# Design of Magnetic Polar Double-Double Perovskite Oxides through Cation
Ordering

Monirul Shaikha<EMAIL_ADDRESS>Duo Wangb Saurabh Ghosha
<EMAIL_ADDRESS>aDepartment of Physics and Nanotechnology, SRM
Institute of Science and Technology, Kattankulathur 603 203, Chennai, India
bFaculty of Applied Sciences, Macao Polytechnic University, Macao SAR 999078,
China

###### Abstract

Commencing from the centrosymmetric MnRMnSbO6 compound, we explore the realm
of magnetic polar double-double perovskite oxides characterized by significant
ferroelectric polarization. Employing symmetry operations, first-principles
methodologies, and Monte Carlo simulations, our investigation delves into the
structural, magnetic, ferroelectric, and electronic attributes of the polar
LaFeMnNiO6 and LaTiMnNiO6 compounds. The structural analysis uncovers that the
paraelectric-ferroelectric phase transition is intricately linked to the
Fe/Ti-displacement of square planar Fe/TiO4. Notably, the magnetic LaFeMnNiO6
and LaTiMnNiO6 compounds demonstrate robust ferroelectric polarizations,
measuring 20.0 $\mu$C/cm2 and 21.8 $\mu$C/cm2, respectively, accompanied by
minimalist forbidden energy gaps of 1.40 eV and 1.18 eV using the GGA+U
method. Furthermore, we pinpoint elevated magnetic transition temperatures for
these compounds. Additionally, our study scrutinizes the energies associated
with diverse spin configurations and identifies potential minimum
decomposition pathways into stable oxides. This comprehensive analysis ensures
the meticulous formation of the LaFeMnNiO6 and LaTiMnNiO6 compounds.

##  I INTRODUCTION

Double perovskite oxides (DPOs) with common chemical formulae A2BB′O6 have
intense research focus due to their implications in sciences and in modern-day
technologies. They exhibit important physical properties, including
ferroelectricity, ferro/ferrimagnetism, and so on [1, 2, 3, 4, 5]. In DPOs
family a framework of corner-shared BO6 octahedra forms a three-dimensional
network and large A-site cations are accommodated at the 12-coordinated
cuboctahedral cavities [6]. In addition, the A- and B-sublattices can be
ordered in various ways, namely layer, columnar, and rock-salt. There are a
good number of perovskite oxides that reportedly show large ferroelectric
distortions with high polarization values but with negligible magnetization
values [7, 8].

Combining cation orderings with chemical substitution in the perovskite oxides
family, is worth exploring for multiferroic properties where both
ferroelectricity, and magnetism can coexist [9, 10, 11]. DPOs AA′BB′O6 where
A, and A′ are alkali metals or rare-earth ions and B, and B′ are transition
metals are predicted and extensively studied for multiferroic properties,
where both ferroelectricity and magnetism can coexist [12, 13, 14, 15, 16].
Due to structural and compositional flexibility in AA′BB′O6 DPOs, both A- and
B-sublattices can host a large number of atomic combinations. Recently, we
realized the importance of cation ordering in both A- and B-sublattices to
design polar magnetic metals/insulators in the DPOs family [17]. Leveraging
machine learning, we have pinpointed key features that contribute to
stabilizing the system, leading to A-site layered B-site rocksalt ordering
[18]. Additionally, our investigation has revealed that the emergence of
ferroelectricity is attributed to an incommensurate cation radius mismatch
between successive AO and A′O layers (within A-site layered ordering) and
structural distortion responsible for ferroelectric switching in DPOs [19].

Double-double perovskite oxides (DDPOs) are characterized by the same formula
as DPOs, i.e., AA′BB′O6, but now are occupied by transition metals (TMs),
represent a potential class of materials where functional properties including
multiferroicity can be explored [20, 21, 22, 23, 24]. A unique feature of
these perovskite oxides is that the different atomic sites i.e., A′, B and B′
can accommodate magnetic transition metals with the possibility of enhanced
magnetic interactions [21, 24]. However, the first reported DDPO is CaFeTi2O6
with only one magnetic (Fe) site [25]. It crystallizes in a tetragonal
centrosymmetric space group P42/nmc with a 10-fold Ca-coordination, a
tetrahedral Fe-coordination, and another square-planer Fe-coordination. In
recent years, A-site ordered CaMnTi2O6 DDPOs with similar coordination of
CaFeTi2O6 at the A-sublattices, attracted attention owing to its ferroelectric
distortion with large bandgap $\sim$ 3.0 eV [20, 22, 23]. A set of similar
DDPOs, MnRMnSbO6 (R = La, Pr, Nd, Sm) with both A-site and B-site cation
ordered phases are synthesized [21]. These DDPOs show large magnetization but
they crystallize in tetragonal centrosymmetric space group $P4_{2}/n$ with no
ferroelectric polarization [21, 24].

In addition to the cation orderings at the A- and B-sublattices, 15 different
tilting patterns lower the symmetry from cubic Pm$\bar{3}$m and give rise to
various fascinating physical properties [26]. Out of which a few Glazer
patterns such as $a^{+}a^{+}a^{+}$, $a^{0}b^{+}b^{+}$, $a^{0}b^{+}b^{-}$,
$a^{-}a^{-}c^{+}$ have been explored for A-site cation ordered DPOs with in-
phase (+), out-of-phase (-), and no (0) rotations of the BO6 octahedra [27,
28]. The $a^{+}a^{+}c^{-}$ rotational pattern [22, 29] with A/A′ cation
ordered DDPOs can lead to ferroelectric distortion into the structure. Thus,
the B-site rocksalt A-site columnar double-double perovskite oxides with
appropriate rotational patterns hold promise for fabricating magnetic
materials with substantial ferroelectric polarization.

In this work, our focus lies on the deliberate design of magnetic
ferroelectrics. We aim to achieve this by introducing $a^{+}a^{+}c^{-}$
octahedra rotations within MnRMnSbO6 double-double perovskite oxides (DDPOs).
This design strategy involves cation ordering at both A- and B-sublattices,
coupled with chemical substitution. Within first principles framework, we
discuss the origin of ferroelectricty, band gaps opening, and high magnetic
transition temperatures in polar LaFeMnNiO6 (LFMNO) and LaTiMnNiO6 (LTMNO)
compounds. We study the stability of these compounds for different spin
configurations, and possible minimum decomposition pathways into the stable
oxides.

##  II COMPUTATIONAL METHODS

Density functional theory (DFT) [30] calculations are performed for
optimization of geometry, total energy, and polarization using the Berry phase
method [31] as implemented in the Vienna $ab-initio$ simulation package (VASP)
[32]. The k-integration in the Brillouin zone is incorporated using
$\Gamma$-centered 4 × 4 × 4 points for geometry optimization and 8 × 8 × 8
points for self-consistent calculations using 520.0 eV energy cut-off. We
considered the generalized gradient approximation (GGA) augmented by the
Hubbard-$U$ corrections (GGA+$U$) [33] to describe the exchange-correlation
effect. To consider $d-d$ Coulomb interactions, we employ $U_{E}$ [34] (=
$U-J_{H}$ where $J_{H}$ is Hund’s exchange parameter) parameters of 0.0 eV for
Ti-$d$ [22], 4.0 eV for both Mn-$d$, and Fe-$d$, and 6.0 eV for Ni-$d$ [17]
electrons. The exchange-correlation part is estimated by Perdew-Burke-
Ernzerhof revision for solids (PBEsol) functional [35]. The total energy and
Hellman-Feynman force are carefully converged for individual atoms down to 1
$\mu eV$ and 1 meV/$\AA$, respectively. To draw, and analyze the geometry of
our three-dimensional systems, we implement the Visualization for Electronic
and Structural Analysis (VESTA) software [36]. We perform phonon calculations
on the fully relaxed $a^{+}a^{+}c^{-}$ rotated LaTMMnNiO6 DDPOs to find polar
structural distortion using the finite difference method as implemented in
VASP [37]. The symmetry operations are performed with the help of the
ISODISTORT tool [38, 39].

Next, we utilize the optimized structure as input for the calculations of
interatomic exchange parameters by means of the magnetic force theorem (MFT)
[40] using full-potential linear muffin-tin orbital (FP-LMTO) in RSPT code
[41]. At last, an effective spin Hamiltonian is constructed, and phase
transition temperatures are obtained by conducting classical Monte Carlo
simulations, as implemented in the UppASD package [42]. Exchange parameters
are calculated using the full-potential linear muffin-tin orbitals code RSPt
[41], and these parameters are then used as input for classical Monte Carlo
simulations, as implemented in the UppASD package [42]. Supercells consisting
of approximately 60,000 magnetic atoms are adopted as the structural model. An
annealing process is simulated by performing calculations that start at high
temperatures and are gradually decreased to 0 K. To ensure that the magnetic
properties obtained at each temperature are in their equilibrium state, an
initial simulation of 50,000 steps is performed.

We calculate the formation energies for each DDPOs proposed in this work by
considering the decomposition reaction of the DDPOs via the most probable
reaction pathways. Many possible reaction sequences can produce these DDPOs,
and each of them could lead us to different formation energies, which could
give erroneous results. Hence, to find the correct formation energy, we find
out the minimum energy reaction sequence to produce these DDPOs.

##  III RESULTS AND DISCUSSIONS

Figure 1: (Color online) (a) Crystal structure of ferroelectric LaFeMnNiO6 DDPO. B-site NiO6 and MnO6 octahedra are arranged in a rock-salt type order whereas A-site La and Fe are ordered in columns parallel to the crystallographic $c$-direction. Further, Fe A-sites govern Fe1O4 and Fe2O4 with square-planar and tetrahedral environments respectively. The La-, Mn-, Ni-, and O-atoms are described by green, magenta, silver, and red balls, respectively. (b) The double-well potential of LFMNO(blue rhombus) and LTMNO(red circles) as a function of Fe/Ti off-center displacements from the square planar environment as is shown in Figure 1 (a). Table 1:  Intriguing physical properties of LFMNO and LTMNO DDPOs. Systems | Space | Band gap | Polarization | Magnetization | TC
---|---|---|---|---|---
| group | Eg (eV) |  $\mu C$/cm2 |  $\mu_{B}$/f.u. | (K)
LFMNO | $P4_{2}$ | 1.40 | 20.0 | 5.0 | 225
LTMNO | $P4_{2}$ | 1.18 | 21.8 | 0.0 | 48
Table 2:  Crystal structure information of LaTMMnNiO6 with TM = Fe and Ti. Systems |  | Lattice parameters ($\AA$) |  | Cell
---|---|---|---|---
| $a$ | $b$ | $c$ | volume ($\AA^{3}$)
LFMNO | 7.55 | 7.55 | 7.74 | 441.20
LTMNO | 7.65 | 7.65 | 7.76 | 454.44
Figure 2: (Color online) The low symmetry ferroelectric $P4_{2}$ phase of LaTMMnNiO6 compounds for $a^{+}a^{+}c^{-}$ rotation is related to the centrosymmetric $P4_{2}/mmc$ reference structure through three major structural distortions; (a) out-of-phase ($a^{0}a^{0}c^{-}$) rotation of Fe/TiO6 octahedra along $c$-axis denoted by $\Gamma_{3}^{+}$ irreducible representation (irrep.), (b) in-phase ($a^{+}a^{+}c^{0}$) rotation of Fe/TiO6 octahedra along two crystallographic $a$\- and $b$-axes represented by M${}_{4}^{-}$ irrep. and (c) ferroelectric displacement of the Fe/Ti-cations from the square plane of Fe/TiO4 described by $\Gamma_{3}^{-}$. (d) The group-subgroup tree for our systems, the ferroelectric $P4_{2}$ phase, is highlighted with blue. The La-, and Fe/Ti-atoms are omitted from the first two structures for clarity. Figure 3: (Color online) All possible spin configurations of LaFeMnNiO6 within the collinear picture. The non-magnetic atoms are not shown for better clarity. The Fe-, Mn-, and Ni- atoms are denoted by blue, magenta, and silver balls respectively. Table 3:  Magnetic moments and charge states of LaTMMnNiO6 DDPOs. Systems |  | Magnetic moments ($\mu_{B}$) of |  | Total magnetic moment |  | Charge state of |
---|---|---|---|---|---|---|---
| A′-site | B-site | B′-site | ($\mu_{B}$/f.u.) | A′-site | B-site | B′-site
LFMNO | 4.21 | 3.16 | 1.71 | 5.00 | 3+ | 4+ | 2+
LTMNO | 0.00 | 3.82 | 1.74 | 0.00 | 4+ | 3+ | 2+
Figure 4: (Color online) All possible spin configurations of LaTiMnNiO6 within the collinear picture. The non-magnetic atoms are not shown for better clarity. The Mn-, and Ni- atoms are denoted by magenta, and silver balls respectively. Table 4: Bond angles, and bond lengths of LaTMMnNiO6 DDPOs. Systems | Average bond angle (∘) | Average bond length (Å)
---|---|---
| O-TMSP-O | Mn-O-Ni | SP TM-O | TH TM-O | Mn-O | Ni-O
LFMNO | 86.8 | 141.7 | 2.02 | 1.94 | 1.96 | 2.03
LTMNO | 84.0 | 139.9 | 1.98 | 1.87 | 2.02 | 2.04

###  III.1 Structural and Ferroelectric Properties

In our pursuit of notable magnetic polar materials, specifically LaFeMnNiO6
(LFMNO)and LaTiMnNiO6 (LTMNO), we have undertaken an exploration of a series
of double-double perovskite oxides (DDPOs) based on 3$d$ transition metals.
These materials feature A′, representing Ti, V, Cr, Mn, Fe, Co, Ni, and Cu in
LaA′MnNiO6 type perovskites. The crystal structure of the A-site ordered DDPO
is illustrated in Figure 1(a), adopting a tetragonal polar (non-
centrosymmetric) $P4_{2}$ symmetry and displaying a substantial ferroelectric
polarization (see TABLE 1). A detailed structural and symmetry analysis
reveals that the polar P42 space group follows $a^{+}a^{+}c^{-}$ (Fe/Ti)O6
octahedra rotation pattern akin to the $P4_{2}/nmc$ structure observed in the
first-synthesized CaFeTi2O6 DDPO [25]. Recently, Ji et al. suggested that
A-site columnar and B-site rocksalt-ordered DDPOs result in a centrosymmetric
$P4_{2}/n$ space group [24]. However, in contrast to their findings, we
present A-site columnar and B-site rocksalt-ordered DDPOs exhibiting a polar
$P4_{2}$ structure. Notably, our design accommodates magnetic transition
metals (TM) at both B- and B′-sublattices, in addition to TM at the A′ site,
adding a layer of versatility to the system. The La-cations acquire a
10-coordinated geometry, while both Mn- and Ni- at the B-sites form BO6
octahedra almost equally tilted along the crystallographic $c$-axis. The TMs
on the other hand at A′-site show two nonequivalent geometry in columns, i.e.
(Fe/Ti)O4 tetrahedra (TH), and (Fe/Ti)O4 square planar (SP) as are shown in
cyan and green respectively in Figure 1 (a). Further, the space group $P4_{2}$
allows both Fe3+ ($d^{5}$) and Ti4+ ($d^{0}$) at the SP-site to move along the
crystallographic $c$-direction, leading to breaking of an inversion center
similar to ref. [20, 22]. The absence of a center of symmetry steers
ferroelectricity into the systems. Detailed information on crystal structures,
magnetic moments, and charge states of these compounds are provided in TABLE
2-4. Nonetheless, total energy as a function of polar distortion for Ti4+
($d^{0}$) in LaTiMnNiO6 is comparable with the PbTiO3 [43]. While Fe3+
($d^{5}$) in LaFeMnNiO6 exhibit a pronounced depth of the double-well as
compared to CaMnTi2O6 as shown in Figure 1 (b) [22].

We then examine the group-subgroup relation that connects between $P4_{2}$
(No. 77) phase with reference $P4_{2}/mmc$ (No. 131) high symmetry structure
(without any distortions), and by implementing ISODISTORT [44]. The
decomposition of the $P4_{2}$ phase with reference to $P4_{2}/mmc$ symmetry
provides us with three contributing structural distortions. These are out-of-
phase rotated $a^{0}a^{0}c^{-}$ (Fe/Ti)O6 octahedra with irreducible
representation (irrep.) $\Gamma_{3}^{+}$, in-phase $a^{+}a^{+}c^{0}$ rotation
of (Fe/Ti)O6 octahedra with irrep. M${}_{4}^{-}$, and off-centering of
(Fe/Ti)-cation of (Fe/Ti)O4 SP from the center-of-symmetry with irep.
$\Gamma_{3}^{-}$ as shown in Figure 2 (a-c). The normalized mode amplitudes
0.34 (0.36) $\AA$, 0.50 (0.53) $\AA$, and 0.16 (0.11) $\AA$ for
$\Gamma_{3}^{+}$, M${}_{4}^{-}$, and $\Gamma_{3}^{-}$ respectively of LTMNO
(LFMNO) explain the depth of the double-wells in Figure 1 (b). Moreover,
symmetry operation indicates a phase-transition chain described in Figure 2
(d). A coupling between in-phase $a^{+}a^{+}c^{0}$ ($\Gamma_{3}^{+}$), and
out-of-phase $a^{0}a^{0}c^{-}$ (M${}_{4}^{-}$) rotated (Fe/TiO)6 octahedra
reduces the symmetry to centrosymmetric $P4_{2}/n$ (No. 86) phase. Further
reduction in symmetry occurs to polar $P4_{2}$ by clubbing $\Gamma_{3}^{-}$
mode into $P4_{2}/n$ phase.

###  III.2 Polar Magnetic Behaviors from Electronic Structure Calculations

Next, we investigate the stability of the working compounds from various spin
configurations. In the case of La3+Fe3+Mn4+Ni2+O6 DDPO, the A′-site Fe-atoms
adapted within the cavity of (Mn/Ni)O6 octahedra. Herein, the nearest-neighbor
distances reduce significantly. Consequently, the structure exhibits a complex
magnetism within the collinear spin configuration. To achieve a proper
magnetic ground state in the collinear spin configuration for LFMNO system, we
consider all possible spin configurations between Fe3+, Mn4+, and Ni2+ and are
described in Figure 3. Out of which a complex ferrimagnetic configuration is
found to be the magnetic ground state as shown in Figure 6 (a). The
corresponding electronic structure is shown in Figure 6 (b). Other collinear
spin configurations are found to be stable within an energy window of $\sim$ 1
eV. A set of similar complex ferrimagnetic compounds were synthesized
previously but all of them were found to be centrosymmetric $P4_{2}/n$ space
group [45]. All the nearest neighbor Fe spins are aligned in opposite
directions, leading to a net zero moment from the Fe site in the system. Thus,
Fe-Fe interactions are antiferromagnetic. An weak AFM Fe3+-O-O-Fe3+
superexchange interaction is found in Fe3+ layers. The Fe3+ ($d^{5}$)
displacement from the square planar surrounding with a
non-$d^{0}$-configuration leads to the breaking of an inversion center similar
to ref. [23]. Further, Fe-spins interact antiferromagnetically with both Mn-,
and Ni-spins. Consequently, ferromagnetic (FM) interactions are found between
Mn and Ni spins. These FM interactions govern a net 5.00 $\mu_{B}$/f.u.
magnetization into the system (TABLE 1). Since the nearest neighbor distances
reduce, we can expect even stronger magnetic exchange interactions as compared
to CaMnTi2O6. The magnitudes of Fe-moments (4.21 $\mu_{B}$/Fe) of SP and TH
FeO4 are found to be almost similar.

The lowest magnetic configuration of La3+Ti4+Mn3+Ni2+O6 is found to be A-type
antiferromagnetic (AFM) ordering as described in Figure 6(c), followed by
ferromagnetic (FM) configuration. The corresponding electronic structure is
shown in Figure 6(d). The AFM ordering with the absence of an inversion center
indicates that this compound is important for multifunctional properties.
Other collinear spin configurations (Figure 4) are found to be stable within
an enhanced energy window ($\sim$ 40 meV) as compared to ref. [22] with
reference to A-type AFM-ordering. This indicates that we may achieve strong
magnetic exchange interactions for LaTiMnNiO6 as compared to CaMnTi2O6.

The magnetic polar behavior of LFMNO and LTMNO makes them superior compounds
in the family. Furthermore, we identify LFMNO, and LTMNO as direct bandgap
semiconductors with energy gaps of 1.40, and 1.18 eV respectively and
presented in Figure 5. This may be facilited them for visible light
absorption. Density of states (DOS) analysis of LFMNO reveals that a local
moment of 4.21 $\mu_{B}$/Fe with filled $d$-orbitals in the up spin channel
(USC) suggests a nominal charge state of Fe3+ ($t_{2g}^{3}e_{g}^{2}$) is in a
high spin state and is shown in Figure 6 (b). The octahedral environment of
Mn-, and Ni-atom show a $t_{2g}$, and $e_{g}$ crystal field splitting.
Ni-$t_{2g}$, and Ni-$e_{g}$ bands in the USC lie between -8 eV energy and the
Fermi energy (EF) and show a strong hybridization with Mn-$d$ and O-$p$. While
Ni-$t_{2g}$ bands in the down spin channel (DSC) are localized between O-$p$
and the Fermi level. The $e_{g}$ levels in the DSC are located above EF. This
DOS along a local moment of 1.71 $\mu_{B}$/Ni indicates a nominal charge state
of Ni2+ ($t_{2g}^{6}e_{g}^{2}$). The filled Mn-$t_{2g}$ bands are located
between Ni-$t_{2g}$ and Ni-$e_{g}$ levels in the USC. Mn-$t_{2g}$ in the DSC
is totally empty. While Mn-$e_{g}$ in both the spin channels are found above
the Fermi level and an insulting phase is obtained analogous to ref. [46].
This suggests a nominal charge state of Mn4+ ($t_{2g}^{3}e_{g}^{0}$) with
local moment of 3.16 $\mu_{B}$/Mn.

Figure 5: (Color online) Calculated electronic band structure from GGA+U
calculation for (a) LaFeMnNiO6 and (b) LaTiMnNiO6 DDPOs. Figure 6: (Color
online) Ground state spin configuration and corresponding electronic structure
of LaFeMnNiO6 (top panel) and LaTiMnNiO6 (bottom panel). The non-magnetic
atoms are not shown for clarity. The Fe-, Mn-, and Ni-atoms are described by
blue, magenta, and silver balls respectively. Figure 7: (Color online) Orbital
resolved electronic structure from GGA+U calculation for LaTiMnNiO6 DDPO.

Investigation on the partial DOS of LTMNO, provides us with Ti-$d$ states to
be mostly empty, suggesting its $d^{0}$-configuration (Ti4+) and is shown in
Figure 6 (d). Due to octahedral surrounding in MnO6, and NiO6 the $d$-orbitals
split up into $t_{2g}$, and $e_{g}$ levels similar to LFMNO. As discussed
earlier the Mn-Ni interaction in LTMNO between layers (A-type AFM ordering) is
antiferromagnetic and hence the DOS is identical in both the spin-channels.
However, if we consider any FM pair of Mn- and Ni-atoms, we find a similar
electronic structure for the Ni2+ ($t_{2g}^{6}e_{g}^{2}$) and is shown in
Figure 7. While the filled Mn-$t_{2g}$ bands are found between Ni-$t_{2g}$ and
Ni-$e_{g}$ levels in the USC. But, Mn-$e_{g}$ lies below and above EF in the
same USC. In the DSC both Mn-$t_{2g}$ and Mn-$e_{g}$ are located above Fermi
energy leading to an insulating solution. This suggests a nominal charge state
of Mn3+ ($t_{2g}^{3}e_{g}^{1}$) with local moment of 3.82 $\mu_{B}$/Mn.

Figure 8: (Color online) (a) Calculated inter-site exchange parameters for
each $TM$-$TM$ pair as a function distance for LFMNO. The positive and
negative values represent ferromagnetic and antiferromagnetic couplings,
respectively. The inset shows the magnetization (black circles) and the
normalized specific heat (red circles) as a function of temperature,
calculated from classical Monte Carlo simulations. A blue vertical line
indicates the magnetization transition temperature. (b) Calculated inter-site
exchange parameters for each $TM$-$TM$ pair as a function distance for LTMNO.
The inset shows the normalized specific heat as a function of temperature,
calculated from classical Monte Carlo simulations. A red vertical line
indicates the magnetization transition temperature.

###  III.3 Monte Carlo Simulations

The calculated inter-site exchange parameters are shown in Figure 8 (a).
First, the magnetic couplings decrease rapidly as the TM-TM distance
increases; only the contributions from the first few nearest neighbours play
an important role in determining the magnetic ground state. Second, among the
couplings between the three transition metal elements, the one between Ni and
Mn is ferromagnetic, while those for the other two pairs contribute to
antiferromagnetic interaction. In addition, different from the nearest Ni-Ni
and Mn-Mn pairs, which are separated by nearly a unit cell’s distance, the two
Fe-Fe pair distances are relatively small and, therefore, exhibit noticeable
antiferromagnetic coupling. This is perfectly consistent with the
ferrimagnetic ground state configuration obtained from our total energy
calculations.

With an average Ni-Mn ferromagnetic coupling of 0.33 mRy, and the Fe-Mn and
Fe-Ni antiferromagnetic couplings of -0.35 mRy and -0.28 mRy, respectively, in
addition to one strong and one weak antiferromagnetic Fe-Fe coupling of -0.42
mRy and -0.06 mRy, the system exhibits a remarkable magnetic transition
temperature of 225 K, as shown in the inset of FIG. 4. In the case of LTMNO,
normalized specific heat as a function of temperature exhibits a magnetic
transition temperature of 48 K calculated from classical Monte Carlo
simulations and is shown in Figure 8 (b).

###  III.4 Computation of Formation Energies

To identify the correct reaction sequence, we follow a theoretical framework
suggested by Akbarzadeh et al. [47] and made appropriate modifications to suit
our problems [17, 48]. We consider all possible stable oxides from the
Materials Project [49] that fall on the energy hull and are listed in TABLE 5.
The reaction energy is calculated by the following expression:

$G=\sum_{i}x_{i}F_{i}$ (1)

Where $G$ is the total reaction energy of the reaction sequences, $i$ includes
a set of all possible stable oxides, $F$ is the free energy (at T = 0 K) of
the $i$th compound, and $x_{i}$ (unknown) is the variable molar fraction of
the $i$th compound at a given composition. To get the minimum energy reaction,
we minimized equation 1 with respect to the molar fraction $x_{i}$, with mass
conservation constraints such as

$f_{s}=\sum_{i}x_{i}n^{s}_{i}=Constant$ (2)

Where $n^{s}_{i}$ is the number of ions of species $s$ in the $i$th compound
per formula unit and $f_{s}$ is the molar fraction of the individual species
$s$ (s = La, Ti, Fe, Mn, Ni, and O). To apply the above formalism, we
considered a set of all possible stable compounds from the Materials Project
database [49] that contains La, Ti, Fe, Mn, Ni, and O as the constituents of
the DDPOs. We calculate the free energy of these compounds using the same
parameters that we used to calculate the energies of DDPOs. Using these
calculated free energies, we minimized the linear equation 1 using a linear
programming solver and calculated the formation energies with respect to these
minima energy reactions.

Table 5:  List of stable oxides considered for computing the formation energies of the double-double perovskites. Stable Oxides | Space group | Stable Oxides | Space group
---|---|---|---
La2O3 | $Ia-3$ | TiNiO3 | $R-3$
La2TiO5 | $Pnma$ | TiMnO3 | $R-3$
La2Ti2O5 | $P2_{1}$ | TiMn2O4 | $P4_{3}2_{2}$
Ti2O | $P-3m_{1}$ | Mn2O3 | $Pbca$
Ti2O3 | $R-3c$ | MnO2 | $I4/m$
Ti3O5 | $C2/m$ | MnO | $Fm-3m$
Ti6O | $P-31c$ | LaNiO3 | $R-3c$
TiO | $P-62m$ | MnNiO3 | $R-3$
NiO | $Fm-3m$ | Fe2NiO4 | $Imma$
Ni3O4 | $Cmmm$ | Mn3O4 | $I41/amd$
LaFeO3 | $R-3c$ | FeO | $C2/m$
FeO | $C2/m$ | Fe2O3 | $R-3c$
Ti3O | $P-31c$ | TiO2 | $C2/m$

The thermodynamic stability is one of the basic requirements to synthesize a
compound for magnetization into the systempractical applications. We,
therefore, examine the formation energies of LFMNO and LTMNO DDPOs. We
calculate the formation energies for these systems by considering the
decomposition reaction of DDPOs via the most probable reaction pathways by
utilizing a linear programming problem clubbed with the grand canonical method
[17]. The detailed methodology is provided in the method section. We consider
the stable oxides that were reported in the inorganic crystal structure
database (ICSD) [50], and in the materials project [49] in our study. The
minimum energy decomposition paths for these compounds are:

$\begin{split}\Delta_{f}E_{5}[LaFeMnNiO_{6}]=E[LaFeMnNiO_{6}]\\\
-(\frac{3}{6}E[La_{2}O_{3}]+\frac{3}{6}E[Fe_{2}O_{3}]+\frac{1}{6}E[Mn_{3}O_{4}]\\\
+\frac{3}{6}E[MnO_{2}]+\frac{2}{6}E[Ni_{3}O_{4}])\end{split}$ (3)
$\begin{split}\Delta_{f}E_{1}[LaTiMnNiO_{6}]=E[LaTiMnNiO_{6}]\\\
-(\frac{3}{6}E[La_{2}Ti_{2}O_{7}]+\frac{2}{6}E[Mn_{3}O_{4}]+\frac{3}{6}E[NiO]\\\
+\frac{1}{6}E[Ni_{3}O_{4}])\end{split}$ (4)

Implementing these equations, the formation energies of LFMNO and LTMNO are
found to be 1.25 eV and 0.89 eV respectively. This is in agreement with the
experimental observations as the DDPOs are synthesized under high pressure
($\sim$ 10-15 GPa) and high-temperature ($\sim$ 1200-1700 oC)[45, 24].

##  IV CONCLUSION

In summary, our investigation based on DFT calculations and symmetry analysis
reveals the presence of direct gap semiconductors, with GGA+U forbidden energy
values of 1.40 eV and 1.18 eV for LFMNO and LTMNO, respectively, suggesting
potential suitability for visible light absorption. We elucidate the origins
of these magnetic polar semiconductors. The observed long-range ferri-magnetic
ordering in LFMNO is attributed to superexchange interactions. Through Monte
Carlo simulations, we determine magnetic transition temperatures of 225K and
48K for LFMNO and LTMNO, respectively, significantly higher than the
approximate 10 K observed in CaMnTi2O6. The calculated spontaneous
polarization we report as 20.0 and 21.8 $\mu C$/cm2 for LFMNO and LTMNO,
respectively. In conclusion, both LFMNO and LTMNO emerge as promising
multiferroics, featuring magnetic transition metals at B- and B′-sublattices,
in addition to TM at the A′-site.

##  V ACKNOWLEDGMENTS

M.S. acknowledges INSPIRE division, Department of Science and Technology, New
Delhi-110 016, Government of India, for his fellowship [IF170335]. S.G.
acknowledges the DST-SERB Core Research Grant (File No. CRG/2018/001728) for
funding. D.W. acknowledges financial support from the Science and Technology
Development Fund from Macau SAR (0062/2023/ITP2) and the Macao Polytechnic
University (RP/FCA-03/2023).

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