# Unusual magnetism of the axion-insulator candidate Eu5In2Sb6

M. C. Rahn<EMAIL_ADDRESS>Institute for Solid State and Materials
Physics, Technical University of Dresden, 01062 Dresden, Germany  Los Alamos
National Laboratory, Los Alamos, New Mexico 87545, USA  M. N. Wilson
Department of Physics, Durham University, South Road, Durham, DH1 3LE, United
Kingdom  Memorial University, Department of Physics and Physical Oceanography,
St. John’s, NL, A1B 3X7, Canada  T. J. Hicken  Department of Physics, Durham
University, South Road, Durham, DH1 3LE, United Kingdom  Laboratory for Muon-
Spin Spectroscopy, Paul Scherrer Institute, CH-5232 Villigen, Switzerland
F.L. Pratt  ISIS Facility, STFC, Rutherford Appleton Laboratory, Chilton,
Didcot, Oxfordshire, OX11 0QX, United Kingdom  C. Wang  Laboratory for Muon-
Spin Spectroscopy, Paul Scherrer Institute, CH-5232 Villigen, Switzerland  F.
Orlandi  ISIS Facility, STFC, Rutherford Appleton Laboratory, Chilton, Didcot,
Oxfordshire, OX11 0QX, United Kingdom  D. D. Khalyavin  ISIS Facility, STFC,
Rutherford Appleton Laboratory, Chilton, Didcot, Oxfordshire, OX11 0QX, United
Kingdom  P. Manuel  ISIS Facility, STFC, Rutherford Appleton Laboratory,
Chilton, Didcot, Oxfordshire, OX11 0QX, United Kingdom  L. S. I. Veiga  London
Centre for Nanotechnology and Department of Physics and Astronomy, University
College London, London WC1E 6BT, United Kingdom  Diamond Light Source Ltd.,
Didcot OX11 0DE, United Kingdom  A. Bombardi  Diamond Light Source Ltd.,
Didcot OX11 0DE, United Kingdom  S. Francoual  Deutsches Elektronen-
Synchrotron (DESY), Hamburg, Germany  P. Bereciartua  Deutsches Elektronen-
Synchrotron (DESY), Hamburg, Germany  A. S. Sukhanov  Institute for Solid
State and Materials Physics, Technical University of Dresden, 01062 Dresden,
Germany  J. D. Thompson  Los Alamos National Laboratory, Los Alamos, New
Mexico 87545, USA  S. M. Thomas  Los Alamos National Laboratory, Los Alamos,
New Mexico 87545, USA  P. F. S. Rosa  Los Alamos National Laboratory, Los
Alamos, New Mexico 87545, USA  T. Lancaster  Department of Physics, Durham
University, South Road, Durham, DH1 3LE, United Kingdom  F. Ronning  Los
Alamos National Laboratory, Los Alamos, New Mexico 87545, USA  M. Janoschek
Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA  Laboratory
for Neutron and Muon Instrumentation, Paul Scherrer Institute, CH-5232
Villigen, Switzerland  Physik-Institut, Universität Zürich, CH-8057 Zürich,
Switzerland<EMAIL_ADDRESS>
###### Abstract

Eu5In2Sb6 is a member of a family of orthorhombic nonsymmorphic rare-earth
intermetallics that combines large localized magnetic moments and itinerant
exchange with a low carrier density and perpendicular glide planes. This may
result in special topological crystalline (wallpaper fermion) or axion
insulating phases. Recent studies of Eu5In2Sb6 single crystals have revealed
colossal negative magnetoresistance and multiple magnetic phase transitions.
Here, we clarify this ordering process using neutron scattering, resonant
elastic X-ray scattering, muon spin-rotation, and magnetometry. The
nonsymmorphic and multisite character of Eu5In2Sb6 results in coplanar
noncollinear magnetic structure with an Ising-like net magnetization along the
$a$ axis. A reordering transition, attributable to competing ferro- and
antiferromagnetic couplings, manifests as the onset of a second commensurate
Fourier component. In the absence of spatially resolved probes, the
experimental evidence for this low-temperature state can be interpreted either
as an unusual double-$q$ structure or in a phase separation scenario. The net
magnetization produces variable anisotropic hysteretic effects which also
couple to charge transport. The implied potential for functional domain
physics and topological transport suggests that this structural family may be
a promising platform to implement concepts of topological antiferromagnetic
spintronics.

##  I Introduction

The guided design and control of topologically protected conduction states on
surfaces and interfaces of bulk insulators and semimetals is one of the most
sought-after results of quantum materials physics. The concept of topological
quantum chemistry Bradlyn  _et al._ (2017) has enabled high-throughput
calculations and predictions of topological band properties in the absence of
electronic correlations Zhang  _et al._ (2019); Vergniory  _et al._ (2019).
While progress on intrinsically magnetic topological matter has been much
slower, the analysis of all magnetic space groups and tabulated magnetic
structures indicates that there is indeed an abundance of candidate strongly
correlated materials with potential for functional band-topology Watanabe  _et
al._ (2018); Xu  _et al._ (2020). However, as magnetic symmetry can currently
not be predicted from first principles, the identification of suitable
material platforms remains an outstanding challenge.

Following the proposal of fourfold degenerate Dirac surface states with
hourglass connectivity in the non-symmorphic orthorhombic Zintl phase
Ba5In2Sb6 Wieder  _et al._ (2018), there has been an increased interest in
magnetic members of this 5-2-6 structural family. The narrow band-gap
semiconductor Eu5In2Sb6 had been previously studied for its outstanding
thermoelectric properties at high temperature Chanakian  _et al._ (2015); Lv
_et al._ (2017). Following the synthesis of single-crystalline samples, the
material revealed highly unusual low-temperature properties, including
colossal magnetoresistance, which has been attributed to the localization of
charge carriers at magnetic polarons Rosa  _et al._ (2020); Souza  _et al._
(2022); Ghosh  _et al._ (2022). In-depth density functional studies of Eu-
based 5-2-6 compounds have since identified the potential for axion insulating
phases and predicted a high sensitivity to chemical and pressure/strain tuning
Varnava  _et al._ (2022). However, both the electronic and magnetic structures
of the 5-2-6 compounds have so far eluded experimental determination.

Figure 1:  (a) The orthorhombic unit cell and $Pbam$ symmetries of Eu5In2Sb6.
The perpendicular glide planes are highlighted in blue and red. (b) Solid
arrows show the spin arrangement inferred for the $\mathit{\Gamma}$ magnetic
component, which forms at $T_{\mathrm{N}}=14.5$ K. The structure is collinear
within the uncertainty of the neutron diffraction data. Shaded arrows
illustrate tilts away from this collinear configuration, which creates a net
magnetization. (c) Corresponding views of the $Z$ magnetic component, which
originates below $T_{Z}=7.5$ K. The X-ray and neutron scattering data do not
allow the distinction between a double-$q$ state and magnetic phase
separation. Both $\mathit{\Gamma}$ and $Z$ components are described by the
same magnetic basis vectors, which generally add to a per-layer magnetization
along the $a$ axis. For the $\mathit{\Gamma}$ component, this results in a
spontaneous macroscopic magnetization, while in the $Z$ component it is
compensated by alternating stacking.

Here, we report on the complex magnetic ordering process of Eu5In2Sb6. Using
muon spin-rotation ($\mu$SR), neutron powder diffraction (NPD) and resonant
elastic x-ray scattering (REXS), we are able to shed light on an unusual two-
step ordering process due to the nonsymmorphic and multi-site character of
Eu5In2Sb6. Unexpectedly, our findings imply the existence of net-magnetized
domains, and angle-dependent electrical resistivity measurements indicate
coupling of magnetism and charge transport.

Fig. 1(a) illustrates the orthorhombic $Pbam$ unit cell of Eu5In2Sb6. The
structure can be understood as layers of coplanar Eu ions (ten per unit cell)
that are interpenetrated by chains of corner-sharing InSb4 tetrahedra along
the $c$ axis. The figure also illustrates the positions of the $b$ glide
(blue), the $a$ glide (red), and the mirror plane (gray). The perpendicular
glide planes endow the Eu-layers with the symmetry of the wallpaper group
$pgg$, encountered in daily life as the herringbone pattern SM . This
combination of non-symmorphic symmetries is the prerequisite for the novel
topological-crystalline phases predicted in this structural family Wieder  _et
al._ (2018). In Eu5$M_{2}X_{6}$ compounds Varnava  _et al._ (2022), this
potential for non-trivial topology combines with the large magnetic moment of
divalent europium ($S=7/2$, $L=0$, $\mu\approx 8\mu_{\mathrm{B}}$).

We find that below the Néel temperature $T_{\mathrm{N}}=14.5$ K, the Eu2+
spins initially form a complex coplanar, non-collinear arrangement
(“$\mathit{\Gamma}$”) that carries a net magnetic moment along the $a$ axis,
as illustrated in Fig. 1(b). Below $T_{Z}=7.5$ K, this phase is either
gradually displaced by a growing volume fraction of another magnetic phase
(“$Z$”), or it forms a double-$q$ state with perpendicular $\mathit{\Gamma}$
and $Z$ components. Crucially, the per-layer spin arrangements of the
$\mathit{\Gamma}$ and $Z$ components obey the same magnetic symmetries – they
differ only in the (parallel vs. alternating) stacking of this motif along the
$c$axis. While all three magnetic states ($\mathit{\Gamma}$, $Z$, double-$q$)
conserve inversion symmetry $\mathcal{I}$, effective time reversal symmetry
$\\{\mathcal{T}|00\frac{1}{2}\\}$ is only conserved in the $Z$ state, which
would make these regions candidate axion insulators Sekine and Nomura (2021);
Bernevig  _et al._ (2022). The phase separation scenario could therefore
provide a platform to explore interfaces of topologically distinct insulating
regimes.

##  II Results and interpretation

Previous measurements of the heat capacity and magnetic susceptibility Rosa
_et al._ (2020), reproduced in Fig. 2(c,d), revealed two continuous magnetic
transitions, which correspond to the Néel temperature
$T_{\mathrm{N}}=14.5\,$K, and the onset of the $Z$ component at
$T_{Z}=7.5\,$K. The magnetic neutron powder diffraction patterns in Fig. 2(a)
illustrate that magnetic Bragg peaks in the intermediate phase
($T_{Z}<10\,\mathrm{K}<T_{\mathrm{N}}$) can be indexed by the propagation
vector $\mathbf{q}_{\mathit{\Gamma}}=(0,0,0)$. The low-temperature dataset
($1.5\,\mathrm{K}<T_{Z}$) features both $\mathbf{q}_{\mathit{\Gamma}}$ and
$\mathbf{q}_{Z}=(0,0,\nicefrac{{1}}{{2}})$ peaks SM .

Figure 2:  (a) Magnetic neutron powder diffraction at 1.5 K and 10 K. The
nuclear intensity (at 25 K) has been subtracted from the data [SM, ]. (b)
Temperature dependence of REXS intensities associated with the propagation
vectors of the Ising weakly-ferromagnetic $\mathit{\Gamma}$ order and the
compensated antiferromagnetic $Z$ component. Below $T_{Z}=7.5$ K,
$\mathit{\Gamma}$ and $Z$ either coexist by phase separation or as a
double-$q$ structure. (c) The consecutive second-order phase transitions at
14.5 K and 7.5 K observed in heat capacity ($C/T$). (d) The anisotropic
magnetic susceptibility $\chi^{a/b/c}(T)$ at 100 mT applied along the
principal axes. Note the small spontaneous magnetization of $\chi^{a}$ in the
$\mathit{\Gamma}$ state, and the change in easy axis below $T_{Z}$.

Using resonant elastic x-ray scattering (REXS) at the Eu $L$ edges, we
characterized the $\mathbf{q}_{Z}$ and $\mathbf{q}_{\mathit{\Gamma}}$ magnetic
order in more detail. Fig. 2(b) shows the thermal evolution of Bragg
intensities associated with the two Fourier components. These intensities are
not necessarily a direct measure of the ordered magnetic moment because the
magnetic order has degrees of freedom that affect the structure factor and may
evolve with temperature. Nevertheless, the rise of $\mathbf{q}_{Z}$ intensity
below $T_{Z}$ appears to be proportional to the reduction of the
$\mathbf{q}_{\mathit{\Gamma}}$ peak, consistent with the displacement of the
$\mathit{\Gamma}$ by the $Z$-type Fourier component. While for a double-$q$
model the $\mathit{\Gamma}$/$Z$ ratio may be different at each Wyckoff site,
the refinement of the 1.5 K neutron powder diffraction data in the phase
separation scenario results in phase volumes of 34% ($Z$) and 66%
($\mathit{\Gamma}$) SM .

Azimuthal scans and full linear polarisation analysis of the REXS signal,
shown in the Supplemental Material SM , reveal the direction of magnetic
structure factor vectors of individual Bragg peaks. Combined with
representational analysis, this information determines the irreducible
representations (irreps) associated with the two magnetic phases SM . The
$\mathit{\Gamma}$ and $Z$ magnetic structures are described by the irreps
$\mathit{\Gamma}_{3}^{+}$ ($Pb^{\prime}am^{\prime}$) and
$Z_{3}^{+}$($P_{b}nma$), and the superposition of both would correspond to
magnetic space group $Pb^{\prime}am^{\prime}$. Crucially, these two irreps
impose the same phase relations (basis vectors) on the relative alignment of
the coplanar magnetic moments – the two magnetic components differ only in the
parallel ($\mathit{\Gamma}$) or alternating ($Z$) stacking of this motif along
the $c$ axis. Given that magnetometry reveals a localized $S=7/2$ ($\approx
8\,\mu_{\mathrm{B}}$) magnetic moment at each site Rosa  _et al._ (2020), a
double-$q$ structure can only form under the special condition that the
Fourier components associated with either irrep must be perpendicular to one
another (else, the magnitude of the moments would alternate between layers).
As indicated by solid arrows in Fig. 1(b,c), we find that the magnetic
components associated with the $\mathit{\Gamma}$ and $Z$ magnetic structures
are close to collinear with the $b$ and $a$ axes, respectively. The shaded
arrows illustrate tilts away from the collinear structures that are allowed by
symmetry and evidenced by magnetometry. The magnitude of the tilts is either
smaller than the uncertainty of the data (neutron scattering) or
$1^{\circ}\sim 10^{\circ}$ (REXS, $Z$ component).

The per-layer arrangement of the ten magnetic moments has several important
implications. The Eu ions occupy one 2-fold and two 4-fold Wyckoff sites, as
indicated by different shades in Fig. 1 (see [SM, ] for more detailed
illustrations). The overall structure is defined by three in-plane spin
orientations (one per site), which may vary continuously with temperature.
Crucially, the components of the magnetic moments (anti-)parallel to the $b$
axis cancel necessarily, but all $a$ axis components per Wyckoff orbit are
parallel. Given the 2/4/4 multiplicity, they can, therefore, in general, not
be compensated between the three sites. This makes each layer a weak (canted)
Ising ferromagnet with a variable net magnetization along the $a$ axis.
Notably, this holds true independently of whether the moments are mostly
aligned with the $a$ or $b$ axis, as shown in Fig. 1(b). For instance, in the
$\mathit{\Gamma}$ component, the net magnetization of layers is small because
the moments are, within the accuracy of our measurements, aligned with the $b$
axis. However, small tilts away from this collinear configuration (indicated
by shaded arrows) produce a small net moment along $a$, which adds up to a
macroscopic magnetization. Conversely, the $Z$ magnetic component carries a
large per-layer magnetization along $a$ (the spins are close to collinear
along $a$), but here this is compensated by the alternating stacking.

Knowledge of this magnetic order explains several unusual observations of bulk
and local magnetic probes. For example, in Fig. 2(d), the $H\parallel a$
magnetic susceptibility $\chi^{a}(T)$ shows a ferromagnet-like increase below
$T_{\mathrm{N}}$. This is not compatible with conventional antiferromagnetism
but can now be explained by the weak $M\parallel a$ ferromagnetism. Below
$T_{Z}$, the bulk susceptibility gradually takes on the character of the $Z$
magnetic structure shown in Fig. 1(b): As the weakly ferromagnetic
($\mathit{\Gamma}$) component recedes, $\chi^{a}(T)$ decreases rapidly.
Eventually (below $\approx 5$ K), the magnetic susceptibility is largest for
$H\parallel c$, as expected of a coplanar antiferromagnet. Given that the
moments of the $Z$ component align largely with the $a$ axis, see Fig. 1(b),
$\chi^{a}$ should be minimal once the $Z$ component dominates. Accordingly,
the gradient of $\chi^{a}(T)$ below $T_{Z}$ is much steeper than that of
$\chi^{b}(T)$.

Figure 3:  (a) Static internal fields (top) and relaxation rates (bottom)
determined by zero-field continuous-source $\mu$SR (GPS instrument). Solid
lines are a guide to the eye. We attribute different components of the $\mu$SR
asymmetry to muon stopping sites that are more sensitive to fields within
(intra) and between (inter) the layers. The inset shows an example of the
$\mu$SR asymmetry measured at GPS (3 K, typical fit indicated by a red line).
(b) Dynamic relaxation rates determined from the decay of the zero-field
$\mu$SR asymmetry up to 20 $\mu$s. Circles and triangles mark values inferred
from data measured at continuous (GPS) and pulsed source (EMU) muon
instruments, respectively. The inset shows a typical fit in this time window,
including both continuous source (black) and pulsed source data (blue).

To probe the magnetic ordering process from a local point of view, we turn to
muon spin-rotation ($\mu$SR) experiments. In order to reveal both static and
dynamic characteristics ($\mu$s timescale), we measured spectra at continuous
(PSI/GPS) and pulsed (ISIS/EMU) muon sources. The observed $\mu$SR asymmetry
can be interpreted by assuming two types of muon stopping sites. Each
contributes an oscillating component associated with a characteristic
relaxation rate $\lambda$ and a dynamical relaxation $\lambda^{\mathrm{dyn}}$
(due to fluctuating fields). One possible interpretation of the fit results is
that one class of stopping sites is more sensitive to magnetic fields within
the planes (intra), and the other is more sensitive to fields between the
planes (inter). The variation of these characteristics throughout the ordering
process, along with exemplary fits to the data, is shown in Fig. 3.

As seen in Fig. 3(a), the signal that we associate with static in-plane fields
$H_{\mathrm{intra}}$ sets in abruptly at $T_{\mathrm{N}}$, as often observed
for materials with two-dimensional correlations. By contrast, the gradual
onset of $H_{\mathrm{inter}}$ is more typical for three-dimensional magnetism.
Interestingly, as seen in Fig. 3(b), neither dynamical relaxation
$\lambda_{\mathrm{inter}}^{\mathrm{dyn}}$, nor
$\lambda_{\mathrm{intra}}^{\mathrm{dyn}}$, which encode magnetic fluctuations,
have strong anomalies at $T_{\mathrm{N}}$. The transition could therefore be
interpreted as a freezing-out of pre-existing short-range magnetic
correlations.

At $T_{Z}$, the absence of discontinuities of the static fields in Fig. 3(a)
is consistent with the picture that the local magnetic structure does not
change abruptly. Since the onset of an alternating stacking ($Z$) component
does not affect the per-layer magnetic symmetry, the static fields within the
layers ($H_{\mathrm{intra}}$, $\lambda_{\mathrm{intra}}$) are not strongly
affected by this transition. By contrast, the onset of the $Z$ component
qualitatively changes the trend of $H_{\mathrm{inter}}$ and causes an anomaly
in $\lambda_{\mathrm{inter}}$. Moreover, as shown in Fig. 3(b), both dynamic
relaxation rates ($\lambda_{\mathrm{inter}}^{\mathrm{dyn}}$,
$\lambda_{\mathrm{intra}}^{\mathrm{dyn}}$) collapse around $T_{Z}$. This
implies that the magnetism overall becomes much more static, possibly because
competition between magnetic correlations is relieved. Below $T_{Z}$,
$H_{\mathrm{inter}}$ has a slight tendency to decrease upon cooling. This
unusual behavior may reflect changes in the domain structure, as has been
previously suggested in GaV4S8 Hicken  _et al._ (2020); Franke  _et al._
(2018). Notably, the relative amplitudes of the components do not change below
$T_{Z}$, as one might expect for varying volume fractions. Although this might
be taken as evidence for the double-$q$ scenario, the nature of the
$\mathit{\Gamma}$ and $Z$ magnetic orders (which share the same magnetic
symmetry per layer) means that we cannot discount the phase separation
picture.

Neither local or bulk magnetic probes nor magnetic scattering allow a clear
distinction between a double-$q$ or phase separation scenario below $T_{Z}$.
In either case, the net magnetization of the $\mathit{\Gamma}$ state implies
that Ising-like magnetic domains and their interplay on the mesoscale may play
an important role in Eu5In2Sb6, as illustrated in Fig. 4. For instance, as
observed in other phase-separated magnets Guo  _et al._ (2022), interfaces of
ferromagnetic domains may provide natural nucleation points for the growth of
antiferromagnetic ($Z$) regimes. As illustrated in Fig. 4(a,b),
$\mathit{\Gamma}$ domains with an odd number of layers could be converted
entirely into a $Z$ region. By contrast, $\mathit{\Gamma}$ domains with an
even number of layers necessarily leave behind solitonic interfaces with net
magnetization, potentially opposite to the original magnetization Papanicolaou
(1995).

Such effects provide a compelling scenario to understand the unusual field and
zero-field cooled (FC/ZFC) characteristics of the $H\parallel a$ magnetic
susceptibility shown in Fig. 4(c). Upon warming, a ZFC sample in which, at low
$T$, the opposite domain configuration happens to be encoded shows the same
(FC) magnetic susceptibility curve of opposite sign until the thermal energy
is sufficient to allow consecutive flips of $\mathit{\Gamma}$ domains (or
domains carrying a $\mathit{\Gamma}$ component) into the field direction. The
observed sign reversal of the magnetic susceptibility might be attributable to
a specific $\mathit{\Gamma}$-component domain structure dictated by
demagnetization fields, which would also be compatible with the double-$q$
scenario. Details of this process should depend on the sample shape and
pinning centres of antiferromagnetic domain walls, which could be verified by
imaging techniques.

Figure 4:  (a) Illustration of the magnetic phase separation scenario below
$T_{Z}$. Each layer of Eu5In2Sb6 is represented by an arrow indicating the
direction of the Ising (per-layer) net magnetic moment. Up (down) domains of
the $\mathbf{q}_{\mathit{\Gamma}}$ state are colored blue (red). Interfaces
between $\mathit{\Gamma}$ domains may provide natural nucleation points of the
$Z$ state (white). $\mathit{\Gamma}$ domains with an odd number of layers can
turn into a continuous $Z$ volume, while those with an even number of layers
leave behind a soliton with a net magnetic moment. (b) Sketch of the thermal
evolution of an Eu5In2Sb6 slab, colored in analogy to (a). (c) Field-cooled
and zero-field cooled (FC/ZFC) curves of the magnetic susceptibility measured
with a weak (20 Oe) bias field along the $a$ axis. Magnetization reversal and
domain flip transitions [indicated by inset illustrations] provide a tentative
explanation for these unusual characteristics. Figure 5:  (a) Pulsed-source
$\mu$SR measurements of the dynamic relaxation rates
$\lambda_{\mathrm{intra}}^{\mathrm{dyn}}$ and
$\lambda_{\mathrm{inter}}^{\mathrm{dyn}}$ indicate an onset of short-ranged
magnetic order at $T^{*}\approx 35$ K. (b) On the same temperature scale, the
resistance $R$ (current $I\parallel c$, log scale) begins to diverge from
activated behavior. (c) Detailed view of the resistance within the
magnetically ordered phase (linear scale). We identify three regimes of charge
transport (I,II,III). Figure 6:  Characteristic anisotropic hysteretic
phenomena in the three regimes of charge transport (I, II, III). (a-d) Polar
plots of the resistance $R$ at 1 T (current $I\parallel\hat{c}$) for varying
field directions $\phi_{H}$ in the $a$–$b$ plane. The three regimes differ not
only in the magnitude of $R$ [cf. Fig. 4(c)], but also in their anisotropy.
(e-h) Respective hysteretic components in the magnetization $\Delta M(H)$ for
fields along $a$ and $b$. Linear slopes have been subtracted from each curve.
(i-l) Corresponding hysteretic contributions $\Delta
R(H)=R(H_{\uparrow})-R(H_{\downarrow})$ to the resistivity. Note both the
quantitative and qualitative variation between the regimes. In regime II (3.0
K$\sim$10 K), $\Delta R(H)$ has a positive slope and is concomitant with
$\Delta M$. It can therefore be associated with the (re-)orientation of
magnetic domains of the $\mathit{\Gamma}$ magnetic order.

The magnetism of Eu5In2Sb6 appears to be coupled to charge transport over a
wide temperature range. Fig. 5(a) illustrates that the $\mu$SR dynamic
relaxation rates $\lambda_{\mathrm{inter}}^{\mathrm{dyn}}$ and
$\lambda_{\mathrm{intra}}^{\mathrm{dyn}}$, associated with slow magnetic
dynamics, increase significantly when cooling below $T^{*}\approx 35$ K. This
temperature scale has previously been associated with the onset of strong
short-ranged antiferromagnetic correlations Rosa  _et al._ (2020). As
illustrated in Fig. 5(b), $T^{*}$ also coincides with the temperature below
which the resistance diverges from activated behavior, $R\propto
T\,\exp(42\,\mathrm{meV}/k_{\mathrm{B}}T)$ Rosa  _et al._ (2020). Taken
together, this suggests that below $T^{*}$, the charge transport in Eu5In2Sb6
starts being affected by magnetic dynamics.

Hysteretic characteristics in Eu5In2Sb6 change throughout the magnetic
ordering process, with a variable effect on charge transport. As illustrated
in Fig. 5(c), we identify three temperature regimes of charge transport: The
CMR regime above $T>10.5$ K (III), the region 3.0 K $<T<10.5$ K where the
resistance is reduced by more than five orders of magnitude (II), and a steep
rise of resistance below $T<$ 3.5 K (I). Figs. 6(a–d) show how the resistance
(current $I\parallel\hat{c}$) varies under an azimuthal rotation in an applied
field of 1 T. The three transport regimes differ not only quantitatively (in
the magnitude of $R$) but also qualitatively.

The corresponding Figs. 6(e–l) show hysteretic effects for fields applied
along the $a$ and $b$ axes, as observed in the magnetization (e–f) and
resistance (i–l). To isolate the hysteretic components $\Delta M$ and $\Delta
R$, a linear slope was subtracted from the $M(H)$ curves, and up- and down-
ramps of $R(H)$ were subtracted. The observations are similar at 5 K and 10 K
(i.e. below and above $T_{Z}$), see Fig. 6(f,g)/(j,k). Specifically, both
$\Delta M$ and $\Delta R$ are negligible for $H\parallel\hat{b}$, but sizable
and of coinciding coercive fields for $H\parallel\hat{a}$. This matches the
expectations for the reversal of weakly ferromagnetic domains – which are
indeed expected below and above $T_{Z}$. Variations of the coercive field and
remanent magnetization between 5 K and 10 K are expected, given the trade-off
between a decreasing ordered magnetic moment and an increasing
$\mathit{\Gamma}$ Fourier component (or possibly $\mathit{\Gamma}$ volume
fraction). The linear $\Delta R(H)\propto\pm H$ at low temperatures
distinguishes the hysteretic effects in regimes (I,II) against the much larger
and non-monotonic $\Delta R(H)$ in the CMR regime (III). Interestingly,
$\Delta R(H)$ changes sign between (I) and (II), but retains its association
with the $a$ axis.

##  III Discussion

Our findings show that the magnetic ordering process in Eu5In2Sb6 is shaped by
its non-symmorphic orthorhombic structure and, specifically, by the fact that
magnetic ions occupy multiple, non-equivalent Wyckoff sites. A search for rare
earth intermetallics with these characteristics indeed yields several
interesting materials with similarly complex magnetism. For instance, the
$R_{5}$Ni2In4 structural family is also of $Pbam$ symmetry and also features a
rare earth $R$ on two 4-fold and one 2-fold site. In contrast to the 5-2-6
family, since the $R$ ions are here trivalent, a range of rare earths can be
substituted. For instance, a similar complex non-collinear magnetism with
multiple phase transitions is found in for $R=$Tm Szytuła  _et al._ (2014),
and similar characteristics, including weak ferromagnetism, can also be
obtained above 100 K by substituting $R=$Dy Provino  _et al._ (2012).

Multiple magnetic (re-)ordering transitions at zero field are a common
observation among these and related orthorhombic rare-earth intermetallics.
This may be attributable to the competition between ferromagnetic and
antiferromagnetic itinerant magnetic exchange interactions and, specifically,
competing ordering tendencies at inequivalent Wyckoff sites. The low symmetry
of these arrangements of localized magnetic ions among anisotropic conduction
bands makes for potentially very complex scenarios. For instance, the
positions of Eu ions in the basal plane of Eu5In2Sb6 can be viewed as a
network of distorted corner-sharing or side-sharing triangles, with 6+1 (in-
plane + out-of-plane) distinct nearest-neighbor distances ranging between 3.8
Å and 4.7 Å SM . The layer-spacing, 4.6 Å, is intermediate to these values,
which emphasizes that competition between in-plane and out-of-plane itinerant
exchange is likely, and magnetic correlations are not necessarily two-
dimensional. Measurements of the magnetic dynamics in these materials would be
of great interest to clarify the hierarchy of correlations.

The small Ising-like net-magnetization of the $\mathit{\Gamma}$-type magnetic
component is a central finding with important consequences. While dipolar
interactions are unlikely a driving force of the magnetic reordering at
$T_{Z}$, the net magnetization must necessarily lift degeneracies of the
electronic structure, which was previously not taken into account in density
functional calculations Rosa  _et al._ (2020); Varnava  _et al._ (2022). The
hysteretic effects imply that magnetic domains play an important role in
Eu5In2Sb6, that the domain structure is coupled to charge transport, and that
this relation changes qualitatively with temperature. Mesoscale transport
studies would now be of interest to clarify whether these observations arise
from intrinsic variations of the transport channels or extrinsic mechanisms
(scattering of charge carriers from domain walls).

The magnetic phase transition at $T_{Z}$ is driven by a competition of ferro-
and antiferromagnetic coupling along the $c$ axis. The two possible scenarios,
phase-separation or double-$q$ magnetic order, impose different constraints on
the spin arrangement but overall have a similar number of degrees of freedom.
Since both models are compatible with magnetic scattering, local and bulk
magnetic probes SM , their distinction will require direct evidence from
spatially resolved techniques. Notably, both possibilities are unusual
results: The double-$q$ structure mixes distinct commensurate structures with
different ratios at the three Eu sublattices but only provides a physical
solution if these components are strictly orthogonal. The concept of phase
separation, as illustrated in Fig. 5(a,b) is more straightforward, but it is
unclear what microscopic mechanism in Eu5In2Sb6 could favor inhomogeneity.

##  IV Conclusion

In this study of the magnetism of the non-symmorphic Zintl phase Eu5In2Sb6, a
combination of local, bulk and momentum-resolved probes of magnetism allowed
us to reveal an unusual magnetic ordering process that is also consistent with
hysteretic phenomena in both magnetisation and charge transport. This
complexity originates in the combination of perpendicular glide symmetries and
the distribution of localized magnetic moments on three magnetic sublattices.
As a consequence, a complex arrangement of ten co-planar Eu spins is endowed
with a net magnetic moment along the $a$ axis, which makes each layer an Ising
weak ferromagnet. The demagnetization energy therefore takes a decisive role
in the bulk magnetism, and a complex behavior of net magnetized domains may be
expected on the mesoscale. At lower temperatures, competing interlayer
exchange couplings induce a compensated antiferromagnetic component that
displaces the weak ferromagnetism. In the absence of microscopic evidence, it
is not possible to distinguish whether this transition occurs in the form of a
double-$q$ structure or phase separation.

It will be of great interest to clarify what bearing the observed magnetic
symmetries have on various proposals for crystalline topological phases in the
5-2-6 Zintls Wieder  _et al._ (2018); Varnava  _et al._ (2022). For instance,
phase separation would imply the presence of $Z$-type magnetic regimes that
conserve effective time-reversal symmetry (time reversal and translation along
$c$), a prerequisite for the axion insulator state. In either case, the net
magnetization of the (variable) $\mathit{\Gamma}$ components will necessarily
have an impact on the valance bands and might also serve as an unexpected,
adjustable (by domain size) parameter of the electronic structure.
Furthermore, 5-2-6 Zintls have also been noted for their chemical tunability,
and density functional studies predict that the putative topological features
in their band structure are extremely sensitive to hydrostatic and uniaxial
pressure Varnava  _et al._ (2022). Among the wider family of nonsymmorphic
orthorhombic intermetallics, there are several structural families in which
bulk probes point to similar complex ordering scenarios. Among these are
compounds with trivalent rare earth sites that provide similar potential for
non-trivial band topology Vergniory  _et al._ (2022) but open up additional
possibilities to tune magnetism. Combining the technological potential of
weakly ferromagnetic domains and band topology in such materials is an
exciting new avenue to manipulate topological transport and thus to realize
specific technological proposals from the emerging field of topological
antiferromagnetic spintronics Šmejkal  _et al._ (2018); Varnava and Vanderbilt
(2018); Varnava  _et al._ (2021).

Note added.
During the preparation of this manuscript, we became aware of a single crystal
neutron diffraction study by Morano et al., Ref. Morano  _et al._ (2023).
Although the datasets are complementary (in terms of the techniques used and
the regime of reciprocal space probed), the type of magnetic structures
inferred are consistent with our findings. As in our case, the refinement of
the single crystal neutron scattering intensities does not allow a clear
distinction between the phase separation and double-$q$ models of the $Z$
magnetic component. However, Morano et al. observe additional
$\mathit{\Gamma}$-type magnetic Bragg intensities with qualitatively different
temperature dependences. This implies that the three Eu sublattices have
different ordering tendencies already below $T_{\mathrm{N}}$– which adds to
the plausibility of the double-$q$ scenario. On the other hand, density
functional calculations by Morano et al. do favor phase separation. Direct
evidence, e.g. from magnetic imaging, will help to clarify this ambiguity.

Methods

Samples

Single crystals of Eu5In2Sb6 were prepared using an In-Sb flux method, as
previously described Rosa  _et al._ (2020). This growth technique yields
prismatic rods of Eu5In2Sb6 (ca. $1\times 0.5\times 0.2$ mm3), along with
thinner needles, which we identified as the Eu11InSb9 phase Fender  _et al._
(2021). To obtain a high-quality powder sample (0.55 g overall), clean
Eu5In2Sb6 crystals were selected from a large number of batches, cleaned, and
ground.

Neutron powder diffraction

Time-of-flight neutron powder diffraction (NPD) was performed at WISH (ISIS,
Rutherford Appleton Laboratory) Chapon  _et al._ (2011), which provides strong
signal-to-noise down to very small momentum transfers. This was essential to
reveal the magnetic scattering from Eu5In2Sb6, given the extreme neutron
absorption cross section of Eu, as well as In. To minimize absorption, the
sample (0.55 g) was filled into a double-walled vanadium can and cooled in a
lHe cryostat (Oxford Instruments). Data was obtained at 1.5 K, 10.0 K and 25.0
K, and was processed using Mantid Arnold  _et al._ (2014) and FullProf
Rodríguez-Carvajal (1993).

Muon-spin relaxation

Zero field $\mu$SR studies of polycrystalline samples were performed on the
EMU spectrometer at the STFC-ISIS Neutron and Muon source (pulsed source,
higher sensitivity to slow relaxation processes), and the GPS spectrometer at
the Swiss Muon Source, Paul Scherrer Institut (continuous source, higher time
resolution to resolve static magnetism). Measurements on GPS were performed
with the muon spin initially rotated 45∘ from the incident beam direction and
made use of a veto detector to remove the signal from muons that missed the
sample. Measurements on EMU were performed with the muon spin anti-parallel to
the incident beam direction. For the EMU measurements, the sample was mounted
on a thick silver plate that stops all muons that miss the sample, yielding a
non-relaxing background signal.

Resonant elastic x-ray scattering

Due to the large Eu2+ ordered magnetic moment and the strong resonant
enhancement at the Eu $L$ absorption edges, hard X-ray resonant elastic
scattering (REXS) is highly suitable for magnetic structure determination in
Eu compounds. REXS studies were performed at the instruments I16 (DIAMOND
Light Source, Rutherford Appleton Laboratory) Collins  _et al._ (2010) and P09
(DESY, Hamburg) Strempfer  _et al._ (2013). In both cases, the photon energy
was tuned to the Eu $L_{3}$ ($2p_{\frac{3}{2}}\leftrightarrow 5d$) absorption
edge (6.970 keV, $\lambda=1.779$ Å). Scattered X-rays were detected either by
a position-sensitive CMOS detector (DECTRIS) or an avalanche photodiode in
combination with a Cu (220) analyzer crystal. The orientation of the magnetic
structure factor vector $\hat{M}(Q)$ was characterized either by azimuthal
scans (I16) or by full linear polarization analysis (P09) Francoual  _et al._
(2013).

Magnetometry and transport measurements

Temperature-dependent magnetic susceptibility and magnetization measurements
were performed in a Quantum Design MPMS3 magnetometer. Resistivity
measurements were performed in a Quantum Design PPMS using the standard four-
point technique and an AC resistance bridge (Lakeshore 372 with 3708 preamp).
The current was applied parallel to the $c$ axis. For the rotation
experiments, the sample was mounted on a motor-driven rotation stage. The axis
of rotation was parallel to the applied current.

###### Acknowledgements.

The authors would like to thank Jens Müller (Goethe-Universität Frankfurt) for
useful discussions. Parts of this research were carried out at DIAMOND Light
Source and at the STFC-ISIS Facility, both institutions of the UK Science and
Technology Research Council (STFC). Experiments were also performed at
S$\mu$S, Paul Scherrer Institut, Switzerland, as well as under beamtimes
I-20190272 and I-20210419 at the synchrotron light source PETRA III at DESY, a
member of the Helmholtz Association (HGF). The UK effort was supported by the
EPSRC Grant no. EP/N032128/1. M.N.W. acknowledges the support of the Natural
Sciences and Engineering Research Council of Canada (NSERC). Work at Los
Alamos was supported by the U.S. Department of Energy, Office of Basic Energy
Sciences, Division of Materials Science and Engineering project “Quantum
Fluctuations in Narrow-Band Systems. Work performed at the Paul Scherrer
Institute, MJ acknowledges support from the Swiss National Science Foundation
(SNSF) [200650]. Work at TU Dresden was supported by the Deutsche
Forschungsgemeinschaft through the CRC 1143, the Würzburg-Dresden Cluster of
Excellence EXC 2147 (ct.qmat), and through the Emmy Noether program of the
German Science Foundation (project number 501391385). L.S.I.V. was supported
by the UK Engineering and Physical Sciences Research Council (EPSRC) (Grants
No. EP/N027671/1 and No. EP/N034694/1). M.C.R. is grateful for Fellowships
provided by the Laboratory Directed Research & Development Program at Los
Alamos and the Humboldt Foundation. Research data from the UK effort will be
made available via Durham Collections. Muon and neutron data collected at
STFC-ISIS are available under DOI:10.5286/ISIS.E.RB2010313 and
DOI:10.5286/ISIS.E.RB1920539.

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