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Predictive Modeling

image of Perovskites

MODELING THE EFFECT OF POINT DEFECTS IN PEROVSKITES

Perovskites
Many electroceramics are based on or related to the perovskite crystal structure.  The generic perovskite oxide formula is generally written as ABO3.  In the ideal cubic form, the A site is coordinated to 12 anions to form cuboctahedral coordination polyhedra.  The B site is coordinated to six anions, forming octahedra.  The anions are coordinated to just two B site cations, as the four nearest A-site cations are about 41% further away. The anion octahedra are corner shared, which is a key feature of all perovskites.

Perovskites abound both in nature and in the laboratory, and their wide compositional range renders a variety of useful properties such that perovskites are encountered in applications as disparate as electroceramics, superconductors, refractories, catalysts, magnetoresistors, and proton conductors.  They are also of interest for use as substrates or buffer layers for compound semiconductor heteroepitaxy.  The design of such advanced materials requires an understanding of the relationship between chemical composition and crystal structure.

Applications
There are currently about 224.3 million smart phone subscribers in the USA, and 4.6 billion cell phone users worldwide, making the cellular phone one of the fastest-selling consumer items in history and the most widely spread technology on the planet. There are three times as many mobile phones than PCs of any kind in the world and more mobile phones than cars. There are over twice as many mobile phone users as internet users, and more mobile phone users than people with a credit card. Twice as many people use SMS text messaging worldwide than use e-mail, with 75,000 messages sent every second in the USA!

Microwave resonators are used extensively in telecommunications equipment, including cellular telephones and satellite links, and are at the heart of this multi-billion dollar market.  Oxide ceramics are critical elements in these devices, and three properties are important in determining their usefulness as a dielectric resonator.  First, the material must have a high dielectric constant (εr) to enable size reduction, the size of a microwave circuit being proportional to εr.  Second, a high quality factor Q (low tanδ) means fine frequency tunability and more channels within a given band.  Third, these ceramic components play a crucial role in compensating for frequency drift because of their low temperature coefficients of resonant frequency (τf).  Optimizing all these properties in a single material is not a trivial problem, and a full understanding of the crystal chemistry of such ceramics is paramount to future development.  Many perovskite-structured ceramic materials are known to have useful microwave dielectric properties, with potential applications in the mobile telecommunications market.

The effect of defects
In this study, a solid-state processing method is used to synthesize single-phase perovskite ceramics with engineered defect concentrations. Powder samples are characterized via X-ray diffraction. The resulting products are then uniaxially pressed and sintered for microstructural analysis. The ultimate goal is to develop a predictive model, based solely on composition, for the effect of point defects on the structure and, by extension, dielectric properties of perovskites.

Point defects like vacancies can have a profound effect on the structure of perovskite ceramics, but the exact mechanisms by which they do this are unclear.  There is some evidence in a variety of perovskite systems that A-site vacancies increase the average A-O bond distance due to mutual electrostatic repulsion of anions across the negatively-charged vacant site, which increases as the number of vacancies increases. A predictive model for the pseudocubic lattice constant which accounts for A-site vacancies and the ionic radii in their correct coordinations (XII, VI, and II for the A, B, and X species respectively) has been recently developed and shown to work adequately in several systems.

Comparison of apc calculated via the model and experimental apc values. The average relative errors for each series are all <0.59%

Fig 1: Comparison of apc calculated via the model and experimental apc values. The average relative errors for each series are all <0.59%

The Jahn-Teller effect, which causes the A-X or B-X bond lengths to either shorten in length or increase in length, is present in several of the perovskite systems that we analyzed. In particular, the model was extended to account for second order Jahn-Teller effects using the perovskite systems Pb1-3xLa2xTiO3 and Pb1-3xLa2x(Zr0.6Ti0.4)O3.

Effect of second order Jahn-Teller distortion on the calculations of the pseudocubic lattice constants a' (left) and a'' (right), where a' is a function of the A-X bond length and a'' is a function of the B-X bond length.

Figs 2 & 3: Effect of second order Jahn-Teller distortion on the calculations of the pseudocubic lattice constants a’ (left) and a” (right), where a’ is a function of the A-X bond length and a” is a function of the B-X bond length.

Another factor that was taken into account by this model is the effect of ordering on the A-site. The model predicts that there should be an increase in the unit cell volume when there is some degree of ordering present on the A-site. This is exactly what was observed in both the Na(1-3x)/2La(1+x)/2TiO3 and Li(1-3x)/2La(1+x)/2TiO3 systems.

Image of a graph

Fig 4: Na[(1-3x)/2]La[(1+x)/2]TiO3 ordering from reported site occupancies where the open data points are extrapolated according to the empirical model. Inset shows the order parameter as a function of rA correction (Angstroms) and can be thought of as an empirical model for ordering (0=disorder, 1=fully ordered)

Inset shows the order parameter as a function of rA correction (Angstroms) and can be thought of as an empirical model for ordering (0=disorder, 1=fully

Fig 5: Li[(1-3x)/2]La[(1+x)/2]TiO3 order parameter and rA correction (Angstroms) where both are functions of x. Ordering from reported site-occupancy numbers. Inset shows the order parameter as a function of rA correction (Angstroms) and can be thought of as an empirical model for ordering (0=disorder, 1=fully ordered)

Recently, a similar model has been developed for the B-site ordered perovskite Ba(Mg1/3Ta2/3)O3 (BMT), which is known for developing long-range 1:2 cation ordering on the B site after sintering or annealing for long periods of time. It is clear from Fig. 6 that the volume shrinkage is directly related to increasing B-site order caused by increased annealing time. Moreover, Fig. 7 shows the inherent relationships between η, ΔrB, and t.  In particular, the inset shows that η decreases as ΔrB increases, corresponding to an increasing unit-cell volume.

Fig 6: Ba(MgTa)O3 unit-cell volume (black curve) and order parameter (red curve) as a function of annealing time

Fig 7: Ba(MgTa)O3 order parameter (red curve) and B-site size adjustment (blue curve) as a function of annealing time. The inset shows the order parameter as a function of ΔrB and can be thought of as an empirical model for ordering (0 = disorder, 1 = fully ordered)

A general model for A-site ordered perovksite titanates was recently developed using data from the layered A-site ordered (NayLi1-y)(1-3x)/2La(1+x)/2TiO3 (NLLT) system. Fig. 8 shows the value of ΔrA and the A-site order parameter, η, as functions of vacancy concentration, x, for (NayLi1-y)(1-3x)/2La(1+x)/2TiO3 (= 0, 0.25, 0.5, 0.75, 1). Fig. 9 shows η as a function of ΔrA. These figures clearly show that the empirical model based on random ionic occupancies increasingly underestimates the unit-cell volume as the degree of cation ordering increases. On the other hand, the model appears surprisingly insensitive to vacancy ordering, as the error in the model decreases to nearly zero as the vacancy concentration increases, as illustrated in Fig. 8.

All the curves in Fig. 8 can be described using Eq. 8 in Smith et. al. 2018.  These curves demonstrate that, for a given x, the magnitude of the ΔrA required decreases as the size of the A-site species increases. Similarly, the curves in Fig. 9 can be described using Eq. 9 in Smith et. al. 2018.

Fig 8: A-site size adjustment factors as functions of composition from experimentally collected data for (NayLi1-y)(1-3x)/2La(1+x)/2TiO3 and A-site order parameter, η, as a function of composition.

Fig 9: A-site order parameters as functions of ΔrA from experimentally collected data for (NayLi1-y)(1-3x)/2La(1+x)/2TiO3.

The curve fits in Fig. 10 demonstrate trigonometric relationships between all the coefficients (A, B, C, P, Q, S) of the general equations for the A-site order parameter, η, and the A-site size correction factor, ΔrA, and the ideal A-site size when x = 0 (rA(id)x=0), which are summarized in Eqns. 10 – 15 in Smith et. al. 2018. Importantly, Eqns. 8 – 15 in Smith et. al. 2018 allow for the prediction of both ΔrA and η using just published ionic radii data.

Fig 10: Coefficients of Eqn. 8 (left) and Eqn. 9 (right), in Smith et. al. 2018, as functions of A-site size, rA(id)x=0.

As the data used for these models have been derived for the NLLT system, 0 ≤ y ≤ 1, these general equations have an accuracy range for titanates in which 1.288 Å ≤ rA ≤ 1.375 Å.  While the model may be more generally applicable, the nature of empirical models makes it impossible to extrapolate beyond this range with certainty; and new data would be required in order to validate/revise the model.