††thanks: Corresponding author email id<EMAIL_ADDRESS>Present
address: Department of Physics, Bar-Ilan University Ramat-Gan, Israel

#  Reentrant spin reorientation transition and Griffiths$\textendash$like
phase in antiferromagnetic TbFe0.5Cr0.5O3

Bhawana Mali  Department of Physics, Indian Institute of Science, Bangalore
560012, India  Harikrishnan S. Nair  Department of Physics, 500 W. University
Ave, The University of Texas at El Paso, TX 79968, USA  T. W. Heitmann
University of Missouri Research Reactor, University of Missouri, Columbia, MO
65211, USA  Hariharan Nhalil  Department of Physics, Indian Institute of
Science, Bangalore 560012, India  Daniel Antonio  Idaho National Laboratory,
Idaho Falls, ID 83415, USA  Krzysztof Gofryk  Idaho National Laboratory, Idaho
Falls, ID 83415, USA  Shalika Ram Bhandari  Central Department of Physics,
Tribhuvan University, Kirtipur, 44613, Kathmandu, Nepal  IFW Dresden,
Helmholtzstr. 20, D-01069, Dresden, Germany  Madhav Prasad Ghimire  Central
Department of Physics, Tribhuvan University, Kirtipur, 44613, Kathmandu, Nepal
IFW Dresden, Helmholtzstr. 20, D-01069, Dresden, Germany  Suja Elizabeth
Department of Physics, Indian Institute of Science, Bangalore 560012, India

###### Abstract

The perovskite TbFe0.5Cr0.5O3 shows two anomalies in the magnetic
susceptibility at $T_{N}$ = 257 K and $T_{\mathrm{SR}}$ = 190 K which are
respectively, the antiferromagnetic and spin reorientation transition that
occur in the Fe/Cr sublattice. Analysis of the magnetic susceptibility reveals
signatures of Griffiths$\textendash$like phase in this compound: the negative
deviation from ideal Curie$\textendash$Weiss law and in
less$\textendash$than$\textendash$unity power$\textendash$law susceptibility
exponents. Neutron diffraction analysis confirms that, as the temperature is
reduced from 350 K, a spin reorientation transition from $\Gamma_{2}$ (Fx, Cy,
Gz) to $\Gamma_{4}$ (Gx, Ay, Fz) occurs at $T_{N}$ = 257 K and subsequently, a
second spin reorientation takes place from $\Gamma_{4}$ (Gx, Ay, Fz) to
$\Gamma_{2}$ (Fx, Cy, Gz) at $T_{\mathrm{SR}}$ = 190 K. The $\Gamma_{2}$ (Fx,
Cy, Gz) structure is stable until 7.7 K where an ordered moment of
7.74(1)$\mu_{\mathrm{B}}$/Fe3+(Cr3+) is obtained from neutron data refinement.
In addition to the long$\textendash$range order of the magnetic structure,
indication of diffuse magnetic scattering at 7.7 K is evident, thereby lending
support to the Griffiths$\textendash$like phase observed in susceptibility. At
7.7 K, Tb develops a ferromagnetic component along the crystallographic $a$
axis. Thermal conductivity, and spin$\textendash$phonon coupling of
TbFe0.5Cr0.5O3 through Raman spectroscopy are studied in the present work. The
magnetic anomalies at $T_{N}$ and $T_{\mathrm{SR}}$ do not reflect in the
thermal conductivity data of TbFe0.5Cr0.5O3; however, it is noticeable that
the application of 9 T magnetic field has no effect on the thermal
conductivity. The $T_{N}$ and $T_{\mathrm{SR}}$ are revealed in the
temperature$\textendash$dependence of
full$\textendash$width$\textendash$at$\textendash$half$\textendash$maximum
curves obtained from Raman intensities. An antiferromagnetic structure with
($\uparrow\downarrow\uparrow\downarrow$) arrangement of Fe/Cr spins is found
in the ground state through first$\textendash$principles energy calculations
which supports the experimental magnetic structure at 7.7 K. The
spin$\textendash$resolved total and partial density of states are determined
showing that TbFe0.5Cr0.5O3 is insulating with a band gap of $\sim 0.12$ (2.4)
eV within GGA (GGA+$U$) functionals

###### pacs:

##  I Introduction

Rare earth orthoferrites and orthochromites with the general formula $RM$O3,
where $R$ = rare earth or yttrium and $M$ = Fe or Cr, crystallize in the
perovskite structure (usually $Pbnm$ space group) with orthorhombic distortion
and an antiferromagnetic ground state White (1969). Rare earth orthoferrites
possess a complex spin structure and have drawn considerable attention due to
their unique physical properties White (1969) and potential applications such
as ultrafast magneto$\textendash$optical recording Kimel et al. (2006),
laser$\textendash$induced thermal spin reorientation Kimel et al. (2004),
precision excitation induced by terahertz pulses Jiang et al. (2013),
inertia$\textendash$driven spin switching Kimel et al. (2009), and
magnetism$\textendash$induced multiferroicity Tokunaga et al. (2009). Most
orthoferrites are G$\textendash$type canted antiferromagnets with a weak
ferromagnetic component due to Dzyaloshinskii$\textendash$Moriya (DM)
interaction and show temperature$\textendash$induced spin reorientation (SR)
from one magnetic symmetry to another. In $R$FeO3, exchange interactions
between Fe3+$\textendash$ Fe3+, $R^{3+}$$\textendash$Fe3+ and
$R^{3+}$$\textendash$$R^{3+}$ play an important role in determining complex
magnetic structures. Isotropic Fe3+$\textendash$Fe3+ exchange interaction
determines the magnetic structure of Fe3+ spins below the antiferromagnetic
ordering temperature. Exchange field due to Fe3+ moment polarizes the $R^{3+}$
spins of the $R$ sublattice and the Fe3+$\textendash$$R^{3+}$ interaction, in
turn, generates effective fields on Fe3+ spins which undergo spin
reorientation transition and align perpendicular to the $R^{3+}$ spins. The
spin reorientation transition might be continuous or abrupt depending on the
$R$ element Yamaguchi (1974).
In TbFeO3, an unusual incommensurate magnetic phase was discovered Artyukhin
et al. (2012) and it was shown that the exchange of spin waves between
extended topological defects could result in novel magnetic phases which draws
parallels with the Yukawa forces that mediate between protons and neutrons in
a nucleus. The Fe3+ moments in TbFeO3 exhibit GxAyFz
($Pb^{\prime}n^{\prime}m$) spin configuration at room temperature Bourée, J.E.
and Hammann, J. (1975); Bertaut et al. (1967a); Tejada et al. (1995) which is
accompanied by a spin reorientation to FxCyGz ($Pbn^{\prime}m^{\prime}$). At 3
K, another spin reorientation occurs to revert to the GxAyFz
($Pb^{\prime}n^{\prime}m$) structure. In recent years, a variety of
interesting properties were achieved by substituting Fe ion by different
transition metal ions Taguchi (1997); Dahmani et al. (2003). According to
Goodenough$\textendash$Kanamori rules Goodenough (1955), Cr3+ is a good choice
to pair with Fe3+ to tune superior magnetic properties due to superexchange
interaction between empty eg orbital of Cr3+ and half filled eg orbital of
Fe3+ ions.
In TbCrO3, the exchange coupling between the nearest neighbour Cr3+ is
predominantly antiferromagnetic and the Cr3+ spins order spontaneously at
$T_{N}$ = 167 K Gordon et al. (1976). Below this temperature, it exhibits weak
ferromagnetism resulting from the canting of Cr3+ magnetic moments. In TbCrO3
the Cr3+ spin structure is GzFx below $T_{N}$ and belongs to $\Gamma_{2}$
configuration which implies that the weak ferromagnet component of the Cr3+
moments orient along the $a$ axis Bertaut et al. (1967b); Bertaut et al.
(1966).

Tb3+ spins order antiferromagnetically at 3.05 K Bertaut et al. (1967b) below
which temperature the Tb3+ spin system exhibits an AxGy structure. In the
temperature range, 3.05 K $<$ T $<$ $T_{N}$ the Tb3+ spin system has FxCy
structure which belongs to $\Gamma_{4}$ representation and is coupled to the
ordered Cr3+ spin system Bertaut et al. (1967b). Spin reorientation,
magnetization reversal and weak ferromagnetism is seen in compounds like
TbFe0.5Mn0.5O3 Nhalil et al. (2015). And reentrant spin reorientation
transition has been seen in compounds like TbFe0.75Mn0.25O3 Fang et al. (2016)
which undergo $\Gamma_{4}$ to $\Gamma_{1}$ transition and then, $\Gamma_{1}$
to $\Gamma_{4}$ transition. Recently, the magnetic structures and spin
reorientation transitions of the mixed orthochromite$\textendash$orthoferrite
perovskites RFe0.5Cr0.5O3, where R = Tb, Dy, Ho, Er have been reported
Bolletta et al. (2018a) using neutron diffraction.
In the present paper, we report a detailed study of magnetic phase transitions
and magnetic structure of TbFe0.5Cr0.5O3 through magnetization, neutron powder
diffraction, Raman scattering and thermal conductivity studies in conjunction
with density functional theory calculations. Our results support reentrant
spin reorientation transition and a Griffiths phase$\textendash$like features
in the title compound. Our neutron scattering study also gives indication of
diffuse magnetic component present below the TN.

##  II Experimental methods

Polycrystalline TbFe0.5Cr0.5O3 was prepared by standard solid state reaction
using high purity ($\geq$3N) Tb4O7, Fe2O3 and Cr2O3 in stoichiometric amounts.
The starting materials were thoroughly mixed and sintered at $1200^{\circ}$C
for 48 h with two times intermediate grinding. The phase purity of sintered
sample was verified by taking powder X$\textendash$ray diffractograms (PXRD)
using Rigaku Smartlab X$\textendash$ray diffractometer with Cu Kα radiation
($\lambda$ = 1.548 Å). Oxidation states of Fe and Cr ions were determined
using X$\textendash$ray photoelectron spectroscopy (XPS) in a AXIS Ultra
spectrometer and the data was analyzed using the CASA XPS spectroscopy
software Fairley (1999). The chemical composition analysis of powder samples
were performed using JEOL$\textendash$JXA$\textendash$8530F electron probe
micro analyzer (EPMA) which yielded the Fe:Cr atomic ratio as 0.48:0.50 (Fe/Cr
= 0.96). Temperature dependent DC magnetization measurements were performed on
sintered pellets using a commercial magnetic property measurement system
(MPMS, Quantum Design) in the temperature range of 5 K $\leq T\leq$ 400 K at
100 Oe and 500 Oe in both zero$\textendash$field cooled (ZFC) and field cooled
(FC) protocols. Additionally, high temperature magnetic susceptibility was
recorded up to 800 K in the high temperature VSM oven option provided with the
physical property measurement system (PPMS). The thermal conductivity of a
parallelopiped sample of TbFe0.5Cr0.5O3 was measured in the temperature range
2 K$\textendash$300 K in 0 T and 9 T magnetic field using a commercial
physical property measurement system (PPMS).
To investigate nuclear and magnetic structure of TbFe0.5Cr0.5O3, neutron
powder diffraction experiments were performed at University of Missouri
Research Reactor (MURR) using the neutron powder diffractometer, PSD. Neutron
powder diffraction patterns of 2 g powder sample were collected at 350 K, 300
K, 215 K, 100 K and 7.7 K using neutrons of wavelength 1.485 Å. The neutron
diffraction data were analyzed using Fullprof suite of programs Rodríguez-
Carvajal (1993) employing the Rietveld method Rietveld (1969). Magnetic
representations belonging to the $Pbnm$ symmetry were determined using the
software SARA$h$ Wills (2000) and the corresponding magnetic structure was
refined using Fullprof. Raman spectra was recorded from 110 K to 300 K
temperature range in the backscattering geometry by using a HORIBA
JOBIN$\textendash$YVON spectrometer with 633 nm laser as an excitation source.
Low temperature was maintained by closed cycle He$\textendash$cryostat
attached to spectrometer.

##  III Computational Details

The electronic and magnetic structure calculations were performed by means of
density$\textendash$functional theory (DFT) approach using the
full$\textendash$potential linearized augmented plane wave plus local orbital
method as implemented in the WIEN2k code Blaha et al. (2001). The
non$\textendash$overlapping muffin$\textendash$tin sphere radii (RMT) of 2.35,
2.0, 1.96, and 1.72 Bohr were used for Tb, Fe, Cr, and O respectively. The
linear tetrahedron method with 500 $k$ points was employed for the
reciprocal$\textendash$space integrations in the whole Brillouin zone (BZ)
that corresponds to 216 $k$$\textendash$points within the irreducible BZ. For
the calculations, the standard generalized$\textendash$gradient approximation
(GGA) in the parameterization of Perdew, Burke, and Ernzerhof
(PBE$\textendash$96) was used Perdew et al. (1996). In order to consider the
strong correlation effects, GGA+$U$ functional with
double$\textendash$counting corrections according to Anisimov et al. Anisimov
et al. (1997) was used. The chosen values of $U$ were 6 eV for
Tb$\textendash$4$f$, 5 eV for Fe$\textendash$3$d$, and 3 eV for
Cr$\textendash$3$d$ states, which are comparable to the values found in
literature Ghimire et al. (2016); Ghimire and Hu (2016); GHIMIRE et al.
(2010); Yuan et al. (2015); Feng et al. (2016, 2019). Calculations were
performed using the lattice parameters obtained from neutron diffraction data
at 7.7 K (see Table 1). The energy and charge convergence was set to 10-6 Ry
and 10-4 of an electron, respectively, for self$\textendash$consistent
calculations. To obtain the magnetic ground states, we have considered five
magnetic configurations by computing their total energies. They are
ferromagnetic (FM$\textendash$$\uparrow\uparrow\uparrow\uparrow$), two
antiferromagnetic (AFM1$\textendash$$\uparrow\downarrow\uparrow\downarrow$ and
AFM2$\textendash$$\uparrow\downarrow\downarrow\uparrow$) and two ferrimagnetic
(FIM1$\textendash$$\uparrow\uparrow\downarrow\downarrow$ and
FIM2$\textendash$$\uparrow\uparrow\uparrow\downarrow$). Here, the spin
arrangements for two inequivalent atoms each of Fe and Cr atoms are arranged
as Fe1, Fe2, Cr1 and Cr2, respectively.

##  IV Results and discussion

Figure 1:  X$\textendash$ray photoelectron spectra of (a) Cr 2$p$, (b) Fe
2$p$, (c) Tb 3$d$ and (d) O 2$p$ are shown in open circles. Solid lines are
fitted peaks, deconvoluted components and background, respectively, in each
graph. Oxidation state of 3+ is inferred for Fe, Cr and Tb from this data.

###  IV.1 X$\textendash$ray photoelectron spectroscopy

Core level X$\textendash$ray photoelectron spectroscopy measurements at room
temperature using Al Kα X$\textendash$ray source was performed to determine
the valence states of cations in TbFe0.5Cr0.5O3. Figure 1 shows the
experimental intensities along with the peak fit obtained using the CASA XPS
software. The core level binding energy was calibrated with carbon (B.E. =
284.8 eV). The Cr $2p_{3/2}$ peak at 576.5 eV is close to the binding energy
of Cr2O3 (576 eV) Allen et al. (1973). However, in oxides, $2p_{3/2}$ peak of
Fe2+ and Fe3+ appear around the binding energy values of 710.3 eV and 711.4
eV, respectively Allen et al. (1974). In TbFe0.5Cr0.5O3, the peak at 711 eV is
close to the binding energy value of Fe3+. Additionally, a satellite peak at 8
eV above the Fe 2$p_{3/2}$ confirms Fe3+ state (Fe2+ gives a satellite peak at
6 eV above the main Fe 2$p_{3/2}$ peak). The XPS spectrum of Tb 3$d_{5/2}$
yields a peak at 1240.8 eV which is very close to that of Tb2O3 peak (1241.2
eV) Padalia et al. (1977). Our XPS results thus indicate 3+ oxidation state in
Tb, Fe and Cr.

Figure 2:  (a) Magnetization as a function of temperature showing a
bifurcation for ZFC and FC curves and the transitions at $T_{N}$$\approx$ 257
K and $T_{\mathrm{SR}}$$\approx$ 190 K. The inset shows the magnetization at
500 Oe. (b) $\chi^{-1}(T)$ at 500 Oe along with Curie$\textendash$Weiss fit
(dashed line). The upper inset shows the derivative $dM$/$dT$ where anomalies
at $T_{N}$ and $T_{\mathrm{SR}}$ are clear. The lower inset shows the inverse
magnetic susceptibility at 100 Oe, 500 Oe and 10 kOe which shows that the
negative curvature vanishes at higher fields. (c) The magnetization isotherms,
$M(H)$, at 100 K, 220 K, 250 K and 300 K. A weak hysteresis that develops
below the $T_{\mathrm{SR}}$ is shown in the upper inset ($T$ = 100 K) and the
lower inset shows a magnified view of a magnetization isotherm above the
$T_{N}$ ($T$ = 340 K).

###  IV.2 Magnetic properties: Spin reorientation and
Griffiths$\textendash$like phase

Figure 2 (a) shows the temperature dependent magnetization, $M(T)$, of
TbFe0.5Cr0.5O3 under ZFC and FC protocol at 100 Oe and 500 Oe external
magnetic field. Two anomalies are seen in the $M(T)$ curve at $\approx$ 257 K
and at 190 K. A bifurcation of the ZFC and FC curves is seen below $\approx$15
K. With the application of of 500 Oe, the bifurcation vanishes (see the inset
of Figure 2(a)). The magnetic phase transition temperatures of TbFe0.5Cr0.5O3
are determined by plotting $dM/dT$ vs $T$ as shown in top inset of Figure 2
(b). $T_{\mathrm{SR}}$= 190 K and $T_{N}$= 257 K are identified in this
manner. The temperature dependent inverse magnetic susceptibility,
$\chi^{-1}(T)$, of TbFe0.5Cr0.5O3 at 500 Oe is plotted in the main panel of
Figure 2 (b) up to 800 K along with a curve fit using the
Curie$\textendash$Weiss (CW) law (dashed line), $\chi^{-1}$ =
$(T\textendash\theta)/C$, where, $C$ =
$N_{A}\mu_{\mathrm{eff}}^{2}/3k_{\mathrm{B}}$ is the Curie constant, $N_{A}$
is the Avogadro’s number, $\mu_{\mathrm{eff}}$ is the effective magnetic
moment, $k_{\mathrm{B}}$ is the Boltzmann constant and $\theta$ is the
Curie$\textendash$Weiss temperature Kittel (1996). The Curie Weiss analysis
yields an effective magnetic moment of $\mu_{\mathrm{eff}}$ = 10.3(2)
$\mu_{\mathrm{B}}$. Assuming 3+ oxidation state for Tb, Fe and Cr, as
determined from the XPS analysis, the theoritically calculated magnetic
moment, $\mu_{\mathrm{theory}}$ in the paramagnetic region using the relation,
$\mu_{\mathrm{theory}}=\sqrt{\mu_{\mathrm{Tb}}^{2}(\mathrm{Tb}^{3+})+0.5\times\mu_{\mathrm{Cr}}^{2}(\mathrm{Cr}^{3+})+0.5\times\mu_{\mathrm{Fe}}^{2}(\mathrm{Fe}^{3+})}$
and considering high spin state of Tb3+ ($\mu_{\mathrm{Tb}}$ = 9.7
$\mu_{\mathrm{B}}$), Fe3+ ($\mu_{\mathrm{Fe}}$ = 5.9 $\mu_{\mathrm{B}}$) and
Cr3+ ($\mu_{\mathrm{Cr}}$ = 3.9 $\mu_{\mathrm{B}}$) is 10.9
$\mu_{\mathrm{B}}$.
A downward deviation of inverse magnetic susceptibility values from the ideal
Curie$\textendash$Weiss law is a signature of Griffith’s phase (GP) Griffiths
(1969a); Nair et al. (2011); Chakraborty et al. (2016). The characteristic
temperature at which the inverse susceptibility deviates from the CW behavior
is known as the Griffiths temperature, $T_{G}$. It is clear from the main
panel of Figure 2 (b) that the inverse susceptibility deviates from CW law
above $T_{N}$ at $T_{G}\approx$ 320 K. The downturn softens with increase in
applied magnetic field as shown in the lower inset of Figure 2 (b), this is a
signature of the presence of Griffiths$\textendash$like phase in the compound.
Figure 2 (c) shows the magnetization isotherms of TbFe0.5Cr0.5O3 at 100 K, 220
K, 250 K, 300 K and 340 K measured upto $\pm$9 T which do not reveal strong
ferromagnetic features. However, at 100 K, an opening of the magnetic
hysteresis loop is observed at low applied field values (upper left inset of
Figure 2 (c)). A magnified view of the isotherm at 340 K ($>$ $T_{N}$) shown
in lower inset of Figure 2 (c) reveals weak hysteresis indicating the presence
of short range magnetism above $T_{N}$. In section IV.3 we present
experimental evidences of spin fluctuation above $T_{N}$ in TbFe0.5Cr0.5O3.
We noted earlier that the downturn softening of the downturn in $\chi^{-1}(T)$
with an increase in applied field, supports GP scenario Pramanik and Banerjee
(2010, 2016); Magen et al. (2006). The suppression of downturn in magnetic
susceptibility at high magnetic fields is due to the rising paramagnetic
background which masks the ferromagnetic signal. Griffiths phase consists of
finite size ferromagnetic (FM) clusters in a paramagnetic matrix well above
the transition temperature in which the spins are ferromagnetically correlated
within those clusters. However, the magnetic system as a whole does not have
long$\textendash$range ordering in GP and thus no spontaneous magnetization
will appear. In GP, the FM clusters will appear with variable sizes having
local ferromagnetic ordering due to which magnetization becomes
non$\textendash$analytic; in the low$\textendash$field region, the magnetic
susceptibility will follow a power law behavior Griffiths (1969b); Pramanik
and Banerjee (2016, 2010) given by, $\chi^{-1}\propto$ $(T\textendash
T^{R}_{c})^{1-\lambda}$, where $T^{R}_{c}$ is the critical temperature of the
FM clusters. Here susceptibility tend to deviate from CW law and $\lambda$ (
$0\leq\lambda\leq 1$) is the exponent which signifies the deviation from CW
behavior due to formation of magnetic clusters in the PM state above the
transition temperature. A power law fit using the above$\textendash$mentioned
equation was administered on the magnetic susceptibility of TbFe0.5Cr0.5O3 as
log($\chi$-1) versus log($T$/$T^{R}_{c}\textendash$ 1) in both PM and GP
regions as shown in Figure 3 (a). Since the value of $\lambda$ is highly
sensitive to $T^{R}_{c}$, we have proceeded to estimate the value of
$T^{R}_{c}$ accurately Pramanik and Banerjee (2016); Jiang et al. (2010). The
critical temperature of ferromagnetic clusters, $T^{R}_{c}$, is always greater
than the transition temperature, so we first estimated the value of
$T^{R}_{c}$ in the purely paramagnetic region. This yields a value of 18 K
which was later used in the curve$\textendash$fitting for the Griffiths phase
regime to obtain $\lambda$ = 0.99. In the high temperature region, we obtained
$\lambda$ as 0.09 which signifies that the system is in paramagnetic phase,
following the CW behavior. These values for $\lambda$ are consistent with the
GP model signifying a Griffiths singularity in TbFe0.5Cr0.5O3.

Figure 3:  (a) Shows the power law fits (red lines) to $\chi^{-1}(T)$ at 100
Oe; plotted in a log$\textendash$log scale. The
reduced$\textendash$temperature is $t_{m}$ = ($T$/$T^{R}_{c}$$\textendash$ 1).
(b) Thermoremanant magnetization, $M_{TRM}$, measured at 50 Oe, 100 Oe and 200
Oe cooling fields, showing the onset of spin reorientation transition $T_{SR}$
and the antiferromagnetic $T_{N}$. The lower and upper insets show magnified
regions neat the $T_{N}$ and $T_{\mathrm{SR}}$, respectively.

Griffiths phase features are experimentally observed in strongly correlated
systems like the layered iron pnictides Inosov et al. (2013) and in
geometrically frustrated antiferromagnets Kumar et al. (2017). In the case of
the former, it is formed by randomly introduced localized magnetic impurities
which form above the quantum critical point associated with the suppression of
the stripe$\textendash$antiferromagnetic spin density wave order. In the
latter, the Griffiths phase was observed to be robust against the oxygen
nonstoichiometry which influenced the antiferromagnetic ordering. Since the
total magnetic susceptibility in the Griffiths phase region contains
contributions from both paramagnetic as well as short$\textendash$range
correlated regions, the downturn observed in the inverse magnetic
susceptibility from ideal CW law is not expected to be sharp in the case of
antiferromagnetically correlated regions. In $R$FeO3, five outer shell
electrons of the Fe3+ ion are in half$\textendash$filled $e_{g}$
($\sigma$$\textendash$bond component) and $t_{2g}$ ($\pi$$\textendash$bond
component) orbitals resulting in superexchange interactions that are
antiferromagnetic. In the case of Cr3+ ions, superexchange interactions in the
half$\textendash$filled $t^{3}\textendash$O$\textendash t^{3}$ induce
antiferromagnetism. Since Fe3+ and Cr3+ are randomly distributed in the
lattice of TbFe0.5Cr0.5O3, it results in stabilization of both ferromagnetic
and antiferromagnetic couplings.
To confirm the GP$\textendash$like scenario in antiferromagnetic
TbFe0.5Cr0.5O3, we employed thermoremanant magnetization protocol to measure
magnetization ($M_{TRM}$), which has been used earlier to study spin glasses
Mathieu et al. (2001). This protocol involves cooling the sample from well
above the magnetic transition temperature in the presence of a magnetic field.
The field is then switched off below $T_{C}$, and the magnetization measured
upon warming in zero field condition. The thermoremanent magnetization
$M_{TRM}$ will exhibit a sharp upturn at the transition temperature. In the
present case of TbFe0.5Cr0.5O3, this protocol was repeated for three different
cooling fields, 50 Oe, 100 Oe and 200 Oe. The zero$\textendash$field
measurements performed here have the advantage that the contributions from the
paramagnetic susceptibility are suppressed compared to an in$\textendash$field
measurement. Figure 3 (b) shows $M_{TRM}$(T) measured at the different cooling
fields. A clear signature of GP$\textendash$like phase is seen in the form of
an upturn in magnetization at temperature well above $T_{N}$. The $T_{G}$
obtained from thermoremanent measurement is 315 K which is close to the value
of 320 K estimated from magnetic susceptibility earlier .

###  IV.3 Neutron diffraction: Reentrant spin reorientation and
short$\textendash$range spin correlations

The macroscopic magnetic measurements explicitly suggest the antiferromagnetic
ordering at $T_{N}$, the possibility of a spin reorientation transition at
$T_{\mathrm{SR}}$ and the presence of Griffiths$\textendash$like phase in
TbFe0.5Cr0.5O3. We now proceed to investigate the magnetic structure of
TbFe0.5Cr0.5O3 in detail so as to understand the spin reorientation process
and to ascertain the magnetic structures above and below the
$T_{\mathrm{SR}}$. For this purpose, neutron diffraction experiments were
carried out on powder samples of TbFe0.5Cr0.5O3 at various temperatures in the
range of 7.7 K to 350 K.

Figure 4:  (a-d) Rietveld refinement of the neutron powder diffraction
patterns of TbFe0.5Cr0.5O3 at 350 K, 215 K, 100 K, and 7.7 K using $Pbnm$
space group model. There is a weak magnetic contribution even at 350 K which
is above the $T_{N}$, observed in magnetometry. (e) Diffuse scattering
intensity at 7.7 K along with the curve fit (solid line) using a Lorentzian
function.

The experimental neutron diffraction patterns at 350 K, 215 K, 100 K and 7.7 K
are shown in Figures 4 (a-d) (red circles). Orthoferrites adopt orthorhombic
structure as observed in a variety of $R$Fe0.5Cr0.5O3 Bolletta et al. (2018b).
For $R$ = Tb, Dy, Ho and Er, the crystal structure belongs to a distorted
perovskite type in the space group $Pbnm$ and ordered antiferromagnetically
below 270 K in $G_{x}$ configuration compatible with the $\Gamma_{4}$
representation. The above mentioned compounds exhibited a spin reorientation
transition from $G_{x}$ ($\Gamma_{4}$) to $G_{z}$ ($\Gamma_{2}$). In present
case of TbFe0.5Cr0.5O3, refinement of the neutron diffraction data at 350 K
was first performed using purely nuclear space groups, $P2_{1}/n$ and $Pbnm$.
The perovskite structure in which cations order crystallographically may adopt
a doubled unit cell with monoclinic $P2_{1}/n$ space group Anderson et al.
(1993). In TbFe0.5Cr0.5O3, Rietveld analysis of the diffraction data at 350 K
resulted in a reasonably good fit assuming $Pbnm$ space group, however, the
intensity of the nuclear Bragg peak position (101) was not fully accounted
for. Even at 350 K, appreciable contribution from magnetic scattering towards
the total scattered intensity was observed.
In order to determine the magnetic structure, we scrutinized the
symmetry$\textendash$allowed magnetic structures for this class of compounds.
It can be seen that for $R$FeO3 compounds in $Pbnm$ space group, there exists
eight irreducible representations $\Gamma_{1}$ through $\Gamma_{8}$ White
(1969). For the $4b$ Wyckoff position, the configurations $\Gamma_{5}$ to
$\Gamma_{8}$ are incompatible with net moment on the Fe, and $\Gamma_{3}$ is
not consistent with the observed strong antiferromagnetic coupling between
nearest Fe neighbours. Hence only the remaining three representations are
possible in orthoferrites White (1969). The $k$$\textendash$search utility in
Fullprof was used for obtaining the propagation vector in the case of
TbFe0.5Cr0.5O3 and subsequently, SARA$h$ was used to obtain the magnetic
representations of the allowed magnetic structures. After testing the three
different magnetic representations along with the nuclear phase in $Pbnm$, a
better visual fit to the experimental data with reasonable agreement factors
were obtained for $\Gamma_{2}$, and was accepted as the solution of the
magnetic structure at 350 K. The goodness$\textendash$of$\textendash$fit for
the magnetic refinement, $R_{\mathrm{mag}}$, for the three representations are
as follows: $\Gamma_{1}$ = 25.4, $\Gamma_{4}$ = 95.2, $\Gamma_{2}$ = 17.4.

Table 1:  Structural parameters and selected bond distances and bond angles of TbFe0.5Cr0.5O3 at different temperatures obtained from neutron diffraction. The nuclear space group is $Pbnm$ where the atomic positions are Tb $4e$ ($x,y,z$), Cr/Fe $4b$ (0.5,0,0.5) and O $4e$ ($x,y,z$). Long ($l$) and short ($s$) bond lengths correspond to $M\textendash$O(2) bonds in the $ab$ plane. Medium ($m$) bond length corresponds to the out$\textendash$of$\textendash$plane $M\textendash$ O(1) apical bond.  |  350 K |  300 K |  215 K |  100 K |  20 K |  7.7 K
---|---|---|---|---|---|---
$a$ (Å) |  5.3111(4) |  5.3125(6) |  5.3100(6) |  5.3121(0) |  5.3140(5) |  5.3124(2)
$b$ (Å) |  5.5548(1) |  5.5560(5) |  5.5517(5) |  5.5442(0) |  5.5403(4) |  5.5391(3)
$c$ (Å) |  7.6117(8) |  7.6111(0) |  7.6039(1) |  7.5963(3) |  7.5929(1) |  7.5917(3)
Fe(Cr)$\textendash$O1 ($m$) (Å) |  1.9885(5) |  1.9902(4) |  1.9886(0) |  1.9866(5) |  1.9858(6) |  1.9850(6)
Fe(Cr)$\textendash$O2 ($l$) (Å) |  2.0106(0) |  2.0115(4) |  2.0120(3) |  2.0083(8) |  2.0076(0) |  2.007(0)
Fe(Cr)$\textendash$O2 ($s$) (Å) |  1.9945(5) |  1.9919(8) |  1.9917(4) |  1.9904(0) |  1.9904(2) |  1.9999(5)
Fe(Cr)$\textendash$O1$\textendash$Fe(Cr)(o) |  146.2(5) |  145.9(0) |  145.8(5) |  145.8(5) |  145.8(3) |  145.9(2)
Fe(Cr)$\textendash$O2$\textendash$Fe(Cr)(o) |  147.2(4) |  147.5(0) |  147.2(3) |  147.1(5) |  147.5(1) |  146.5(4)

Figure 4 (a) shows the neutron diffraction patterns at 350 K along with the
refinement patterns using $Pbnm$ nuclear space group and the magnetic
structure according to $\Gamma_{2}$ representation. The nuclear space group of
TbFe0.5Cr0.5O3 at all temperatures till 7.7 K was found to be $Pbnm$. The
refined values of the lattice and bond parameters at different temperatures
are given in Table 1. Here, three different $M\textendash O$ bond lengths are
listed. Long ($l$) and short ($s$) bond lengths correspond to $M\textendash
O(2)$ bonds in the $ab$ plane while the medium ($m$) bond length corresponds
to the out of plane $M\textendash O(1)$ apical bond length which is almost
parallel to the $c$ axis.

Figure 5:  (a, b, c) The magnetic structures of the Fe/Cr sublattice at 300 K
($\Gamma_{2}$), 215 K ($\Gamma_{4}$) and 100 K ($\Gamma_{2}$) respectively.
The $\Gamma_{2}$ structure remains stable down to 7.7 K, which was the lowest
probed temperature by neutrons in this study. (d and e) The magnetic
structures of Tb3+ at 20 K and 7.7 K. The Fe/Cr atoms are represented as
yellow spheres and Tb as blue

As understood from the magnetization data presented in Figure 2 (a), a
magnetic phase transition occurs in TbFe0.5Cr0.5O3 at $T_{N}$ = 257 K.
Refinement of the diffraction pattern suggests that the nuclear structure is
$Pbnm$ and the magnetic structure is $\Gamma_{4}$ ($Pb^{\prime}n^{\prime}m$)
at 215 K. Thus, the magnetic structure changes from
$\Gamma_{2}\rightarrow\Gamma_{4}$ at $T_{N}$. The refined neutron diffraction
pattern at 215 K is presented in Figure 4 (b). Interestingly, we observe a
second spin reorientation transition back to the $\Gamma_{2}$
($Pbn^{\prime}m^{\prime}$) structure at 100 K. This temperature is below
$T_{\mathrm{SR}}$ (190 K) which is identified through the derivative of
magnetization curve. The $\Gamma_{2}$ magnetic structure remains stable down
to 7.7 K. In Figure 4 (e), the neutron diffraction intensity of TbFe0.5Cr0.5O3
at 7.7 K is presented after subtracting the contribution of the empty vanadium
can that was used as the sample holder. Preliminary evidence for diffuse
magnetism is present in difference curves. This observation lends support to
the short$\textendash$range magnetic fluctuations with a close link to the
features similar to those of Griffiths phase in the magnetization of
TbFe0.5Cr0.5O3. We attempted to analyze the diffuse intensity by fitting to
the Lorentzian curve, which is shown as a red solid line in Figure 4 (e). The
fit enabled us to extract a correlation length of approximately 9 Å. Elastisc
and inelastic experiments are underway in single crystals of TbFe0.5Cr0.5O3 to
study the diffuse signatures.
Further, the rare earth magnetic atoms in $R$Fe0.5Cr0.5O3 is reported to
develop magnetic ordering at low temperature below 15 K Bolletta et al.
(2018b). Our data is in agreement to this observation. Our refinement at 7.7 K
are consistent with the picture that the Tb3+ moments are magnetically ordered
in FxCy magnetic structure with a ferromagnetic component along the $a$ axis.
It is reported in a recent work Bolletta et al. (2018a) on TbFe0.5Cr0.5O3 that
only the Cy part remains whereas the ferromagnetic interactions disappear with
the spin reorientation at 1.9 K. As a result, diffuse magnetic scattering
features emerge; this is well$\textendash$captured in our work as can be seen
in Figure 4 (e). The magnetic structures of the transition metal and rare
earth moments as a function of temperature are shown in Figure 5.

###  IV.4 Thermal conductivity and Raman spectroscopy

Figure 6 shows the thermal conductivity, $\kappa_{t}(T)$ of TbFe0.5Cr0.5O3
measured in zero and in an applied magnetic field of 9 T. The overall
magnitude and temperature dependence of the thermal conductivity suggest that
the lattice thermal transport dominates in this material. As can be seen from
the figure, there is no appreciable change in $\kappa_{t}(T)$ with the
application of magnetic field. The relatively low value of $\kappa_{t}(T)$
might point to the presence of disorder in this material. In the context of
the presence of the atomic disorder in UZr2 and its impact to heat transport
behavior, it is helpful to compare the measured thermal conductivity to the
theoretically achievable minimum of the lattice contribution (fully disordered
structure). In this model the $\kappa_{t(min)}(T)$ dependence can be
calculated by using Debye approximation and assuming that the transverse and
longitudinal acoustic phonon modes are indistinguishable Cahill and Pohl
(1989). The results obtained for TbFe0.5Cr0.5O3 using the Debye temperature,
$\theta_{D}$ = 380 K Vagadia et al. (2018) and number of atoms per unit
volume, n = 4.7528 m-3, are shown in Figure 6 by the blue solid line.

Figure 6:  Variation of thermal conductivity, $\kappa_{\mathrm{t}}(T)$ of
TbFe0.5Cr0.5O3 as a function of temperature. As seen, there is no appreciable
change in $\kappa_{t}(T)$ with the application of 9 T. The magnetic anomalies
at $T_{N}$ and $T_{\mathrm{SR}}$ seen in the derivative of magnetization are
not observed in $\kappa_{\mathrm{t}}(T)$ or in the derivative (not shown). The
blue solid line represents the minimum thermal conductivity (see the text). In
the inset, solid and the dashed lines represent $T^{1.7}$ and $T^{3}$
dependence of $\kappa_{\mathrm{t}}(T)$ respectively.

The magnetic anomalies that occur at $T_{N}$ and $T_{\mathrm{SR}}$ (seen in
the derivative of magnetization) are absent in the derivative of
$\kappa_{t}(T)$ (not shown here). In general, the behavior of $\kappa_{t}(T)$
of TbFe0.5Cr0.5O3 is similar to the thermal conductivity variation in other
$R$FeO3 compounds like YFeO3, GdFeO3 and DyFeO3 Zhao et al. (2017). However,
in the work by Zhao et al. Zhao et al. (2017), single crystal samples of
orthoferrites were studied in the milli$\textendash$Kelvin temperature range
and in external magnetic fields up to 14 T. In earlier studies of GdFeO3 and
DyFeO3, anomalies that occur in the vicinity of the magnetic transitions were
reflected in the thermal conductivity response also Zhao et al. (2011, 2014).
Significantly, low $c$ axis thermal conductivity was observed in the case of
YFeO3, GdFeO3 and DyFeO3, considering that the present sample is a
polycrystalline pellet, in comparison, we observed higher values of thermal
conductivity in TbFe0.5Cr0.5O3. The total thermal conductivity could be
compared to the $T^{3}$ boundary scattering limit of phonons Berman and
Klemens (1978). In the inset of Figure 6, the temperature dependence of the
$T^{3}$ form of $\kappa_{t}(T)$ is shown as a dashed line. The solid line is
fitted to the $\kappa_{t}(T)\propto T^{n}$ expression where $n$ is varied as a
free parameter. Here, a value of 1.7(3) was obtained for $n$. The
$\kappa_{t}(T)$ curve of DyFeO3 shows a weak curvature at low temperature
(below 3 K) which is attributable to the magnonic contribution of Dy spin
system Zhao et al. (2014). Such a concave structure is not readily observed in
the present case, however, the beginning of such a curvature could be
discernible near 2 K.
Raman spectroscopy was carried out at different temperatures in order to
understand the phonon behaviour across the magnetic transitions observed in
TbFe0.5Cr0.5O3. Raman spectra was recorded from 110 K to 300 K as shown in
Figure 7 (top panel) with the most intense mode assignment matching with
$R$FeO3 Weber et al. (2016) and $R$CrO3 Bhadram et al. (2013). TbFe0.5Cr0.5O3
is an orthorhombically distorted perovskite with $Pbnm$ space group symmetry.
The irreducible representations corresponding to the phonon modes at the
Brillouin zone center Venugopalan et al. (1985) can be defined as, $\Gamma$ =
7$A_{g}$ \+ 7$B_{1g}$ \+ 5$B_{2g}$ \+ 5$B_{3g}$ \+ 8$A_{u}$ \+ 10$B_{1u}$ \+
8$B_{2u}$ \+ 10$B_{3u}$. Here, $A_{g}$, $B_{1g}$, $B_{2g}$, $B_{3g}$ are the
Raman active modes, $B_{1u}$, $B_{2u}$, $B_{3u}$ are the infrared modes, and
$A_{u}$ is inactive mode. Among them, the modes which are above 300 cm-1 are
related to the vibrations of oxygen, and the modes below 300 cm-1 are
associated with the rare earth ions Singh et al. (2008). However, the Raman
vibrational modes corresponding to an orthorhombic structure are $A_{g}$ \+
$B_{1g}$ and 2$B_{2g}$ \+ 2$B_{3g}$, which are symmetric and antisymmetric
modes, respectively.

Figure 7:  Top panel: Raman spectrum of TbFe0.5Cr0.5O3 at different
temperatures along with the most intense phonon modes assigned. (a) Shows
temperature dependence of phonon frequency obtained from the fit of the
spectral profile with Lorentzian function. Red solid line shows anharmonic
function fitting. (b) Shows temperature dependence of phonon linewidth.
Vertical violet dashed line shows position of $T_{N}$and $T_{\mathrm{SR}}$.

In contrast, $A_{g}$ \+ 2$B_{1g}$ \+ $B_{3g}$, 2$A_{g}$ \+ 2$B_{2g}$ \+
$B_{1g}$ \+ $B_{3g}$, and 3$A_{g}$ \+ $B_{2g}$ \+ 3$B_{1g}$ \+ $B_{2g}$ are
associated with the bending modes, rotation and tilt mode of the octahedra,
and for the rare earth vibrations, respectively Iliev et al. (1998). Raman
modes generally shifts to low frequency as the temperature increases
accompanied by a monotonic increase in
full$\textendash$width$\textendash$at$\textendash$half$\textendash$maximum
(FWHM) Nonato et al. (2014).

Figure 8:  Total and partial DOS of TbFe0.5Cr0.5O3 in AFM1 configuration:Total
DOS within GGA (top) and GGA+U (bottom) (left) functionals; partial DOS
contributions from Tb$\textendash$4f, Fe$\textendash$3d, Cr$\textendash$3d and
O$\textendash$2p states, respectively within GGA (middle) and GGA+U (right)
functionals for the spin$\textendash$up and spin$\textendash$down channels.

This is mainly due to the expansion of lattice as thermal energy increases.
Absence of any extra peak indicates that the spectral symmetry remains same at
all measured temperature thus confirming that the magnetic transitions are not
associated with any structural phase transition. We analyzed
well$\textendash$resolved Raman modes in detail. These modes were fitted with
Lorentzian function. The temperature variation of the phonon frequencies of
modes $A_{g}$(2) (141 cm-1), $A_{g}$(4) (333 cm-1) and $B_{2g}$(4) (678 cm-1)
along with the fit assuming the standard anharmonic dependence Balkanski et
al. (1983) of phonon modes are shown in Figure 7 (a).
The anharmonic dependence of the modes is given by, $\omega_{\mathrm{anh}}$(T)
= $\omega_{0}$$\textendash$C(1 + (2/($e^{(\hbar\omega/k_{B}T)}\textendash
1)))$ where $\omega_{0}$ is temperature$\textendash$independent part of
linewidth, C is a constant determined from the fitting, $\hbar\omega$ is the
phonon energy, and $k_{B}$ is the Boltzmann constant. The sudden change in the
phonon frequency near $T_{N}$ and $T_{\mathrm{SR}}$ can be clearly seen in
Figure 7 (a). A similar kind of anomaly in Raman modes near the magnetic
transition was reported in $R$CrO3 Bhadram et al. (2013) compounds.
Magnetostriction can also give rise to similar anomalous behaviour in phonon
frequency by modyfying unit cell volume Nonato et al. (2014). But in that
case, FWHM remains unchanged as it corresponds to phonon lifetime which is not
affected by subtle change in lattice volume caused by magnetostriction. But,
from Figure 7 (b), it can be seen that FWHM abruptly drops near the magnetic
transitions. The anomalous change in the mode frequencies and linewidths near
the magnetic transition establishes the spin$\textendash$phonon coupling in
TbFe0.5Cr0.5O3. A similiar signature of spin$\textendash$phonon coupling was
reported in $R$CrO3 Bhadram et al. (2013) and DyFe0.5Cr0.5O3 Yin et al.
(2014). The possible coupling mechanism involved is the phonon modulation of
superexchange integral below the magnetic ordering temperature Granado et al.
(1999).

###  IV.5 Density functional theory calculations

From the total energy calculations for five different collinear magnetic
configurations, AFM1 ($\uparrow\downarrow\uparrow\downarrow$) is found to be
most stable with the lowest energy. The AFM1 spin structure is found to be
consistent with our experimental observation for the $\Gamma_{2}$ state at 7.7
K. Similarly, the first excited AFM2 configuration is consistent with the spin
structure for the $\Gamma_{4}$ state at 300 K whose total energy is $\sim 36$
meV per formula unit higher compared to the AFM1 state of TbFe0.5Cr0.5O3. The
order of relative stability of the magnetic states are AFM1 $>$ AFM2 $>$ FIM2
$>$ FIM1 $>$ FM. This may be an indication of the competing ground state
between AFM1 and AFM2 observed as a Griffiths phase transition from
$\Gamma_{2}$ to $\Gamma_{4}$ and, subsequently, the reentrant to $\Gamma_{2}$
phase as seen in Figure 5. The magnetic anisotropy energy calculated is $\sim$
4.68 meV per formula unit of TbFe0.5Cr0.5O3 with in$\textendash$plane easy
axes. In TbFe0.5Cr0.5O3, the lanthanide Tb takes the charge state $3+$ with
4f8 configuration. Likewise, the transition element Fe nominally takes the
charge state $3+$ with 3d5 and Cr with charge state $3+$ should take the 3d3
configurations, respectively. In the stable AFM1 state, the calculated spin
moment at each site of Tb, Fe and Cr are $\pm$ 5.9 $\mu_{B}$, $\pm$ 3.65
$\mu_{B}$, and $\pm$ 2.36 $\mu_{B}$, respectively. Their respective orbital
moments are $\pm 1.03\mu_{B}$, $\pm 0.05\mu_{B}$, and $\mp 0.034\mu_{B}$
respectively. With GGA+$U$ effects, the spin moment of Tb, Fe and Cr turns out
to $\pm 5.97\mu_{B}$, $\pm 4.14\mu_{B}$, and $\pm 2.57\mu_{B}$ respectively.
The total magnetic moment compensates to zero as Tb, Fe and Cr couples
antiferromagnetically among each other as observed in Figure 5.
We now proceed to the electronic structure of TbFe0.5Cr0.5O3 in AFM1 state
within GGA and GGA+$U$, respectively. The spin$\textendash$resolved total and
partial density of states (DOS) are shown in Figure 8. TbFe0.5Cr0.5O3 is found
to be insulating with a band gap of $\sim 0.12$ (2.4) eV within GGA (GGA$+U$).
The correlation effects ${}^{\prime}U^{\prime}$ significantly changes the
electronic behaviour. As seen in the partial DOS, the main contributions from
Tb$\textendash$$4f$ states that were observed around $E_{\rm F}$ are shifting
away from each other. Those states that are fully occupied shift deep in the
valence region while the un-occupied state moves far away in the conduction
region. Similar features were observed also for Fe$\textendash$$3d$ states
around $E_{\rm F}$. On the other hand, Cr$\textendash$$3d$ states are
contributing at and around $E_{\rm F}$ hybridizing strongly with the
O$\textendash$$2p$ orbitals (see partial DOS in Figure 8). This is mainly due
to the hybridization between the $3d$ states of Cr and Fe with the
O$\textendash$$2p$ states. From the partial DOS contributions of
Fe$\textendash$$3d$, three $t_{2g}$ and two $e_{g}$ are fully occupied by five
electrons in spin$\textendash$up but in Cr$\textendash$$3d$, three $t_{2g}$
are fully occupied in spin up channel while $e_{g}$ bands are empty.

##  V Conclusions

We observe an antiferromagnetic transition $T_{N}$ at 257 K and a spin
reorientation transition $T_{\mathrm{SR}}$ at 190 K in the orthoferrite
TbFe0.5Cr0.5O3. Interestingly, a reentrant spin reorientation is seen in this
compound, where the spins reorient again at 100 K. Through detailed neutron
diffraction experiments and analysis we find that the spin structure changes
from the $\Gamma_{2}$ representation at 350 K to $\Gamma_{4}$ at 215 K and
then reverts to $\Gamma_{2}$ at 100 K. This structure remains stable until 7.7
K. A clear signature of Griffith phase is observed in the magnetization
response of TbFe0.5Cr0.5O3 and also short$\textendash$range spin fluctuations
that extend up to high temperature. The value of thermal conductivity is low
in TbFe0.5Cr0.5O3 which is not affected by the application of magnetic field
of 9 T. The magnetic anomalies at $T_{N}$ and $T_{\mathrm{SR}}$ are not
directly seen in the thermal conductivity data, but the latter is dominated by
the phonon contributions. Raman spectroscopic investigation reveals clear
evidence of spin$\textendash$phonon coupling in this compound.

##  VI Acknowledgments

The authors acknowledge Center for Nano Science and Engineering (CeNSE),
Indian Institute of Science, Bengaluru. B.M. acknowledges financial support
from University Grants commission (UGC), India for Senior research fellowship
(SRF). H.S.N. acknowledges faculty start-up grant from UTEP and Rising Stars
award. Work at INL was supported by DOE’s Early Career Research Program.
M.P.G. acknowledges the Higher Education Reform Project (HERP DLI-7B) of
Tribhuvan University, Kirtipur, Nepal for the start-up grant, and Alexander
von Humboldt Foundation, Germany for the partial support as return fellowship.
S.R.B. thanks NAST, Nepal for the PhD fellowship, and IFW-Dresden for funding
during research stay in Germany. M.P.G. and S.R.B. thanks Manuel Richter for
the fruitful discussion and Ulrike Nitzsche for the technical assistance.

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