# Charge density wave-templated spin cycloid in topological semimetal
NdSbxTe2-x-δ

Tyger H. Salters  Department of Chemistry, Princeton University, Princeton,
New Jersey 08544, United States  Fabio Orlandi  ISIS Neutron Pulsed Facility,
Science and Technology Facilities Council, Rutherford Appleton Laboratory,
Oxford OX11 0QX, UK  Tanya Berry  Department of Chemistry, Princeton
University, Princeton, New Jersey 08544, United States  Jason F. Khoury
Department of Chemistry, Princeton University, Princeton, New Jersey 08544,
United States  Ethan Whittaker  Department of Chemistry, Princeton University,
Princeton, New Jersey 08544, United States  Pascal Manuel  ISIS Neutron Pulsed
Facility, Science and Technology Facilities Council, Rutherford Appleton
Laboratory, Oxford OX11 0QX, UK  Leslie M. Schoop<EMAIL_ADDRESS>Department of Chemistry, Princeton University, Princeton, New Jersey 08544,
United States

###### Abstract

Magnetic topological semimetals present open questions regarding the interplay
of crystal symmetry, magnetism, band topology, and electron correlations.
LnSbxTe2-x-δ (Ln= lanthanide) is a family of square-net-derived topological
semimetals that allows compositional control of band filling, and access to
different topological states via an evolving charge density wave (CDW)
distortion. Previously studied Gd and Ce members containing a CDW have shown
complex magnetic phase diagrams, which implied that spins localized on Ln
interact with the CDW, but to this date, no magnetic structures have been
solved within the CDW regime of this family of compounds. Here, we report on
the interplay of the CDW with magnetism in NdSbxTe2-x-δ, by comparing the
undistorted square net member NdSb0.94Te0.92 with the CDW-distorted phase
NdSb0.48Te1.37, via single-crystal x-ray diffraction, magnetometry, heat
capacity, and neutron powder diffraction. NdSb0.94Te0.92 is a collinear
antiferromagnet with TN $\sim$ 2.7 K, where spins align antiparallel to each
other, but parallel to the square net of the nuclear structure. NdSb0.48Te1.37
exhibits a nearly five-fold modulated CDW (qCDW=0.18), isostructural to other
LnSbxTe2-x-δ at similar x. NdSb0.48Te1.37 displays more complex magnetism with
TN = 2.3 K, additional metamagnetic transitions, and an elliptical cycloid
magnetic structure with qmag=-0.41b∗. The magnitudes of qCDW and qmag exhibit
a integer relationship, $1+2q_{mag}=q_{CDW}$, implying a coupling between the
CDW and magnetic structure. Given that the CDW is localized within the
nonmagnetic distorted square net, we propose that conduction electrons
“template” the spin modulation via the Ruderman–Kittel–Kasuya–Yosida
interaction.

##  I Introduction

In the search for new topological matter, materials with a square-net
structural motif have been heavily studied as a popular platform to host
topological electronic states.[1, 2, 3, 4, 5, 6, 7, 8] The $p_{x}$ and $p_{y}$
orbitals that contribute much of the bonding in the square-net yield band
structures with multiple topological band crossings, and thus the materials
have been found to host Dirac, Weyl, and nodal line topological semimetals
(TSMs), as well as topological insulators[9, 10, 11]. From a chemical
standpoint, the bonding within the square net can be described as
“hypervalent”, which means that it is delocalized over the square net, as the
electrons occupy an exactly half-filled $p_{x}$/$p_{y}$ band. Deviation from a
half-filled electron count will disturb this type of bonding, causing electron
localization that is accommodated by distortions of the net.[12] Such a
deviation is reflected in the band structure by the appearance of nesting at
the Fermi surface, wherein pockets coincide by translation of some wavevector
q.[13] Such an instability can be alleviated by formation of a charge density
wave (CDW), in which bond localization, also influenced by electron-phonon
coupling, leads to a periodic distortion of the structure with a modulation
wavevector reflecting the nesting vector q.[14]

The canonical nodal line semimetal ZrSiS, as well as several isostructural
relatives, contains a nodal line at the Fermi level and a non-symmorphic Dirac
node above $E_{F}$.[11] Members of the family crystallize in the tetragonal
P4/nmm space group, in which, if using ZrSiS as an example, the Si square net
lies in the ab plane. The square net is sandwiched by layers of Zr and S along
the c axis, in a puckered rock-salt layer between each square net. One
isostructural subgroup of the ZrSiS family, LnSbTe (Ln=Lanthanide), affords
chemical tuning of $E_{F}$ by substitution of Te into the Sb square net[15],
allowing access to the non-symmorphic Dirac node; such solid solutions are
largely overlooked by popular databases used to predict TSMs due to the limits
of the computational methods which they are built upon.[16, 17, 18, 19]
Magnetic rare earth ions also make the LnSbTe family attractive to study the
interplay of topological states with magnetism. This allows for opportunities
to observe new topological states through the breaking of time-reversal
symmetry, leveraging magnetic order as another tunable parameter to control
electronic structure and access multiple topologically distinct
phases.LnSbxTe2-x-δ ($\delta$ indicating vacancy concentration in the square
net) materials also exhibit charge density waves (CDWs), which are coupled
with structural distortions in the square-net that evolve continuously with
Sb/Te composition, and are a result of changes in band filling within the
square net.[20, 21, 22] We have recently reported that the CDW is responsible
for the gapping of topologically trivial states in GdSb0.46Te1.48, yielding an
ideal nonsymmorphic Dirac semimetal band structure free of all other states
about EF, separated by hundreds of meV.[15] Depending of the doping level, the
CDW can have one or multiple q-vectors that describe the structure modulation.
The single q-vector region usually lies in the intermediate doping range.
Here, the q-vector is in-plane and agrees with a nesting vector that connects
the two sheets of the diamond-shaped Fermi surface that is typical to such
square-net materials.[15, 20] GdSbxTe2-x-δ also displays highly complex
magnetism in which magnetic phase diagrams evolve with the compositionally
driven CDW.[23] In-plane magnetic anisotropy has been reported in mechanically
detwinned GdSb0.46Te1.48 crystals, indicating that the unidirectional CDW
distortion yields especially complex H-T magnetic phase diagrams along the
direction of the distortion. Notably, magnetoentropic analysis of this
composition also suggests the signature of an antiferromagnetic skyrmion phase
at low applied field.[23] Across all distorted phases, the CDW does not
significantly distort the puckered magnetic LnTe layers in the parent
structure. However, the evolving distortion and composition of the nonmagentic
(Sb,Te) square net is suggested to influence couplings through
Ruderman–Kittel–Kasuya–Yosida (RKKY) interaction. RKKY interactions are
sensitive to the also-evolving density of states, yielding a Sb-concentration
dependent magnetic phase diagram.

The highly neutron-absorbing nature of Gd complicates the feasibility of
determining the magnetic structures of GdSbxTe2-x-δ with neutron diffraction.
In an effort to study evolving magnetism and topology with neutron-friendly
analogues, we have previously synthesized isostructural CeSbxTe2-x-δ. [22] The
stoichiometric parent compound, CeSbTe, is capable of switching between Dirac
and Weyl semimetal states by changing the strength of an applied magnetic
field.[24] Single-crystal x-ray diffraction (SCXRD) has shown that CDWs are
continuously tunable in CeSbxTe2-x-δ and exhibit modulated structures
consistent with GdSbxTe2-x-δ at similar Sb compositions. Distortions and
calculations on CeSb0.51Te1.40 suggest that the approach of driving CDWs to
discover “clean” TSMs can be extended to several members of the LnSbxTe2-x-δ
family[22]. In the single-q CDW region of the phase diagram, the magnetism of
CeSbxTe2-x-δ does not exhibit the same complexity, but does yield a
composition and distortion dependent magnetic phase diagram. Spins align out-
of-plane in CeSbxTe2-x-δ, perpendicular to the square net, which is consistent
with neutron diffraction studies of the parent compound, CeSbTe,[24] and
divergent from GdSbxTe2-x-δ, which exhibits an in-plane easy axis. Thus, in
the Ce system, the spins do not align with the CDW q-vector, at least in the
single q-vector region, which might be the cause of its less complex
magnetism. We now direct our efforts to a different neutron-friendly analog,
NdSbxTe2-x-δ, where spins are oriented in-plane, similarly to GdSbxTe2-x-δ.

NdSbTe has been reported to exhibit metamagnetic transitions and field-
dependent transport, implying an interaction of magnetism with electronic
structure [25]. Herein we report the magnetism and magnetic structures of
NdSb0.91Te0.89 and NdSb0.48Te1.37, two members of NdSbxTe2-x-δ, which we
studied by magnetometry, single crystal x-ray diffraction, and neutron powder
diffraction. NdSb0.91Te0.89 is found to have a crystal structure consistent
with other tetragonal LnSbxTe2-x-δ phases, and exhibits a collinear magnetic
structure in which spins align within the ab plane. NdSb0.48Te1.37 is nearly
isostructural to GdSb0.46Te1.48 with a single-q CDW modulation; and displays a
complex elliptical cycloid magnetic structure. The magnitude of the magnetic
propagation vector qmag is found to have an integer relationship with that of
qCDW, $1+2q_{mag}=q_{CDW}$, which implies the interaction of conduction
electrons from the distorted square net with the unpaired electrons in Nd that
lead to magnetism, which are considered to sit at energy levels far below the
Fermi energy. This is the first magnetic structure solved for a CDW
LnSbxTe2-x-δ phase, and may give insight into the interaction between
magnetism and topological bands in these “ideal” TSMs.

Table 1: Easy-axis and neutron absorptivity in LnSbxTe2-x-δ (Ln in LnSbxTe2-x-δ) | Spin alignment | Neutron absorption cross-section (barn)[26]
---|---|---
Gd |  $\rightarrow(\parallel ab)$[23] | 49700
Ce |  $\uparrow(\perp ab)$[22] | 0.63
Nd |  $\rightarrow(\parallel ab)$${}^{this~{}work}$ | 50.5

##  II Experimental

###  II.1 Synthesis of NdSb0.91Te0.89 and NdSb0.48Te1.37

Single crystals of NdSb0.91Te0.89 and NdSb0.48Te1.37 were prepared using a
chemical vapor transport (CVT) method. Prior to CVT growth, precursor powders
with nominal compositions of NdSb0.5Te1.5 and NdSbTe were prepared from Nd
pieces (Sigma Aldrich, 99.9%), Sb shot (Alfa Aesar, 99.999%), and Te pieces
which were purified in-house after purchase from a commercial source (Sigma
Aldrich, 99.999%). The elements were loaded into a 12 mm fused quartz tube and
flame sealed under vacuum with an Ar backfill at approximately 60 mTorr,
heated to 1000 ∘C over 12 hr, held for 2d and cooled over 12 hr in a box
furnace. The resulting powders were ground and loaded into a 16 mm tube with
120 mg I2 as a vapor transport agent, sealed under vacuum resulting in tubes
of 6-8 cm in length. CVT growth was carried out by placing the tubes in a
single zone tube furnace, with the powder sides placed in the middle of the
furnace to create a thermal gradient at the sink side. The tubes were heated
to 1000 ∘C over 12 hr, held for 7d and cooled over 12 hr, resulting in
crystals formed at the sink end. The elemental composition and approximate
stoichiometry of the structure were confirmed via scanning electron microscopy
(SEM)/Energy-Dispersive Spectroscopy (EDS). In attempted preparation of
NdSbTe, Sb loss was frequently observed during CVT growth, confirmed by
composition analysis.

Powder phase samples for neutron diffraction were prepared from the elements
using a vapor-assisted technique. Samples were sealed in quartz tubes with I2
identically to the previously described procedure. The tubes were heated in a
box furnace to 950 ∘C over 12 hr, held for 7d and cooled over 12 hr, yielding
highly crystalline powders. Single crystals of NdSb0.48Te1.37, of size and
quality for magnetic and specific heat measurement were also obtained from
this method in the same tube.

###  II.2 Single-crystal X-ray Diffraction and Magnetic Measurements

SCXRD data for NdSb0.91Te0.89 and NdSb0.48Te1.37 were collected on a Bruker D8
VENTURE with a PHOTON 3 CPAD detector and a graphite-monochromatized
Mo-$K_{\alpha}$ radiation source. Integrated data were corrected with a
multiscan absorption correction using SADABS. The structure of NdSb0.91Te0.89
was solved via SHELXT[27] using intrinsic phasing and refined with SHELXL[28]
using the least squares method, as implemented in the OLEX2 program.[29]
Satellite peaks, indicative of a structural modulation were observed in
NdSb0.48Te1.37. The satellite peaks were indexed to an incommensurate
modulation of q=0.18b∗. Refinement was thus carried out using the superspace
approach, where the displacive distortion of atomic positions is expressed by
a periodic modulation function, yielding a 3 + 1 dimensional superspace group.
Refinements with the superspace approach were carried out in JANA2006.[30]
This approach included positional, ADP and site occupancy modulations in the
refinement. Determination of Sb/Te occupancy and ordering within the square-
net was limited by near-identical scattering power due to their closeness in
atomic number (Z = 51 and 52, respectively). At laboratory-accessible X-ray
wavelengths, the two species were indistinguishable if present on the same
crystallographic site. Thus, for all refinements, atomic occupancies within
the Sb/Te square-net were constrained to the stoichiometry derived from EDS.

Magnetic measurements were conducted using the vibrating sample magnetometer
(VSM) option of either a Quantum Design DynaCool Physical Property Measurement
System (PPMS) or a Magnetic Property Measurement System (MPMS) 3 SQUID
magnetometer between 1.8 and 300 K. Single crystals were fixed between quartz
paddles in a brass sample holding using GE Varnish, with field applied
parallel and perpendicular to the ab plane. Magnetic phase diagrams were
generated by collecting temperature-dependent magnetization curves during
warming, with varying fields, up to 0.1 T in 0.01 T increments, then up to 5.0
T in 0.1 T increments. Critical temperatures and phase boundaries are derived
from peaks in the temperature derivative of the DC magnetic susceptibility,
d$\chi$/dT. d$\chi$/dT curves were plotted to build the magnetic phase diagram
in the H-T space.

###  II.3 Heat Capacity

Heat Capacity (CP) measurements of NdSb0.48Te1.37 were carried out using a
Quantum Design DynaCool PPMS using the semiadiabatic method. The single-
crystal sample was fixed to the stage using Apiezon N grease, and measurements
were carried out between 1.8 and 300 K. Variable-field measurements were
conducted by applying a magnetic field perpendicular to the ab plane.

###  II.4 Neutron powder diffraction

Neutron powder diffraction data were collected on the WISH instrument at the
ISIS Muon and Neutron Source (UK). [31] The samples were contained in sealed
Vanadium cans and the neutron diffraction data were collected in the 1.5 – 5 K
temperature range. The diffraction data were collected in five detectors banks
each covering 32 degrees of the scattering plane. The Rietveld refinement of
the powder data were conducted with the help of the JANA2020 software. [30,
32] Group theoretical calculations and symmetry analysis were performed with
the help of the ISOTROPY [33] and ISODISTORT[34] software and of the Bilbao
crystallographic server [35, 36, 37].

##  III Results and discussion

###  III.1 Crystal Structure

Figure 1: Crystal structures of (a,b) NdSb0.91Te0.89 and (c, d)
NdSb0.48Te1.37, as determined by single-crystal x-ray diffraction with bond
distances given. Bonds in the distorted net of NdSb0.48Te1.37 are drawn for
distances equal to or shorter than the determined Sb-Sb bond length in
undistorted NdSb0.91Te0.89, emphasizing the loss of bond delocalization upon
CDW. Bond distances in d) are shown as their average lengths $\pm$ distance
modulation, with errors given in parenthesis.

The crystal structures of NdSb0.91Te0.89 and NdSb0.48Te1.37, as determined by
SCXRD at 100 K are shown in Figure 1. The structure and lattice parameters of
NdSb0.91Te0.89 are consistent with other reported tetragonal LnSbxTe2-x-δ
members, where no CDW ordering is observed. The Sb-Sb bond length within the
square net is 3.0574(2) Å. The Sb-Sb bond lengths in the square net are
significantly longer than what would be expected in a localized and covalent
Sb-Sb single bond, as it appears for example in elemental Sb (approximately
2.9Åat 78 K)[38]. The elongated, symmetric bonding can be rationalized by
“hypervalent” bonding, where a -1 charge on Sb is stabilized by electron
delocalization across the entire net, as detailed by Papoian and Hoffmann.
[12] The NdTe layer of the structure displays the aformentioned puckered rock
salt structure, with Nd-Te bond length nearly parallel to ab (within the unit
cell) held constant at 3.1931(2) Å, and bond length parallel to c (across unit
cells) held at 3.2788(6) Å.

The structure of NdSb0.48Te1.37 displays a CDW distortion, as evidenced by the
appearance of low intensity satellite peaks visible in the precession images
of the diffraction patterns (Figure S2). The pattern can be indexed as an
incommensurately modulated structure, with a modulation wavevector q=0.18b∗.
This modulation wavelength corresponds to an approximate five-fold expansion
of the unit cell along b∗. Using the superspace approach to structure solution
and refinement, NdSb0.48Te1.37 is found to crystallize in the
$Pmmn$(0$\beta$0)00$s$ superspace group, and when visualized as a commensurate
five-fold approximant, is isostructural to CeSb0.51Te1.40 and GdSb0.46Te1.48,
where the (Sb,Te) square net is found to distort into a pattern of zigzag
chains and more isolated atoms. This is congruent with our previous studies on
the evolving CDWs observed in GdSbxTe2-x-δ and CeSbxTe2-x-δ. This same
structure was found to give rise to an ideal nonsymmorphic Dirac semimetal
band structure.[15]

Using the Sb-Sb bond length in tetragonal NdSb0.91Te0.89 as a reference for an
undistorted square net with fully delocalized bonding, we discuss the
localization and symmetry breaking of the square net in NdSb0.48Te1.37. The
square net breaks into a repeating pattern of parallel chains separated by
lines of more isolated (Sb,Te) atoms (Fig 1d). Intra-chain Sb-Sb bonding
varies from 2.92105(1) to 2.9593(1) Å, and remains constant within each chain.
The nearest distance from a chain atom to an isolated atom ranges from
3.1098(1) to 3.1706(1) Å. Inter-chain atom distances are the largest, ranging
from 3.2655(1) to 3.2872(1) Å. The significant deviations from the hypervalent
bond length found in the undistorted square-net imply a strong localization of
bonding along the chain directions, with minimal bonding interactions between
the chains. The chain-to-isolated atom distance sits in an intermediate range,
larger than the expected value for delocalized bonding, but may carry more
bonding interaction than inter-chain interactions.

The heavy distortion of the square net due to the CDW in NdSb0.48Te1.37 is
contrasted with minimal changes to bond length within the NdTe layer. NdTe
bond length nearly parallel to ab held constant at 3.2122(1) Å, and bond
length parallel to c is 3.2699(2) Å. These bond lengths are quite similar to
that in undistorted NdSb0.91Te0.89, with the deviation well explained by
differences in the parent cell lattice parameters between the two. Figure S3a
shows the modulation amplitude of bond distances in the NdTe structural unit
and within the distorted square net. Nd-Te bonds do not vary more than 0.01 Å,
whereas the amplitude of bond modulations within the distorted square net vary
as much as 0.4 Å. (Figure S3b) This feature is common to all of the modulated
LnSbxTe2-x-δ we have studied[21, 22], and consistent with our understanding of
bonding in the square net. The CDW distortion is driven by electronic
instability that arises from adding Te (and additional electrons) into the
square net, breaking the bond delocalization. We consider the electrons
involved in bonding in the NdTe layer to lie well below the Fermi energy, and
thus will be unaffected by the CDW. This lack of structural interaction with
the CDW will become interesting to consider in our discussion of the magnetic
structure and its connection to the CDW.

###  III.2 Magnetic measurements and specific heat

Figure 2: Magnetic phase diagrams (H vs T) of NdSb0.91Te0.89 and
NdSb0.48Te1.37. Plots were generated from the first order derivative of $\chi$
over temperature (d$\chi$/dT).

In Figure 2, we present the magnetic phase diagrams of NdSb0.91Te0.89 (Fig 2.
a, b) and NdSb0.48Te1.37 (Fig 2. c, d), constructed by plotting the first-
order derivative of $\chi$ ($T$) curves measured along the crystallographic ab
plane and c axis. In the case of NdSb0.91Te0.89, a Neel temperature TN = 2.7 K
is observed. A magnetic field of 3 T eventually suppresses the AFM order when
applied along the ab plane(Fig 2a); the system presumably transitions to a
fully spin-polarized state at this field. If the field is applied along the c
axis (Fig 2b), no metamagnetic transition is clearly visible from the variable
field moment versus temperature measurements used to construct the phase
diagram. Field-dependent magnetization curves show a subtle change in slope at
3 T in the derivative curve (Fig S4b), and a smaller magnetic moment at all
fields compared to in-plane measurements (Fig S2a). Thus, the feature at 3 T
may be an in-plane contribution due to imperfect alignment of the crystal in
the applied field. Significantly different moments at the maximum applied
field ($M_{7T,\textit{ab}}\gtrsim 2M_{7T,\textit{c}}$) along in- and out-of-
plane directions, further implies magnetic anisotropy (Fig S4a). These results
are similar to previously reported magnetic measurements on tetragonal
NdSbxTe2-x-δ, where x = 0.8.[25]

In NdSb0.48Te1.37, the observed TN is 2.3 K, lower than the tetragonal
compound. This contrasts behavior observed in Ce and Gd members of
LnSbxTe2-x-δ. CeSbxTe2-x-δ displays an increase in TN with decreasing x;
GdSbxTe2-x-δ shows nonlinear behavior where TN increases to a maximum at x =
0.48, and decreases at lower Sb compositions. [22, 23] Measurements with
fields applied within the ab plane (Figure 2c) detect a metamagnetic
transition at $\sim$0.6 T, with a region of enhanced $\chi$ that persists to 3
T, where another metamagnetic transition occurs, presumably to a fully spin-
polarized state. Both phase boundaries are corroborated by magnetization vs
field measurements (Figure S4d) collected at 1.8 K. Measurements with fields
applied along the c axis (Figure 2d) indicate a pair of metamagnetic
transitions at $\sim$0.7 and $\sim$1.2 T, which is also observed in the
magnetization curves at 1.8 K as a shoulder in the derivative curve followed
by a peak with increasing field. The presence of more metamagnetic transitions
at lower critical fields in both directions, implies that spins are more
readily polarized along either crystallographic direction in comparison to
NdSb0.91Te0.89. Field-dependent magnetization also shows a smaller difference
between in- and out-of-plane saturating moments in NdSb0.48Te1.37 (Fig S4c),
altogether indicating that the CDW phase displays less magnetic anisotropy
than its undistorted relative. In addition, the CDW phase exhibits more
complex magnetism as compared with the tetragonal one, as evidenced by the
multiple metamagnetic transitions.

The magnetic phase diagram of NdSb0.48Te1.37 can be compared to the
isostructural CDW-containing relatives GdSb0.46Te1.48 and CeSb0.51Te1.40.[23,
22] GdSb0.46Te1.48 exhibits a significantly more complex phase diagram than
its tetragonal family member GdSb0.85Te1.15[21], along all measured
directions, but particularly under fields applied along the ab plane, a
behavior also observed in NdSb0.48Te1.37. In the latter case the magnetic
ordering temperature is much lower and so the various magnetic transitions may
be more difficult to resolve, whereas the Gd compound shows multiple
temperature-induced phase transitions. The observed multiple in-plane
transitions contrast with CeSb0.51Te1.40, where no metamagnetic transitions
are observed if a field is applied along ab, but multiple are observed for
fields along c. This corresponds with the out-of-plane magnetic structure of
its stoichiometric parent, CeSbTe.

Figure 3: Heat capacity measurements. a) Temperature dependence of the heat
capacity CP for NdSb0.48Te1.37 at variable applied fields. Inset focuses on
low-temperature behavior at applied fields, highlighting the suppression of TN
with applied field. b)Debye fit of CP/T with fitted phonon contribution shown.
c) Magnetic entropy Sm calculated from heat capacity data.

To corroborate the observed TN and field dependence of the magnetic
transitions in NdSb0.48Te1.37, we performed heat capacity measurements on a
single crystal. At zero applied field, the observed TN is identical to that
observed in the $\chi$ plots, and is suppressed with applied field in a manner
consistent with them. The spin state of NdSb0.48Te1.37 can also be determined
by integrating the magnetic component of the total heat capacity over
temperature ($\frac{C_{P}}{T}$) to calculate the change in the magnetic
entropy ($\Delta S_{mag}$). To determine the magnetic heat capacity,
$C_{mag}$, the phonon contribution to heat capacity was subtracted using a
single Debye mode as described in the equation shown below:

$C_{D}({\theta_{D},T})=9sR\left(\frac{T}{\theta_{D}}\right)^{3}\int\limits_{0}^{\frac{\theta_{D}}{T}}\frac{(\theta/T)^{4}e^{\theta/T}}{[e^{\theta/T}-1]^{2}}d\frac{\theta}{T}$
(1)

where $s$ is the oscillator strength, $R$ is the molar Boltzmann constant, and
$\theta_{D}$ is the Debye temperature. The parameters extracted from the
least-squares fit between $T$ = 25 - 265K is $s$ = 3.007(8) and $\theta_{D}$ =
182.8(4) K respectively. The total oscillator strength is in reasonable
agreement with the total number of atoms per formula unit in NdSb0.48Te1.37.
In the phonon fit, an Einstein mode was not incorporated because there was no
$T_{max}$ in the $\frac{C_{P}}{T^{3}}$ plot. Thus, a single Debye mode was
sufficient for the phonon fit. The $\Delta S_{mag}$ with respect to
temperature is found to saturate at $\simeq$4.99 J/mol-K, slightly below what
would be expected for a $S=\frac{1}{2}$ spin state ($R$ln2). The slight
decrease in the $\Delta S_{mag}$ result from additional entropy release below
$T$ = 2 K that is not captured by a linear extrapolation of $C_{p}/T$
$\rightarrow$ 0 at $T$ = 0 K. The presence of artifacts and magnetic entropy
beyond the phonon subtraction region are viable possibilities; thus the heat
capacity analysis had some limitations. Our results notably contrast with
entropy observed by Pandey et. al. in NdSbxTe2-x-δ, where x = 0.8, which more
closely indicates a $S=\frac{3}{2}$ state consistent with a free Nd3+ ion.
[25]

###  III.3 Neutron diffraction and magnetic structures

The magnetic ground state in zero applied magnetic field for both compounds
has been investigated through powder neutron diffraction in the 5 – 1.5 K
temperature range. The paramagnetic data set of NdSb0.94Te0.92, collected at 5
K, is in good agreement with the structural model obtained on single crystal
specimens (see Figure S5 and Table S3 and S4 in the supporting information for
details). On cooling below $T_{N}\approx 2K$, broad magnetic reflections are
observed in the diffraction pattern, which indicates a magnetic correlation
length significantly shorter than the nuclear one. The peaks can be indexed
with the propagation vector k=(0 1/2 1/2). The possible magnetic isotropy
subgroups consistent with both, the parent structure symmetry and the observed
propagation vector, were obtained with the help of the ISODISTORT
software.[34] The best agreement with the experimental data is obtained for
the magnetic space group Ab21/c (Pc21/c, basis=(0,1,1),(-1,0,0),(0,-2,0)
origin=(1,1,0) in standard settings, see table S3 in the SI for details). The
refined magnetic structure, together with the magnetic Rietveld plot, are
shown in figure 4a and 4b. The spin ordering consists of AFM chains that run
along the b axis, where the spins lie parallel to b and antiparallel to each
other along the chain. A refined moment of 1.674(8)$\mu_{B}$ per Nd was
obtained. The two AFM chains within a NdTe layer are shifted by half a unit
cell in plane, due to the presence of the n-glide symmetry operation in the
parent structure, which leads to one ferromagnetic ($J_{1}$) and one
antiferromagnetic ($J_{2}$) Nd-Nd interaction. This geometrical frustration of
the intra-chain interactions could be the reason of the very broad magnetic
reflections observed in the neutron data. In fact, in VOCl, which is
structurally closely related to tetragonal NdSb0.94Te0.92, a lattice
distortion, which lifts this frustration, is observed at the magnetic ordering
temperature.[39] Since no such distortion is observed NdSb0.94Te0.92, the
frustration between the intra-chain interaction remains, resulting in the
broad magnetic Bragg peaks. In general, the observed magnetic structure is
typical for PbFCl-type compounds, as VOCl adopts a very similar one. [39]
Still, it is different than that observed in CeSbTe, where spins point along
the c axis and the AFM coupling doubles the unit cell.[24]

The paramagnetic diffraction pattern of modulated NdSb0.48Te1.37, collected at
5 K, is in agreement with the structural model obtained from the single
crystal data (see Table S7-8 and Figure S6 in the SI for details). The
refinement has been conducted in the $Pmnm(00\gamma)000$ superspace group and
due to the small intensity of the modulation satellite peaks, only the first
order positional modulation parameters have been refined. The obtained
structural parameters are reported in the SI and the refined value of the
modulation vector is $q_{CDW}$=(0,0.183(1),0) (with respect to the parent
P4/nmm structure). On cooling below TN=2.0(1)K, sharp magnetic reflections are
observed in the diffraction pattern. These can be indexed with the propagation
vector $q_{mag}$=(0,-0.408(1),0), the minus sign and the relation with the
nuclear modulation vector will be discussed in detail later. The possible
magnetic superspace groups consistent with the observed modulation vector and
the tetragonal $P4/nmm$ parent structure have been calculated with the help of
the ISOTROPY and ISODISTORT software.[34, 33] The best agreement with the
experimental data has been achieved with the $Pmnm.1^{\prime}(00\gamma)s00s$
superspace group and the obtained spin structure and the relative magnetic
Rietveld plots are shown in Figure 4c,d (see table S9 and S10 in the SI for
detail). The magnetic structure can be refined as an elliptical cycloid with a
sinusoidal in plane amplitude that is $\approx 3$ times larger than the out of
plane cosinusoidal component. The rotation of the moments between adjacent
unit cells along the propagation vector is of $\pi-\phi$ (the sign choice of
the propagation vector component change the rotation angle from $\pi-\phi$ to
$\pi+\phi$). Due to the presence of the $\\{n|0000\\}$ symmetry element in the
superspace group, the two cycloids on the Nd atoms in the NdTe layer are
rotating in opposite directions. This suggests that the magnetic interaction
between these two positions will average to zero over the period of the
magnetic structure. The solution shown in Figure 4 is the only model that is
able to fit the observed intensity correctly; attempts to constrain a constant
moment structure with a circular cycloid return worse fits of the data, as
shown in Figure S7. The same is true for attempts using a positive propagation
vector i.e. a cycloid with a rotation angle of $\pi+\phi$ (Figure S8).

Figure 4: Magnetic structures determined by neutron diffraction. a) Magnetic
structure of NdSb0.94Te0.92. b) Magnetic structure of NdSb0.48Te1.37, with
exchange coupling terms $J_{1}$ and $J_{2}$ shown. In parenthesis, standard
crystallographic axes used in Rietveld model are given. c) Magnetic Rietveld
fit for NdSb0.94Te0.92, with magnetic scattering at 1.5 K plotted by
subtraction of nuclear scattering collected at 5 K. d) Magnetic Rietveld fit
for NdSb0.48Te1.37, with magnetic scattering at 1.5 K plotted by subtraction
of nuclear scattering collected at 5 K.

###  III.4 Coupling between the CDW and the elliptical cycloid

It is worth underlining the strong coupling between the observed elliptical
magnetic structure and the charge density wave, which occurs above the
magnetic transition. By observing the value of the modulation vectors, one can
notice that they are related by $1+2\nu_{2}=\nu_{1}$ where $\nu_{1}$ and
$\nu_{2}$ are the components of the CDW and cycloidal modulation vectors,
respectively. Indeed, this relation can be rewritten, in vectorial form, as
qCDW+2qmag=$(0\bar{1}0)$ which suggest that the nuclear modulation can be
considered the second harmonic of the magnetic distortion, with the caveat
that with respect to the parent $P4/nmm$ structure the two propagation vectors
have opposite sign (the sign choice being a consequence of symmetry
decomposition from the parent structure).

With respect to the parent $P4/nmm$ structure the nuclear modulation
transforms as the $\Delta_{3}$ irreducible representation with the order
parameter $(\delta,\delta^{*};\eta,\eta^{*})$, defined in a complex basis, and
modulation vector $q_{CDW}=(0,\nu_{1},0)$. On the other hand, the elliptical
magnetic structure transform as the time-even irreducible representation
$m\Delta_{2}$ with the order parameter $(\mu,\mu^{*};\xi,\xi^{*})$ and
propagation vector $q_{CDW}=(0,\nu_{2},0)$. By inspecting the transformation
matrices of the generators of the $P4/nmm.1^{\prime}$ space group for these
two irreducible representations (reported in table S9), it is possible to
write a third degree free energy invariant that couples these two distortions.
The coupling term is
$\delta^{*}\mu^{2}+\delta\mu^{*2}+\eta^{*}\xi^{2}+\eta\xi^{*2}$ and it is
allowed only if the components of the propagation vector of the two
distortions verify the condition $1+2\nu_{2}=\nu_{1}$. This coupling term
gives a natural explanation why the system decides to order magnetically with
the observed propagation vector and the observed symmetry. In fact, by
choosing the $m\Delta_{2}$ symmetry for the magnetic order parameter and the
value of the magnetic propagation vector such that $1+2\nu_{2}=\nu_{1}$, the
system lowers its free energy by coupling to the nuclear modulation already
present above the magnetic transition. The coupling term also provides the
system a mechanism to relieve the magnetic frustration as can be appreciated
from the sharper magnetic peaks in NdSb0.48Te1.37 compared to NdSb0.94Te0.92.
The coupling term alone does not justify the elliptical character of the
cycloidal structure, which is likely the result of the competition between the
magnetic anisotropy observed in the magnetization data and the magnetoelastic
coupling term with the CDW. The energy gain offered by activating the coupling
term could resolve the frustration of the Nd-Nd interactions by stabilizing
the elliptical cycloid structure instead of the collinear structure observed
in tetragonal NdSb0.94Te0.92.

To further confirm the coupling term, the diffraction data collected at 1.5 K
were refined in the $Pmnm.1^{\prime}(00\gamma)s00s$ magnetic superspace group,
refining the magnetic modulation as first order harmonics and the nuclear
modulation as second order harmonics. The results of the refinement are
summarised in table S9 and S10 and the corresponding Rietveld plots are shown
in Figure S9. As can be appreciated, the refined nuclear modulation is in good
agreement with the refined structure at 5 K, as well as the single crystal
data.

###  III.5 Discussion

Both charge- and spin-density waves are phenomena driven by Fermi surface
nesting, and are frequently considered as competing phase transitions, as they
both result in the opening of an energy gap to relieve the electronic
instability of the nesting.[13] The coexistence of both in a system implies
the coupling of charge and spin degrees of freedom, which has been observed in
metallic Cr [40, 41], FeTe1-y[42], and in layered nickelate La4Ni3O10[43], and
recently, KMn6Bi5[44]. In a majority of these cases, CDW and SDW onset
coincide at the same transition temperature, frequently attributed to
magnetoelastic coupling, and the CDW can be considered as a secondary order
parameter induced by the SDW. [45] In NdSb0.48Te1.37, the CDW is observed at
room temperature, well above the appearance of the spin modulation. In this
case, the spin modulation is caused by the RKKY interaction, where the
ordering of localized Nd moments is mediated by the conduction electrons that
come from the (Sb,Te) square net, which carry the information of the lattice
modulation to the spins. Thus, the CDW plays a role in “templating” the
resulting spin wave below TN, as well as relieving the magnetic frustration
apparent by comparison to NdSb0.94Te0.92. A similar “templating” effect has
been observed in $CaMn_{7}O_{12}$ [46] in which the observed orbital density
wave [47] influence the exchange interactions stabilizing the intermediate
temperature SDW, [46] and in $\zeta-Mn_{2}O_{3}$ where the commensurate
orbital order drives the selection of the magnetic ground state.[48] The
“templating” effect of the CDW and the subsequent relief of the exchange
frustration explains why the temperature induced antiferromagnetic transitions
occur more sharply in NdSb0.48Te1.37. The lower critical field for
metamagnetic transitions, both in-plane (ab) and in the out-of-plane direction
(c), agree well with the magnetic structure, since the elliptical cycloid lies
in the bc plane.

The coupling term constructed for NdSb0.48Te1.37 also implies that for a CDW
phase involving 2 arms of the star of q, the resultant magnetic structures
will have the same number of arms involved. Given that multi-q magnetic states
are known to give rise skyrmion phases and other related spin textures, it is
possible that CDW can be used as a tool to access them via RKKY interactions
in addition to the more-common Dzyaloshinskii–Moriya interaction which drives
their presence in other systems. Indeed, it has been shown that densely packed
skyrmion lattices can exist in centrosymmetric crystals, where the skyrmions
are believed to originate from frustrated RKKY interactions. [49, 50, 51, 52]
Recent powder neutron diffraction on stoichiometric HoSbTe and TbSbTe,
tetragonal members of LnSbxTe2-x-δ, reveal complex, multi-q magnetic
structures, requiring multiple commensurate propagation vectors in the ground
state, which transform to incommensurate upon warming while still below
TN.[53] Exploring the synthesis of their CDW-containing derivatives may hold
potential to access such skyrmion lattices. In previously mentioned
GdSb0.46Te1.48, a low-field region of the magnetic phase diagram in the out-
of-plane direction shows possible signatures of an antiferromagnetic skyrmion
phase by magnetoentropic analysis. The multiple low-field transitions present
in the out-of-plane magnetization of NdSb0.48Te1.37 may bear relation to the
magnetic phases observed in GdSb0.46Te1.48, also hinting to the role that the
CDW may play therein. A recent single-crystal neutron diffraction study on
single-q, near-6-fold CDW phase GdSb0.71Te1.22 reports the appearance of two
magnetic phases which arise subsequently below TN and below the final
transition at 5 K, respectively. [54] The multi-q CDW phase CeSb0.10Te1.79
shows complex out-of-plane metamagnetic transitions in the form of a “devil’s
staircase”[22] in the out-of-plane magnetization, where it is also suggested
that the CDW couples to a type of spin wave. However, in this system, the CDW
wavevectors contain out-of-plane components, which may allow interaction with
the expected out-of-plane spin structure.

##  IV Conclusion

In conclusion, we have investigated the crystal structure, magnetic
properties, and magnetic structure of the undistorted square-net TSM
NdSb0.94Te0.92 and its CDW-distorted relative NdSb0.48Te1.37 to understand the
interactions between the charge density wave and magnetism in LnSbxTe2-x-δ
systems. NdSb0.94Te0.92 is isostructural to other stoichiometric LnSbTe, where
delocalized bonding stabilizes the Sb square net. NdSb0.48Te1.37 exhibits a
CDW distortion that breaks the square net into zig-zag chains with isolated
atoms, isostructural to other LnSbxTe2-x-δ with similar x and nearly
commensurate (q=0.18b∗). While NdSb0.94Te0.92 behaves as a typical collinear
antiferromagnet with a TN of 2.7 K , NdSb0.48Te1.37 displays additional
complex magnetism, with a lower TN of 2.3 K and additional metamagnetic
transitions. NdSb0.48Te1.37 has an elliptical cycloid magnetic structure with
a propagation vector q=-0.41b∗. The near-integer multiple of the CDW vector
provides a rare example of CDW inducing complex spin order through coupling of
propagation vectors, as well as the most direct evidence of CDW coupling with
magnetism in LnSbxTe2-x-δ to date. This is also the first example of a
magnetic structure of a CDW phase within the LnSbxTe2-x-δ family. The
existence of the CDW above the magnetic transition temperature suggests that
it plays a role in “templating” the spin interactions between Ln atoms via
RKKY interaction. Furthermore the complexity of the magnetism and the richness
of CDW phases accessible in LnSbxTe2-x-δ phases, suggest that it is possible
to achieve RKKY interaction-induced skrymions in these materials.

###### Acknowledgements.

This work is supported by an NSF CAREER grant (DMR-2144295) to LMS. Additional
support was provided by the Air Force office of Scientific Research under
award number FA9550-20-1-0246. This work was further supported by the Gordon
and Betty Moore Foundation (EPiQS Synthesis Award) through grant GBMF9064, the
David and Lucile Packard Foundation, the Alfred P. Solan foundation, and the
Arnold and Mabel Beckman foundation through a BYI grant awarded to LMS and an
AOB postdoctoral fellowship awarded to JFK. TB acknowledges support from NSF-
MRSEC through the Princeton Center for Complex Materials NSF-DMR-2011750. THS
is grateful to Marceline Martineau for helpful discussions. F.O. and P.M. are
grateful to Dr. Dmitry Khalyavin for the very fruitful discussion. For the
purpose of open access, the author has applied a creative commons attribution
(CC BY) licence to any author accepted manuscript version arising.

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