Advances in Ma teri als Physics and Chemistry, 2012, 2, 1-4
doi:10.4236/ampc.2012.24B001 Published Online December 2012 (http://www.Sci RP.org/journal/ampc)
A First-Principles Study of Structure-Property Correlation
and the Origin of Ferrimagnetism in Gallium Ferrite
Amritendu Roy1, Ashish Garg1, Rajendra Prasad2, Sushil Auluck3
1Department of M ater ial s Science an d Engi neering, Indian Institute of Technology Kanpur, Kanpur, India
2Department of Physics, Indian Institute of Technology Kanpur, Kanpur, India
3National Physical Laboratory, Dr. K. S. Krishnan Marg, New Delhi, India
Received 2012
ABSTRACT
A fir st-prin ciples study of structure property correlation and the origin of ferrimagnetism is presented based on LSDA+U method. In
particu lar, the resu lts for the grou nd state stru cture, electron ic band structure, den sity of states, Born effecti ve charges, spontaneous
pol arization and cationic disorder are discussed. The calculations were done using Vienna ab-initio simulation package (VASP) with
projector augmented wave method. We find that the ground state structure is orthorhombic and insulating having A-type antiferro-
magnetic sp in configuration. The cationic disorder is found to play an important role. Although the cationic site disorder is not spon-
taneous in the ground state, interchange of octahedrally coordinated Ga2 and Fe2 sites is most favored. We find that ferrimagnetism
in gallium ferrite is p rimaril y due to this exch ange between Ga-Fe si tes such th at Fe spins at Ga1 and Ga2 si tes are antiferro magneti-
cally aligned while maintaining ferromagnetic coupling between Fe spins at Ga1 and Fe1 sites as well as between Fe spins at Ga2
and Fe2 sites. Further, the partial density of states shows noticeable hybridization of Fe 3d, Ga 4s, Ga 4p and O 2p states indicating
some covalent character of Ga/Fe-O bonds. However, the charge density and electron localization functions show largely the ionic
character of these bonds. Our calculation predicts spontaneous polarization of ~59 μC/cm2 along b-axis.
Keywords: Gallium Ferrite; LSDA+U; Spontaneous Polarization; Ferrimagnetism; Cation Site Disorder
1. Introduction
Gallium ferrite (GaFeO3 or GFO) is a room temperature pie-
zoelectric an d a ferrimagnet who se magnetic transit ion temper-
ature (TC) is slightly lower than room temperature [1] but tuna-
ble to room temperature and beyond by tailoring the composi-
tion [1,2] and processing conditions.[1-3] Thus, composition-
ally modulated GFO is a promising candidate for room temper-
ature magn etoelectric applications.
Early studies on GFO [2] predicted simultaneous piezoelec-
tricity and ferromagnetism. Structural characterization using
x-ray [1,4,5] and neutron [1,4,6] diffraction techniques con-
cluded an orthorhombic structure (Space Group: Pc21n) with
eight formula units per unit cell, is stable over a wide tempera-
ture domain (4K-700K)[1,7]. The unit cell comprises of two
types of Ga (Ga1 and Ga2) and Fe (Fe1 and Fe2) ions and six
types of O (O1, O2,….O6) ions.[1] The above studies also
suggest substantial cationic site disorder i.e. some of the Ga
sites are always occupied by Fe ions and vice-versa.[1] How-
ever, magnetic behavior of GFO had been a matter of uncer-
tainty for a long time. Initial prediction of ferromagnetic order-
ing [2] was challen ged by Franke l et al.[8] who using high field
Mössbauer spectroscopy and macroscopic magnetic measure-
ments proved collinear ferrimagnetism in GFO. Ferrimagnetic
ordering has been further demonstrated by almost all subse-
quent studies using neutron diffraction technique.[1,4] Piezoe-
lectricity in GFO, on the other hand, has been hardly studied
with few exceptions showing that the piezoelectric constants
are almost double to that of quartz [9] which is attributed to the
asymmetric Ga1-O tetrahedron in the GFO unit cell. [10]
In spite of extensive experimental studies, first-principles cal-
culations on GFO, have not been carried out much, partly due to
the co mplex cr ystal st ru ctu re with substantially large number of
ions in the unit cell having partial site occupancies of the cations.
However, such type of studies have been quite successful in
predicting and analyzing structure-property correlations in
complex mat erial s ystems an d in th is regard, a detailed study on
GFO would be particularly interesting to probe the hitherto
disputed issues such as the ground state structure, magnetic
structure and the piezoelectric response of the material. Our
study, using first-principles calculati ons, shows that the ground
state structure of GFO is A-type antiferromagnetic. The calcu-
lations indicate the presence of large spontaneous polarization
(Ps) of ~ 59 µC/cm2 along crystallographic b-axis. Finally, we
predi ct that th e observed ferri magnetis m is solely due the inhe-
rent cation site disorder in the material.
2. Calculation Methodolo gy
We used Vienna ab -initio simulation package (VASP ) [11] with
projector augmented wave method (PAW) [12] in our work.
The Kohn-Sham equation [13] was solved using local spin
density approximation (LSDA+U) [14] with Hubbard parame-
ter, U = 5 eV, and the exchange interacti on, J = 1 eV. Our cal-
culations for the determination of the ground states structure are
based on the stoichiometric GFO assuming no partial occupan-
cies of the co nstituent ions with startin g parameters taken fro m
previo us literatur e.[1] We con sidered 3 valence elect rons of Ga

A. ROY ET AL.
2
(4s24p1), 8 for F e (3d74s1) and 6 for O (2s22p4) ions. Structural
optimization was carried out using Monkhorst-Pack [15] 7 × 7
× 12 mesh. Born effective charges, and spontaneous polariza-
tion for the ground state structure were calculated using Berry
phase m e t hod. [ 16 ]
3. Results and Discussions
3.1. Structural Optimization: Ground State Structure
Though previous literature predicts the ground state structure of
GFO to be antiferromagnetic [1,17], possible antiferromagnetic
configurations have not been explored. Therefore, we started
with four possible antiferromagnetic spin configurations as
shown schematically in Figure 1(a)-(d) i.e. AFM-1, AFM-2,
AFM-3 and AFM-4, to arrive at the ground state spin ordering.
It should be noted that other possible antiferromagnetic spin
configurations were found to be equivalent to one of the above
configurations. Computation of the total energies of the above
structures shows, that the energetically, AFM-3 > AFM-4 >
AFM-2 > AFM -1 . Therefore, we can co n clude th at AFM-1 spi n
configuration is the most favored configuration and all the fur-
ther calculations were performed o n AFM-1 structure.
Structural optimization by relaxing the ionic positions, lattice
parameters and unit cell shape further reveals that the ground
state structure retains the original Pc21n symmetry observed
experimentally [1,18]. The calculated ground state lattice pa-
rameters: a = 8.6717 Å, b = 9.3027 Å and c = 5.0403 Å are in
good agreement with previous experimental results.[1,7,18]
Calculated ionic positions show that Fe1 and Fe2 ions are lo-
cated on alternate planes parallel to ac-plane. Since Fe1 and
Fe2 have antiparallel spin configurations and are situated on
alternate parallel planes, we conclude (see Figure 1) that the
ground state magnetic structure of GFO is A-type antiferro-
magnetic. Using the ground state structural data, we further
calculat ed catio n-oxygen bond length and Fe/Ga-O-Fe/Ga bond
angles. We found that cation-oxygen polyhedra are significantly
distorted which could contribute to the observed piezoelectric
behavior. Calculation of Fe-O-Fe, on the other hand, could be
linked with the super-exchange interaction between O and
neighboring Fe3+ ions. In general, larger the Fe-O-Fe bond an-
gle results in stronger antiferromagnetic super-exchange.[19]
The maximum value of Fe1-O1 -Fe2, bond angle is ~168.54o
while other angles are: Fe1-O3-Fe2, 123.13o and Fe1-O5-Fe2,
12 6.23o, respectively. Such large Fe-O- Fe bond angles (larger
than 90°) results in antiferromagnetic ordering of Fe1 and Fe2
ions and lead to not iceable super-exchange i nteraction b etween
Fe and O ions which is reflected in significantly large magnetic
moments of O. Similar large bond angles among Fe1-O-Ga2
and Fe2-O-Ga1 ensure that Fe ions occupying Ga sites due to
site disorder would therefore form strong antiferromagnetic
spin arrangement between Fe at Ga1 with Fe2 and Fe at Ga2
with Fe1, respectively. Therefore we can conclude that any Fe
ion, due to partial site occupancy, occupying Ga1 site would
have antiferromagnetic ordering with Fe2 and Fe at Ga2 site
and would be antiferromagnetically coupled with Fe1 site.
3.2. Electronic Band Struc ture, Density of States and
Bonding
Figure 2 shows the electronic band structure along high sym-
metry directions and total density of states of GFO. It is found
that GFO possesses a direct band gap (Eg) of ~2.0 eV (Γ- Γ).
However, experimental studies reports a band gap of 2.7-3.0
eV.[20] The difference bet ween ou r results and the exper imen-
tal results is primarily due to underestimation of band gap by
the LSDA techn ique which is very co mmon in electron ic str uc-
ture calculation of oxides. [21] Cation site disorder, inherent to
the experimental structure of GFO might also contribute to the
effect of enhancing the band gap. [22] The angular momentum
character of the bands over different energy domains can be
identified by computing the partial density of states. Our calcu-
lations shows that the upper most part of the valence band is
mainly composed of Fe 3d and O 2p states. Beyond the Fermi
level, a narrow energy band (1.77 eV to 2.45 eV) comprises
mainly o f Fe 3d character, while the h ighest en ergy window has
contributions from Fe 3d, Ga 4s, Ga 4p and O 2p states. Calcu-
lations of the partial density of states demonstrate significant
hybridization of Fe 3d and O 2p states throughout the upper-
most part of the valence band which further indicates presence
of significant covalent character between Fe-O bonds.
Figure 1. Schematic representation of different antiferromagnetic
ordering (a) AFM-1, (b) AFM-2, (c) AFM-3 and (d) AFM-4
Figure 2 . Electronic band structure (lef t) and total densi ty of states
of GFO with orthorhom b ic sym metry.

A. ROY ET AL.
3
Analysis of the chemical bonding can further be carried out
by plotting electron localization function (ELF) which gives a
measure of the local influence of the Pauli repulsion on the
behavior of the electrons. A large value of ELF function indi-
cates space with anti-parallel spin configuration .[23] Our cal-
culation of ELF distribution in GFO unit cell (not shown here)
depicts maximum ELF value at O sites and small values at Fe
and Ga sites i nd icati ng ch arge tra n sfer fro m Fe/ Ga to O sites . A
complete charge transfer was found between Fe2 and O3 ions.
Similar charge transfer was also noticed between Fe1 and O1,
O2 ions. Thus, we can predict that Fe-O bonds in GFO are
mostly ionic. In comparison, finite value of ELF between O and
Ga1 and Ga2 i ndicat e s ome degree of covalent characteris tics.
3.3. Cation Site Disorder and Ferrimagnetism
Calculation of the magnetic moments of the constituent ions in
the ground state shows that Fe1 and Fe2 ions have magnetic
moments of + 4.05 µB and - 4.04 µB, resp ectively. We find that
the magnitude of moments agrees reasonably well with the
experimental data. [1] It was also found that the oxygen ions
surrounding the Fe ions manifest small but finite magnetic
moments attributed to super-exchange interactions with the
surrounding Fe ions.
Till now, our calculations have been based on the ground
state structure containing no partial site occupancies of the
cations which is however, in contrast with the experimental
repor ts showing significant Fe occu pancies at the Ga sites. Thus,
to probe the effect of cation site disorder on the magnetic cha-
racteristi cs of GFO, we selectively interchanged F e and Ga sites
and computed total energy of the system. Since, GFO unit-cell
contains four ions of each type of cation, such an interchange
would result in ¼th site occupancy of Fe ions at Ga sites and
vice-versa. Calculations of total energy of these disordered
structures show that partial site occupancy is not favored in the
ground state, also observed previously by Han et al. [17] How-
ever, it was found that among various possible cases of site
disorders, Fe2 ions preferentially occupying Ga2 sites is most
probable since ∆E, the energy difference with respect to the
ground state in that case is minimum. Although these energy
differences may be affected by the computational methodology,
the magnitude of the available thermal energy at room temper-
ature (kT ~25 meV) is of the order of the energy difference for
Fe2-Ga2 site disorder indicating towards the role of thermally
originated defects. An important observation of the inclusion of
cation site disorder in the calculation would be on the modifica-
tion of the local magnetic moments. It was observed that upon
interch angin g Fe1 and Ga1 s ites, the avera ge magneti c moment
of Fe ion at Ga1 site becomes 3.99 µB. On the other hand, the
magnetic momen t of Fe ion at G a2 site becomes 4 .11 µB when
Fe2 and Ga2 sites are interchanged. In order to analyze the
observed ferrimagnetism as against the antiferromagnetic or-
dering in the ground state, we further calculated the total mag-
netic moment of the disordered structure, using the partial site
occupancies from the Rietveld refinement data [24] and taking
the magnetic moments for different cation sites, from our cal-
culation with site disorder. We estimated net magnetic mo ment
of 0.24 µB/ Fe site which agrees quite well with experimental
results. [25] Therefore, we conclude that ferrimagnetism in
GFO is solely due to site disorder in the structure.
3.4. Born Effectiv e C harge and Spontaneous
Polariza t ion
Born effective charges (BEC or Z*), are important quantities in
characterizing the piezoelectrics, ferroelectrics and multifer-
roics since they relate the lattice displacements and electric
field and therefo re give a measure long range Coulomb interac-
tion, whose competition with the short range forces leads to the
ferroelectric transition. Recent first-principles calculations
show anomalously large BECs for some ions in common fer-
roelectrics [21] which are often explained as manifestation of
strong covalent char acter of bonds bet ween the sp ecific ion s. In
GFO, fro m the E LF pl ots, we find th at charge sharin g between
the Ga/Fe and O ions in cation-oxygen bonds is insignificant
while the structural data indicates large distortion of the ca-
tion-oxygen polyhedra. Since ferroelectric and/ piezoelectric
responses are combined manifestations of structural distortions
and effecti ve charges of const ituent ions [26] it is imperati ve to
calculate the Born effective charges of ions in GFO. Here we
calculate the BEC tensors of nonequivalent ions of GFO by
slightly displacing each ion, one at a time, along three axes of
Cartesian co-ordinates and then calculating the resulting dif-
ference in p ol arization, using Berry phase method. Table 1 lists
the diagonal elements of BEC tensor for each ion. It is noticed
that the principal elements of BEC tensor for Ga1 are close to
the nomin al ionic ch arge of Ga i.e., +3 . Thus we pred ict that all
the Ga1-O bonds are mainly ionic in nature. On the other hand,
Ga2 develops a maximum effective charge of 3.53, ~ 18%
higher with respect to its nominal charge. In contrast, both Fe1
and Fe2 ions show much higher increase in the effective
charges, 36% and 27% respectively. Oxygen ions show a
maximum reduction of 39.5% with respect to the nominal ionic
charge.
Structural analysis of GFO shows that while the constituent
ion s possess inversi on symmetry i n the crystallographic a and c
directions, the inversion symmetry is lost in crystallographic
b-direction. Since existence of spontaneous polarization (Ps) is
manifestation of the lack of inversion symmetry, we argue that
the direction of spontaneous polarization in GFO is only along
the b-direction. Using the Born effective charges from Tabl e 1,
we further calculated the magnitude of Ps of GFO as ~ 58.63
µC/cm2 along the b-direction which is an order of magnitude
larger than that predicted by Arima et al.[1] who neglected the
role of effective charges and other ions in determining the
magnitude of Ps. A calculation on the relative contribution of
Table 1. Principal elements of Born effective charge tensors of
constituent ions along with their nominal charges, in GFO.
Ion Nominal ionic
charge (e) Z* (e)
Zxx Zyy Zzz
Ga1
+3
3.01
3.11
2.99
Ga2 +3 3.5 7 3.16 3.53
Fe1
+3
3.66
3.78
4.08
Fe2 +3 3.68 3 .3 8 3.82
O1
-2
-2.29
-2.58
-2.79
O2 -2 -2.45 -2.29 -2.41
O3
-2
-2.54
-2.30
-2.75
O4 -2 -2.27 -2.85 -2.17
O5
-2
-2.50
-2.16
-2.79
O6 -2 -2.32 -2.08 -2.40

A. ROY ET AL.
4
individual ions shows that the contribution of Ga1 is largest.
However, it is balanced by opposite contributions from Fe1, O1,
O2 and O6. Structural data further substantiate that these ions
are the most as ymmetrical ly placed around the i nversion center
whereas Ga2 and Fe2 cations maintain nearly centrosymmetric
configuration and contribute minimum to the total polarization.
Therefore, we predict that Ps in GFO is primarily contributed
from Ga1, Fe1, O1, O2 and O6 ions.
4. Summary
To summarize, we show, using first-principles calculations
using LSDA+U that orthorhombic Pc21n symmetry with
A-type antiferromagnetic spin configuration is the ground state
structure of gallium ferrite. The calculated ground state lattice
parameters, bond strength and bond angles agree well with the
reported as well as our experimental results. Electronic density
of states showed hybridization among Fe 3d, Ga 4s, Ga 4p and
O 2p states. From the electron localization function (ELF) cal-
culation, we find almost complete charge transfer between Fe2
and O3 and F e1 and O1 , O2 ions suggesti ng th at F e-O bonds in
GFO have mostly ionic character. Calculations also showed a
spontaneous polarization of ~ 59 µC/cm2 along b-direction i.e.
[010]-axis of GFO, attributed to the non-centrosymmetry and
effective charges of Ga1, Fe1, O1, O2 and O6 ions. We find
that the cation site disorder, although not preferred in the
ground state, is the most favored configuration in the disordered
state. An examination of the role of cation site disorder on
magnetic structure of GFO shows modification of the local
magnetic str uctur e with alter ed magnetic moments of Fe ions at
Ga site. This suggests that ferrimagnetism in GFO is solely due
to the site disorder.
5. Acknowledgements
The work was supported by Department of Science and Tech-
nology, Govt. of India through project number SR/S2/CMP-
0098/2010. SA thanks NPL for the J C Bose Fellowship.
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