††thanks: These two authors contributed equally.††thanks: These two authors
contributed equally.

# High-temperature magnetism and crystallography of a YCrO3 single crystal

Yinghao Zhu  Joint Key Laboratory of the Ministry of Education, Institute of
Applied Physics and Materials Engineering, University of Macau, Avenida da
Universidade, Taipa, Macao SAR 999078, China.  Si Wu  Joint Key Laboratory of
the Ministry of Education, Institute of Applied Physics and Materials
Engineering, University of Macau, Avenida da Universidade, Taipa, Macao SAR
999078, China.  Bao Tu  Joint Key Laboratory of the Ministry of Education,
Institute of Applied Physics and Materials Engineering, University of Macau,
Avenida da Universidade, Taipa, Macao SAR 999078, China.  Department of
Materials Science and Engineering, Shenzhen Key Laboratory of Full Spectral
Solar Electricity Generation (FSSEG), Southern University of Science and
Technology, No. 1088, Xueyuan Rd., Shenzhen 518055, Guangdong, China.
Shangjian Jin  State Key Laboratory of Optoelectronic Materials and
Technologies, School of Physics and Engineering, Sun Yat-sen University,
Guangzhou 510275, China.  Ashfia Huq  Oak Ridge National Laboratory, Oak
Ridge, Tennessee 37831, USA.  J$\ddot{\textrm{o}}$rg Persson  J$\ddot{u}$lich
Centre for Neutron Science JCNS-2 and Peter Gr$\ddot{u}$nberg Institut PGI,
JARA-FIT, Forschungszentrum J$\ddot{u}$lich GmbH, D-52425 J$\ddot{u}$lich,
Germany.  Haoshi Gao  Joint Key Laboratory of the Ministry of Education,
Institute of Applied Physics and Materials Engineering, University of Macau,
Avenida da Universidade, Taipa, Macao SAR 999078, China.  State Key Laboratory
of Quality Research in Chinese Medicine, Institute of Chinese Medical Sciences
(ICMS), University of Macau, Avenida da Universidade, Taipa, Macao SAR 999078,
China.  Defang Ouyang  State Key Laboratory of Quality Research in Chinese
Medicine, Institute of Chinese Medical Sciences (ICMS), University of Macau,
Avenida da Universidade, Taipa, Macao SAR 999078, China.  Zhubing He
Department of Materials Science and Engineering, Shenzhen Key Laboratory of
Full Spectral Solar Electricity Generation (FSSEG), Southern University of
Science and Technology, No. 1088, Xueyuan Rd., Shenzhen 518055, Guangdong,
China.  Dao-Xin Yao<EMAIL_ADDRESS>State Key Laboratory of
Optoelectronic Materials and Technologies, School of Physics and Engineering,
Sun Yat-sen University, Guangzhou 510275, China.  Zikang Tang
<EMAIL_ADDRESS>Joint Key Laboratory of the Ministry of Education, Institute
of Applied Physics and Materials Engineering, University of Macau, Avenida da
Universidade, Taipa, Macao SAR 999078, China.  Hai-Feng Li
<EMAIL_ADDRESS>Joint Key Laboratory of the Ministry of Education,
Institute of Applied Physics and Materials Engineering, University of Macau,
Avenida da Universidade, Taipa, Macao SAR 999078, China.

###### Abstract

Magnetization measurements and time-of-flight neutron powder-diffraction
studies on the high-temperature (300–980 K) magnetism and crystal structure
(321–1200 K) of a pulverized YCrO3 single crystal have been performed.
Temperature-dependent inverse magnetic susceptibility coincides with a
piecewise linear function with five regimes, with which we fit a Curie-Weiss
law and calculate the frustration factor $f$. The fit results indicate a
formation of magnetic polarons between 300 and 540 K and a very strong
magnetic frustration. By including one factor $\eta$ that represents the
degree of spin interactions into the Brillouin function, we can fit well the
applied-magnetic-field dependence of magnetization. No structural phase
transition was observed from 321 to 1200 K. The average thermal expansions of
lattice configurations (_a_ , _b_ , _c_ , and _V_) obey well the
Gr$\ddot{\textrm{u}}$neisen approximations with an anomaly appearing around
900 K, implying an isosymmetric structural phase transition, and display an
anisotropic character along the crystallographic _a_ , _b_ , and _c_ axes with
the incompressibility $K^{a}_{0}>K^{c}_{0}>K^{b}_{0}$. It is interesting to
find that at 321 K, the local distortion size $\Delta$(O2) $\approx$
1.96$\Delta$(O1) $\approx$ 4.32$\Delta$(Y) $\approx$ 293.89$\Delta$(Cr). Based
on the refined Y-O and Cr-O bond lengths, we deduce the local distortion
environments and modes of Y, Cr, O1, and O2 ions. Especially, the Y and O2
ions display obvious atomic displacement and charge subduction, which may shed
light on the dielectric property of the YCrO3 compound. Additionally, by
comparing Kramers Mn3+ with non-Kramers Cr3+ ions, it is noted that being a
Kramers or non-Kramers ion can strongly affect the local distortion size,
whereas, it may not be able to change the detailed distortion mode.

##  I I. Introduction

Figure 1: (a) Measured magnetization (_M_) of chromium ions in single-crystal
YCrO3 compound (circles) with an increase of temperature at $\mu_{0}H=$ 0.3 T
($\sim 40008$ circles overlap each other so most of the error bars are
embedded into the symbols). The solid line is a fit by Eq. (1) as described in
the text. (b) Corresponding inverse magnetic susceptibility $\chi^{-1}$
(circles) of chromium ions in single-crystal YCrO3 compound versus
temperature. The solid line indicates a CW behavior of the data as described
by Eq. (2) from 300 to 980 K, which was extrapolated to $\chi^{-1}=0$ (dashed
line) to show the PM Curie temperatures $\theta_{\textrm{CW}}$.

The cooperating and competing interactions between charge, spin, lattice, and
orbital degrees of freedom in condensed-matter science result in a variety of
exotic macroscopic properties such as multiferroicity that continues to be an
exciting field of research Fiebig2002 ; Kimura2003 ; Hur2004 ; Cheong2007 ;
Kagawa2010 ; Valencia2011 ; Tokunaga2012 , urging experimental and theoretical
scientists to unravel the complicated electron correlations and obtain a full
understanding of the origin of the enduring phenomena.

Multiferroic materials are so called because multiple long-range orders such
as magnetic and ferroelectric within them all coexist and/or interact with
each other to give the substance extraordinary properties. They are
potentially valuable materials for promising multifunctional magnetic and
electric devices in information storage Cheong2007 ; Kagawa2010 ; Valencia2011
; Tokunaga2012 ; Jones2014 ; Zhao2014  and provide a mode for information
writing and reading Eerenstein2006 ; Nan2008 ; Dong2015 ; Spaldin2019 .
Multiferroic materials can be classified into two types, based on the detailed
couplings. In the first type, magnetism and ferroelectricity have different
microscopic origins and thus a weak interaction existing between them. In such
kinds of materials like YMnO3, the ferroelectricity was usually caused by
symmetry breaking Aken2004 ; Gibbs2011 . By contrast, in the second type of
multiferroics, magnetism and ferroelectricity are strongly coupled with each
other, and a particular magnetic order may lead to the appearance of
ferroelectricity, like in the case of TbMnO3 Kenzelmann2005 . As is well
known, the magnetization in multiferroic materials is localized and originates
from unpaired electrons in the partially filled _d_ \- or _f_ -shell. Previous
studies report many microscopic mechanisms for the formation of
ferroelectricity such as charge order Choi2008 , lone electron pairs Hill200 ,
structural geometry Aken2004 , magnetoelectric coupling Mostovoy2006 , and
oxygen vacancy Aschauer2013 .

The ABO3 system with a perovskite-type crystalline structure constructs a
significant part of the multiferroic materials. Among them, the BiFeO3 thin
film has been believed to be a typical representative in the research field of
multiferroics. It simultaneously holds both a ferroelectric transition
($T_{E}$ = 1103 K) and a large remanent polarization ($P_{\textrm{r}}$ $\sim$
55 $\mu$C/cm2) above room temperature, which was attributed to the ordering of
lone electron pairs Wang2003 . The YMnO3 compound accommodates both a
ferroelectric transition at $\sim$ 914 K and an antiferromagnetic (AFM)
transition at $\sim$ 76 K Katsufuji2001 , and its ferroelectric polarization
(_P_ $\sim$ 6 $\mu$C/cm2) was suggested to be caused by the geometric
variation, i.e., the tilting and distortion of MnO5 blocks modifies the
interaction between Y and O ions and thus forms the unbalanced dipoles of Y
ions Aken2004 ; Fennie2005 ; Gibbs2011 . A very strong magnetoelectric effect
exits in the TbMnO3 compound and its charge polarization was induced by the
specific magnetic order Kimura2003 . By changing chemical pressure, i.e.,
substituting the elements located at the crystallographic A or B site, a wide
range of crystalline structures with potential multiferroic property could be
realized Fedorova2018 .

Ferroelectricity is hard to coexist theoretically with magnetism in one
perovskite so understanding the incompatible factors that are responsible for
both properties has been a central topic in the area of multiferroics
Cheong2007 ; Ederer2011 ; Balke2012 . A perovskite-type YCrO3 compound was
reported in 1954 Looby1954 . This compound is likewise a rare system to retain
ferroelectricity and magnetism simultaneously. The YCrO3 compound shows a
magnetic phase transition at $T_{\textrm{N}}\approx$ 141.5 K; the applied-
field dependent magnetization shows an unsaturated magnetic moment of mere
$\sim$ 0.096 $\mu$${}_{\textrm{B}}$ per Cr3+ ion at 2 K and 7 T, which
indicates that the YCrO3 compound may behave like a canted AFM state below the
_T_ ${}_{\textrm{N}}$; with regard to the low-temperature magnetic structure
as well as the corresponding spin interaction parameters, the Dzyaloshinskii-
Moriya interaction mechanism has been considered in a further report on our
inelastic neutron scattering study of the YCrO3 compound Jin2019 . A
dielectric anomaly of the YCrO3 compound was observed at $\sim$ 473 K with
polarization _P_ $\sim$ 2 $\mu$C/cm2, which was ascribed to the local Cr-ion
off-centering displacement Serrao2005 . Electric polarization was reported in
the orthorhombic GdCrO3 polycrystalline samples below the canted AFM phase
transition temperature _T_ ${}_{\textrm{N}}$ $\sim$ 167 K Rajeswaran2012 ,
which was accompanied by anomalies of both the Gd-Cr and Gd-O bond lengths
Mahana2017 ; Mahana2018 . The ferroelectricity of the GdCrO3 compound (_P_
$\sim$ 0.7 ${\mu}$C/cm2) Rajeswaran2012  was proposed to be induced by the
magnetic coupling between 4 _f_ and 3 _d_ electrons, as well as the atomic
off-center displacement of a Gd site Mahana2017 ; Mahana2018 . It is pointed
out that most of the previous research works on YCrO3 compounds were focused
on polycrystals due to the difficulty in growing large and high-quality
single-crystalline samples, and the high-temperature magnetism and
crystallographic information were seldom studied, though such kind of
information is essential in completely understanding the previously observed
dielectric anomaly Serrao2005 .

Figure 2: (a) Inverse magnetic susceptibility $\chi^{-1}$ (circles) of
chromium ions in single-crystal YCrO3 compound versus temperature. The solid
lines indicate CW behaviors of the data as described by Eq. (2) at respective
temperature regimes of 300–400 K and 750–980 K. They were extrapolated down to
$\chi^{-1}=0$ (dash-dotted lines) to show the PM Curie temperatures
$\theta_{\textrm{CW}}$ and up to 1020 K (dash-dotted line). The fit results
are listed in Table I. (b) Inverse magnetic susceptibility $\chi^{-1}$
(circles) of chromium ions in single-crystal YCrO3 compound versus
temperature. The solid lines indicate CW behaviors of the data as described by
Eq. (2) at respective temperature regimes of 300–400 K, 400–540 K, 540–640 K,
640–750 K, and 750–980 K. They were extrapolated to $\chi^{-1}=0$ (dashed
lines) to show the PM Curie temperatures $\theta_{\textrm{CW}}$. The fit
results are listed in Table I.

In this paper, the magnetization, crystal structure, thermal expansion, and
local crystalline distortion of a pulverized YCrO3 single crystal have been
studied by PPMS DynaCool characterizations (300–980 K) and time-of-flight
neutron powder-diffraction studies (321–1200 K). With our modified Brillouin
function that includes a parameter $\eta$ representing the degree of magnetic
interactions and a Curie-Weiss (CW) law, we uniquely determine the detailed
magnetic parameters such as the effective paramagnetic (PM) moment, PM CW
temperature, the parameter $\eta$, and the frustration factor _f_ to
quantitatively understand the magnetism in YCrO3 compounds. The space group of
the crystal structure keeps $Pmnb$ in the entire temperature range.
Anisotropic thermal expansion exists along the crystallographic _a_ , _b_ ,
and _c_ axes with the largest incompressibility $K_{0}$ along the $a$ axis,
demonstrated by our fitting with the first-order Gr$\ddot{\textrm{u}}$neisen
function taking into account only the phonon contribution for an insulator.
The Y, O1, and O2 ions show very large local distortion size $\Delta$. We
extract the detailed local distortion modes of Y, Cr, O1, and O2 ions. It is
noted that distinct atomic displacement and a large charge subduction exiting
for the Y and O2 ions are indicative of their important roles in producing the
dielectric anomaly of YCrO3 compound.

##  II II. Experimental

Polycrystalline samples of YCrO3 were prepared from stoichiometric mixtures of
raw Y2O3 (ALFA AESAR, 99.9%) and Cr2O3 (ALFA AESAR, 99.6%) compounds by
traditional solid-state reaction method Li2008 . After milling and mixing by a
Vibratory Micro Mill (FRITSCH PULVERISETTE 0), the mixture was heated at 1000
∘C for 24 h with an increasing and decreasing temperature speed of 200 ∘C/h in
air to perform the process of pre-reaction. A similar heating procedure was
carried out at 1100 ∘C. After that, the resultant green mixture was
isostatically pressed into a $\sim$ 12 cm cylindrical rod with a pressure of
70 MPa. The rod was then sintered once at 1300 ∘C for 36 h in air. With above
firing steps and milling and mixing with a ball of 50 mm in diameter after
each heating process, we finally obtained a dense and homogenous pure
polycrystalline YCrO3 phase. High-quality single crystals of YCrO3 were grown
by the floating-zone (FZ) technique Li2008 ; Li2009  with a laser diode FZ
furnace (Model: LD-FZ-5-200W-VPO-PC-UM) Zhu2019 .

The DC magnetization was measured with a temperature increasing speed of 1
K/min at 0.3 T in the temperature range from 300 to 980 K on a Quantum Design
Physical Property Measurement System (PPMS DynaCool instrument). The _M_
curves at 300, 500, 700, and 900 K versus applied magnetic field ($\mu_{0}H$)
up to 14 T were also recorded.

Figure 3: Measured magnetization per chromium ion in single-crystal YCrO3
compound (circles) as a function of applied magnetic fields up to 14 T at 300,
500, 700, and 900 K. The dashed lines are fits to Eq. (3). See detailed
analysis in the text. Error bars are standard deviations and embedded into the
circles because collected data points ($\sim$ 1600) at respective temperatures
overlap each other. Table I: Theoretical quantum numbers for YCrO3 compound:
spin _S_ and the Land$\acute{\textrm{e}}$ factor $g_{J}$. We summarize the
theoretical (theo) and measured (meas) (Fig. 1) values of effective (eff)
chromium moment, $\mu_{\textrm{eff}}$, and PM Curie temperature,
$\theta_{\textrm{CW}}$. $R^{2}$ represents the goodness of fit.
$T_{\textrm{N}}=141.5$(1) K was extracted from our low-temperature (5–300 K)
magnetization measurement. We also calculated the magnetic frustration factor
_f_. The $\eta$ factor is from Eq. (3). See detailed analyses in the text. The
numbers in parenthesis are the estimated standard deviations of the last
significant digit. A YCrO3 single crystal
---
$S$ | 3/2
$g_{J}$ | 2
$\mu_{\textrm{eff{\\_}theo}}$ $(\mu_{\textrm{B}})$ | 3.873
_M_ versus _T_ @ 0.3 T
_T_ (K) | 300–400 | 400–540 | 540–640 | 640–750 | 750–980
$\mu_{\textrm{eff{\\_}meas}}$ $(\mu_{\textrm{B}})$ | 4.09(1) | 3.97(1) | 3.86(1) | 3.70(1) | 3.47(1)
$\theta_{\textrm{CW}}$ (K) | –331.6(1) | –290.3(1) | –245.7(2) | –172.0(2) | –64.8(2)
$f=|\theta_{\textrm{CW}}|/T_{\textrm{N}}$ | 2.34(1) | 2.05(1) | 1.74(1) | 1.22(1) | 0.46(1)
$R^{2}$ | 0.999 | 0.999 | 0.999 | 0.999 | 0.999
_M_ versus $\mu_{\textrm{0}}H$ @ _T_
_T_ (K) | 300 | 500 | 700 | 900 |
$\eta$ | 0.50(1) | 0.63(1) | 0.70(1) | 0.72(1) |

We got the powder sample for structural study by carefully pulverizing one as-
grown YCrO3 single crystal. High-resolution time-of-flight neutron powder-
diffraction studies were carried out at the POWGEN diffractometer (SNS, USA)
from 321 to 1200 K to explore the detailed temperature-dependent structural
information. A large $Q$ ($=2\pi/d$) band was measured from 0.759 to 3.695
Å-1. Neutrons are very sensitive to oxygen atoms in comparsion to x rays, and
time-of-flight neutrons naturally overcome the $\lambda/2$ problem. Therefore,
the band at high _Q_ regime is able to monitor all possible structural
modifications. We use the FULLPROF SUITE Carvajal1993  to refine all collected
time-of-flight neutron powder-diffraction data. The scale factor, lattice
constants, zero epithermal shift, and background contribution determined using
a linear interpolation between automatically selected 30 points, peak profile
shape, atomic positions, isotropic thermal parameters, and preferred
orientation were all refined in the final analysis step.

Figure 4: Observed (circles) and calculated (solid lines) time-of-flight
neutron powder-diffraction patterns of a pulverized YCrO3 single crystal,
collected on the POWGEN diffractometer (SNS, USA) at 321, 750, and 1200 K. The
vertical bars mark the positions of nuclear Bragg reflections (space group
$Pmnb$). The lower curves represent the difference between observed and
calculated patterns. Figure 5: Crystal structure (space group $Pmnb$) with one
unit cell (solid line) of the YCrO3 single crystal within the present
experimental accuracy at the studied temperature regime of 321–1200 K. The
solid balls were labeled as Y, Cr, O1, and O2 ions, respectively.

##  III III. Results and discussion

###  III.1 A. Magnetization versus temperature

Table II: Representative refined structural parameters, including lattice
constants, unit-cell volume, atomic positions, isotropic thermal parameters
(_B_), bond lengths, bond angles, and distortion parameter ($\Delta$) of CrO6
octahedra, of the pulverized single-crystal YCrO3 compound by the FULLPROF
SUITE Carvajal1993  with crystal structure (_Pmnb_) as shown in Fig. 5 at 321,
750, and 1200 K. The Wyckoff site of each ion and the relevant reliability
factors were also listed. The numbers in parenthesis are the estimated
standard deviations of the last significant digit. Pulverized YCrO3 single
crystal
---
(Orthorhombic, space group $Pmnb$, $Z=4$)
$T$ (K) | 321 | 750 | 1200
$a$ (Å) | 7.5332(3) | 7.5615(3) | 7.5976(3)
$b$ (Å) | 5.5213(2) | 5.5280(2) | 5.5367(3)
$c$ (Å) | 5.2418(2) | 5.2636(2) | 5.2921(2)
$\alpha(\beta,\gamma)$ $(^{\circ})$ | 90 | 90 | 90
$V$ (Å3) | 218.02(1) | 220.02(2) | 222.61(2)
Y | 4 _c_ | 4 _c_ | 4 _c_
_x_ | 0.25 | 0.25 | 0.25
_y_ | 0.0689(8) | 0.0675(9) | 0.0649(10)
_z_ | –0.0169(6) | –0.0155(7) | –0.0142(8)
_B_ (Å2) | 4.4(13) | 5.7(17) | 5.4(19)
Cr | 4 _b_ | 4 _b_ | 4 _b_
$(x,y,z)$ | (0, 0, 0.5) | (0, 0, 0.5) | (0, 0, 0.5)
_B_ (Å2) | 3.8(12) | 4.8(15) | 4.5(16)
O1 | 4 _c_ | 4 _c_ | 4 _c_
_x_ | 0.25 | 0.25 | 0.25
_y_ | 0.4654(7) | 0.4660(9) | 0.4668(10)
_z_ | 0.1053(8) | 0.1039(9) | 0.1017(11)
_B_ (Å2) | 3.9(14) | 5.2(18) | 5.1(19)
O2 | 8 _d_ | 8 _d_ | 8 _d_
_x_ | 0.0545(4) | 0.0531(5) | 0.0523(6)
_y_ | 0.3039(5) | 0.3039(5) | 0.3044(6)
_z_ | –0.3082(5) | –0.3074(6) | –0.3070(7)
_B_ (Å2) | 3.9(14) | 5.2(18) | 5.1(19)
Y-O11 (Å) | 2.232(5) | 2.238(6) | 2.250(7)
Y-O12 (Å) | 2.281(6) | 2.291(7) | 2.308(8)
Y-O21 (Å) ($\times 2$) | 2.269(4) | 2.282(5) | 2.287(6)
Y-O22 (Å) ($\times 2$) | 2.487(4) | 2.507(5) | 2.533(6)
$<$Y-O$>$ (Å) | 2.338(2) | 2.351(2) | 2.366(3)
Cr-O1 (Å) ($\times 2$) | 1.972(1) | 1.977(1) | 1.983(2)
Cr-O21 (Å) ($\times 2$) | 1.988(3) | 1.989(3) | 1.993(4)
Cr-O22 (Å) ($\times 2$) | 1.999(3) | 2.003(3) | 2.010(3)
$<$Cr-O$>$ (Å) | 1.986(1) | 1.990(1) | 1.995(1)
$\angle$Cr-O1-Cr $(^{\circ})$ | 145.54(5) | 145.98(6) | 146.66(7)
$\angle$Cr-O2-Cr $(^{\circ})$ | 145.46(11) | 145.96(12) | 146.15(14)
$\Delta$(Cr) $(\times 10^{-4})$ | 0.306 | 0.285 | 0.329
$R_{\textrm{p}}$ | 7.83 | 7.44 | 6.88
$R_{\textrm{wp}}$ | 8.12 | 8.09 | 7.58
$R_{\textrm{exp}}$ | 4.49 | 4.31 | 4.06
$\chi^{2}$ | 3.26 | 3.52 | 3.49

To explore possible magnetoelectric coupling effect Zheng2004 ; Lilienblum2015
in a single-crystal YCrO3 compound, we performed a magnetization measurement
at DC field of 0.3 T from 300 to 980 K. That broad temperature regime is far
above the weak ferromagnetic (FM) transition temperature, $\sim$ 140 K, so we
could ignore magnetic anisotropy, and covers the temperature point of $\sim$
473 K, at which the dielectric anomaly was reported Serrao2005 . As shown in
Fig. 1(a), we transferred the measured magnetization into magnetic moment
$\mu_{\textrm{B}}$ per Cr3+ ion. With an increase of temperature, the
magnetization decreases smoothly in the whole temperature range. The data
collected in pure PM state can be fit to

$\displaystyle\emph{M}=\frac{m}{T-\theta_{\textrm{CW}}},$ (1)

where _m_ is a constant, and $\theta_{\textrm{CW}}$ is the PM CW temperature.
The diamagnetism of Cr3+ ions is a temperature-independent constant, $\sim$
–1.1 $\times$ 10-5 emu/mol Bain2008 , which is quite tiny so it can be
neglected reasonably. Actually, the measured Cr3+ magnetization at 980 K and
0.3 T is equal to 4.33(1) emu/mol in this study, by far larger than its
diamagnetism. We tried to fit the measured data, shown as the solid line in
Fig. 1(a). This results in $\theta_{\textrm{CW}}=-264.0(1)$ K [Fig. 1(b)], and
the goodness of fit $\chi^{2}=68.629$. As shown in Fig. 1, it is obvious that
the fit is not comparable to the collected data, indicating that the CW
temperature should have multiple components as a change in temperature. To
confirm this, we included a constant, $M_{\textrm{Cr}}$, into the Eq. (1) to
express the temperature-independent ordered magnetism of Cr3+ ions Yin2015 ;
Dash2017 , though the YCrO3 compound stays in a pure PM state from 300 to 980
K. With the modified equation, we fit our temperature-dependent magnetization
data and the resultant $M_{\textrm{Cr}}=$ –2.329(2) emu/mol, much smaller than
the Cr3+ diamagnetism, which is physical nonsense.

The linear increase of the inverse magnetic susceptibility
$\chi^{-1}=\frac{\mu_{0}H}{M}$ with temperature in the pure PM state obeys
well the molar susceptibility by CW law Li2009 ; Li2014 ; Li2015 ,

$\displaystyle\chi^{-1}(T)=\frac{3k_{B}(T-\theta_{\textrm{CW}})}{N_{A}\mu^{2}_{\textrm{eff}}},$
(2)

where $k_{B}$ = 1.38062 $\times$ 10-23 J/K is the Boltzmann constant, $N_{A}$
= 6.022 $\times$ 1023 mol-1 is the Avogadro’s number, and $\mu_{\textrm{eff}}$
= $g\mu_{\textrm{B}}\sqrt{J(J+1})$ is the effective PM moment. Here $J=S$ for
the YCrO3 compound. As shown in Fig. 2(a), we first fit the data in the
temperature regimes of 300–400 K and 750–980 K (solid lines) and extrapolated
them to the whole temperature range (dash-dotted lines). The two lines
intersect at $T\sim$ 620 K and are not able to cover all features of the data.
In addition, there exists no anomaly in crystallographic information at 620 K
as discussed below. Therefore, it is evident that $\chi^{-1}$ versus _T_ obeys
a piecewise linear function as shown in Fig. 2(b). Only if we separated the
inverse magnetic susceptibility into five regimes were we able to model all
features and fit well the data by Eq. (2). The corresponding fit results as
well as the goodness of fit, $R^{2}$, are listed in Table I. With increasing
the temperature ranges, the measured effective PM moment,
$\mu_{\textrm{eff{\\_}meas}}$, decreases from 4.09(1) (300–400 K) to 3.47(1)
$\mu_{\textrm{B}}$ (750–980 K), whereas the corresponding PM CW temperature,
$\theta_{\textrm{CW}}$, increases from –331.6(1) (300–400 K) to –64.8(2) K
(750–980 K). In the temperature ranges of 300–400 K and 400–540 K, the
measured effective PM moments $\mu_{\textrm{eff{\\_}meas}}=$ 4.09(1) and
3.97(1) $\mu_{\textrm{B}}$, respectively. Both values are larger than the
calculated theoretical one $\mu_{\textrm{eff{\\_}theo}}=$ 3.873
$\mu_{\textrm{B}}$ supposing that all valence states are ionic and integer.
The enhancement may be attributed to unquenched orbital angular momentum or
the existence of local FM clusters with short-range spin interactions, i.e.,
magnetic polarons Li2009 ; Teresa1997 ; Nogues2001 ; Li2012 . In the
temperature range 540–640 K, $\mu_{\textrm{eff{\\_}meas}}=$ 3.86(1)
$\mu_{\textrm{B}}$ is consistent with $\mu_{\textrm{eff{\\_}theo}}$. At
elevated temperature regimes of 640–750 and 750–980 K,
$\mu_{\textrm{eff{\\_}meas}}$ becomes smaller than
$\mu_{\textrm{eff{\\_}theo}}$. Based on these observations, it is deduced that
the above hypothesis on the formation of magnetic polarons is more reasonable.
The increase of temperature easily destroys magnetic interactions, i.e., the
forming ground of magnetic polarons.

It is interesting that in the whole studied temperature range, all deduced PM
CW temperatures, $\theta_{\textrm{CW}}$, are negative, indicating an AFM
interaction, and largely deviated from the weak FM transition temperature
$T_{\textrm{N}}=141.5$(1) K that was obtained from our magnetization study on
the same single crystal at low temperatures of 5–300 K. The coappearance of
AFM and FM phenomena may indicate a canted AFM state with strong magnetic
frustration that can be characterized by a frustration parameter
$f=\frac{|\theta_{\textrm{CW}}|}{T_{\textrm{N}}}$. The larger (than 1) the
value of $f$, the stronger the corresponding magnetic frustration is
Ramirez2001 . As listed in Table I, the _f_ values are all larger than 1
except for the one in the temperature range 750–980 K, implying an existence
of strong magnetic frustration and a complex low-temperature magnetic
structure.

The values of $f$ for the RMnO3 compounds were reported to be 10.1 (YMnO3),
10.3 (LuMnO3), and 7.8 (ScMnO3) Katsufuji2001 . Within these compounds, the Mn
ions form a triangular lattice in the hexagonal structure, therefore, there
exists a geometrical spin frustration. It is pointed out that the
crystallographic structure of YCrO3 compound doesn’t accommodate any geometric
frustration Ramirez1994 ; Diep2004 ; Shores2005 ; HFLi2014 ; Li2015 . The
magnetic frustration in YCrO3 compound originates from the observation of AFM
and FM behaviors, which necessitates a determination of the detailed low-
temperature magnetic structure.

###  III.2 B. Magnetization versus applied magnetic field

In this paper, we focus on the measurement of high-temperature ($\geq$ 300 K)
magnetic properties. As shown in Fig. 3, we monitored the magnetization as a
function of applied magnetic field up to 14 T at 300, 500, 700, and 900 K with
a small piece of randomly orientated single crystal ($\sim$ 14.91 mg). While
increasing applied magnetic field, the magnetization increases linearly at all
temperatures. The measured magnetic moment at 300 K and 14 T is 0.079(1)
$\mu_{\textrm{B}}$/Cr, mere $\sim$ 2.63% of the theoretical saturated (sat)
Cr3+ moment $\mu_{\textrm{sat{\\_}theo}}=g_{J}S\mu_{\textrm{B}}=$ 3
$\mu_{\textrm{B}}$ (Table I).

For theoretically isolated atoms, the change of magnetization with applied
magnetic field at high temperatures obeys a Brillouin function given by
Stephen2001

$\displaystyle M(\mu_{\textrm{0}}H)=\eta
M^{\textrm{sat}}_{\textrm{theo}}B_{J}(x),\ \ \ \ \ \ \ \textrm{with}$
$\displaystyle
B_{J}(x)=\frac{2J+1}{2J}\textrm{coth}{{{(}}}\frac{2J+1}{2J}x{{{)}}}-\frac{1}{2J}\textrm{coth}{{{(}}}\frac{1}{2J}x{{{)}}},$
(3)

where $M^{\rm sat}_{\rm theo}=g_{J}J\mu_{\textrm{B}}=$ 3 $\mu_{\textrm{B}}$ is
the theoretical value of the saturated mole moment, $J=\frac{3}{2}$ is the
total angular momentum, and $x=\frac{g_{J}J\mu_{\rm
B}\mu_{\textrm{0}}H}{k_{B}T}$. Equations (3) were used to fit the collected
data as shown in Fig. 3 (dashed lines). The corresponding $\eta$ values at
respective temperatures are listed in Table I. When $\eta=$ 1, i.e., the
theoretical case, there exists no magnetic exchange. With a decrease of
temperature, spin interactions and the resultant magnetic order become
possible. Since rotating and aligning spin moments of an antiferromagnet
necessitate a very strong applied magnetic field, depending on detailed
exchange parameters Tian2010 ; Toft2012 ; HFLi2016 , measured magnetization
will deviate from the value of the saturation moment, leading to the $\eta$
value getting smaller than 1 and becoming smaller and smaller with an increase
of AFM domains. As temperature decreases, the $\eta$ values decrease, in
agreement with the increase of _f_ factors (Table I).

Figure 6: Temperature dependence of the lattice constants, _a_ , _b_ , and _c_
, of the pulverized YCrO3 single crystal (void symbols), which was extracted
from our FULLPROF Carvajal1993  refinements based on the time-of-flight
neutron powder-diffraction data collected on POWGEN diffractometer (SNS, USA)
between 321 and 1200 K. The solid lines are theoretical estimates of the
variation of structural parameters at the respective temperature regimes of
321–900 K and 900–1200 K, using the Gr$\ddot{\textrm{u}}$neisen model [Eqs.
(4) and (5)] with Debye temperature $\theta_{\textrm{D}}$ = 580 K, and
extrapolated to the entire temperature range of 321–1200 K (dashed lines).
Error bars are standard deviations obtained from our FULLPROF Carvajal1993
refinements in the _Pmnb_ symmetry. Figure 7: Temperature-dependent unit-cell
volume, _V_ , of the YCrO3 single crystal (void symbols). This was extracted
from our FULLPROF Carvajal1993  refinements based on the time-of-flight
neutron powder-diffraction data collected on POWGEN diffractometer (SNS, USA)
between 321 and 1200 K. The solid lines are theoretical estimates of the
variation of _V_ within respective temperature regimes using the
Gr$\ddot{\textrm{u}}$neisen model [Eqs. (4) and (5)] with Debye temperature
$\theta_{\textrm{D}}$ = 580 K and extrapolated to the whole temperature range
(dashed lines). Error bars are standard deviations obtained from our FULLPROF
Carvajal1993  refinements in the _Pmnb_ symmetry.

###  III.3 C. Time-of-flight neutron powder diffraction

Table III: Fit parameters of the lattice configuration $\epsilon$ ($a,b,c$, and $V$) of YCrO3 compound with Eqs. (4) and (5), where $N=$ 5 and $k_{B}=1.38062\times 10^{-23}$ J/K, at the respective temperature regimes of 321–900 K and 900–1200 K [Figs. 6 and 7].  $T$ (K) | 321–900 | 900–1200
---|---|---
$\Theta_{D}$ (K) | 580 | 580
$\epsilon^{a}_{0}$ (Å) | 7.52180(2) | 7.51080(7)
$\epsilon^{b}_{0}$ (Å) | 5.51790(4) | 5.51580(1)
$\epsilon^{c}_{0}$ (Å) | 5.23320(3) | 5.22330(5)
$\epsilon^{V}_{0}$ (Å3) | 217.190(1) | 216.350(9)
$K^{a}_{0}$ (Å/J) | $3.380(7)\times 10^{17}$ | $4.160(5)\times 10^{17}$
$K^{b}_{0}$ (Å/J) | $8.390(1)\times 10^{16}$ | $9.910(8)\times 10^{16}$
$K^{c}_{0}$ (Å/J) | $2.590(4)\times 10^{17}$ | $3.300(8)\times 10^{17}$
$K^{V}_{0}$ (Å3/J) | $2.400(7)\times 10^{19}$ | $3.000(2)\times 10^{19}$

To explore possible structural phase transitions of YCrO3 compounds above room
temperature, we carried out a time-of-flight neutron powder-diffraction study
from 321 to 1200 K. Three representative neutron powder-diffraction patterns
are shown in Fig. 4. We carefully checked the Bragg peak shape, especially for
the peaks located in the low-_d_ regime. All collected Bragg peaks were well
indexed with the space group _Pmnb_ (orthorhombic structure). No additional
peak and possible peak splitting were observed. We therefore conclude that
within the present experimental accuracy, there exists no structural phase
transition for the YCrO3 compound from 321 to 1200 K. The refined crystal
structure in one unit cell was depicted in Fig. 5. The refined structural
parameters at 321, 750, and 1200 K, including lattice constants, unit-cell
volume, atomic positions, and thermal parameters are all listed in Table II,
where the extracted bond lengths, bond angles, and the distortion parameter of
CrO6 octahedra are also displayed. The low values of the reliability factors
validate the goodness of our refinements.

###  III.4 D. Anisotropic thermal expansion

Figure 8: Temperature-dependent bond lengths of Cr-O1, Cr-O21, and Cr-O22 as
well as the averaged bond length of Cr-O, i.e., $<$Cr-O$>$, of the single-
crystal YCrO3 compound (void symbols). This was extracted from our time-of-
flight neutron powder-diffraction study. Error bars are standard (for the
Cr-O1, Cr-O21, and Cr-O22 bond lengths)/combined (for the Cr-O bond length)
deviations. Solid lines are linear fits. Figure 9: Lengths of Y-O11, Y-O12,
Y-O21, and Y-O22 bonds of the single-crystal YCrO3 compound versus temperature
varying from 321 to 1200 K (void symbols), which was extracted from our time-
of-flight neutron powder-diffraction study. Error bars are standard
deviations. Solid lines are linear fits. Figure 10: Temperature variation of
the distortion parameter $\Delta$ of Y, Cr, O1, and O2 ions of the single-
crystal YCrO3 compound (void symbols), calculated by Eq. (6) from the refined
structural parameters between 321 and 1200 K. The error bar was estimated
based on the propagation law of errors Zhu2019-E . The solid lines are
tentative linear fits.

Figure 6 shows our refined lattice constants, _a_ , _b_ , and _c_ , at all
temperatures from 321 to 1200 K. The corresponding change in unit-cell volume
(_V_) was depicted in Fig. 7. For the insulating YCrO3 compound, the
contribution of lattice vibrations to the thermal expansion of the lattice
configuration ($\varepsilon$) is much larger than that of electrons.
Therefore, the temperature-dependent nonmagnetic contribution component of the
thermal expansion is mainly from phonons. This can approximately be calculated
based on the Gr$\ddot{\textrm{u}}$neisen function at zero pressure with the
first-order fashion Li2012 ; Vocadlo2002 ; Wallace1998 ,

$\displaystyle\varepsilon(T)=\varepsilon_{0}+K_{0}U,$ (4)

where $\varepsilon_{0}$ is the lattice configuration at 0 K, $K_{0}$ is a
constant that reflects the incompressibility of the compound, and the internal
energy _U_ can be estimated based on the Debye approximations,

$\displaystyle
U(T)=9Nk_{B}T{{{(}}}\frac{T}{\Theta_{D}}{{{)}}}^{3}\int^{\frac{\Theta_{D}}{T}}_{0}\frac{x^{3}}{e^{x}-1}dx,$
(5)

where _N_ = 5 is the number of atoms per formula unit, and $\Theta_{D}=$ 580 K
is the Debye temperature that can be determined by the upturn point of the
$\varepsilon-T$ curve Li2012 . With the above Eqs. (4) and (5), we fit the
lattice configurations (_a_ , _b_ , _c_ , and _V_) of YCrO3 compound in two
temperature ranges of 321–900 K and 900–1200 K (solid lines) and extrapolated
the fits to the whole temperature regime (dashed lines) as shown in Figs. 6
and 7. The fit results are shown in Table III. Within each temperature regime,
as a whole, the temperature-dependent lattice configurations agree well with
the theoretical estimations. However, at the boundary of the two temperature
regimes, i.e., around 900 K, there exists an anomaly.

For lattice constants _a_ , _b_ , and _c_ , the incompressibility $K_{0}$ in
the temperature range 900–1200 K is larger than that in $T=$ 321–900 K, which
may be attributed to more developed phonon modes above 900 K. In both
temperature regimes, $K^{a}_{0}>K^{c}_{0}>K^{b}_{0}$ indicates an anisotropic
thermal expansion along the three crystallographic directions. This jointly
results in the unit-cell volume expansion as shown in Fig. 7.
($K^{a}_{0(900{-}1200\textrm{K})}$ –
$K^{a}_{0(321{-}900\textrm{K})}$)/($K^{a}_{0(321{-}900\textrm{K})}$) = 22.71%,
($K^{b}_{0(900{-}1200\textrm{K})}$ –
$K^{b}_{0(321{-}900\textrm{K})}$)/($K^{b}_{0(321{-}900\textrm{K})}$) = 18.24%,
and ($K^{c}_{0(900{-}1200\textrm{K})}$ –
$K^{c}_{0(321{-}900\textrm{K})}$)/($K^{c}_{0(321{-}900\textrm{K})}$) = 27.80%,
which implies that the temperature-dependent relative increase of $K_{0}$ is
the largest one along the crystallographic _c_ axis.

For the YMnO3 compound, a clear structural phase transition with a change in
space group from centrosymmetric $P6_{3}/mmc$ to $P6_{3}cm$ occurs at 1258(14)
K Jeong2007 ; Gibbs2011 ; Howard2013 . This structural phase transition
temperature is much higher than its improper ferroelectricity transition
temperature around 914 K and its AFM transition temperature around 76 K
Katsufuji2001 . Clear anomalies were observed in the lattice constants _a_ and
_c_ and unit-cell volume _V_ as well as the distance between Y1 and Y2 ions,
the displacements of O3 and O4 ions, the tilting angle of apical O1-Mn-O2, and
the lengths of Y1-O bonds, accompanying the structural phase transition
Gibbs2011 . Moreover, an isosymmetric phase transition was found at $\sim$ 900
K, accompanied by a sharp decrease in polarization and anomalies in physical
properties, which was attributed to a Y-O hybridization Gibbs2011 . In our
study, for the YCrO3 compound, structural anomalies were observed in lattice
configurations (_a_ , _b_ , _c_ , and _V_) as well as Cr-O and Y-O bond
lengths around 900 K (as shown below). However, no clear change in the space
group was distinguished. The preservation of the $Pmnb$ space group of YCrO3
compound may suggest that an isosymmetric structural phase transition happens
around 900 K Gibbs2011 ; Christy1995 .

###  III.5 E. Bond lengths of Cr-O and Y-O

To analyze detailed local crystalline environments of Cr and Y ions in the
YCrO3 compound, we extracted and plotted the lengths of Cr-O21, Cr-O22, and
Cr-O1 [Fig. 8] and Y-O21, Y-O22, Y-O11, and Y-O12 [Fig. 9] bonds. The averaged
Cr-O bond length, i.e., $<$Cr-O$>$, was calculated and plotted in addition
[Fig. 8]. We tentatively refined the bond lengths with a piecewise linear
function in the entire temperature regime with two ranges of 321–900 K and
900–1200 K. As shown in Figs. 8 and 9, the fit results (shown as solid lines)
clearly display an anomaly around 900 K, consistent with our observation in
the lattice configurations of _a_ , _b_ , _c_ , and _V_ [Figs. 6 and 7]. From
the temperature range of 321–900 K to 900–1200 K, the slope of the bond length
versus $T$ curve obviously increases for the Cr-O21, Cr-O1, Y-O21, Y-O22, and
Y-O12 bonds, e.g., for the Y-O22 bond, it increases from $4.42\times 10^{-5}$
to $7.23\times 10^{-5}$ Å/K, by $\sim$ 63.57%, whereas the slope sharply
decreases for the Cr-O22 and Y-O11 bonds, e.g., for the Cr-O22 bond, it
decreases by $\sim$ 62.59%.

By comparing all Cr-O and Y-O bonds, we find that only the length of Cr-O21
bond increases in an inconspicuous way from 321 to 900 K, implying a small
contribution to the thermal expansion of the lattice configuration. From 900
to 1200 K, the upturn of the lattice configuration (_a_ , _b_ , _c_ , and _V_)
was attributed to an increase of the lengths of Cr-O21, Cr-O1, Y-O21, Y-O22,
and Y-O12 bonds. It is reasonable to deduce that the anisotropic thermal
expansion in lattice constants _a_ , _b_ , and _c_ is from different increases
in lengths along the different directions of Cr-O and Y-O bonds.

Figure 11: Local pentahedron environment of Y ions in the single-crystal YCrO3
compound, which was extracted based on our FULLPROF refinements Carvajal1993 .
The Y, O11, O12, O21, and O22 ions are labeled as displayed. Detailed bond
lengths of Y-O11, Y-O12, Y-O21 ($\times 2$), and Y-O22 ($\times 2$) are listed
in Table II. The arrows sitting on the Y-O bonds schematically show the
deduced pentahedron distortion configuration. Figure 12: (a) Local octahedral
environment of Cr ion in the single-crystal YCrO3 compound, which was
extracted based on our FULLPROF refinements Carvajal1993 . The arrows drawn
through the oxygen ions ($2\times$O1, $2\times$O21, and $2\times$O22)
schematically show the deduced octahedral distortion mode. Representative
refined bond lengths of Cr-O1, Cr-O21, and Cr-O22 at 321, 750, and 1200 K are
listed in Table II. (b) In such octahedral geometry, we schematically drew the
approximate 3$d_{yz}$ orbital shape in real space. Figure 13: Local
tetrahedral environments of O1 (a) and O2 (b) ions in the single-crystal YCrO3
compound, which was extracted based on our FULLPROF refinements Carvajal1993 .
The Y, Cr, O1, and O2 ions are labeled as displayed. Detailed bond lengths of
O1-Cr ($\times 2$), O1-Y, O2-Cr, and O2-Y at 321 K were marked. The arrows
sitting on the O-Y and O-Cr bonds schematically show the deduced tetrahedral
distortion modes.

###  III.6 F. Local distortion modes of Y, Cr and O ions

Identifying a detailed local crystalline environment is essential in
determining electronic structure, spin configuration, orbital degeneracy and
crystal field effect of 3 _d_ -, 4 _d_ -, 5 _d_ -, or 4 _f_ \- compounds
Li2008 . A quantitative measurement of the magnitude of a local crystalline
environment can be evaluated by the local distortion parameter $\Delta$
defined as Li2008 ; Li2009 ; Li2007-1 ; Li2007-2

$\displaystyle\Delta=\frac{1}{n}\sum_{i=1}^{n}{{{(}}}\frac{d_{n}-\langle
d\rangle}{\langle d\rangle}{{{)}}}^{2},$ (6)

where _n_ is the coordination number, $d_{n}$ is the bond length along one of
the _n_ coordination directions, and $\langle d\rangle$ is the averaged bond
length. With Eq. (6), we calculated the local distortion parameter $\Delta$ as
a function of temperature for the Y, Cr, O1, and O2 ions as shown in Fig. 10.
For the 3 _d_ Cr3+ ions in single-crystal YCrO3 compound, it is interesting
that the local distortion parameter $\Delta$ keeps almost a constant in the
whole temperature range, displaying no response to the previously reported
dielectric anomaly Serrao2005 . The averaged $\Delta$ value of Cr3+ ions from
321 to 1200 K is of $\sim$ $3.3\times 10^{-5}$ that is approximately two
orders of magnitude lower than that of 3 _d_ Kramers Mn3+ ions in the Jahn-
Teller (JT) distorted regime of single-crystal
La${}_{\frac{7}{8}}$Sr${}_{\frac{1}{8}}$MnO3 compound Li2009 . This sharp
comparison dramatically demonstrates the importance of local crystalline
environment.

It is noted that the Y, O1, and O2 ions show huge $\Delta$ values, almost two
orders of magnitude larger than that of Cr ions. At 321 K, $\Delta$(O2)
$\approx$ 1.96$\Delta$(O1) $\approx$ 4.32$\Delta$(Y) $\approx$
293.89$\Delta$(Cr), indicating a significant local crystalline environmental
effect of O2, O1, and Y ions on the property of a YCrO3 compound. This urges
us to figure out the local distortion modes of Y, Cr, and O ions, as
schematically drawn in Figs. 11–13, respectively.

Figure 11 shows the local pentahedron environment of Y ions. It is worth
mentioning that there are six Y-O bonds at the neighbor of the Y atom, i.e.,
with a coordination number 6, two Y-O1 and four Y-O2 bonds. Among them,
2$\times$Y-O22 bonds are stretched, and Y-O11, Y-O12, and 2$\times$Y-O21 bonds
are shortened (Table II). This results in both the O11-O21-O22-O12 planes
being bent outward, shifting the charge weight center of Y ions upward, whilst
holding the 2$\times$O21 and the 2$\times$O22 ions within one O21-O22-O22-O21
plane. The extracted pentahedron distortion mode of Y ion was schematically
displayed by the arrows sitting on the six Y-O bonds.

The Cr ion in YCrO3 compound has three electrons in the unfilled 3 _d_ shell,
therefore, Cr3+ is a non-Kramers ion, in principle, without JT effect. That is
why Cr3+ ions have a very small local distortion parameter $\Delta$ [Fig. 10].
From our refined Cr-O bond lengths as listed in Table II, we deduced the
octahedral distortion mode of Cr3+ ions (coordination number: 6) in YCrO3
compound, as shown in Fig. 12(a). This distortion mode, as schematically
displayed by the arrows sitting on oxygen ions, results from stretched Cr-O21
and Cr-O22 and shortened Cr-O1 bonds, behaving like a cooperative JT
distortion. The local crystalline environment of Cr3+ ions in YCrO3 compound
coincides with the $t_{\textrm{2g}}$ orbital shapes ($d_{xy}$, $d_{zx}$,
$d_{yz}$). As shown in Fig. 12(b), the 3$d_{yz}$ orbital shape in real space
was accommodated into the CrO6 octahedron. Because the two Cr-O1/O21/O22 bonds
locate along the plane/body diagonal direction and have the same length, the
sum effect of electric-lattice interactions is canceled out. Therefore, within
the present crystal symmetry, no displacement happens to the Cr ions.

Figure 13 illustrates the local distortion environments of O1 [Fig. 13(a)] and
O2 [Fig. 13(b)] ions in single-crystal YCrO3 compound. Both O1 and O2 ions
have a coordination number 4, forming a tetrahedron. By comparing the refined
Cr-O and Y-O bond lengths (referring to Table II), we deduced two tetrahedral
distortion modes for the O1 and O2 ions, respectively, as displayed by the
arrows sitting on the Cr-O and Y-O bonds. For example, for the O1 ions [Fig.
13(a)], the 2$\times$Cr-O bonds are shortened, whereas the two Y-O bonds are
stretched. This pushes the O1 ions towards to the Cr ions. So do the O2 ions
[Fig. 13(b)]. Within the two oxygen tetrahedrons, the longest Y-O bond is Y-O2
= 2.487(4) Å (at 321 K) as displayed in Fig. 13(b). This drives the O2 ions
very close to the bottom Y-Cr-Cr plane, a large negative charge displacement.
Compared to the O1 ions, the local crystalline environment of O2 ions is much
more distorted in agreement with our calculations as shown in Fig. 10 where
the local distortion parameter $\Delta$ of O2 ions is approximately two times
larger than that of O1 ions.

The distortion parameter $\Delta$, to some extent, is a criterion for what
magnitude a certain atom displaces with its surrounding ligands Li2008 . For
example, Kramers ions usually show a JT distortion whose magnitude can be
expressed by the size of the distortion parameter $\Delta$. The JT effect
occurring in 3 _d_ transition metal oxides can lead to the degeneracy of _d_
orbitals accompanied by lowering the structural symmetry to release the
electronic occupied energy. That can result in charge/orbital ordering and
magnetic transition and may shed light on the colossal magnetoresistance
effect Li2008 ; Millis1996 . For 3 _d_ 3 Cr3+ ions in YCrO3 compounds, no JT
effect is expected, and thus the distortion parameter $\Delta$ of Cr3+ ions is
quite small, which is hard to break the centrosymmetry of the center Cr3+
ions, therefore, no ferroelectricity is expected from the Cr-sites. However,
as shown in Figs. 11 and 13(b), the centrosymmetries of Y3+ and O${}_{2}^{2-}$
ions could be broken by their large local environmental distortions, which may
induce the geometric ferroelectricity. Similar observation was also reported
on the orthorhombic GdCrO3 compound, in which the Gd3+ and O2+ ions move
toward different directions and produce a huge charge density in the Gd-O
bonds Mahana2018 .

Figure 14: Temperature variation of the bond valence states (BVSs) of Y, Cr,
O1, and O2 ions in the single-crystal YCrO3 compound, calculated from our
refined structural parameters between 321 and 1200 K by the FULLPROF SUITE
Carvajal1993 . For a clear comparsion, we also calculated the average BVSs of
O1 and O2 ions, i.e., (O1 + O2)/2. Error bars are combined standard
deviations.

###  III.7 G. Bond valence states of Y, Cr and O ions

It is well known that bond valences are strongly correlated to bond distances,
and some empirical relationships were previously proposed Carvajal1993 ;
Brown1985 ; Brese1991 ; Mohri2000 . With our refinements, we extracted the
bond valence states (BVSs) of Y, Cr, and O ions as shown in Fig. 14. As
temperature increases, the BVSs of Y, Cr, and O ions almost decrease linearly.
The calculated BVS values for the Cr and O1 ions are 2.956(7)+ and 2.079(11)–
at 321 K, close to the ideal 3+ and 2–, respectively. However, for the Y and
O2 ions, the calculated BVS values, BVS(Y) = 2.636(15)+ and BVS(O2) =
1.757(8)–, differ largely from the respective perfect values of 3+ and 2– in
pure ionic model. Therefore, there exit positive and negative charge
displacements from Y and O2 ions that coincide with their strongly distorted
local crystalline environments. In the $Pmnb$ symmetry, there are two non-
equivalent crystallographic sites for the O1 and O2 ions. As shown in Fig. 14,
the charge difference between O1 and O2 ions is approximate 0.317(15)$e^{-}$
with $\sim$ 21 standard deviations, indicative of a big degree of the charge
disproportion. It is of interest to notice that our extracted charge
difference between O1 and O2 ions is even much bigger than the charge
difference between Mn3+ and Mn4+ ions in the charge/orbital ordered states of
manganites Li2009 ; Martin2004  where for the
La${}_{\frac{7}{8}}$Sr${}_{\frac{1}{8}}$MnO3 compound Li2009 , the charge
difference is 0.11(5) $e^{-}$, and for the half-doped
Nd${}_{\frac{1}{2}}$Sr${}_{\frac{1}{2}}$MnO3 Martin2004  it is 0.16 $e^{-}$.
It is thus reasonable to deduce that the Y and O2 ions play an important role
in forming the dielectric anomaly of YCrO3 compound.

Finally, the temperature-dependent isotropic thermal parameters $B$ of Y, Cr,
and O1/O2 (constrained to be the same) ions of YCrO3 compound were present in
Fig. 15 where almost no change exists from 321 to 1200 K within the
experimental accuracy.

Figure 15: Temperature variation of the isotropic thermal parameters, _B_ , of
Y, Cr, and O1/O2 ions in the single-crystal YCrO3 compound. During our
FULLPROF Carvajal1993  refinements, we constrained the _B_ sizes of O1 and O2
ions to the same value. Error bars are standard deviations.

As is well known, the contribution of magnetism in YCrO3 compound comes from
the Cr3+ (3 _d_ 3) ions supposing that the YCrO3 compound forms pure ionic
bonds and thus there is no magnetic contribution from oxygen sites Li2008 ;
Li2009 . As in the foregoing discussion, the dielectric anomaly Serrao2005  of
YCrO3 compound may be ascribed to the obvious atomic displacement and charge
subduction of Y and O2 ions. The Y3+ ion is nonmagnetic because the ground-
state electronic configuration of neutral Y is [Kr]4 _d_ 15 _s_ 2. Therefore,
the structural parameters extracted from our time-of-flight neutron powder-
diffraction study don’t display a response to the previously observed
dielectric anomaly. On the other hand, that doesn’t mean that there is no
magnetoelectric coupling existing in YCrO3 compound. To unravel this coupling
necessitates measurements with extremely high applied magnetic fields. By
comparison, the YMnO3 compound belongs to a hexagonal symmetry with the
structural building block of MnO5, and its ferroelectric polarization is due
to opposing unequal dipoles of the two Y sites as well as the tilting and
distortion of the MnO5 blocks Aken2004 ; Fennie2005 ; Gibbs2011 . The Gd3+
(4$f^{7}$) ions in GdCrO3 compounds has a very strong magnetic contribution.
It is expected that the Gd-Cr coupling in GdCrO3 compounds is much stronger
than the Y-Cr coupling in YCrO3 compound. As foregoing remarks, the previously
reported dielectric anomaly of YCrO3 compound may be due to large atomic
displacement and charge subduction of Y and O2 ions, whereas, only Cr ions
contribute to its magnetism. That may be the reason why the structural
parameters don’t show any anomaly around 473 K.

It is stressed that the space group $Pmnb$ belongs to a centrosymmetric
structure in which the sum of negative and positive charge shifts has to be
zero. Our interesting observations, e.g., the extremely large local distortion
parameter $\Delta$ of Y, O1, and O2 ions, the obvious displacement of Y and O2
ions and the charge subduction of Y and O2 ions, indicate that the actual
structural symmetry of YCrO3 compounds may be much lower than $Pmnb$. As a
foregoing discussion, even though Kramers ions like Mn3+ theoretically and
experimentally exhibit a larger local distortion, e.g., a JT/cooperative JT
effect, than non-Kramers ions like Cr3+, their detailed distortion modes can
be the same, as demonstrated in this study. Additionally this distortion mode
can even be applicable to the 4$f$ ions HFLi2014 .

##  IV IV. Conclusions

To summarize, we have quantitatively investigated the high-temperature
(300–980 K) magnetism and structural information (321–1200 K) of a single-
crystalline YCrO3 compound. The high-temperature magnetization can only be fit
by the CW law with multiple effective PM moments and multiple PM CW
temperatures with strong magnetic frustration, implying a complicated low-
temperature magnetic structure. The magnetization versus applied-magnetic-
field curve obeys well our modified Brillouin function with an inclusion of a
factor $\eta$ denoting the strength of magnetic interactions. We refined the
crystal structure with $Pmnb$ symmetry in the entire studied temperature range
within the present experimental accuracy. Detailed structural information
including lattice constants, unit-cell volume, atomic positions, thermal
parameters, bond lengths, local distortion parameter, bond angles, local
distortion modes, and BVSs, were extracted. The thermal expansions along the
_a_ , _b_ , and _c_ axes are anisotropic with an anomaly appearing around 900
K. We attribute this anomaly to an isosymmetric structural phase transition.
The local distortion parameter $\Delta$ of Cr ions is about two orders of
magnitude lower than that of Y and O ions. We find that both Y and O2 ions
produce a clear atomic displacement and a large charge deviation from
theoretical ones. We thus suggest that the Y and O2 ions may play an important
role in forming the previously reported dielectric anomaly of the YCrO3
compound. The present results make YCrO3 a particularly significant compound
for theoretical and further experimental studies on $t_{\textrm{2g}}$ physics.
The magnetic structure and dielectric property would be further explored in
combination with theoretical calculations.

##  V Acknowledgements

S.J. and D.-X.Y. acknowledge support from NKRDPC-2018YFA0306001,
NKRDPC-2017YFA0206203, NSFC-11974432, NSFG-2019A1515011337, and Leading Talent
Program of Guangdong Special Projects. Z.T. acknowledges the start-up research
grant (Grant No. SRG2016-00002-FST) at the University of Macau and the
financial support from the Science and Technology Development Fund, Macau SAR
(File No. 063/2016/A2). D.O acknowledges financial support from the Science
and Technology Development Fund, Macau SAR (File No. 0029/2018/A1). H.-F.L
acknowledges the start-up research grant (Grant No. SRG2016-00091-FST) at the
University of Macau and financial support from the Science and Technology
Development Fund, Macau SAR (File No. 064/2016/A2, File No. 028/2017/A1, and
File No. 0051/2019/AFJ). This research used resources at the Spallation
Neutron Source, a Department of Energy Office of Science User Facility
operated by Oak Ridge National Laboratory.

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