Atomic Scale Design of Polar Perovskite Oxides without Second-Order

Article pubs.acs.org/cm

Atomic Scale Design of Polar Perovskite Oxides without Second-Order Jahn−Teller Ions Joshua Young and James M. Rondinelli* Department of Materials Science & Engineering, Drexel University, Philadelphia, Pennsylvania 19104, USA S Supporting Information *

ABSTRACT: Demands for low-power and high-efficiency electronic devices have spurred an increased interest in new ferroelectric oxides, which display spontaneous electric polarizations. There are only a few mechanisms, however, capable of producing ordered dipoles in solid-state materials. Using first-principles density functional calculations, we extend the current repertoire and identify the required rotational patterns conducive to “geometric” ferroelectricity in (A,A′)B2O6 perovskite oxides with A cation order along [001]-, [111]-, and [110]-directions. For the polar oxides, we show that electric polarizations arise through a geometric, “rotation-induced” mechanism and are greater than those induced by spindriven mechanisms. We also discuss the energetics of each ordered arrangement and explain how competing centrosymmetric phases can lead to potential complications in thin-film growth of these materials. Finally, we generalize these results to a simple set of structural chemistry guidelines, which may be used to design other artificial oxides without inversion symmetry. KEYWORDS: perovskites, cation order, ferroelectrics, improper, density functional theory, rotations

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INTRODUCTION Many useful material properties, such as ferroelectricity, piezoelectricity, and nonlinear optical behavior, arise from the absence of inversion symmetry in a crystal structure.1 Developing new strategies to deterministically lift the operation of spatial parity is critical to engineering compounds with such “acentric” properties. One such route common in inorganic chemistry relies on crystal engineering.2−5 One selects the optimal combination of cations and anions to form (dis)connected polyhedral, e.g., tetrahedra in the spinel or octahedra in the perovskite structure, so that the metal centers undergo cooperative off-centering displacements within their coordination environments.6 In general, however, the tendency for the cations to displace in a polar fashion, the so-called second order Jahn−Teller (SOJT) effect,7 is limited to particular electronic valence configurations, i.e, d0 transition metals or lone-pair active cations.8,9 In BaTiO3, for example, the change in hybridization of the titanium 3d-states with the oxygen 2p-states from the Ti4+ displacements stabilizes its off-centering in its octahedral coordination, leading to a net electric polarization. The atomic structure and many functional properties of perovskite oxides, however, are controlled by B cations with nonzero electron counts; for example, the size mismatch with the A cation often produces BO6 octahedral rotations10,11 and not inversion symmetry lifting displacements.12 It is therefore desirable to find design routes that yield polar oxides with nontoxic A and earth-abundant B cations through “geometric” mechanisms,13−17 i.e., those that arise due to cooperative atomic displacements that are independent of the cation’s electronic state, yielding minor changes in chemical bonding. In this context, recent searches for ferroelectric oxides in perovskite16,18−23 (or perovskite-derived24−26) crystals have © 2013 American Chemical Society

focused on electric polarizations induced by coupling of polar cation displacements to cooperative rotations of oxygen octahedra.27,28 The essential ingredients for obtaining polar structures relies on the interaction between cation order or disconnected octahedra and nominally centrosymmetric rotations or “tilts” of the oxygen cages. The cation arrangements (or the two-dimensionality, as in the case of the Ruddlesden−Popper oxides described in ref 26) shift the location of the inversion center in the crystal to a position permitting the BO6 rotations to remove the parity operation. Primary efforts on A-site ordered double perovskites, (A,A′)B2O6, have established design guidelines for the flavors of octahedral rotations19 that will produce large electric polarizations and small ferroelectric switching barriers when the A and A′ cations alternate along the [001]-direction.26 In other words, do the optimal rotations have in-phase (+), out-ofphase (−), or mixed Glazer tilt systems29 along the various Cartesian directions? The [001] ordered arrangement is particularly attractive because (i) it is the one most conducive to layer-by-layer growth methods of artificial oxides,30 since the geometry is equivalent to an ultrashort period perovskite superlattice, i.e., (ABO3)/(A′BO3), and (ii) the frequently observed orthorhombic a−a−b+ rotation pattern fulfills the necessary symmetry and energetic criteria outlined to produce this type of rotation-induced ferroelectricity. Owing to the directional control of cation order through advanced solid-state synthetic chemistry methods31,32 and the accessibility of perovskite substrates with (110) and (111) Received: July 28, 2013 Revised: October 16, 2013 Published: October 16, 2013 4545

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surface terminations, these previous observations beckon the question: Does atomic scale 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 previously identified? In this work, we carry out density functional theory (DFT) calculations on three (A,A′)B2O6 perovskite compositions to show that rotations of oxygen octahedra can induce ferroelectric polarizations if the A cation ordering occurs along either the [001] or [111] pseudocubic direction. We find that useful electric polarizations occur in the absence of any SOJT-active cations. The symmetry constraints on the flavor of the rotation are specified by the direction of the orderingnot all rotations are equally capable of lifting inversion. Indeed, we find no combination of octahedral rotations and [110]-directed cation order that will produce a polar crystal structure alone. Finally, we describe the crystal−chemistry requirements to rationally tune the polar distortions and show that these oxides exhibit electric polarizations comparable to conventional ferroelectric ceramics.

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Table 1. Summary of PBEsol Calculated Structural Parameters for the Orthorhombically Distorted Perovskitesa bulk phase

t

θxy (deg)

θz (deg)

δxy (Å)

LaGaO3 NdGaO3 SrZrO3 CaZrO3 SrHfO3 CaHfO3

0.956 0.936 0.942 0.891 0.949 0.897

12.6 15.5 13.4 18.9 11.7 17.4

6.96 10.3 8.59 11.3 7.15 10.7

0.168 0.299 0.207 0.317 0.157 0.289

a

The tolerance factor (t) gives a qualitative measure of the tendency of a cubic perovskite to distort due to the A- and B-site size differential; the further t is from 1, the more distorted the structure becomes. t is calculated using bond lengths obtained from the bond valence model.44 θxy defines the magnitude of the out-of-phase rotations, while θz describes the in-phase rotations, following the conventions given in ref 45 and depicted in Figure 1. δxy is a measure of the planar A cation displacements relative to the aristotype cubic perovskite.

as (La,Nd)Ga2O6, (Sr,Ca)Zr2O6, and (Sr,Ca)Hf2O6. The A cation ordering is taken along three different crystallographic planes, all which maintain identical compositions: (001), (111), or (110) planes, relative to the pseudocubic perovskite directions (Figure 1). For simplicity, we differentiate them using the

COMPUTATIONAL METHODS

First-Principles Calculations. All investigations were performed by employing density functional theory33 using projector augmented-wave potentials34 within PBEsol35 as implemented in the Vienna ab initio Simulation Package (VASP).36,37 PBEsol provides an improved description of the lattice parameters when compared to LDA. We used a 600 eV plane-wave cutoff and a 6 × 6 × 6 Monkhorst-Pack mesh.38 We determined the phonon band structure of each superlattice in its paraelectric phase using density functional perturbation theory.39 When performing density functional perturbation theory calculations, we used an increased plane wave-cutoff of 800 eV. We then used linear combinations of the unstable modes found to generate plausible atomic displacement patterns, resulting in 15 candidate structures for each ordered compound. We then performed a full relaxation, including the lattice and atomic degrees of freedom, on each to determine the ground state. The charge density plot was created from a difference map of a nonself consistent charge density obtained by a linear superposition of atomic orbitals to emphasize changes in bonding. The total polarization was calculated using the Berry phase method40,41 as implemented in VASP.

Figure 1. Three types of A/A′ cation ordering investigated: (a) layered, (b), columnar, and (c) rock salt arrangements. The amplitude of the outof-phase and in-phase angles are given as θxy = (180 − Θxy)/2 and θz = (90 − Θz)/2, respectively. In the case illustrated in (a), θxy = θz = 0. Oxygen atoms are omitted for clarity.

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following scheme, with symmetries specified for the ideal structures without any atomic displacements: (001) : Layered, P4/mmm (110) : Columnar, P4/mmm (111) : Rocksalt, Fm3̅m

RESULTS AND DISCUSSION To address our main question, we first determine the ground state crystal structures of six nonpolar perovskite dielectrics from which ordered double perovskites will be formed: LaGaO3, NdGaO3, SrZrO3, CaZrO3, SrHfO3, and CaHfO3. We selected these compounds because they are band insulators and never exhibit polar ground states. Yet, each adopts the Pnma space group with large tilts (θxy) and rotations (θz) of the octahedra (Table 1) and the mixed-tilt system (a−a−b+) previously determined19 critical for the octahedral rotations to break inversion symmetry in the presence of [001] A cation order. Consistent with the available experimental results for the gallates,42 we find sizable displacements (δxy) of the A cations from their high symmetry positions in the xy-plane in the Pnma structure, with Nd more so than La due to its smaller size as reflected by its smaller tolerance factor (t).43 The same is true in the zirconates and hafnates, with Ca displacing more than Sr. Note that, for a fixed B cation, the A cation size also correlates with the amplitude of the octahedral rotations. These features are critical to rotation-induced ferroelectrics, as the spontaneous polarization largely arises from the nonzero canceling of AO layer dipoles.26 We use the bulk perovskites as components to assemble ordered double perovskites with continuous B cation sublattices

The lowest energy ground state structures are obtained using density functional calculations with the PBEsol exchangecorrelation functional through a symmetry-restricted softphonon search (cf. Computational Methods). For all compositions explored, the space groups adopted by the layered, rock salt, and columnar orderings are the polar Pmc21 (space group No. 26) and Pmn21 (No. 31) and nonpolar P21/m (No. 11), respectively (Tables S1−S3 of the Supporting Information). Each structure contains A cation displacements within the xyplane and the a+b−c− BO6 tilt pattern,46 consistent with the preferred bulk tilt patterns. The low symmetry phases also show substantial energy gains relative to the high-symmetry configurations (Table 2), largely from the octahedral rotations. A group theoretical analysis47 of the atomic displacements in terms of symmetry modes involved in the structural transition referenced relative to a disordered A cation (5-atom) cubic perovskite26 reveals three main modes participate: M+3 (describing in-phase rotations), R+4 (out-of-phase rotations), and X+5 (antipolar A-site displacements). There is a small secondary, symmetry-allowed distortion for the polar structures (Γ−4 ), which 4546

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barriers than (La,Nd)Ga2O6, as changing the direction of the polarization in these types of materials involves a complete change in the sense of one of the BO6 octahedral rotation patterns. This is also in agreement with previous results showing that the ferroelectric switching barrier decreases with increasing average tolerance factor.26 Note that determining a quantitative estimate for the switching feasibility is a much more complicated task,48 as the real experimental switching path need not pass through the high-symmetry paraelectric phase. Nonetheless, a meaningful comparison can still be made in this way as each of the equivalently ordered polar compounds exhibits the same ground state crystal structure. On the basis of the fact that the number of orientational twin and antiphase domains must be identical, we thus anticipate that the domain dynamics and microstructure would likely be similar for all compounds explored. For this reason, ΔER could reasonably serve as a qualitative descriptor of the ferroelectric switching barrier. We now examine the stability and energetic trends of each A cation order for a given chemistry. We find that the rock salt ordering is always more stable than the other A cation orderings in the structurally distorted phases containing octahedral rotations, followed by the layered and then columnar arrangements (Table 2, ΔEC). In the paraelectric phases, however, the layered ordering is the lowest in energy, which is followed by columnar and then rock salt (highest energy). To explain this behavior, we examine the Coulombic contribution to the undistorted paralelectric structure (Figure 1) stability by calculating lattice sums at a fixed geometric configuration and varying the charge of the cations from A3+/A3+ (B3+) to A2+/A2+ (B4+). These combinations correspond to the (La,Nd)Ga2O6 and (Sr,Ca)Zr2O6 [or (Sr,Ca)Hf2O6] structures, respectively. The Madelung energy of the rock salt structure is lower than the layered structure regardless of the valence state of the cations. This is true in both the paraelectric and polar ground state structures, contrary to our DFT results, which only finds this to be the case in the low-symmetry distorted structures. This discrepancy can be understood using Pauling’s fifth rule of parsimony, which states that it is most favorable for each ion to be in similar environments. Because the paraelectric rock salt structure offers slightly more homogeneous surroundings than the layered or columnar structures (as the A and A′ atoms coordinate the anions in a trans rather than cis fashion), it minimizes the energetic penalties of cation order. Once the crystal symmetry is enforced, however, the paraelectric layered and columnar structures (P4/mmm) become lower in energy than the rock salt structure (Fm3̅m), because the P4/mmm space group allows for oxygen displacements, while the Fm3̅m does not, i.e., the oxygen atoms lie on inversion centers.31 Therefore, the oxygen atoms in the layered and columnar structures can shift

Table 2. Summary of Atomic and Electronic Ground State Properties of the Ordered (A,A′)B2O6 Perovskitesa structure

ordering

S.G.

ΔER

ΔEC

7 (μC/cm2)

(La,Nd)Ga2O6

layered rock salt columnar layered rock salt columnar layered rock salt columnar

Pmc21 Pmn21 P21/m Pmc21 Pmn21 P21/m Pmc21 Pmn21 P21/m

−498 −952 −940 −1306 −1364 −1301 −983 −1025 −1002

0.839 0 6.71 6.12 0 17.1 9.42 0 19.1

5.22 1.78  3.61 2.84  3.09 2.29 

(Sr,Ca)Zr2O6

(Sr,Ca)Hf2O6

a

The energy stability (ΔER) is given relative to the fully relaxed highsymmetry reference phase without octahedral distortions (Figure 1) at a single cation arrangement. ΔEC gives the relative energy difference for the ground state structures within a single composition with respect to cation ordering. Positive values indicate higher energy configurations; energy differences are given in units of meV/f.u. The ‘’ indicates the polarization, 7 , is required to be zero by symmetry.

lifts inversions symmetry and produces the ionic contribution to the electric polarizations described later (Supporting Information Table S4). Note that the presence of layered or rock salt cation order and the superposition of M+3 ⊕ R+4 , which describes the a−a−b+ tilt system, are sufficient to lift inversion symmetry in the absence of any polar atomic displacements. In all configurations, the M+3 and R+4 modes describing the octahedral rotations significantly lower the energy relative to the high-symmetry paraelectric structure, e.g., in rock salt (La,Nd)Ga2O6 the gains are 394 meV/f.u. for M+3 and 761 meV/f.u. for R+4 . The X+5 mode marginally lowers or increases the total energy independent of the other modes. Furthermore, the combination of in-phase and out-of-phase rotations is found to cooperatively lower the energy of each systemmore so than each octahedral rotation mode individually. Note that, in the absence of these rotations, displacements of the A or B cations that result in layer dipoles (discussed later) are energetically unfavorable. These calculations indicate the phenomenological origin for the polarization is through a hybrid improper ferroelectric mechanism (Figure 2) and not driven by the SOJT-mechanism. The energy minima are deeper for (Sr,Ca)Zr2O6 and (Sr,Ca)Hf2O6 than that for (La,Nd)Ga2O6 (Figure 2). The energy minima in the (Sr,Ca)-ordered compounds also appear at larger rotation angles than (La,Nd)Ga2O6. This is consistent with the tolerance factors presented in Table 1; a smaller tolerance factor results in larger octahedral rotations and typically a deeper minimum. From these observations, we infer that (Sr,Ca)Zr2O6 and (Sr,Ca)Hf2O6 would exhibit higher ferroelectric switching

Figure 2. Calculated two-dimensional energy surface contours for the (A,A′)B2O6 perovskites with respect to the amplitude of the out-of-phase (abscissa) and in-phase (ordinate) rotation angles. For all compositions, the layered and rock salt ordering have energy minima located at nonzero values of the a0a0c+ and a−a−c0 tilt patterns, indicating the microscopic driving force for inversion symmetry breaking originates from the octahedral rotations. 4547

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Each ordered structure can be thought of as having either columns of A and A′ atoms (or combinations thereof) along the z-direction parallel to the axis about which the GaO6 octahedra rotate in-phase (Figure 4a−c). On the basis of the specific atomic scale ordering, these columns are either mixed, i.e., containing both A and A′, or chemically homogeneous. Moving along this direction, the A cations are found to alternate their displacements in the perpendicular direction along [001]. The fact that the A and A′ atoms in each chemically heterogeneous column displace by different amounts, owing to ionic size, mass, and bond valence optimization requirements, yields the net macroscopic electric polarizations in the layered and rock salt structures (cf. Table 2). Thus, the same atomic displacements that stabilize the rock salt and layered structures produce the spontaneous electric polarizations. Using an ionic charge model and layer polarizations only, we find this estimate is within 85% of the DFT result, indicating there is not a substantial electronic-only contribution to the polarization, and thus the polar A cation displacements are not driven by chemical bond formation or SOJT activity. These features are characteristic of hybrid improper ferroelectrics.26 Interestingly, in both the layered and rock salt arrangements, the electric dipoles are constrained to planes but oriented in opposite directions. In the columnar ordering, each column is made up of chemically equivalent atoms; therefore, the displacement amplitudes are identical there is no net polarization. In each configuration explored, atoms contained in the (A/A′)O2 plane orthogonal to the direction of in-phase rotations displace, regardless of the chemical species that comprise that plane. Critically, if the direction of the [001]-ordering in the layered structure is no longer collinear with the in-phase octahedral rotation axis, the electric polarization goes to zero. For example, if the direction of in-phase rotations lies in the same plane as the ordered layers, then each “column” of A-sites is made up of chemically identical atoms, resulting in zero polarization as in the columnar arrangement. This effect manifests in the crystal symmetry, whereby the polar Pmc21 would transform to a centrosymmetric P21/m space group if the in-phase rotation axis occurs in the [100]-direction, while the layered ordering is maintained along [001]. Our calculations on the energetic stability of this layered P21/m structure in (La,Nd)Ga2O6 show that the nonpolar structure is ∼0.89 meV/f.u. higher in energy than the polar structure. The fact that these two structures are essentially degenerate could result in the formation of Pmc21/ P21/m domains during synthesis, leading to a diminished ferroelectric polarization. Such microstructures have been previously

in response to the A/A′ size differential, while the atoms in the rock salt structure cannot, resulting in an energetic cost. In the polar layered and rock salt structures, however, there are no inversion centers owing to the combination of octahedral rotations and cation order. As a result the A cations and oxygen atoms are free to move to maximize the Coulombic energy and minimize any repulsive interactions, which results in an enhanced covalency of (A/A′)−O bonds (Figure 3). Indeed, the enhanced

Figure 3. Electron charge density difference plot for layered (La,Nd)Ga2O6 in the (110) plane reveals stronger hybridization between (a) Nd and O than that of (b) La and O.

covalent interactions are greater for the Nd−O bonds than for the La−O bonds in the (110) plane of (La,Nd)Ga2O6. For the same reasons mentioned previously, this lowers the energy of the rock salt ordered compound below that of the layered arrangement. The columnar structure is now highest in energy, because its distorted structure (space group P21/m) does not allow for these same energy-lowering polar displacements. Using the modern theory of polarization,40,41 we now calculate the electric polarization for the six compositions which exhibit a nonzero polarization in their ground state (Table 2). For each polar structure, the electric polarization is constrained by symmetry to the [100] direction orthogonal to the in-phase rotation axis. At a fixed composition, we find that the layered A cation order always gives the highest electric polarization (∼4 μC/cm2) followed by the rock salt arrangement, which ranges from 30 to 80% lower in magnitude. To understand this behavior, we examine the local atomic displacements of the A and A′ cations that connect the highsymmetry phases to the low-symmetry ground state crystal structures. We find a nontrivial influence of the ordering on the magnitude and direction of the cation displacements (Figure 4).

Figure 4. Ground state crystal structures of the [001] (layered), [111] (rock salt), and [110] (columnar) atomically ordered (La,Nd)Ga2O6. The A cations displace in the plane orthogonal to the in-phase rotation axis [001]. 4548

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are greater in magnitude than those induced by electronic mechanisms such as spin ordering, charge or orbital ordering, or unusual spin arrangements,50 which exhibit polarizations ranging from 10−3 to 1 μC/cm2. We also identified that out-of-phase rotation patterns alone are capable of producing polar space groups in A/A′ [111]-ordered perovskites through a traditional improper ferroelectric mechanism. These results were then generalized into a set of chemistry-independent guidelines necessary for engineering acentric oxides. We hope the framework resulting from these guidelines will be useful in guiding the selection and synthesis of new oxides displaying enhanced electronic and optical properties.

reported in the growth of layered perovskite superlattice thinfilms.49 Finally, our microscopic displacement and symmetry analysis reveals that although both in-phase and out-of-phase rotations break the inversion symmetry needed to have a polar space group in the layered structure, only out-of-phase rotations are required in the rock salt structure. Inversion is lifted in this way through a traditional improper mechanism, rather than a hybrid higher order anharmonic interaction, since the out-of-phase rotational pattern is only described by one mode. An examination of each structure with only out-of-phase rotations shows why this is the case (Figure 5). First, the ligands comprising the BO6 octahedron

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ASSOCIATED CONTENT

S Supporting Information *

Complete crystallographic structure data used in the calculations and additional tables. This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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− − 0

Figure 5. Out-of-phase octahedral rotational pattern (such as the a a b pattern shown here) is not always sufficient to lift inversion symmetry. The ordering of the A-sites in a layered fashion preserves the inversion centers (black spheres) in the crystal (a), observed by the antialignment (red arrows) of the LaOn polyhedra. Rock salt A-cation order breaks the inversion symmetry, discerned by the aligned arrows (b). Only LaOn polyhedra have been shown for clarity.

ACKNOWLEDGMENTS J.Y. and J.M.R. were supported by ARO (W911NF-12-1-0133) and thank the Fennie Group at Cornell University for insightful discussions. Calculations were carried out at the high-performance computing cluster (CARBON) of the Center for Nanoscale Materials (Argonne National Laboratory) supported by the U.S. DOE, Office of Science, Office of Basic Energy Sciences, under Contract No. DE-AC02-06CH11357.

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and AO12 cuboctahedron in the perovskite structure are the same oxygen atoms; therefore, the displacement of any one oxygen atom distorts both polyhedra. In other words, the rotation of the BO6 octahedra described by a Glazer tilt system will simultaneously distort the A-site cation coordination topology. While normally 12-fold coordinated, the A-site polyhedra is more accurately described by an acentric 9-sided elongated square pyramid polyhedron with a well-defined base and vertex when only out-of-phase rotations occur. Tiling these AO9/A′O9 units on a primitive lattice along [001] or [111] to produce the layered and rock salt ordering, respectively, creates the chemical environments shown in Figure 5. By drawing a vector from the base to vertex of each AO9 polyhedron, it becomes clear that the disconnected A cation polyhedra are antialigned relative to each other and maintain inversion symmetry for layered ordering [Figure 5a]. Indeed, inversion centers are found at the 2a, 2b, 2c, and 2d Wyckoff positions in this space group (Pmma) if only outof-phase octahedral rotations occur. For this reason, [001]ordering requires an additional (in-phase) rotation mode to lift inversion; thus, the hybrid-improper mechanism is active. In contrast, the AO9 polyhedra are cooperatively aligned for rock salt orderinga single out-of-phase tilt is sufficient to produce a noncentrosymmetric structure through a conventional improper mechanism.

REFERENCES

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CONCLUSIONS In summary, we have shown that A cation ordering in perovskite oxides along the [001]- and [111]-direction, in addition to an orthorhombic a−a−b+ tilt pattern, is sufficient to produce polar structures with useful electric polarizations. These polarizations 4549

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dx.doi.org/10.1021/cm402550q | Chem. Mater. 2013, 25, 4545−4550