 4b. Fig. 4c shows that indeed there is a pocket in
the K–M–K path and with energies between 10 and 12 meV from which bulk magnons
are absent, with or without DMI. Fig. 4d shows the modified magnon band
structure upon truncating the crystal along the $(01\bar{1}0)$ direction,
i.e., making a horizontal cut through the lattice shown in Fig. 4a. We indeed
find that magnon surface states do appear in the identified region where bulk
magnon bands were absent. Without the DMI, these surface magnons are
disconnected and gapped from each other. The DMI restructures the band
connectivity and leads to a crossing that resembles a distorted Weyl-like
crossing. The crossing is not located at the M-point as the slab loses the
$ac$-mirror plane, retaining instead a twofold rotation around the $c$-axis.
There are other surface magnons at around 3 meV and 18 meV that are only
weakly affected by the DMI, and result from the reduced coordination number
introduced by creating the surface. Lastly, the identified surface magnons
were found to extend only a couple of unit cells towards the interior of the
slab, confirming their localization at the surface.

The experimental detection of the predicted surface magnons is quite
challenging, as the scattering volume is too small for a straightforward
detection using INS. Other techniques such as Brillouin light scattering (for
magnons near the $\Gamma$-point) or electron energy loss spectroscopy could be
considered for this purpose Zakeri (2016).

##  III Discussion

We now briefly discuss some outstanding points. Firstly, we return to the
quantitative disagreement between theory and experiment concerning the magnon
bands. The main issue seems to be an overestimation of the magnitude of the
magnetic interactions in the DFT calculations, which has also been found in
other studies. A recent example from the literature is Co3Sn2S2 Zhang  _et
al._ (2021), where two different DFT approaches are compared with experiment,
with disagreements also in the $\Gamma-\text{M}-\text{K}-\Gamma$ plane but not
in the $\Gamma-\text{A}$ direction. Another issue is that the simplified spin
model does not adequately capture the relative intensities of the two peaks
around the K-point, despite providing a reasonable description of the
experimental magnon energies. This is likely due to the model assumptions,
namely treating the Mn2 sites as a single effective spin and neglecting the
atomic magnetic form factor, as well as employing a simple Lorentzian
broadening for the peaks. This shows the need for further developments on the
theoretical side. We also noticed that the ratio of the intensities of the two
peaks varies slightly in different experimental setups, see Figs. 3b-c. This
might be due to changes in the domain state of the sample arising from the
measurement history.

Our INS measurements for Mn5Ge3 revealed a peculiar dispersion for the second
mode in the $\Gamma-\text{K}$ direction. Such a steep linear-like dispersion
close to the $\Gamma$-point resembles an AFM magnon or a phonon mode. We
restate that the material is FM and the measured excitations are of magnetic
nature as verified by measurements above $T_{\text{C}}$ (Fig. 3a and
Supplementary Figs. 4-8 com ), so that both simple explanations are ruled out.
However, it is possible to have formation of hybrid collective modes. INS can
identify the magnetic and phononic component of such excitations, which are
referred as magnon-polarons (magnetoelastic modes), and are reported in
several magnetic materials Petit  _et al._ (2007); Sukhanov  _et al._ (2019).
Avoided crossings are the common signature of magnetoelastic interactions, but
they also underpin the magnetovibrational scattering term in the INS
scattering cross section Squires (2012); Raymond  _et al._ (2019). Additional
discussion is given in the Supplementary Information, Section I.C com . We
propose that Mn5Ge3 is also an interesting 3D FM candidate material for
detailed investigation of magnetoelastic effects Zhang  _et al._ (2019); Go
_et al._ (2019).

The key interaction responsible for opening the magnon gap at the K-point and
thus endowing the Dirac magnons with a topological character is the DMI. There
are several possible routes by which this interaction could be engineered, so
that the magnon properties can be tuned for specific purposes for magnonic
devices operating in a broad temperature range. Firstly, chemical substitution
of Ge by Si has been explored in the literature to connect to the
multifunctional AFM Mn5Si3 Sürgers  _et al._ (2014); Biniskos  _et al._
(2018); dos Santos  _et al._ (2021); Biniskos  _et al._ (2022). Substituting
Ge by Si reduces $T_{\text{C}}$ Zhao  _et al._ (2006), but the impact on the
magnons and on the DMI is unknown. On the other hand, carbon implantation is
demonstrated to enhance $T_{\text{C}}$ Kraft  _et al._ (2020); Michez  _et
al._ (2022) for which the imprint on the Heisenberg exchange interactions has
been theoretically established Slipukhina  _et al._ (2009), but once again the
effect on the magnons and the DMI remains unexplored. Lastly, Mn5Ge3 can also
be grown in thin film form Kraft  _et al._ (2020); Michez  _et al._ (2022).
This could modify the DMI by epitaxial strain, which would also be interesting
in connection to potential magnetoelastic effects, by interfaces to other
materials, or by quantum confinement effects if the thickness is just a few
nanometers.

In conclusion, we presented a combined theoretical and experimental study of
the magnons in the centrosymmetric 3D FM Mn5Ge3. Despite the inversion
symmetry, significant DMI has been theoretically identified on Mn-Mn bonds
which do not contain an inversion center. This DMI is responsible for opening
a gap in the magnon spectrum at the K-point, where otherwise symmetry would
enforce a Dirac-like crossing of the magnon bands. INS measurements of the
magnon spectrum show qualitative agreement with the main points predicted by
theory, and confirm the expected gap at the K-point. We experimentally observe
the closing of the gap by rotating the magnetization from the $c$-axis to the
$a^{*}$-axis with a magnetic field. This both validates the gap generation
mechanism and the topological nature of the magnons at the K-point, thus
establishing Mn5Ge3 as a realization of a gapped Dirac magnon material in
three dimensions. The ability to control the gap at the K-point with an
external magnetic field will also impact topological magnon surface states,
and deserves further study. As the macroscopic magnetic properties of Mn5Ge3
can be tuned by chemical substitution of Ge with Si or by carbon implantation,
and it can also be grown as thin films in spintronics heterostructures, we
foresee that the features of the newly-discovered topological magnons can be
engineered and subsequently integrated in novel device concepts for magnonic
applications. Looking beyond Mn5Ge3, the physical mechanism leading to the
formation of topological magnons at the K-point should be present in many
other bulk centrosymmetric hexagonal materials, which opens an exciting avenue
for future investigations.

##  IV Methods

###  IV.1 Experimental methods

Inelastic neutron scattering (INS) experiments have been carried out on a cold
(IN12) and a thermal (IN22) triple axis spectrometer at the Institut Laue-
Langevin, in Grenoble, France. We use the hexagonal coordinate system and the
scattering vector $\mathbf{Q}$ is given in reciprocal lattice units (r.l.u.).
The wave-vector $\mathbf{q}$ is related to $\mathbf{Q}$ through
$\mathbf{Q}=\mathbf{q}+\mathbf{G}$, where $\mathbf{G}$ is a Brillouin zone
center. Inelastic scans were performed with constant $|\mathbf{k}_{f}|$, where
$\mathbf{k}_{f}$ is the wave-vector of the scattered neutron beam. Data were
collected holding either the energy (constant-$E$) or the scattering vector
(constant-$\mathbf{Q}$) fixed. Further details on the experimental procedures
and additional measurements can be found in the Supplementary Information,
Section I com .

###  IV.2 Theoretical methods

The theoretical results were obtained with DFT calculations and the extracted
spin Hamiltonian. The unit cell parameters and the atomic positions were
optimized with the DFT code Quantum Espresso Giannozzi  _et al._ (2009). The
magnetic parameters were computed with the DFT code juKKR Bauer (2014); juk ,
which are then used to solve a spin Hamiltonian in the linear spin wave
approximation dos Santos  _et al._ (2018). Further details on all these
aspects can be found in the Supplementary Information, Section II com .

##  V Data availability

The authors declare that the data supporting the findings of this study are
available within the paper, its supplementary information file and in the
following repositories dat (2021a, b, c, d); dos Santos Dias  _et al._ (2023).

##  VI Code availability

The DFT simulation packages Quantum Espresso and juKKR are publicly available
(see Methods). The code for the solution of the linear spin wave problem is
available from the corresponding authors upon request.

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##  VII Acknowledgements

We thank S. Raymond for discussions and comments. The work of M.d.S.D. made
use of computational support by CoSeC, the Computational Science Centre for
Research Communities, through CCP9. N.B. acknowledges the support of JCNS
through the Tasso Springer fellowship. F.J.d.S. acknowledges support of the
European H2020 Intersect project (Grant No. 814487), and N.M. of the Swiss
National Science Foundation (SNSF) through its National Centre of Competence
in Research (NCCR) MARVEL. This work was also supported by the Brazilian
funding agency CAPES under Project No. 13703/13-7 and the Deutsche
Forschungsgemeinschaft (DFG) through SPP 2137 “Skyrmionics” (Project LO
1659/8-1). We gratefully acknowledge the computing time granted through JARA
on the supercomputer JURECA Jülich Supercomputing Centre (2021) at
Forschungszentrum Jülich GmbH and by RWTH Aachen University. The neutron data
collected at the Institut Laue-Langevin are available at Refs. dat (2021a, b,
c, d).

##  VIII Author contributions

MdSD, NB and FJdS contributed equally to this work. MdSD and NB conceived the
project together with TB and SL. MdSD performed most DFT calculations and the
spin-wave modelling, with additional calculations performed by FJdS. JP grew
the Mn5Ge3 single crystal. NB performed the experimental measurements and the
corresponding data analysis. KS and FB were local contacts of IN12 and IN22
and provided instrument support. The theoretical aspects of the work were
discussed with NM, SB and SL. All authors participated in the discussion of
the results. MdSD, NB and FJdS wrote the manuscript with input from all
authors.

##  IX Competing interests

The authors declare no competing interests.