Tetrahedral triple-Q magnetic ordering and large spontaneous Hall conductivity in the metallic triangular antiferromagnet Co1/3TaS2

Abstract

The triangular lattice antiferromagnet (TLAF) has been the standard paradigm of frustrated magnetism for several decades. The most common magnetic ordering in insulating TLAFs is the 120° structure. However, a new triple-Q chiral ordering can emerge in metallic TLAFs, representing the short wavelength limit of magnetic skyrmion crystals. We report the metallic TLAF Co1/3TaS2 as the first example of tetrahedral triple-Q magnetic ordering with the associated topological Hall effect (non-zero σxy(H = 0)). We also present a theoretical framework that describes the emergence of this magnetic ground state, which is further supported by the electronic structure measured by angle-resolved photoemission spectroscopy. Additionally, our measurements of the inelastic neutron scattering cross section are consistent with the calculated dynamical structure factor of the tetrahedral triple-Q state.

Full text

Article https://doi.org/10.1038/s41467-023-43853-4
Tetrahedral triple-Q magnetic ordering and
large spontaneous Hall conductivity in the
metallic triangular antiferromagnet
Co
1/3
TaS
2
Pyeongjae Park
1,2
, Woonghee Cho
1,2
,ChaebinKim
1,2
,YeochanAn
1,2
,
Yoon-Gu Kang
3
, Maxim Avdeev
4,5
, Romain Sibille
6
,KazukiIida
7
,
Ryoichi Kajimoto
8
, Ki Hoon Lee
9
, Woori Ju
10
, En-Jin Cho
10
,Han-JinNoh
10
,
Myung Joon Han
3
, Shang-Shun Zhang
11
,CristianD.Batista
12,13
&
Je-Geun Park
1,2,14
The triangular lattice antiferromagnet (TLAF) has been the standard paradigm
of frustrated magnetism for several decades. The most common magnetic
ordering in insulating TLAFs is the 120° structure. However, a new triple-Q
chiral ordering can emerge in metallic TLAFs, representing the short wave-
length limit of magnetic skyrmion crystals. We report the metallic TLAF Co
1/
3
TaS
2
as the first example of tetrahedral triple-Qmagnetic ordering with the
associated topological Hall effect (non-zero σ
xy
(H=0)).Wealsopresenta
theoretical framework that describes the emergence of this magnetic ground
state, which is further supported by the electronic structure measured by
angle-resolved photoemission spectroscopy. Additionally, our measurements
of the inelastic neutron scattering cross section are consistent with the cal-
culated dynamical structure factor of the tetrahedral triple-Qstate.
Discovering magnetic orderings with novel properties and function-
alities is an important goal of condensed matter physics. While ferro-
magnets are still the best examples of functional magnetic materials
due to their vast spectrum of technological applications, antiferro-
magnets are creating a paradigm shift for developing new spintronic
components1,2. A more recent case is the discovery of Skyrmion crys-
tals induced by a magnetic field in different classes of materials. These
chiral states, which result from a superposition of three spirals with
ordering wave vectors that differ by a ± 120° rotation about a high-
symmetry axis, can produce a substantial synthetic magnetic field that
couples only to the orbital degrees of freedom of conduction
electrons3,4. When conduction electrons propagate through a sky-
rmion spin texture, they exhibit a spontaneous Hall effect. The origin
of this effect becomes transparent in the adiabatic limit. Due to the
Received: 12 April 2023
Accepted: 22 November 2023
Check for updates
1
Center for Quantum Materials, Seoul National University, Seoul 08826, Republic of Korea.
2
Department of Physics & Astronomy, Seoul National University,
Seoul 08826, Republic of Korea.
3
Department of Physics, KAIST,Daejeon 34141, Republic of Korea.
4
Australian Nuclear Science and Technology Organisation
(ANSTO), New Illawarra Road, Lucas Heights,NSW 2234, Australia.
5
School of Chemistry, The University of Sydney, Sydney, NSW 2006, Australia.
6
Laboratory
for Neutron Scattering and Imaging, Paul Scherrer Institut, 5232 Villigen, Switzerland.
7
Comprehensive Research Organization for Science and Society
(CROSS), Tokai, Ibaraki 319-1106, Japan.
8
Materials and Life Science Division, J-PARC Center, Japan Atomic Energy Agency, Tokai, Ibaraki 319-1195, Japan.
9
Department of Physics, Incheon National University, Incheon 22012, Republic of Korea.
10
Department of Physics, Chonnam National University, Gwangju
61186, Republic of Korea.
11
School of Physics and Astronomy and William I. Fine Theoretical Physics Institute, University of Minnesota, Minneapolis, MN 5 5455,
USA.
12
Department of Physics and Astronomy, The University of Tennessee, Knoxville, TN 37996, USA.
13
Quantum Condensed Matter Division and Shull-
Wollan Center, Oak Ridge National Laboratory,Oak Ridge, TN 37831, USA.
14
Institute of Applied Physics, Seoul National University, Seoul 08826, Republic of
Korea. e-mail: cbatist2@utk.edu;jgpark10@snu.ac.kr
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exchange interaction, the underlying local moment texture aligns the
spin of a conduction electron that moves in a loop, inducing a Berry
phase in its wavefunction equal to half of the solid angle spanned by
the local moments enclosed by the loop. This phase is indistinguish-
able from the Aharonov-Bohm phase induced by a real magnetic flux,
and each skyrmion generates a flux quantum in the corresponding
magnetic unit because the spins span the full solid angle of the
sphere (4π).
The triangular lattice Heisenberg model is a textbook example
that can host diverse quantum states with small variations of short-
range exchange interactions. The generic ground state for nearest-
neighbor (NN) antiferromagnetic interactions is the three-sublattice
120° structure shown in Fig. 1a. This spiral structure is characterized by
an ordering wave vector located at ± K-points of the hexagonal Bril-
louin zone (Fig. 1d). Adding a relatively small second NN anti-
ferromagnetic interaction gives rise to the two-sublattice collinear
stripe spin configuration shown in Fig. 1b, whose ordering wave vector
is one of the three M-points of the Brillouin zone (see Fig. 1e).
Remarkably, for S= ½, these two phases seem to be separated by a
quantum spin liquid state5–11 whose nature is not yet fully understood.
However, small effective four-spin interactions can induce a fun-
damentally different chiral antiferromagnetic order in triangular lat-
tice antiferromagnets. This state is the triple-Qversion of the stripe
order, where the three different M-ordering wave vectors (see Fig. 1f)
coexist in the same phase giving rise to a non-coplanar four-sublattice
magnetic ordering (see Fig. 1c). The spins of each sublattice point
along the all-in or all-out principal directions of a regular tetrahedron.
Theoretical studies suggest that this state can appear naturally in
metallic TLAFs, where effective four-spin interactions arise from the
exchange interactionbetween conduction electronsand localized spin
degrees of freedom12,13. This state was predicted to appear in the Mn
monolayers on Cu(111) surfaces by density functional theory14,andit
was observed in the hcp Mn monolayers on Re(0001) using spin-
polarized scanning tunneling microscopy15,16. However, it has not been
reported yet in bulk systems.
The tetrahedral ordering is noteworthy for its topological nature,
as it can be viewed as the short-wavelength limit of a magnetic sky-
rmion crystal17. The three spins of each triangular plaquette span one-
quarter of the solid angle of a sphere, implying that each skyrmion
(one flux quantum) is confined to four triangular plaquettes. As
illustrated in Fig. 1c, the two-dimensional (2D) magnetic unit cell of the
tetrahedral ordering consists of eight triangular plaquettes, meaning
there are two skyrmions per magnetic unit cell. In other words, the
tetrahedral triple-Qordering creates a very strong, effective magnetic
field of one flux quantum divided by the area of 4 triangular pl aquettes
in the adiabatic limit. Notably, this ordering does not have any net spin
magnetization. However, the emergent magnetic field couples to the
orbitaldegrees of freedom of the conduction electrons, giving rise to a
uniform orbital magnetization and a large topological Hall effect
characterized by scalar spin chirality (χijk =hSiSj×Ski)3,4.Thus,this
spin configuration is the simplest textbook example where non-trivial
band topology is induced in the absence of relativistic spin-orbit
coupling (the non-coplanar configuration generates an effective gauge
field that couples to the orbital degrees of freedom of the conduction
electrons). Moreover, the tetrahedral ordering can provide a potential
route to realize the antiferromagnetic Chern insulator by properly
adjusting the Fermi level of the system, as suggested in Refs. 12,13,18.
This work reports a four-sublattice tetrahedral triple-Qordering
as the only scenario known to the authors that is consistent with our
data for the metallic triangular antiferromagnet Co
1/3
TaS
2
.Ourkey
observations are the coexistence of long-range antiferromagnetic
ordering with wave vector q
m
= (1/2, 0, 0), a weak ferromagnetic
moment (M
z
ðH=0Þ), and a non-zero AHE (σxyðH=0Þ)belowT
N2
,which
rules out the possibility of single-Q and double-Q ordering19–22.Based
on the crystalline and electronic structure of Co
1/3
TaS
2,
we also provide
a theoretical conjecture about the origin of the observed ordering
,
consistent with our angle-resolved photoemission spectroscopy
(ARPES) data. Moreover, the calculated low-energy magnon spectra of
the tetrahedral ordering agree with the spectra measured by inelastic
neutron scattering. Finally, we discuss the robustness of the tetra-
hedral ordering against an applied magnetic field in Co
1/3
TaS
2
.
Results and discussion
Co
1/3
TaS
2
is a Co-intercalated metal comprising triangular layers of
magnetic Co2+ ions (Fig. 1g). Previous studies on Co
1/3
TaS
2
in the
1980s reported the bulk properties of a metallic antiferromagnet
with S= 3/2 (a high-spin d7configuration of Co2+)23–25, including a
neutron diffraction study that reported an ordering wave vector
q
m
= (1/3, 1/3, 0) characteristic of a 120° ordering25. More recently, an
experimental study on single-crystal Co
1/3
TaS
2
observed a significant
Fig. 1 | The Tetrahedral triple-Q state and crystal structure of Co
1/3
TaS
2
.
a–cThree fundamental antiferromagnetic orderings for a triangular lattice system.
The red-shaded regions denote each magnetic unit cell. d–fPositions of the mag-
netic Bragg peaks (red circles) in momentum space generated by a–cA black (red)
hexagon corresponds to a crystallographic (magnetic) Brillouin zone. The green
and blue circles in d–edenote the magnetic Bragg peaks from the other two
magnetic domains. gA crystallographic unit cell of Co
1/3
TaS
2
.hThe temperature-
dependent magnetization of single-crystal Co
1/3
TaS
2
with H//c.
Article https://doi.org/10.1038/s41467-023-43853-4
Nature Communications | (2023) 14:8346 2
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anomalous Hall effect (AHE) comparable to those in ferromagnets
below 26.5 K, which is the second transition temperature (T
N2
)ofthe
two antiferromagnetic phase transitions at T
N1
= 38 K and T
N2
=26.5K
(see Fig. 1h)26. Based on the 120° ordering reported in ref. 25 and a
symmetry argument, the authors of ref. 26 suggested that the
observed AHE, σxyðH=0Þ≠0, can be attributed to a ferroic order of
cluster toroidal dipole moments. However, our latest neutron scat-
tering data reported in this work reveals an entirely different picture:
Co
1/3
TaS
2
has a magnetic structure with ordering wave vectors of the
M-points (q
m
= (1/2, 0, 0) and symmetry-related vectors) instead of
q
m
= (1/3, 1/3, 0). The most likely scenario for such distinct outcomes
would be a difference in Co composition; while we confirmed that
our experimental results are consistently observed in Co
x
TaS
2
with
0.299(4) < x< 0.325(4), the reported composition value for ref. 25’s
sample is x= 0.29 (see ref. 24). Additional Co disorder, beyond var-
iations in composition, could also be a contributing factor. However,
the limited data available for the sample in ref. 25 prevents us from
forming a definitive assessment in this regard (see Supplementary
Notes). In any case, the theoretical analysis in ref. 26 based on ref. 25
is not valid for q
m
= (1/2, 0, 0) since a different ordering wave vector
leads to a qualitatively different scenario. This forced us to conduct a
more extensive investigation and come up with a scenario consistent
with our comprehensive experimental datasets.
Figure 2a, b show the neutron diffraction patterns of powder and
single-crystal Co
1/3
TaS
2
for T<T
N
.Magneticreflections appear at the
M-points of the Brillouin zone for both T<T
N2
and T
N2
<T<T
N1
,
implying that the ordering wave vector is q
m
=(1/2, 0, 0) or its sym-
metrically equivalent wave vectors. It is worth noting that this obser-
vation is inconsistent with the previously reported wave vector q
m
=(1/
3, 1/3, 0)25; i.e., Co
1/3
TaS
2
does not possess a 120° magnetic ordering.
The magnetic Bragg peaks at the three different M points connected
by the three-fold rotation along the c-axis (C
3z
) have equivalent
intensities within the experimental error (Fig. 2c). This result suggests
either single-Qor double-Qordering with three equally weighted
magnetic domains or triple-Qordering.
First, we analyzed the neutron diffraction data based on group
representation theory and Rietveld refinement, assuming a single-Q
magnetic structure: Mν
i=Δνcos ðqν
mriÞwith ν=1,2,and3(seeSup-
plementary Text and Supplementary Tables 2 and 3). As a result, we
found that the spin configurations for T<T
N2
and T
N2
<T<T
N1
corre-
spond to Fig. 2g, h, respectively, where each configuration belongs to
the Γ2+Γ4(αVν
22 +βVν
41, see Supplementary Table 3) and Γ2(Vν
22)
representations—see Supplementary Notes. The refinement result
yielded an ordered moment of 1.27(1)μB/Co2+ at 3 K.
However, a single-Qspin configuration with q
m
= (1/2, 0, 0) pos-
sesses time-reversal symmetry (TRS) combined with lattice translation
(τ1aT,seeFig.2g, h), which strictly forbids the finite σxy ðH=0Þand
MzðH=0Þobserved at T<T
N2
(Fig. 2e, f). Similarly, double-Qspin
configurations in a triangular lattice are not compatible with finite
σxyðH=0Þand MzðH=0Þdue to its residual symmetry relevant to
chirality cancellation19.Triple-Qordering is, therefore, the only possi-
ble scenario that can resolve this contradiction since it allows for finite
σxyðH=0Þand MzðH=0Þdue to broken τ1aTsymmetry19–21.Ingeneral,
determining whether the magneticstructureofasystemisasingle/
double-Qphase with three magnetic domains or a triple-Qordering
requires advanced experiments. However, q
m
= (1/2, 0, 0) is a special
case where a triple-Qstate can be easily distinguished from the other
possibilities by probing non-zero TRS-odd quantities incompatible
with the symmetry of single/double-Qstructures.
The most symmetric triple-Qordering that produces the same
neutron diffraction pattern as that of Fig. 2h is illustrated in Fig. 2i. This is
precisely the four-sublattice tetrahedral ordering shown in Fig. 1c,
except that Co
1/3
TaS
2
hasanadditional3DstructurewithanABstacking
pattern. Such a triple-Qcounterpart can be obtained through a linear
combination of three symmetrically equivalent single-Qstates
(Mν
i=Δ
νcos ðqν
mriÞwith ν= 1, 2, 3) connected by the three-fold rota-
tion about the caxis22 (see Supplementary Notes for more explanation).
However, an arbitrary linear combination of these three components
yields a triple-Qspin configuration with a site-dependent magnitude of
ordered moments on the Co sites, i.e., |Mtri
i| depends on i(see Supple-
1
210
1
201
1
203
1
200
Fig. 2 | Magnetic ground state of Co
1/3
TaS
2
revealed by neutron diffraction.
aThe powder neutron diffraction pattern of Co
1/3
TaS
2
, measured at 60 K (Gray
squares) and 3K (Red circles). A weak λ/2 signal is highlighted explicitly (see
Methods). The solid black lineis the diffractionpattern of Co
1/3
TaS
2
simulated with
the spin configuration shown in h, or equivalently i. The gray (red) vertical solid
lines denote the position of nuclear (magnetic) reflections. The full diffraction data
can be found in Supplementary Fig. 3. bThe single-crystal neutron diffraction
pattern at 5K, demonstrating magnetic Bragg peaks located at the M points. cThe
intensities of the three magnetic Bragg peaks originating from three different
ordering wave vectors. dThe temperature-dependent intensities of some magnetic
Bragg peaks in single-crystal Co
1/3
TaS
2
.e,fThe temperature dependence of
σxy(H=0)andMz(H=0)inCo
1/3
TaS
2
,measuredafterfield cooling under 5 T.
g,hThe refined magnetic structures for g26.5 K < T< 38 K and hT<26.5K.iThe
triple-Qcounterpart (tetrahedral) of the single-Q(stripe) ordering shown in h.Note
that the spin configurations in hand Igive the same powder diffraction pattern.
Article https://doi.org/10.1038/s41467-023-43853-4
Nature Communications | (2023) 14:8346 3
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mentary Fig. 7). A uniform |Mtri
i| is obtained only when the Fourier
components Δνof the three wave vectors qν
mare orthogonal to each
other12, i.e., Mtri
i=Pν= 1,3Δνcosðqν
mriÞwith Δ
v
⊥Δ
v′
for ν≠ν0
:We note,
however, that the magnitude of the ordered moments (|Mtri
i|) does not
have to be the same for quantum mechanical spins. Nevertheless, as
explained in detail in the Supplementary Information, only non-coplanar
triple-Qorderings corresponding to equilateral ( Δν


=Δν0


,Fig.2h) or
non-equilateral ( Δν


Δν0


) tetrahedral configurations are consistent with
our Rietveld refinement of neutron diffraction data.
In the high-temperature ordered phase at T
N2
<T<T
N1
, the triple-
Qordering that yields the same neutron diffraction pattern as that of
the single-Qordering shown in Fig. 2g is collinear, giving rise to highly
nonuniform |Mtri
i| (see Supplementary Fig. 7). Such a strong modula-
tion of |Mtri
i|isunlikelyforS= 3/2 moments weakly coupled to con-
duction electrons (see discussion below). In addition, the collinear
triple-Qordering allows for finite M
z
(H=0) and σxy ðH=0Þ19–22,while
they are precisely zero within our measurement error for T
N2
<T<T
N1
.
On the other hand, the single-Qordering shown in Fig. 2gismore
consistent with M
z
(H=0)=σxyðH=0Þ= 0 in the temperature range
T
N2
<T<T
N1
due to its τ1aTsymmetry. Therefore, the combined neu-
tron diffraction and anomalous transport data indicate that a transi-
tion from a collinear single-Qto non-coplanar triple-Qordering occurs
at T
N2
. Interestingly, as we will see below, our theoretical analysis
captures this two-step transition process.
We now examine the feasibility of the tetrahedral triple-Qground
state in Co
1/3
TaS
2
. Notably, in contrast to typical triple-Qorderings
reported in other materials27–29, this state emerges spontaneously in
Co
1/3
TaS
2
without requiring an external magnetic field. It was pro-
posed on theoretical grounds that this state could arise in a 2D metallic
TLAF formulated by the Kondo lattice model12,13,18,30:
H=tXhi,jicy
iαcjαJXiSicy
iασαβciβ:ð1Þ
From a crystal structure perspective, Co
1/3
TaS
2
is an ideal candi-
date to be described by this model. The nearest Co-Co distance
(5.74 Å) is well above Hill’slimit,andtheCoS
6
octahedra are fully
isolated ((Fig. 3a). This observation suggests that the Co 3d bands
would retain their localized character, while itinerant electrons mainly
arise from the Ta 5d bands. Our density functional theory (DFT) cal-
culations confirm this picture and reveal that the density of states near
the Fermi energy has a dominant Ta 5d orbital character (Supple-
mentary Fig. 11). In this situation, a magnetic Co2+ ion can interact with
another Co2+ ion only via the conduction electrons in the Ta 5d bands.
Thus, in a first approximation, the Co2+ 3d electrons can be treated as
localized magnetic moments interacting via exchange with the Ta 5d
itinerant electrons23,25 (see Fig. 3b). Indeed, the Curie-Weiss behavior
observed in Co
1/3
TaS
2
provides additional support for this picture; the
magnitude of the fitted effective magnetic moment indicates S~1.35,
close to the single-ion limit of Co2+ in the high spin S=3/2
configuration26. However, our neutron diffraction measurement indi-
cates a significant suppression of the ordered magnetic moment
S~ 0.64 (assuming that a g-factor is 2). Similar reductions of the
ordered moment have been reported in other metallic magnets com-
prising 3d transition metal elements, and they are generally attributed
to a partial delocalization of the magnetic moments. In addition,
interactions between Co local moments and itinerant electrons from
Ta 5d band can lead to partial screening of the local moments. Another
possible origin of the reduction of the ordered moment is quantum
spin fluctuations arising from the frustrated nature of the effective
spin-spin interactions.
Tetrahedral ordering can naturally emerge when the Fermi sur-
face (FS) is three-quarters(3/4) filled12,13,18,31 becausethe shape of the FS
is a regular hexagon (for a tight-binding model with nearest-neighbor
hopping), whose vertices touch the M-points of the first Brillouin zone
(Fig. 3c). In this case, thereare three nesting wave vectors connecting
the edges of the regular hexagon and the van Hove singularities at
Fig. 3 | Stabilization mechanism and dynamical properties of the tetrahedral
order in Co
1/3
TaS
2
.aThe in-plane crystal structure of Co
1/3
TaS
2
demonstrates
isolated CoS
6
octahedrons (purple-colored) and a long NN Co-Co distance. bThe
exchange interaction between Co local moments and conduction electrons from
TaS
2
layers leads to an effective RKKY interaction between the local moments.
cThe Fermi surface of a 2D TLAF with 3/4 filling (shaded hexagons) and the Fermi
surface of Co
1/3
TaS
2
measured by ARPES. dThe magnon spectraof Co
1/3
TaS
2
at 5 K
along the(00 L) direction.E
i
= 7.9 and 14 meV da ta are plotted. eAntiferromagnetic
NN interlayer coupling (J
c
)ofCo
1/3
TaS
2
, which is necessary for explaining the data
in dand the refined spin configuration (Fig. 2g–i). fConst-Ecuts of the INS data
measuredat 5 K (<T
N2
). An energy integration range for each plot is ± 0.2 meV. The
E= 1 and 1.5 meV (2.0~3.0 meV) plots are based on the E
i
= 5 (7.9) meV data. In
additionto bright circular spots centered atsix M points (=linear modes), a weak,
diffuse ring-like scattering which we interpret as the quadratic mode predicted by
spin-wave theory, appears for E>1.5 meV. gThe calculated INScross-section of the
tetrahedral triple-Qordering with J
1
S2= 3.92 meV, J
c
S2=2.95meV,J
2
/J
1
=0.19, and
K
bq
/J
1
= 0.02 (see Supplementary Materials). hThe calculated INS cross-section of
the single-Qordering with three domains, using J
1
S2=3.92meV,J
c
S2= 2.95 meV, J
2
/
J
1
=0.1,andK
bq
/J
1
= 0. The line-shaped signal in hhas a much higher intensity than in
for g. The simulations in g,hinclude resolution convolution (see Supplementary
Fig. 9), and their momentum andenergy integration range are the same as f.iLow-
energy magnon spectra measured with E
i
= 3.5 meV at 5 K, showing the energy gap
of the linear magnon mode.
Article https://doi.org/10.1038/s41467-023-43853-4
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different M points, leading to magnetic susceptibility of the conduc-
tion electrons (χðq,EFÞ)thatdivergesaslog
2qqν
m


at q2M-points30.
This naturally results in a magnetic state with ordering wave vectors
corresponding to the three M points (i.e., triple-Q). While perfect
nesting conditions are not expected to hold for real materials, χðq,EFÞ
is still expected to have a global maximum at the three M points for
Fermi surfaces with the above-mentioned hexagonal shape18.Asshown
on the right side of Fig. 3c, our ARPES measurements reveal that this is
indeed the case of the FS of Co
1/3
TaS
2
, indicating that the filling frac-
tion is close to 3/4 and that the effective interaction between the Co2+
magnetic ions is mediated by the conduction electrons.
The above-described stabilization mechanism based on the shape
of the Fermi surface suggests that the effective exchange interaction
between the Co2+ magnetic ions and the conduction electrons is weak
compared to the Fermi energy. Under these conditions, it is possible to
derive an effective RKKY spin Hamiltonian by applying degenerate
second-order perturbation theory in J/t30,31. Since the RKKY model
includes only bilinear spin-spin interactions, the single-Q(stripe) and
triple-Q(tetrahedral) orderings remain degenerate in the classical
limit. Moreover, by tuning the mutually orthogonal vector amplitudes
Δνwhile preserving the norm Δ1



2+Δ2



2+Δ3



2, it is possible to
continuously connect the single-Q(stripe) and triple-Q(tetrahedral)
orderings via a continuous manifold of degenerate multi-Qorderings.
The classical spin model’s accidental ground state degeneracy leads to
a gapless magnon mode with quadratic dispersion and linear Gold-
stone modes. The accidental degeneracy is broken by effective four-
spin exchange interactions that gap out the quadratic magnon mode,
which naturally arises from Eq. (1) when effective interactions beyond
the RKKY level are taken into consideration14,17,30,31. The simplest
example of a four-spin interaction favouring the triple-Qordering is
the bi-quadratic term KbqðSiSjÞ2with Kbq >0. This was explicitly
demonstrated using classical Monte-Carlo simulations with a simpli-
fied phenomenological J
1
-J
2
-J
c
-K
bq
model (see Fig. 3aand3e), as shown
in Supplementary Fig. 8. This model successfully captures the tetra-
hedral ground state and manifests a two-step transition process at T
N2
and T
N1
with an intermediate single-Qordering, which is consistent
with our conclusion based on experimental observations. The latter
outcome can be attributed to thermal fluctuations favouring the col-
linear spin ordering32,33. In addition, previous DFT studies on TM
1/
3
NbS
2
(TM = Fe, Co, Ni), which is isostructural to Co
1/3
TaS
2
, also
revealed that a non-coplanar state could be energetically more favor-
able than collinear and co-planar states34.
Before analyzing the magnon spectra in detail, it is worth con-
sidering the interlayer network of Co
1/3
TaS
2
with AB stacking. Along
with the refined magnetic structure (Fig. 2g–i), the steep magnon
dispersion shown in Fig. 3d indicates non-negligible antiferromagnetic
NN interlayer exchange: JcS2∼2.95 meV (see Fig. 3e). However, the
finite value of J
c
does not change the competition between stripe and
tetrahedral orderings analyzed in the 2D limit. More importantly, the
antiferromagnetic exchange J
c
forces the tetrahedral spin configura-
tion of the B layer to be the same as that of the A layer (see Supple-
mentaryFig. 8a). Therefore, all triangularlayers have the same sign of
the scalar chirality χijk (or skyrmion charge), resulting in the realization
of 3D ferro-chiral ordering. In this 3D structure, each m agnetic unit cell
of Co
1/3
TaS
2
includes four skyrmions.
The low-energy magnon spectra (<3 meV) of Co
1/3
TaS
2
measured
by INS are presented in Fig. 3f. In addition to the linear (Goldstone)
magnon modes, which appear as bright circular signals centered at the
M points, a ring-like hexagonal signal connecting six M points was
additionally observed with weaker intensity. This can be interpreted as
the traceof the quadratic modes since they shouldbe present together
with linear modes at low energy (see Supplementary Fig. 9). Since the
signal is only present for E> 1.5 meV, we infer that the quadratic
mode is slightly gapped. We used linear spin-wave theory to compare
the measured INS spectra with the theoretical spectra of both single-Q
and triple-Qorderings. Despite the simplicity of the J
1
-J
2
-J
c
-K
bq
model,
the calculated magnon spectra of the tetrahedral ordering (Fig. 3g)
successfully describe the measured INS spectra. The magnon spectra
of the stripe order are also presented in Fig. 3h for comparison. The
intensity of the quadratic magnon mode is much stronger than that of
the triple-Qspectra, in apparent disagreement with our INS data. A
complete comparison between our data and the two calculations is
shown in Supplementary Fig. 10. Additionally, the linear magnon
modes of Co
1/3
TaS
2
are also slightly gapped (~0.5 meV, see Fig. 3i). This
feature can be explained for tetrahedral ordering only by considering
both exchange anisotropy and higher order corrections in the 1/S
expansion (see Supplementary Notes).
Finally, we discuss the behavior of the tetrahedral ordering in
Co
1/3
TaS
2
in response to out-of-plane and in-plane magnetic fields.
Figure 4cillustratesthefield dependence of the measured anomalous
Hall conductivity (σAHE
xy )andM
z
for H//c, showing evident hysteresis
with a sign change at ± Hc1 . Indeed, as already discussed in ref. 26,M
z
cannot characterize the observed σAHE
xy . Instead, as explained in the
introduction, it is χijk that characterizes σAHE
xy for the tetrahedral
ordering. Therefore, based on our triple-Q scenario (see Fig. 4a, b), the
sign change at ± Hc1 can be interpreted as the transition between
tetrahedral orderings with positive and negative values of the scalar
chirality χijk . In addition, the coexistence of weak ferromagnetic
moment and large σxyðH=0Þis expected because, as we explained
before, the real-space Berry curvature of the tetrahedral triple-Q
ordering generates both the orbital ferromagnetic moment (of con-
duction electrons) and spontaneous Hall conductivity. However, the
experimental techniques used in this study cannot discriminate
between the spin and orbital contributions to M
z
(H= 0). While it will be
interesting to identify the nature of this weak ferromagnetic moment,
this is left for future studies.
We also compared the field dependence of the measured σAHE
xy and
χijk calculated from our M
z
(H) data based on the canting expected in
the tetrahedral order of Co
1/3
TaS
2
(blue and orange arrows in Fig. 4). As
shown in Fig. 4d, σAHE
xy decreases slightly in response to both positive
and negative magnetic fields, consistent with the calculated χijk of the
tetrahedral ordering with mild canting. However, the model cannot
capturethe sudden decrease of σAHE
xy due to a meta-magnetic transition
at ± Hc2, indicating that this transition changes the tetrahedral spin
configuration. Additional neutron diffraction is required to identify the
new ordering for |H|>H
c2
.
The effect of an in-plane magnetic field was investigated using
single-crystal neutron diffraction. Figure 4eshowsfield-dependent
(H//q
1
m= (1/2, 0, 0)) intensities of the three magnetic Bragg peaks,
each originating from three different qν
mof the tetrahedral ordering.
Interestingly, the equal intensity of the three peaks remains almost
unchanged by a magnetic field up to 10T, indicating robustness of the
tetrahedral ordering against an in-plane magnetic field. This should
be contrasted with triple-Qstates found in other materials, which are
induced by a finite magnetic field and occupy narrow regions of the
phase diagram due to the ferromagnetic nature of the dominant
exchange interaction27–29.
In summary, we have reported a tetrahedral triple-Qordering in
Co
1/3
TaS
2
, as the only magnetic ground state compatible with our
bulk properties (non-zero σxyðH=0Þand M
z
(H= 0)) and neutron
diffraction data (q
m
= (1/2, 0, 0) or its symmetry-equivalent pairs).
Moreover, we provide a complete theoretical picture of how this
exotic phase can be stabilized in the triangular metallic magnet Co
1/
3
TaS
2
, which is further corroborated by our measurements of elec-
tronic structure and long-wavelength magnetic excitations using
ARPES and INS. We further investigated the effect of external mag-
netic fields on this triple-Qordering, which demonstrates its resi-
lience against the fields. Our study opens avenues for exploring chiral
magnetic orderings with the potential for spontaneous integer
quantum Hall effect30.
Article https://doi.org/10.1038/s41467-023-43853-4
Nature Communications | (2023) 14:8346 5
Content courtesy of Springer Nature, terms of use apply. Rights reserved

Note added: After the submission of this work, we came across
another neutron diffraction work on Co
1/3
TaS
2
published recently35.
While they have drawn the same conclusion of magnetic structure as
our tetrahedral triple-Qmagnetic ordering, our study further delves
into themicroscopic aspects of this magnetic ground state, specifically
in relation to electronic structure and the properties of a geometrically
frustrated triangular lattice. We also substantiate our findings through
the ARPES and INS measurements.
Methods
Sample preparation and structure characterization
Polycrystalline Co
1/3
TaS
2
was synthesized by the solid-state reaction. A
well-ground mixture of Co (Alfa Aesar, >99.99%), Ta (Sigma Aldrich,
>99.99%), and S (Sigma Aldrich, >99.999%) was sealed in an evacuated
quartz ampoule and then sintered at 900 °C for 10 days. A molar ratio
of the three raw materials was α:3:6 with 1.05 < α<1.1, which is
necessary to obtain the resultant stoichiometric ratio of Co close to
1/3. Single-crystal Co
1/3
TaS
2
was grown by the chemical vapor trans-
port method with an I
2
transport agent (4.5 mg I
2
/cm3). The pre-
reacted polycrystalline precursor and I
2
were placed in an evacuated
quartz tube and then were heated in a two-zone furnace with a tem-
perature gradient from 940 to 860 °C for 10~14 days.
We measured the powder X-ray diffraction (XRD) pattern of Co
1/
3
TaS
2
using a high-resolution (Smartlab, Rigaku Japan) diffractometer,
which confirmed the desired crystal structure without any noticeable
disorder26. In particular, the intercalation profile of Co atoms was
carefully checked by the (10 L) superlattice peak pattern in the low-2θ
region; see ref. 26. The powder neutron diffraction experiment further
corroborated this result (see the relevant subsection below). Finally,
single crystals were examined by XRD (XtaLAB PRO, Cu K
α
, Rigaku
Japan) and Raman spectroscopy (XperRam Compact, Nanobase
Korea), confirming the high quality of our crystals, e.g., the sharp
Raman peak at 137 cm−1(ref. 26).
The composition xof Co
x
TaS
2
single-crystal was confirmed pri-
marily by energy-dispersive X-ray (EDX) spectroscopy (Quantax 100,
Bruker USA & EM-30, Coxem Korea). We measured 12 square areas of
300μm× 300μm wide for every single piece, yielding homogeneous x
centered at ~0.320 with a standard deviation of ~0.004. The homo-
geneity of xwas further verified by the spatial profile (~1.5 μmreso-
lution) of the EDX spectra, which is very uniform, as shown in
Supplementary Fig. 1. The obtained xfrom EDX was again cross-
checked by the composition measured by inductively coupled plasma
(ICP) spectroscopy (OPTIMA 8300, Perkin-Elmer USA), which is almost
the same within a measurement error bar26.
01
211
2
1
21
1
201
Fig. 4 | The effect of the out-of-plane and in-plane magnetic field on the tet-
rahedraltriple-Q ordering in Co
1/3
TaS
2
.a,bA time-reversal pair of the tetrahedral
spin configuration, having χijk opposite to each other. The blue and orange arrows
in adepictthe generic canting of the tetrahedralordering in Co
1/3
TaS
2
by an out-of-
plane magnetic field. cComparison between the measured σAHE
xy (orange) and M
z
(red) underthe out-of-planefield at 3 K. dComparison between the measured σAHE
xy
(orange) at 3 K and χijk (red) calculated from the M
z
data in c.eIntensities of the
three magnetic Bragg peaks in Fig. 2c under an externalmagnetic field along the a*
direction. Error bars in erepresent the standard deviation of the measured
intensity.
Article https://doi.org/10.1038/s41467-023-43853-4
Nature Communications | (2023) 14:8346 6
Content courtesy of Springer Nature, terms of use apply. Rights reserved

Bulk property measurements
We measured the magnetic properties of Co
1/3
TaS
2
using MPMS-XL5
and PPMS-14 with the VSM option (Quantum Design USA). To measure
the spontaneous magnetic moment (Fig. 3d), a sample was field-cooled
under 5 T and then measured without a magnetic field. Transport
properties of Co
1/3
TaS
2
were measured by using four systems: our
home-built set-up, PPMS-9 (Quantum Design, USA), PPMS-14 (Quan-
tum Design, USA), and CFMS-9T (Cryogenic Ltd, UK). To observe the
temperature dependence of the anomalous Hall effect (Fig. 3d), we
field-cooled the sample under ±9 T and then measured the Hall voltage
without a magnetic field. The measured Hall voltage was anti-
symmetrized to remove any longitudinal components. Hall con-
ductivity (σxy) was derived using the following formula:
σxy =ρxy
ρxx2+ρxy 2:ð2Þ
Anomalous Hall conductivity σAHE
xy was derived by first subtracting
a normal Hall effect from measured ρxy and then using Eq. 2.
Powder neutron diffraction
We carried out powder neutron diffraction experiments of Co
1/3
TaS
2
using the ECHIDNA high-resolution powder diffractometer
(λ= 2.4395 Å) at ANSTO, Australia. To acquire clear magnetic signals of
Co
1/3
TaS
2
, which are weak due to the small content of Co and its small
ordered moment, we used 20 g of powder Co
1/3
TaS
2
. The quality of the
sample was checked before the diffraction experiment by measuring
its magnetic susceptibility and high-resolution powder XRD. The
neutron beam at ECHIDNA contains weak λ/2 harmonics (~0.3%),
yielding additional nuclear Bragg peaks (Supplementary Fig. 3). We
also performed Rietveld refinement and magnetic symmetry analysis
using Fullprof software36. The results are summarized in Supplemen-
tary Tables 1–3 and Supplementary Figs. 2–4.
Single-crystal neutron diffraction
We carried out single-crystal neutron diffraction under a magnetic
field using ZEBRA thermal neutron diffractometer at the Swiss spalla-
tion neutron source SINQ. We used one single-crystal piece (~16mg)
for the experiment, which was aligned in a 10 T vertical magnet
(Oxford instruments) with (HHL) horizontal (Supplementary Fig. 5a).
The beam of thermal neutrons was monochromatic using the (220)
reflection of germanium crystals, yielding a neutron wavelength of
λ= 1.383 Å with >1% of λ/2 contamination. Bragg intensities as a func-
tion of temperature and magnetic field were measured using a single
3He-tube detector in front of which slits were appropriately adjusted.
Angle-resolved photoemission spectroscopy (ARPES)
measurements
The ARPES measurements were performed at the 4A1 beamline of the
Pohang Light Source with a Scienta R4000 spectrometer37.Single
crystalline samples with a large AHE were introduced into an ultra-high
vacuum chamber and were cleaved insitu by a top post method at the
sample temperature of ~20 K under the chamber pressure of
∼5:0×10
11 Torr. A liquid helium cryostat maintained the low sample
temperature. The photon energy was set to 90 eV to obtain a high
photoelectron intensity for the states of Co 3d characters. The total
energy resolution was ~30 meV, and the momentum resolution was
~0.025 A1in the measurements.
Single-crystal inelastic neutron scattering
We performed single-crystal inelastic neutron scattering of Co
1/3
TaS
2
using the 4SEASONS time-of-flight spectrometer at J-PARC, Japan38.For
the experiment, we used nearly 60 pieces of the single-crystal with a
total mass of 2.2 g, which were co-aligned on the Al sample holder with
an overall mosaicity of ~1.5° (Supplementary Fig. 5b). We mountedthe
sample with the geometry of the (H0L) plane horizontal and per-
formed sample rotation during the measurement. The data were col-
lected with multiple incident neutron energies (3.5, 5.0, 7.9, 14.0, and
31.6 meV) and the Fermi chopper frequency of 150 Hz using the
repetition-rate-multiplication technique39. We used the Utsusemi40 and
Horace41 software for the data analysis. Based on the crystalline sym-
metry of Co
1/3
TaS
2
, the data were symmetrized into the irreducible
Brillouin zone to enhance statistics.
Classical Monte-Carlo simulations
To investigate the magnetic phase diagram of our spin model, we
performed a classical Monte-Carlo (MC) simulation combined with
simulated annealing. We used Langevin dynamics42 for the sampling
method of a spin system. To search for a zero-temperature ground
state, a large spin system consisting of 30 × 30 × 4 unit cells
(7200 spins) with periodic boundary conditions was slowly cooled
down from 150 K to 0.004 K. The thermal equilibrium was reached at
each temperature by evolving the system through 5000 Langevin time
steps, with the length of each time-step defined as dt = 0.02/(J
1
S2).
Finally, we adopted the final spin configuration at 0.004 K as the
magnetic ground state, which was further confirmed by checking
whether the same result was reproduced in the second trial.
For finite-temperature phase diagrams, we used a
30 × 30 × 6 supercell (10,800 spins). After waiting for 600~2000 Lan-
gevin time steps for equilibration, 600,000~2,000,000 Langevin time
steps were used for the sampling. Since thermalization and decorr-
elation time strongly depend on the temperature, the length of equi-
libration and sampling time steps were set differently for different
temperatures. From this result, we calculated heat capacity (C
V
),
staggered magnetization (M
stagg
), and total scalar spin chirality ðχijk Þ
using the following equations:
CV=ϵ2

ϵhi
2
kBT2,ð3Þ
Mstagg =gμB
NX
i
1ðÞ
2πðqν
mriÞSi

,ð4Þ
χijk =1
S3
P
Δ
SΔ1ðSΔ2SΔ3Þ

Nt
,ð5Þ
where Ahiis an ensemble average of A estimated by the sampling, ϵis
the totalenergy per Co2+ ion, gis the Lande g-factor, qν
m(ν= 1,2,3) is the
ordering wave vector of two-sublattice stripe order (see Supplemen-
tary Note), Δis the index for a single triangular plaquette on a Co
triangular lattice consisting of three sites (Δ1,Δ2,Δ3), and N(N
t
)isthe
total number of Co2+ ions (triangular plaquettes). When calculating
M
stagg
, one should first identify which ordering wave vector the spin
system chose among the three qν
m(ν= 1,2,3), and then use proper qν
m
to calculate Mstagg . For all the simulations, we used S= 3/2. The results
of the simulations are shown in Supplementary Fig. 8.
Spin-wave calculations
We calculated the magnon dispersion and INS cross-section of Co
1/
3
TaS
2
using linear spin-wave theory. For this calculation, we used the
SpinW library43. Since we did not align positive and negative χijk
domains of the tetrahedral ordering in our INS experiments, we aver-
aged the INS cross-section of both cases.
DFT calculations
We performed first-principles calculations using ‘Vienna ab initio
simulation package (VASP)’44–46 based on projector augmented wave
(PAW) potential47 and within Perdew-Burke-Ernzerhof (PBE) type of
Article https://doi.org/10.1038/s41467-023-43853-4
Nature Communications | (2023) 14:8346 7
Content courtesy of Springer Nature, terms of use apply. Rights reserved

GGA functional48 (see Supplementary Fig. 11). DFT + Umethod49,50 was
adopted to take into account localized Co-3dorbitals properly, where
U=4.1eVandJ
Hund
= 0.8eV were used as obtained by the constrained
RPA for CoO and Co51. For the 2 × 2 magnetic unit cell, we used the Γ-
centered 6 × 6 × 6 k-grid. We obtained and used the optimized crystal
structure with the force criterion for the relaxation fixed at 1 meV/Å.
The plane-wave energy cutoff was set to 500eV.
Data availability
The authors declare that data supporting the findings of this study are
available within the paper and the Supplementary Information. Further
datasets are available from the corresponding author upon request.
Code availability
Custom codes used in this article areavailable from the corresponding
author upon request.
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Acknowledgements
WeacknowledgeS.H.Lee,S.S.Lee,Y.Noda,andM.Mostovoyfortheir
helpful discussions and M. Kenzelmann for his help with the experiments
at SINQ. The Samsung Science & Technology Foundation supported this
work (Grant No. SSTF-BA2101-05). The neutron scattering experiment at
the Japan Proton Accelerator Research Complex (J-PARC) was performed
under the user program (Proposal No. 2021B0049). One of the authors
(J.-G.P.) is partly funded by the Leading Researcher Program of the
National Research Foundation of Korea (Grant No. 2020R1A3B2079375).
This work is based on experiments performed at the Swiss spallation
neutron source SINQ, Paul Scherrer Institute, Villigen, Switzerland. C.D.B.
acknowledges support from the U.S. Department of Energy, Office of
Science, Office of Basic Energy Sciences, under Award No. DE-
SC0022311.
Author contributions
J.-G.P. initiated and supervised the project. P.P. synthesized the poly-
crystalline and single-crystal samples. P.P. performed all the bulk
characterizations. M.A. carried out the powder neutron diffraction
experiment. P.P. and R.S. performed the single-crystal neutron diffrac-
tion experiment at ZEBRA. P.P. analyzed the neutron diffraction data
together with M.A.. P.P., C.K., Y.A., K.I., and R.K. conducted the single-
crystal inelastic neutron scattering experiment at 4SEASONS. W.J., E.-
J.C., and H.-J.N. conducted the ARPES experiment. Y.-G.K. and M.J.H.
performed the DFT calculations. P.P., W.H.C., S.-S.Z., and C.D.B. con-
ducted spin model calculations. P.P., W.H.C., S.-S.Z., K.H.L., Y.-G.K.,
M.J.H, H.-J.N., C.D.B., and J.-G.P. contributed to the theoretical analysis
and discussion. P.P., C.D.B., and J.-G.P. wrote the manuscript with con-
tributions from all authors.
Competing interests
The authors declare no competing interests.
Additional information
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Correspondence and requests for materials should be addressed to
Cristian D. Batista or Je-Geun Park.
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Article https://doi.org/10.1038/s41467-023-43853-4
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