1. Atomic Scale Structural Design
Strategies for Artificial Polar Oxides
Joshua Young
Materials Theory and Design Group, Drexel University
joshua.young@drexel.edu

2. Introduction
• Many useful material properties, such as
ferroelectricity, arise because of symmetry breaking in
the materials ground state.
• If we understand what breaks the symmetry in the
systems, we can purposefully engineer these qualities to
get new, interesting properties.
• The key to understanding this is how atoms form
repeating polyhedral units in crystal systems (tetrahedra
in diamond, octahedra in perovskites, etc.).

3. Structure of perovskites
• Perovskite oxides have the general formula ABO3, where A is an
alkali metal or rare earth element, and B is a transition metal.
• The structures can be thought of as supramolecular assemblies of
two polyhedral building blocks.
The basic building blocks are
BO6 and AO12 polyhedra. The
interplay of the size, shape,
A
B
= and connectivity of these units
leads to the various properties
of different perovskites.
O

4. Structure of perovskites
• Perovskites display several types of structural distortions which can change their
properties, the most common of which are rotations of BO6 octahedral units.
• The octahedra can rotate in-phase or out-of-phase along different crystallographic
directions. Combinations of these rotation types also exist in different materials. The
notation used to describe octahedral rotations is known as Glazer notation.
• If the octahedra tilt in-phase along a certain direction, that direction is given a +
sign. If they tilt out-of-phase, they are given a – sign. If there is no tilting along a
direction, it is given a 0. The three letters, in order, correspond to the x, y, or z
direction.
a0a0c+ a-a-c0 a+ b - b -

5. Chemistry of perovskites
• There are an enormous number of possible perovskite oxides. In
addition to the typical ABO3 structure, it is possible to have two types
of A atoms, B atoms, or both in one compound.
With approximately 24 possible A- B-sites
sites and 30 B-sites, there are:
• 720 ABO3 structures
• 16,560 (A,A’)BO3 structures
• 20,880 A(B,B’)O3 structures
• 480,240 (A,A’)(B,B’)O3 structures!
A-sites

6. Motivation
• It is simply not feasible to experimentally produce and characterize
the vast amount of perovskite oxides that are possible. Using first-
principle calculations, we can streamline the search for new
functional materials.
• Although perovskites display an enormous variety of different
properties, this work focuses on identifying new ferroelectrics.
• A ferroelectric is a material which displays a spontaneous electric
polarization that is reversible with an external electric field, and they
have many applications in the electronics industry, such as in
tunable capacitors or new types of RAM.

7. Motivation
• An inversion center is a symmetry element that maps any point (x, y, z) to
(-x, -y, -z). Materials with this property are “centrosymmetric”. Examples of
centrosymmetric objects are the AO12 and BO6 octahedra in perovskite oxides
shown previously.
• Inversion plays a fundamental role in determining many material properties,
including ferroelectricity. In a typical ferroelectric, for example, the B-site
displaces off-center, breaking inversion symmetry and resulting in a net
polarization.
Absence of an inversion center
centrosymmetric non-centrosymmetric

8. Objective
• Previous work has shown that the combination of two out-of-phase
rotations and one in-phase rotation, in addition to layered A-site
cation ordering, breaks inversion symmetry in perovskite oxides.
• This leads to the question driving my research: Does cation ordering
along alternative directions in the perovskite structure also lead to
the loss of inversion symmetry, and does it require the same
rotational pattern?
•In addition, can we engineer ferroelectricity from the extended
structure of the octahedral units?
J. M. Rondinelli and C. J. Fennie, Adv. Mater. 24, 1961 (2012).

9. Density functional theory
• The experiments in this project were performed using density functional
theory, a quantum mechanical modeling method.
• The ground state properties of a material are obtained through calculation
of its electron density n(r), which uniquely defines the energy, E, of the
system:
interacting non-interacting
electrons electrons
kinetic energy, Hartree terms, quantum mechanical the atomic structure and any external fields
(exchange-correlation) and electromagnetic terms are contained in this term
P. Hohenberg and W. Kohn, Phys. Rev. 136 B864 (1964).
W. Kohn and L. Sham, Phys. Rev. 140 A1133 (1965).

10. Experimental methods
• First, I determined the ground state structure of four different non-polar
perovskite oxides: LaGaO3, NdGaO3, SrZrO3, and CaZrO3. There all exhibited the
desired Pnma space group and a+b-b- tilt pattern.
• Next, La and Nd atoms were chemically ordered along the [100], [110], and
[111] crystallographic directions to create three different types of nanoscale
(La,Nd)1/2GaO3 superlattices. The same was done for the zirconates for a total
of six superlattices.
+
NdGaO3 LaGaO3
[100] ordered [110] ordered [111] ordered
“Layered” “Columnar” “Rock Salt”

11. Experimental methods
• Using density functional perturbation theory, the phonon band
structure of each superlattice was computed. From this, linear
combinations of unstable modes were used to determine the global
ground state structure of each superlattice.
Each of the arrows represents
a series of atomic distortions,
such as octahedral rotations
or A-site displacements, that
lead to a lower energy
structure. The final ground
state of each superlattice is
some combination of these
“modes.”
Phonon band structure of paraelectric
rock salt ordered (La,Nd)1/2GaO3.

12. Results and discussion
• The ground state structures all exhibit A-site cation displacements, as
well as an a+b−c− octahedral rotation pattern, with large energy gains over
the paraelectric phases.
• The layered and rock salt ordered superlattices exhibit net electric
polarizations. The columnar ordered ones do not.
Summary of ground state structures and
polarization of each ordered superlattice.

13. Results and discussion LaGaO3/NdGaO3 Layered
The polarization in the rock salt and layered Nd
structures arises from the fact that differing La
displacements of A and A’ atoms along -6 -4 -2 0 2 4 6
chemically heterogeneous columns do not LaGaO3/NdGaO3 Rocksalt
cancel. The chemically homogeneous columns Nd
in the columnar structure result in zero net La
polarization. This can be seen below. -2 -1.5 -1 -0.5 0 0.5 1 1.5 2
Layered Rock Salt Columnar LaGaO3/NdGaO3 Columnar
Nd
La
Nd
La
-6 -4 -2 0 2 4 6
Polarization (μC/cm^2)
The fact that the smaller A-site atom (Nd
or Ca, respectively) moves more than the
larger one results in the polarization seen
heterogeneous columns homogeneous column these superlattices.

14. Results and discussion
• The energetics describing the phase tran-
sition from paraelectric to polar in each
superlattice can be written as a combination
of different modes. an M3+ mode (describing
in-phase rotations), an R4− (describing out-
of-phase rotations and A-site off-
centering), and a polar Γ5− mode.
• Note how in each type of superlattice, the
in-phase and out-of-phase rotations greatly
lower the energy of system. It is also
possible to determine the amount each
mode contributes to the final system. This
can be used to further quantify differences
between the superlattices. The amount of
Γ5− is negligible.

15. Results and discussion
• Further investigation into the energetics of
the system reveals that the combination of
both the in-phase and out-of-phase rotations
is found to cooperatively lower the energy of
the system, more so than each mode
individually.
• The blue dots on the plot to the right show
that the minimum energy lies at some
combination of the R4− and M3+ modes.
• This means that the polarization is coupled to
both types of octahedral rotation. The fact that
this occurs in both kinds of superlattices
(La/Nd and Sr/Ca) shows that this property is
independent of the chemistry of the system. Energy as a function of in-phase and out-of-phase
rotations in layered (La,Nd)1/2GaO3.
N. A. Benedek and C. J. Fennie, Phys. Rev. Lett. 106, 107204 (2011).

16. Results and discussion
• The octahedral rotations and A-site displacements present in the superlattices are
stabilized via increased covalency between the oxygens and A-site cations.
• As a perovskite deviates from a cubic lattice, the A-sites displace in order to form
stronger bonds. It is energetically more favorable for the A-site to move and form a few
strong bonds and a few weak bonds than to have many "medium strength" bonds.
• To show this, I calculated the charge density maps for each compound, one of which is
shown below. Increased charge density can be seen between La-O than between Nd-O.
Charge density map of Nd-O layers (left) and
La-O layers (right) of layered (La,Nd)1/2GaO3.

17. Results and discussion
• The last piece of evidence that shows these Layered (La,Nd)1/2GaO3
systems are ferroelectric in nature is given by their
band structures. When any system goes from a
paraelectric to ferroelectric structure, there is a
increase in the band gap.
ξ (mode amplitude)
• The transition from the paraelectric to polar
states can be characterized by the symbol ξ, where
ξ = 0 describes the high symmetry phase and ξ = 1
describes the low symmetry phase. Any value
between 0 and 1 is some intermediate structure.
• The figure to the right shows the band gap for the
paraelectric phase, polar phase, and a phase
halfway between. The top of the valence band
decreases in the polar, resulting in a 2 eV increase
in the band gap.

18. Design rules
• Now, using these results, it is possible to develop a set of general rules describing
how to break inversion symmetry in perovskites.
• The cubic perovskite structure is very symmetric (space group 221, Pm-3m), with
inversion centers existing inside polyhedral units (B-sites in BO6 octahedra), as well
as between units (A-sites).
• Breaking inversion between each unit can be done by ordering two different types
of atoms. Now there are A-sites that exist which can not be mapped onto each other
through the inversion center on the B-site, thereby breaking it.
Two ordered (La,Nd)1/2GaO3
structures: rock salt and layered.
The structures exhibit space
or group 225 (Fm-3m) and 123
(P4/mmm) respectively. Both of
these space groups are
centrosymmetric.

19. Design rules
• To break the other inversion symmetry present in the crystal, octahedral rotation
patterns are inserted to distort the environment around the A-site. Octahedral rotation is
controlled by selecting A-sites of different sizes.
• Examination of how rotations distort the A-site environment has shown that, by
themselves, in-phase rotations cannot break the inversion symmetry. This is because the
A-O polyhedra created are centrosymmetric.
• In contrast, out-of-phase rotations creates asymmetric A-O polyhedra. This seems like it
should be sufficient to break inversion in both types of ordered superlattices, but it is
actually not.
Undistorted A-site environment A-site environment +
A-site + in-phase rotations out-of-phase rotations
environment
H. T. Stokes, E. H. Kisi, D. M. Hatch, and C. J. Howard, Acta. Crys., B58, 934 (2002).

20. Design Rules
• Out-of-phase rotations can break inversion symmetry in the rock salt ordered, but not
the layered. This is solely an effect of the cation ordering.
• As can be seen below, the out-of-phase A-O polyhedra tend to “point” in one direction.
The rock salt order causes all of them to point the same direction, leading to no inversion
symmetry. In the layered structure, the ordering causes each polyhedra to be a balanced
by an opposite facing one, leading to a preservation of inversion.
• Therefore, a combination of both in-phase and out-of-phase rotations is needed to
break the inversion in layered superlattices. The polyhedra created by this pattern is
shown below.
Rock salt (left) and layered
(right) with out-of-phase
rotations. All atoms except
lanthanum have been
A-O polyhedra created
removed for clarity.
from in-phase and out-of-
phase rotations.

21. Summary
• In conclusion, I have identified new ferroelectric
systems, cation ordered (La,Nd)1/2GaO3 and (Sr,Ca)1/2ZrO3.
The net polarization arises from inversion symmetry
breaking in these materials ground states. By using these as
models, I have elucidated a series of design rules that can be
used to construct other polar materials.
• All of the rules put forth previously are summarized in the
chart to the right. Although atomic ordering and octahedral
rotations were used in the case studies to break inversion
symmetry in each system, other distortions can be used in
different systems.
• These principles can be applied to many systems in order to
intelligently design new materials with interesting properties.
Having a series of steps to follow can help scientists streamline
and improve their search for new materials, as opposed to
trying a random assortment.

22. Summary
The two questions proposed earlier have now been answered. To summarize:
1. Does cation ordering along alternative directions in the perovskite structure also
lead to the loss of inversion symmetry, and does it require the same rotational
pattern?
Answer: Yes. [100] and [111] (layered and rock salt) A-site cation ordering, in
combination with BO6 octahedral rotations, can break inversion symmetry. In rock salt
ordered superlattices, out-of-phase rotations are sufficient to lift inversion. Layered
superlattices require both in-phase and out-of-phase rotations.
2. Can we engineer ferroelectricity from the extended structure of the octahedral units?
Answer: Yes. The fact that the A-sites are different sizes means that they displace
different amounts in the polar structures. This leads to a net polarization.

23. Future work
• I have begun investigating this phenomenon in aluminates, such as
(La,Nd)1/2AlO3, (La,Pr)1/2AlO3, and (La,Gd)AlO3. These compounds represent a variety of
tilt patterns (a-a-a-, a0b-b-, and a+b-b-, respectively), allowing for study of what makes
certain tilt patterns stable in different compounds.
• In addition, I am investigating the role of strain in stabilizing new phases in these
compounds. For example, rock salt ordered (La,Nd)1/2GaO3 showed a chiral phase that is
slightly higher in energy than the ground state. Through epitaxial strain, it could be
possible to stabilize this structure, leading to new properties, such as optical activity.

24. Acknowledgements
Many people and organizations have contributed to making this research a
success, and I would especially like to thank the following:
• The Materials Theory and Design Group, in particular my advisor, Dr. James Rondinelli
• All of our collaborators, both at Drexel and elsewhere.
• Drexel University Office of the Provost, the Office of Naval Research, and Defense
Advanced Research Projects for financial support.
• The Argonne National Laboratory Center for Nanoscale Materials, the National Science
Foundation XSEDE, and the Department of Defense High Performance Computing Program
for computational support.