# Breathing pyrochlore magnet CuGaCr4S8: Magnetic, thermodynamic, and
dielectric properties

Masaki Gen<EMAIL_ADDRESS>Department of Advanced Materials Science, The
University of Tokyo, Kashiwa 277-8561, Japan  RIKEN Center for Emergent Matter
Science (CEMS), Wako 351-0198, Japan  Hajime Ishikawa  Institute for Solid
State Physics, The University of Tokyo, Kashiwa 277-8581, Japan  Atsushi
Miyake  Institute for Solid State Physics, The University of Tokyo, Kashiwa
277-8581, Japan  Takeshi Yajima  Institute for Solid State Physics, The
University of Tokyo, Kashiwa 277-8581, Japan  Harald O. Jeschke  Research
Institute for Interdisciplinary Science, Okayama University, Okayama 700-8530,
Japan  Hajime Sagayama  Institute of Materials Structure Science, High Energy
Accelerator Research Organization, Tsukuba 305-0801, Japan  Akihiko Ikeda
Institute for Solid State Physics, The University of Tokyo, Kashiwa 277-8581,
Japan  Department of Engineering Science, University of Electro-
Communications, Chofu, Tokyo 182-8585, Japan  Yasuhiro H. Matsuda  Institute
for Solid State Physics, The University of Tokyo, Kashiwa 277-8581, Japan
Koichi Kindo  Institute for Solid State Physics, The University of Tokyo,
Kashiwa 277-8581, Japan  Masashi Tokunaga  Institute for Solid State Physics,
The University of Tokyo, Kashiwa 277-8581, Japan  Yoshimitsu Kohama  Institute
for Solid State Physics, The University of Tokyo, Kashiwa 277-8581, Japan
Takashi Kurumaji  Department of Advanced Materials Science, The University of
Tokyo, Kashiwa 277-8561, Japan  Yusuke Tokunaga  Department of Advanced
Materials Science, The University of Tokyo, Kashiwa 277-8561, Japan  Taka-hisa
Arima  Department of Advanced Materials Science, The University of Tokyo,
Kashiwa 277-8561, Japan  RIKEN Center for Emergent Matter Science (CEMS), Wako
351-0198, Japan

###### Abstract

We investigate the crystallographic and magnetic properties of a chromium-
based thiospinel CuGaCr4S8. From a synchrotron x-ray diffraction experiment
and structural refinement, Cu and Ga atoms are found to occupy the tetrahedral
A-sites in an alternate way, yielding breathing pyrochlore Cr network.
CuGaCr4S8 undergoes a magnetic transition associated with a structural
distortion at 31 K in zero magnetic field, indicating that the spin-lattice
coupling is responsible for relieving the geometrical frustration. When
applying a pulsed high magnetic field, a sharp metamagnetic transition takes
place at 40 T, followed by a 1/2-magnetization plateau up to 103 T. These
phase transitions accompany dielectric anomalies, suggesting the presence of
helical spin correlations in low-field phases. The density-functional-theory
calculations suggest that CuGaCr4S8 is dominated by antiferromagnetic and
ferromagnetic exchange couplings within small and large tetrahedra,
respectively, in analogy with CuInCr4S8. We argue that A-site-ordered Cr
thiospinels serve as an excellent platform to explore diverse magnetic phases
along with pronounced magnetoelastic and magnetodielectric responses.

##  I Introduction

Magnetic materials of the pyrochlore lattice, a three-dimensional network of
corner-sharing tetrahedra, have been a central research subject in the context
of frustrated magnetism 2010_Gar ; 2021_Rei . In recent years, the breathing
pyrochlore lattice, where up- and down-pointing tetrahedra differ in size, has
attracted growing interest from the viewpoint of ground state control 2013_Oka
. The key concept of this spin model lies in the introduction of inequivalent
exchange couplings $J$ and $J^{\prime}$ in the small and large tetrahedra,
respectively [Fig. 1(a)]. Depending on the signs and magnitudes of $J$ and
$J^{\prime}$ as well as the nature of spins, various exotic magnetic states
and emergent phenomena have been theoretically predicted: e.g., unconventional
spin-liquid states and excitations 2015_Ben ; 2016_Sav ; 2016_Li ; 2019_Iqb ;
2020_Yan ; 2022_Han , spin-lattice-coupled superlattice long-range orders
(LROs) 2019_Aoy ; 2021_Aoy , and a magnetic hedgehog-lattice 2021_Aoy_HG ;
2022_Aoy_HG .

A representative realization of the breathing pyrochlore system is a quantum
Heisenberg antiferromagnet Ba3Yb2Zn5O11, whose magnetism is governed by Yb3+
ions with pseudospin-1/2 2014_Kim . In this compound, the breathing ratio
$r^{\prime}/r$, where $r$ ($r^{\prime}$) represents the nearest-neighbor (NN)
bond length in small (large) tetrahedra, amounts to approximately 2, resulting
in $J\gg J^{\prime}>0$, i.e., close to the decoupled tetrahedron limit
2014_Kim . The magnetization, specific heat, and inelastic neutron scattering
suggested the singlet formation without any signs of a magnetic LRO at low
temperatures 2014_Kim ; 2016_Hak , though the development of intertetrahedral
correlations is also pointed out 2016_Rau ; 2022_Dis .

For the larger spin case, A-site-ordered chromium-based spinels AA’Cr4X8,
where Cr3+ ions with spin-3/2 form a breathing pyrochlore lattice, offer a
fertile playground to address an effective spin Hamiltonian with various sets
of $J$ and $J^{\prime}$ 2019_Gho . Due to the difference in the ionic radius
between A+ and A’3+ cations, their crystallographic ordering like the zinc-
blende-type arrangement should modulate the chemical pressure and as a
consequence induce the breathing bond-alternation in the Cr pyrochlore network
[Fig. 1(b)]. This material design was first proposed by Joubert and Durif in
1966 1966_Jou . They prepared two oxides, LiGaCr4O8 and LiInCr4O8, and found
the lack of an inversion center in their crystal structures, signaling the
ordering of Li and Ga/In atoms. Subsequently, Pinch et al. synthesized Cr
thiospinels AA’Cr4S8 with various combinations of A and A’ atoms: A=Li, Cu,
Ag; A’=Al, Ga, In 1970_Pin . Among them, the A-site ordering was confirmed for
LiGaCr4S8, LiInCr4S8, CuAlCr4S8, and CuInCr4S8. Structural refinements and
detailed physical property measurements on these compounds have been actively
performed in the last decade 2015_Nil ; 2016_Lee ; 2016_Sah ; 2018_Oka ;
2018_Pok ; 2019_Gen ; 2020_Gen ; 2020_Kan ; 2020_Pok ; 2021_Gao ; 2022_Sha ;
2022_Gen , triggered by the renewed interest by Okamoto et al. in 2013
2013_Oka .

It is noteworthy that a peculiar combination of antiferromagnetic (AFM) $J$
and ferromagnetic (FM) $J^{\prime}$ can be realized in AA’Cr4X8 due to the
competition between the AFM Cr–Cr direct exchange and FM Cr–X–Cr superexchange
interactions 2019_Gho . Such a spin Hamiltonian can be effectively mapped on
the spin-6 Heisenberg antiferromagnet on the face-centered-cubic (FCC) lattice
if $J^{\prime}$ is strong. Indeed, a cluster excitation was observed in
CuInCr4S8 at low temperatures 2021_Gao , indicating the development of FM
correlations within each large tetrahedron. Another characteristic feature is
the intrinsic strong spin-lattice coupling (SLC) arising from the sensitivity
of the strength of $J$ ($J^{\prime}$) against the NN Cr–Cr bond length, i.e.,
large $|dJ/dr|$ ($|dJ^{\prime}/dr^{\prime}|$). The SLC can act as a principal
perturbation to lift the macroscopic degeneracy and bring about
magnetostructural transitions at low temperatures 2015_Nil ; 2016_Lee ;
2016_Sah ; 2020_Kan  and in an applied magnetic field 2019_Gen ; 2020_Gen ;
2021_Aoy . Interestingly, CuInCr4S8 exhibits a fascinating magnetic-field-
versus-temperature ($H$-$T$) phase diagram including a robust 3-up-1-down
phase associated with a 1/2-magnetization plateau 2020_Gen  as well as a
thermal equilibrium phase pocket 2022_Gen  reminiscent of a skyrmion lattice
2009_Muh ; 2012_Sek ; 2012_Oku .

In this work, we report the structural, magnetic, thermodynamic, and
dielectric properties of CuGaCr4S8. Although CuGaCr4S8 was previously
synthesized in Refs. 1970_Pin ; 1976_Wil ; 2005_Kes ; 2018_Ami , no conclusive
remark on the A-site ordering was given because the close proximity of the
scattering factors between Cu+ and Ga3+ made it challenging to judge the
presence or absence of reflections forbidden for the A-site disordered case,
i.e., $(hk0)$ peaks with $h+k=4n+2$. Our synchrotron powder x-ray diffraction
(XRD) measurement confirms the presence of 200 and 420 peaks of structural
origin. The Rietveld analysis shows that the A-site-ordered spinel structure
with the breathing pyrochlore Cr network ($F{\overline{4}}3m$ space group)
provides better refinement than the normal spinel structure
($Fd{\overline{3}}m$ space group). CuGaCr4S8 undergoes a magnetic transition
at $T_{\rm N}=31$ K in zero magnetic field. We observe a crystal symmetry
lowering below $T_{\rm N}$, which is compatible with the reported low-symmetry
incommensurate helical structure and was overlooked in the previous neutron
diffraction study 1976_Wil . A series of pulsed high-field experiments reveal
that CuGaCr4S8 exhibits a rich $H$-$T$ phase diagram associated with
magnetoelastic and magnetodielectric effects, similar to that of CuInCr4S8
2020_Gen ; 2022_Gen . The effective spin Hamiltonian of CuGaCr4S8 is discussed
on the basis of the density-functional-theory (DFT) energy mapping as well as
magnetoelastic theory 2020_Gen .

Figure 1: (a) Schematic of a breathing pyrochlore lattice, where small and
large tetrahedra with the nearest-neighbor bond lengths, $r$ and $r^{\prime}$,
are characterized by the exchange interactions, $J$ and $J^{\prime}$,
respectively. (b) Crystal structure of the A-site-ordered Cr spinels. The
present focus is CuGaCr4S8, where nonmagnetic Cu+ and Ga3+ ions are found to
be arranged in the zinc-blende-type structure, yielding a breathing pyrochlore
Cr network. The illustrations are drawn with VESTA software 2011_Mom .

##  II Methods

Polycrystalline samples of CuGaCr4S8 were synthesized by the conventional
solid-state reaction method. Starting ingredients were high-purity gallium
ingots (99.999%) and copper (99.99%), chromium (99.99%), and sulfur (99.99%)
powders. They were mixed in the stoichiometric ratio, sealed in an evacuated
quartz tube, and heated at 400∘C for 24 h and then at 800∘C for 48 h in a box
furnace. Then, the sintering was repeated twice at 900∘C for 96 h after
grinding and pelletizing the sintered products.

A powder synchrotron x-ray diffraction (XRD) profile was collected at room
temperature on Photon Factory BL-8A. The wavelength was
$\lambda=0.689739~{}\AA$. The Rietveld analysis was performed using the
RIETAN-FP program RIETAN . The temperature evolution of the powder XRD pattern
was measured between 4 and 300 K using a commercial x-ray diffractometer
(SmartLab, Rigaku) at the Institute for Solid State Physics (ISSP), University
of Tokyo. The incident x-ray beam was monochromatized by a Johansson-type
monochromator with a Ge(111) crystal to select only Cu-$K\alpha$1 radiation.

Figure 2: Synchrotron XRD pattern of CuGaCr4S8 powder samples (red open
circle) and calculated XRD pattern obtained by the Rietveld analysis (black
solid line). The Rietveld refinement was performed in a range of
$5^{\circ}<2\theta<120^{\circ}$, where several tiny peaks originating from
Cr3S4 and unknown impurity phases are excluded. Blue vertical bars indicate
the nuclear Bragg reflections, and the green line is the difference between
the experimental and calculated patterns. Insets show enlarged views of the
experimental XRD profiles focusing on 200 and 640 peaks, which are forbidden
for $Fd{\overline{3}}m$ but allowed for $F{\overline{4}}3m$.

Magnetization up to 7 T was measured using a SQUID magnetometer (MPMS, Quantum
Design). Magnetization up to 14 T was measured using a vibrating sample
magnetometer installed in a physical property measurement system (PPMS,
Quantum Design) equipped with a superconducting magnet. Magnetization up to 57
T was measured by the induction method in a non-destructive (ND) pulsed magnet
($\sim$4 ms duration). Magnetization up to 140 T was measured by the induction
method using a coaxial-type pickup coil in a horizontal single-turn-coil (STC)
megagauss generator ($\sim$8 $\mu$s duration) 2020_Gen . Thermal expansion was
measured by the fiber-Bragg-grating (FBG) method using an optical sensing
instrument (Hyperion si155, LUNA) in a cryostat equipped with a
superconducting magnet (Spectromag, Oxford) at zero field. Longitudinal
magnetostriction up to 54 T was measured by the FBG method in a ND pulsed
magnet ($\sim$36 ms duration), where the optical filter method was employed to
detect the relative sample-length change $\Delta L/L_{\rm 0T}$ 2018_Ike . The
fiber was attached to a rod-shaped sintered sample with epoxy Stycast1266.
Dielectric constant at zero field was measured at a frequency of 10 kHz by
using an LCR meter (E4980A, Agilent) in a PPMS. Dielectric constant along the
field direction ($E\parallel B$) up to 48 T was measured at a frequency of 50
kHz by using a capacitance bridge (1615-A, General Radio) in a ND pulsed
magnet ($\sim$36 ms duration) 2020_Miy . Silver paste was painted on the two
large surfaces of a disk-shaped sintered sample to form electrodes. Heat
capacity was measured by the thermal relaxation method in the PPMS at zero
field. All the pulsed high-field experiments were performed at ISSP.

In our DFT energy mapping 2021_Yam ; 2021_Hei ; 2022_Her , we worked with the
full potential local orbital basis set 1999_Koe  and generalized gradient
approximation (GGA) type exchange and correlation functional 1996_Per . Strong
electronic correlations on Cr $3d$ orbitals were treated with DFT+$U$
corrections 1995_Lie , where we varied the onsite correlation strength $U$ and
fix the Hund’s rule coupling to $J_{H}=0.72$ eV 1996_Miz . We use a $2\times
2\times 1$ supercell of CuGaCr4S8 with $Pm$ symmetry and twelve inequivalent
Cr3+ positions to extract the exchange couplings up to the fifth NN.

##  III Basic physical properties

###  III.1  Structural analysis

We first show that the crystal structure of CuGaCr4S8 belongs to the
$F{\overline{4}}3m$ space group and consists of the breathing pyrochlore
lattice, similar to the previously reported A-site-ordered Cr thiospinels
2018_Oka ; 2022_Sha . The synchrotron powder XRD pattern of CuGaCr4S8 at room
temperature is depicted by red open circles in Fig. 2. All major peaks can be
indexed to the FCC symmetry characteristic of the spinel structure. We observe
small additional peaks at $2\theta=7.98^{\circ}$ and $29.04^{\circ}$ (insets
of Fig. 2), indexed as the 200 and 640 reflections, respectively. Again,
$(hk0)$ peaks with $h+k=4n+2$ are forbidden for $Fd{\overline{3}}m$ but
allowed for $F{\overline{4}}3m$. The possibility of impurity contribution to
these peaks is excluded as long as we examine the known phases in the database
(ICSD). The effect of multiple diffraction 1980_Tok ; 1981_Ste  is minimal in
our XRD measurement using the randomly oriented powder sample. The anisotropy
of the local structure around the transition metal atoms could in principle
break the forbidden rule of the $Fd{\overline{3}}m$ space group without the
A-site ordering 2004_Sub , while the incident x-ray energy is sufficiently
large compared to the resonance energies of the composition elements to
suppress this factor.

Table 1: Structural parameters of CuGaCr4S8 at room temperature assuming the $F{\overline{4}}3m$ space group, where Cu and Ga atoms occupy the $4a$ and $4d$ sites, respectively, and the atomic position $x$ of Cr is less than 0.375. The lattice constant is $a=9.92036(8)$ Å. Reliability factors are $R_{\rm wp}=2.826$, $R_{\rm p}=1.755$, $R_{\rm e}=1.711$, $S=1.6518$. |  | $x$ | $y$ | $z$ |  Occup. | B (Å)
---|---|---|---|---|---|---
Cu |  $4a$ |  0 |  0 |  0 | 1 |  0.86(10)
Ga |  $4d$ |  3/4 |  3/4 |  3/4 | 1 |  0.83(8)
Cr |  $16e$ |  0.37042(15) |  $x$ |  $x$ | 1 |  0.63(2)
S1 |  $16e$ |  0.13316(26) |  $x$ |  $x$ | 1 |  0.79(5)
S2 |  $16e$ |  0.61679(21) |  $x$ |  $x$ | 1 |  0.65(5)

Using the present XRD data, we performed the Rietveld analysis assuming
several types of structural models. Details of the analysis are found in
Appendix A. We confirm that the $F{\overline{4}}3m$ model, assuming perfect
A-site ordering and allowing the breathing distortion of the Cr pyrochlore
lattice, yields better refinement than the $Fd{\overline{3}}m$ model with
random distribution of Cu and Ga atoms. The fitting result and structural
parameters obtained for the $F{\overline{4}}3m$ space group are shown in Fig.
2 and Table 1, respectively. The lattice constant is $a=9.92036(8)~{}\AA$, in
accord with Ref. 1970_Pin . The NN Cr–Cr bond lengths are $r=3.377(4)~{}\AA$
and $r^{\prime}=3.638(4)~{}\AA$ in small and large tetrahedra, respectively.
The breathing ratio $r^{\prime}/r=1.077$ is larger than those in oxides, 1.035
(LiGaCr4O8) and 1.049 (LiInCr4O8) 2013_Oka , and comparable to those in other
sulfides, 1.074 (LiGaCr4S8), 1.089 (LiInCr4S8) 2018_Oka , 1.066 (CuAlCr4S8)
2022_Sha , and 1.084 (CuInCr4S8) 2022_Gen . Although it is difficult to
evaluate the ratio of site mixing between Cu and Ga, the refined value of
$r^{\prime}/r$ is robust to the incorporation of the site mixing in the
Rietveld analysis. We hence infer that Cu and Ga atoms are almost perfectly
ordered in CuGaCr4S8, like in Li(Ga, In)Cr4S8 and Cu(Al, In)Cr4S8 2018_Oka ;
2022_Sha .

Figure 3: Temperature dependence of (a) magnetic susceptibility $M/H$ at 1 T
and (b) heat capacity divided by temperature $C/T$ at 0 T. The inverse
magnetic susceptibility $H/M$ and the Curie-Weiss fit above 160 K (black) are
displayed in the right axis of (a). The solid line in (b) denotes the
estimated lattice heat capacity based on the Debye model with the Debye
temperature of $\Theta_{\rm D}=430$ K.

###  III.2  Magnetic and structural transitions at low temperatures

Figures 3(a) and 3(b) show the magnetic susceptibility $M/H$ measured at 1 T
and the heat capacity divided by temperature $C/T$ measured at 0 T as a
function of temperature, respectively. The inverse susceptibility $H/M$
exhibits linear temperature dependence between 160 and 300 K [right axis of
Fig. 3(a)], following the Curie-Weiss law with the Weiss temperature of
$\Theta_{\rm W}=-103$ K and the effective moment $p_{\rm eff}=4.08$ $\mu_{\rm
B}$. The large negative $\Theta_{\rm W}$ indicates dominant AFM exchange
couplings, like Cu(Al, In)Cr4S8 2022_Sha ; 2022_Gen  but unlike Li(Ga,
In)Cr4S8 2018_Oka ; 2018_Pok . The estimated $p_{\rm eff}$ is slightly larger
than the theoretical value of 3.87 $\mu_{\rm B}$ expected for $S=3/2$ with
quenched orbital moments, ensuring a nearly isotropic spin for CuGaCr4S8. In
order to estimate lattice contributions to $C/T$, we employ the Debye model
with the Debye temperature of $\Theta_{\rm D}=430$ K as shown by a black line
in Fig. 3(b). The calculated curve fits well to the experimental data above
$\sim$120 K.

Figure 4: Temperature dependence of (a) magnetic susceptibility $M/H$ at 7 T,
(b) magnetic heat capacity divided by temperature $C_{\rm mag}/T$ at 0 T, (c)
thermal expansion $\Delta L/L_{\rm 2K}$ at 0 T, and (d) dielectric constant
$\varepsilon^{\prime}$ at a frequency of 10 kHz at 0 T. The insets of (a) and
(d) show enlarged views of $M/H$ and $\varepsilon^{\prime}$ around $T_{\rm
N}$, respectively. The magnetic entropy $S_{\rm mag}$ calculated by
integrating $C_{\rm mag}/T$ with respect to temperature is displayed in the
right axis of (b). Black arrows represent the direction of the temperature-
sweeping process.

In the measured temperature range, we observe several anomalies in the
$M/H$-$T$ curve [Fig. 3(a)]. $M/H$ exhibits a steplike anomaly at $\sim$160 K,
below which $H/M$ deviates from the linear temperature dependence. This could
be attributed to the onset of short-range AFM correlations in the main phase
or to an AFM transition in an unidentified impurity phase, which we will leave
open for future interpretation. The inverse susceptibility $H/M$ further
exhibits a concave behavior below $\sim$120 K, where $C/T$ deviates from the
estimated lattice heat capacity [Fig. 3(b)], indicating the development of a
magnetic short-range order. The similar feature was also observed for
CuInCr4S8 2018_Oka ; 2022_Gen . On further cooling, an abrupt $M/H$ drop and a
sharp $C/T$ peak are observed at $T_{\rm N}=31$ K, indicating the onset of a
magnetic LRO. The transition temperature is consistent with the previously
reported values 1970_Pin ; 2005_Kes ; 2018_Ami .

Figure 5: Waterfall plots of the temperature evolution of the powder XRD
pattern focusing on (a) 440, (b) 444, and (c) 800 reflections indexed for the
cubic $F{\overline{4}}3m$ space group. The data were obtained using a
laboratory x-ray diffractometer with monochromatized Cu-$K\alpha$1 radiation.
Triangles denote the peak positions obtained by the (multi-) Lorentzian fit to
each peak profile.

Figure 4 summarizes the temperature dependence of various physical quantities
in the vicinity of $T_{\rm N}$. As shown in the inset of Fig. 4(a), $M/H$
shows a clear hysteresis, indicating that the magnetic transition is first
order. Figure 4(b) shows the magnetic heat capacity divided by temperature
$C_{\rm mag}/T$, which is obtained by subtracting the estimated lattice
contributions from the experimental $C/T$ as shown in Fig. 3(b). By
integrating $C_{\rm mag}/T$ with respect to temperature, the magnetic entropy
$S_{\rm mag}$ is found to reach $\sim$5 J/(K mol-Cr) just above $T_{\rm N}$
comment . We obtain $S_{\rm mag}$ at 120 K to be $\sim$13 J/(K mol-Cr) (not
shown), which roughly agrees with the theoretical value $R{\rm ln}4$ = 11.5
J/(K mol-Cr) for the $S=3/2$ spin system. Notably, the thermal expansion
$\Delta L/L_{\rm 2K}$ rapidly decreases below 35 K with decreasing
temperature, suggesting a significant volume contraction across $T_{\rm N}$.
This behavior is similar to Li(Ga,In)Cr4O8 2020_Kan , where a crystal symmetry
lowering was observed 2015_Nil ; 2016_Sah . As shown below, we confirm the
crystal symmetry lowering for CuGaCr4S8 from the powder XRD measurement at low
temperatures. Moreover, the dielectric constant $\varepsilon^{\prime}$
exhibits a steplike anomaly at $T_{\rm N}$ [inset of Fig. 4(d)] as observed in
Li(Ga,In)Cr4O8 2016_Lee ; 2016_Sah . A previous powder neutron diffraction
study 1976_Wil  proposes that the magnetic structure below $T_{\rm N}$ is an
incommensurate spiral state. Thus, the observed dielectric anomaly would be of
magnetic origin; in other words, CuGaCr4S8 would be a type-II multiferroic
2009_Kho .

Figure 6: (a) Definition of the exchange couplings up to the fifth-NN path in
the breathing pyrochlore lattice. (b) Exchange parameters of CuGaCr4S8 at room
temperature obtained by the DFT energy mapping as function of the onsite
interaction strength $U$. The corresponding Weiss temperature is denoted by
crosses in the right axis. The vertical line indicates the $U$ value where the
exchange couplings match the experimental Weiss temperature $\Theta_{\rm
W}=-103$ K.

To reveal the structural change across the magnetic transition, we investigate
the powder XRD patterns at low temperatures. Figure 5 shows the temperature
evolution of peak profile of some Bragg reflections. Peak splitting is clearly
observed for many reflections below $T_{\rm N}$, signaling that the magnetic
transition accompanies a structural transition. We note that the peak
splitting was not observed in the previous powder neutron diffraction study
1976_Wil  because the expected splitting is much smaller than the experimental
peak width, $\sim$0.2∘ and $\sim$2∘, respectively, which is not sufficient to
resolve the symmetry lowering below $T_{\rm N}$. Remarkably, $hhh$ reflections
split into two peaks whereas no splitting or broadening is observed for $h00$
reflections [Figs. 5(b) and 5(c)]. These observations can be accounted for by
rhombohedral distortion, as opposed to tetragonal or orthorhombic distortion
in other Cr spinels 2006_Ued ; 2007_Lee ; 2009_Yok ; 2009_Kan ; 2015_Nil ;
2016_Sah . However, the proposed magnetic modulation vector
${\mathbf{Q}}=(0.18,0,0.80)$ 1976_Wil  may not directly cause rhombohedral
distortion, but is compatible with monoclinic distortion. Indeed, the peak
profile of 440 reflection below $T_{\rm N}$ can be well fitted by the
superposition of three Lorentzian functions rather than two [see Fig. 5(a) and
Appendix B]. This suggests that the crystal symmetry below $T_{\rm N}$ is
lower than rhombohedral. More detailed crystallographic and magnetic structure
analysis is necessary to settle the issue.

Table 2: Exchange couplings up to third-NN in four kinds of A-site-ordered Cr thiospinels estimated by the DFT energy mapping. | $J/k_{\rm B}$ | $J^{\prime}/k_{\rm B}$ | $J_{2}/k_{\rm B}$ | $J_{3a}/k_{\rm B}$ | $J_{3b}/k_{\rm B}$ | Ref.
---|---|---|---|---|---|---
LiGaCr4S8 |  $-7.7$ K |  $-12.2$ K |  1.2 K |  6.1 K |  3.0 K |  2019_Gho
LiInCr4S8 |  $-0.3$ K |  $-28.0$ K |  0.7 K |  5.3 K |  2.4 K |  2019_Gho
CuGaCr4S8 |  9.8 K |  $-11.4$ K |  2.3 K |  5.9 K |  4.4 K |  This work
CuInCr4S8 |  14.7 K |  $-26.0$ K |  1.1 K |  6.4 K |  4.5 K |  2019_Gho

###  III.3  DFT energy mapping

Based on the A-site-ordered crystal structure at room temperature shown in
Table 1, we performed the DFT calculations to estimate the exchange couplings
of CuGaCr4S8. More details of the calculations are found in Appendix. C.
Figure 6(b) shows the DFT energy mapping up to the fifth-NN exchange
couplings, which are defined in the Heisenberg Hamiltonian of the form
${{\mathcal{H}}}=\sum_{i<j}J_{ij}{\mathbf{S}}_{i}\cdot{\mathbf{S}}_{j}$ [see
Fig. 6(a)]. Exchange interactions monotonically evolve with onsite Coulomb
interaction strength $U$. The vertical dashed line indicates the $U$ value for
which the exchange couplings match the experimental Weiss temperature of
$\Theta_{\rm W}=-103$ K (see also Appendix A). The obtained parameter set is
$J/k_{\rm B}=9.8(7)$ K, $J^{\prime}/k_{\rm B}=-11.4(6)$ K, $J_{2}/k_{\rm
B}=2.3(5)$ K, $J_{3a}/k_{\rm B}=5.9(3)$ K, $J_{3b}/k_{\rm B}=4.4(3)$ K,
$J_{4}/k_{\rm B}=-1.0(3)$ K, and $J_{5}/k_{\rm B}=0.4(3)$ K, where
$k_{\mathrm{B}}$ is the Boltzmann’s constant, representing that CuGaCr4S8 is
characterized by strong AFM $J$ and FM $J^{\prime}$. Since the strengths of
the exchange couplings should be modified below $T_{\rm N}$ due to the lower-
symmetry crystal structure, the ground state cannot be simply represented by
these parameters. Nevertheless, our magnetization measurements suggest that
the low-temperature magnetism of CuGaCr4S8 can be understood based on the
AFM-$J$-FM-$J^{\prime}$ picture, as discussed in Sec. IV.1.

Table 2 compares the exchange parameters among four kinds of A-site-ordered Cr
thiospinels estimated in the previous works 2019_Gho  and this work. One can
find that $J$ and $J^{\prime}$ are strongly dependent on the types of
nonmagnetic cations. The occupation of Li atoms at the $4a$ site leads to FM
$J$, whereas that of Cu atoms leads to AFM $J$. $J^{\prime}$ is always FM, and
its strength is enhanced when In atoms occupy the $4d$ site. These tendencies
are reasonable because monovalent A+ (trivalent A’3+) cations are surrounded
by S1 (S2) atoms connecting the short (long) NN Cr–Cr bonds [Fig. 1(b)]. The
Hamiltonian of CuGaCr4S8 is qualitatively similar to that of CuInCr4S8 except
that $|J^{\prime}|$ is much smaller. In other words, the effects of further-
neighbor interactions, especially $J_{3a}$ and $J_{3b}$, are more important
for CuGaCr4S8 than for CuInCr4S8. This may be responsible for the difference
in the ground state at zero field; a commensurate ${\mathbf{Q}}=(1,0,0)$ state
with an $S=6$ spin cluster in the large tetrahedron is realized for CuInCr4S8
2021_Gao , whereas an incommensurate spiral state in which four spins in the
large tetrahedra are not parallel with each other for CuGaCr4S8 1976_Wil .

##  IV  Magnetic-field induced phase transitions

###  IV.1  Magnetization curves

We here move on to the field-induced properties of CuGaCr4S8 revealed by
pulsed high-field experiments. Figure 7(a) shows magnetization curves measured
at various initial temperatures $T_{\rm ini}$ using the ND pulsed magnet.
Although they appear simple at a glance, anomalies in the field derivatives
$dM/dH$ show a complicated temperature dependence, as shown in Fig. 7(b). For
$T_{\rm ini}=4.2$ K, a drastic magnetization jump is observed at around 40 T
accompanied by a large hysteresis, where $dM/dH$ exhibits a sharp peak at
$\mu_{0}H_{\rm c1}=40.4$ T and a shoulder-like anomaly at $\mu_{0}H_{\rm
c2}=42.1$ T in the field-increasing process. The shoulder-like anomaly does
not change its position while the peak moves to a lower field at 37.4 T in the
field-decreasing process. After the metamagnetic transition, $M$ reaches
$\sim$1.2 $\mu_{\rm B}$/Cr, which is much smaller than the expected saturation
value of $\sim$3 $\mu_{\rm B}$/Cr. As $T_{\rm ini}$ increases, $H_{\rm c1}$
shifts to a lower field and the hysteresis becomes smaller. Notably, a broad
shoulder-like structure appears on the low-field side of the $dM/dH$ peak at
$H_{\rm c1}^{\prime}$, as denoted by open triangles in Fig. 7(b). As shown in
Sec. IV.2, this magnetization anomaly is accompanied by a pronounced
dielectric response. Even for $T_{\rm ini}=36$ K ($>T_{\rm N}$), a weak
metamagnetic transition from the paramagnetic phase is observed at
$\mu_{0}H_{\rm p}=38.3$ T, indicating that the field-induced phase is robust
against thermal fluctuations.

Figure 7: (a)(b) Magnetic-field dependence of (a) magnetization $M$ and (b)
its field derivative $dM/dH$ measured at various initial temperatures $T_{\rm
ini}$ in a non-destructive (ND) pulsed magnet. Thick (thin) lines correspond
to the data in field-increasing (decreasing) processes. The curves except for
$T_{\rm ini}=1.4$ K are shifted upward for clarity. The inset of (b) is an
enlarged view of $dM/dH$ around the lowest-field phase transition for $T_{\rm
ini}=1.4$ K. (c) Magnetic-field dependence of $M$ (left) and $dM/dH$ (right)
in the field-increasing process measured at $T_{\rm ini}\sim 5$ K in a single-
turn-coil (STC) system. The absolute value of $M$ is calibrated by fitting
with the $M$–$H$ curve for $T_{\rm ini}=4.2$ K obtained in a ND pulsed magnet
(gray).

For all the measured $T_{\rm ini}$’s below $T_{\rm N}$, a subtle slope change
in the magnetization curve is observed at $\mu_{0}H_{\rm c0}\approx 10$ T,
which is visible as a $dM/dH$ cusp [inset of Fig. 7(b)]. This phase transition
is also confirmed by the magnetization measurement up to 14 T in a static
magnetic field (see Fig. 11 in Appendix D). We ascribe these anomalies to a
spin-flop transition with the reorientation of magnetic domains of the helical
state, as observed for CdCr2O4 2007_Mat_1 ; 2011_Bha ; 2019_Ros ; 2020_Ros ;
2005_Chu . For $T_{\rm ini}=1.4$ K, $dM/dH$ exhibits two peaks around 40 T in
the field-increasing process [Fig. 7(b)]. We tentatively view this behavior as
a splitting of the peak at $H_{\rm c1}$ observed for $T_{\rm ini}\geq 4.2$ K,
though its origin is unclear at present.

Table 3: Critical fields of the successive phase transitions in CuGaCr4S8 and CuInCr4S8 at $\sim$5 K. $H_{\rm c0}$ indicates a spin-flop transition in a low-field region observed only for CuGaCr4S8. $H_{\rm c1}\sim H_{\rm c3}$ correspond to the termination fields of X, Y, and C phases, which can be assigned to canted 2:2, canted 2:1:1, and 3-up-1-down phases, respectively, based on the magnetoelastic theory 2020_Gen . $H_{\rm sat}$ indicates the expected saturation field deduced from Eq. (2) ($J^{\prime}$: FM) using the exchange couplings based on the DFT calculations 2019_Gho . | $\mu_{0}H_{\rm c0}$ | $\mu_{0}H_{\rm c1}$ | $\mu_{0}H_{\rm c2}$ | $\mu_{0}H_{\rm c3}$ | $\mu_{0}H_{\rm sat}$ | Ref.
---|---|---|---|---|---|---
CuGaCr4S8 |  10.8 T |  40.4 T |  42.1 T |  103 T |  $170$ T |  This work
CuInCr4S8 |  — |  32 T |  56 T |  112 T |  180 T |  2020_Gen ; 2022_Gen

To get a whole picture of the field-induced phase transitions, we further
measure the magnetization at $T_{\rm ini}\sim 5$ K up to $\sim$140 T using the
STC system. As shown in Fig. 7(c), a plateau-like behavior is observed between
$\mu_{0}H_{\rm c1}=40$ T and $\mu_{0}H_{\rm c3}=103$ T in the field-increasing
process. Note that the transition at $H_{\rm c2}$ is not resolved due to an
electromagnetic noise. Judging from the magnitude of $M$ in this field range
($\sim$1.5 $\mu_{\rm B}$/Cr), a 3-up-1-down state with a 1/2-magnetization
plateau is expected to appear as in Cr spinel oxides 2006_Ued ; 2008_Koj ;
2011_Miy_JPSJ ; 2014_Miy ; 2019_Gen ; 2007_Mat_2 ; 2010_Mat  and CuInCr4S8
2020_Gen . Above $H_{\rm c3}$, $M$ rapidly increases up to the applied maximum
field of 140 T, where $M$ reaches $\sim$2.8 $\mu_{\rm B}$/Cr. This indicates
that the saturation field $H_{\rm sat}$ is a bit higher than 140 T.

Theoretically, AFM-$J$-AFM-$J^{\prime}$ or AFM-$J$-FM-$J^{\prime}$ breathing
pyrochlore magnets with the spin-lattice coupling can host a 1/2-magnetization
plateau 2021_Aoy ; 2020_Gen . If we consider the mean-field approximation and
neglect the spin-lattice coupling, $\Theta_{\rm W}$ and $H_{\rm sat}$ are
related to the exchange couplings as

$\begin{split}-\frac{S(S+1)}{k_{\rm B}}\Theta_{\rm
W}=J+J^{\prime}+{\overline{J}}_{\rm AFM}+{\overline{J}}_{\rm FM}\end{split}$
(1)

and

$\frac{g\mu_{\rm B}}{4S}\mu_{0}H_{\rm
sat}=\left\\{\begin{array}[]{ll}J+J^{\prime}+{\overline{J}}_{\rm
AFM}&(J^{\prime}:{\rm AFM})\\\ J+{\overline{J}}_{\rm AFM}&(J^{\prime}:{\rm
FM}),\end{array}\right.$ (2)

where ${\overline{J}}_{\rm AFM}$ and ${\overline{J}}_{\rm FM}$ are the
summation of AFM and FM further-neighbor exchange couplings, respectively, and
$g\approx 2.1$ is the Landé $g$ factor estimated from the Curie-Weiss fit
[Fig. 3(a)]. For CuGaCr4S8, if we assume AFM-$J$-AFM-$J^{\prime}$ and
${\overline{J}}_{\rm FM}\approx 0$, the saturation field is estimated to
$\mu_{0}H_{\rm sat}\approx 117$ T. This value is significantly underestimated
compared to the experimentally expected value ($>140$ T), ensuring that the
AFM-$J$-FM-$J^{\prime}$ picture is appropriate, as suggested by the DFT
calculations (Sec. III.1). If we substitute the exchange parameters shown in
Table 2 into Eq. (2), where ${\overline{J}}_{\rm
AFM}=4J_{2}+2J_{3a}+2J_{3b}+2J_{5}$ and ${\overline{J}}_{\rm FM}=2J_{4}$, the
saturation field is estimated to $\mu_{0}H_{\rm sat}\approx 170$ T. This seems
to be in good agreement with the experimental magnetization curve at 5 K [Fig.
7(c)], given that $H_{\rm sat}$ would be suppressed by the spin-lattice
coupling and thermal fluctuations.

The overall magnetization curve of CuGaCr4S8 at 4.2 or 5 K is similar to that
of CuInCr4S8 2020_Gen ; 2022_Gen . For both the compounds, an intermediate-
field phase appears just below the 1/2-magnetization plateau. The magnetic
structure in the intermediate-field phase is expected to be a canted 2:1:1
state according to the magnetoelastic theory assuming the effective $S=6$ FCC-
lattice model 2020_Gen , which predicts successive phase transitions from a
canted 2:2 to canted 2:1:1, 3-up-1-down, canted 3:1, and a fully polarized
phase. Note that a canted 2:1:1 phase is also observed for ZnCr2O4
2011_Miy_JPSJ  and MgCr2O4 2014_Miy  in a narrow field range. An
incommensurate spiral component would coexist in the canted spin states in
CuGaCr4S8 due to the presence of sizable further-neighbor exchange couplings
and/or the DM interaction, as proposed for CdCr2O4 2007_Mat_1 ; 2020_Ros . In
the following, we call the field-induced phases of CuGaCr4S8 at 5 K X, Y, C,
and C’ phases in the ascending order of the field [Fig. 7(c)]. Table 3
summarizes the critical field of each phase transition in the field-increasing
process for CuGaCr4S8 and CuInCr4S8 2020_Gen ; 2022_Gen . Theoretically, as
the SLC gets stronger, the 1/2-magnetization plateau expands while the canted
2:1:1 phase gets narrower 2020_Gen . Considering $H_{\rm c3}/H_{\rm c2}\sim
2.5$ and 2.0 for CuGaCr4S8 and CuInCr4S8, respectively, it can be said that
the 3-up-1-down state is more stable in CuGaCr4S8 than in CuInCr4S8. Besides,
the field range between $H_{\rm c1}$ and $H_{\rm c2}$ (Y phase) is much
narrower in CuGaCr4S8 than in CuInCr4S8. These trends suggest that the SLC is
stronger in CuGaCr4S8 than in CuInCr4S8. The strong SLC in CuGaCr4S8 is
consistent with the observation of a first-order phase transition at $T_{\rm
N}$.

Figure 8: Magnetic-field dependence of relative changes in (a) the sample
length along the field direction $\Delta L/L_{\rm 0T}$ and (b) dielectric
constant $\Delta\varepsilon^{\prime}/\varepsilon^{\prime}$ measured at various
initial temperatures $T_{\rm ini}$. Data presentation is the same as Fig. 7.

###  IV.2  Magnetostrictive and magnetodielectric effects

Figure 8(a) shows the field dependence of the longitudinal magnetostriction
and the longitudinal dielectric constant, respectively, measured on a sintered
sample at various $T_{\rm ini}$’s using the ND pulsed magnet. Here, $\Delta
L/L_{\rm 0T}$ and $\Delta\varepsilon^{\prime}/\varepsilon^{\prime}_{\rm 0T}$
represent the relative changes normalized by the zero-field values at each
$T_{\rm ini}$. For $T_{\rm ini}=36$ K, $\Delta L/L_{\rm 0T}$ shows a parabolic
field dependence at low fields as expected in the paramagnetic state. For
$T_{\rm ini}$ below $T_{\rm N}$, on the other hand, $\Delta L/L_{\rm 0T}$
remains almost constant or exhibits negative magnetostriction in the spiral
phase and X phase. This tendency is in contrast to Li(Ga, In)Cr4S8 2020_Kan
and CuInCr4S8 2022_Gen , where $\Delta L/L_{\rm 0T}$ gradually increases with
respect to a magnetic field. A slight slope change at around 10 T observed for
$12\leq T_{\rm ini}\leq 28$ K would reflect the spin-flop transition at
$H_{\rm c0}$. On entering the 1/2-magnetization plateau above $H_{\rm c1}$, a
drastic lattice expansion is observed in analogy with other Cr spinels
2022_Gen ; 2007_Tan ; 2019_Ros , suggesting that the 3-up-1-down collinear
state is stabilized by the exchange striction.

Interestingly, diverse dielectric responses are observed as shown in Fig.
8(b). For $T_{\rm ini}=1.4$ and 4.2 K,
$\Delta\varepsilon^{\prime}/\varepsilon^{\prime}_{\rm 0T}$ shows a sharp peak
with a magnitude of $\sim$4% between X and Y phases at $H_{\rm c1}$. As
$T_{\rm ini}$ increases toward $T_{\rm N}$, the dielectric anomaly around
$H_{\rm c1}$ transforms into a broad valley shape with a reduction of as large
as $\sim$8%. For $T_{\rm ini}=12$ K, a valley structure is seen only in the
field-decreasing process, presumably suggesting that the actual sample
temperature would be higher than $T_{\rm ini}$ in the field-decreasing process
due to the magnetocaloric effect 2022_Gen ; 2022_Kim . Of particular note is
the 20-K data, in which a tiny
$\Delta\varepsilon^{\prime}/\varepsilon^{\prime}$ peak coexist with the valley
structure. This supports the occurrence of another phase transition other than
that from X to Y phase, as suggested by the magnetization data [Fig. 7(b)]. We
hereafter call the additional higher-temperature phase “Z phase”.

The magnetoelectric effect of the A-site-ordered spinel has long been a
subject of interest in view of the inversion symmetry breaking of the crystal
structure 2014_Ter ; 2021_Sun . However, the reported dielectric anomalies
associated with magnetic transitions are weak for Li(Ga, In)Cr4O8 2016_Lee ;
2016_Sah , where a collinear 2-up-2-down magnetic LRO is induced by the SLC at
low temperatures. We stress that the observed dielectric anomaly in CuGaCr4S8
is much larger than in the oxide cases thanks to the emergence of
incommensurate magnetic LROs 1976_Wil . A single-crystal study on CuGaCr4S8
would be a promising route to seek for significant electric polarization
changes.

Figure 9: $H$-$T$ phase diagram of CuGaCr4S8 based on physical property
measurements performed in the present work. Note that another high-field phase
(C’ phase) appears above $\mu_{0}H_{\rm c3}=103$ T at $\sim$5 K. According to
the magnetoelastic theory 2020_Gen , the magnetic structure of C phase is
expected to be a 3-up-1-down state.

###  IV.3  H-T phase diagram

Based on a series of physical property measurements, we construct an $H$-$T$
phase diagram of CuGaCr4S8, as shown in Fig. 9. As discussed in Sec. IV.1, the
magnetic structures of X, Y, C phases are possibly canted 2:2, canted 2:1:1,
and 3-up-1-down states, respectively, based on the magnetoelastic theory
2020_Gen , though in reality an incommensurate spiral component would coexist
with the commensurate collinear component. We also observe a weak spin-flop
transition at around 10 T, so that the X phase may be just a flopped spiral
state with ${\mathbf{Q}}=(0.18,0,0.80)$ 1976_Wil . An additional magnetization
measurement using the STC system reveals that the C phase terminates at
$\mu_{0}H_{\rm c3}=103$ T, followed by the C’ phase with a possible canted 3:1
state up to at least 140 T at $\sim$5 K.

The identified magnetic phase diagram is basically shared with those of Cr
spinel oxides 2006_Ued ; 2019_Ros ; 2008_Koj ; 2011_Miy_JPSJ ; 2014_Miy ;
2022_Kim  and CuInCr4S8 2020_Gen ; 2022_Gen  where a robust 1/2-magnetization
plateau (C phase) intervenes between spin-canted phases on lower- and higher-
field sides. A characteristic feature of CuGaCr4S8 is the appearance of a
field-induced high-temperature phase (Z phase) immediately below the C phase,
in common with another AFM-$J$-FM-$J^{\prime}$ breathing pyrochlore compound
CuInCr4S8, which hosts an A phase in a closed $H$-$T$ regime around 25–40 T
and 10–35 K 2022_Gen . In the case of CuInCr4S8, the appearance of the A phase
is accompanied by negative magnetostriction and the enhancement of
magnetocapacitance at a lower phase boundary 2022_Gen . These features are not
the case for the Z phase in CuGaCr4S8. Thus, we infer that the Z phase in
CuGaCr4S8 is different from the A phase in CuInCr4S8. The identification of
these field-induced high-temperature phases in AFM-$J$-FM-$J^{\prime}$
breathing pyrochlore systems would be an intriguing issue left for future
works.

##  V Summary

We synthesized CuGaCr4S8 polycrystalline samples and demonstrated the
formation of a breathing pyrochlore Cr network by the synchrotron XRD
measurement and the Rietveld analysis. The DFT energy mapping suggests that
the spin Hamiltonian is characterized by AFM $J$, FM $J^{\prime}$, and
relatively strong further-neighbor exchange couplings, so that the system
harbors both geometrical frustration and bond frustration. Remarkably, the
comparison of exchange parameters among four AA’Cr4S8 compounds (Table 2)
provides insights into the relationship between the chemical composition and
the spin Hamiltonian in breathing pyrochlore chromium thiospinels. We also
unveil diverse magnetic phases associated with prominent magnetostrictive and
magnetodielectric effects at low temperatures and in high magnetic fields.
These observations in CuGaCr4S8 could be attributed to the magnetic
frustration and strong SLC, highlighting the complex interplay of these
factors.

## Acknowledgements

We appreciate Y. Okamoto, M. Mori, and S. Kitou for helpful discussions. This
work was financially supported by the Japan Society for the Promotion of
Science (JSPS) KAKENHI Grants-In-Aid for Scientific Research (No. 20J10988).
M.G. was a postdoctoral research fellow of the JSPS.

##  Appendix A  Rietveld analysis assuming various structural models

Figure 10: Rietveld refinement on the synchrotron XRD pattern based on the
$Fd{\overline{3}}m$ space group.

Here, we show the results of the Rietveld analysis assuming several types of
structural models. Basically, the normal spinel structure with the
$Fd{\overline{3}}m$ space group, in which Cu and Ga atoms are randomly
distributed at the $8a$ site with an occupancy of 0.5, yields satisfactory
refinement with relatively small reliability factors, as shown in Fig. 10 and
Table 4. However, upon relaxing the constraint on the atomic position of Cr,
$x({\rm Cr})$, while maintaining the random distribution of Cu and Ga and the
atomic position of S, $x({\rm Cr})$ tends to deviate from 0.375, suggesting
that the $F{\overline{4}}3m$ space group is more likely. Such symmetry
lowering should originate from the crystallographic ordering of Cu+ and Ga3+
with different ionic radii (0.60 $\AA$ and 0.47 $\AA$, respectively, in the
four-fold coordination). Accordingly, we consider two types of perfectly
A-site-ordered structures with $x({\rm Cr})<0.375$: Cu (Ga) occupies the $4a$
($4d$) site for type #1, whereas Cu (Ga) occupies the $4d$ ($4a$) site for
type #2. Note that there can be two inequivalent atomic sites of S for the
$F{\overline{4}}3m$ space group 2013_Oka ; 2018_Oka ; 2022_Sha . The
refinement results for the type #1 and #2 structures are shown in Tables 1 and
5, respectively. In the both refinements, the reliability factors are smaller
compared to the case of the $Fd{\overline{3}}m$ model. Although the resultant
structural parameters are almost the same within errors between type #1 and
#2, the thermal parameter of Ga is unusually large for type #2, suggesting
that type #1 is more reasonable. This is further supported by the calculation
of DFT total energies; type #1 is 293 meV per formula unit lower in energy
than type #2.

Table 4: Structural parameters of CuGaCr4S8 at room temperature assuming the $Fd{\overline{3}}m$ space group, where Cu and Ga atoms are randomly distributed at the $8a$ site with an occupancy of 0.5. The lattice constant is $a=9.91910(9)$ Å. Reliability factors are $R_{\rm wp}=2.967$, $R_{\rm p}=1.897$, $R_{\rm e}=1.706$, $S=1.7390$. |  | $x$ | $y$ | $z$ |  Occup. | B (Å)
---|---|---|---|---|---|---
Cu |  $8a$ |  0 |  0 |  0 | 0.5 |  0.80(2)
Ga |  $8a$ |  0 |  0 |  0 | 0.5 |  0.80(2)
Cr |  $16d$ |  0.375 |  $x$ |  $x$ | 1 |  0.85(2)
S |  $32e$ |  0.38382(6) |  $x$ |  $x$ | 1 |  0.61(2)
Table 5: Structural parameters of CuGaCr4S8 at room temperature assuming the $F{\overline{4}}3m$ space group, where Cu and Ga atoms occupy the $4d$ and $4a$ sites, respectively, and the atomic position $x$ of Cr is less than 0.375. The lattice constant is $a=9.92035(8)$ Å. Reliability factors are $R_{\rm wp}=2.838$, $R_{\rm p}=1.773$, $R_{\rm e}=1.711$, $S=1.6588$. |  | $x$ | $y$ | $z$ |  Occup. | B (Å)
---|---|---|---|---|---|---
Cu |  $4d$ |  3/4 |  3/4 |  3/4 | 1 |  0.69(6)
Ga |  $4a$ |  0 |  0 |  0 | 1 |  1.00(7)
Cr |  $16e$ |  0.37041(14) |  $x$ |  $x$ | 1 |  0.64(2)
S1 |  $16e$ |  0.13340(22) |  $x$ |  $x$ | 1 |  0.79(4)
S2 |  $16e$ |  0.61683(17) |  $x$ |  $x$ | 1 |  0.62(4)

##  Appendix B  Lorentzian fit on the peak profile of 440 reflection at low
temperatures

Figure 11 shows an enlarged view of the powder XRD pattern of CuGaCr4S8 around
the 440 reflection at 4 K. This peak can be fitted by the superposition of
three Lorentzian functions with peak positions of 52.08∘, 52.19∘, and 52.34∘.

Figure 11: Multi-Lorentzian fit (black) to the peak profile of 440 reflection
at 4 K (pink), which is obtained by the superposition of three Lorentzian
functions (blue).

##  Appendix C  Details of the DFT energy mapping

Our DFT calculations are performed using the all electron full potential FPLO
code 1999_Koe . We create a $2\times 2\times 1$ supercell with $Pm$ symmetry
and twelve symmetry inequivalent Cr3+ ions. This allows us to go slightly
beyond the minimal set of exchange interactions $J$, $J^{\prime}$, $J_{2}$,
$J_{3a}$, and $J_{3b}$, which is known to describe the breathing pyrochlore
thiospinels well 2019_Gho ; 2020_Gen ; we also resolve $J_{4}$ which has been
shown to noticeably affect the inelastic neutron spectrum of CuInCr4S8
2021_Gao  and ZnCr2Se4 2022_Gao . We calculate 30 spin configurations and fit
them with the classical Heisenberg Hamiltonian energies; the good quality of
the fit is shown in Fig. 12. We select the onsite interactions strength $U$
which is suitable for the description of CuGaCr4S8 by calculating the Weiss
temperature for the obtained exchange interactions via Eq. (1), where
${\overline{J}}_{\rm AFM}=4J_{2}+2J_{3a}+2J_{3b}+2J_{5}$ and
${\overline{J}}_{\rm FM}=2J_{4}$. For an interpolated $U=1.70$ eV, the set of
couplings exactly matches the experimental value $\Theta_{\rm W}=-103$ K. This
$U$ value falls into the range of $U$ values $1.4~{}{\rm eV}<U<2.0~{}{\rm eV}$
that describe many other chromium spinels 2019_Gho .

Figure 12: DFT energies of CuGaCr4S8, calculated within GGA+$U$ at
$J_{H}=0.72$ eV, $U=1.70$ eV and $6\times 6\times 6$ $k$ points in a $2\times
2\times 1$ supercell for 30 different spin configurations, compared with a fit
to the Heisenberg model. The good fit indicates that the extracted exchange
interactions represent the magnetic properties of CuGaCr4S8 well.

##  Appendix D  Low-field phase transition observed in a static magnetic field

Figure 13 shows the magnetization curves measured at various temperatures
using a vibrating sample magnetometer in a PPMS. A cusp structure is clearly
observed in the $dM/dH$ data below 25 K, indicating a spin-flop transition.
The transition field $H_{\rm c0}$ gradually increases as the temperature
increases. This trend is in line with the typical $H$-$T$ phase diagram of the
spin-flop transition in antiferromagnets.

Figure 13: Magnetic-field dependence of (a) magnetization $M$ and (b) its
field derivative $dM/dH$ measured at various temperatures in a static magnetic
field. The curves except for 4.2 K are shifted upward for clarity. Dashed
lines in (a) are guides to the eye to make it easier to see the slope change
in the $M$–$H$ curves.

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