# Origins of multi-sublattice magnetism and superexchange interactions in
double-double perovskite CaMnCrSbO6

Rakshanda Dhawan1,‡, Padmanabhan Balasubramanian2,†,‡, Tashi Nautiyal1
<EMAIL_ADDRESS>
###### Abstract

The recently reported double-double perovskite compound CaMnCrSbO6 offers an
interesting possiblity for the theoretical exploration of exchange
interactions between the multiple magnetic sublattices in the system. We have
deployed density functional theory, Wannier function analysis and mean-field
calculations to investigate this compound. The crystallographically non-
equivalent Mn atoms in the unit cell have tetrahedral and planar oxygen
coordinations (labelled as Mn(1) and Mn(2)), while the Cr atom is in the
centre of distorted oxygen octahedron. While the bulk magnetization and
neutron diffraction suggest a simpler ferrimagnetic order ($T_{C}$${\sim}$49
K) between Mn2+ and Cr3+ spins, the exchange interactions are more complex
than that expected from a two sublattice magnetic system. The separations
between the on-site energies of the $d$-orbitals of Mn(1), Mn(2) and Cr
obtained from Wannier function analysis are in agreement with the expected
crystal field splittings. The electronic structure calculations yield a
ferrimagnetic insulating ground state even in absence of Hubbard $U$ which
persists for a wide range of $U$.
Due to the ${\sim}$90o superexchange, nearest neighbour exchange constants
obtained from DFT are an order of magnitude smaller than those reported for
various magnetic perovskite and double-perovskite compounds. The Mn(1)-O-Mn(2)
(out of plane and in-plane), Mn(1)-O-Cr and Mn(2)-O-Cr superexchange
interactions are found to be anti-ferromagnetic, while the Cr-O-O-Cr super-
superexchange is found to be ferromagnetic. The Mn(2)-O-Cr superexchange is
weaker than the Mn(1)-O-Cr super-exchange, thus effectively resulting in
ferrimagnetism. From a simple 3-site Hubbard model, we derived expressions for
the antiferromagnetic superexchange strength $J_{AFM}$ and the weaker
ferromagnetic $J_{FM}$. The relative strengths of $J_{AFM}$ for the various
superexchange interactions are in agreement with those obtained from DFT. The
expression for Cr-O-O-Cr super-superexchange strength (${\tilde{J}}_{SS}$),
which is derived considering a 4-site Hubbard model, predicts a ferromagnetic
exchange in agreement with DFT. Finally, our mean field calculations reveal
that assuming a set of four magnetic sub-lattice for Mn2+ spins and a single
magnetic sublattice for Cr3+ spins yields a much improved $T_{C}$, while a
simple two magnetic sublattice model yields a much higher $T_{C}$.

1Department of Physics, Indian Institute of technology, Roorkee-247667,
Uttarakhand, India.
2Department of Physics, Graphic Era University, Dehra Dun, Uttarakhand 248
002, India. 111Both the authors have made equal contribution.

##  1 Introduction

The perovskite oxide materials are known to exhibit a wide range of properties
like complex magnetic structures, multiferroicity, metal-insulator transition
and colossal magnetoresistance. The basic perovskite ($A$$B$O3 : $A$-divalent
element, $B$-transition metal (TM)) has a cubic structure (space
group:$Pm{\bar{3}}m$) e.g. SrCrO3, SrTiO3 [1, 2]. When $A$ is a trivalent ion
(e.g. Rare earth or Y), structural distortions occur which include tilting of
the $B$O6 octahedra and the Jahn-Teller effect, due to which the crystal
structure becomes orthorhombic. Depending on the choice of $B$, diverse
properties are observed, e.g. multiferroicity, orbital and charge ordering in
manganites, high spin to low spin transition in cobaltates, spin reorientation
in orthoferrites and orthochromates[3, 4, 5, 6, 7].
Addition of another transition metal atom at $B$-site in a regular pattern, in
the cubic perovskite unit cell results in doubling of the unit cell. The new
compound, $A_{2}$$BB^{\prime}O_{6}$ with the space group $Fm{\bar{3}}m$, is
known as the double perovskite. Most of the double perovskite compounds have a
rock-salt arrangement of the $B/B^{\prime}$ atoms. Some examples are Ba2MnWO6,
Sr2MnWO6, Ba2PrIrO6 and Ba2YIrO6[8, 9, 10]. The cubic double-perovskite
structure is unstable for most of the elements at the $B/B^{\prime}$ sites.
Increase in the difference in the size of the $A$\- and $B$-site cations
reduces the tolerance factor below unity, thereby resulting in tetragonal
($I4/m$)[11], ($P4_{2}/n$)[12], rhombohedral($R{\bar{3}}$)[13], and the most
common monoclinic space group ($P2_{1}/n$)[14, 15] structure. An example of
the double perovskites with monoclinic structure is the well-studied
$R_{2}$MnNiO6 series[16]. All the members of this series are ferromagnetic
insulators with a wide range of Curie temperatures from 80 to 280 K. The
ferromagnetic insulating nature for these is also predicted through density
functional theory calculations. When $A$ is a divalent element,
$B$/$B^{\prime}$ belong to $3d$ and $4d$/$5d$ groups, respectively, the
compounds show ferrimagnetism with a Curie temperature above 300 K, and also a
large magnetoresistance[17, 18, 19].
When $B^{\prime}$ is a non-magnetic $p$-element (e.g.: Sb), the magnetism
arises entirely due to the $B$-sublattice. Due to a greater separation between
the $B$-site atoms. there occurs a drastic reduction in the magnetic
transition temperature along with possible occurance of magnetic frustrations
[20, 21]. For instance, Ca2CrSbO6 is a ferromagnet with a $T_{C}$ of 16 K
[20], while Sr2CrSbO6 is an antiferromagnet with a $T_{N}$ value of 12 K [20,
22], eventhough both the compounds crystallize in the monoclinic ($P2_{1}/n$)
structure. Density functional theory studies on Ca2CrSbO6 reveal the role of
greater structural distortion in stabilizing the ferromagnetic ground
state[23].
Recently, double perovskites of the form Mn2$M$SbO6 and Mn2$M$ReO6 ($M$: Sc,
Cr, Fe, Co and Mn)[24, 25, 26] have been studied experimentally. These
compounds also have a monoclinic ($P2_{1}/n$) structure with the Mn atom at
the $A$-site in the 2+ ionic state and a four-fold oxygen coordination.
Combining Mn and Ca results in the solid solution Ca2-xMnx$M$SbO6 which
retains the single phase monoclinic structure for 0$<$$x$$<$0.17 and
1.73$<$$x$$<$2\. For 0.74$<$$x$$<$1.1, the compositions crystallize in the
tetragonal $P4_{2}/n$ structure while the remaining composition ranges show a
two-phase co-existence[27]. We have chosen the composition with $x$$=$1, also
known as the double-double perovskite compound, with the general chemical
formula $AA^{\prime}BB^{\prime}O_{6}$, as the subject of the present study.
These compounds show a dual cationic order, with a columnar arrangement of the
$A/A^{\prime}$ atoms and a rocksalt arrangement of the $B/B^{\prime}$ atoms.
The lattice parameters of these compounds are close to that of the cubic
$Fm{\bar{3}}m$ unit cell[27]. Similar to Mn2$M$SbO6, double-double perovskites
also require high pressures for synthesis.
The present work is a density functional theory (DFT) based study on a
recently synthesized double-double perovskite CaMnCrSbO6[28], belonging to the
Ca2-xMnxCrSbO6 series. The two end compounds of this series, Ca2CrSbO6 and
Mn2CrSbO6 have a monoclinic structure ($P2_{1}/n$), but show entirely
different magnetic behaviour. The former, shows a long range ferromagnetic
order[20] while Mn2CrSbO6 orders antiferromagnetically at $T_{N}$$=$55 K,
though with a more complex propagation vector ${\vec{k}}$=(0.5,0,0.5)[24].
Thus with the presence of two different magnetic ions in the unit cell, one
could expect that the multiple exchange interactions may lead to a complex
long range magnetic structure or spin glass like behaviour in CaMnCrSbO6.
Additionally, with 50${\%}$ replacement of Mn by Ca, and a larger separation
between the Cr atoms, one may also expect a considerable reduction in the
transition temperature. However, magnetic neutron diffraction and bulk
magnetization studies on CaMnCrSbO6 confirm a collinear ferrimagnetic
structure with oppositely aligned Mn2+ and Cr3+ spins having an ordering
temperature, $T_{C}$$=$49 K, which is close to that of Mn2CrSbO6. The
structural neutron diffraction studies also show that due to similar ionic
radii, anti-site disorder occurs between the Cr/Mn and Cr/Sb sites, which
affects the magnetic properties of the system [28]. The different ligand
symmetries of two non-equivalent Mn atoms in the unit cell could result in
exchange interactions of different magnitudes between Mn and Cr, thus
affecting the transition temperature along with the nature of magnetic order.
Thus, to understand the magnetic structure in the structurally “ordered” unit
cell, it is essential to estimate the magnitude and signs of various exchange
constants ($J$) present in the system, which forms an important part of the
present work. The origins for these exchange interactions are analyzed in
terms of hopping between the different metal-and ligand-Wannier orbitals.
Using a simpler formalism of a 3-site Hubbard model with a ground state
comprising of single occupied $d$-orbial at each transition metal ion site and
a doubly occupied O $p$-orbital, we have estimated the Mn-O-Mn, Mn-O-Cr
superexchange interaction strengths. Similarly, the more complex Cr-O-O-Cr
super-superexchange strength is also estimated from a 4-site Hubbard model
with similar ground state as above. In order to determine $T_{C}$, we consider
a model with four Mn2+ magnetic sub-lattices and one magnetic sub-lattice for
the Cr3+ spins. The results thus obtained provide good agreement with the
experimental observations and offer a comprehensive picture of the magnetic
structure of CaMnCrSbO6.
This paper is organized as follows: after presenting the computational details
in Section 2, we discuss in Section 3, the crystal structure and the local
point group symmetry of the two non-equivalent Mn atoms and of the Cr atom in
the unit cell. Section 4.1 comprises of the non-magnetic density of states
along with the Wannier function analysis. In Section 4.2, we discuss the spin-
polarized electronic structure, effect of Hubbard $U$ and the magnetic
structures along with the anisotropy of the system. The effects of anti-site
disorder on the electronic structure and magnetism are discussed in Section
4.3. We have extracted the various exchange interactions in Section 4.4, while
Section 4.5 comprises of discussion on various superexchange mechanisms in
terms of hopping between various metal-ligand hopping integrals, with
expression for $J$ derived using 3-site and 4-site Hubbard model. Finally in
Section 4.6, we present a detailed calculation on estimation of the magnetic
transition temperature using the mean field theory.

##  2 Computational details

The calculations have been performed using DFT as implemented in the Vienna
${\it ab}$ ${\it initio}$ simulation package (VASP) which uses the projector
augmented wave (PAW) method[29]. The Perdew-Burke-Ernzerhof (PBE) potential
has been employed within the GGA (Generalized Gradient Approximation) and
GGA+$U$[30, 31]. A cut-off energy of 500 eV is used in the expansion of the
plane waves. A 9$\times$9$\times$9 Monkhorst-Pack $k$-mesh centered at
${\Gamma}$ point in Brillouin zone (BZ) is used for performing the BZ
integrations. The electrons from Cr/Mn: $3d$, $4s$, O: $2s$, $2p$ and Sb:
$4f$, $5p,5d$, $6s$ and Ca: $3p$, $4s$ states were considered as the valence
electrons. Non-magnetic as well as spin-polarized self-consistent calculations
have been performed. In addition, non-collinear calculations were also
performed within the GGA+U+SO formalism.
Further, to obtain the local orbital representation, real-space Wannier
functions derived from Mn and Cr $3d$ orbitals are constructed in the non-
magnetic (NM) phase. Disentanglement has been achieved by employing a suitable
energy window around the Fermi energy. The crystal-field-split onsite energies
and hopping amplitudes have also been derived.

##  3 Crystal Structure and local symmetry

Figure 1: Crystal structure of CaMnCrSbO6. The inequivalent Mn atoms are
marked as Mn1 and Mn2.

The double-double perovskite compound CaMnCrSbO6 crystallizes in the
tetragonal space group $P4_{2}n$. The unit cell (Fig. 1) comprises of four
formula units: 24 O, 4 Mn, 4 Cr, 4 Sb and 4 Ca atoms. The site symmetries of
the atoms are as follows: Ca - $4e$, O - $8g$, Sb - $4c$, Cr - $4c$, Mn(1) -
$2a$ and Mn(2) - $2b$, respectively. The Mn(1) atoms are coordinated by four
oxygen atoms in tetrahedral symmetry (Fig. 2d), while the Mn(2) atoms are
coordinated by four oxygen atoms in a square-planar geometry (Fig. 2e). The Cr
atoms occupy the $4c$-site with a six-fold coordination of oxygen atoms in a
highly distorted octahedra, showing both, orthorhombic and GdFeO3-like
distortions(Fig. 2f). In this structural arrangement, the Mn(1), Mn(2) and Cr
ions have formal valency of +2, +2 and +3, respectively.

##  4 Results and discussion

###  4.1 Non-magnetic electronic structure, Wannier orbitals and on-site
energies

Figure 2: (a)-(c) Site-resolved non-magnetic partial $d$ DOS of CaMnCrSbO6.
The Fermi energy is at 0 eV. (d)-(f) The local point group symmetries along
with corresponding crystal field splitting schemes for (d) Mn(1), (e) Mn(2)
and (f) Cr.

The non-magnetic electronic structure calculations yield a metallic state for
CaMnCrSbO6 as seen from the density of states (DOS) (Figs. 2(a-c)). All the
five $d$ orbitals show significant spectral weight at the Fermi energy
($E_{F}$), while the O $p$ orbitals have a smaller spectral weight in the DOS
(not shown here). The effect of the crystal-field splittings on the different
magnetic ions is seen in the partial DOS. Due to tetrahedral crystal field,
the $d$-orbitals of Mn(1)2+ ion split into a two-fold degenerate $E_{g}$
($d_{x^{2}-y^{2}}$ and $d_{3z^{2}-r^{2}}$) orbitals which are at lower energy,
while the three-fold degenerate $T_{2g}$($d_{xy}$, $d_{xz}$ and $d_{yz}$)
orbitals are at higher energy(Fig. 2d). This splitting is in reverse order
with respect to the usual $d$-orbital splitting in cubic crystalline fields.
Fig. 2a, shows a clear separation between the $E_{g}$ and $T_{2g}$ DOS of the
Mn(1) atom with peaks at ${\sim}$-0.5 and +0.5 eV respectively in agreement
with the local symmetry (Fig. 2d).
The $d$-orbitals of the Mn(2)2+ ion, which experience a planar crystal field,
split into a set of four levels. The lowest level is made of the two-fold
degenerate $E_{g}$ comprising of $d_{xz}$/$d_{yz}$ orbitals. The $d_{xy}$
orbital, belonging now to the $B_{2g}$ representation, lies above the
$d_{xz}$/$d_{yz}$ orbitals (Fig. 2e). The $d_{x^{2}-y^{2}}$ and
$d_{3z^{2}-r^{2}}$ orbitals are non-degenerate and belong to the $B_{1g}$ and
$A_{1g}$ representations, respectively. The $d_{x^{2}-y^{2}}$ orbital has the
highest energy. The $d$ DOS of Mn(2) atom (Fig. 2b) shows features in
accordance with the planar geometry. With almost identical spectral features,
the $d_{xy}$ and $d_{xz}$/$d_{yz}$ DOS of the Mn(2) atom are close in energy.
The DOS of all the three states pass through $E_{F}$ with a peak at -0.2 eV.
While the $d_{3z^{2}-r^{2}}$ DOS shows a large peak at $E_{F}$, the
$d_{x^{2}-y^{2}}$ corresponds to the highest energy level with a peak in DOS
at +1 eV.
For the Cr3+ ion, being at the centre of an orthorhombically distorted
octahedron, the $T_{2g}$ ($d_{xz}$, $d_{yz}$ and $d_{xy}$) and $E_{g}$
($d_{x^{2}-y^{2}}$ and $d_{3z^{2}-r^{2}}$) orbitals undergo additional
splitting, lifting the degeneracy completely. As shown in Fig. 2f, the
$T_{2g}$ orbitals of Cr3+ split to form a set of three non-degenerate $B_{2g}$
orbitals, while the degenerate $E_{g}$ orbitals form a linear combination of
$d_{x^{2}-y^{2}}$ and $d_{3z^{2}-r^{2}}$ orbitals resulting in two non-
degenerate $A_{1g}$ and $B_{1g}$ orbitals. The $d$ DOS of Cr (Fig. 2c)
comprises of partially occupied $d_{xy}$, $d_{xz}$ and $d_{yz}$ states at
$E_{F}$, while the $d_{x^{2}-y^{2}}$ and $d_{3z^{2}-r^{2}}$ states occur
mostly above $E_{F}$, with broad peaks around +1.2 and +2.0 eV, respectively.
Thus these are in agreement with the expected crystal-field splitting scheme.

Figure 3: (a1)-(e1), (a2)-(e2) and (a)-(e) correspond to the
$d_{3z^{2}-r^{2}}$, $d_{xz}$, $d_{yz}$, $d_{xy}$ and $d_{x^{2}-y^{2}}$ Wannier
$d$\- orbitals of Mn(1), Mn(2) and Cr, respectively, in CaMnCrSbO6. The $x$,
$y$ and $z$ axes correspond to the $a$, $b$ and $c$ axes of the crystal.

In order to estimate the crystal-field splitting in the transition metal (TM)
ions, along with the various hopping integrals, the Bloch states obtained in
terms of augmented plane wave basis have been expressed in terms of real space
Wannier orbitals, considering the $d$-orbitals of both, Mn and Cr along with
the $p$-orbitals of oxygen. In Fig. 3, we show the Wannier orbitals of Mn(1),
Mn(2) and Cr atoms. For the Mn(1) atom (Fig. 3(a1-e1) the $z$-axis is along
the crystallographic $c$-direction, while the $x$\- and $y$-axes are diagonal
to the $a$ and $b$-axes of the crystal. In case of the Mn(2) atom, (Fig.
3(a2-e2)), the $x$\- and $y$-axes are along the lines joining the centres of
the Mn(2) and planar O atoms, while the $z$-axis, similar to the case of Mn(1)
is along the crystallographic $c$-direction. The local coordinate axes for the
Cr atom are along the lines joining the centres of the the Cr and O atoms of
the octahedron and thus tilted w.r.t. the global crystallographic axes (Fig.
3(a-e)).
From the Wannier analysis, we obtain the real-space matrix elements of the
Hamiltonian, i.e. the hopping amplitudes between the TM ion and oxygen atoms.
The diagonal elements, which correspond to the the on-site energies of Mn(1),
Mn(2) and Cr $d$ states, are listed in Table 1. The on-site energies of the
$d$ states in Mn(1), Mn(2) and Cr follow the crystal-field splitting scheme as
is shown in Fig. 2(d-f).It should be mentioned that in the Mn(2) atom, the on-
site energy of the $d_{xy}$ state obtained is slightly lower than
$d_{xz}$/$d_{yz}$ states by 0.06 eV, unlike as shown in Fig. 2e. The total
crystal-field splitting is the smallest for Mn(1), as is expected for an
Mn(1)2+ ion placed in a tetrahedral environment, while the Cr3+ ion placed in
a highly distorted octahedral environment exhibits the maximum splitting.
These hopping amplitudes would be utilized later, in the context of
superexchange interactions.

Table 1: On-site energies (eV) of the crystal field levels of Mn(1)2+, Mn(2)2+ and Cr3+. 00 | $d_{xy}$ | $d_{xz}$ | $d_{yz}$ | $d_{x^{2}-y^{2}}$ | $d_{3z^{2}-r^{2}}$ |  $E_{max}$$-$$E_{min}$ (eV)
---|---|---|---|---|---|---
0Mn(1)2+ | 6.37 | 6.32 | 6.31 | 6.14 | 6.03 |  --0.34
0Mn(2)2+ | 6.32 | 6.38 | 6.38 | 6.82 | 6.51 |  --0.50
0Cr3+ | 6.11 | 6.06 | 6.13 | 6.90 | 6.66 |  --0.84

###  4.2 Spin-polarized electronic structure and effect of correlations

The spin polarized calculations have been carried out for the ferrimagnetic
(FiM) configuration, with the Mn2+ and Cr3+ spins arranged as spin-up and
spin-down, respectively. Additionally, the calculations have also been
performed for ferromagnetic (FM) and various anti-ferromagnetic (AFM)
arrangements, viz. A-type, C-type, and G-type. The calculations, for all the
magnetic configurations, have been performed for different values of
$U_{eff}$($=$$U$$-$$J_{H}$; $U$ is the Coulomb interaction energy of the
$d$-orbitals, $J_{H}$ is the Hund’s exchange energy). The $U$ value was varied
between 0 and 6 eV, with $J_{H}$$=$1 eV for both, Mn and Cr. Calculations were
also performed for unequal values of $U_{eff}$ in Mn and Cr. For AFM
calculations, either the Mn or the Cr sublattice was fixed in FM
configuration, while the other sublattice was arranged as A-type, C-type and
G-type. As seen from Table 2, at $U_{eff}$$=$0 eV, the FiM configuration is
prefered by a large margin, while the FM configuration has the highest energy.
We note that with the increase in $U_{eff}$, the difference between the
energies of various configurations gets reduced and at $U_{eff}$$=$5 eV, the
FiM no longer corresponds to the lowest energy configuration.

Table 2: Total energy per unit cell (in meV) w.r.t that for the FM configuration for various magnetic configurations and different $U_{eff}$$=$$U$-$J_{H}$ (in eV) values. The values of $U$ and $J$ are same for Mn and Cr, unless mentioned otherwise. |  |  | Relative energy (meV)
---|---|---|---
Magnetic |  |  $U$$=$0 |  $U$$=$4 |  $U$$=$6(Mn), 5.5(Cr) |  $U$$=$6
Configuration |  |  $J_{H}$$=$0 |  $J_{H}$$=$1 |  $J_{H}$$=$1.5(Mn), 1(Cr) |  $J_{H}$$=$1
|  |  $U_{eff}$$=$0 |  $U_{eff}$$=$3 |  $U_{eff}$$=$4.5(Mn, Cr) |  $U_{eff}$$=$5
FM |  | 0.0 | 0.0 |  0.0 | 0.0
FiM |  | -227.0 | -73.4 |  -20.8 | -10.8
Cr-A(AFM) |  | -62.9 | -6.0 |  13.9 | 17.5
Cr-C(AFM) |  | -62.8 | -3.5 |  16.6 | 31.3
Cr-G(AFM) |  | -62.8 | -3.5 |  16.6 | 31.3
Mn-A(AFM) |  | -127.2 | -53.7 |  -16.5 | -15.4
Mn-C(AFM) |  | -131.9 | -48.9 |  -15.4 | -13.0
Mn-G(AFM) |  | -94.5 | -42.5 |  -14.1 | -14.1

Figure 4: Ferrimagnetic DOS of CaMnCrSbO6 for $U_{eff}$=0 eV (left panel) and
$U_{eff}$=3 eV.

In Figs. 4(a-c), we show the spin polarized DOS of CaMnCrSbO6 for the FiM
configuration corresponding to $U_{eff}$$=$0 eV. Even in the absence of
Hubbard $U$, FiM CaMnCrSbO6 shows an insulating behaviour with a fairly large
band gap of ${\sim}$ 0.7 eV. As seen from Figs. 4(a) and (b), the spin-up $d$
states of Mn(1) and Mn(2) are fully occupied, while spin-down $d$ states are
completely empty. The energy scheme of the crystal field split levels in the
NM DOS case (Fig. 2) is found to be applicable in the FiM configuration as
well. In the Mn(1) $d$-DOS (Fig. 4a), the $d_{x^{2}-y^{2}}$ has the lowest
energy, with a sharp peak at ${\sim}$ -1.8 eV. The $d_{xz}$/$d_{yz}$ states
retain their near-degenerate character and are close to $E_{F}$, while the
$d_{xy}$ gets shifted to -0.5 eV, preceded by the $d_{3z^{2}-r^{2}}$, which
shows a peak at ${\sim}$ -1 eV.
In Mn(2) spin-up DOS, the $d_{x^{2}-y^{2}}$ occurs in isolation near $E_{F}$
with a peak at ${\sim}$ -0.3 eV, while the spin-up DOS of the remaining four
$d$ states shows sharp peaks between -1.7 and -2.0 eV. Above $E_{F}$, the peak
for $d_{3z^{2}-r^{2}}$ along with $d_{xz}$ and $d_{yz}$ spin-down DOS occurs
at ${\sim}$ +2.1 eV, while the peak for $d_{x^{2}-y^{2}}$ spin-down DOS occurs
at ${\sim}$ +3.2 eV. In Fig. 4c, the spin-down states of Cr are partially
occupied, with the $d_{xz}$ and $d_{yz}$ spin-down DOS showing sharp peaks at
${\sim}$ -0.5 eV, while the $d_{xy}$ DOS shows a broader feature, with a peak
at ${\sim}$ -0.2 eV. The $d_{3z^{2}-r^{2}}$ and $d_{x^{2}-y^{2}}$ spin-down
DOS occur above $E_{F}$, overlapping with the unoccupied $T_{2g}$ spin-up DOS
above +1.5 eV.
In Figs. 4(d-f), we show the FiM partial DOS of CaMnCrSbO6 for $U_{eff}$$=$3
eV. The effect of $U$ is different for each atom, though there occurs a
broadening, in general, for each of the DOS. For the Mn(1) atom, the
separation between the spin-up $d_{x^{2}-y^{2}}$ and $d_{3z^{2}-r^{2}}$ DOS
shows an increase, whereby the former now shows a peak at ${\sim}$ -2 eV. The
spin-up $d_{xz}$/$d_{yz}$ states are nearest to $E_{F}$. For Mn(2) atom, the
spin-up $d_{x^{2}-y^{2}}$ DOS (Fig. 4e ) retains its proximity to $E_{F}$. For
the remaning $d$ states of Mn(2), the DOS is spread in the range -1 to -5 eV,
with reduced spectral weight as compared to $U_{eff}$=0 eV case. However, the
spectral width of the spin-down DOS remains uneffected, though the peaks are
almost rigidly shifted by ${\sim}$1.5 eV.The Cr spin-down DOS(Fig. 4f) below
$E_{F}$ partly retain their features in presence of the correlations, though
with increased width and diminished intensity of peaks. The unoccupied Cr
spin-down $d_{x^{2}-y^{2}}$ and $d_{3z^{2}-r^{2}}$ DOS are shifted to +2 eV
with accentuated peaks, thus resulting in a larger bandgap (${\sim}$ 1.7 eV)
for CaMnCrSbO6 with $U_{eff}$$=$3 eV.

Figure 5: Variation of (a) energy gap and (b) magnetic moments with $U_{eff}$.
Mn(1) and Mn(2) show the same variation on this scale.

In Fig. 5(a), we show the variation of bandgap in CaMnCrSbO6 as a function of
$U_{eff}$. There occurs a bandgap of ${\sim}$0.7 eV for $U_{eff}$$=$0 eV.
After a sudden jump, with further increase in $U_{eff}$, the bandgap shows
almost a linear variation with correlation energy. Thus CaMnCrSbO6 shows the
typical characteristics of a Mott-Hubbard insulator. In Fig. 5(b), we also
show the variation of magnetic moment of Mn and Cr. The variation in moments
of Mn(1) and Mn(2) are identical. While Cr attains the fully saturated moment
of 3 ${\mu}_{B}$ for $U_{eff}$$=$3 eV, Mn(1) and Mn(2) show a slight
reduction, from the expected value of 5 ${\mu}_{B}$ in their magnetic moments.

####  4.2.1 Non-collinear calculation and anisotropy

Studies reveal that below the Neel temperature, CaMnCrSbO6 orders magnetically
in the ${\Gamma}_{3}$ representation[28]. In this, the $x$ and $y$ components
of the spins within the Mn(1)2+ and Mn(2)2+ sub-lattices are ferromagnetically
coupled, though there is no restriction on the inter-sublattice coupling.The
four Cr3+ spins in the unit cell are anti-ferromagnetically coupled in the
$x$\- and $z$-directions, and ferromagnetically coupled in the $y$-direction.
It may be noted that experimentally, it is not resolved, whether the spins are
along the $x$ or the $y$-directions. To probe the anisotropy of the compound
using DFT, non-collinear calculations have been performed by including the
effects of spin-orbit coupling. We used $U$$=$6 eV and $J_{H}$$=$1.5 eV for Mn
and $U$$=$5.5 eV and $J_{H}$$=$1 eV for Cr. Calculations were performed for
various orientations of the spins. We find that the configuration with the
Mn3+ and Cr3+ spins in the $x$-direction correspond to the lowest energy,
while the spins in the $y$ and $z$-directions yield higher energy by 0.26 and
0.29 meV per unit cell respectively, thus conforming to the experimental
results.

###  4.3 Effect of anti-site disorders

Structural studies in CaMnCrSbO6 using neutron diffraction reveal that upto 20
${\%}$ of the Mn(1) and Mn(2) sites are occupied by Cr atoms and vice versa,
while upto 5 ${\%}$ of the Cr(Sb) sites are occupied by Sb(Cr) atoms, thus
resulting in formation of anti-site disorders (ASD) [28]. The reduced magnetic
moment of 3.8 ${\mu}_{B}$, obtained from magnetic neutron diffraction for both
Mn2+ spins, has been attributed to the ASDs [28]. We have made an effort to
probe the effect of ASD on the magnetic and electronic structure of
CaMnCrSbO6. For this, we have adopted a simple picture and performed
additional spin-polarized calculations with inclusion of ASDs. The full-
fledged calculations incorporating the reported switching between atoms
(${\sim}$ 20${\%}$, 20${\%}$ and 5${\%}$ for Mn(1), Mn(2), Sb, respectively,
with Cr atoms) would be computationally expensive requiring a very large
supercell. Hence, instead, we tried to assess the effect of ASD in a simple
manner by carrying out three independent calculations (in the FiM
configuration, with $U_{eff}$$=$5 eV for Mn and Cr): one, where there is no
ASD (labelled as “Ordered”), other with an exchange between Mn and Cr
(labelled as ASD1), and the third where Sb and Cr are exchanged (labelled as
ASD2). A supercell with lattice parameters ($a$, $a$, $2c$) was employed in
the FiM configuration was used for all the three cases. In this supercell
there are 8 atoms each of Mn, Cr and Sb. In ASD1, one Cr atom is switched with
an Mn atom (we chose Mn(1) site for this), while in ASD2, one Cr atom is
switched with an Sb atom. The resulting percentage disorders (12.5${\%}$ in
ASD1 and ASD2, each) within the supercell are fair representatives of those
from structural analysis.

Table 3: Magnetic moments and band gap in supercells of FiM CaMnCrSbO6 for
$U_{eff}$$=$5 eV in ordered and ASD configurations. The prime corresponds to
the position after switching of the atoms. The total energy w.r.t. ordered
supercell is also mentioned.

  * 00 |  | Atom(Position) | Magnetic | Band | Total Energy |  |  |
---|---|---|---|---|---|---|---|---
00 |  |  | Moment(${\mu}_{B}$) | gap (eV) |  -(eV) |  |  |
| Ordered | Mn(1)(0.25, 0.25, 0.25) |  -4.71 |  |  |  |  |
|  | Mn(2)(0.75, 0.75, 0.25) |  -4.73 |  -2.0 |  -0 |  |  |
|  | Cr(0, 0, 0) |  --3.10 |  |  |  |  |
|  | Cr(0.5, 0.5, 0) |  --3.10 |  |  |  |  |
|  | Sb(0, 0, 0.5) |  -0.00 |  |  |  |  |
| ASD1 | Mn(2)(0.75, 0.75, 0.25) |  -4.71 |  |  |  |  |
|  | Mn’(0, 0, 0) |  --3.96 |  -0.1 |  -+2.18 |  |  |
|  | Cr(0.5, 0.5, 0.0) |  --3.10 |  |  |  |  |
|  | Cr’(0.25, 0.25, 0.25) |  -3.96 |  |  |  |  |
|  | Sb(0, 0, 0.5) |  -0.00 |  |  |  |  |
| ASD2 | Mn(1)(0.25, 0.25, 0.25) |  -4.71 |  |  |  |  |
|  | Mn(2)(0.75, 0.75, 0.25) |  -4.73 |  -0.7 |  -+2.31 |  |  |
|  | Cr(0.5, 0.5, 0) |  --3.10 |  |  |  |  |
|  | Cr’(0, 0, 0.5) |  --3.06 |  |  |  |  |
|  | Sb’(0, 0, 0) |  -0.00 |  |  |  |  |

Figure 6: FiM DOS of CaMnCrSbO6 for $U_{eff}$$=$5 eV, in (a) ordered, (b) ASD1
and (c) ASD2 structures.

We show the total $d$-orbital DOS of FiM CaMnCrSbO6 with $U_{eff}$$=$5 eV in
the regular structure (Fig. 6a) and ASD structures (Figs. 6b and 6c). The DOS
of the regular structure is similar to that obtained for $U_{eff}$$=$3 eV,
except for a larger band gap now, i.e. ${\sim}$2.1 eV. The inclusion of ASD
drastically affects the DOS. In ASD1, the spin-up and spin-down DOS of the
unswitched Mn and Cr atoms, are about 1 eV below $E_{F}$, as seen from Fig.
6b. In comparision to the regular Cr and Mn atoms, the switched atoms ie Mn’
and Cr’, show drastically different spectral features in the DOS. The spin-up
DOS just below $E_{F}$ is dominated by the $d$ DOS of the Cr’ atom, which
shows sharp peaks at ${\sim}$-0.2 and ${\sim}$-0.5 eV. Similarly, the spin-
down DOS just below and above $E_{F}$ comprises of sharp peaks due to $d$ DOS
of the Mn’ atom, thereby making the band-gap almost negliblible in ASD1. The
ASD2 does not affect the magnetic moments, even though interestingly it
partially affects the DOS and reduces the band gap. As seen in Fig. 6c, in
ASD2 structure, the band gap has reduced to ${\sim}$ 1 eV, due to the switched
Cr’ $d$-states, which contribute to the spin-down DOS just below $E_{F}$.
The total $d$-electron occupation number obtained for Mn(1), Mn(2) and Cr in
the ordered supercell are 5.06, 5.03 and 4.23 respectively. While in Mn(1) and
Mn(2), the $d$-occupation number is quite in agreement with the formal valency
of +2 (3$d^{5}$), the $d$-occupancy of Cr is considerably higher than that
expected from a valency of +3 (3$d^{3}$), thus indicating a greater charge
transfer nature. In ASD1, the occupancies of the switched atoms, Mn’ and Cr’,
show significant changes. The occupancy of Mn’ has decreased to 4.88, while
that of Cr’ has increased to 4.40. The changes in occupation is thus reflected
in the magnetic moment of the switched atoms. As seen from Table 3, the
magnitude of Mn’ magnetic moment has reduced to 3.96 ${\mu}_{B}$, while Cr’
shows an enhanced magnetic moment of magnitude 3.96 ${\mu}_{B}$. These
absolute values of magnetic moments indicate that the Mn’ and Cr’ atoms
possess formal valencies of +3 and +2 respectively. In ASD2, though the
occupancy of Cr’ increases to 4.34, a similar increase in Cr magnetic moment
is not observed. As mentioned in Table 3, the magnitude of Cr’ moment has
slightly reduced to 3.06 ${\mu}_{B}$. The inclusion of ASDs result in an
overall decrease in the magnetic moment of Mn within the supercell, in
accordance with the experiments.

###  4.4 Determination of exchange constants

Figure 7: Cr/Mn lattice with paths marked for various exchange interactions.

The stability of the FiM ground state can be understood in a better manner
with the help of the various exchange constants in the system. Since
CaMnCrSbO6 is found to be insulating from our electronic structure
calculations, with a strong Mott Hubbard character, the exchange interactions
between the two TM ions arise predominantly through the intervening O2-
$p$-orbitals, thus resulting in superexchange interactions. The exchange
interactions ($J$’s) were evaluated by mapping the total energy of the various
magnetic configurations (Table 2) to the Ising model Hamiltonian,

$H=-\sum_{i,j}J_{ij}S_{i}^{z}S_{j}^{z}$ (1)

where $J_{ij}$ is the exchange interaction strength, while $S_{i}^{z}$ and
$S_{j}^{z}$ correspond to Mn2+ or Cr3+ spins, as per applicability. The $J$’s
were obtained for the following Hubbard parameters, $U$$=$5.5 eV, $J_{H}$$=$1
eV for Cr and $U$$=$6 eV, $J_{H}$$=$1.5 eV for Mn. The role of oxygen and
other non-magnetic atoms in the lattice has been neglected here. Due to
presence of two different transition metals, there occur multiple exchange
paths, as compared to the case of single and double perovskites. The various
exchange mechanisms can be understood from the magnetic lattice, whose
projection onto the $b$$-$$c$ plane is shown in Fig. 7. In Table 4, we list
the various $J$’s in order of increasing distance ‘$d$’ for different sets of
TMs in the magnetic lattice.

Table 4: Exchange constants (meV) between the magnetic ions in CaMnCrSbO6. The ’+’ and ’-’ sign indicate the FM or AFM nature of the exchange interaction. |  | Path |  $d(\AA$) | Method I | Method II | Method III
---|---|---|---|---|---|---
|  |  |  |  -(meV) |  -(meV) |  -(meV)
| $J_{1a}^{\prime}$ | Mn(1)-Cr | 3.3031 |  --0.41 |  --- |  --0.46
| $J_{1b}^{\prime}$ | Mn(2)-Cr | 3.3031 |  --0.06 |  --- |  --0.14
| $J_{1}$ | Mn(1)-Mn(2)($c$-axis) | 3.8104 |  --0.07 |  --0.11 |  --0.11
| $J_{2}$ | Mn(1)-Mn(2)($a$$-$$b$) | 5.3966 |  --0.03 |  --0.05 |  --0.05
| $J_{2}^{\prime}$ | Cr-Cr | 5.3966 |  -+0.32 |  -+0.37 |  -+0.28

Considering only the magnetic atoms, we note that Cr atoms are the nearest
neighbours to both, Mn(1) and Mn(2) atoms. Each Cr atom has two Mn(1) and two
Mn(2) atoms as the nearest neighbours. Due to different ligand environment of
the Mn(1) and Mn(2) atoms, we expect the Mn(1)-Cr and Mn(2)-Cr exchange
interactions as distinct. The Mn(1) atoms are flanked by two Mn(2) atoms along
the $c$-axis as nearest neighbours and vice versa. In the $a$$-$$b$ plane,
each Mn(1) is surrounded by four Mn(2) atoms and vice-versa. The Mn(1)-Mn(2)
separation in the $a$$-$$b$ plane is much larger and is the same as the Cr-Cr
separation. For each Cr atom, there are twelve next nearest neighbour Cr
atoms, with four Cr atoms in the (110), (101) and (011) planes, each. The
corresponding Cr-Cr interactions are denoted as $J_{2}^{110^{\prime}}$,
$J_{2}^{101^{\prime}}$ and $J_{2}^{011^{\prime}}$, respectively. Substituting
$S_{Mn}^{z}$$=$${\pm}$5/2 and $S_{Cr}^{z}$$=$${\pm}$3/2 for the Mn2+ and Cr3+
spins, respectively, the Ising Hamiltonian is expanded for the different
magnetic configurations. After a careful enumeration, the total energy in the
FM and FiM configurations can be expanded as,

$E^{FM}=-25J_{1}-50J_{2}-18J_{2}^{110^{\prime}}-18J_{2}^{101^{\prime}}-18J_{2}^{011^{\prime}}-30J_{1a}^{\prime}-30J_{1b}^{\prime}$
(2)
$E^{FiM}=-25J_{1}-50J_{2}-18J_{2}^{110^{\prime}}-18J_{2}^{101^{\prime}}-18J_{2}^{011^{\prime}}+30J_{1a}^{\prime}+30J_{1b}^{\prime}$
(3)

To obtain Mn(1)-Cr and Mn(2)-Cr exchange constants, i.e. $J_{1a}^{\prime}$ and
$J_{1b}^{\prime}$, unambiguously, two additional calculations were performed.
In both the calculations, among the four Cr3+ spins in the unit cell, a single
Cr3+ spin was fixed as spin-down. In the first configuration (denoted as
configuration $M$), a single Mn(1)2+ spin was kept as spin-down, and in the
second case (denoted as configuration $N$), a single Mn(2)2+ spin was fixed as
spin-down. Due to cancellation of most of the terms, the total energy for
configurations $M$ and $N$ reduces to,

$\begin{split}E_{M}&=-15J_{1b}^{\prime}\\\
E_{N}&=-15J_{1a}^{\prime}\end{split}$ (4)

Our calculations show that the energies of both the configurations $M$ and $N$
are higher than that of FM by +8.33 and +4.93 meV per unit cell, respectively.
From equations (2), (3) and (4), we obtain $J_{1a}^{\prime}$$=$-0.41 meV and
$J_{1b}^{\prime}$$=$-0.06 meV. The negative values indicate that these
exchange interations are anti-ferromagnetic in nature. Interestingly, the
differences in the local environment of both the Mn2+ ions result in a large
difference between Mn(1)-Cr and Mn(2)-Cr interaction strengths. To obtain the
exchange interactions within the individual Mn and Cr sublattices, we refer to
the calculations in which a) Mn sublattice was kept fixed in FM configuration
and magnetic configuration of the Cr sublattice was varied, and b) Cr
sublattice was kept fixed in FM configuration, while the magnetic
configuration of the Mn sublattice was varied (Table 2). The set of equations
for the A-, C- and G-type arrangement of the Mn-sublattice is,

$E_{Mn}^{A}=25J_{1}-50J_{2}-18J_{2}^{110^{\prime}}-18J_{2}^{101^{\prime}}-18J_{2}^{011^{\prime}}$
(5)
$E_{Mn}^{C}=-25J_{1}+50J_{2}-18J_{2}^{110^{\prime}}-18J_{2}^{101^{\prime}}-18J_{2}^{011^{\prime}}$
(6)
$E_{Mn}^{G}=25J_{1}+50J_{2}-18J_{2}^{110^{\prime}}-18J_{2}^{101^{\prime}}-18J_{2}^{011^{\prime}}$
(7)

Similarly, the set of equations for A-, C- and G-type arrangement of the Cr-
sublattice is,

$E_{Cr}^{A}=-25J_{1}-50J_{2}-18J_{2}^{110^{\prime}}+18J_{2}^{101^{\prime}}+18J_{2}^{011^{\prime}}$
(8)
$E_{Cr}^{C}=-25J_{1}-50J_{2}+18J_{2}^{110^{\prime}}-18J_{2}^{101^{\prime}}+18J_{2}^{011^{\prime}}$
(9)
$E_{Cr}^{G}=-25J_{1}-50J_{2}+18J_{2}^{110^{\prime}}+18J_{2}^{101^{\prime}}-18J_{2}^{011^{\prime}}$
(10)

After solving the equations (2)-(10), it is found that all the Mn(1)-Mn(2)
interactions are anti-ferromagnetic and smaller as compared to the Cr-Cr
interactions which are found to be ferromagnetic. While,
$J_{2}^{110^{\prime}}$$=$+0.37 meV, both $J_{2}^{101^{\prime}}$, and
$J_{2}^{011^{\prime}}$ are found to be +0.29 meV, suggesting a small
anisotropy. However, we only consider the average of the three values, denoted
as $J_{2}^{\prime}$. Inspite of having nearly the same separation distance,
the Cr-Cr interaction strength in our system is much greater than Cr-Cr
exchange strength of +0.05 eV, obtained in Ca2CrSbO6[23]. In Table (4) we list
all these exchange constants in the column under Method I.
To assertain the values of $J$’s obtained above, we have independently
obtained the exchange parameters, using two additional methods, labelled as
Method II and III. In Method II, one magnetic element is entirely replaced by
a non-magnetic element. Hence, to obtain the Cr-Cr and Mn-Mn exchange
interactions, the Mn(Cr) atoms are replaced by isovalent Zn(Sc) atoms.
However, in this method, the inter sublattice i.e. Mn-Cr exchange interaction
strength cannot be determined. The intra-sublattice exchange interactions thus
obtained are shown in Table 4 in the column Method II. In Method III, except
for the two magnetic atoms whose exchange interaction strength is to be
determined, rest of the magnetic atoms are replaced by the isovalent non-
magnetic equivalents[32]. The corresponding exchange interaction strength is
then proportional to the energy difference between the FM and AFM arrangement
of the two spins and are listed in Table 4 under Method III. The sign of $J$
for the various exchange interactions, obtained from the three methods are the
same, and the magnitudes almost equal, though discrepencies upto 90 ${\mu}$eV
are found in the case of Cr-Cr interactions. It is noted that these values of
nearest neighbour $J$ from DFT studies for this double-double perovskite are
an order of magnitude smaller than those found in single and double perovskite
compounds for similar values of $U$ and $J_{H}$[32, 33, 34]. The Mn(1)-Mn(2)
exchange interactions which are found to be anti-ferromagnetic, appears
contrary to the experimental studies. The ferrimagnetic behaviour obtained
from bulk magnetization and neutron diffraction suggests a parallel alignment
of Mn2+ spins which is possible only when the Mn(1)-Mn(2) exchange interaction
is ferromagetic. To understand the origin and sign of the various exchange
interactions, we have investigated the superexchange mechanisms in the
preceeding section.

Figure 8: Magnitude of hopping integrals between the MLWFs of $d$ orbitals of
Mn(1), Mn(2) and Cr with the O $p$ orbitals. Hopping integrals correspond to
following superexchanges, (a)-(b) Mn(1)-O-Mn(2) (along the $c$-direction),
(b)-(c) Mn(1)-O-Cr, (d)-(e) Mn(2)-O-Cr, (f)-(h) Cr-O-O-Cr ‘super-
superexchange’ (i)-(j) Mn(1)-O-Mn(2) ($a$$-$$b$ plane). The coordinates of the
transition metal and Oxygen atoms are mentioned for each panel.

###  4.5 Superexchange from hopping integrals

The exchange interaction strengths obtained in the preceding section can be
understood in terms of the hopping amplitudes, between the Wannier orbitals of
Mn(1), Mn(2) and Cr via the O $p$-orbitals. The $J$’s have been calculated
from the overlap integrals between the metal and the ligand Wannier functions.
As shown in Fig. 2(d-f) and Fig. 3, due to the differences of the point group
symmetry and the local coordinate axes of Mn(1)2+, Mn(2)2+ and Cr3+ ions, the
nearest neighbour $d$-$p$ hoppings show ${\sigma}$ as well as ${\pi}$
character.
In Fig. 8(a-e), we show the magnitudes of the $d$$-$$p$ hopping integrals
($t_{pd}$), which give rise to the superexchange interaction between the
nearest neighbour Mn(1)-Mn(2), Mn(1)-Cr and Mn(2)-Cr spin pairs, while Fig.
8(i) and 8(j) correspond to $d$$-$$p$ hopping associated with the in-plane
Mn(1)-Mn(2) superexchange interactions. Fig. 8(f-h) corresponds to the Cr-Cr
exchange occuring via two intermediate O2- ions, and is labelled as ’super-
superexchange’. Thus, while Figs. 8(f) and 8(h) correspond to the hopping
between the adjacant Cr-O pairs, Fig. 8(g) represents the hopping between the
two O2- ions, denoted by $t_{pp}$. The coordinates of the TM and the
intermediate oxygen atoms are also mentioned in the sub-plots. The labels
(1a), (2a),.. have been placed only for the purpose of identification of the
various O atoms. Based on these amplitudes of the $d$$-$$p$ hoppings, we
illustrate some of the possible paths for the nearest neighbour
Mn(1)-O(1a)-Mn(2), Mn(1)-O(1a)-Cr, Mn(2)-O(2a)-Cr superexchange interactions
in Fig. 9(a-e). The Mn(1)-O-Mn(2) has a bond-angle of ${\sim}$100o, while the
Mn(1)-O-Cr and Mn(2)-O-Cr bond angles are ${\sim}$105o; these can thus be
understood in the formalism of the 90o superexchange according to the
Goodenough-Kanamori rules[35]. With the ${\sigma}$ as well as ${\pi}$ hopping
contributing to the superexchange process, the 90o superexchange is more
complex, though weaker than the 180o antiferromagnetic superexchange which is
seen in the perovskite systems[35].
In general, when the hopping between the $d$-orbitals of two TM ions occurs
through a single doubly occupied O2- $p$-orbital, the resultant superexchange
is antiferromagnetic. Within a simple formalism, considering only single
$d$-orbital at the TM site the expression for $J$, derived from a three site
Hubbard model (Appendix, equations (A1)-(A6)) is,

$J_{AFM}=-2(t_{pd1}t_{pd2})^{2}\left[\frac{1}{{\tilde{U}}_{1}^{2}(U_{1}+{\Delta}{\epsilon})}+\frac{1}{{\tilde{U}}_{2}^{2}(U_{2}-{\Delta}{\epsilon})}+\frac{1}{{\bar{U}}}\left(\frac{1}{{\tilde{U}}_{1}}+\frac{1}{{\tilde{U}}_{2}}\right)^{2}\right]$
(11)

Here, the different hopping strengths, $U$ and charge trasfer
energies(${\Delta}_{pd}$) associated with two different TM ions are taken into
account. The values of Hubbard $U$ for Mn and Cr are fixed at 6 and 5.5 eV,
respectively. The various terms and symbols used are explained in the
Appendix. Though valid for metal-ligand-metal in 180o geometry, the hopping
parameters take into account the deviation from the ideal geometry. We now
explain the various superexchange processes.

Figure 9: Exchange paths between selected atomic Mn, Cr $d$ orbitals via O $p$
orbitals, obtained from Wannier function. The nature of the bonding
(${\sigma}$ or ${\pi}$) is also highlighted. (a) Nearest neighbour (along
$c$-direction) Mn(1)-O(1a)-Mn(2) AFM superexchange; (b) and (c) AFM
superexchange from Mn(1)-O(1a)-Cr and Mn(2)-O(2a)-Cr. (d) and (e) FM
superexchange involving two oxygen orbitals and (f) FM superexchange between
the two Cr atoms involving two intermediate O atoms.

The superexchange between Mn(1)$d_{yz}$ and Mn(2)$d_{xz}$ via O(1a)$p_{z}$ is
shown in Fig. 9a. Since, the Mn(2)-O(1a) bondlength (${\sim}$2.83 Å) is
greater than the Mn(1)-O(1a) bondlength (${\sim}$2.03 Å), the
Mn(2)$d$-O(1a)$p$ hopping amplitude is smaller, thus resulting in a small
Mn(1)-O(1a)-Mn(2) exchange strength. Using equation (11), we obtain the anti-
ferromagnetic $J_{AFM}$ value of -0.5 meV for this particular superexchange
path, which is the maximum value for the Mn(1)-O-Mn(2) superexchange, along
the $c$-direction. A comparable $J_{AFM}$ value of -0.4 meV is obtained for
the Mn(1)$d_{yz}$-O(1a)$p_{z}$-Mn(2)$d_{3z^{2}-r^{2}}$ superexchange. The
antiferromagnetic $J_{AFM}$ calculated for a few combinations of Mn(1), Mn(2)
$d$ and O(1a) $p$-orbitals are listed in Fig. 10a. The superexchange
interaction between Mn(1) and Mn(2) atoms, situated in-plane, occurs through
the intermediate O(2b) atom. As seen from Figs.(8i and 8j), the Mn(1)-O(2b)
hopping amplitude is smaller than that for Mn(2)-O(2b). The $J_{AFM}$ values
for various possible Mn(1)-O-Mn(2) in-plane superexchange paths are shown in
Fig. 10b. The highest value of $J_{AFM}$$=$-0.12 meV is obtained for the
Mn(1)$d_{x^{2}-y^{2}}$-O(2b)$p_{x}$-Mn(2)$d_{x^{2}-y^{2}}$ superexchange which
is close to the value obtained from DFT.

Figure 10:  Magnitude of calculated exchange interaction energies ($x$-axis)
for various kinds of superexchange interactions calculated from equations
(11)-(13). The corresponding exchange paths are mentioned on the $y$-axis.

The Mn(1)-Cr exchange can be understood through the superexchange via O(1a)
$p$-orbitals. Due to ${\sigma}$-bond, between Cr$d_{x^{2}-y^{2}}$ and
O(1a)$p_{x}$, the hopping amplitude is the maximum (Fig. 8c). Out of the many
possible superexchange mechanisms involving the different $d$-orbitals of
Mn(1) and Cr and the $p$-orbitals of O(1a), we show the path
Mn(1)$d_{xz}$-O(1a)$p_{x}$-Cr$d_{x^{2}-y^{2}}$ (Fig. 9b), which yields a large
antiferromagnetic superexchange strength ($J_{AFM}$$=$-4.2 meV). Similar value
of $J_{AFM}$ is obtained, for instance, in the case of
Mn(1)$d_{x^{2}-y^{2}}$-O(1a)$p_{x}$-Cr$d_{x^{2}-y^{2}}$ superexchange (Fig.
10d). Thus the maximum $J_{AFM}$ values (Fig. 10d) are an order of magnitude
higher than the exchange strength ($J_{1a}^{\prime}$) directly obtained from
DFT (Table 4).
The Mn(2)-Cr exchange can be understood through the superexchange via the
$p$-orbitals of O(2a). The Mn(2)-O(2a) hopping amplitudes are smaller, as
compared to those for Cr-O(2a), as shown in Figs. 8(e) and 8(d), respectively.
In Fig. 9c we show the path Mn(2)$d_{xz}$-O(1a)$p_{x}$-Cr$d_{3z^{2}-r^{2}}$,
which yields an antiferromagnetic superexchange strength ($J_{AFM}$$=$-1.5
meV). In Fig. 10e, we show $J_{AFM}$ for some of the AFM superexchange
interactions for Mn(2)-O-Cr. For all these possible cases considered, the
maximum $J_{AFM}$ for Mn(2)-O(2a)-Cr is nearly one-third of the maximum
$J_{AFM}$ obtained for Mn(1)-O(1a)-Cr (Fig. 10d), thus in agreement with our
DFT calculations.
In a 90o superexchange as is present in our system, ferromagnetic
superexchange between the TM ions also occurs due to hopping via two different
$p$-orbitals of the single O2- ion[35]. Considering the additional $p$-orbital
and an exchange coupling between the two O$p$-orbitals, the exchange
interaction strength (Appendix B) is,

$J_{FM}=\frac{\left(\frac{t_{pd1}^{2}}{{\tilde{U}}_{1}}+\frac{t_{pd2}^{2}}{{\tilde{U}}_{2}}\right)^{2}}{{\tilde{U}}-J_{p}}+\frac{\left({t_{pd1}^{2}}+{t_{pd2}^{2}}\right)^{2}}{J_{p}^{2}-{\tilde{U}}^{2}}\begin{bmatrix}\frac{{\tilde{U}}}{{\tilde{U}}_{1}^{2}}&\frac{-J_{p}}{{\tilde{U}}_{1}{\tilde{U}}_{2}}\\\
\frac{-J_{p}}{{\tilde{U}}_{1}{\tilde{U}}_{2}}&\frac{{\tilde{U}}}{{\tilde{U}}_{2}^{2}}\\\
\end{bmatrix}$ (12)

Using the above formula, the ferromagnetic superexchange strength for
Mn(1)$d_{yz}$ and Cr$d_{x^{2}-y^{2}}$ occurring via O(1a) $p_{x}$ and
$p_{y}$-orbitals (Fig. 9d) yields a value, $J_{FM}$$=$ 0.5 meV (Fig. 10d)
which is nearly an order of magnitude smaller than the largest $J_{AFM}$. The
Mn(2)$d_{x^{2}-y^{2}}$-O(2a)$p_{x}$-O(2a)$p_{z}$-Cr$d_{3z^{2}-r^{2}}$
superexchange (Fig. 9e), which involves hopping through two ${\sigma}$
orbitals, yields $J_{FM}$$=$0.5 meV which however is comparable to $J_{AFM}$
of Mn(2)-O(2a)-Cr (Fig. 10e). Similarly we obtain $J_{FM}$$=$0.1 meV for the
Mn(1)$d_{yz}$-O(1a)$p_{z}$-O(1a)$p_{y}$-Mn(2)$d_{3z^{2}-r^{2}}$ ferromagnetic
superexchange(Fig. 10a). Though ${J_{FM}}$ is always smaller than $J_{AFM}$,
this can cause a net decrease in the overall strength of the antiferromagnetic
interactions present in our system.
Finally, we discuss the ‘super-superexchange’ between two Cr3+ ions, which in
principle is weaker than the rest of the interactions. Due to the identical
orthorhombic ligand symmetry, the Cr(0,0,0)-O(1b) hopping is almost identical
to the Cr(0.5,0.5,0)-O(3a) hopping (Fig. 8f and h). As shown in Fig. 8g,
maximum hopping occurs between the $p_{x}$ orbitals of O(1a) and O(3b), thus
acting as a bridge for the ‘super-superexchange’ process. Using sixth order
perturbation theory, Toyoda et al.[36, 37] have calculated the exchange
constant for the super-superexchange process
($J_{SS}$$=$-$t_{pd}^{4}$$t_{pp}^{2}$${\Delta}_{pd}^{-4}$(${\Delta}_{pd}^{-1}$+$U^{-1}$)).
Considering only a single path Cr$d_{xy}$-O(1b)$p_{x}$-O(3a)$p_{x}$-Cr$d_{xy}$
(Fig. 9f) which has maximum hopping amplitudes throughout, we obtain $J_{SS}$
$=$-0.25 meV. So this formula predicts a correct order of magnitude, however,
the sign of $J_{SS}$ is anti-ferromagnetic. This discrepency is due to the
fact that the expression is obtained from the assumption that the ground state
is made up of empty ligand orbital and a single hole at each TM ion site. This
being appropriate for the cuprate superconductors (Cu2+, $d^{9}$) or
Cr2+($d^{4}$) systems, but not suitable for our system[38]. Hence, similar to
the case of first order superexchange, we extended the formalism to include
the second O2- ion and obtained expression for the ‘super-superexchange’
constant ${\tilde{J}_{SS}}$ (Appendix C) as,

${\tilde{J}_{SS}}=\frac{2t_{pd}^{2}(t_{pd}^{2}+t_{pp}^{2})}{{\tilde{U}}({\tilde{U}}+t_{pd})^{2}}-\frac{2t_{pd}^{4}t_{pp}^{2}}{U{\tilde{U}}^{4}}-\frac{2t_{pd}^{4}}{{\tilde{U}}^{3}}+\frac{4t_{pd}^{4}t_{pp}^{2}}{{\tilde{U}}^{5}}$
(13)

Using equation(13), we obtain $J_{SS}$$=$0.075 meV for
Cr$d_{xy}$-O(1b)$p_{x}$-O(3a)$p_{x}$-Cr$d_{xy}$ path, which though smaller has
the right ferromagnetic character. However, on neglecting $t_{pd}$ in
comparision to ${\tilde{U}}$ in the first term, we obtain a higher value
of${\tilde{J}_{SS}}$$=$0.32 meV, which is close to the value obtained from DFT
and hence can be considered as an upper limit. The maximum ${\tilde{J}_{SS}}$
and $J_{SS}$ are shown in Fig. 10c.

###  4.6 Determination of Curie temperature from mean field theory

Finally, based on the $J$-values obtained from our DFT calculations, we have
determined the Curie temperature of CaMnCrSbO6 using mean field theory.
Similar to Section 4.4, we describe the Mn/Cr sublattices as a system of Ising
spins. Due to the anti-ferromagnetic Mn(1)-Mn(2) interactions within the
$a$$-$$b$ plane and $c$-directions, a two-sublattice system followed by
Gauvin-Ndiay et al.[34] for double perovskites does not seem valid for the
double-double perovskite CaMnCrSbO6, a relatively more complex system. Hence,
we need to regard each Mn2+ spin in a unit cell as belonging to a different
sublattice. Thus, there are four Mn2+ sublattices and one Cr3+ sublattice in
the unit cell. We consider the entire spin lattice to be consisting of $2N$
spins, with $N$ Cr3+ spins and $N$/4 spins in each magnetic sublattice of Mn2+
spins. We denote the $z$-components of the four different Mn2+ sublattice
spins and the Cr3+ spins as $S_{M1}^{z}$, $S_{M2}^{z}$, $S_{M3}^{z}$,
$S_{M4}^{z}$ and $S_{C}^{z}$, respectively. It may be noted that $S_{M1}^{z}$
and $S_{M4}^{z}$ correspond to Mn(1) atoms, while $S_{M2}^{z}$ and
$S_{M3}^{z}$ correspond to Mn(2) atoms. Using the same notation of $J$‘s as in
Table 4 and Fig. 7, the exchange Hamiltonian of the equation (1) is now
written as,

$\begin{split}H&=-J_{1}{\sum\limits_{i,j=1}^{N/4}{{{S_{M1i}^{z}}{S_{M2j}^{z}}}}}-J_{1}{\sum\limits_{i,j=1}^{N/4}{{{S_{M3i}^{z}}{S_{M4j}^{z}}}}}-J_{1a}^{\prime}{\sum\limits_{i,j=1}^{N/4}{{{S_{Ci}^{z}}{(S_{M1j}^{z}+S_{M4j}^{z})}}}}-J_{1b}^{\prime}{\sum\limits_{i,j=1}^{N/4}{{{S_{Ci}^{z}}{(S_{M2j}^{z}+S_{M3j}^{z})}}}}\\\
&-J_{2}{\sum\limits_{i,j=1}^{N/4}{{{S_{M1i}^{z}}{S_{M3j}^{z}}}}}-J_{2}{\sum\limits_{i,j=1}^{N/4}{{{S_{M2i}^{z}}{S_{M4j}^{z}}}}}-J_{2}^{\prime}{\sum\limits_{i,j=1}^{N}{{{S_{Ci}^{z}}{S_{Cj}^{z}}}}}\end{split}$
(14)

In the mean field approximation, we substitute
$S_{ai}^{z}$$=$($<$$S_{ai}^{z}$$>$+${\delta}$$S_{ai}^{z}$) ($a$$=$$M1$, ..,
$M4$ and $C$), where ${\delta}$$S_{ai}^{z}$=$S_{ai}^{z}$-$<$$S_{ai}^{z}$$>$
and neglect the second order terms (${\delta}$$S_{ai}^{z})^{2}$. The average
sublattice magnetization is defined as,
$M_{k}$$=$$(g{\mu}_{B}N/4)$$<$$S_{Mk}^{z}$$>$;
$M_{C}$$=$$g{\mu}_{B}N<S_{C}^{z}>$ where $M_{k}$($k$ = 1 to 4) correponds to
the sub-lattice magnetization of the four Mn2+ spins, while $M_{C}$ is the
sub-lattice magnetization of Cr3+ spins. Using the mean field approximation
(hereafter dropping the limits of summation), we obtain the mean field
Hamiltonian,

$\begin{split}H_{MF}&=\frac{-J_{1}}{g{\mu}_{B}N}\left(2M_{2}{\sum
S_{M1i}^{z}}+2M_{1}{\sum S_{M2i}^{z}}+2M_{4}{\sum S_{M3i}^{z}}+2M_{3}{\sum
S_{M4i}^{z}}\right)\\\ &\frac{-J_{2}}{g{\mu}_{B}N}\left(4M_{3}{\sum
S_{M1i}^{z}}+4M_{4}{\sum S_{M2i}^{z}}+2M_{1}{\sum S_{M3i}^{z}}+2M_{2}{\sum
S_{M4i}^{z}}\right)\\\
&-\frac{1}{g{\mu}_{B}N}\left(J_{1a}^{\prime}(M_{1}+M_{4})+J_{1b}^{\prime}(M_{2}+M_{3})\right){\sum
S_{Ci}^{z}}\\\ &-\frac{M_{C}}{g{\mu}_{B}N}\left(J_{1a}^{\prime}({\sum
S_{M1i}^{z}}+{\sum S_{M4i}^{z}})+J_{1b}^{\prime}({\sum S_{M2i}^{z}}+{\sum
S_{M3i}^{z}})\right)\\\ &-\frac{24J_{2}^{\prime}M_{C}}{g{\mu}_{B}N}{\sum
S_{Ci}^{z}}\\\ \end{split}$ (15)

In terms of effective mean fields ($h_{M}$ or $h_{C}$) acting on each sub-
lattice atom,

$H_{MF}={{{h_{M1}}\sum{S_{M1i}^{z}}}}+{{{h_{M2}}\sum{S_{2i}^{z}}}}+{{{h_{M3}}\sum{S_{3i}^{z}}}}+{{{h_{M4}}{\sum
S_{M4i}^{z}}}}+{{{h_{C}}{\sum{S_{Ci}^{z}}}}}$ (16)

where the exchange(mean) field acting on each sublattice is,

$\begin{split}h_{M1}&=\frac{-1}{g{\mu}_{B}N^{\prime}}(2J_{1}M_{2}+4J_{2}M_{3}+J_{1a}^{\prime}M_{C})\\\
h_{M2}&=\frac{-1}{g{\mu}_{B}N^{\prime}}(2J_{1}M_{1}+4J_{2}M_{4}+J_{1b}^{\prime}M_{C})\\\
h_{M3}&=\frac{-1}{g{\mu}_{B}N^{\prime}}(2J_{4}M_{2}+4J_{1}M_{3}+J_{1b}^{\prime}M_{C})\\\
h_{M4}&=\frac{-1}{g{\mu}_{B}N^{\prime}}(2J_{1}M_{3}+4J_{2}M_{2}+J_{1a}^{\prime}M_{C})\\\
h_{C}&=\frac{-1}{g{\mu}_{B}N}(J_{1a}^{\prime}(M_{1}+M_{4})+J_{1b}^{\prime}(M_{2}+M_{3})+24J_{2}^{\prime}M_{C})\\\
\end{split}$ (17)

where $N^{\prime}$$=$$N$/4. The total Hamiltonian, which comprises of the
Zeeman and mean field terms is,

$\begin{split}H_{total}&=-g{\mu}_{B}\\{H_{0}+\frac{h_{M1}}{g{\mu}_{B}}\\}\sum
S_{M1i}^{z}-g{\mu}_{B}\\{H_{0}+\frac{h_{M2}}{g{\mu}_{B}}\\}\sum S_{M2i}^{z}\\\
&-g{\mu}_{B}\\{H_{0}+\frac{h_{M3}}{g{\mu}_{B}}\\}\sum
S_{M3i}^{z}-g{\mu}_{B}\\{H_{0}+\frac{h_{M4}}{g{\mu}_{B}}\\}\sum S_{M4i}^{z}\\\
&-g{\mu}_{B}\\{H_{0}+\frac{h_{C}}{g{\mu}_{B}}\\}\sum S_{Ci}^{z}\end{split}$
(18)

where $H_{0}$ is the applied magnetic field. The term within the curly
brackets denoted as $h_{a}^{eff}$ ($a$=M1,..M4 and Cr), corresponds to the
total effective field. In terms of the effective fields ahead in equation
(19), the thermal average of sub-lattice magnetization for the Mn2+ and Cr3+
spins is,

$\begin{split}M_{Mk}&=gS_{M}{\mu}_{B}N^{\prime}B_{s}\left(\frac{gS_{M}{\mu}_{B}h_{Mk}^{eff}}{k_{B}T}\right)(k=1,..,4)\\\
M_{C}&=gS_{C}{\mu}_{B}NB_{s}\left(\frac{gS_{C}{\mu}_{B}h_{C}^{eff}}{k_{B}T}\right)\end{split}$
(19)

where
$B_{s}(x)$(=$\frac{2S+1}{2S}$coth($\frac{2S+1}{2S}$$x$)-$\frac{1}{2S}$coth($\frac{x}{2S}$))
is the Brillouin function. For $x$${<}$${<}$1, i.e. when
${\mu}_{B}h_{a}^{eff}$${<}$${<}k_{B}T$ ($a$: M or C) which is true for large
temperatures, the sub-lattice magnetization for Mn2+ and Cr3+ spins can be
written as,

$\begin{split}M_{k}&=g^{2}{\mu}_{B}^{2}N^{\prime}S_{M}(S_{M}+1)\frac{h_{Mk}^{eff}}{3k_{B}T}\\\
M_{C}&=g^{2}{\mu}_{B}^{2}NS_{C}(S_{C}+1)\frac{h_{C}^{eff}}{3k_{B}T}\\\
\end{split}$ (20)

Setting $H_{0}$$=$0, from equation (18)-(20), we obtain a system of linear
equations,

$\begin{split}M_{1}&=\frac{S_{M}(S_{M}+1)}{3k_{B}T}(2J_{1}M_{2}+4J_{2}M_{3}+J_{1a}^{\prime}M_{C})\\\
M_{2}&=\frac{S_{M}(S_{M}+1)}{3k_{B}T}(2J_{1}M_{1}+4J_{2}M_{4}+J_{1b}^{\prime}M_{C})\\\
M_{3}&=\frac{S_{M}(S_{M}+1)}{3k_{B}T}(2J_{4}M_{2}+4J_{1}M_{3}+J_{1b}^{\prime}M_{C})\\\
M_{4}&=\frac{S_{M}(S_{M}+1)}{3k_{B}T}(2J_{1}M_{3}+4J_{2}M_{2}+J_{1a}^{\prime}M_{C})\\\
M_{C}&=\frac{S_{C}(S_{C}+1)}{3k_{B}T}(J_{1a}^{\prime}(M_{1}+M_{4})+J_{1b}^{\prime}(M_{2}+M_{3})+24J_{2}^{\prime}M_{C})\\\
\end{split}$ (21)

The transition temperature $T_{C}$ is obtained from the homogeneous solution
of equation (21) i.e.

$\begin{vmatrix}1&\frac{-2J_{1}\bar{S}_{1}}{y}&\frac{-4J_{2}\bar{S}_{1}}{y}&0&\frac{-J_{1a}^{\prime}\bar{S}_{1}}{y}\\\
&\\\
\frac{-2J_{1}\bar{S}_{1}}{y}&1&0&\frac{-4J_{2}\bar{S}_{1}}{y}&\frac{-J_{1b}^{\prime}\bar{S}_{1}}{y}\\\
&\\\
\frac{-4J_{2}\bar{S}_{1}}{y}&0&1&\frac{-2J_{1}\bar{S}_{1}}{y}&\frac{-J_{1b}^{\prime}\bar{S}_{1}}{y}\\\
&\\\
0&\frac{-4J_{2}\bar{S}_{1}}{y}&\frac{-2J_{1}\bar{S}_{1}}{y}&1&\frac{-J_{1a}^{\prime}\bar{S}_{1}}{y}\\\
&\\\
\frac{-J_{1a}^{\prime}\bar{S}_{2}}{y}&\frac{-J_{1b}^{\prime}\bar{S}_{2}}{y}&\frac{-J_{1b}^{\prime}\bar{S}_{2}}{y}&\frac{-J_{1a}^{\prime}\bar{S}_{1}}{y}&1-\frac{24J_{2}^{\prime}\bar{S}_{2}}{y}\\\
\end{vmatrix}=0$ (22)

where, $y$$=$3$k_{B}$$T_{C}$, ${\bar{S}_{1}}$$=$$S_{M}(S_{M}+1)$,
${\bar{S}_{2}}$$=$$S_{C}(S_{C}+1)$. Solving equation (22), we obtain
$T_{C}$$=$ 99 K, which is closer to the experimental value of 49 K, while the
two sublattice model yields a considerably higher value of ${\sim}$ 188 K.

##  5 Summary and conclusions

We have investigated thoroughly the electronic and magnetic structure along
with magnetic exchange interactions in the recently reported double-double
perovskite CaMnCrSbO6. Amongst the various arrangements of Mn2+ and Cr3+
spins, the lowest energy corresponds to ferrimagnetic configuration, in
agreement with the experimental observation. For $U_{eff}$$=$0 eV, i.e.
without inclusion of electronic correlations, a band gap of ${\sim}$0.7 eV is
obtained which increases on increasing the $U_{eff}$, thereby suggesting that
the system has a Mott-Hubbard like character. Our non-collinear calculations
reveal that the combined Mn2+/Cr3+ spins show anisotropy in the
crystallographic $x$-direction, thus resolving the ambiguity (in the
experimental results) about the preferred direction of the spins.The inclusion
of the anti-site disorder explains the reduced value of the Mn2+ magnetic
moments in this compound.
The various competing exchange interactions in the system have been evaluated
by mapping the total energies obtained from DFT to the Ising model. The
nearest neighbour (along $c$-direction) Mn(1)-O-Mn(2) superexchange
interactions are anti-ferromagnetic and weaker than the ferromagnetic Cr-O-O-
Cr super-superexchange interactions, while the in-plane Mn(1)-O-Mn(2)
superexchange is the weakest in the lattice. The Mn-O-Cr interactions are also
predominantly antiferromagnetic. A simple two sublattice model (Mn(1)2+ and
Mn(2)2+ spin-up, Cr3+ spin-down) thus cannot explain the magnetism of this
system. Due to difference in the ligand symmetry between Mn(1) and Mn(2), the
Mn(1)-O-Cr superexchange interaction is stronger than the Mn(2)-O-Cr
interaction, thereby establishing a “ferrimagnetic” order.
The origins and the strengths of the various superexchange interactions have
been calculated in terms of the hopping between the Wannier orbitals of Mn(1),
Mn(2) and Cr via the intermediate O $p$-orbitals. Using a 3-site Hubbard model
with singly occupied $d$-orbital at both TM sites and a doubly occupied O
$p$-orbital, we obtain expression for $J_{AFM}$. The formula predicts correct
relative strengths for the various anti-ferromagnetic interactions. While the
maximum $J_{AFM}$ for Mn(1)-O-Mn(2) (out of plane and in-plane) is closer to
the DFT results, the maximum $J_{AFM}$ for Mn-O-Cr superexchange is
overestimated by an order of magnitude. Similarly, considering hopping via two
different $p$-orbitals, we obtain expression for $J_{FM}$, which also exists
in a 90o superexchange. Though much smaller than $J_{AFM}$, it can result in
an overall decrease in the antiferromagnetic superexchange strengths. Using a
4-site Hubbard model, we obtain an expression for the Cr-O-O-Cr super-
superexchange ${\tilde{J}}_{SS}$, which is found to be ferromagnetic, with a
maximum value in close agreement with the DFT results. We show that
considering a set of five magnetic sub-lattices, using the mean field theory
one gets a Curie temperature of 99 K, which is closer to the experimental
value of 49 K unlike the two-sublattice model which yields a higher Curie
temperature.

##  6 Acknowledgments

The authors wish to thank the Institute Compute Centre of IIT Roorkee for
providing the computational facility.

##  7 Appendix

Appendix A. AFM Super-exchange interaction
The Hubbard Hamiltonian for the first order neighbour superexchange between
two TM ions and a single oxygen ion can be written as,

$H=-\sum_{{\sigma}={\uparrow},{\downarrow}}[{\epsilon}_{d1}{\hat{n}_{1{\sigma}}}+{\epsilon}_{d2}{\hat{n}_{2{\sigma}}}+{\epsilon}_{p}{\hat{n}_{p{\sigma}}}-\sum_{i=1,2}t_{pdi}(d_{i{\sigma}}^{+}c_{p{\sigma}}+c_{p{\sigma}}^{+}d_{i{\sigma}})]+U_{1}{\hat{n}_{1{\uparrow}}}{\hat{n}_{1{\downarrow}}}+U_{2}{\hat{n}_{2{\uparrow}}}{\hat{n}_{2{\downarrow}}}$
(A1)

where ${\epsilon}_{da}$($a$=1,2) correspond to the on-site energies of the
$d$-orbitals of the two TM ions under consideration, ${\epsilon}_{p}$ is the
on-site energy of the O $p$-orbital, $t_{pd}$ is the hopping integral between
the TM $d$\- O $p$-orbitals, $U_{1}$ and $U_{2}$ are the Hubbard $U$
parameters for the first and second TM ions respectively. Due to the distinct
TM ions, (Mn(1)2+, Mn(2)2+ and Cr3+), the various parameters (on-site, hopping
and Hubbard) are different, hence we use additional index $i$. The ground
state comprises of singly occupied $d$-orbitals (parallel or anti-parallel).
The Hamiltonian matrix for the ferromagnetic (spin-triplet) case is given as,

$H_{FM}=\begin{bmatrix}0&t_{pd1}&t_{pd2}\\\ t_{pd1}&{\tilde{U}}_{1}&0\\\
t_{pd2}&0&{\tilde{U}}_{2}\\\ \end{bmatrix}$ (A2)

The first term which corresponds to
${\epsilon}_{d1}$+${\epsilon}_{d2}$+2${\epsilon}_{p}$ is set to be 0. We
define, ${\Delta}_{pdi}$$=$${\epsilon}_{di}$-${\epsilon}_{p}$ as the charge
transfer energy. In the above matrix, we have made the following
abbreviations, ${\tilde{U}}_{1}$$=$$U_{1}$+${\Delta}_{pd1}$,
${\tilde{U}}_{2}$$=$$U_{2}$+${\Delta}_{pd2}$.

The corresponding matrix for anti-ferromagnetic (spin singlet) case is,

$H_{AFM}=\begin{bmatrix}0&0&t_{pd}^{1}&t_{pd}^{2}&0&0&0&0&0\\\
0&0&0&0&t_{pd}^{1}&t_{pd}^{2}&0&0&0\\\
t_{pd}^{1}&0&{\tilde{U}}_{1}&0&0&0&-t_{pd}^{2}&0&-t_{pd}^{2}\\\
t_{pd}^{2}&0&0&{\tilde{U}}_{2}&0&0&0&-t_{pd}^{1}&-t_{pd}^{1}\\\
0&t_{pd}^{1}&0&0&{\tilde{U}}_{1}&0&t_{pd}^{2}&0&t_{pd}^{2}\\\
0&t_{pd}^{1}&0&0&0&{\tilde{U}}_{2}&0&t_{pd}^{1}&t_{pd}^{1}\\\
0&0&-t_{pd}^{2}&0&t_{pd}^{2}&0&U_{1}+{\Delta}{\epsilon}&0&0\\\
0&0&0&-t_{pd}^{1}&0&t_{pd}^{1}&0&U_{2}-{\Delta}{\epsilon}&0\\\
0&0&-t_{pd}^{2}&-t_{pd}^{1}&t_{pd}^{2}&t_{pd}^{1}&0&0&{\bar{U}}\\\
\end{bmatrix}$ (A3)

where, ${\bar{U}}$$=$${\tilde{U}}_{1}$+${\tilde{U}}_{2}$ and
${\Delta}{\epsilon}$$=$${\epsilon}_{d1}$-${\epsilon}_{d2}$. Using the method
of downfolding, we obtain an effective Hamiltonian,

$H_{eff}=-T_{01}H_{11}^{-1}T_{10}-T_{01}H_{11}^{-1}T_{12}H_{22}^{-1}T_{21}H_{11}^{-1}T_{10}$
(A4)

For the FM case, only the first term in equation (A4) exists.

$H_{eff}^{FM}=-\frac{t_{pd1}^{2}}{{\tilde{U}}_{1}}-\frac{t_{pd2}^{2}}{{\tilde{U}}_{2}}$
(A5)
$\begin{split}H_{eff}^{AFM}&=\left(-\frac{t_{pd}^{1}}{{\tilde{U}}_{1}}-\frac{t_{pd}^{2}}{{\tilde{U}}_{2}}\right)\begin{bmatrix}1&0\\\
0&1\\\
\end{bmatrix}-(t_{pd1}t_{pd2})^{2}\left[\frac{1}{{\tilde{U}}_{1}^{2}(U_{1}+{\Delta}{\epsilon})}+\frac{1}{{\tilde{U}}_{2}^{2}(U_{2}-{\Delta}{\epsilon})}+\frac{1}{{\bar{U}}}\left(\frac{1}{{\tilde{U}}_{1}}+\frac{1}{{\tilde{U}}_{2}}\right)^{2}\right]\\\
{\times}&\begin{bmatrix}1&-1\\\ -1&1\\\ \end{bmatrix}\end{split}$ (A6)

Subtracting (A5) from (A6) yields the antiferromagnetic exchange constant

$J_{AFM}=-2(t_{pd1}t_{pd2})^{2}\left[\frac{1}{{\tilde{U}}_{1}^{2}(U_{1}+{\Delta}{\epsilon})}+\frac{1}{{\tilde{U}}_{2}^{2}(U_{2}-{\Delta}{\epsilon})}+\frac{1}{{\bar{U}}}\left(\frac{1}{{\tilde{U}}_{1}}+\frac{1}{{\tilde{U}}_{2}}\right)^{2}\right]$
(A7)

The sign of antiferromagnetic exchange is negative.
Appendix B. FM superexchange interaction
The ferromagnetic superexchange which involves two $p$-orbitals of the
intermediate O2- ion can be described by the Hamiltonian,

$\begin{split}H&=\sum_{{\sigma}={\uparrow},{\downarrow}}[{\epsilon}_{d1}{\hat{n}_{1{\sigma}}}+{\epsilon}_{d2}{\hat{n}_{2{\sigma}}}+{\epsilon}_{p}({\hat{n}_{p1{\sigma}}}+{\hat{n}_{p2{\sigma}}})-J_{p}{\hat{n}_{p1,{\sigma}}}{\hat{n}_{p2,-{\sigma}}}\\\
&-\sum_{i=1,2}t_{pdi}(d_{i{\sigma}}^{+}c_{p{\sigma}}+c_{p{\sigma}}^{+}d_{i{\sigma}})]+U_{1}{\hat{n}_{1{\uparrow}}}{\hat{n}_{1{\downarrow}}}+U_{2}{\hat{n}_{2{\uparrow}}}{\hat{n}_{2{\downarrow}}}\\\
\end{split}$ (B1)

The Hamiltonian now involves occupancy of the two $p$-orbitals, while
$J_{p1,p2}$ corresponds to the exchange term between the two $p$-orbitals. The
on-site repsulsion at the O2- ion is neglected. Similarly, no hopping occurs
between the two $p$-orbitals of the same anion. The matrix representation for
the triplet ground state is,

$H_{FM}=\begin{bmatrix}0&t_{pd1}&t_{pd2}&0\\\
t_{pd1}&{\tilde{U}}_{1}&0&t_{pd1}\\\ t_{pd2}&0&{\tilde{U}}_{2}&t_{pd2}\\\
0&t_{pd1}&t_{pd2}&{\tilde{U}}-J_{p}\\\ \end{bmatrix}$ (B2)

The corresponding Hamiltonian matrix for the singlet state is

$H_{AFM}=\begin{bmatrix}0&0&t_{pd}^{1}&0&t_{pd}^{2}&0&0&0\\\
0&0&0&t_{pd}^{1}&0&t_{pd}^{2}&0&0\\\
t_{pd}^{1}&0&{\tilde{U}}_{1}&0&0&0&-t_{pd}^{2}&0\\\
0&t_{pd}^{2}&0&{\tilde{U}}_{2}&0&0&0&-t_{pd}^{1}\\\
t_{pd}^{1}&0&0&0&{\tilde{U}}_{1}&0&t_{pd}^{2}&0\\\
0&t_{pd}^{1}&0&0&0&{\tilde{U}}_{2}&0&t_{pd}^{1}\\\
0&0&-t_{pd}^{2}&0&-t_{pd}^{2}&0&{\tilde{U}}&-J_{p}\\\
0&0&0&-t_{pd}^{1}&0&t_{pd}^{1}&-J_{p}&{\tilde{U}}\\\ \end{bmatrix}$ (B3)

The effective Hamiltonian for the triplet is given by,

$H_{eff}^{FM}=-\left(\frac{t_{pd1}^{2}}{{\tilde{U}}_{1}}+\frac{t_{pd2}^{2}}{{\tilde{U}}_{2}}\right)-\frac{\left(\frac{t_{pd1}^{2}}{{\tilde{U}}_{1}}+\frac{t_{pd2}^{2}}{{\tilde{U}}_{2}}\right)^{2}}{{\tilde{U}}-J_{p}}$
(B4)

while for the singlet,

$H_{eff}^{AFM}=-\left(\frac{t_{pd1}^{2}}{{\tilde{U}}_{1}}+\frac{t_{pd2}^{2}}{{\tilde{U}}_{2}}\right)-\frac{\left({t_{pd1}^{2}}+{t_{pd2}^{2}}\right)^{2}}{J_{p}^{2}-{\tilde{U}}^{2}}\begin{bmatrix}\frac{{\tilde{U}}}{{\tilde{U}}_{1}^{2}}&\frac{-J_{p}}{{\tilde{U}}_{1}{\tilde{U}}_{2}}\\\
\frac{-J_{p}}{{\tilde{U}}_{1}{\tilde{U}}_{2}}&\frac{{\tilde{U}}}{{\tilde{U}}_{2}^{2}}\\\
\end{bmatrix}$ (B5)

The exchange interaction strength for the ferromagnetic superexchange is
obtained as,

$J_{FM}=\frac{\left(\frac{t_{pd1}^{2}}{{\tilde{U}}_{1}}+\frac{t_{pd2}^{2}}{{\tilde{U}}_{2}}\right)^{2}}{{\tilde{U}}-J_{p}}-\frac{\left({t_{pd1}^{2}}+{t_{pd2}^{2}}\right)^{2}}{J_{p}^{2}-{\tilde{U}}^{2}}\begin{bmatrix}\frac{{\tilde{U}}}{{\tilde{U}}_{1}^{2}}&\frac{-J_{p}}{{\tilde{U}}_{1}{\tilde{U}}_{2}}\\\
\frac{-J_{p}}{{\tilde{U}}_{1}{\tilde{U}}_{2}}&\frac{{\tilde{U}}}{{\tilde{U}}_{2}^{2}}\\\
\end{bmatrix}$ (B6)

The sign of $J_{FM}$ is positive.
Appendix C. Super-superexchange interaction
The exchange interaction between the Cr3+ spins occurs via hopping through two
O2- ions, thus the Hamiltonian can be written as,

$H=-\sum_{{\sigma}={\uparrow},{\downarrow}}\sum_{i=1,2}[{\epsilon}_{di}{\hat{n}_{i{\sigma}}}+{\epsilon}_{pi}{\hat{n}_{pi{\sigma}}}+U_{i}{\hat{n}_{i{\uparrow}}}{\hat{n}_{i{\downarrow}}}]-t_{pd}(d_{1{\sigma}}^{+}c_{p1{\sigma}}+d_{2{\sigma}}^{+}c_{p2{\sigma}})-t_{pp}c_{p2{\sigma}}^{+}c_{p1{\sigma}}+\text{h.c}$
(C1)

In the above equation, $t_{pp}$ corresponds to the hopping amplitude between
the two O2- $p$-orbitals. The matrix corresponding to the triplet state is,

$H_{FM}=\begin{bmatrix}0&t_{pd}&t_{pd}&0&0&0\\\
t_{pd}&{\tilde{U}}&t_{pd}&t_{pp}&0&t_{pd}\\\
t_{pd}&t_{pd}&{\tilde{U}}&0&t_{pp}&t_{pd}\\\ 0&t_{pp}&0&{\tilde{U}}&0&0\\\
0&0&t_{pp}&0&{\tilde{U}}&0\\\ 0&t_{pd}&t_{pd}&0&0&2{\tilde{U}}\\\
\end{bmatrix}$ (C2)

After downfolding (using equation (A4)), the effective Hamiltonian for the
triplet state is,

$H_{eff}^{FM}=-\frac{2t_{pd}^{2}}{{\tilde{U}}+t_{pd}}-\frac{2t_{pd}^{2}(t_{pd}^{2}+t_{pp}^{2})}{{\tilde{U}}({\tilde{U}}+t_{pd})^{2}}\begin{bmatrix}1&1\\\
1&1\\\ \end{bmatrix}$ (C3)

The results for the singlet state are more complex yielding a 16x16 matrix,
which in block form can be written as,

$H_{AFM}=\begin{bmatrix}H_{00}&T_{01}&T_{02}&T_{03}\\\
T_{10}&H_{11}&T_{12}&T_{13}\\\ T_{20}&T_{21}&H_{22}&T_{23}\\\
T_{30}&T_{31}&T_{32}&H_{33}\\\ \end{bmatrix}$ (C4)

$H_{00}$ is a 2x2 diagonal matrix corresponding to the ground state on-site
energy 2${\epsilon}_{d}$+4${\epsilon}_{p}$ which is set to zero. The other
block matrices along the diagonal are,

$H_{11}=\begin{bmatrix}{\tilde{U}}&0&0&0&0&0&t_{pp}&0\\\
0&{\tilde{U}}&0&0&0&0&0&t_{pp}\\\ 0&0&{\tilde{U}}&0&t_{pp}&0&0&0\\\
0&0&0&{\tilde{U}}&0&t_{pp}&0&0\\\ 0&0&t_{pp}&0&{\tilde{U}}&0&0&0\\\
0&0&0&t_{pp}&0&{\tilde{U}}&0&0\\\ t_{pp}&0&0&0&0&0&{\tilde{U}}&0\\\
0&t_{pp}&0&0&0&0&0&{\tilde{U}}\\\ \end{bmatrix},\quad
H_{22}=\begin{bmatrix}U&0\\\ 0&U\\\ \end{bmatrix},\quad
H_{33}=\begin{bmatrix}2{\tilde{U}}&0&t_{pp}&t_{pp}\\\
0&2{\tilde{U}}&-t_{pp}&-t_{pp}\\\ t_{pp}&-t_{pp}&2{\tilde{U}}&0\\\
t_{pp}&-t_{pp}&0&2{\tilde{U}}\\\ \end{bmatrix}$ (C5)

where ${\tilde{U}}$$=$$U$+${\Delta}_{pd}$. The off-diagonal block matrices are
given below

$T_{10}=\begin{bmatrix}t_{pd}&0\\\ 0&t_{pd}\\\ t_{pd}&0\\\ 0&t_{pd}\\\ 0&0\\\
0&0\\\ 0&0\\\ 0&0\\\ \end{bmatrix},\quad T_{12}=\begin{bmatrix}0&0\\\ 0&0\\\
0&0\\\ 0&0\\\ 0&t_{pd}\\\ 0&-t_{pd}\\\ t_{pd}&0\\\ -t_{pd}&0\\\
\end{bmatrix},\quad T_{13}=\begin{bmatrix}t_{pd}&0&0&0\\\ 0&t_{pd}&0&0\\\
t_{pd}&0&0&0\\\ 0&t_{pd}&0&0\\\ 0&0&t_{pd}&0\\\ 0&0&-t_{pd}&0\\\
0&0&0&t_{pd}\\\ 0&0&0&-t_{pd}\\\ \end{bmatrix}$ (C6)

$T_{01}$$=$$T_{10}^{\dagger}$, $T_{21}$$=$$T_{12}^{\dagger}$ and
$T_{31}$$=$$T_{13}^{\dagger}$. The rest of the block matrices are null
matrices. Using the method of downfolding, the effective Hamiltonian is given
as,

$H_{eff}^{AFM}=-T_{01}H_{11}^{-1}T_{10}-T_{01}H_{11}^{-1}T_{12}H_{22}^{-1}T_{21}H_{11}^{-1}T_{10}-T_{01}H_{11}^{-1}T_{13}H_{33}^{-1}T_{31}H_{11}^{-1}T_{10}$
(C7)

The corresponding effective Hamiltonian of the singlet state is,

$H_{eff}^{AFM}=\frac{2t_{pd}^{2}{\tilde{U}}}{t_{pp}^{2}-{\tilde{U}}^{2}}-\frac{2t_{pd}^{4}t_{pp}^{2}}{U(t_{pp}^{2}-{\tilde{U}}^{2})^{2}}\begin{bmatrix}1&1\\\
1&1\\\
\end{bmatrix}+\frac{2t_{pd}^{4}{\tilde{U}}}{(t_{pp}-{\tilde{U}})^{3}(t_{pp}+{\tilde{U}})^{3}}\begin{bmatrix}{\tilde{U}}^{2}+t_{pp}^{2}&t_{pp}^{2}\\\
t_{pp}^{2}&{\tilde{U}}^{2}+t_{pp}^{2}\\\ \end{bmatrix}$ (C8)

The final expression for the super-superexchange interaction strength, which
is the difference between equation(A21) and (A16), is

$J_{SS}=\frac{2t_{pd}^{2}(t_{pd}^{2}+t_{pp}^{2})}{{\tilde{U}}({\tilde{U}}+t_{pd})^{2}}-\frac{2t_{pd}^{4}t_{pp}^{2}}{U{\tilde{U}}^{4}}-\frac{2t_{pd}^{4}}{{\tilde{U}}^{3}}+\frac{4t_{pd}^{4}t_{pp}^{2}}{{\tilde{U}}^{5}}$
(C9)

The first terms in the R.H.S of equations (C3) and (C8) correspond to a direct
exchange process and hence not considered. Equation (C9) is obtained after
making the approximation $t_{pp}$ ${<}{<}$ ${\tilde{U}}$ in equation (C8).

##  8 References

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