Coupling between magnetic ordering and structural instabilities in perovskite
biferroics: A first-principles study
Nirat Ray and Umesh V. Waghmare
Theoretical Sciences Unit
Jawaharlal Nehru Centre for Advanced Scientific Research
Jakkur PO, Bangalore 560 064, India
(Dated: November 13, 2018)
We use first-principles density functional theory-based calculations to investigate structural insta-
bilities in the high symmetry cubic perovskite structure of rare-earth (R = La, Y, Lu) and Bi-based
biferroic chromites, focusing on Γ and R point phonons of states with para-, ferro-, and antiferro-
magnetic ordering. We find that (a) the structure with G-type antiferromagnetic ordering is most
stable, (b) the most dominant structural instabilities in these oxides are the ones associated with ro-
tations of oxygen octahedra, and (c) structural instabilities involving changes in Cr-O-Cr bond angle
depend sensitively on the changes in magnetic ordering. The dependence of structural instabilities
on magnetic ordering can be understood in terms of how super-exchange interactions depend on the
Cr-O-Cr bond angles and Cr-O bond lengths. We demonstrate how adequate buckling of Cr-O-Cr
chains can favour ferromagnetism. Born effective charges (BEC) calculated using the Berry phase
expression are found to be anomalously large for the A-cations, indicating their chemical relevance
to ferroelectric distortions.
I. INTRODUCTION
A ferroic is a material which exhibits spontaneous and
switchable ordering of electric polarization or magnetiza-
tion or elastic strain. Materials exhibiting more than one
of such orderings termed ‘multiferroics’- have recently be-
come the focus of much research1. Most of the biferroics
investigated in recent years are ABO3 oxides with per-
ovskite structure. The d0−ness or the zero occupancy
of tranistion metal B cation is known chemically to fa-
vor ferroelectricity2. Hence, the availability of transition
metal d -electrons in the perovskite oxides necessary for
magnetism, reduces the tendency for off-centering ferro-
electric distortions2 making multiferroics relatively rare.
How the ordering of d−electronic spins of the B cation in-
fluence ferroelectric or other competing structural insta-
bilities has not yet been explored and understood. The
coupling between magnetic ordering and structural in-
stabilities is expected to involve interesting physics and
is of direct relevance to technological applications3 such
as multiple state memory elements and novel memory
media.
There are at least three families of high tempera-
ture biferroic materials. Bi- based perovskite oxides,
like BiMnO3
4, BiCrO3
5, and BiFeO3
6,7 are known to
be promising biferroics. Ferroelectricity in these materi-
als arises from the stereochemical activity of the 6s lone
pair electrons of Bi. Hexagonal rare earth manganates
LnMnO3 and InMnO3 are biferroics which exhibit im-
proper or geometric ferroelectricity8,9,10,11. Rare earth
chromites LnCrO3
12,13 (with Ln = Ho, Er, Tm, Yb, Lu
or Y) have been recently shown to be biferroic; YCrO3
has been shown12 to exhibit canted antiferromagnetic be-
havior below 140 K and a ferroelectric transition around
473 K. Similarly, LuCrO3 becomes a canted antiferro-
magnet below 115 K, and is ferroelectric below 488 K13.
The absence of any ferroelctricity in LaCrO3, has been
attributed to the large size of the La3+ ion, in comparison
with Y3+.
However, small values of polarization reported for
these materials (about 2µC/cm2 for YCrO3
12 and
6µC/cm2 of BiFeO3
7) inspite of large A-cation off-
centering distortions remain a puzzle. More recently14,
a new concept of ‘local non-centrosymmetry’ in YCrO3
has been proposed to account for the small value of
polarization observed. Perovskite oxides are known to
have many competing structural instabilities15 and this
competition is further enriched by the magnetic insta-
bilities. The coupling and competition between various
magnetic and structural ordering can be partly respon-
sible for weak ferroelectricity or the possibility of local
non-centrosymmetry. Our goal here is to investigate this
issue through determination of various structural insta-
bilities for different magnetic orderings, with a focus on
rare earth chromites.
We present results of detailed electronic structure and
frozen phonon (at the Γ and R points) calculations for a
set of five materials (LuCrO3, YCrO3, LaCrO3, BiCrO3
and YFeO3) in the cubic phase with three different mag-
netic orderings (para-, ferro- and antiferromagnetic). In
Section II, we briefly describe the methods used in cal-
culations here. In section III, we report results for struc-
tural energetics, the electronic density of states (DOS)
and the Born Effective charges (BEC) for the cubic phase
of these materials with different magnetic orderings. In
Section IV, we report results for structural instabilities
in chromites and compare them with those in a related
compound YFeO3. Since the many-electron correlations
are important in magnetic oxides, we estimate their ef-
fects on the structural instabilities through use of the
Hubbard parameter U16. Using our results for structural
instabilities, we show how certain nonpolar structural
instabilities can cooperatively stabilize ferromagnetism.
Our work reveals how the structural instabilities of these
http://arxiv.org/abs/0704.1251v1
biferroic oxides depend on the size of A-cation (with the
same B-cation), and a change in B-cation. We interpret
the results using arguments based on superexchange17,
and the well-known Goodenough-Kanamori rules18. Fi-
nally, we summarize in Section V.
FIG. 1: Perovskite structure: Cell doubled along the 〈111〉
direction, to represent G-type antiferroamgnetic ordering.
II. METHODOLOGY
Our calculations are based on first-principles pseu-
dopotential based Density Functional Theory within a
generalized gradient approximation (GGA)19 as imple-
mented in the PWSCF package20. The interaction be-
tween ions and electrons is approximated with ultra-
soft pseudopotentials21 treating explicitly 11 electrons
[(n-1)s2 (n-1)p6 (n-1)d1 and ns2] in the valence shell of
Lu(n=6), La(n=6) and Y(n=5). We consider 6 valence
electrons for Oxygen [2s22p4] and 14 electrons for Cr
[3s23p63d54s1]. We used a plane wave basis with kinetic
energy cut off of 25 Ryd (150 Ryd) to represent wave
functions (density). For cubic structures, we sample the
Brillouin zone using a 5 × 5 × 5 Monkhorst Pack Mesh22,
and a denser mesh (6 × 6 × 6) and higher energy cut-off
(30 Ryd) for energy differences. Phonon frequencies cal-
culated with these larger parameters do not differ much
from those calculated with a lower cut-off. We perform
spin polarised calculations by initializing different spins
on neighbouring magnetic ions; For paramagnetic order-
ing we initialise a zero spin on Cr ions. To represent anti-
ferromagnetic ordering, the unit cell is doubled along the
〈111〉 direction (see Fig1). To investigate structural in-
stabilities in the prototype cubic structure, we determine
its dynamical matrix using frozen phonon calculations.
We use a finite difference form of the first derivative to
compute an element of the force constant matrix:
Kiαjβ = −
TABLE I: Lattice constants of various oxides in the cubic
perovskite structure with experimental unit cell volumes.
Stress(GPa)
a (Å) PM FM AFM
LuCrO3 3.77 -4.712 3.302 1.429
YCrO3 3.79 -3.875 3.883 2.240
LaCrO3 3.88 -7.043 -0.083 -1.564
YFeO3 3.83
a -15.8 -5.9 2.5
BiCrO3 3.85
b -15.4 -9.7 -11.2
aRef.25.
bRef.4.
= −Fiα(ujβ = ∆)− Fiα(ujβ = −∆)
where Fiα is the Hellman-Feynman force acting on the
ith atom in α direction, and, ujβ the displacement of the
jth atom in β direction with respect to the equilibrium
structure. We used ∆=0.04 Å, about 1% of the lattice
constant. The dynamical matrix is then calculated from
the force constant matrix,
Diαjβ =
Kiαjβ√
, (2)
whose eigenvalues correspond to the square of the phonon
frequencies (ω2).
In periodic systems, the dynamical charge tensor or
Born effective charge tensor can be defined23 as the co-
efficient of proportionality between the macroscopic po-
larization created in direction β and a rigid displacement
of the sublattice of atoms j in direction α,
Z∗j,αβ = Ωo
∂P el
∂uj,α
, (3)
Ωo being the unit cell volume. The polarization is deter-
mined using the berry phase formalism24 as implemented
in the PWSCF package.
III. PROPERTIES OF THE CUBIC
PEROVSKITE STRUCTURE
We have determined electronic structure of RCrO3
compounds (R=Y, Lu, La) in the high symmetry cu-
bic structure with different magnetic orderings. This is
accomplished through calculations with different initial
guesses for atomic spin polarization and optimizing with
respect to spin density. All our calculations are for the ex-
perimental unit cell volumes, as ferroelectricity is known
to be sensitive to lattice constants or pressure (see lat-
tice constants listed in Table I). In many magnetic com-
pounds, a change of magnetic ordering causes a stress
which induces a structural distortion26. This concept of
‘magnetic stress’ was introduced to describe structural
phase transitions that are induced by magnetic order-
ing, and applied to materials with degenerate (usual eg)
orbitals27. In these materials as well, a change in mag-
netic ordering (with fixed lattice parameters) produces a
change in stress (see Table I). For LuCrO3 and YCrO3,
the introduction of spin polarization produces a change
from compressive stress in the paramagnetic phase to a
tensile stress, with stress being minimum in the antifer-
romagnetic structure (the lowest energy ordering). In
LaCrO3, the stress remains compressive with all three
magnetic orderings.
A. Electronic structure of YCrO3, LuCrO3 and
LaCrO3
1. Cubic Paramagnetic(PM) Structure
FIG. 2: Density of States for cubic PM YCrO3, LuCrO3 and
LaCrO3. The fermi level (indicated by a dashed line) has
been set to zero in all the three cases.
First, we present results for the highest symmetry cu-
bic structure with no spin polarization. Although this
state is experimentally inaccessible, it provides a useful
reference for understanding the spin-polarized structures
discussed later in the paper. The plotted energy range
is from -10 to 4 eV, and the lower lying semicore states
have been omitted for clarity. In the PM cubic YCrO3,
LuCrO3 and LaCrO3, (see Fig 2) there is high density
of electronic states at the Fermi level, driving the system
towards a Stoner instability28. This suggests that this
phase should be unstable with respect to spin polariza-
tion and/or structural distortions. The contribution of
various orbitals to the DOS can be understood better by
examining the orbital projected density of states (see Fig
3) which show that, the contribution between -8 to -3 eV
is mainly from Oxygen 2p orbitals. Cr d-orbitals con-
tribute predominantly to the peaks at the Fermi level.
In contrast, the contribution from the Lu d-orbitals is
substantial only 2eV above the Fermi level.
2. Cubic Ferromagnetic(FM) Structure
Ferromagnetic cubic structure is simulated by initial-
izing spins on both Cr ions in the same direction. In
FIG. 3: Orbital resolved density of states for cubic PM
LuCrO3. The high density of states at the Fermi level hints
that it is an unstable phase.
FIG. 4: Total and orbital resolved Density of states for cubic
ferromagnetic YCrO3, LuCrO3 and LaCrO3 with the Fermi
level set to zero in all the three cases.
all chromites studied here, the ferromagnetic structures
have a magnetic moment of 3 µB in accordance with the
Hund’s rule value expected for a d3 configuration. The
majority spins are represented by the solid line on the
positive Y axis, and the minority spins on the negative
Y axis. The introduction of spin polarization reduces
the energy by approximately 2 eV per unit cell. The
source of stabilization is clear from the density of states
(see Fig 4) which reveals opening of a gap at the Fermi
level. The states corresponding to non-magnetic atoms
are unchanged in comparison with PM ordering. The
down-spin Cr 3d states are split off from the O 2p states
creating a wide gap for the minority states. The up-spin
Cr 3d states hybridize with the O 2p states and there
is a very small gap for the majority carriers. The den-
sity of electronic states at the Fermi level is still finite
having a small contribution from the Cr d-orbitals. This
hints that either the ferromagnetic phase may not be the
most stable, and that either an antiferromagnetic (AFM)
spin arrangement could lower the energy of the system,
or that the cubic structure is unstable and a structural
distortion will lower the energy of the system. Since Cr3+
is a d3 ion, it is Jahn-Teller inactive, and the structural
distortions (if any) probably involve the A-cation (at the
corners), or the oxygen anions.
3. Antiferromagnetic Structure
We simulated antiferromagnetic structure by initializ-
ing antiparallel spins on the two Cr ions in the super-
cell. It is well known that, the superexchange between
eg orbitals of adjacent ions connected through oxygen
with a 180o metal-oxygen-metal bond angle, is much
stronger than the interaction between the correspond-
ing t2g orbitals, since the former is mediated by stronger
dpσ bonds as compared to the weaker dpπ bonds in the
latter17,18. So, we expect a superexchange interaction in
which there is a weak coupling between the t2g orbitals
of the adjacent Cr atoms giving rise to an antiferromag-
netic interaction. Further, this coupling will be stronger
in the cubic structure as the bond angle between Cr-O-Cr
is 1800, as compared to a distorted structure. From su-
FIG. 5: Density of states for antiferromagnetic YCrO3,
LuCrO3 and LaCrO3.
perexchange arguments applied to d3 configurations, the
structure with G-type antiferromagnetic ordering hav-
ing rhombohedral symmetry should be most stable. We
consider collinear spins assuming that the canting of the
spins would be small. We find that the AFM structure
is lower in energy by about 0.4 eV than the FM phase.
We note that this gain in energy by FM ordering with
respect to PM ordering is much more (around 2eV) than
the gain in energy in going from the FM to the AFM
structure (see Table III). LuCrO3, like YCrO3, is also
found to be insulating with the introduction of a gap at
the Fermi level (see Fig. 5). Both spin channels have
identical density of states consistent with an AFM spin
arrangement.
From the orbital projected density of states for AFM
arrangement, we find that the t2g orbitals of Cr are fully
occupied (and constitute the HOMO), whereas the eg
orbitals are unoccupied. Although the LUMO consists
of Cr d-orbitals (about 2eV above the Fermi level), the
Lu d states also appear within the same energy range
(see Fig. 6).
FIG. 6: Orbital resolved density of states for antiferromag-
netic LuCrO3.
B. Born Effective Charge (BEC)
The effective charge tensors have been calculated from
polarization differences between the perfect and distorted
structures in the AFM phase. The anomalous values of
Z* so obtained indicate that a large force is felt by a given
ion due to small macroscopic electric field, thus favoring a
tendency for off-centering and toward a polarized ground
state. The effective charge of the A-cation (see Table
II) is about the same for the three cases. For LaCrO3,
we find a larger BEC on Cr and one of the oxygen atom
moving along the bond. This is possibly because of larger
Cr-O bond length (arising from larger size of La cation,
see Table III) and correspondingly greater contribution
from the long-range charge transfer. We expect from this,
that the superexchange interaction in LaCrO3 should be
stronger as well.
TABLE II: The XX component of Born effective charge tensor
for AFM LaCrO3, LuCrO3 and YCrO3. Nominal charges are
indicated in brackets.
Z*A Z*B Z*Ox Z*Oy,z
LuCrO3 4.42(3) 3.43(3) -2.56(-2) -2.62(-2))
YCrO3 4.45(3) 3.44(3) -2.62(-2) -2.66(-2))
LaCrO3 4.5(3) 3.76(3) -3.82(-2) -2.22(-2))
TABLE III: Relative Energies of different Magnetic phases,
Cr-O bond lengths, and Neel’s temperatures for cubic
LuCrO3, YCrO3 and LaCrO3 (Energy of the PM phase has
been set to zero).
PM FM G-AFM Cr-O Bond length TN
LuCrO3 0.0 1.89 eV 2.3 eV 1.88(Å) 115 K
YCrO3 0.0 2.02 eV 2.4 eV 1.90(Å) 140 K
LaCrO3 0.0 2.04 eV 2.5 eV 1.94(Å) 282 K
aNeel’s temperature taken from Ref. 12,13.
bRef. 4.
IV. STRUCTURAL INSTABILITIES
A. Coupling with Magnetic Ordering
In order to represent G-type antiferromagnetic order-
ing which has been shown to be most favourable energet-
ically, we use a unit cell doubled along the 〈111〉 direc-
tion. We determine structural instabilities in this struc-
ture with different magnetic orderings. A single unit cell
has 10 atoms which results in 30 phonon branches: 3 ac-
coustic (which have zero frequency at k=(0,0,0)) and 27
optical, some of which are triply degenerate. We are in-
terested mainly in optical modes with imaginary phonon
frequencies corresponding to instabilities in the structure.
Doubling the unit cell along the 〈111〉 direction, gives us
access to zone boundary phonon modes (R-point) which
form the dominant structural instabilities in this struc-
ture, along with the zone-center modes.
In the paramagnetic phase, both YCrO3 and LuCrO3
exhibit a zone center instability at 116.5 and 144.8 cm−1
respectively, which is a polar mode (with Γ15 symme-
try) involving mainly the off centering of A-cation. This
instability, however, is absent in LaCrO3. We find two
instabilities in the FM phase: Γ15 and Γ25 modes. The
non-polar Γ25 mode involves oxygen displacements only,
and is strongly unstable in the FM phase. The Γ15 mode
involves the A-cation (rare earth ion) moving in a direc-
tion opposite to that of the oxygen cage and Cr atom
resulting in a ferroelectric polar structural distortion.
Note that the Cr atom moves in the same direction as
the oxygen ion, in contrast to the behavior of Ti ion in
BaTiO3
15 and PbTiO3
29, but similar to the behavior of
Mn in BiMnO3
11 and Cr in BiCrO3
With AFM spin arrangement, for LuCrO3, we find
three triply degenerate instabilities, at 339.5 cm−1, 145
cm−1 and the weakest at around 60 cm−1. For YCrO3,
the corresponding instabilities are at 309 cm−1, 140 cm−1
and 80 cm−1 respectively. The strongest instability (at
around 300 cm−1 for the two materials) has R25 symme-
try and corresponds to rotation of the corner connected
oxygen octahedra. The next instability is the ferroelec-
tric Γ15 mode (around 140 cm
−1). The weakest insta-
bility (60 cm−1 for LuCrO3 and 80 cm
−1 for YCrO3)
has R15 symmetry and involves displacement of the A-
cations (Lu and Y for our case) and small oxygen dis-
placements; these are antiparallel in neighbouring unit
cells. In LaCrO3, we find only two triply degenerate in-
stabilities. The first instability at around 220 cm−1 cor-
responding to the oxygen rotations (the R25 mode) and
the second close to 18 cm−1 having Γ15 symmetry (see
Fig 7).
FIG. 7: Eigenvectors of the unstable Γ point phonon modes:
Γ15 and Γ25 modes
For the rare earth chromites studied in this paper, un-
stable R-point modes in the AFM phase do not change
significantly with magnetic ordering. Γ point instabili-
ties in contrast depend strongly on magnetic ordering.
Only the high frequency phonons are affected in going
from para- to antiferromagnetic phase and all Γ15 and
R15 modes are softer in the FM phase.
The Γ25 mode which brings about a significant change
in the Cr-O-Cr as well as O-Cr-O bond angle shows a
spectacular change with magnetic ordering. This mode,
highly unstable in the ferromagnetic phase (at around
200 cm −1) becomes stable at around 50 cm−1 with
para- and antiferromagnetic orderings. This behavior,
although not as pronounced, is also seen for this mode
in LaCrO3. The R25 instability, also involving a change
in the Cr-O-Cr bond angle, is not affected by the change
in magnetic ordering possibly because the O-Cr-O bond
angle still remains unchanged.
Another significant change is observed for the R’25
mode, which involves a movement of the two B-cations
(Cr in our case) in opposite directions. After introduction
of spin-polarization, the R25’ mode (close to 250 cm
−1 for
PM phase) becomes more stable at around 400 cm−1 for
the FM and AFM phases.
We compare our results with the Bi-based biferroic
chromite, BiCrO3 and find a similar behavior of the Γ25
mode here as well. We thus attribute this behavior to
the B cation (Cr ion for the chromites) and expect it to
be their general behavior. To interpret the general trend
in phonon frequencies (see Fig. 8), the following rules
apply:
1. The modes which involve a change in the Cr-O-
Cr bond angle (as well as O-Cr-O bond angle) are more
stable in the AFM phases and are relatively less, or un-
stable, in the FM phase. This behaviour is seen in the
Γ25 and Γ15 modes for LuCrO3.
2. Secondly, the modes which involve a change in the
Cr-O bond length tend to harden with the introduction
of spin polarization, as observed for the R’25 and R’12
modes.
FIG. 8: Changes in phonon frequencies of cubic LuCrO3 with
different magnetic orderings. Insets show similar curves for
LaCrO3 and BiCrO3.
In order to study the effect of change in B-cation (see
Fig 9), we compare instabilities in YFeO3 with YCrO3.
We find that a change in magnetic ordering has an op-
posite effect on the unstable modes involving a change
in Fe-O-Fe bond angle. A spectacular change is the R’2
mode, an oxygen breathing mode, which softens in the
FM phase as compared to the PM and AFM phases. The
Γ25 mode also shows a different behaviour, showing a sta-
bilization with FM ordering. These differences are due
to the filled eg orbitals in Fe
3+, which are unoccupied in
Cr3+, leading to a much stronger superexchange inter-
action mediated by the eg orbitals, and have a different
geometry dependence.
B. Stabilization of Ferromagnetic Ordering
The spectacular change in the frequency of the Γ25
mode with FM ordering, prompts us to discuss whether
ferromagnetism can be stabilized in these chromites by
varying the Cr-O-Cr bond angle. Since the Γ25 mode is
unstable only in the FM phase, we want to study the
effect of freezing in a distortion of this mode. We have
calculated the total energy as a function of Γ25 displace-
ments in the 〈111〉 and 〈100〉 directions, for the FM and
AFM phases (see Fig 10). For LuCrO3, we observe a
FIG. 9: Changes in phonon frequencies with changes in B-
cation: YCrO3 and YFeO3.
crossover, at a displacement of 0.407 Å, beyond which the
FM phase is energetically favoured. The Cr-O-Cr bond
angle, at the crossover point is found to be 153◦, which is
significantly different from, the value suggested by Good-
enough and Kanamori for ferromagnetic superexchange
interaction (130◦) in 1951. On examining the Density of
states beyond the crossover point (see Fig. 11), we find
more significant hybridization between Cr and oxygens
for the FM phase. Secondly, the FM phase so stabilized
is found to be insulating.
FIG. 10: Energy vs displacement corresponding to the Γ25
mode for FM and AFM orderings in LuCrO3. FM becomes
more stable than the AFM state for rhombohedral distortions
greater than 0.407 Å.
C. Effect of Correlations
As mentioned earlier in the paper, the LDA+U method
has been successfully applied to describe the electronic
structure of sytems containing localized d and f elec-
trons where LDA sometimes leads to incorrect results16,
and recently it has been applied to obtain structural pa-
rameters that are in better agreement with experimental
FIG. 11: Density of states for FM and AFM phases at Γ25
displacement of 0.407 Å (see Fig. 10, where the FM state is
more stable).
results than LDA or GGA.7,30. In this work, we use a
value of U = 3.0 eV, adapted from work on full struc-
tural optimization of BiCrO3
7. With the introduction
of correlations through U parameter, we find that the
modes in FM phase do not change significantly. Our re-
sults for YCrO3 (see Fig 12) bear that only Γ25, R’2 and
R’12 modes are noticeably affected. Correlations lead to
softening of the R’2 and R’12 modes by 50 cm
−1 , and
tend to harden the Γ25 mode in the PM phase. In an
AFM spin arrangement, these R-point modes are hard-
ened by approximately 20 cm−1 and the Γ25 instability
is softened.
FIG. 12: Effect of correlations on the phonon modes of YCrO3
with different magnetic orderings. For each mode, data on the
left of the vertical dashed line represents estimates with GGA
and the data on the right represents estimates with GGA+U
(U=3.0 eV).
V. SUMMARY
In conclusion, we have determined structural instabil-
ities of LaCrO3, LuCrO3, YCrO3 and BiCrO3 in their
cubic perovskite structures with different magnetic or-
derings. Our finding that the G-type antiferromagnetic
ordering is most stable can be explained with superex-
change arguments. Ferroelectric structural instabilities
in the cubic structures involve A-cation (Lu or Y) dis-
placements, as indicated by the eigenvectors of the fer-
roelectric Γ15 modes and an anomalous BEC of the A-
cation. We find that certain phonon frequencies de-
pend sensitively on magnetic ordering: the modes in-
volving a change in bond-angle are stable (harder) with
the antiferro- and paramagnetic ordering than in the FM
state; on the other hand, the modes involving a change in
Cr-O bond length are softer in the paramagnetic phase
and comparable in the FM and AFM states. The Γ25
oxygen mode brings about a significant change in the Cr-
O-Cr bond angle and is highly unstable in the FM phase,
and corresponding structural distortion leads to stabiliza-
tion of ferromagnetic ordering in these chromites. Among
the competing structural instabilities the antiferrodis-
tortive instability (R25 mode) is the strongest. Electron
correlations are found to have little effect on the unstable
phonon modes, but result in a slight change in a few of
the stable phonon modes in the PM phase. We note that
the effects of magnetic ordering on structural instabilities
are quite different (in fact, opposite sometimes) in YFeO3
with respect to chromites, and a more detailed study is
required to understand such couplings in ferrites. Origin
of small polarization and/or local non-centrosymmetry14
is probably from the relatively weak ferroelectric instabil-
ities and their competition with various structural mag-
netic instabilities, and our work should be useful in for-
mulating a phenomenological analysis of the same.
VI. ACKNOWLEDGMENTS
Nirat Ray thanks JNCASR for Summer Research fel-
lowship Programme and Joydeep Bhattacharjee for dis-
cussions. UVW is thankful to Professor C N R Rao for
stimulating discussions and encouragement for this work
and acknowledges use of central computing facility and
financial support from the Centre for Computational Ma-
terials Science at JNCASR.
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  We use first-principles density functional theory-based calculations to
investigate structural instabilities in the high symmetry cubic perovskite
structure of rare-earth (R $=$ La, Y, Lu) and Bi-based biferroic chromites,
focusing on $\Gamma$ and $R$ point phonons of states with para-, ferro-, and
antiferromagnetic ordering. We find that (a) the structure with G-type
antiferromagnetic ordering is most stable, (b) the most dominant structural
instabilities in these oxides are the ones associated with rotations of oxygen
octahedra, and (c) structural instabilities involving changes in Cr-O-Cr bond
angle depend sensitively on the changes in magnetic ordering. The dependence of
structural instabilities on magnetic ordering can be understood in terms of how
super-exchange interactions depend on the Cr-O-Cr bond angles and Cr-O bond
lengths. We demonstrate how adequate buckling of Cr-O-Cr chains can favour
ferromagnetism. Born effective charges (BEC) calculated using the Berry phase
expression are found to be anomalously large for the A-cations, indicating
their chemical relevance to ferroelectric distortions.

Coupling between magnetic ordering and structural instabilities in perovskite
biferroics: A first-principles study
Nirat Ray and Umesh V. Waghmare
Theoretical Sciences Unit
Jawaharlal Nehru Centre for Advanced Scientific Research
Jakkur PO, Bangalore 560 064, India
(Dated: November 13, 2018)
We use first-principles density functional theory-based calculations to investigate structural insta-
bilities in the high symmetry cubic perovskite structure of rare-earth (R = La, Y, Lu) and Bi-based
biferroic chromites, focusing on Γ and R point phonons of states with para-, ferro-, and antiferro-
magnetic ordering. We find that (a) the structure with G-type antiferromagnetic ordering is most
stable, (b) the most dominant structural instabilities in these oxides are the ones associated with ro-
tations of oxygen octahedra, and (c) structural instabilities involving changes in Cr-O-Cr bond angle
depend sensitively on the changes in magnetic ordering. The dependence of structural instabilities
on magnetic ordering can be understood in terms of how super-exchange interactions depend on the
Cr-O-Cr bond angles and Cr-O bond lengths. We demonstrate how adequate buckling of Cr-O-Cr
chains can favour ferromagnetism. Born effective charges (BEC) calculated using the Berry phase
expression are found to be anomalously large for the A-cations, indicating their chemical relevance
to ferroelectric distortions.
I. INTRODUCTION
A ferroic is a material which exhibits spontaneous and
switchable ordering of electric polarization or magnetiza-
tion or elastic strain. Materials exhibiting more than one
of such orderings termed ‘multiferroics’- have recently be-
come the focus of much research1. Most of the biferroics
investigated in recent years are ABO3 oxides with per-
ovskite structure. The d0−ness or the zero occupancy
of tranistion metal B cation is known chemically to fa-
vor ferroelectricity2. Hence, the availability of transition
metal d -electrons in the perovskite oxides necessary for
magnetism, reduces the tendency for off-centering ferro-
electric distortions2 making multiferroics relatively rare.
How the ordering of d−electronic spins of the B cation in-
fluence ferroelectric or other competing structural insta-
bilities has not yet been explored and understood. The
coupling between magnetic ordering and structural in-
stabilities is expected to involve interesting physics and
is of direct relevance to technological applications3 such
as multiple state memory elements and novel memory
media.
There are at least three families of high tempera-
ture biferroic materials. Bi- based perovskite oxides,
like BiMnO3
4, BiCrO3
5, and BiFeO3
6,7 are known to
be promising biferroics. Ferroelectricity in these materi-
als arises from the stereochemical activity of the 6s lone
pair electrons of Bi. Hexagonal rare earth manganates
LnMnO3 and InMnO3 are biferroics which exhibit im-
proper or geometric ferroelectricity8,9,10,11. Rare earth
chromites LnCrO3
12,13 (with Ln = Ho, Er, Tm, Yb, Lu
or Y) have been recently shown to be biferroic; YCrO3
has been shown12 to exhibit canted antiferromagnetic be-
havior below 140 K and a ferroelectric transition around
473 K. Similarly, LuCrO3 becomes a canted antiferro-
magnet below 115 K, and is ferroelectric below 488 K13.
The absence of any ferroelctricity in LaCrO3, has been
attributed to the large size of the La3+ ion, in comparison
with Y3+.
However, small values of polarization reported for
these materials (about 2µC/cm2 for YCrO3
12 and
6µC/cm2 of BiFeO3
7) inspite of large A-cation off-
centering distortions remain a puzzle. More recently14,
a new concept of ‘local non-centrosymmetry’ in YCrO3
has been proposed to account for the small value of
polarization observed. Perovskite oxides are known to
have many competing structural instabilities15 and this
competition is further enriched by the magnetic insta-
bilities. The coupling and competition between various
magnetic and structural ordering can be partly respon-
sible for weak ferroelectricity or the possibility of local
non-centrosymmetry. Our goal here is to investigate this
issue through determination of various structural insta-
bilities for different magnetic orderings, with a focus on
rare earth chromites.
We present results of detailed electronic structure and
frozen phonon (at the Γ and R points) calculations for a
set of five materials (LuCrO3, YCrO3, LaCrO3, BiCrO3
and YFeO3) in the cubic phase with three different mag-
netic orderings (para-, ferro- and antiferromagnetic). In
Section II, we briefly describe the methods used in cal-
culations here. In section III, we report results for struc-
tural energetics, the electronic density of states (DOS)
and the Born Effective charges (BEC) for the cubic phase
of these materials with different magnetic orderings. In
Section IV, we report results for structural instabilities
in chromites and compare them with those in a related
compound YFeO3. Since the many-electron correlations
are important in magnetic oxides, we estimate their ef-
fects on the structural instabilities through use of the
Hubbard parameter U16. Using our results for structural
instabilities, we show how certain nonpolar structural
instabilities can cooperatively stabilize ferromagnetism.
Our work reveals how the structural instabilities of these
http://arxiv.org/abs/0704.1251v1
biferroic oxides depend on the size of A-cation (with the
same B-cation), and a change in B-cation. We interpret
the results using arguments based on superexchange17,
and the well-known Goodenough-Kanamori rules18. Fi-
nally, we summarize in Section V.
FIG. 1: Perovskite structure: Cell doubled along the 〈111〉
direction, to represent G-type antiferroamgnetic ordering.
II. METHODOLOGY
Our calculations are based on first-principles pseu-
dopotential based Density Functional Theory within a
generalized gradient approximation (GGA)19 as imple-
mented in the PWSCF package20. The interaction be-
tween ions and electrons is approximated with ultra-
soft pseudopotentials21 treating explicitly 11 electrons
[(n-1)s2 (n-1)p6 (n-1)d1 and ns2] in the valence shell of
Lu(n=6), La(n=6) and Y(n=5). We consider 6 valence
electrons for Oxygen [2s22p4] and 14 electrons for Cr
[3s23p63d54s1]. We used a plane wave basis with kinetic
energy cut off of 25 Ryd (150 Ryd) to represent wave
functions (density). For cubic structures, we sample the
Brillouin zone using a 5 × 5 × 5 Monkhorst Pack Mesh22,
and a denser mesh (6 × 6 × 6) and higher energy cut-off
(30 Ryd) for energy differences. Phonon frequencies cal-
culated with these larger parameters do not differ much
from those calculated with a lower cut-off. We perform
spin polarised calculations by initializing different spins
on neighbouring magnetic ions; For paramagnetic order-
ing we initialise a zero spin on Cr ions. To represent anti-
ferromagnetic ordering, the unit cell is doubled along the
〈111〉 direction (see Fig1). To investigate structural in-
stabilities in the prototype cubic structure, we determine
its dynamical matrix using frozen phonon calculations.
We use a finite difference form of the first derivative to
compute an element of the force constant matrix:
Kiαjβ = −
TABLE I: Lattice constants of various oxides in the cubic
perovskite structure with experimental unit cell volumes.
Stress(GPa)
a (Å) PM FM AFM
LuCrO3 3.77 -4.712 3.302 1.429
YCrO3 3.79 -3.875 3.883 2.240
LaCrO3 3.88 -7.043 -0.083 -1.564
YFeO3 3.83
a -15.8 -5.9 2.5
BiCrO3 3.85
b -15.4 -9.7 -11.2
aRef.25.
bRef.4.
= −Fiα(ujβ = ∆)− Fiα(ujβ = −∆)
where Fiα is the Hellman-Feynman force acting on the
ith atom in α direction, and, ujβ the displacement of the
jth atom in β direction with respect to the equilibrium
structure. We used ∆=0.04 Å, about 1% of the lattice
constant. The dynamical matrix is then calculated from
the force constant matrix,
Diαjβ =
Kiαjβ√
, (2)
whose eigenvalues correspond to the square of the phonon
frequencies (ω2).
In periodic systems, the dynamical charge tensor or
Born effective charge tensor can be defined23 as the co-
efficient of proportionality between the macroscopic po-
larization created in direction β and a rigid displacement
of the sublattice of atoms j in direction α,
Z∗j,αβ = Ωo
∂P el
∂uj,α
, (3)
Ωo being the unit cell volume. The polarization is deter-
mined using the berry phase formalism24 as implemented
in the PWSCF package.
III. PROPERTIES OF THE CUBIC
PEROVSKITE STRUCTURE
We have determined electronic structure of RCrO3
compounds (R=Y, Lu, La) in the high symmetry cu-
bic structure with different magnetic orderings. This is
accomplished through calculations with different initial
guesses for atomic spin polarization and optimizing with
respect to spin density. All our calculations are for the ex-
perimental unit cell volumes, as ferroelectricity is known
to be sensitive to lattice constants or pressure (see lat-
tice constants listed in Table I). In many magnetic com-
pounds, a change of magnetic ordering causes a stress
which induces a structural distortion26. This concept of
‘magnetic stress’ was introduced to describe structural
phase transitions that are induced by magnetic order-
ing, and applied to materials with degenerate (usual eg)
orbitals27. In these materials as well, a change in mag-
netic ordering (with fixed lattice parameters) produces a
change in stress (see Table I). For LuCrO3 and YCrO3,
the introduction of spin polarization produces a change
from compressive stress in the paramagnetic phase to a
tensile stress, with stress being minimum in the antifer-
romagnetic structure (the lowest energy ordering). In
LaCrO3, the stress remains compressive with all three
magnetic orderings.
A. Electronic structure of YCrO3, LuCrO3 and
LaCrO3
1. Cubic Paramagnetic(PM) Structure
FIG. 2: Density of States for cubic PM YCrO3, LuCrO3 and
LaCrO3. The fermi level (indicated by a dashed line) has
been set to zero in all the three cases.
First, we present results for the highest symmetry cu-
bic structure with no spin polarization. Although this
state is experimentally inaccessible, it provides a useful
reference for understanding the spin-polarized structures
discussed later in the paper. The plotted energy range
is from -10 to 4 eV, and the lower lying semicore states
have been omitted for clarity. In the PM cubic YCrO3,
LuCrO3 and LaCrO3, (see Fig 2) there is high density
of electronic states at the Fermi level, driving the system
towards a Stoner instability28. This suggests that this
phase should be unstable with respect to spin polariza-
tion and/or structural distortions. The contribution of
various orbitals to the DOS can be understood better by
examining the orbital projected density of states (see Fig
3) which show that, the contribution between -8 to -3 eV
is mainly from Oxygen 2p orbitals. Cr d-orbitals con-
tribute predominantly to the peaks at the Fermi level.
In contrast, the contribution from the Lu d-orbitals is
substantial only 2eV above the Fermi level.
2. Cubic Ferromagnetic(FM) Structure
Ferromagnetic cubic structure is simulated by initial-
izing spins on both Cr ions in the same direction. In
FIG. 3: Orbital resolved density of states for cubic PM
LuCrO3. The high density of states at the Fermi level hints
that it is an unstable phase.
FIG. 4: Total and orbital resolved Density of states for cubic
ferromagnetic YCrO3, LuCrO3 and LaCrO3 with the Fermi
level set to zero in all the three cases.
all chromites studied here, the ferromagnetic structures
have a magnetic moment of 3 µB in accordance with the
Hund’s rule value expected for a d3 configuration. The
majority spins are represented by the solid line on the
positive Y axis, and the minority spins on the negative
Y axis. The introduction of spin polarization reduces
the energy by approximately 2 eV per unit cell. The
source of stabilization is clear from the density of states
(see Fig 4) which reveals opening of a gap at the Fermi
level. The states corresponding to non-magnetic atoms
are unchanged in comparison with PM ordering. The
down-spin Cr 3d states are split off from the O 2p states
creating a wide gap for the minority states. The up-spin
Cr 3d states hybridize with the O 2p states and there
is a very small gap for the majority carriers. The den-
sity of electronic states at the Fermi level is still finite
having a small contribution from the Cr d-orbitals. This
hints that either the ferromagnetic phase may not be the
most stable, and that either an antiferromagnetic (AFM)
spin arrangement could lower the energy of the system,
or that the cubic structure is unstable and a structural
distortion will lower the energy of the system. Since Cr3+
is a d3 ion, it is Jahn-Teller inactive, and the structural
distortions (if any) probably involve the A-cation (at the
corners), or the oxygen anions.
3. Antiferromagnetic Structure
We simulated antiferromagnetic structure by initializ-
ing antiparallel spins on the two Cr ions in the super-
cell. It is well known that, the superexchange between
eg orbitals of adjacent ions connected through oxygen
with a 180o metal-oxygen-metal bond angle, is much
stronger than the interaction between the correspond-
ing t2g orbitals, since the former is mediated by stronger
dpσ bonds as compared to the weaker dpπ bonds in the
latter17,18. So, we expect a superexchange interaction in
which there is a weak coupling between the t2g orbitals
of the adjacent Cr atoms giving rise to an antiferromag-
netic interaction. Further, this coupling will be stronger
in the cubic structure as the bond angle between Cr-O-Cr
is 1800, as compared to a distorted structure. From su-
FIG. 5: Density of states for antiferromagnetic YCrO3,
LuCrO3 and LaCrO3.
perexchange arguments applied to d3 configurations, the
structure with G-type antiferromagnetic ordering hav-
ing rhombohedral symmetry should be most stable. We
consider collinear spins assuming that the canting of the
spins would be small. We find that the AFM structure
is lower in energy by about 0.4 eV than the FM phase.
We note that this gain in energy by FM ordering with
respect to PM ordering is much more (around 2eV) than
the gain in energy in going from the FM to the AFM
structure (see Table III). LuCrO3, like YCrO3, is also
found to be insulating with the introduction of a gap at
the Fermi level (see Fig. 5). Both spin channels have
identical density of states consistent with an AFM spin
arrangement.
From the orbital projected density of states for AFM
arrangement, we find that the t2g orbitals of Cr are fully
occupied (and constitute the HOMO), whereas the eg
orbitals are unoccupied. Although the LUMO consists
of Cr d-orbitals (about 2eV above the Fermi level), the
Lu d states also appear within the same energy range
(see Fig. 6).
FIG. 6: Orbital resolved density of states for antiferromag-
netic LuCrO3.
B. Born Effective Charge (BEC)
The effective charge tensors have been calculated from
polarization differences between the perfect and distorted
structures in the AFM phase. The anomalous values of
Z* so obtained indicate that a large force is felt by a given
ion due to small macroscopic electric field, thus favoring a
tendency for off-centering and toward a polarized ground
state. The effective charge of the A-cation (see Table
II) is about the same for the three cases. For LaCrO3,
we find a larger BEC on Cr and one of the oxygen atom
moving along the bond. This is possibly because of larger
Cr-O bond length (arising from larger size of La cation,
see Table III) and correspondingly greater contribution
from the long-range charge transfer. We expect from this,
that the superexchange interaction in LaCrO3 should be
stronger as well.
TABLE II: The XX component of Born effective charge tensor
for AFM LaCrO3, LuCrO3 and YCrO3. Nominal charges are
indicated in brackets.
Z*A Z*B Z*Ox Z*Oy,z
LuCrO3 4.42(3) 3.43(3) -2.56(-2) -2.62(-2))
YCrO3 4.45(3) 3.44(3) -2.62(-2) -2.66(-2))
LaCrO3 4.5(3) 3.76(3) -3.82(-2) -2.22(-2))
TABLE III: Relative Energies of different Magnetic phases,
Cr-O bond lengths, and Neel’s temperatures for cubic
LuCrO3, YCrO3 and LaCrO3 (Energy of the PM phase has
been set to zero).
PM FM G-AFM Cr-O Bond length TN
LuCrO3 0.0 1.89 eV 2.3 eV 1.88(Å) 115 K
YCrO3 0.0 2.02 eV 2.4 eV 1.90(Å) 140 K
LaCrO3 0.0 2.04 eV 2.5 eV 1.94(Å) 282 K
aNeel’s temperature taken from Ref. 12,13.
bRef. 4.
IV. STRUCTURAL INSTABILITIES
A. Coupling with Magnetic Ordering
In order to represent G-type antiferromagnetic order-
ing which has been shown to be most favourable energet-
ically, we use a unit cell doubled along the 〈111〉 direc-
tion. We determine structural instabilities in this struc-
ture with different magnetic orderings. A single unit cell
has 10 atoms which results in 30 phonon branches: 3 ac-
coustic (which have zero frequency at k=(0,0,0)) and 27
optical, some of which are triply degenerate. We are in-
terested mainly in optical modes with imaginary phonon
frequencies corresponding to instabilities in the structure.
Doubling the unit cell along the 〈111〉 direction, gives us
access to zone boundary phonon modes (R-point) which
form the dominant structural instabilities in this struc-
ture, along with the zone-center modes.
In the paramagnetic phase, both YCrO3 and LuCrO3
exhibit a zone center instability at 116.5 and 144.8 cm−1
respectively, which is a polar mode (with Γ15 symme-
try) involving mainly the off centering of A-cation. This
instability, however, is absent in LaCrO3. We find two
instabilities in the FM phase: Γ15 and Γ25 modes. The
non-polar Γ25 mode involves oxygen displacements only,
and is strongly unstable in the FM phase. The Γ15 mode
involves the A-cation (rare earth ion) moving in a direc-
tion opposite to that of the oxygen cage and Cr atom
resulting in a ferroelectric polar structural distortion.
Note that the Cr atom moves in the same direction as
the oxygen ion, in contrast to the behavior of Ti ion in
BaTiO3
15 and PbTiO3
29, but similar to the behavior of
Mn in BiMnO3
11 and Cr in BiCrO3
With AFM spin arrangement, for LuCrO3, we find
three triply degenerate instabilities, at 339.5 cm−1, 145
cm−1 and the weakest at around 60 cm−1. For YCrO3,
the corresponding instabilities are at 309 cm−1, 140 cm−1
and 80 cm−1 respectively. The strongest instability (at
around 300 cm−1 for the two materials) has R25 symme-
try and corresponds to rotation of the corner connected
oxygen octahedra. The next instability is the ferroelec-
tric Γ15 mode (around 140 cm
−1). The weakest insta-
bility (60 cm−1 for LuCrO3 and 80 cm
−1 for YCrO3)
has R15 symmetry and involves displacement of the A-
cations (Lu and Y for our case) and small oxygen dis-
placements; these are antiparallel in neighbouring unit
cells. In LaCrO3, we find only two triply degenerate in-
stabilities. The first instability at around 220 cm−1 cor-
responding to the oxygen rotations (the R25 mode) and
the second close to 18 cm−1 having Γ15 symmetry (see
Fig 7).
FIG. 7: Eigenvectors of the unstable Γ point phonon modes:
Γ15 and Γ25 modes
For the rare earth chromites studied in this paper, un-
stable R-point modes in the AFM phase do not change
significantly with magnetic ordering. Γ point instabili-
ties in contrast depend strongly on magnetic ordering.
Only the high frequency phonons are affected in going
from para- to antiferromagnetic phase and all Γ15 and
R15 modes are softer in the FM phase.
The Γ25 mode which brings about a significant change
in the Cr-O-Cr as well as O-Cr-O bond angle shows a
spectacular change with magnetic ordering. This mode,
highly unstable in the ferromagnetic phase (at around
200 cm −1) becomes stable at around 50 cm−1 with
para- and antiferromagnetic orderings. This behavior,
although not as pronounced, is also seen for this mode
in LaCrO3. The R25 instability, also involving a change
in the Cr-O-Cr bond angle, is not affected by the change
in magnetic ordering possibly because the O-Cr-O bond
angle still remains unchanged.
Another significant change is observed for the R’25
mode, which involves a movement of the two B-cations
(Cr in our case) in opposite directions. After introduction
of spin-polarization, the R25’ mode (close to 250 cm
−1 for
PM phase) becomes more stable at around 400 cm−1 for
the FM and AFM phases.
We compare our results with the Bi-based biferroic
chromite, BiCrO3 and find a similar behavior of the Γ25
mode here as well. We thus attribute this behavior to
the B cation (Cr ion for the chromites) and expect it to
be their general behavior. To interpret the general trend
in phonon frequencies (see Fig. 8), the following rules
apply:
1. The modes which involve a change in the Cr-O-
Cr bond angle (as well as O-Cr-O bond angle) are more
stable in the AFM phases and are relatively less, or un-
stable, in the FM phase. This behaviour is seen in the
Γ25 and Γ15 modes for LuCrO3.
2. Secondly, the modes which involve a change in the
Cr-O bond length tend to harden with the introduction
of spin polarization, as observed for the R’25 and R’12
modes.
FIG. 8: Changes in phonon frequencies of cubic LuCrO3 with
different magnetic orderings. Insets show similar curves for
LaCrO3 and BiCrO3.
In order to study the effect of change in B-cation (see
Fig 9), we compare instabilities in YFeO3 with YCrO3.
We find that a change in magnetic ordering has an op-
posite effect on the unstable modes involving a change
in Fe-O-Fe bond angle. A spectacular change is the R’2
mode, an oxygen breathing mode, which softens in the
FM phase as compared to the PM and AFM phases. The
Γ25 mode also shows a different behaviour, showing a sta-
bilization with FM ordering. These differences are due
to the filled eg orbitals in Fe
3+, which are unoccupied in
Cr3+, leading to a much stronger superexchange inter-
action mediated by the eg orbitals, and have a different
geometry dependence.
B. Stabilization of Ferromagnetic Ordering
The spectacular change in the frequency of the Γ25
mode with FM ordering, prompts us to discuss whether
ferromagnetism can be stabilized in these chromites by
varying the Cr-O-Cr bond angle. Since the Γ25 mode is
unstable only in the FM phase, we want to study the
effect of freezing in a distortion of this mode. We have
calculated the total energy as a function of Γ25 displace-
ments in the 〈111〉 and 〈100〉 directions, for the FM and
AFM phases (see Fig 10). For LuCrO3, we observe a
FIG. 9: Changes in phonon frequencies with changes in B-
cation: YCrO3 and YFeO3.
crossover, at a displacement of 0.407 Å, beyond which the
FM phase is energetically favoured. The Cr-O-Cr bond
angle, at the crossover point is found to be 153◦, which is
significantly different from, the value suggested by Good-
enough and Kanamori for ferromagnetic superexchange
interaction (130◦) in 1951. On examining the Density of
states beyond the crossover point (see Fig. 11), we find
more significant hybridization between Cr and oxygens
for the FM phase. Secondly, the FM phase so stabilized
is found to be insulating.
FIG. 10: Energy vs displacement corresponding to the Γ25
mode for FM and AFM orderings in LuCrO3. FM becomes
more stable than the AFM state for rhombohedral distortions
greater than 0.407 Å.
C. Effect of Correlations
As mentioned earlier in the paper, the LDA+U method
has been successfully applied to describe the electronic
structure of sytems containing localized d and f elec-
trons where LDA sometimes leads to incorrect results16,
and recently it has been applied to obtain structural pa-
rameters that are in better agreement with experimental
FIG. 11: Density of states for FM and AFM phases at Γ25
displacement of 0.407 Å (see Fig. 10, where the FM state is
more stable).
results than LDA or GGA.7,30. In this work, we use a
value of U = 3.0 eV, adapted from work on full struc-
tural optimization of BiCrO3
7. With the introduction
of correlations through U parameter, we find that the
modes in FM phase do not change significantly. Our re-
sults for YCrO3 (see Fig 12) bear that only Γ25, R’2 and
R’12 modes are noticeably affected. Correlations lead to
softening of the R’2 and R’12 modes by 50 cm
−1 , and
tend to harden the Γ25 mode in the PM phase. In an
AFM spin arrangement, these R-point modes are hard-
ened by approximately 20 cm−1 and the Γ25 instability
is softened.
FIG. 12: Effect of correlations on the phonon modes of YCrO3
with different magnetic orderings. For each mode, data on the
left of the vertical dashed line represents estimates with GGA
and the data on the right represents estimates with GGA+U
(U=3.0 eV).
V. SUMMARY
In conclusion, we have determined structural instabil-
ities of LaCrO3, LuCrO3, YCrO3 and BiCrO3 in their
cubic perovskite structures with different magnetic or-
derings. Our finding that the G-type antiferromagnetic
ordering is most stable can be explained with superex-
change arguments. Ferroelectric structural instabilities
in the cubic structures involve A-cation (Lu or Y) dis-
placements, as indicated by the eigenvectors of the fer-
roelectric Γ15 modes and an anomalous BEC of the A-
cation. We find that certain phonon frequencies de-
pend sensitively on magnetic ordering: the modes in-
volving a change in bond-angle are stable (harder) with
the antiferro- and paramagnetic ordering than in the FM
state; on the other hand, the modes involving a change in
Cr-O bond length are softer in the paramagnetic phase
and comparable in the FM and AFM states. The Γ25
oxygen mode brings about a significant change in the Cr-
O-Cr bond angle and is highly unstable in the FM phase,
and corresponding structural distortion leads to stabiliza-
tion of ferromagnetic ordering in these chromites. Among
the competing structural instabilities the antiferrodis-
tortive instability (R25 mode) is the strongest. Electron
correlations are found to have little effect on the unstable
phonon modes, but result in a slight change in a few of
the stable phonon modes in the PM phase. We note that
the effects of magnetic ordering on structural instabilities
are quite different (in fact, opposite sometimes) in YFeO3
with respect to chromites, and a more detailed study is
required to understand such couplings in ferrites. Origin
of small polarization and/or local non-centrosymmetry14
is probably from the relatively weak ferroelectric instabil-
ities and their competition with various structural mag-
netic instabilities, and our work should be useful in for-
mulating a phenomenological analysis of the same.
VI. ACKNOWLEDGMENTS
Nirat Ray thanks JNCASR for Summer Research fel-
lowship Programme and Joydeep Bhattacharjee for dis-
cussions. UVW is thankful to Professor C N R Rao for
stimulating discussions and encouragement for this work
and acknowledges use of central computing facility and
financial support from the Centre for Computational Ma-
terials Science at JNCASR.
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