Atomic Scale Structural Design Strategies for Artificial Polar Oxides

Joshua YoungMaterials Theory and Design Group, Drexel University

joshua.young@drexel.edu

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.).

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.

A

B

O

=

The basic building blocks are BO6 and AO12 polyhedra. The interplay of the size, shape, and connectivity of these units leads to the various properties of different perovskites.

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-

Chemistry of perovskites

A-sites

B-sites

• 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-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!

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.

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

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).

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:

P. Hohenberg and W. Kohn, Phys. Rev. 136 B864 (1964).W. Kohn and L. Sham, Phys. Rev. 140 A1133 (1965).

interactingelectrons

non-interactingelectrons

the atomic structure and any external fields are contained in this term

kinetic energy, Hartree terms, quantum mechanical (exchange-correlation) and electromagnetic terms

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.

+

[100] ordered“Layered”

[111] ordered“Rock Salt”

[110] ordered“Columnar”

NdGaO3 LaGaO3

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.

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.

Inserting different tilt patterns changes the energetics of the system.

Results and discussionThe polarization in the rock salt and layered structures arises from the fact that differing displacements of A and A’ atoms along chemically heterogeneous columns do not cancel. The chemically homogeneous columns in the columnar structure result in zero net polarization. This can be seen below.

La

Nd

La

Nd

-6 -4 -2 0 2 4 6

LaGaO3/NdGaO3 Columnar

Polarization (μC/cm^2)

La

Nd

-6 -4 -2 0 2 4 6

LaGaO3/NdGaO3 Layered

La

Nd

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

LaGaO3/NdGaO3 Rocksalt

The fact that the smaller A-site atom (Nd or Ca, respectively) moves more than the larger one results in the polarization seen these superlattices.heterogeneous columns homogeneous column

Layered Rock Salt Columnar

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.

€

Γ5−

€

M3+

€

R4−

• 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.

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.

N. A. Benedek and C. J. Fennie, Phys. Rev. Lett. 106, 107204 (2011).

Energy as a function of in-phase and out-of-phase rotations in layered (La,Nd)1/2GaO3.

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.

Results and discussion• The last piece of evidence that shows these 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.

• 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.

ξ (mode amplitude)

Layered (La,Nd)1/2GaO3

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 group 225 (Fm-3m) and 123 (P4/mmm) respectively. Both of these space groups are centrosymmetric.

or

Design rules

H. T. Stokes, E. H. Kisi, D. M. Hatch, and C. J. Howard, Acta. Crys., B58, 934 (2002).

• 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 + in-phase rotations

A-site environment + out-of-phase rotations

Design rules• Out-of-phase rotations can break inversion symmetry in the rock salt ordered structure, 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 removed for clarity.

A-O polyhedra created from in-phase and out-of-phase rotations.

Summary

• 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.

• 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.

SummaryThe 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.

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.

Acknowledgements

• 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.

Many people and organizations have contributed to making this research a success, and I would especially like to thank the following: