# The Duel of Magnetic Interactions & Structural Instabilities: Itinerant
Frustration in the Triangular Lattice Compound LiCrSe2

E. Nocerino  KTH Royal Institute of Technology, Department of Applied Physics,
Alba Nova University Center, Stockholm, SE-114 21, Sweden<EMAIL_ADDRESS>S.
Kobayashi  Japan Synchrotron Radiation Research Institute (JASRI), 1-1-1
Kouto, Sayo 679-5198, Japan  C. Witteveen  Department of Quantum Matter
Physics, University of Geneva, 24 Quai Ernest-Ansermet, 1211 Geneva 4,
Switzerland  Department of Physics, University of Zürich, Winterthurerstr.
190, 8057 Zürich, Switzerland  O. K. Forslund  Chalmers University of
Technology, Department of Physics, G$\ddot{o}$teborg, SE-412 96, Sweden  N.
Matsubara  KTH Royal Institute of Technology, Department of Applied Physics,
Alba Nova University Center, Stockholm, SE-114 21, Sweden  C. Tang  Diamond
House, Harwell Science and Innovation Campus, Fermi Ave, Didcot OX11 0DE, UK
T. Matsukawa  Frontier Research Center for Applied Atomic Sciences, Ibaraki
University, 162-1 Shirakata, Tokai, Ibaraki 319-1106, Japan  A. Hoshikawa
Frontier Research Center for Applied Atomic Sciences, Ibaraki University,
162-1 Shirakata, Tokai, Ibaraki 319-1106, Japan  A. Koda  Organization (KEK),
Tsukuba, Ibaraki 305-0801, Japan  Department of Materials Structure Science,
The Graduate University for Advanced Studies, Tsukuba, Ibaraki 305-0801, Japan
K. Yoshimura  Department of Chemistry, Graduate School of Science, Kyoto
University, Kyoto 606-8502, Japan  I. Umegaki  Muon Science Laboratory,
Institute of Materials Structure Science, KEK, Tokai, Ibaraki 319-1106, Japan
Y. Sassa  Chalmers University of Technology, Department of Physics,
G$\ddot{o}$teborg, SE-412 96, Sweden  F. O. von Rohr  Department of Quantum
Matter Physics, University of Geneva, 24 Quai Ernest-Ansermet, 1211 Geneva 4,
Switzerland  V. Pomjakushin  Laboratory for Neutron Scattering and Imaging,
Paul Scherrer Institute, 5232 Villigen PSI, Switzerland  J. H. Brewer
Department of Physics and Astronomy, University of British Columbia,
Vancouver, British Columbia, Canada V6T 1Z1  TRIUMF, 4004 Wesbrook Mall,
Vancouver, British Columbia, Canada V6T 2A3  J. Sugiyama  Neutron Science and
Technology Center, Comprehensive Research Organization for Science and Society
(CROSS), Tokai, Ibaraki 319-1106, Japan  Advanced Science Research Center,
Japan Atomic Energy Agency, Tokai, Ibaraki 319-1195, Japan  M. Månsson  KTH
Royal Institute of Technology, Department of Applied Physics, Alba Nova
University Center, Stockholm, SE-114 21, Sweden<EMAIL_ADDRESS>
###### Abstract

The recent synthesis of the chromium selenide compound LiCrSe2 constitutes a
valuable addition to the ensemble of two-dimensional triangular lattice
antiferromagnets (2D-TLA). In this work we present the very first
comprehensive study of the combined low temperature nuclear and magnetic
structure established in this material. Details on the connection between Li-
ion dynamics and structural changes are also presented along with a direct
link between atomic structure and spin order via a strong magnetoelastic
coupling. LiCrSe2 was found to undergo a first order structural transition
from a trigonal crystal system with space group $P\bar{3}m1$ to a monoclinic
one with space group $C2/m$ at $T_{\rm s}=30$ K. Such restructuring of the
lattice is accompanied by a magnetic transition at $T_{\rm N}=30$ K, with the
formation of a complex spin arrangement for the Cr3+ moments. Refinement of
the magnetic structure with neutron diffraction data and complementary muon
spin rotation analysis reveal the presence of two incommensurate magnetic
domains with a up-up-down-down arrangement of the spins with ferromagnetic
(FM) double chains coupled antiferromagnetically (AFM). In addition to this
unusual arrangement, the spin axial vector is modulated both in direction and
modulus, resulting in a spin density wave-like order with periodic suppression
of the Cr moment along the chains. This behavior is believed to appear as a
result of strong competition between direct exchange AFM and superexchange FM
couplings established between both nearest neighbor and next nearest neighbor
Cr3+ ions. We finally conjecture that the resulting magnetic order is
stabilized via subtle vacancy/charge order within the Li layers, potentially
causing a mix of two co-existing magnetic phases within the sample.

###### keywords:

triangular lattice antiferromagnets, magnetic frustration, neutron
diffraction, muon spin rotation

## Introduction

In periodic crystalline lattices containing magnetic atoms, long range
ordering of their electronic magnetic moments can be stabilized by the
exchange couplings among them at low temperatures. In some systems however,
due to peculiar structural properties, not all such interactions can be
satisfied simultaneously, leading to unconventional magnetic ground states,
which possess unusual physical properties. One example of such systems is
provided by geometrically frustrated triangular lattice antiferromagnets
(TLA). Their crystal structure consists of two-dimensional triangular lattice
(2D-TL) layers, in which each corner of the triangular lattice is occupied by
a magnetic atom, with nearest neighbor antiferromagnetic (AFM) interactions
that cannot be stabilized [1]. Some prominent examples of 2D-TLA are found
amid transition metal oxides (TMO), such as NaCoO2 [2, 3, 4, 5, 6] or
Li/Na/KCrO2 [7, 8]. Among the currently available chromium based 2D-TLA,
compounds of the type $A$Cr$X_{2}$, where $A$ is a monovalent atom and $X$ is
a chalcogen element, have been extensively investigated as geometrically
frustrated $S=3/2$ Heisenberg spin systems. The interest around these
materials lies in the fact that they manifest an extremely luscious
smörgåsbord of complex magnetic ground states [9, 10, 11, 12, 13, 14, 15, 16]
and attractive technological properties (e.g. multiferroicity) [17, 18, 19].
These systems entail higher energy spin configurations, therefore, in the low
temperature regime, often a compromise between spin and lattice degrees of
freedom has to be reached. The compromise can be realized through a first-
order energetically beneficial distortion of the lattice leading to a new
symmetry of the crystal, in which new interaction paths are realized and a
magnetic order can be stabilized. When such an interplay between strongly
correlated spins and geometrical degrees of freedom is realized, it is said
that the system manifests magnetoelastic coupling. The appeal for materials
undergoing these kind of phenomena is twofold, as they constitute a unique
playground for fundamental scientific research [9, 10, 11, 12, 13, 14, 15, 16,
20], but also lend themselves to potential novel technical applications [21,
22].

The recent synthesis of LiCrSe2, via Li intercalation in CrSe2 by Kobayashi
$et\leavevmode\nobreak\ al.$ [23, 24], constitutes a valuable addition to the
ensemble of TLAs that manifest magnetoelastic coupling since, probably due to
the difficulty in manufacturing alkali chromium selenides, the most of the
studies in this framework are performed on oxides and sulfides. Beyond the
fundamental physics interest surrounding LiCrSe2, recent theoretical works
foresee remarkable electronic properties, such as the presence of tunable
field-free topological spin textures, which make this material appealing for
applications in spintronics circuitry [25, 26, 27]. The room temperature
crystal structure of LiCrSe2 was identified as trigonal by synchrotron X-ray
diffraction, with space group $P\bar{3}m1$. Here, the Cr3+ ions create 2DTL
layers, stacked along the $c$ axis, by the connection of edge-sharing CrSe6
octahedra in the $ab-$plane [28]. Indications of a first order transition
towards a lower symmetry crystal system were observed around $T=30$ K with in-
house X-ray diffraction [24]; here it is suggested a transition towards a
monoclinic structure. From bulk magnetic characterization methods, LiCrSe2 was
found to undergo a first order-like magnetic transition, with an AFM
character, also around $T_{\rm N}=30$ K, which is a hint of magnetoelastic
coupling. This behavior is similar to the one observed in the magnetoelastic
TLA-Cr sulfides AuCrS2 and AgCrS2 [10, 15]. However, remarkable qualitative
differences in the structural distortions and magnetic properties between
LiCrSe2 and the sulfides, indicate that LiCrSe2 would select a different
magnetic ground state with respect to the one realized in AuCrS2 and AgCrS2.
Indeed, the small negative value of the LiCrSe2 Curie-Weiss temperature
$\Theta=-28$ K and the small frustration index $f=\frac{|\Theta|}{T_{\rm
N}}\approx 1$ estimated in reference [24], are suggestive of a weak nearest
neighbor magnetic interaction (weaker than the one found in the sulfides) in
favor of a strong further neighbor magnetic interaction (stronger than the one
found in the sulfides). These results indicate that the further neighbor
interactions have a leading role in the determination of the magnetic ground
state in LiCrSe2, making it a promising candidate for the realization of
unconventional magnetic structures.

In this work we investigated the temperature evolution of the crystal and
magnetic structure of polycrystalline LiCrSe2 by means of neutron diffraction
(ND), synchrotron X-ray diffraction (XRD) and muon spin rotation
($\mu^{+}$SR). Our observations confirm the conjectures of Kobayashi and
coworkers, as a complex magnetic structure seems to be established in the
sample, with two incommensurate magnetic domains with periodic modulation in
direction and modulus of the Cr3+ spin axial vector. The two domains
contribute differently to the neutron diffraction pattern, suggesting that one
is larger than the other, as also indicated by the $\mu^{+}$SR analysis. A
solution for the main and secondary magnetic structures is proposed, together
with a possible interpretation of such a spin ordering in terms of the
magnetic exchange interactions for the main phase.

## Results

The experimental results and data analysis are presented below in the
respective sub-sections. First the low-temperature $\mu^{+}$SR investigation
is presented, followed by the NPD study and complementary synchrotron XRD
measurements as a function of temperature.

Figure 1: (a) Weak transverse field (wTF) $\mu^{+}$SR time spectra at selected
temperatures with best fit to Eq. 1 shown as solid lines. (b) Temperature
dependence of the wTF asymmetry with the continuous red line representing a
fit to a sigmoid function giving the AFM transition temperature $T_{\rm
N}=29.11\pm 0.06$ K. (c,d) Zero-field (ZF) $\mu^{+}$SR time spectrum at $T=2$
K with the fit to Eq. 3 (solid magenta line) shown in (c) long and (d) short
time domain, respectively. (e) Fast Fourier Transform (FFT) of the spectrum
shown in (d). (f) ZF muon time spectra at selected temperatures $T=25$ K, 28.5
K and 29.2 K in the short time domain. The spectra are shifted along the
y-axis for clarity of display. The black continuous lines are fits to Eq. 3.
(g,h) Temperature dependencies of the ZF fit parameters extracted from Eq. 3
for the two oscillating components of the signal plus the slow tail component.

###  $\mu^{+}$SR Results

A zero-field (ZF) and weak transverse field (wTF) bulk $\mu^{+}$SR experiment
has been performed for different temperatures, in order to observe the
evolution of the sample’s internal magnetic field distribution. In the wTF
mode an external magnetic field of weak magnitude is applied (here, wTF = 30
G), with its flux lines perpendicular to the initial direction of the muon
spin polarization. In this set-up it is possible to follow the magnetic phase
transition, while lowering the temperature of the system, as the muon spin
precession (Larmor precession) is more and more affected by the sample’s
strong internal field, induced by long range magnetic ordering, at the expense
of the externally applied weak field (for more details about the $\mu^{+}$SR
experimental technique refer to [29]). The wTF time spectra are well fitted
using the following function:

$\displaystyle A_{0}\,P_{\rm TF}(t)\;=\;A_{\rm TF}\cos(2\pi\nu_{\rm
TF}t\;+\;\frac{\pi\phi}{180})\cdot{}e^{(-\lambda_{\rm TF}t)}\;+\;A_{\rm
tail}\cdot{}e^{(-\lambda_{\rm tail}t)}$ (1)

were $A_{0}$ is the initial asymmetry at time zero, $P_{\rm TF}$(t) is the
muon spin polarization function, $A_{\rm TF}$ is the oscillating asymmetry
while $A_{\rm tail}$ is the asymmetry of a slow relaxing exponential term due
to the components of the internal field which are parallel to the initial
polarization direction of the muons spins. This "tail" component, is also
sometimes called the "powder average" signal when measuring polycrystalline
(powder) samples, which then ideally constitute 1/3 of the total asymmetry.
Further, $\nu_{\rm TF}$ is the muon spin precession frequency having the phase
$\phi$ (which in this case has a value of $\approx$ 12∘ in the entire
temperature range), and finally $\lambda_{\rm TF}$ and $\lambda_{\rm tail}$
are the depolarisation rates for the respective polarization components. wTF
$\mu^{+}$SR time spectra at selected temperatures are shown in Fig. 1(a) with
best fits to Eq. 1 as solid lines. Further fitting of the full wTF temperature
dependence and extracting the wTF asymmetry as a function of temperature
($A_{\rm TF}(T)$) allow us to estimate the magnetic transition temperature of
LiCrSe2. The result is shown in Fig. 1(b) where $A_{\rm TF}(T)$ is fitted to a
sigmoid function giving $T_{\rm N}=29.11\pm 0.06$ K, which is in very good
agreement with previously published bulk characterizations [24]. From the wTF
data it is also possible to deduce that more or less 100% of the sample enters
the magnetically ordered state. Further, and contrary to our previous
$\mu^{+}$SR studies of the related LiCrTe2 compound [30], we do not seem to
have any clear missing asymmetry fraction above $T_{\rm N}$, i.e. absence of
any evident FM impurities in the current LiCrSe2 sample.

In the ZF configuration, the muon spin only senses the local internal magnetic
field of the sample and if long-range magnetic order is established, a clear
precession signal (oscillation) will appear. The ZF $\mu^{+}$SR time spectrum
at base temperature ($T=2$ K) in LiCrSe2 is displayed in Fig. 1(c,d).

At a first look, the ZF signal in the short time domain [Fig. 1(d)] presents
very clear oscillations, indicating the occurrence of AFM spin order. However,
initial fitting attempts, using two simple exponentially relaxing cosine
functions, gave poor results. Several problems were found, e.g., the full
asymmetry could not be recovered, the fit could only work if the phase of the
cosine functions would be left free and acquire very large un-physical values,
and the chosen model could not fully capture the features of the Fast Fourier
transformed (FFT) signal [Fig. 1(e)]. After testing several combinations of
different oscillatory functions, the best fit to the data was achieved with a
combination of two spherical Bessel functions $j_{0}$:

$\displaystyle A_{0}\,P_{\rm ZF}(t)\;=$ $\displaystyle A_{\rm
Bess1}j_{0}(2\pi\nu_{\rm Bess1}t\;+\;\frac{\pi\phi_{\rm
Bess1}}{180})\cdot{}e^{(-\lambda_{\rm Bess1}t)}\;+\;A_{\rm
Bess2}j_{0}(2\pi\nu_{\rm Bess2}t\;+\;\frac{\pi\phi_{\rm
Bess2}}{180})\cdot{}e^{(-\lambda_{\rm Bess2}t)}\;+$ (2)
$\displaystyle+\;A_{\rm tail}\cdot{}e^{(-\lambda_{\rm tail}t)}\;+\;A_{\rm
BG}\cdot{}e^{(-\lambda_{\rm BG}t)}.$ (3)

Here $A_{0}$ is the initial asymmetry, $P_{\rm TF}$(t) is the muon spin
polarization function, $A_{\rm Bess1}$ and $A_{\rm Bess2}$ are the asymmetries
of the oscillating part of the signal, $A_{\rm BG}$ is a small background
component possibly coming from muons stopping in the sample holder, and
$A_{\rm tail}$ is the asymmetry of the tail component. The frequency of the
Larmor precession is given by $2\pi\cdot\nu_{\rm Bess}$, and $\phi$ is the
relative phase of the oscillating signals. Here $\phi\approx 0$ for both the
Bessel functions in the entire temperature range. Finally, $\lambda_{\rm
Bess}$, $\lambda_{\rm tail}$ and $\lambda_{\rm BG}=0$ are the depolarisation
rates for the respective polarization components. Figure 1(d,f) shows the ZF
raw data in the short time domain, for different temperatures, with the
fitting curves as solid lines. As seen from these results, as well as the
accurate fit to the FFT at $T=2$ K in a large frequency range in Fig. 1(e),
Eq. 3 provides a good model to the experimental data.

The temperature dependence of the relevant ZF fitting parameters are shown in
Fig 1(g,h). For both the relaxation rates ($\lambda$) as well as oscillation
frequencies ($\nu_{\rm Bess}$) it is clear that an abrupt, first order like,
transition occurs for $T_{\rm N}\approx 30$ K. In more detail, the
depolarization rate, and especially $\lambda_{\rm Bess2}$ display a very
typical and strong increase just below $T_{\rm N}$, indicating the onset of
spin dynamics/fluctuations as the magnetic phase transition approaches. The
trend for $\lambda_{\rm Bess1}$ is similar, however, the absolute values are
much smaller. Here it is also important to emphasize that $A_{\rm
Bess2}\approx 12\cdot A_{\rm Bess1}$, i.e. have more than an order of
magnitude larger volume fraction. What such differences means is hard to
rigorously deduce using only the $\mu^{+}$SR data. Nevertheless, it indicates
that the two oscillating components originate from muons experiencing two
rather different local magnetic environments (this point will be further
discussed below). Finally, the temperature dependence of the oscillation
frequencies are shown in Fig. 1(h), which reflect the magnetic order parameter
of the phase transition. The behavior of the two components is in fact very
similar, the oscillation frequencies have similar values, and follow the same
temperature dependence. The latter shows an approximately constant trend below
$T_{\rm N}$, with a slow downturn of both $\nu_{\rm Bess1}$ and $\nu_{\rm
Bess1}$. This is then abruptly interrupted at $T_{\rm N}\approx 30$ K which,
we know from previous studies [24], coincides with a structural phase
transition at $T_{\rm s}$.

The presence of large non-zero phases ($\phi$) for cosine functions and
replacement with a damped $j_{0}$ Bessel is commonly used in the $\mu^{+}$SR
community to model incommensurate magnetic structures. This solution has been
well established and proven, both theoretically and experimentally, to well
describe incommensurate spin density waves associated to single $q-$vectors
[31, 32, 33]. The presence of two distinct Bessel functions with two
frequencies of similar value in LiCrSe2 could be due to the presence of two
different muon stopping sites in the lattice, thereby probing the same
incommensurate magnetic structure in different fashion. Another option would
be that we have two slightly different magnetic phases present in the sample.
Additional discussions on this matter will follow below.

Figure 2: 2D color maps showing the temperature evolution of the NPD data.
(a-d) The nuclear Bragg peak splitting in different $Q-$ranges below $T_{\rm
s}\approx 30$ K indicates the structural phase transition from $P\bar{3}m1$ to
$C2/m$. (e,f) Magnetic Bragg peaks, plotted together with some (non splitted)
nuclear peaks, appear below $T_{\rm N}\approx 30$ K. Please note the nonlinear
temperature scale (y-axis), having a higher density of points in proximity of
the transition, and that $T_{\rm s}=T_{\rm N}$. The Miller indices of the
reflections in the high temperature structure are reported in red for clarity
of display.

### XRD and NPD Results: Structural Refinement

The high and low temperature nuclear structure of LiCrSe2, was determined from
high-resolution neutron powder diffraction (NPD) studies [see Fig. 2(a-f) and
Fig. 3(c-f)] along with complementary synchrotron X-ray diffraction (XRD) data
[see Fig. 3(a,b)]. The refinement of the diffraction patterns was carried out
by means of the Rietveld refinement method. A global fit was employed, which
included all the detector banks while keeping the cell parameters, the atomic
positions and the basis vectors of the magnetic moment as common parameters.
The calculated NPD/XRD pattern consists of a majority LiCrSe2 phase plus a
very tiny Li2Se non mangetic impurity phase ($<5\%$, consistent with the
$\mu^{+}$SR data). For the XRD an additional Al phase is included to account
for the sample holder/capillary. Further, for NPD the magnetic phase is
present below $T_{\rm N}$ in the low $Q$-range, while disregarded for large Q
due to the magnetic form factor [34] (by means of the command MDLIM in
FullProf).

Figure 3:  a,b) X-ray diffraction patterns with the corresponding calculated
patterns resulting from structural refinement at room and base temperature.
c-f) Neutron diffraction patterns (detector bank BS) with the corresponding
calculated patterns resulting from structural refinement at room and base
temperature. The peak splitting is clearly visible in the close-up plot in
figure f. g) Graphic representation of the crystal structure in
polycrystalline LiCrSe2 at 300 K and 2 K oriented along the $c$ axis for clear
visualization of the $ab-$plane. h) Temperature dependence of the lattice
parameters in LiCrSe2. The y axis of the lattice parameter $c$ is plotted on
the right for clarity of display. The values of the lattice parameters below
the transition are reported in the hexagonal setting for correct comparison
with the higher temperature values.

The room temperature crystal structure [see Fig. 3(g)] was found to be well
modeled by the trigonal $P\bar{3}m1$ space group (no. 164), with lattice
parameters $a=b=3.65425(3)$ Å, $c=6.28911(4)$ Å, $\alpha=\beta=90^{\circ}$,
$\gamma=120^{\circ}$, in which Cr3+ ions occupy the vertices of the hexagonal
unit cell, hereby creating the 2D-TLA. Below $T\approx 30$ K $=T_{\rm
s}=T_{\rm N}$, the diffraction pattern undergoes a dramatic change, as clearly
displayed in Fig. 2. Several nuclear Bragg peaks split into double peaks [Fig.
2(a-d)] and, at the same time, the magnetic Bragg peaks appear [Fig. 2(e,f)].
First, we focus on the structural change. The increased number of Bragg
reflections in the diffraction pattern, due to the peak splitting, is
suggestive of a symmetry loss in the system, possibly owing to the
disappearance of the 3-fold symmetry axis. Moreover, no clear super-lattice
reflection peaks seem to appear in the diffraction pattern. Under the light of
these observations, and from a group/subgroup relationship argument, the most
probable final space group for such a structural transition would be the
monoclinic $C2/m$ (no. 12). This is because $C2/m$ is the only monoclinic
maximal subgroup of the parent space group $P\bar{3}m1$ with a single step in
its symmetry reduction path (see Bärnighausen tree for $P\bar{3}m1$ from the
Bilbao Crystallographic Server [35]). Indeed, Rietveld refinement of the low
temperature diffraction pattern with $C2/m$ provided a good agreement with the
experimental data, as it was able to correctly index all the additional new
peaks [Fig. 3(b,d,f)]. Nevertheless, in the light of the first-order nature of
the structural transition, the possibility that the group subgroup
relationship might not hold in this particular case must be considered.
Several other monoclinic space groups were tested with a trial an error
approach; among them $Cm$ (no. 8) provided the best matching between the
calculated and the observed pattern, which was however comparable to $C2/m$.
The fact that the loss of the 2-fold symmetry axis from $C2/m$ to $Cm$ did not
improve the refinement, made $C2/m$ a more reasonable choice. This space group
was also suggested from previous in-house XRD measurements [24], and we have
now confirmed such suggestion with high-resolution experimental methods. The
base temperature ($T=2$ K) crystal structure was therefore modeled with the
monoclinic $C2/m$ space group, with lattice parameters $a$ = 6.32904(8) Å, $b$
= 3.64287(5) Å, $c$ = 6.25345(7) Å, $\alpha$ = $\gamma$ = 90∘, and $\beta$ =
90.08(13)∘. Here the Cr3+ ions occupy the vertices of a unit cell, which is
nearly rectangular in the $ab-$plane, with one additional Cr atom in the
central point (0.5, 0.5, z) [see Fig. 3(g)]. The detailed result of the
refinement is displayed in Fig. 3(e,f), with a plot of the diffraction
patterns (observed and calculated) at high and low temperature, in the high
resolution BS detector bank. It is important to emphasize that in this bank
(and $Q-$range) any potential low-temperature magnetic peak is virtually
invisible, and does not interfere with the structural refinement. The
complementary XRD patterns are also shown [Fig. 3(a-b)], and the structural
refinement for XRD/NPD measurements is fully consistent, further allowing us
to effectively discern between nuclear and magnetic contributions. It should
also be noted that refinement of fractional occupancies did not improve the
goodness of the fit, which allow us to conclude that the LiCrSe2 sample has a
stoichiometric composition very close to the ideal one.

The temperature evolution of the lattice parameters, resulting from the
structural refinement, is displayed in Fig. 3(h). Table 1 further summarizes
the refined structural parameter along with the Bragg agreement $R-$factors,
whose low values highlight the goodness of the refinement model. Since the Cr
atoms sit at the vertices of the unit cell, the evolution of the lattice
parameters reflects directly the evolution of the in-plane Cr-Cr distances.
This is a factor of major importance in the determination of the magnetic
structure for 2D-TLAs [36]. On cooling, the size of the crystallographic axes
is reduced, due to thermal contraction, down to the magnetic transition
temperature $T_{\rm N}$. As seen in Fig. 3(h), the contraction of the $c-$axis
occurs at a much higher rate with respect to the $a-$ and $b-$axes, which is
phenomenologically compatible with the 2D-TLA nature of the compound. At
$T_{\rm N}=T_{\rm s}$ the first-order structural transition is clearly visible
as a sharp change in the temperature dependence of the axes, with the major
effect observed in the $ab-$plane. The synchronized nuclear and magnetic phase
transition appearing at $T=30$ K indicated a strong spin-lattice coupling.
Here, the ordering of the Cr magnetic moments seems to be the driving
mechanism of a modification of the crystal lattice, aimed at stabilizing the
competing coupling mechanisms simultaneously established among them. To
further understand such synergic effect, a detailed understanding of the spin
structure is required. A dedicated discussion around the structural evolution
of the system in concomitance with the evolution of the magnetic structure
will be presented in the next subsection.

Table 1: Lattice parameters, atomic positions, isotropic thermal displacement parameters and Bragg $R-$factors from refinement of neutron powder diffraction data on polycrystalline LiCrSe2 | 300 K | 2 K
---|---|---
Space Group | $P\bar{3}m1$ | $C2/m$
$a$ Å | 3.65425(3) | 6.32904(8)
$b$ Å | 3.65425(3) | 3.64287(5)
$c$ Å | 6.28911(4) | 6.25345(7)
$\alpha$ |  90∘ |  90∘
$\beta$ |  90∘ |  90.08(13)∘
$\gamma$ |  120∘ |  90∘
Li (x, y, z) |  (0, 0, $\frac{1}{2}$) |  (0, 0, $\frac{1}{2}$)
Cr (x, y, z) | (0, 0, 0) | (0, 0, 0)
Se (x, y, z) |  ($\frac{1}{3}$, $\frac{2}{3}$, 0.226(1)) | (0.667(3), 0, 0.773(1))
BLi (Å2) | 1.98(9) | 0.85(7)
BCr (Å2) | 0.63(5) | 0.23(3)
BSe (Å2) | 0.39(1) | 0.018(7)
$R_{B(nuc)}$ (%) | 4.2 | 4.5

### NPD Results: Magnetic Refinement

Several temperature dependent magnetic Bragg peaks appear below 30 K, as
clearly seen in Fig. 2(e,f). These peaks are assumed to be magnetic. Such
transition is presented in higher detail in Fig. 4(a,b), which displays a
comparison between NPD patterns (SE $\rightarrow$ 90∘ and LA35 $\rightarrow$
135∘ detector banks) acquired at $T=35$ K (just above $T_{\rm N}$) and at base
temperature ($T=2$ K). The square root of the integrated intensity of the
large magnetic peak in the lower $Q-$region ($Q=0.66$ Å-1) is also plotted as
a function of temperature [see Fig. 4(c)]. The latter quantity reflects the
order parameter of the magnetic transition which, as clearly seen from the
plot in Fig. 4(c), has a first-order character. The fit of the plot to a
sigmoid function [solid line in Fig. 4(c)], well follows the sharp profile of
the data and provides the magnetic transition temperature $T_{\rm N}=29.94\pm
0.09$ K (defined as the midpoint of the sigmoid curve). The inset of Fig. 4(c)
shows the normalized integrated intensities of the large magnetic peak
together with the two neighboring small satellites, which all clearly manifest
the same temperature dependent behavior. Nevertheless, as discussed further
below, these peaks do not belong to the same magnetic phase. Here a
distinction is necessary between a $main$ and a $secondary$ magnetic phase. In
the immediate following, we will describe the main magnetic phase.

Figure 4: Neutron diffraction patterns (raw data) at $T=35$ K and $T=2$ K
acquired using the (a) LA35 and (b) SE detector banks, respectively. The
appearance of magnetic Bragg peaks below $T_{\rm N}\approx 30$ K is evidenced
by the magenta arrows (orange arrow indicates the strongest among the peaks
un-indexed by the main magnetic phase, see text). (c) Square root of the
integrated intensity of the strongest magnetic peak at $Q=0.66$ Å-1, visible
in the LA35 Bank. The continuous line is a fit to a sigmoid function for
precise determination of the magnetic transition temperature. Inset: plot of
the normalized integrated intensities of the three lower$-Q$ magnetic peaks.

The magnetic propagation vector $q_{\rm main}$ was determined by a very
extensive trial and error procedure with the software K-Search. As expected
from our $\mu^{+}$SR data, it was found to be incommensurate: $q_{\rm main}$ =
(0.045, $\approx\frac{1}{4}$, $\approx\frac{1}{2}$). Such propagation vector
can successfully index the high intensity magnetic peak and most of the lower
intensity ones as well. The remaining weak peaks were still un-indexed [e.g.
at $Q=0.74$ Å-1, orange arrow in Fig. 4(a)], this issue will be addressed in
the next section.

Figure 5: Main Magnetic Phase: Overview and close-up neutron diffraction
pattern with the corresponding calculated pattern resulting from magnetic
refinement at $T=2$ K using the (a,b) LA35 and (c,d) SE, detector banks,
respectively. (e) Value of the spin only Cr moment and (f) the components of
the axial spin vector C1 (left y-axis) and C2 (right y-axis). Both plotted
along the $a-$axis as a function of the $x_{\rm Cr}$ atomic coordinate. (g)
Graphic representation of the main magnetic structure in polycrystalline
LiCrSe2 oriented along the main axes of the unit cell (4 cells in the
$ac-$plane, 12 cells in the $ab-$plane, and finally 22 cells along the
$a-$axis at the very bottom). The up-up-down-down arrangement of the spins
along the $b-$axis is highlighted by the dashed magenta ellipses. See also
Table 2 for refined parameters.

The LiCrSe2 magnetic ($q_{\rm main}$) phase, refined by the Rietveld Method,
includes a single magnetic Cr atom in its crystallographic site Cr(0 0 0). The
scale factor and structural parameters were constrained to be equal to their
counterparts in the nuclear LiCrSe2 phase, for correct estimation of the Cr
magnetic moment $\mu_{\rm Cr}$. The software BasIreps provided only one
possible irreducible representation of the propagation vector group $G_{k}$
for $q_{\rm main}$, compatible with the space group $C2/m$ (IRrep
$\Gamma_{1}$), consisting of three basis vectors BsV, with no imaginary
components, parallel to the main crystallographic axes:

  * •

BsV(1): Re (1 0 0),

  * •

BsV(2): Re (0 1 0),

  * •

BsV(3): Re (0 0 1).

The refinement of the coefficients of the basis vectors C1, C2 and C3 provided
a large component in the $a$ direction for the spin axial vector and a
negligibly small component along the $b$ direction [Fig. 5(a-g)].

Using IRrep $\Gamma_{1}$ in the magnetic structure refinement [shown in Fig.
5(a-d)], it was possible to well capture the temperature dependent magnetic
peaks of the diffraction pattern indexed by the k-vector $q_{\rm main}$. The
resulting magnetic structure presents a very large magnetic unit cell, with
AFM coupled layers stacked along the $c-$axis with a peculiar "breathing"
arrangement of the moments within the $ab-$plane [Fig. 5(e-g)]. The intra-
layer orientation of the Cr moments along the $b-$axis is reminiscent of an
alternate up-up-down-down sequence [see dashed magenta ellipses in Fig. 5(g)],
built up by double FM-coupled Cr spin chains running along the $a-$axis. This
peculiar arrangement of the Cr moments is very similar to what was found for
AuCrS2 [10]. However, in this case the orientation of the spins in the FM-
coupled chains undergoes a change of direction, accompanied by a size
modulation of the magnetic moment while moving along the $a-$axis [see plot
(e) and bottom of panel (g) in Fig. 5]. As a result, a periodic suppression of
the magnetic moment appears along the chain. The modulation interval between
two consecutive spin flips along the $a-$axis is 11 unit cells. Figure 5(e,f)
displays the calculated value of the Cr moment $\mu_{\rm Cr}$ as a function of
the atomic coordinate $x_{\rm Cr}$. This is obtained as the modulus of the
refined spin axial vector expressed in units of Bohr magneton. The periodic
"breathing" feature of the magnetic moment modulation is clearly visible. The
highest value for $\mu_{Cr}$ in the chain is 2.25(3)$\mu_{B}$, lower than the
value previously estimated from bulk magnetic measurements $\approx$ 4
$\mu_{B}$ [24], which is closer to the ideal spin-only value $\mu_{3/2}$ =
2$\sqrt{S(S+1)}$ = 3.87 $\mu_{B}$ for an isolated Cr3+ ion in the spin state
$S=3/2$. However, in a covalently bonded system, a reduction in the value of
the magnetic moment of a transition metal element with respect to the ideal
value is expected, due to the competition between covalency and magnetism
[37]. Further, when geometric frustration is present, the appearance of
staggered magnetic moments and a following reduction of $\mu_{\rm Cr}$ is very
common [7, 15, 10, 38, 39, 40]. Therefore, in agreement with systems analogous
to LiCrSe2 [10], the value $\mu_{\rm Cr}$ = 2.25(3)$\mu_{B}$ estimated by NPD
in the current work can be considered reliable, and consistent with a $S$ =
3/2 spin system scenario. The resulting parameters obtained from the
refinement of the magnetic unit cell for the LiCrSe2 magnetic phase are
summarized in Table 2, together with their $R_{Bragg}$ agreement factors. The
observed profiles are in very good agreement with the calculated models, as
clearly seen in Fig. 5(a-d). The goodness of the model is underlined by the
value of the reliability $R-$factor for the magnetic phase, not exceeding a
few percent. All the values reported in Tables 1 and 2 are obtained from the
global procedure involving the simultaneous Rietveld refinement of diffraction
patterns from the 3 detector banks BS, SE and LA35.

Table 2: Main Magnetic Phase: Magnetic refinement parameters and agreement factors for the neutron powder diffraction data from LiCrSe2 at $T=2$ K Space Group | $P1$
---|---
am (Å) | 75.9486(1)
bm (Å) | 14.57148(2)
cm (Å) | 12.50691(2)
$\alpha_{m}$ |  90∘
$\beta_{m}$ |  90.0762(13)∘
$\gamma_{m}$ |  90∘
$R_{B(mag)}$ (%) | 2.91
$\mu_{Cr}$ ($\mu_{B}$) | 2.25(3)

## Discussion

The resulting incommensurate magnetic order, with itinerant frustration of the
Cr moment [41, 42], found in LiCrSe2 is indeed rather peculiar. While the
inter-planar magnetic ordering along the $c-$axis can be explained with an AFM
coupling mediated by super-exchange interactions through the path Cr-Se-Se-Cr,
to explain the complex intra-planar magnetic structure the coexistence of
competing magnetic interactions must be considered [Fig. 6]. In ternary
chromium chalcogenide compounds, the magnetic interactions between two
neighboring Cr3+ ions can be of two types, as for the Goodenough-Kanamori-
Anderson rules [43, 44, 45]. Either a FM super-exchange interaction is
established, by hybridization of the $p$ orbitals of an anion with the $d$
orbitals 111Specifically with the half-filled $t_{2g}$ orbital of one Cr ion
and the empty $e_{g}$ orbital of the other one of the Cr ions forming a 90∘
bond of the type Cr-Anion-Cr; or an AFM direct exchange interaction is
established between the half-filled $t_{2g}$ orbitals of two adjacent Cr3+
ions. In the FM coupling, the Cr-Cr distance $d(Cr-Cr)$ is unimportant as long
as the angle in the bond Cr-Anion-Cr is close to 90∘. On the other hand, for
the AFM coupling, $d(Cr-Cr)$ has a major relevance. Rosenberg and coworkers
identified a cutoff nearest neighbor (NN) Cr-Cr distance of $\approx$ 3.6 Å in
the Cr layers of ternary chromium chalcogenides, for one mechanism to be
dominant over the other [36]. In particular, for $d(Cr-Cr)$ $<$ 3.6 Å the AFM
interaction would be preferred, while for $d(Cr-Cr)$ $>$ 3.6 Å the FM
interaction would be preferred. In LiCrSe2 a dramatic structural change is
observed, in concomitance to the magnetic transition at $T_{\rm N}$, leading
to a modification of the Cr-Cr distances and of the angles $\widehat{Cr-Se-
Cr}$, which are summarized in Table 3.

Figure 6: Pictorial representation of the magnetic couplings established between Cr3+ atoms in the $ab-$plane of LiCrSe2. The FM interactions $J_{1}$ and $J_{2}$ are displayed in orange while the AFM interactions $J^{\prime}_{1}$ and $J^{\prime}_{2}$ are displayed in purple. The Cr-Cr distances and angles established from the present diffraction studies are also shown (for the $C2/m$ space group at $T=2$ K, see also Table 3). Table 3:  NN and NNN Cr-Cr distances and Cr-Se-Cr bonding angles just above and below the magnetic transition in LiCrSe2 | 35 K | 2 K
---|---|---
Space Group | $P\bar{3}m1$ | $C2/m$
NN-$d_{1}(Cr-Cr)_{a-b}$ (Å) | 3.65059(1) | 3.64287(6)
NN-$d_{2}(Cr-Cr)_{a-b}$ (Å) | - | 3.65128(4)
NNN-$d_{1}(Cr-Cr)_{a-b}$ (Å) | 6.32301(1) | 6.31449(7)
NNN-$d_{2}(Cr-Cr)_{a-b}$ (Å) | - | 6.32905(9)
$d(Cr-Cr)_{c}$ (Å) | 6.25113(1) | 6.25345(8)
$\widehat{Cr-Se-Cr}_{1}$ |  91.8863(1)∘ |  91.66(5)∘
$\widehat{Cr-Se-Cr}_{2}$ |  91.8863(1)∘ |  91.92(3)∘

In the trigonal phase just above the magnetic transition, the distance
NN-$d(Cr-Cr)_{a-b}$ is very close to the critical value 3.6 Å and the angle
$\widehat{Cr-Se-Cr}$ is very close to 90∘. In these conditions the AFM and FM
couplings are equally likely to be established between NN Cr3+ ions. As a
consequence, the system is subjected to a geometrical frustration, which is
removed through the energetically beneficial structural transition. This
results in two non-equivalent NN and two non-equivalent next nearest neighbor
(NNN) Cr3+ distances, $d_{1}(Cr-Cr)_{a-b}$ and $d_{2}(Cr-Cr)_{a-b}$. In the
low temperature monoclinic phase small distortions of the lattice occur,
indeed both the NN distances are still very close to the critical value 3.6 Å
(see Table 3), and the angles $\widehat{Cr-Se-Cr}$ are still very close to
90∘. Nevertheless, as usually happens in frustration driven structural
transitions, this distortion is enough to lift the degeneracy of the magnetic
ground state. Here, a strong competition between AFM and FM interactions is
established, which leads to an overall weakening of the exchange mechanism
between NN Cr3+ ions. At this point, the NNN Cr3+ interactions must come into
play [see Fig. 6] to stabilize the magnetic structure. In particular, given
the results of our magnetic refinement [Fig. 5], we propose that a FM super-
exchange interaction $J_{2}$ is always established through the Cr-Se-Se-Cr
path between the NNN Cr3+ atoms connected by the shorter distance
NNN-$d_{1}(Cr-Cr)_{a-b}$. Further, the AFM super-exchange interaction
$J^{\prime}_{2}$ is always established through the Cr-Se-Se-Cr path between
the NNN Cr3+ atoms connected by the longer distance NNN-$d_{2}(Cr-Cr)_{a-b}$.
Concerning the nearest neighbor interactions, a FM super-exchange interaction
$J_{1}$ through the bond Cr-Se-Cr and an AFM direct exchange interaction
$J^{\prime}_{1}$ are established for both the NN-$d(Cr-Cr)_{a-b}$ distances,
in an alternate arrangements between adjacent cells. This is not surprising
since the geometric conditions for the occurrence of both couplings are
fulfilled on both Cr-Cr paths, and there is no reason for the predominance of
one mechanism over the other. As a result, these magnetic couplings,
exemplified in Fig. 6, are equiprobable on the same crystallographic sites. In
addition, exchange interaction established among further than NNN Cr ions
might be relevant in the determination of the ground state of LiCrSe2. Indeed,
it was found that in delafossite structures with S as chalcogenide ligand
(e.g., AuCrS2 and AgCrS2), the NNN exchange interaction $J_{2}$ is typically
comparable with the 3rd neighbor AFM exchange interaction $J_{3}$ [46]. Such
mechanism of super-super-exchange, realized via two $p$-orbitals of the S
ligands, seems to drive the magnetic transition and determine the magnetic
ground state in these materials [46]. The third neighbor interaction was
recently found to be relevant also for the determination of the magnetic
ground state in the triangular Cr-oxide PdCrO2 [47]. The $p$-orbitals of the
Se element have a a larger spatial extension with respect to the ones in S and
O. Therefore, in qualitative agreement with the aforementioned systems [46,
47], $J_{3}$ could be expected to be even larger than $J_{2}$ in LiCrSe2.
However, first principle calculations would be needed to estimate the
relevance of $J_{3}$ in the determination of the magnetic order on this
material.

Figure 7: Secondary Magnetic Phase: Neutron diffraction pattern with the
corresponding calculated pattern resulting from magnetic refinement at $T=2$ K
for the two detector banks (a) LA35 and (b) SE, respectively. Both the main
(magenta) and secondary (organge) magnetic phases are characterized by
vertical tick bars of different color. (c) Pictorial representation of the
spin structure for the secondary magnetic domain. See also Table 4 for refined
parameters.

The magnetic structure induced by such competing interactions presents
ferromagnetic double chains running along the $a-$axis, which are AFM coupled
along the $b-$axis, similarly to AuCrS2 and AgCrS2 [10, 15]. The only
difference is that for LiCrSe2 the orientation of the Cr moment along the
chains overturns periodically with a consequent modulation of its modulus. The
alternating diagonal AFM-FM interactions between NN Cr3+ atoms are probably
responsible for this peculiar behavior. The proposed magnetic structure is
consistent with the observation that magnetic correlations in compounds
similar to LiCrSe2 display significant further neighbor interactions involving
Cr-Cr distances of 6–7 Å [10, 15]. Additionally, our results are in very good
agreement with the findings of Kobayashi $et\leavevmode\nobreak\ al.$ [24],
who observed an AFM character of the signal in magnetic susceptibility
measurements. This suggested that further neighbor exchange interactions in
LiCrSe2 are in fact stronger than the nearest neighbor ones and of utmost
importance in the determination of the magnetic ground state of LiCrSe2. In
reference [24] it was also hypothesized that LiCrSe2 is likely to possess a
magnetic ground state different from other triangular lattice Cr chalcogenides
with a similar structure and behavior [10, 15, 9]. The results of our work
confirm this conjecture and provide a qualitative explanation for the
difference between LiCrSe2 and the other systems: in AuCrS2 and AgCrS2 and
CuCrS2 the in-plane Cr-Cr distances are always 3-4$\%$ different from the
critical value found by Rosenberg. On the other hand, for LiCrSe2 the in-plane
Cr-Cr distances are just $\approx$ 1$\%$ different from the critical value 3.6
Å. The Cr-Anion-Cr angles are very close to 90∘ in all cases. Therefore, in
LiCrSe2 distinctive geometric conditions are realized, which lead to a
configuration of the overlapping electronic orbitals that allow the
conflictual coexistence of FM and AFM exchange interactions over the same
crystallographic sites, making this material unique in the 2D-TLA family.
Future modeling efforts would be very beneficial to systematically understand
the complex spin interactions and order in LiCrSe2. The detailed information
presented in this work, concerning the atomic/spin arrangement across the
transition, will most likely be a valuable input for such modeling efforts.

Going back to the un-indexed weak Bragg peaks found in the low-temperature
$C2/m$ phase, it is important to emphasize that we clearly do not believe that
these peaks are from an impurity phase. This is because there is no trace of
magnetic impurities in the room temperature diffraction patterns, there is no
sign of phase separation in the wTF $\mu^{+}$SR data, and the additional peaks
appear exactly at the same structural/magnetic transition temperature of the
LiCrSe2 main phase ($T_{\rm s}=T_{\rm N}=30$ K). In our refinement efforts, a
huge number of different magnetic propagation vectors in combination with
different crystal structures and super-cells, including higher-order harmonics
of the parent cell, were attempted for the purpose of simultaneous indexing of
all the peaks in the diffraction pattern. However, these efforts did not
produce any outcome and, even if the $C2/m$ nuclear structure combined with
magnetic propagation vector $q_{\rm main}$ gives a very nice result, a few
weak Bragg peaks were still left un-indexed [e.g. $Q=0.74$ Å-1 being the
strongest un-indexed peak, see Fig. 4(a) and Fig. 5(b)]. Considering the
coinciding structural and magnetic phase transition occuring at $T\approx 30$
K, it is indeed challenging to identify whether new Bragg peaks appearing at
low temperature have nuclear or magnetic origin. In this work we therefore
combined NPD with synchrotron XRD experiments in order to improve the
identification process. Indeed, the low-temperature peak at $Q=0.74$ Å-1 is
not visible in the $T=11$ K XRD data and would then be assumed to have
magnetic origin (but we will get back to this issue). Considering that the ZF
$\mu^{+}$SR results indicated the presence of two slightly different magnetic
phases we decided to try to index the few weak remaining low-temperature peaks
with a secondary magnetic phase. Considering the fact that the ZF asymmetry of
the the first Bessel component, $A_{\rm Bess1}$, was so much smaller than
$A_{\rm Bess2}$, it seems reasonable to assume that the weak un-indexed
magnetic Bragg peaks could be associated with $A_{\rm Bess1}$.

To deduce the secondary magnetic phase (consistent with the presence of a
magnetic supercell) we utilized a similar procedure as for the identification
of the main magnetic structure. As a result we find an incommensurate
propagation vector $q_{\rm secondary}$ = (0.7249(5), -0.0885(3), -0.1292(9))
and only one possible irreducible representation of the propagation vector
group $G_{k}$ for $q_{\rm secondary}$, compatible with the space group $C2/m$.
Again, this IRrep consists of three basis vectors, with no imaginary
components, parallel to the main crystallographic axes, BsV(1): Re (1 0 0),
BsV(2): Re (0 1 0), BsV(3): Re (0 0 1) with coefficients (0.45(11), 0,
1.26(5)) and a canting angle of $\sim 20^{\circ}$ from the $c-$axis for the
spin axial vector. The resulting secondary magnetic structure [refinement in
Fig. 7(a,b)] appears similar yet complementary to the main structure; here the
canted modulated FM chains run along the $c-$axis while the up-up-down-down
arrangement of the Cr spins develops along the $a-$axis [Fig. 7(c)]. The
refined model inclusive of this additional magnetic phase is able to well
capture most of the smaller magnetic satellites in the experimental data
[orange tick-marks in Fig. 7(a,b)]. The ordered Cr moment of the secondary
domain is refined as $\mu_{Cr}=1.33(2)\mu_{B}$. The resulting parameters
obtained from refinement of the secondary magnetic unit cell are summarized in
Table 4.

Figure 8: Temperature dependence of (a) $B_{\rm iso}$ parameters from NPD data, and (b) Li-ion hopping rate ($\nu_{\rm Li}$) from $\mu^{+}$SR measurements. Inset in (b) show the layered structure where Li-ions diffuse (hop) from site to site above $T_{\rm D}\approx 180$ K. $T_{\rm si}\approx 45$ K indicate the onset of structural instability. Solid and dashed lines are guides to the eye.  Table 4: Secondary Magnetic Phase: Magnetic refinement parameters and agreement factors for the neutron powder diffraction data from LiCrSe2 at $T=2$ K Space Group | $P1$
---|---
am (Å) | 6.32905(1)
bm (Å) | 40.0716(2)
cm (Å) | 50.0276(2)
$\alpha_{m}$ |  90∘
$\beta_{m}$ |  90.0762(13)∘
$\gamma_{m}$ |  90∘
$R_{B(mag)}$ (%) | 25.7
$\mu_{Cr}$ ($\mu_{B}$) | 1.33(2)

Comparing the magnetic peaks visible in the LA35 bank [see Fig. 7(a)], the
magnitude of the integrated intensity of the largest peak belonging to the
main magnetic phase (corresponding to the (0 0 1) and (0 0 0) reflections) is
a factor of $\sim 8$ larger than the integrated intensity of the largest peak
belonging to the secondary phase (corresponding to the (0 0 0) reflection).
This ratio is roughly consistent with the calculated values of the ordered
moments in the two phases, since the integrated intensity of the magnetic
Bragg peak is proportional to $\mu_{Cr}^{2}$. The presence of two distinct
incommensurate magnetic phases is also consistent with the results of the ZF
$\mu^{+}$SR analysis. Here it was found that the volume fractions of the two
incommensurate components were identified as $A_{\rm Bess2}\approx 12\cdot
A_{\rm Bess1}$, which is compatible with the refinement of the main:secondary
phases. Moreover, the clear canting angle found in the secondary magnetic
phase is suggestive of the presence of asymmetric magnetic coupling, possibly
due to the presence of Dzyaloshinskii-Moriya interaction (DMI). This would be
consistent with a recent theoretical work [25], which indeed predicts the
occurrence of large DMI. Nevertheless, the rather large value of the
reliability $R-$factor for the secondary magnetic phase ($R_{\rm B(mag)}^{\rm
secondary}=25.7\%$) makes this interpretation a bit uncertain. Finally, we
speculate about the electronic origins of the magnetic peaks and the very
unusual spin arrangement realized in LiCrSe2. The spin density modulation
along the FM chains and the incommensurate positions of the magnetic peaks
might be originated by a combination of localized and itinerant moments, with
a phenomenology similar to both the NaxCoO2 compound [48] (see also below) as
well as the intermetallic material ErPd2Si2. The latter in fact also manifests
two distinct incommensurate spin states [49]. Looking at the size of the
magnetic peaks belonging to the two structures, it can be argued that the
moments for $q_{\rm main}$ have mostly a localized character, while $q_{\rm
secondary}$ are mainly itinerant. Although being consistent with the
experimental observations, single crystal magnetic field dependent studies
would be necessary to confirm or deny such hypothesis.

Another option is that the small un-indexed low-temperature Bragg peaks are in
fact not magnetic but instead have nuclear origin related to charge ordering.
Indeed, we could consider the appearance of a superstructure induced by e.g.
Li(-vacancy) ordering. In principle such superstructure peaks should be
visible also in the XRD, however, if they are indeed originating from the
light Li atoms (low $Z$) the peaks might be difficult to discern (or even
invisible) for X-rays. If we look more closely into the structural refinement
from NPD, and specifically on the isotropic thermal displacement parameters
$B_{\rm iso}$ shown in Fig. 8(a) there are some indications. First of all, the
structural transition at $T_{\rm s}=30$ K is clearly visible. It is also clear
that $B_{\rm iso}$ for all atoms actually deviate from the general trend
already above the transition, indicating the onset of a structural instability
(si) at around $T_{\rm si}\approx 45$ K. This would actually explain the broad
transition found in the wTF $\mu^{+}$SR data [Fig. 1(b)], just above the first
order downturn. It is also clear that the Li atoms undergo the highest thermal
displacement (cf. Cr and Se). At high temperature $B_{\rm Li}$ display an
almost linear temperature dependence. However, below approximately 180 K a
clear deviation appears. If we consider the similarity of LiCrSe2 with several
other well-known layered TMO compounds (e.g. LiCoO2 [50]), and their
utilization in rechargeable batteries [51], we should consider the possibility
of, and effects from, Li-ion dynamics. In fact our team has developed and
pioneered an experimental method to investigate ion dynamics by $\mu^{+}$SR. A
detailed description of the method and selected examples can be found in Refs.
[52, 53, 54]. Briefly, the muon spin is so sensitive to internal fields that
it is capable of even probing static and dynamic changes in the nuclear
moment. When implanted in these compounds in their higher temperature
paramagnetic (PM) phase, the muon will mainly ’feel’ the nuclear moment of the
alkali metal (here Li), measured in both ZF and applied magnetic field
parallel to the initial direction of the muon spin polarization
(LF-$\mu+^{+}$SR). If the Li-ion is static, only a static nuclear field
distribution ($\Delta$) is probed. However, if Li-ion dynamics sets in, an
additional field fluctuation rate ($\nu$) component will be present. In the
simplest interpretation, $\nu$ would directly translate into a Li-ion hopping
rate that can be directly probed as a function of, e.g. temperature. In fact
our team recently conducted initial test measurements (still to be completed)
of Li-ion dynamics in LiCrSe2 and $\nu(T)$ was successfully extracted at a few
selected temperatures. As shown in Fig. 8(b), $\nu(T)$ show a clear
exponential increase starting around $T_{\rm D}\approx 180$ K, typical for a
thermally activated diffusion process. This means that the Li-ions are fully
mobile at least down to $T_{\rm D}\approx 180$ K, which would indicate the
possibility for Li order to appear during cool-down of the sample. This would
clearly also explain the anomaly found for $B_{\rm Li}$ at $T_{\rm D}$. In
fact a smaller kink is also visible in $B_{\rm Se}$, which is reasonable
considering that the Se sits next to the Li layers in the LiCrSe2 structure
[see inset of Fig. 8(b)]. Interestingly, this behavior is different with
respect to the one observed in the directly related compound LiCrO2, which was
proven to be electrochemically inactive up to 480 K, despite its striking
structural similarity with LiCrSe2. Therefore, the diffusion properties of
this material could provide a hint on how to increment the Li-ion mobility in
layered structures.

In fact, the dynamic phenomenology manifested by LiCrSe2 is reminiscent of the
behavior of the rather well-known and strongly related NaxCoO2 (NCO) compound
[2, 3, 4, 5, 6]. From very extensive studies of NCO it has been shown that Na-
ions (vacancies) can order in many different types of configurations [55, 56,
57]. It was also clearly shown that the Na-ion are in fact mobile even below
room temperature and that the dynamics allow the Na-ions to arrange themselves
in different patterns at lower temperatures [58, 59, 5]. Even in the case of
NCO with the heavier Na-ions, it was not straightforward to identify the
superstructure ordering. Several of the successful studies were performed
using neutron scattering and, in more or less all cases, the superstructures
could only be identified using single crystalline samples rather than powder.
It was also in single crystals where the most recent study of the in-plane as
well as out-of-plane Na ordering was studied, this time using XRD [60]. In the
case of LiCrSe2 it is therefore not improbable that the small satellites found
in NPD could be nuclear superstructures and still not visible in the XRD
pattern.

The comparison with NCO actually opens up an even more interesting
possibility. Considering that both LiCrSe2 and NCO have their magnetic ions
sitting on the triangular (frustrated) lattice with competing magnetic
interactions, the spins are highly susceptible to any additional perturbation.
In the case of NCO it was clearly shown that the Na-ion ordering actually
creates a periodic coulomb potential (charge) landscape that is able to tune
the electronic states and magnetic properties within the TMO layers [48]. It
has even been proven that, by controlling the cooling rate of the sample,
thereby gaining control of dynamics and vacancy order, different magnetic
ground states could be achieved for the NCO compound [4, 60]. Further, by
electrochemical cycling of NCO inside a battery cell (i.e. changing the Na
content $x$ for NaxCoO2), very detailed studies could be performed on the
nuclear structure [61] as well as how this strongly affects the magnetic phase
diagram [62, 6]. The conditions for such phenomenology could in fact be
verified also for LiCrSe2. We could imagine that if Li-ions organize
themselves in an ordered pattern, their periodic potential could affect the
spin order within the triangular lattice Cr planes. Such effect could be the
origin for the peculiar main magnetic phase ($q_{\rm main}$) firmly
established in LiCrSe2, as well as for the potential secondary magnetic domain
($q_{\rm secondary}$). Nevertheless, NCO and LiCrSe2 present some differences,
in NCO there are two Na sites and neither is fully occupied, while in LiCrSe2
there is only one Li site. Additionally, we found in our refinement that all
the atomic sites in our sample are fully occupied (i.e. no vacancies). On the
other hand, the refined magnetic (main) phase has a very long periodicity
(large unit cell), which would fit well with a Li-vacancy order based on very
few existing vacancies that we did not discern in our refinement. Finally, it
is of course possible to have Li order without vacancies, e.g. in the form of
a charge-density-wave (CDW). Future more detailed studies of Li-ion ordering
and dynamics in LiCrSe2 coupled to the magnetic properties would clearly be
very interesting.

In summary, we present a few possible scenarios attempting explain the un-
indexed Bragg peaks found in our NPD data. To ultimately discern if the
additional peaks are of nuclear or magnetic origin, single crystal polarized
neutron diffraction would be the optimal approach. However, to the best of our
knowledge no large enough LiCrSe2 single crystals are presently available.

## Conclusions

The magnetic and crystal structure of the triangular antiferromanget LiCrSe2
has been solved at high and low temperature by means of neutron diffraction,
synchrotron X-ray diffraction and muon spin rotation experimental techniques.
The unique arrangement of the Cr atoms in this material allows the
simultaneous presence of competing nearest neighbor ferromagnetic and
antiferromagnetic exchange couplings over the same crystallographic sites,
leading to a predominance of the next nearest neighbor interactions (which are
also both FM and AFM) in the determination of the magnetic ground state in
LiCrSe2. As a result, a structural transition from trigonal $P\bar{3}m1$ to
monoclinic $C2/m$ crystal system induced by strong magnetoelastic coupling
occurs, accompanied by the formation of two up-up-down-down magnetic
structures (a main one and a secondary one) with itinerant modulation of the
Cr spin axial vector, in direction and modulus, and incommensurate $q-$vectors
$q_{main}$ = (0.045, $\approx\frac{1}{4}$, $\approx\frac{1}{2}$) and
$q_{secondary}$ = (0.7249(5), -0.0885(3), -0.1292(9)). Such arrangement of the
Cr spins is reminiscent of the magnetic ordering established in the Cr sulfide
triangular antiferromagnets AuCrS2 and AgCrS2 which also manifest strong
magnetoelastic coupling. However, the unprecedented geometric conditions
realized in LiCrSe2 provide access to a unique combination for the competing
magnetic mechanisms, resulting in a much more complex magnetic ground state in
which the magnetic moment is periodically suppressed and changes direction.
This peculiar behavior might be originated by a dual itinerant/localized
nature of the electrons in this system. From the results of this work LiCrSe2
was proven to be an extremely fascinating study case, which lends itself to
very promising future single crystal investigations, and constitutes an
important step forward in understanding the spin-lattice relationship in
strongly correlated spin systems.

## Methods

Polycrystalline LiCrSe2 was prepared by direct reaction of the Li, Cr, and Se
elements. Further details on the sample preparation can be found in reference
[24].

The low-temperature $\mu^{+}$SR spectra have been acquired, as a function of
temperature, at the multi purpose muon beam-line M20, at the Canada’s particle
accelerator centre TRIUMF. $\sim$ 200 mg of sample in powder was packed in a
1$\times$1 cm2 area Aluminum-coated Mylar tape (0.05 mm thickness), in order
to reduce the background signal. This envelope was attached to a low
background sample holder inserted in a helium exchange gas cryostat
(temperature range: 2 K to 300 K).

The high-temperature $\mu^{+}$SR data were obtained using the surface muon
beamline S1 at the high intensity proton accelerator facility J-PARC, in
Japan. The LiCrSe2 powder was pressed into a pellet introduced into
hermetically sealed non-magnetic titanium cells. The surface of the pellet
facing the muon beam was covered using a very thin (50 $\mu$m) titanium
window. By using a closed-cycle refrigerator (CCR), data was collected in the
temperature range $T=50-375$ K. A more detailed description of the sample cell
and experimental setup can be found in Ref. [40].

The XRD measurements were collected at the instrument I11 of the Diamond Light
Source in Didcot (UK). The sample was mounted by filling and sealing an Al
capillary with internal diameter 0.7 mm, which was in turn attached to a
phoenix cryostat, with a 10∘ sample rocking stage, which allows measurements
in a temperature range from 11 K to 300 K.

The neutron diffraction measurements were performed at the time-of-flight
(ToF) powder diffractometer iMATERIA [63], at J-PARC, in Japan. The powder
samples ($\sim$ 500 mg) were mounted into cylindrical vanadium cells with
diameters and 5 mm and sealed with an aluminium cap, aluminium screws and
indium wire. The cell was mounted on a closed cycle refrigerator to reach
temperatures between 2 K and 300 K. iMATERIA allows for the simultaneous
adoption of different detector banks for wide d-range ($Q-$range) coverage.
The backscattering bank (BS), suitable for detailed structural
characterization, has a resolution $\Delta d/d$ = 0.16$\%$ and allows a
d-range from 0.181 Å up to 5.09 Å, the 90-degree bank (SE) has a resolution
$\Delta d/d$ = 0.5$\%$ and allows a d-range from 0.255 Å up to 7.2 Å, and the
low angle bank (LA35), ideal for the detection of low $Q-$range magnetic Bragg
peaks, has a d-range from 0.25 Å up to 40 Å.

Due to high sensitivity of the sample to air and moisture, the sample
preparations for all the measurements were carried out in a controlled
environment using Ar and/or He glove-boxes.

The crystal and magnetic structure determination was carried out with the help
of the Bilbao crystallographic server [64, 65], and the FullProf software
suite [66] was employed for the data refinement. All images involving crystal
structure were made with the VESTA software [67], the parameter plots and
fitting are produced with the software Igorpro [68] and the $\mu^{+}$SR data
were fitted using the software package musrfit [69].

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## Acknowledgements

The $\mu^{+}$SR measurements were performed at the instrument M20 of the muon
source TRIUMF (beamtime proposal: M1673) as well as at the S1 beamline of
J-PARC (beamtime proposal: 2019A0327). The XRD measurements were performed at
the instrument I11 of the synchrotron facility Diamond (beamtime proposal:
CY23840). The NPD measurements were performed at the instrument iMATERIA of
the neutron spallation source J-PARC (beamtime proposal: 20180056A). The
authors wish to thank Hiroshi Nozaki for his support during the $\mu^{+}$SR
experiment, and Juan Rodriguez-Carvajal for the assistance he provided on the
neutron diffraction data refinement, along wit the staff of TRIUMF, Diamond
and J-PARC for the help in the experimental measurements. This research is
funded by the Swedish Foundation for Strategic Research (SSF) within the
Swedish national graduate school in neutron scattering (SwedNess), as well as
the Swedish Research Council VR (Dnr. 2021-06157 and Dnr. 2017-05078), and the
Carl Tryggers Foundation for Scientific Research (CTS-18:272). J.S. is
supported by the Japan Society for the Promotion Science (JSPS) KAKENHI Grant
No. JP18H01863 and JP20K21149. Y.S. and O.K.F. are funded by the Chalmers Area
of Advance - Materials Science. S.K. and K.Y. are supported by JSPS KAKENHI
Grant No.18KK0150. The work at the University of Zürich and the University of
Geneva was supported by the Swiss National Science Foundation under Grant No.
PCEFP2-194183.

## Author contributions statement

E.N., J.S., and M.M. conceived the experiments. E.N., O.K.F., N.M., C.T.,
T.M., A.O., A.K., I.U., Y.S., J.S. and M.M. conducted the experiments. E.N,
J.S., N.M., O.K.F., J.H.B., V.P, and M.M. analyzed the results. The samples
were synthesized by S.K. and K.Y., as well as by C.W. and F.O.v.R.; they also
conducted the initial sample characterizations. E.N and M.M. made all the
figures. E.N. created the first draft, and all co-authors reviewed and revised
the manuscript.

## Data availability statement

All the data of this work are available from the corresponding authors on
request.

Competing interests

The authors declare no competing interests.

The corresponding author is responsible for submitting a competing interests
statement on behalf of all authors of the paper.