# First-principles study on phase stability and physical properties of B-site
ordered Nd2CrFeO6 double perovskite

M. D. I. Bhuyan,a Rana Hossain,∗b Ferdous Ara c and M. A.
Basith${}^{\textit{${}^{a}$}}$<EMAIL_ADDRESS>
<EMAIL_ADDRESS>aNanotechnology Research Laboratory,
Department of Physics, Bangladesh University of Engineering and Technology,
Dhaka 1000, Bangladesh.
bDepartment of Mechanical Science and Bioengineering, Osaka University, Osaka
560-8531, Japan,
cInstitute of Multidisciplinary Research of Advanced Materials, Tohoku
University, 2-1-1, Katahira, Aoba-ku, Sendai 980-0877, Japan
DOI: 10.1039/D1CP03523A

###### Abstract

Here, the first-principles predictions on the structural stability, magnetic
behavior and electronic structure of B-site ordered double perovskite
Nd2CrFeO6 have been reported. Initially, the ground state of the parent single
perovskites NdCrO3 and NdFeO3 have been studied to determine the relevant
Hubbard U parameter to investigate the properties of Nd2CrFeO6. The
thermodynamic, mechanical, and dynamic stability analyses suggest the
possibility of the synthesis of Nd2CrFeO6 double perovskite at ambient
pressure. The compound shows ferrimagnetic (FiM) nature with 2 $\mu_{B}$ net
magnetic moment and the magnetic ordering temperature has been estimated to be
$\sim$265 K. Electronic structure indicates higher probability of direct
photon transition over the indirect transition with a band gap of $\sim$1.85
eV. Additional effect of Nd (4f) spin and spin-orbit coupling (SOC) on the
band edges have been found to be negligible for this 4f-3d-3d spin system.
This first-principles investigation predicts that due to the ferrimagnetic
nature and significantly lower band gap compared to its antiferromagnetic
parent single perovskites, B-site ordered Nd2CrFeO6 double perovskite could be
a promising material for spintronic and visible-light driven energy
applications.

##  I Introduction

Figure 1:  (a) Cation order in AB${}^{{}^{\prime}}$O3 perovskite. (b) Random
distribution of B${}^{{}^{\prime}}$ and B${}^{{}^{\prime\prime}}$ cations in
disordered AB${}^{{}^{\prime}}_{0.5}$B${}^{{}^{\prime\prime}}_{0.5}$O3
perovskite. (c) Partial ordering of B${}^{{}^{\prime}}$ and
B${}^{{}^{\prime\prime}}$ cations in disordered
AB${}^{{}^{\prime}}_{0.5}$B${}^{{}^{\prime\prime}}_{0.5}$O3 perovskite. (d)
Perfectly B${}^{{}^{\prime}}$ and B${}^{{}^{\prime\prime}}$ sites ordered
double perovskite A2B${}^{{}^{\prime}}$B${}^{{}^{\prime\prime}}$O6.

The potential of B-site ordered double perovskites
A2B${}^{{}^{\prime}}$B${}^{{}^{\prime\prime}}$O6 (A: alkaline or rare-earth
ions e.g., Ca, Sr, La, Nd, Eu, Gd, Dy, etc.; B${}^{{}^{\prime}}$,
B${}^{{}^{\prime\prime}}$: transition metals like Cr, Mn, Fe, Co, Ni, etc.)
over AB${}^{{}^{\prime}}$O3 and AB${}^{{}^{\prime\prime}}$O3 single
perovskites for magnetic and semiconductor-based applications have long been
recognized. Vasala and Karppinen (2015); Evans _et al._ (2021); Mazumdar and
Das (2021); Kangsabanik _et al._ (2018); Bhuyan _et al._ (2021) Double
perovskites provide a wide range of structural flexibility as well as freedom
of choosing a suitable combination of B${}^{{}^{\prime}}$ and
B${}^{{}^{\prime\prime}}$ cations from the periodic table and provides
opportunity to control their electronic structure, magnetic behavior, etc. for
practical application. Kim _et al._ (2019) To realize the opportunities of
A2B${}^{{}^{\prime}}$B${}^{{}^{\prime\prime}}$O6, the major challenge is to
achieve the long-range B-site (B${}^{{}^{\prime}}$ and
B${}^{{}^{\prime\prime}}$ sites) ordering, which is governed by kinetic and
thermodynamic factors of order-disorder reactions during synthesis of these
materials.Mandal and Gopalakrishnan (2005); Shimada _et al._ (2003) Fig. 1
shows a schematic illustration of possible B${}^{{}^{\prime}}$ and
B${}^{{}^{\prime\prime}}$ sites distribution when B${}^{{}^{\prime\prime}}$ is
inserted into AB${}^{{}^{\prime}}$O3 in order to form
A2B${}^{{}^{\prime}}$B${}^{{}^{\prime\prime}}$O6 double perovskite. For a
certain material, if the synthesis conditions are not optimized for a certain
material, it may form a randomly distributed B${}^{{}^{\prime}}$ and
B${}^{{}^{\prime\prime}}$ phase or a partially ordered phase. The physical
properties especially magnetism is strongly dependent on the degree of cations
ordering in double perovskite. Yoshimatsu _et al._ (2019) For example,
ReB${}^{{}^{\prime}}_{0.5}$B${}^{{}^{\prime\prime}}_{0.5}$O3 perovskites,
containing rare-earth (Re) elements, have orthorhombic (Pnma) crystal
structure similar to ReB${}^{{}^{\prime}}$O3, but demonstrate enhanced
magnetization because of the additional
B${}^{{}^{\prime}}$-O-B${}^{{}^{\prime\prime}}$ net ferromagnetic (FM)
exchange interaction along with B${}^{{}^{\prime}}$-O-B${}^{{}^{\prime}}$,
B${}^{{}^{\prime\prime}}$-O-B${}^{{}^{\prime\prime}}$, Re-
O-B${}^{{}^{\prime}}$/B${}^{{}^{\prime\prime}}$, and Re-O-Re antiferromagnetic
(AFM) interactions.Gray _et al._ (2010) These materials are generally weak FMs
due to the competing AFM and FM superexchange interactions arising from the
B-site disordering as shown in Fig. 1(b). Qiao _et al._ (2017); Yoshimatsu _et
al._ (2019); Bolletta _et al._ (2018); Nair _et al._ (2014); Hossain _et al._
(2021); Mazumdar and Das (2021) Yuan et al.Yuan _et al._ (2014) studied
several orthorhombic ReFe0.5Cr0.5O3 systems and reported that magnetization
improves greatly upon degree of B-site ordering of Fe and Cr atoms via the
Fe3+(d5)–O–Cr3+(d3) interaction. Interestingly, due to the presence of
multiple magnetic interactions, multiple magnetic phase transitions with
variation in temperature have been observed corresponding to the activation of
each interaction. It is also reported Yin _et al._ (2014) that the
DyFe0.5Cr0.5O3 perovskite shows an anomaly in temperature dependent
magnetization curve near 261 K due to the Fe3+(d5)–O–Cr3+(d3) interaction and
another at 120 K, related to the Cr3+(d3)–O–Cr3+(d3) interaction which is
expected for DyCrO3. Indeed, DyFe0.5Cr0.5O3 also show another anomaly at 640 K
corresponding to the onset of Fe3+(d5)–O–Fe3+(d5) interaction, which also
observed for DyFeO3.Nair _et al._ (2014) Similar multiple magnetic phase
transitions have been reported for NdFe0.5Cr0.5O3 disordered
perovskite.Sharannia _et al._ (2017); Hou _et al._ (2021) From these reports
it is evident that
ReB${}^{{}^{\prime}}_{0.5}$B${}^{{}^{\prime\prime}}_{0.5}$O3 retains the
properties of their parent compounds and the strength of
B${}^{{}^{\prime}}$-O-B${}^{{}^{\prime\prime}}$ interaction is much weaker
than that of B${}^{{}^{\prime}}$-O-B${}^{{}^{\prime}}$ and
B${}^{{}^{\prime\prime}}$-O-B${}^{{}^{\prime\prime}}$ AFM
interaction.Rodrigues _et al._ (2020); Coutinho and Barrozo (2018); Selvadurai
_et al._ (2015) For the technological applications, long range
B${}^{{}^{\prime}}$-O-B${}^{{}^{\prime\prime}}$ interaction is more demanded.
Hossain _et al._ (2018) Therefore, to achieve a very high degree of ordering
in B-sites, a number of synthesis techniques like solid-state reaction, soft
chemical-based synthesis routes, and thin-film deposition techniques have been
proposed.Mandal and Gopalakrishnan (2005); Shimada _et al._ (2003); M. Seikh
_et al._ (2013); Kleibeuker _et al._ (2017); Yoshimatsu _et al._ (2019); Dass
and Goodenough (2003) Notably, in B-site ordered double perovskites, one may
find three patterns (rock salt, columnar and layered) for B${}^{{}^{\prime}}$
and B${}^{{}^{\prime\prime}}$ depending on the choice of cations and synthesis
condition. King and Woodward (2010); Wang _et al._ (2018) Rock salt ordering
is most symmetric and common as the array of B${}^{{}^{\prime}}$ and
B${}^{{}^{\prime\prime}}$ cations is equivalent to the anion and cation
positions as shown in Fig. 1(d).King and Woodward (2010); Wang _et al._ (2015)

It is also noteworthy that among the double perovskites, rare-earth (Re)
containing magnetic semiconductors
Re2B${}^{{}^{\prime}}$B${}^{{}^{\prime\prime}}$O6 have gained significant
research interest because of their wide range of technological applications
and rich physical properties. Leng _et al._ (2020); Chen _et al._ (2019) For
instance, Re2NiMnO6 systems have been studied extensively due to their
ferromagnetism, low band gap, large magneto-capacitance, magneto-resistance,
and relaxor ferroelectricity.Nasir _et al._ (2019); Lekshmi _et al._ (2014)
Notably, Re2CrFeO6 is another class of double perovskites which is expected to
be a hub of ferrimagnetic semiconductorsSun _et al._ (2020), however, they are
less studied either by experimental or computational investigation. Double
perovskite La2CrFeO6 is an example of this class that has been synthesized
using the pulsed laser deposition technique with a high degree of Cr and Fe
ordering Lv _et al._ (2013); Chakraverty _et al._ (2011) and the electronic
properties of this compound have been studied by the first-principles
calculations.Lee and Ahn (2012) Unlike the randomly distributed Cr and Fe
disordered AFM perovskite LaCr0.5Fe0.5O3, the ordered La2CrFeO6 double
perovskite demonstrated a ferrimagnetic (FiM) nature with a saturation
magnetization of $\sim$ 2$\mu_{B}$ per formula unit (f.u.).Lv _et al._ (2013);
Chakraverty _et al._ (2011) Recently Majumder et al.Mazumdar and Das (2021)
reported a saturation magnetization of $\sim 0.8$ $\mu$B f.u.-1 in partially
ordered Pr2FeCrO6 bulk sample. Undoubtedly, a complete B-site ordered phase is
difficult to form in double perovskites by conventional solid-state or sol-gel
techniques. Interestingly, a perfect ordering was achieved Booth _et al._
(2009) in several Re2NiMnO6 double perovskites by repeated sintering where the
number of required sintering steps and sintering temperature varied with
different Re cations. Therefore, extensive investigation is required to
optimize different steps for the synthesis of B-site ordered Re2CrFeO6 double
perovskites. Moreover, the experimental investigation of the material
properties of these double perovskites at the electronic scale are
unattainable due to the unavailability of the required experimental
facilities. However, these limitations can be overcome significantly by
executing the density function theory (DFT) based first-principles
calculation.Yu _et al._ (2020); Ali _et al._ (2021)

In this investigation, the physical properties of B-site, ordered Nd2CrFeO6
double perovskite have been studied by generalized gradient approximation
(GGA)+U calculations. The parent single perovskites NdCrO3 and NdFeO3 were
studied initially to understand the effect of Hubbard parameter U on magnetic
and electronic properties. Our calculations infer that if the experimental
properties (e.g. band gap and/or atomic magnetic moments) of the parent single
perovskites are known, then the desired properties of the double perovskites
might be investigated by optimizing the U values of the parent compounds. This
strategy could be used to understand the properties of other
Re2B${}^{{}^{\prime}}$B${}^{{}^{\prime\prime}}$O6 double perovskites using
computationally cheaper GGA+U calculations. Our calculation demonstrated that
double perovskite Nd2CrFeO6 satisfied the thermodynamic, mechanical, and
dynamic criteria as a stable compound. This compound showed ferrimagnetism and
significantly lower band gap compared to its parent single perovskites NdCrO3
and NdFeO3.

##  II Computational details

The Vienna ab-initio simulation software (VASP),Kresse and Furthmüller (1996,
1996); Kresse and Hafner (1994) was used to carry out all the calculations by
using projector-augmented wave (PAW) approach. Blöchl (1994); Kresse and
Joubert (1999) The generalized gradient approximation (GGA)Perdew _et al._
(1996, 1997) was adopted to evaluate the electronic exchange-correlation
energy using the improved Perdew–Burke–Ernzerhof (PBE) functional. Since, the
GGA approach underestimates the band gap of insulators and semiconductors, the
on-site Coulomb interaction was introduced using spin-polarized GGA+U approach
(Dudarev et al.Dudarev _et al._ (1998)) to properly describe the localized d
electrons of Fe and Cr atoms. According to the Dudarev’s approach, the total
energy functional is of the form

$\displaystyle\small
E_{LDA(GGA)+U}=E_{LDA(GGA)}+\frac{U-J}{2}\sum_{\sigma}{Tr\rho^{\sigma}-Tr(\rho^{\sigma}\rho^{\sigma}),}$

where $\rho^{\sigma}$ is the density matrix of d(f)-state with spin
$\sigma$.Dudarev _et al._ (1998); Himmetoglu _et al._ (2014) Since the on-site
Coulomb interaction U and exchange interaction $J$ parameters are not
considered separately and only the difference ($U-J$) is meaningful in
Dudarev’s approach, one single parameter U (with $J$=0) was considered in this
study for simplicity.Wang _et al._ (2012) For all calculations, Fe’s
3s23p63d64s2, Cr’s 3s23p63d54s1 and O’s 2s22p4 electrons were considered as
valence electrons. Because partly filled f-state are not adequately
represented by current DFT approaches and often fail to converge, the 4f
electrons of the rare earth Nd atom were taken into account as core electrons
in structural relaxation. Singh _et al._ (2020); Zhao _et al._ (2014); Sun _et
al._ (2020) However, Nd’s 4f electrons were considered as valence electrons to
investigate the effect of 4f spin on the electronic structure, and spin–orbit
coupling (SOC). Integration over the Brillouin-zone was performed in a
$6\times 6\times 4$, $\Gamma$-centered Monkhorst–Pack $k$-point mesh. A plane-
wave cutoff was set at 520 eV, and the self-consistent convergence criteria
for energy was taken to 10-8 eV. The structure was completely relaxed until
the forces fell below 1 meV Å-1.

To investigate the thermodynamic stability of Nd2CrFeO6, the total energies of
the other stable phases bounded in Nd-Fe-Cr-O phase diagram have been
considered. Initial structures of the stable phases (as shown in Fig. 4) are
taken from Materials Project database.Jain _et al._ (2013) Structural
relaxation was carried out for all the phases (with U=3.5 eV for the phases
containing Fe/Cr). Phase diagram analysis has been done by using the phase
diagram module the Python Materials Genomic (pymatgen) package. Ong _et al._
(2013, 2008) The symmetry of the optimized structures has been calculated by
spglib package. Togo and Tanaka (2018) The transition dipole moments have been
studied by Vaspkit. Wang _et al._ (2021) The phonon dispersion was calculated
by the Phonopy code Togo and Tanaka (2015) using a $2\times 2\times 1$
supercell with a $2\times 2\times 2$ k-mesh.

##  III Effect of $U$ on the ground-state

Figure 2:  Possible spin arrangements of Fe and Cr atoms in B-site ordered
Nd2CrFeO6 double perovskite: (a) FM, (b) A-type AFM (A-AFM), (c) C-type AFM
(C-AFM), and (d) G-type AFM (G-AFM). In A-AFM orientation, the intra-plane
coupling is FM while inter-plane coupling is AFM, whereas in C-AFM, this is
opposite i.e., the intra-plane coupling is AFM while inter-plane coupling is
FM and in G-AFM, both intra-plane and inter-plane coupling are AFM. Figure 3:
Effect of U parameter on different ground-state properties. (a) Variation of
ground-state energy as a function of U for possible spin orders in Nd2CrFeO6
compared to the FM order. (b) Differences in spin magnetic moment ($\Delta$m)
of Fe site in AFM spin orders compared to the FM order. (c) Calculated band
gap (for spin down channel) for different spin orders considered for
Nd2CrFeO6. (d) Comparative plot of band gap for NdCrO3, NdFeO3 single
perovskite, and Nd2CrFeO6 double perovskite considering their low energy spin
(G-AFM) state.

To determine the lowest energy of spin order, the FM, A-AFM, C-AFM, and G-AFM
arrangements of spin (shown in Fig. 2) were considered. The structure and
atomic positions were fully relaxed for a set of U ranging from 0–9 eV for 3d
electrons of Fe and Cr atoms. Fig. 3 (a) shows the total energy difference,
$\Delta$E (meV f.u.-1) of the ground states of Nd2CrFeO6 for various spin
orders (as shown in Fig. 2) compared to the FM order and the variation
corresponding to the change in U values. The total energy of all spin orders
increases with U as the on-site occupation number in d orbital (and spin
magnetic moment) of Cr/Fe found to be increased, which is consistent with
Dudarev’s formalism.Dudarev _et al._ (1998) It should be noted that Fe/Cr spin
magnetic moments increase while average induced magnetic moment in O atoms
decrease from 0.09 $\mu_{B}$ to 0.07 $\mu_{B}$ with increasing U indicating
the attenuation of the strength of the magnetic exchange coupling via Fe-O-Cr
interactions for larger U.Song _et al._ (2019) However, $\Delta$E between FM
and AFM spin orders decreases as U increases. For U<6 eV, G-AFM arrangement of
Cr and Fe spins are energetically favorable while for larger U values (>6 eV)
FM order become the lowest energy state. The mechanism could be understood
from the variation of the difference in spin magnetic moments as shown in Fig.
3(b). The variation and turning of $\Delta$m curves are identical to the
$\Delta$E curves for AFM orders. It can be clearly seen that spin magnetic
moments of AFM orders are approaching towards the FM baseline with increasing
U and overtaking for U>6 eV. A similar variation has been observed for Cr. The
calculated band gap for FM and AFM spin orientations are shown in Fig. 3(c)
and increases smoothly with increment in U except for FM order at U=0 eV. At
U=0 eV, the FM spin arrangement yields metallic behavior. This is the well-
known artifact of GGA in determining electronic gap of semiconductors. Prior
to investigate the physical properties of Nd2CrFeO6 double perovskite, it is
essential to choose a physically reasonable value of U which can adequately
describe the electronic structure. Therefore, the parent single perovskites
were studied, since the experimental band gaps of these magnetic
semiconductors already have been reported.

For single perovskites NdCrO3 and NdFeO3, the G-AFM arrangement of spins shows
minimum energy for all values of U with orthorhombic (Pnma) structure and are
in agreement with experimental observations. Bora and Ravi (2015); Chakraborty
_et al._ (2014); Yamaguchi (1974) Since GGA (U=0) underestimates the band gap,
the experimental band gaps (E${}_{g}^{exp.}$) were compared with the
calculated band gaps for different U values. Das _et al._ (2021) Fig. 3(d)
shows the variation in calculated band gap of NdCrO3, NdFeO3, and Nd2CrFeO6.
It is found that for U=3.50 eV (for Cr/Fe), the calculated band gaps for
NdCrO3 and NdFeO3 matched well with experimental values.Mannepalli _et al._
(2017); Nguyen _et al._ (2021); Aparnadevi _et al._ (2016) For the same value,
i.e. UCr=UFe=3.50 eV, a band gap of 1.82 eV has been obtained for double
perovskite Nd2CrFeO6. Table 1 shows the calculated lattice parameters,
symmetry of the optimized structures as well as the band gap values obtained
by spin-polarized GGA and GGA+U calculations. From the Fig. 3(d) and Table 1,
it may be inferred that the band gap of double perovskite Nd2CrFeO6
(E${}_{g}^{DFT}$=1.82 eV) is lower than its parent single perovskites NdCrO3
(E${}_{g}^{DFT}$=3.0 eV) and NdFeO3 (E${}_{g}^{DFT}$=2.12 eV). To understand
the reason of the low band gap in Nd2CrFeO6, valence band maxima (VBM),
conduction band minima (CBM), and the charge transfer (CT) from Cr/Fe cations
to O anion have been calculated and shown in Table 1. Compared to NdFeO3, the
CT for Fe is found to increase in case of Nd2CrFeO6, while for Cr it is
similar to that of NdCrO3. The higher ionicity of Fe may push the Fe-3d states
in the CBM to a lower-energy position together with the almost unshifted O-2p
states in VBM, resulting in the band gap reduction. A previous investigation
Quattropani _et al._ (2018) on Bi2FeCrO6 (band gap 1.5 eV) double perovskite
also demonstrated band gap reduction from their parent single perovskites
following similar mechanism. It should be noted that, the experimental band
gaps of the synthesized materials depend on many factors e.g. synthesis
condition, particle size, presence of vacancies, secondary phases,
etc.,Geneste _et al._ (2019); Mocherla _et al._ (2013); Wang _et al._ (2019)
therefore, the experimental band gap owing to these effects could vary from
the DFT-calculated results for perfect system. Wu _et al._ (2020) In the
subsequent investigation on Nd2CrFeO6 double perovskite, we have used G-AFM
spin orientation as it gives the lowest energy ground state and U=3.50 eV for
the 3d electrons of Cr and Fe atoms since this value can satisfactorily
describe the electronic structure of the parent perovskite compounds.

Table 1:  GGA calculated lattice parameters and band gaps for lowest energy
(G-AFM) state of NdCrO3, NdFeO3 and Nd2CrFeO6. Parenthesis is showing the
results for (UCr=UFe=3.50 eV) calculations. GGA+U-calculated position of VBM,
CBM, and CT values for Cr/Fe cations to the oxygen anions obtained by Bader
charge analysis.Henkelman _et al._ (2006)

|  NdCrO3 |  NdFeO3 |  Nd2CrFeO6
---|---|---|---
a | 5.561 (5.586) | 5.664 (5.676) | 5.439 (5.465)
b | 7.724 (7.770) | 7.797 (7.824) | 5.615 (5.627)
c | 5.417 (5.449) | 5.455 (5.472) | 7.754 (7.802)
angle ($\beta$) | 90 (90) | 90 (90) | 90.006 (90.012)
space-group | Pnma | Pnma |  P21/n
E${}_{g}^{DFT}$ (eV) | 1.40 (3.00) | 0.643 (2.12) | 0.33 (1.82)
E${}_{g}^{exp.}$ (eV) |  3.10Mannepalli _et al._ (2017) |  2.06Nguyen _et al._ (2021) | –
VBM (eV) | 8.30 | 6.90 | 6.94
CBM (eV) | 5.30 | 4.78 | 5.12
CT (eV/atom) | Cr=+1.73 | Fe=+1.67 | Cr=+1.73, Fe=+1.70

##  IV Phase stability

Figure 4:  (a) The DFT-calculated phase diagram for the Nd–Cr–Fe–O system at
T=0 K. The red circles are indicating the stable phases, these are connected
by the blue tie-lines that forming the convex hull. (b) The pseudo-ternary
compound phase diagram of Nd2O3–Cr2O3–Fe2O3 systems, where the black tie-lines
that forming the convex hull.

Fig. 4(a) and 4(b) show the constructed phase diagram for Nd-Cr-Fe-O systems
and the compound phase diagram for Nd2O3–Cr2O3–Fe2O3 systems, respectively. It
can be observed that the Nd2CrFeO6 phase is on the tie-line of the convex hull
indicating this phase is thermodynamically stable. Further, the formation
energy per atom of Nd2CrFeO6 has been calculated using the following equation,

$\small
E^{form}_{atom}=\frac{1}{20}E^{Nd_{2}CrFeO_{6}}_{bulk}-2\times\frac{1}{2}E^{Nd}_{bulk}\\\
-\frac{1}{2}E^{Fe}_{bulk}-\frac{1}{2}E^{Cr}_{bulk}-3\times\frac{1}{8}E^{O_{2}}_{bulk}.$
(1)

The calculated value of the formation energy, E${}^{form}_{atom}$, is found
-1.4 eV per atom. The negative value of E${}^{form}_{atom}$ indicates that
this compound is formed by an exothermic process that needs less energy than
the breaking of chemical bonds. Emery and Wolverton (2017) To evaluate the
thermal stability of Nd2CrFeO6 double perovskite, the phase decomposition
energies have been calculated for the two most possible paths by

$Nd_{2}CrFeO_{6}\xrightarrow{E_{DP1}}NdFeO_{3}+NdCrO_{3}.$
$Nd_{2}CrFeO_{6}\xrightarrow{E_{DP2}}Nd_{2}O_{3}+\frac{1}{2}Fe_{2}O_{3}+\frac{1}{2}Cr_{2}O_{3}.$

If the decomposition energy for any pathway (EDP) is negative, it is not
possible to observe a stable region in the phase diagram. Han _et al._ (2018)
The estimated decomposition energies for Nd2CrFeO6 double perovskite in both
pathways are found to be 2.52 eV f.u.-1 ($E_{DP1}$) and 0.86 eV f.u.-1
($E_{DP2}$) suggesting that this compound will not be decomposed
spontaneously. This is implying that the requirements for the thermodynamic
stability of Nd2CrFeO6 double perovskite have been satisfied.

To investigate the mechanical stability, elastic constants were calculated
using the finite strain method.Qi _et al._ (2018) The calculated elastic
constants are summarized in Table 2. For monoclinic crystal, the independent
elastic stiffness tensor is reduced to thirteen components, C11, C22, C33,
C44, C55, C66, C12, C13. C15, C23, C25, C35 and C46 in the Voigt notation.
Vajeeston and Fjellvåg (2017) For a mechanically stable structure, Cij has to
satisfy Born–Huang criteria. Vajeeston and Fjellvåg (2017); Mouhat and Coudert
(2014) The mechanical stability criteria is given by:

$C_{11}>0,C_{22}>0,C_{33}>0,C_{44}>0,C_{55}>0,C_{66}>0$
$C_{33}C_{55}-C_{35}^{2}>0,C_{44}C_{66}-C_{46}^{2}>0,C_{22}+C_{33}-2C_{23}>0$
$[C_{22}(C_{33}C_{55}-C_{35}^{2})+2C_{23}C_{25}C_{35}-C_{23}^{2}C_{55}-C_{25}^{2}C_{33})]>0$
$2[C_{15}C_{25}(C_{33}C_{12}-C_{13}C_{23})+C_{15}C_{35}(C_{22}C_{13}-C_{12}C_{23})$
$+C_{25}C_{35}(C_{11}C_{23}-C_{12}C_{13})]-h+C_{55}g>0$

where,

$g=C_{11}C_{22}C_{33}-C_{11}C_{23}^{2}-C_{22}C_{13}^{2}-C_{33}C_{12}^{2}+2C_{12}C_{13}C_{23}$

$h=C_{15}^{2}(C_{22}C_{33}-C_{23}^{2})+C_{25}^{2}(C_{11}C_{33}-C_{13}^{2})+C_{35}^{2}(C_{11}C_{22}-C_{12}^{2})$

The Nd2CrFeO6 double perovskite meets all essential stability criteria for the
monoclinic system.

Table 2:  The calculated elastic constants, $C_{ij}$ (GPa) of Nd2CrFeO6 double perovskite. $C_{ij}$ | 1 | 2 | 3 | 4 | 5 | 6
---|---|---|---|---|---|---
1 | 281.7 | 131.8 | 117.0 | 0.0 | 2.0 | 0.09
2 | 131.8 | 238.8 | 113.9 | 0.0 | 2.0 | 0.0
3 | 117.0 | 113.9 | 275.0 | 0.0 | -4.0 | 0.0
4 | 0.0 | 0.0 | 0.0 | 101.0 | 0 | 3.0
5 | 2.0 | 2.0 | -4.0 | 0.0 | 85.0 | 0.0
6 | 0.0 | 0.0 | 0.0 | 3.0 | 0.0 | 81.0

The phonon dispersion curve of Nd2CrFeO6 double perovskite has shown in Fig. 5
to assess the dynamical stability. The structure may be considered dynamically
stable if there are no imaginary phonon frequencies in the whole Brillouin
zone (BZ). Cheng _et al._ (2014); Chu _et al._ (2015) In this case, we did not
observe any imaginary phonon frequency in the whole BZ which indicates that
the structure is dynamically stable.

Figure 5:  The phonon dispersion curves of Nd2CrFeO6 double perovskite.

##  V Magnetic properties

Table 3 shows the ground-state energy comparison of different spin
arrangements (as shown in Fig. 2) relative to the FM spin arrangement and
magnetic moment per atom. As discussed above Nd2CrFeO6 double perovskite and
its parent single perovskites NdCrO3 and NdFeO3 have G-type AFM order at
ground-state. Due to the G-type AFM spin arrangement with
Cr${}_{\uparrow}^{3+}$-O-Cr${}_{\downarrow}^{3+}$ and
Fe${}_{\uparrow}^{3+}$-O-Fe${}_{\downarrow}^{3+}$ super-exchange interactions
in NdCrO3 and NdFeO3 respectively, they do not possess any net magnetic
moment. However, Nd2CrFeO6 is quite interesting than NdCrO3 and NdFeO3 as it
possess a net magnetic moment of $\sim$2 $\mu_{B}$ f.u.-1 due to the ordered
distribution of Cr and Fe cations mediated by
Fe${}_{\uparrow}^{3+}$-O-Cr${}_{\downarrow}^{3+}$ super-exchange interaction.

Figure 6:  In and out of plane bonds ($d_{1}$ and $d_{2}$) in CrFe6 octahedron and magnetic exchange interactions (nearest neighbors, $J_{1}$ and next nearest neighbors $J_{2}$) in Nd2CrFeO6 double perovskite Table 3:  Ground-state energy comparison $\Delta$E (meV/f.u.) of FM, A-AFM, C-AFM and G-AFM states relative to FM state for UCr,Fe=3.50 eV. Order |  NdCrO3 |  NdFeO3 |  Nd2CrFeO6
---|---|---|---
FM | 0 | 0 | 0
A-AFM | -13 | -86 | -14
C-AFM | -25 | -168 | -27
G-AFM | -36 | -235 | -34
M (Cr/Fe)($\mu_{B}$) | 2.66 | 3.99 | -2.58/4.05
Mtotal ($\mu_{B}$/f.u.) | 0 | 0 | 2

Furthermore, the calculated total energies have been used to estimate the
magnetic ordering temperature (Tc) by mapping to the classical Heisenberg
Hamiltonian:Pajda _et al._ (2001)

$H=-\sum_{i,j}J_{ij}S_{i}S_{j}$

where $\left|S_{i}\right|=\sqrt{S_{i}(S_{i}+1)}$ is the spin at site $i$ and
$J_{ij}$ is the spin exchange interaction parameter between $i$ and $j$ sites.
Here, we have taken interactions up to the nearest neighbors $J_{1}$ and
second nearest neighbors $J_{2}$. It should be noted that two types of bonds
(in plane and out of plane) with different lengths are present due to the
distortion from the ideal $B^{{}^{\prime}}B^{{}^{\prime\prime}}_{6}$
octahedron (as shown in Fig. 6). Therefore, the total energies per magnetic
species may be expressed as

$E_{0}+\sqrt{S_{1}(S_{1}+1)}\sqrt{S_{2}(S_{2}+1)}(2J_{1}+4J_{2})=E_{FM}$
$E_{0}+\sqrt{S_{1}(S_{1}+1)}\sqrt{S_{2}(S_{2}+1)}(-2J_{1}+4J_{2})=E_{C-AFM}$
$E_{0}+\sqrt{S_{1}(S_{1}+1)}\sqrt{S_{2}(S_{2}-1)}(-2J_{1}-4J_{2})=E_{G-AFM}$

where $E_{0}$ is the spin independent part. The exchange parameters
$J_{1}$=0.62 meV and $J_{2}$=1.19 meV were calculated by the above equations
using $S_{1}$=5/2 and $S_{2}$=3/2 for the Fe3+ and Cr3+ spins. The positive
values of exchange interactions are necessarily indicating the AFM nature of
Fe-O-Cr interactions. For Nd2CrFeO6, magnetic phase transition temperature has
been estimated by the mean-field approximation,Chen _et al._ (2019)

$T_{c}=\frac{2}{3k_{B}}\sqrt{S_{1}(S_{1}+1)}\sqrt{S_{2}(S_{2}+1)}(2J_{1}+4J_{2})$

Since in B-site ordered double perovskite Nd2CrFeO6, the Fe-O-Cr interaction
will be dominating, the calculated transition temperature 265 K can be
compared with experimentally observed magnetic transition at around 250 K due
to Fe-O-Cr interaction in NdFe0.5Cr0.5O3 perovskite. Sharannia _et al._
(2017); Wan _et al._ (2016) It should be noted that the classical mean-field
approach can give a rough estimation on transition temperature. Wu _et al._
(2016); Kabiraj _et al._ (2020) To reveal the whole magnetic phase diagram, to
understand the microscopic dynamic mechanism and to give a more qualitative
picture of the magnetic propertiesSasani _et al._ (2021), further
investigation by classical atomistic spin dynamics method based on the DFT
calculated exchange parameters is necessary.

##  VI Electronic structure

Figure 7: Calculated band structure (top panel) and transition dipole moment
(bottom panel) for UCr=UFe=0 in (a) spin-up and (b) spin-down channels. (c)
Spin-up and (d) spin-down band structure and transition dipole moment for
UCr=UFe=3.5 eV Figure 8: (a) TDOS calculated by GGA+U corresponds to the G-AFM
ground state. (b-d) PDOS of Fe (3d), Cr (3d) and O (2p) orbitals respectively.
Positive and negative DOS values stand for the up and down spin channel. Fermi
level is set at 0 eV. Figure 9: (a) Total charge and (b) magnetization density
plots. (c) Cross-sectional and (d) perspective view of three dimensional (3D)
magnetization density iso-surface plot at the same iso-value of $\pm$0.01
eV/Å3. The light yellow and light blue colors represent the spin up and spin
down states, respectively. Figure 10: Effect of Nd’s 4f electrons and SOC on
the electronic structure of Nd2CrFeO6. The calculated PDOS by considering; (a)
4f electron as core electron and (b) 4f electron as valence electron. The
obtained PDOS including SOC by considering; (c) 4f electron as core electron
and (d) 4f electron as valence electron

Fig. 7 shows spin polarized band structures of Nd2CrFeO6 for UCr=UFe=0 and 3.5
eV, along the high symmetry directions in the first Brillouin zone for both
spin up and spin down channels. The Fermi level (set to 0 eV) was shown by the
horizontal line between the valence and conduction bands. The insulator nature
of Nd2CrFeO6 has been predicted in both spin channels. For U=0 eV (Fig. 7(a)
and 7(b)), VBM and CBM are located at the same symmetry point ($Y$) in both
spin up and down channels, indicating direct band gap of 2.19 eV and 0.32 eV,
respectively. On the other hand, for UCr=UFe=3.5 eV, in spin down channel, VBM
is found to be shifted to the $E$ points, mainly indicating the indirect
nature with a 1.82 eV gap. Notably, it can be seen that in down spin channel,
VBM at $Y$ point is located just 0.03 eV below the $E$ point. Therefore, in
Nd2CrFeO6, the indirect band gap of 1.82 eV ($E\xrightarrow[]{}Y$) and a
direct band gap of $\sim$ 1.85 eV ($Y\xrightarrow[]{}Y$) is comparable. The
probability of transition either from $E$ or $Y$ points can be understood by
the calculated square of the transition dipole moment (P2, transition
probabilities between two states Wang _et al._ (2021)) and is shown in the
bottom panel of Fig. 7(d). At the $E$ point the transition probability is near
to zero, whereas a strong probability can be seen around $Y$ point. Therefore,
for double perovskite Nd2CrFeO6, the optical absorption could be initiated by
direct $Y\xrightarrow[]{}Y$ transition.

To understand the features of VBM and CBM, the total density of states (TDOS)
and atom projected orbital decomposed density of states (PDOS) have been
analyzed. Fig. 8(a)-(d) display the TDOS of Nd2CrFeO6 double perovskite and
PDOS of Fe (3d) Cr (3d) and O (2p) orbitals, respectively. The Fe (3d) and Cr
(3d) orbitals are found to be split into $t_{2g}$ and $e_{g}$ levels due to
the FeO6 and CrO6 octahedral environment. It should be noted that following
the local rotation of Fe/CrO6 octahedra in the unit cell orientation, the
d${}_{{x^{2}}-{y^{2}}}$, dyz and dxz orbitals belong to the $t_{2g}$ states.
On the other hand, d${}_{z^{2}}$ and dxy orbitals belong to $e_{g}$ states.
Ong _et al._ (2008) In the valence band, the DOS in the range from -1 eV to 0
eV in the spin-up channel is dominated by O (2p) orbitals, whereas in the
spin-down channel, the VBM is made up of the hybridization of $t_{2g}$ of Cr
(3d) and O (2p) states. The CBM was constructed by the hybridization of
unoccupied Fe $t_{2g}$ and O (2p) orbitals. From Fig. 8(b), it is evident that
both $t_{2g}$ and $e_{g}$ orbitals of Fe (3d) are filled by electrons in the
spin-up channel, while in the spin-down channel $t_{2g}$ and $e_{g}$ states
are completely empty. This demonstrates
Fe3+($t^{3}_{2g\uparrow}e^{2}_{g\uparrow}$) valence state Das _et al._ (2008)
in Nd2CrFeO6 as illustrated in the inset of Fig. 8(b). In case of Cr (3d),
both $t_{2g}$ and $e_{g}$ orbitals are found to be empty in spin-up channel
and in spin-down channel only the $t_{2g}$ state is occupied, suggesting
Cr3+($t^{3}_{2g\downarrow}e^{0}_{g\downarrow}$) configuration (inset of Fig.
8(c)). These results indicate the strong AFM superexchange between
Cr3+($t^{3}_{2g\downarrow}e^{0}_{g\downarrow}$) and
Fe3+($t^{3}_{2g\uparrow}e^{2}_{g\uparrow}$) cations via O2- anions. For a
deeper understanding of the bonding as well as magnetism, charge density
analysis have been carried out. Fig. 9(a) shows the total charge density
distribution plot, which indicates the dominant Fe-O and Cr-O covalent bonding
due to the presence of bond charges along the path between Fe–O and Cr–O. The
magnetization density (Fig. 9(b)) displays the antiparallel alignment of spins
but with unequal magnitudes demonstrating FiM nature of Nd2CrFeO6. Fig. 9(c)
and 9(d) show the cross-sectional and perspective view of the 3D iso-surface
of magnetization density. In analogy to the DOS, here the spin density is also
mainly contributed by Fe (3d) and Cr (3d) orbitals. The spherical shape around
Fe atoms are indicating Fe3+($t^{3}_{2g\uparrow}e^{2}_{g\uparrow}$) with 3d5
orbital character occupied by spin-up electrons.Kharkwal _et al._ (2020) On
the other hand, only $t^{3}_{2g}$ orbital are visible around Cr atoms are
showing Cr3+($t^{3}_{2g\downarrow}$) character.Nazir (2021) In addition, tiny
spin densities have also appeared on the O atoms, due to the
Fe${}_{\uparrow}^{3+}$-O2--Cr${}_{\downarrow}^{3+}$ superexchange coupling. To
investigate the effects of Nd’s 4f spin and SOC, further calculations have
been carried out considering Nd’s 4f electrons as valence electrons with UNd=
6.5 eV. Including Nd’s 4f electrons in valence shell, the G-AFM ordering
remain as the lowest energy spin state, which is in agreement with the
previous reports on ReFeO3 and ReCrO3 perovskites.Zhu _et al._ (2017); Zhao
_et al._ (2017) Fig. 10(a) and 10(b) show the comparative DOS plots using 4f
electrons as core and valence electrons, respectively. No contribution of Nd
(4f) orbital can be seen either in VBM and CBM from Fig. 10(b). Moreover, the
band gap change (<0.8 eV) was found to be negligible. Similar results have
been found from the calculations with including SOC effect as shown in Figs.
10(c) and 10(d). Therefore, we think that effects of Nd’s 4f spin and SOC are
not so significant that change our conclusion with the results obtained by
considering Nd’s 4f electron as core electrons without SOC.

Before concluding, it could be worthwhile to compare the calculated magnetic
properties and band gap of Nd2CrFeO6 with other similar B-site ordered double
perovskite oxides. Table 4 shows the magnetic order, transition temperature
and band gaps of some Re containing B-site ordered double perovskites along
with our DFT-calculated results. The obtained ground state is in agreement
with Re2CrFeO6 and band gap is within the visible range like other candidates.
Therefore, it is anticipated that this compound may be a potential candidate
in various intriguing applications including spintronic memory devices,
memristive high-performance data storage devices, light-emitting diodes,
cryogenic magnetic cooling devices, solar cell and visible light driven
photocatalysis.

Table 4: Magnetic order (M), transition temperature (Tc) and optical band gap (Eg) of some Re2B${}^{{}^{\prime}}$B${}^{{}^{\prime\prime}}$O6 materials. Compound | B-site | M |  Tc (K) |  Eg (eV) | Ref.
---|---|---|---|---|---
La2MnCoO6 | ordered | FM | 226 | 1.93 | Gauvin-Ndiaye _et al._ (2018); Chang _et al._ (2017)
La2NiMnO6 | ordered | FM | 270 | 1.42 | Nasir _et al._ (2019)
Nd2NiMnO6 | ordered | FM | 195 | 1.57 | Nasir _et al._ (2019)
Dy2NiMnO6 | ordered | FM | 105 | 1.62 | Chanda _et al._ (2015)
La2FeCrO6 | ordered | FiM | 45 | 1.6 | Ohtomo _et al._ (2013)
Pr2FeCrO6 | partial | FiM | 245 | 2.13 | Gaikwad _et al._ (2019)
Nd2CrFeO6 | ordered | FiM | 265 | 1.85 | This work

##  VII Conclusions

This article focused on the phase stability, magnetic properties, and
electronic band structure of B-site ordered Nd2CrFeO6 double perovskite based
on DFT calculations. The value of on-site Coulomb interaction U was chosen to
be 3.5 eV for 3d-electrons of Cr and Fe by applying a range of U values within
GGA+U calculations that can reproduce the band gaps and magnetic ground state
of parent single perovskites NdCrO3 and NdCrO3 as well as double perovskite
Nd2CrFeO6. The material has been predicted to be stable by means of thermal,
mechanical, and dynamic stability criteria. Compared to the AFM nature of the
parent single perovskites counterparts, B site ordered double perovskite
Nd2CrFeO6 demonstrates a ferrimagnetic ground-state with a total magnetization
of 2 $\mu_{B}$ per formula unit due to the
Fe${}_{\uparrow}^{3+}$-O-Cr${}_{\downarrow}^{3+}$ super-exchange interaction.
The band structure and density of states calculations confirm the
semiconducting nature of this material with a direct band gap of $\sim$1.85 eV
which is significantly smaller than the band gaps of its parent single
perovskites. Notably, the probability of direct transition from VBM to CBM has
been found to be higher by using both GGA and GGA+U approaches, which is
essential for many optical applications such as solar cell, LASER etc. Due to
ferrimagnetism along with the direct nature of optical band gap, B-site
ordered Nd2CrFeO6 double perovskite could be a potential candidate for the
spintronic and visible light-mediated optical applications. Notably, the
crystallographic structure and physical properties of most of the single
perovskite materials are experimentally well established, however, double
perovskites are comparatively less investigated. Therefore, the experimental
data for various double perovskites are always not available. Our present
investigation might pave the way for the theoretical prediction of the
physical properties of any new double perovskite materials by employing the
experimental parameters of their parent single perovskites and optimizing the
effects of on-site d-d Coulomb interaction energy within GGA+U calculations.

## Conflicts of interest

“There are no conflicts to declare”.

## Acknowledgements

The computational facility provided by ’Ogata Laboratory (theoretical solid
mechanics), Department of Mechanical Science and Bioengineering, Osaka
University, Japan’ is sincerely acknowledged. The authors would also like to
acknowledge the Committee for Advanced Studies and Research (CASR), BUET for
financial assistance.

## References

  * Vasala and Karppinen (2015) S. Vasala and M. Karppinen, _Prog. Solid. State Ch._ , 2015, 43, 1–36.
  * Evans _et al._ (2021) H. A. Evans, L. Mao, R. Seshadri and A. K. Cheetham, _Annu. Rev. Mater. Res._ , 2021, 51, 351–380.
  * Mazumdar and Das (2021) D. Mazumdar and I. Das, _Phys. Chem. Chem. Phys._ , 2021, 23, 5596.
  * Kangsabanik _et al._ (2018) J. Kangsabanik, V. Sugathan, A. Yadav, A. Yella and A. Alam, _Phys. Rev. Mater._ , 2018, 2, 055401.
  * Bhuyan _et al._ (2021) M. D. I. Bhuyan, S. Das and M. A. Basith, _J. Alloys Compd._ , 2021, 878, 160389.
  * Kim _et al._ (2019) M. K. Kim, J. Y. Moon, S. H. Oh, D. G. Oh, Y. J. Choi and N. Lee, _Sci. Rep._ , 2019, 9, 5456.
  * Mandal and Gopalakrishnan (2005) T. K. Mandal and J. Gopalakrishnan, _Chem. Mater._ , 2005, 17, 2310–2316.
  * Shimada _et al._ (2003) T. Shimada, J. Nakamura, T. Motohashi, H. Yamauchi and M. Karppinen, _Chem. Mater._ , 2003, 15, 4494.
  * Yoshimatsu _et al._ (2019) K. Yoshimatsu, J. Ishimaru, K. Watarai, K. Yamamoto, Y. Hirata, H. Wadati, Y. Takeda, K. Horiba, H. Kumigashira, O. Sakata and A. Ohtomo, _Phys. Rev. B_ , 2019, 99, 235129.
  * Gray _et al._ (2010) B. Gray, H. N. Lee, J. Liu, J. Chakhalian and J. W. Freeland, _Appl. Phys. Lett._ , 2010, 97, 013105.
  * Qiao _et al._ (2017) Y. Qiao, Y. Zhou, S. Wang, L. Yuan, Y. Du, D. Lu, G. Che and H. Che, _Dalton Trans._ , 2017, 46, 5930.
  * Bolletta _et al._ (2018) J. P. Bolletta, F. Pomiro, R. D. Sánchez, V. Pomjakushin, G. Aurelio, A. Maignan, C. Martin and R. E. Carbonio, _Phys. Rev. B_ , 2018, 98, 134417.
  * Nair _et al._ (2014) V. G. Nair, L. Pal, V. Subramanian and P. N. Santhosh, _J. Appl. Phys._ , 2014, 115, 17D728.
  * Hossain _et al._ (2021) R. Hossain, A. Billah, M. Ishizaki, S. Kubota, F. Hirose and B. Ahmmad, _Dalton Trans._ , 2021, 50, 9519.
  * Yuan _et al._ (2014) L. Yuan, K. Huang, C. Hou, W. Feng, S. Wang, C. Zhou and S. Feng, _New J. Chem._ , 2014, 38, 1168–1172.
  * Yin _et al._ (2014) L. H. Yin, J. Yang, R. R. Zhang, J. M. Dai, W. H. Song and Y. P. Sun, _Appl. Phys. Lett._ , 2014, 104, 032904.
  * Sharannia _et al._ (2017) M. P. Sharannia, S. De, R. Singh, A. Das, R. Nirmala and P. N. Santhosh, _J. Magn. Magn. Mater._ , 2017, 430, 109–113.
  * Hou _et al._ (2021) L. Hou, L. Shi, J. Zhao, R. Tong and Y. Xin, _J. Phys. Chem. C_ , 2021, 125, 7950.
  * Rodrigues _et al._ (2020) A. P. G. Rodrigues, M. A. Morales, R. B. Silva, D. R. A. B. Lima, R. L. B. A. Medeiros, J. H. Araújo and D. M. A. Melo, _J. Phys. Chem. Solids_ , 2020, 141, 109334.
  * Coutinho and Barrozo (2018) P. V. Coutinho and P. Barrozo, _Appl. Phys. A_ , 2018, 124, 668.
  * Selvadurai _et al._ (2015) A. P. B. Selvadurai, V. Pazhanivelu, C. Jagadeeshwaran, R. Murugaraj, I. P. Muthuselvam and F. C. Chou, _J. Alloys Compd._ , 2015, 646, 924–931.
  * Hossain _et al._ (2018) A. Hossain, P. Bandyopadhyay and S. Roy, _J. Alloys Compd._ , 2018, 740, 414–427.
  * M. Seikh _et al._ (2013) M. M. Seikh, V. Pralong, O. I. Lebedev, V. Caignaert and B. Raveau, _J. Appl. Phys._ , 2013, 114, 013902.
  * Kleibeuker _et al._ (2017) J. E. Kleibeuker, E. M. Choi, E. D. Jones, T. M. Yu, B. Sala, B. A. MacLaren, D. Kepaptsoglou, D. H. Maldonado, Q. M. Ramasse, L. Jones, J. Barthel, I. MacLaren and J. L. M. Driscoll, _NPG Asia Mater._ , 2017, 9, e406.
  * Dass and Goodenough (2003) R. I. Dass and J. B. Goodenough, _Phys. Rev. B_ , 2003, 67, 014401.
  * King and Woodward (2010) G. King and P. M. Woodward, _J. Mater. Chem._ , 2010, 20, 5785–5796.
  * Wang _et al._ (2018) W. Wang, F. Liu, X. Zhang, X. Shen, Y. Yao, Y. Wang, B. Liu, X. Liu and R. Yu, _Inorg. Chem. Front._ , 2018, 5, 217–224.
  * Wang _et al._ (2015) J. Wang, X. Hao, Y. Xu, Z. Li, N. Zu, Z. Wu and F. Gao, _RSC Adv._ , 2015, 5, 50913–50918.
  * Leng _et al._ (2020) K. Leng, Q. Tang, Y. Wei, L. Yang, Y. Xie, Z. Wu and X. Zhu, _AIP Adv._ , 2020, 10, 120701.
  * Chen _et al._ (2019) X. Chen, J. Xu, Y. Xu, F. Luo and Y. Du, _Inorg. Chem. Front._ , 2019, 6, 2226.
  * Nasir _et al._ (2019) M. Nasir, S. Kumar, N. Patra, D. Bhattacharya, S. N. Jha, D. R. Basaula, S. Bhatt, M. Khan, S. W. Liu, S. Biring and S. Sen, _ACS Appl. Electron. Mater._ , 2019, 1, 141–153.
  * Lekshmi _et al._ (2014) P. N. Lekshmi, M. Vasundhara, M. R. Varma, K. G. Suresh and M. Valant, _Phys. Rev. B Condens. Matter_ , 2014, 448, 285–289.
  * Sun _et al._ (2020) M. Sun, Y. Xuan, G. Liu, Y. Liu, F. Zhang, J. Ren and M. Chen, _J. Magn. Magn. Mater._ , 2020, 504, 166670.
  * Lv _et al._ (2013) S. Lv, M. Saito, Z. Wang, C. Chen, S. Chakraverty, M. Kawasaki and Y. Ikuhara, _J. Appl. Phys._ , 2013, 114, 113705.
  * Chakraverty _et al._ (2011) S. Chakraverty, A. Ohtomo, D. Okuyama, M. Saito, M. Okude, R. Kumai, T. Arima, Y. Tokura, S. Tsukimoto, Y. Ikuhara and M. Kawasaki, _Phys. Rev. B_ , 2011, 84, 064436.
  * Lee and Ahn (2012) K. W. Lee and K. H. Ahn, _Phys. Rev. B_ , 2012, 85, 224404.
  * Booth _et al._ (2009) R. Booth, R. Fillman, H. Whitaker, A. Nag, R. Tiwari, K. Ramanujachary, J. Gopalakrishnan and S. Lofland, _Mater. Res. Bull._ , 2009, 44, 1559–1564.
  * Yu _et al._ (2020) M. Yu, S. Yang, C. Wu and N. Marom, _Npj Comput. Mater._ , 2020, 6, 180.
  * Ali _et al._ (2021) M. S. Ali, S. Das, Y. F. Abed and M. A. Basith, _Phys. Chem. Chem. Phys._ , 2021, 23, 22184–22198.
  * Kresse and Furthmüller (1996) G. Kresse and J. Furthmüller, _Comput. Mater. Sci._ , 1996, 6, 15–50.
  * Kresse and Furthmüller (1996) G. Kresse and J. Furthmüller, _Phys. Rev. B_ , 1996, 54, 11169.
  * Kresse and Hafner (1994) G. Kresse and J. Hafner, _Phys. Rev. B_ , 1994, 49, 14251.
  * Blöchl (1994) P. E. Blöchl, _Phys. Rev. B_ , 1994, 50, 17953.
  * Kresse and Joubert (1999) G. Kresse and D. Joubert, _Phys. Rev. B_ , 1999, 59, 1758.
  * Perdew _et al._ (1996) J. P. Perdew, K. Burke and M. Ernzerhof, _Phys. Rev. Lett._ , 1996, 77, 3865.
  * Perdew _et al._ (1997) J. P. Perdew, K. Burke and M. Ernzerhof, _Phys. Rev. Lett._ , 1997, 78, 1396.
  * Dudarev _et al._ (1998) S. L. Dudarev, G. A. Botton, S. Y. Savrasov, C. J. Humphreys and A. P. Sutton, _Phys. Rev. B_ , 1998, 57, 1505–1509.
  * Himmetoglu _et al._ (2014) B. Himmetoglu, A. Floris, S. de Gironcoli and M. Cococcioni, _Int. J. Quantum Chem._ , 2014, 114, 14–49.
  * Wang _et al._ (2012) B. T. Wang, W. Yin, W. D. Li and F. Wang, _J. Appl. Phys._ , 2012, 111, 013503.
  * Singh _et al._ (2020) A. Singh, S. Rajput, P. Balasubramanian, M. Anas, F. Damay, C. M. N. Kumar, G. Eguchi, A. Jain, S. M. Yusuf, T. Maitra and V. K. Malik, _Phys. Rev. B_ , 2020, 102, 144432.
  * Zhao _et al._ (2014) H. J. Zhao, W. Ren, Y. Yang, J. Íñiguez, X. M. Chen and L. Bellaiche, _Nat. Commun._ , 2014, 5, 4021.
  * Jain _et al._ (2013) A. Jain, S. P. Ong, G. Hautier, W. Chen, W. D. Richards, S. Dacek, S. Cholia, D. Gunter, D. Skinner, G. Ceder and K. a. Persson, _APL Mater._ , 2013, 1, 011002.
  * Ong _et al._ (2013) S. P. Ong, W. D. Richards, A. Jain, G. Hautier, M. Kocher, S. Cholia, D. Gunter, V. L. Chevrier, K. A. Persson and G. Ceder, _Comput. Mater. Sci._ , 2013, 68, 314–319.
  * Ong _et al._ (2008) S. P. Ong, L. Wang, B. Kang and G. Ceder, _Chem. Mater._ , 2008, 20, 1798–1807.
  * Togo and Tanaka (2018) A. Togo and I. Tanaka, _arXiv preprint arXiv:1808.01590_ , 2018.
  * Wang _et al._ (2021) V. Wang, N. Xu, J. C. Liu, G. Tang and W. T. Geng, _Comput. Phys. Commun._ , 2021, 267, 108033.
  * Togo and Tanaka (2015) A. Togo and I. Tanaka, _Scr. Mater._ , 2015, 108, 1–5.
  * Song _et al._ (2019) G. Song, Y. Chen, G. Li and B. Gao, _RSC Adv._ , 2019, 9, 2143–2151.
  * Bora and Ravi (2015) T. Bora and S. Ravi, _J. Supercond. Nov. Magn._ , 2015, 28, 869–872.
  * Chakraborty _et al._ (2014) K. R. Chakraborty, S. Mukherjee, S. D. Kaushik, S. Rayaprol, C. L. Prajapat, M. R. Singh, V. Siruguri, A. K. Tyagi and S. M. Yusuf, _J. Magn. Magn. Mater._ , 2014, 361, 81–87.
  * Yamaguchi (1974) T. Yamaguchi, _J. Phys. Chem. Solids_ , 1974, 35, 479–500.
  * Das _et al._ (2021) S. Das, M. D. I. Bhuyan and M. A. Basith, _J. Mater. Res. Technol._ , 2021, 13, 2408–2418.
  * Mannepalli _et al._ (2017) V. R. Mannepalli, M. M. S. Mohan and R. Ranjith, _Bull. Mater. Sci._ , 2017, 40, 1503–1511.
  * Nguyen _et al._ (2021) T. A. Nguyen, T. L. Pham, I. Y. Mittova, V. O. Mittova, T. L. T. Nguyen, H. V. Nguyen and V. X. Bui, _Nanomaterials_ , 2021, 11, 937.
  * Aparnadevi _et al._ (2016) N. Aparnadevi, K. Saravana Kumar, M. Manikandan, D. Paul Joseph and C. Venkateswaran, _J. Appl. Phys._ , 2016, 120, 034101.
  * Quattropani _et al._ (2018) A. Quattropani, D. Stoeffler, T. Fix, G. Schmerber, M. Lenertz, G. Versini, J. L. Rehspringer, A. Slaoui, A. Dinia and S. Colis, _J. Phys. Chem. C_ , 2018, 122, 1070–1077.
  * Geneste _et al._ (2019) G. Geneste, C. Paillard and B. Dkhil, _Phys. Rev. B_ , 2019, 99, 024104.
  * Mocherla _et al._ (2013) P. S. V. Mocherla, C. Karthik, R. Ubic, M. S. Ramachandra Rao and C. Sudakar, _Appl. Phys. Lett._ , 2013, 103, 022910.
  * Wang _et al._ (2019) J. Wang, L. Luo, C. Han, R. Yun, X. Tang, Y. Zhu, Z. Nie, W. Zhao and Z. Feng, _Materials_ , 2019, 12, 1444.
  * Wu _et al._ (2020) H. Wu, Z. Pei, W. Xia, Y. Lu, K. Leng and X. Zhu, _J. Alloys Compd._ , 2020, 819, 153007.
  * Henkelman _et al._ (2006) G. Henkelman, A. Arnaldsson and H. Jónsson, _Comput. Mater. Sci._ , 2006, 36, 354–360.
  * Emery and Wolverton (2017) A. A. Emery and C. Wolverton, _Sci. Data_ , 2017, 4, 170153.
  * Han _et al._ (2018) D. Han, T. Zhang, M. Huang, D. Sun, M. H. Du and S. Chen, _APL Mater._ , 2018, 6, 084902.
  * Qi _et al._ (2018) X. Qi, N. Cai, T. Chen, S. Wang and B. Li, _J. Appl. Phys._ , 2018, 124, 075902.
  * Vajeeston and Fjellvåg (2017) P. Vajeeston and H. Fjellvåg, _RSC Adv._ , 2017, 7, 16843.
  * Mouhat and Coudert (2014) F. Mouhat and F. X. Coudert, _Phys. Rev. B_ , 2014, 90, 224104.
  * Cheng _et al._ (2014) X. Y. Cheng, X. Q. Chen, D. Z. Li and Y. Y. Li, _Acta Crystallogr. C Struct. Chem._ , 2014, 70, 85–103.
  * Chu _et al._ (2015) B. Chu, D. Li, F. Tian, D. Duan, X. Sha, Y. Lv, H. Zhang, B. Liu and T. Cui, _Sci. Rep._ , 2015, 5, 10500.
  * Pajda _et al._ (2001) M. Pajda, J. Kudrnovský, I. Turek, V. Drchal and P. Bruno, _Phys. Rev. B_ , 2001, 64, 174402.
  * Chen _et al._ (2019) W. Chen, J. George, J. B. Varley, G. M. Rignanese and G. Hautier, _Npj Comput. Mater._ , 2019, 5, 72.
  * Wan _et al._ (2016) X. Wan, H. C. Ding, S. Y. Savrasov and C. G. Duan, _Sci. Rep._ , 2016, 6, 22743.
  * Wu _et al._ (2016) C. Wu, Z. Yu, K. Sun, J. Nie, R. Guo, H. Liu, X. Jiang and Z. Lan, _Sci. Rep._ , 2016, 6, 36200.
  * Kabiraj _et al._ (2020) A. Kabiraj, M. Kumar and S. Mahapatra, _Npj Comput. Mater._ , 2020, 6, 35.
  * Sasani _et al._ (2021) A. Sasani, J. Iñiguez and E. Bousquet, _Phys. Rev. B_ , 2021, 104, 064431.
  * Ong _et al._ (2008) K. P. Ong, P. Blaha and P. Wu, _Phys. Rev. B_ , 2008, 77, 073102.
  * Das _et al._ (2008) H. Das, U. V. Waghmare, T. Saha-Dasgupta and D. D. Sarma, _Phys. Rev. Lett._ , 2008, 100, 186402.
  * Kharkwal _et al._ (2020) K. C. Kharkwal, R. Roy, H. Kumar, A. K. Bera, S. M. Yusuf, A. K. Shukla, K. Kumar, S. Kanungo and A. K. Pramanik, _Phys. Rev. B_ , 2020, 102, 174401.
  * Nazir (2021) S. Nazir, _Sci. Rep._ , 2021, 11, 1240.
  * Zhu _et al._ (2017) X. H. Zhu, X. B. Xiao, X. R. Chen and B. G. Liu, _RSC Adv._ , 2017, 7, 4054–4061.
  * Zhao _et al._ (2017) H. J. Zhao, L. Bellaiche, X. M. Chen and J. Íñiguez, _Nat. Commun._ , 2017, 8, 14025.
  * Gauvin-Ndiaye _et al._ (2018) C. Gauvin-Ndiaye, T. E. Baker, P. Karan, É. Massé, M. Balli, N. Brahiti, M. Eskandari, P. Fournier, A. S. Tremblay and R. Nourafkan, _Phys. Rev. B_ , 2018, 98, 125132.
  * Chang _et al._ (2017) H. Chang, Y. Gao, F. Liu, Y. Liu, H. Zhu and Y. Yun, _J. Alloys Compd._ , 2017, 690, 8–14.
  * Chanda _et al._ (2015) S. Chanda, S. Saha, A. Dutta and T. P. Sinha, _Mater. Res. Bull._ , 2015, 62, 153–160.
  * Ohtomo _et al._ (2013) A. Ohtomo, S. Chakraverty, H. Mashiko, T. Oshima and M. Kawasaki, _J. Mater. Res._ , 2013, 28, 689–695.
  * Gaikwad _et al._ (2019) V. M. Gaikwad, M. Brahma, R. Borah and S. Ravi, _J. Solid State Chem._ , 2019, 278, 120903.