Tri-layer superlattices: A route to magnetoelectric multiferroics?
Alison J. Hatt and Nicola A. Spaldin
Materials Department, University of California
(Dated: October 22, 2018)
We explore computationally the formation of tri-layer superlattices as an alternative approach for
combining ferroelectricity with magnetism to form magnetoelectric multiferroics. We find that the
contribution to the superlattice polarization from tri-layering is small compared to typical polar-
izations in conventional ferroelectrics, and the switchable ferroelectric component is negligible. In
contrast, we show that epitaxial strain and “negative pressure” can yield large, switchable polar-
izations that are compatible with the coexistence of magnetism, even in materials with no active
ferroelectric ions.
PACS numbers:
The simultaneous presence of ferromagnetism and fer-
roelectricity in magnetoelectric multiferroics suggests
tremendous potential for innovative device applications
and exploration of the fundamental physics of coupled
phenomena. However, the two properties are chemically
contra-indicated, since the transition metal d electrons
which are favorable for ferromagnetism disfavor the off-
centering of cations required for ferroelectricity [1]. Con-
tinued progress in this burgeoning field rests on the iden-
tification of alternative mechanisms for ferroelectricity
which are compatible with the existence of magnetism
[2, 3]. Mechanisms discovered to date include the incor-
poration of stereochemically active lone pair cations, for
example in BiMnO3 [4, 5] and BiFeO3 [6, 7], geometric
ferroelectricity in YMnO3 [8], BaNiF4 [9, 10] and related
compounds, charge ordering as in LuFe2O4 [11, 12], and
polar magnetic spin-spiral states, of which TbMnO3 is
the prototype [13]. However, there are currently no single
phase multiferroics with large and robust magnetization
and polarization at or near room temperature [14].
The study of ferroelectrics has been invigorated over
the last few years by tremendous improvements in the
ability to grow high quality ferroelectric thin films with
precisely controlled composition, atomic arrangements
and interfaces. In particular, the use of compositional
ordering that breaks inversion symmetry, such as the
layer-by-layer growth of three materials in an A-B-C-
A-B-C... arrangement, has produced systems with en-
hanced polarizations and large non-linear optical re-
sponses [15, 16, 17, 18]. Here we explore computation-
ally this tri-layering approach as an alternative route to
magnetoelectric multiferroics. Our hypothesis is that the
magnetic ions in such a tri-layer superlattice will be con-
strained in a polar, ferroelectric state by the symmetry of
the system, in spite of their natural tendency to remain
centrosymmetric. We note, however, that in previous
tri-layering studies, at least one of the constituents has
been a strong ferroelectric in its own right, and the other
constituents have often contained so-called second-order
Jahn-Teller ions such as Ti4+, which have a tendency to
off-center. Therefore factors such as electrostatic effects
from internal electric fields originating in the strong fer-
roelectric layers [19], or epitaxial strain, which is well
established to enhance or even induce ferroelectric prop-
erties in thin films with second-order Jahn-Teller ions
[6, 20, 21], could have been responsible for the enhanced
polarization in those studies.
We choose a [001] tri-layer superlattice of perovskite-
structure LaAlO3, LaFeO3 and LaCrO3 as our model
system (see Fig. 1, inset.) Our choice is motivated by
three factors. First, all of the ions are filled shell or filled
sub-shell, and therefore insulating behavior, a prerequi-
site for ferroelectricity, is likely. Second, the Fe3+ and
Cr3+ will introduce magnetism. And third, none of the
parent compounds are ferroelectric or even contain ions
that have a tendency towards ferroelectric distortions, al-
lowing us to test the influence of trilayering alone as the
driving force for ferroelectricity. For all calculations we
use the LDA+U method [22] of density functional the-
ory as implemented in the Vienna Ab-initio Simulation
Package (VASP) [23]. We use the projector augmented
wave (PAW) method [24, 25] with the default VASP po-
tentials (La, Al, Fe pv, Cr pv, O), a 6x6x2 Monkhorst-
Pack mesh and a plane-wave energy cutoff of 450 eV. Po-
larizations are obtained using the standard Berry phase
technique [26, 27] as implemented in VASP. We find that
U/J values of 6/0.6 eV and 5/0.5 eV on the Fe and Cr
ions respectively, are required to obtain insulating band
structures; smaller values of U lead to metallic ground
states. These values have been shown to give reasonable
agreement with experimental band gaps and magnetic
moments in related systems [28] but are somewhat lower
than values obtained for trivalent Fe and Cr using a con-
strained LDA approach [29]. We therefore regard them as
a likely lower limit of physically meaningful U/J values.
(Correspondingly, since increasing U often decreases the
covalency of a system, our calculated polarizations likely
provide upper bounds to the experimentally attainable
polarizations).
We begin by constraining the in-plane a lattice con-
stant to the LDA lattice constant of cubic SrTiO3 (3.85
Å) to simulate growth on a substrate, and adjust the out-
of-plane c lattice constant until the stress is minimized,
with the ions constrained in each layer to the ideal, high-
symmetry perovskite positions. We refer to this as our
reference structure. (The LDA (LDA+U) lattice con-
http://arxiv.org/abs/0704.0781v3
stants for cubic LaAlO3 (LaFeO3, LaCrO3) are 3.75, 3.85
and 3.84 Å, respectively. Thus, LaAlO3 is under tensile
strain and LaFeO3/LaCrO3 are unstrained.) The cal-
culated total density of states, and the local densities of
states on the magnetic ions, are shown in Figure 2; a band
gap of 0.32 eV is clearly visible. The polarization of this
reference structure differs from that of the corresponding
non-polar single-component material (for example pure
LaAlO3) at the same lattice parameters by 0.21 µC/cm
. Note, however, that this polarization is not switch-
able by an electric field since it is a consequence of the
tri-layered arrangement of the B-site cations. Next, we
remove the constraint on the high symmetry ionic posi-
tions, and relax the ions to their lowest energy positions
along the c axis by minimizing the Hellmann-Feynman
forces, while retaining tetragonal symmetry. We obtain
a ground state that is significantly (0.14 eV) lower in en-
ergy than the reference structure, but which has a simi-
lar value of polarization. Two stable ground states with
different and opposite polarizations from the reference
structure, the signature of a ferroelectric, are not ob-
tained. Thus it appears that tri-layering alone does not
lead to a significant switchable polarization in the ab-
sence of some additional driving force for ferroelectricity.
In all cases, the magnetic ions are high spin with negligi-
ble energy differences between ferro- and ferri-magnetic
orderings of the Fe and Cr ions; both arrangements lead
to substantial magnetizations of 440 and 110 emu/cm3
respectively. Such magnetic tri-layer systems could prove
useful in non-linear-optical applications, where a break-
ing of the inversion center is required, but a switchable
polarization is not.
Since epitaxial strain has been shown to have a strong
influence on the polarization of some ferroelectrics (such
as increasing the remanent polarization and Curie tem-
perature of BaTiO3 [20] and inducing room temperature
ferroelectricity in otherwise paraelectric SrTiO3 [21]) we
next explore the effect of epitaxial strain on the polar-
ization of La(Al,Fe,Cr)O3. To simulate the effects of epi-
taxial strain we constrain the value of the in-plane lat-
tice parameter, adjust the out of plane parameter so as
to maintain a constant cell volume, and relax the atomic
positions. The volume maintained is that of the calcu-
lated fully optimized structure, 167 Å3, which has an
in-plane lattice constant of 3.82 Å. As shown in Figure
3, we find that La(Al,Fe,Cr)O3 undergoes a phase transi-
tion to a polar state at an in-plane lattice constant of 3.76
Å, which corresponds to a (compressive) strain of -0.016
(calculated from (a‖−a0)/a0 where a‖ is the in-plane lat-
tice constant and a0 is the calculated equilibrium lattice
constant). A compressive strain of -0.016 is within the
range attainable by growing a thin film on a substrate
with a suitably reduced lattice constant.
We find that significant ferroelectric polarizations can
be induced in La(Al,Fe,Cr)O3 at even smaller strain val-
ues by using negative pressure conditions. We simulate
negative pressure by increasing all three lattice constants
and imposing the constraint a=b=c/3; such a growth
condition might be realized experimentally by growing
the film in small cavities on the surface of a large-lattice-
constant substrate, such that epitaxy occurs both hori-
zontally and vertically. As in the planar epitaxial strain
state, the system becomes strongly polar; this time the
phase transition to the polar state occurs at a lattice con-
stant of 3.85 Å, at which the strain is a negligible 0.001
relative to the lattice constant of the fully optimized sys-
In Fig. 1 we show the calculated energy versus dis-
tortion profile and polarization for negative pressure
La(Al,Fe,Cr)O3 with in-plane lattice constant = 3.95 Å,
well within the ferroelectric region of the phase diagram
shown in Fig. 3. The system has a characteristic ferro-
electric double well potential which is almost symmetric
in spite of the tri-layering; the two ground states have
polarizations of 38.9 and -39.9 µC cm−2 respectively, rel-
ative to the reference structure at the same lattice con-
stant. Since the energies of the two minima are almost
identical, the effective electric field Eeff=∆E/∆P, intro-
duced in Ref [15], is close to zero and there is no tendency
to self pole. The origin of the symmetry is seen in the
calculated Born effective charges (3.6, 3.5 and 3.3 for Al,
Fe and Cr respectively) which show that the system is
largely ionic, with the ions showing very similar trivalent
cationic behavior. A similar profile is observed under
planar epitaxial strain, although the planar strained sys-
tem is around 0.15 eV lower in energy than the negative
pressure system for the same in-plane lattice constant.
To decouple the effects of interfacial strain and tri-
layering we calculate the polarization as a function of
strain and negative pressure for the individual compo-
nents, LaAlO3, LaFeO3 and LaCrO3. We find that all
three single-phase materials become polar at planar epi-
taxial strains of -0.03 (LaAlO3), -0.02 (LaFeO3), and
-0.01 (LaCrO3). Likewise, all three components be-
come polar at negative pressure, under strains of +0.03
(LaAlO3), +0.001 (LaFeO3), and +0.001 (LaCrO3).
(The higher strains required in LaAlO3 reflect its smaller
equilibrium lattice constant.)
These results confirm our earlier conclusion that the
large polarizations obtained in strained and negative
pressure La(Al,Fe,Cr)O3 are not a result of the tri-
layering. We therefore suggest that many perovskite ox-
ides should be expected to show ferroelectricity provided
that two conditions imposed in our calculations are met:
First, the ionic radii of the cation sites in the high sym-
metry structure are larger than the ideal radii, so that
structural distortions are desirable in order to achieve an
optimal bonding configuration. This can be achieved by
straining the system epitaxially or in a “negative pres-
sure” configuration. And second, non-polar structural
distortions, such as Glazer tiltings [30], are de-activated
relative to polar, off-centering distortions. These have
been prohibited in our calculations by the imposition of
tetragonal symmetry; we propose that the symmetry con-
straints provided experimentally by hetero-epitaxy in two
or three dimensions should also disfavor non-polar tilting
and rotational distortions. A recent intriguing theoretical
prediction that disorder can be used to disfavor cooper-
ative tilting modes is awaiting experimental verification
[31].
Finally, we compare the tri-layered La(Al,Fe,Cr)O3
with the polarization of its individual components. Cal-
culated separately, the remnant polarizations of LaAlO3,
LaFeO3 and LaCrO3, all at negative pressure with
a=c=3.95 Å, average to 40.4 µC cm−2. This is only
slightly larger than the calculated polarizations of the
heterostructure, 38.9 and 39.9 µC cm−2, indicating that
tri-laying has a negligible effect on the polarity. This sur-
prizing result warrants further investigation into how the
layering geometry modifies the overall polarization.
In conclusion, we have shown that asymmetric layering
alone is not sufficient to produce a significant switchable
polarization in a La(Al,Fe,Cr)O3 superlattice, and we
suggest that earlier reports of large polarizations in other
tri-layer structures may have resulted from the intrinsic
polarization of one of the components combined with epi-
taxial strain. We find instead that La(Al,Fe,Cr)O3 and
its parent compounds can become strongly polar under
reasonable values of epitaxial strain and symmetry con-
straints, and that tri-layering serves to modify the re-
sulting polarization. Finally, we suggest “negative pres-
sure” as an alternative route to ferroelectricity and hope
that our prediction motivates experimental exploration
of such growth techniques.
This work was funded by the NSF IGERT program,
grant number DGE-9987618, and the NSF Division of
Materials Research, grant number DMR-0605852. The
authors thank Massimiliano Stengel and Claude Ederer
for helpful discussions.
[1] N. A. Hill, J. Phys. Chem. B 104, 6694 (2000).
[2] C. Ederer and N. A. Spaldin, Curr. Opin. Solid State
Mater. Sci. 9, 128 (2005).
[3] M. Fiebig, J. Phys. D: Appl. Phys. 38, R1 (2005).
[4] R. Seshadri and N. A. Hill, Chem. Mater. 13, 2892
(2001).
[5] A. M. dos Santos, S. Parashar, A. R. Raju, Y. S. Zhao,
A. K. Cheetham, and C. N. R. Rao, Solid State Commun.
122, 49 (2002).
[6] J. Wang, J. B. Neaton, H. Zheng, V. Nagarajan, S. B.
Ogale, B. Liu, D. Viehland, V. Vaithyanathan, D. G.
Schlom, U. V. Waghmare, et al., Science 299, 1719
(2003).
[7] J. B. Neaton, C. Ederer, U. V. Waghmare, N. A. Spaldin,
and K. M. Rabe, Phys. Rev. B 71, 014113 (2005).
[8] B. B. van Aken, T. T. M. Palstra, A. Filippetti, and N. A.
Spaldin, Nat. Mater. 3, 164 (2004).
[9] D. L. Fox and J. F. Scott, J. Phys. C 10, L329 (1977).
[10] C. Ederer and N. A. Spaldin, Physical Re-
view B (Condensed Matter and Materials
Physics) 74, 024102 (pages 8) (2006), URL
http://link.aps.org/abstract/PRB/v74/e024102 .
[11] N. Ikeda, H. Ohsumi, K. Ohwada, K. Ishii, T. Inami,
K. Kakurai, Y. Murakami, K. Yoshii, S. Mori, Y. Horibe,
et al., Nature 436, 1136 (2005).
[12] M. A. Subramanian, H. Tao, C. Jiazhong, N. S. Rogado,
T. G. Calvarese, and A. W. Sleight, Adv. mater. 18, 1737
(2006).
[13] T. Kimura, T. Goto, H. Shintani, K. Ishizaka, T. Arima,
and Y. Tokura, Nature 426, 55 (2003).
[14] R. Ramesh and N. A. Spaldin, Nat. Mater. 6, 21 (2007).
[15] N. Sai, B. Meyer, and D. Vanderbilt, Phys. Rev. Lett.
84, 5636 (2000).
[16] H. N. Lee, H. M. Christen, M. F. Chisholm, C. M.
Rouleau, and D. H. Lowndes, Nature 433, 395 (2005).
[17] M. P. Warusawithana, E. V. Colla, J. N. Eckstein, and
M. B. Weissman, Phys. Rev. Lett. 90, 036802 (2003).
[18] Y. Ogawa, H. Yamada, T. Ogasawara, T. Arima,
H. Okamoto, M. Kawasaki, and Y. Tokura, Phys. Rev.
Lett. 90, 217403 (2003).
[19] J. B. Neaton and K. M. Rabe, Appl. Phys. Lett. 82, 1586
(2003).
[20] K. J. Choi, M. Biegalski, Y. L. Li, A. Sharan, J. Schubert,
R. Uecker, P. Reiche, Y. B. Chen, X. Q. Pan, V. Gopalan,
et al., Science 306, 1005 (2004).
[21] J. H. Haeni, P. Irvin, W. Chang, R.Uecker, P. Re-
iche, Y. L. Li, S. Choudhury, W. Tian, M. E. Hawley,
B. Craigo, et al., Nature 430, 758 (2004).
[22] V. I. Anisimov, F. Aryasetiawan, and A. I. Liechtenstein,
J. Phys.: Condens. Mat. 9, 767 (1997).
[23] G. Kresse and J. Furthmüller, Phys. Rev. B 54, 11169
(1996).
[24] P. E. Blöchl, Phys. Rev. B 50, 17953 (1994).
[25] G. Kresse and D. Joubert, Phys. Rev. B 59, 1758 (1999).
[26] R. D. King-Smith and D. Vanderbilt, Phys. Rev. B 47,
1651 (1993).
[27] D. Vanderbilt and R. D. King-Smith, Phys. Rev. B 48,
4442 (1993).
[28] Z. Yang, Z. Huang, L. Ye, and X. Xie, Phys. Rev. B 60,
15674 (1999).
[29] I. Solovyev, N. Hamada, and K. Terakura, Phys. Rev. B
53, 7158 (1996).
[30] A. M. Glazer, Acta Crystallogr. B 28, 3384 (1972).
[31] D. I. Bilc and D. J. Singh, Phys. Rev. Lett. 96, 147602
(pages 4) (2006).
http://link.aps.org/abstract/PRB/v74/e024102
FIG. 1: Energy and polarization as a function of displacement
from the centrosymmetric structure for La(Al,Fe,Cr)O3 under
negative pressure with a = c/3 = 3.95 Å. Inset: Schematic
representation of the centrosymmetric unit cell (center) and
displacements of the metal cations corresponding to the en-
ergy minima. Displacements are exaggerated for clarity.
FIG. 2: Density of states for Fe and Cr ions in La(Al,Fe,Cr)O3
with U/J values of 6/0.6 eV and 5/0.5 eV respectively. The
dashed line at 0 eV indicates the position of the Fermi energy.
FIG. 3: Calculated polarizations of negative pressure (cir-
cles) and epitaxially strained (triangles) La(Al,Fe,Cr)O3 as a
function of change in (a) in-plane and (b) out-of-plane lattice
constants relative to the lattice constants of the fully relaxed
structures. The polarizations are reported relative to the ap-
propriate corresponding reference structures in each case.

  We explore computationally the formation of tri-layer superlattices as an
alternative approach for combining ferroelectricity with magnetism to form
magnetoelectric multiferroics. We find that the contribution to the
superlattice polarization from tri-layering is small compared to typical
polarizations in conventionalferroelectrics, and the switchable ferroelectric
component is negligible. In contrast, we show that epitaxial strain and
``negative pressure'' can yield large, switchable polarizations that are
compatible with the coexistence of magnetism, even in materials with no active
ferroelectric ions.

Tri-layer superlattices: A route to magnetoelectric multiferroics?
Alison J. Hatt and Nicola A. Spaldin
Materials Department, University of California
(Dated: October 22, 2018)
We explore computationally the formation of tri-layer superlattices as an alternative approach for
combining ferroelectricity with magnetism to form magnetoelectric multiferroics. We find that the
contribution to the superlattice polarization from tri-layering is small compared to typical polar-
izations in conventional ferroelectrics, and the switchable ferroelectric component is negligible. In
contrast, we show that epitaxial strain and “negative pressure” can yield large, switchable polar-
izations that are compatible with the coexistence of magnetism, even in materials with no active
ferroelectric ions.
PACS numbers:
The simultaneous presence of ferromagnetism and fer-
roelectricity in magnetoelectric multiferroics suggests
tremendous potential for innovative device applications
and exploration of the fundamental physics of coupled
phenomena. However, the two properties are chemically
contra-indicated, since the transition metal d electrons
which are favorable for ferromagnetism disfavor the off-
centering of cations required for ferroelectricity [1]. Con-
tinued progress in this burgeoning field rests on the iden-
tification of alternative mechanisms for ferroelectricity
which are compatible with the existence of magnetism
[2, 3]. Mechanisms discovered to date include the incor-
poration of stereochemically active lone pair cations, for
example in BiMnO3 [4, 5] and BiFeO3 [6, 7], geometric
ferroelectricity in YMnO3 [8], BaNiF4 [9, 10] and related
compounds, charge ordering as in LuFe2O4 [11, 12], and
polar magnetic spin-spiral states, of which TbMnO3 is
the prototype [13]. However, there are currently no single
phase multiferroics with large and robust magnetization
and polarization at or near room temperature [14].
The study of ferroelectrics has been invigorated over
the last few years by tremendous improvements in the
ability to grow high quality ferroelectric thin films with
precisely controlled composition, atomic arrangements
and interfaces. In particular, the use of compositional
ordering that breaks inversion symmetry, such as the
layer-by-layer growth of three materials in an A-B-C-
A-B-C... arrangement, has produced systems with en-
hanced polarizations and large non-linear optical re-
sponses [15, 16, 17, 18]. Here we explore computation-
ally this tri-layering approach as an alternative route to
magnetoelectric multiferroics. Our hypothesis is that the
magnetic ions in such a tri-layer superlattice will be con-
strained in a polar, ferroelectric state by the symmetry of
the system, in spite of their natural tendency to remain
centrosymmetric. We note, however, that in previous
tri-layering studies, at least one of the constituents has
been a strong ferroelectric in its own right, and the other
constituents have often contained so-called second-order
Jahn-Teller ions such as Ti4+, which have a tendency to
off-center. Therefore factors such as electrostatic effects
from internal electric fields originating in the strong fer-
roelectric layers [19], or epitaxial strain, which is well
established to enhance or even induce ferroelectric prop-
erties in thin films with second-order Jahn-Teller ions
[6, 20, 21], could have been responsible for the enhanced
polarization in those studies.
We choose a [001] tri-layer superlattice of perovskite-
structure LaAlO3, LaFeO3 and LaCrO3 as our model
system (see Fig. 1, inset.) Our choice is motivated by
three factors. First, all of the ions are filled shell or filled
sub-shell, and therefore insulating behavior, a prerequi-
site for ferroelectricity, is likely. Second, the Fe3+ and
Cr3+ will introduce magnetism. And third, none of the
parent compounds are ferroelectric or even contain ions
that have a tendency towards ferroelectric distortions, al-
lowing us to test the influence of trilayering alone as the
driving force for ferroelectricity. For all calculations we
use the LDA+U method [22] of density functional the-
ory as implemented in the Vienna Ab-initio Simulation
Package (VASP) [23]. We use the projector augmented
wave (PAW) method [24, 25] with the default VASP po-
tentials (La, Al, Fe pv, Cr pv, O), a 6x6x2 Monkhorst-
Pack mesh and a plane-wave energy cutoff of 450 eV. Po-
larizations are obtained using the standard Berry phase
technique [26, 27] as implemented in VASP. We find that
U/J values of 6/0.6 eV and 5/0.5 eV on the Fe and Cr
ions respectively, are required to obtain insulating band
structures; smaller values of U lead to metallic ground
states. These values have been shown to give reasonable
agreement with experimental band gaps and magnetic
moments in related systems [28] but are somewhat lower
than values obtained for trivalent Fe and Cr using a con-
strained LDA approach [29]. We therefore regard them as
a likely lower limit of physically meaningful U/J values.
(Correspondingly, since increasing U often decreases the
covalency of a system, our calculated polarizations likely
provide upper bounds to the experimentally attainable
polarizations).
We begin by constraining the in-plane a lattice con-
stant to the LDA lattice constant of cubic SrTiO3 (3.85
Å) to simulate growth on a substrate, and adjust the out-
of-plane c lattice constant until the stress is minimized,
with the ions constrained in each layer to the ideal, high-
symmetry perovskite positions. We refer to this as our
reference structure. (The LDA (LDA+U) lattice con-
http://arxiv.org/abs/0704.0781v3
stants for cubic LaAlO3 (LaFeO3, LaCrO3) are 3.75, 3.85
and 3.84 Å, respectively. Thus, LaAlO3 is under tensile
strain and LaFeO3/LaCrO3 are unstrained.) The cal-
culated total density of states, and the local densities of
states on the magnetic ions, are shown in Figure 2; a band
gap of 0.32 eV is clearly visible. The polarization of this
reference structure differs from that of the corresponding
non-polar single-component material (for example pure
LaAlO3) at the same lattice parameters by 0.21 µC/cm
. Note, however, that this polarization is not switch-
able by an electric field since it is a consequence of the
tri-layered arrangement of the B-site cations. Next, we
remove the constraint on the high symmetry ionic posi-
tions, and relax the ions to their lowest energy positions
along the c axis by minimizing the Hellmann-Feynman
forces, while retaining tetragonal symmetry. We obtain
a ground state that is significantly (0.14 eV) lower in en-
ergy than the reference structure, but which has a simi-
lar value of polarization. Two stable ground states with
different and opposite polarizations from the reference
structure, the signature of a ferroelectric, are not ob-
tained. Thus it appears that tri-layering alone does not
lead to a significant switchable polarization in the ab-
sence of some additional driving force for ferroelectricity.
In all cases, the magnetic ions are high spin with negligi-
ble energy differences between ferro- and ferri-magnetic
orderings of the Fe and Cr ions; both arrangements lead
to substantial magnetizations of 440 and 110 emu/cm3
respectively. Such magnetic tri-layer systems could prove
useful in non-linear-optical applications, where a break-
ing of the inversion center is required, but a switchable
polarization is not.
Since epitaxial strain has been shown to have a strong
influence on the polarization of some ferroelectrics (such
as increasing the remanent polarization and Curie tem-
perature of BaTiO3 [20] and inducing room temperature
ferroelectricity in otherwise paraelectric SrTiO3 [21]) we
next explore the effect of epitaxial strain on the polar-
ization of La(Al,Fe,Cr)O3. To simulate the effects of epi-
taxial strain we constrain the value of the in-plane lat-
tice parameter, adjust the out of plane parameter so as
to maintain a constant cell volume, and relax the atomic
positions. The volume maintained is that of the calcu-
lated fully optimized structure, 167 Å3, which has an
in-plane lattice constant of 3.82 Å. As shown in Figure
3, we find that La(Al,Fe,Cr)O3 undergoes a phase transi-
tion to a polar state at an in-plane lattice constant of 3.76
Å, which corresponds to a (compressive) strain of -0.016
(calculated from (a‖−a0)/a0 where a‖ is the in-plane lat-
tice constant and a0 is the calculated equilibrium lattice
constant). A compressive strain of -0.016 is within the
range attainable by growing a thin film on a substrate
with a suitably reduced lattice constant.
We find that significant ferroelectric polarizations can
be induced in La(Al,Fe,Cr)O3 at even smaller strain val-
ues by using negative pressure conditions. We simulate
negative pressure by increasing all three lattice constants
and imposing the constraint a=b=c/3; such a growth
condition might be realized experimentally by growing
the film in small cavities on the surface of a large-lattice-
constant substrate, such that epitaxy occurs both hori-
zontally and vertically. As in the planar epitaxial strain
state, the system becomes strongly polar; this time the
phase transition to the polar state occurs at a lattice con-
stant of 3.85 Å, at which the strain is a negligible 0.001
relative to the lattice constant of the fully optimized sys-
In Fig. 1 we show the calculated energy versus dis-
tortion profile and polarization for negative pressure
La(Al,Fe,Cr)O3 with in-plane lattice constant = 3.95 Å,
well within the ferroelectric region of the phase diagram
shown in Fig. 3. The system has a characteristic ferro-
electric double well potential which is almost symmetric
in spite of the tri-layering; the two ground states have
polarizations of 38.9 and -39.9 µC cm−2 respectively, rel-
ative to the reference structure at the same lattice con-
stant. Since the energies of the two minima are almost
identical, the effective electric field Eeff=∆E/∆P, intro-
duced in Ref [15], is close to zero and there is no tendency
to self pole. The origin of the symmetry is seen in the
calculated Born effective charges (3.6, 3.5 and 3.3 for Al,
Fe and Cr respectively) which show that the system is
largely ionic, with the ions showing very similar trivalent
cationic behavior. A similar profile is observed under
planar epitaxial strain, although the planar strained sys-
tem is around 0.15 eV lower in energy than the negative
pressure system for the same in-plane lattice constant.
To decouple the effects of interfacial strain and tri-
layering we calculate the polarization as a function of
strain and negative pressure for the individual compo-
nents, LaAlO3, LaFeO3 and LaCrO3. We find that all
three single-phase materials become polar at planar epi-
taxial strains of -0.03 (LaAlO3), -0.02 (LaFeO3), and
-0.01 (LaCrO3). Likewise, all three components be-
come polar at negative pressure, under strains of +0.03
(LaAlO3), +0.001 (LaFeO3), and +0.001 (LaCrO3).
(The higher strains required in LaAlO3 reflect its smaller
equilibrium lattice constant.)
These results confirm our earlier conclusion that the
large polarizations obtained in strained and negative
pressure La(Al,Fe,Cr)O3 are not a result of the tri-
layering. We therefore suggest that many perovskite ox-
ides should be expected to show ferroelectricity provided
that two conditions imposed in our calculations are met:
First, the ionic radii of the cation sites in the high sym-
metry structure are larger than the ideal radii, so that
structural distortions are desirable in order to achieve an
optimal bonding configuration. This can be achieved by
straining the system epitaxially or in a “negative pres-
sure” configuration. And second, non-polar structural
distortions, such as Glazer tiltings [30], are de-activated
relative to polar, off-centering distortions. These have
been prohibited in our calculations by the imposition of
tetragonal symmetry; we propose that the symmetry con-
straints provided experimentally by hetero-epitaxy in two
or three dimensions should also disfavor non-polar tilting
and rotational distortions. A recent intriguing theoretical
prediction that disorder can be used to disfavor cooper-
ative tilting modes is awaiting experimental verification
[31].
Finally, we compare the tri-layered La(Al,Fe,Cr)O3
with the polarization of its individual components. Cal-
culated separately, the remnant polarizations of LaAlO3,
LaFeO3 and LaCrO3, all at negative pressure with
a=c=3.95 Å, average to 40.4 µC cm−2. This is only
slightly larger than the calculated polarizations of the
heterostructure, 38.9 and 39.9 µC cm−2, indicating that
tri-laying has a negligible effect on the polarity. This sur-
prizing result warrants further investigation into how the
layering geometry modifies the overall polarization.
In conclusion, we have shown that asymmetric layering
alone is not sufficient to produce a significant switchable
polarization in a La(Al,Fe,Cr)O3 superlattice, and we
suggest that earlier reports of large polarizations in other
tri-layer structures may have resulted from the intrinsic
polarization of one of the components combined with epi-
taxial strain. We find instead that La(Al,Fe,Cr)O3 and
its parent compounds can become strongly polar under
reasonable values of epitaxial strain and symmetry con-
straints, and that tri-layering serves to modify the re-
sulting polarization. Finally, we suggest “negative pres-
sure” as an alternative route to ferroelectricity and hope
that our prediction motivates experimental exploration
of such growth techniques.
This work was funded by the NSF IGERT program,
grant number DGE-9987618, and the NSF Division of
Materials Research, grant number DMR-0605852. The
authors thank Massimiliano Stengel and Claude Ederer
for helpful discussions.
[1] N. A. Hill, J. Phys. Chem. B 104, 6694 (2000).
[2] C. Ederer and N. A. Spaldin, Curr. Opin. Solid State
Mater. Sci. 9, 128 (2005).
[3] M. Fiebig, J. Phys. D: Appl. Phys. 38, R1 (2005).
[4] R. Seshadri and N. A. Hill, Chem. Mater. 13, 2892
(2001).
[5] A. M. dos Santos, S. Parashar, A. R. Raju, Y. S. Zhao,
A. K. Cheetham, and C. N. R. Rao, Solid State Commun.
122, 49 (2002).
[6] J. Wang, J. B. Neaton, H. Zheng, V. Nagarajan, S. B.
Ogale, B. Liu, D. Viehland, V. Vaithyanathan, D. G.
Schlom, U. V. Waghmare, et al., Science 299, 1719
(2003).
[7] J. B. Neaton, C. Ederer, U. V. Waghmare, N. A. Spaldin,
and K. M. Rabe, Phys. Rev. B 71, 014113 (2005).
[8] B. B. van Aken, T. T. M. Palstra, A. Filippetti, and N. A.
Spaldin, Nat. Mater. 3, 164 (2004).
[9] D. L. Fox and J. F. Scott, J. Phys. C 10, L329 (1977).
[10] C. Ederer and N. A. Spaldin, Physical Re-
view B (Condensed Matter and Materials
Physics) 74, 024102 (pages 8) (2006), URL
http://link.aps.org/abstract/PRB/v74/e024102 .
[11] N. Ikeda, H. Ohsumi, K. Ohwada, K. Ishii, T. Inami,
K. Kakurai, Y. Murakami, K. Yoshii, S. Mori, Y. Horibe,
et al., Nature 436, 1136 (2005).
[12] M. A. Subramanian, H. Tao, C. Jiazhong, N. S. Rogado,
T. G. Calvarese, and A. W. Sleight, Adv. mater. 18, 1737
(2006).
[13] T. Kimura, T. Goto, H. Shintani, K. Ishizaka, T. Arima,
and Y. Tokura, Nature 426, 55 (2003).
[14] R. Ramesh and N. A. Spaldin, Nat. Mater. 6, 21 (2007).
[15] N. Sai, B. Meyer, and D. Vanderbilt, Phys. Rev. Lett.
84, 5636 (2000).
[16] H. N. Lee, H. M. Christen, M. F. Chisholm, C. M.
Rouleau, and D. H. Lowndes, Nature 433, 395 (2005).
[17] M. P. Warusawithana, E. V. Colla, J. N. Eckstein, and
M. B. Weissman, Phys. Rev. Lett. 90, 036802 (2003).
[18] Y. Ogawa, H. Yamada, T. Ogasawara, T. Arima,
H. Okamoto, M. Kawasaki, and Y. Tokura, Phys. Rev.
Lett. 90, 217403 (2003).
[19] J. B. Neaton and K. M. Rabe, Appl. Phys. Lett. 82, 1586
(2003).
[20] K. J. Choi, M. Biegalski, Y. L. Li, A. Sharan, J. Schubert,
R. Uecker, P. Reiche, Y. B. Chen, X. Q. Pan, V. Gopalan,
et al., Science 306, 1005 (2004).
[21] J. H. Haeni, P. Irvin, W. Chang, R.Uecker, P. Re-
iche, Y. L. Li, S. Choudhury, W. Tian, M. E. Hawley,
B. Craigo, et al., Nature 430, 758 (2004).
[22] V. I. Anisimov, F. Aryasetiawan, and A. I. Liechtenstein,
J. Phys.: Condens. Mat. 9, 767 (1997).
[23] G. Kresse and J. Furthmüller, Phys. Rev. B 54, 11169
(1996).
[24] P. E. Blöchl, Phys. Rev. B 50, 17953 (1994).
[25] G. Kresse and D. Joubert, Phys. Rev. B 59, 1758 (1999).
[26] R. D. King-Smith and D. Vanderbilt, Phys. Rev. B 47,
1651 (1993).
[27] D. Vanderbilt and R. D. King-Smith, Phys. Rev. B 48,
4442 (1993).
[28] Z. Yang, Z. Huang, L. Ye, and X. Xie, Phys. Rev. B 60,
15674 (1999).
[29] I. Solovyev, N. Hamada, and K. Terakura, Phys. Rev. B
53, 7158 (1996).
[30] A. M. Glazer, Acta Crystallogr. B 28, 3384 (1972).
[31] D. I. Bilc and D. J. Singh, Phys. Rev. Lett. 96, 147602
(pages 4) (2006).
http://link.aps.org/abstract/PRB/v74/e024102
FIG. 1: Energy and polarization as a function of displacement
from the centrosymmetric structure for La(Al,Fe,Cr)O3 under
negative pressure with a = c/3 = 3.95 Å. Inset: Schematic
representation of the centrosymmetric unit cell (center) and
displacements of the metal cations corresponding to the en-
ergy minima. Displacements are exaggerated for clarity.
FIG. 2: Density of states for Fe and Cr ions in La(Al,Fe,Cr)O3
with U/J values of 6/0.6 eV and 5/0.5 eV respectively. The
dashed line at 0 eV indicates the position of the Fermi energy.
FIG. 3: Calculated polarizations of negative pressure (cir-
cles) and epitaxially strained (triangles) La(Al,Fe,Cr)O3 as a
function of change in (a) in-plane and (b) out-of-plane lattice
constants relative to the lattice constants of the fully relaxed
structures. The polarizations are reported relative to the ap-
propriate corresponding reference structures in each case.
