Predicting the Structural Stability and Formability of

ABO3-Type Perovskite Compounds Using Artificial

Neural Networks

Y M Zhanga, R Ubicb, D F Xuec and S Yangd*

a Ningbo Institute of Material Technology & Engineering,

Chinese Academy of Sciences, Ningbo, Zhejiang Province, P.R. China.

b Department of Materials Science & Engineering, Boise State University

1910 University Drive, Boise, Idaho 83725, USA

c State Key Laboratory of Rare Earth Resource Utilization, Changchun Institute

of Applied Chemistry, Chinese Academy of Sciences, Changchun 130022,

China

d Engineering Sciences, Faculty of Engineering and the Environments,

University of Southampton, Southampton, SO17 1BJ, UK.

* Corresponding author. Email: s.yang@soton.ac.uk

Abstract

Information on crystal structure is a prerequisite for materials modelling. Several methods

have been developed in the last few years to determine crystal structures of perovskites, an

important structure with diverse physical/chemistry properties. In this work, an artificial

neural networks (ANNs) modelling method is used to make predictions of global instability

index (GII) from bond-valence based tolerance factors tBV. It is found that, by normalizing

with the valence of A-site cations, it is possible to deduce a unified correlation between GII

and tBV for general ABO3-type perovskites. The ANNs were also used to make predictions

about the formability of perovskites by using A-O and B-O bond distances. This method

improves prediction accuracy. In the future it is hoped that the introduction of more

parameters will increase the prediction accuracy, as ANN has a unique ability to make

predictions in high dimensional spaces.

Key words: crystal structure, perovskite, data mining, artificial neural networks, global

instability index, tolerance factors

1

1.0 Introduction

1.1 Crystal structure determination

Crystal-structure and materials properties are intimately linked together, so obtaining

information on crystal structures continues to be an area of ongoing research. Presently,

crystallographic information is a prerequisite for any extensive materials modelling, making

knowledge of crystal structures a more practically-pressing problem (Morgan and Ceder,

2005).

There are three main different methodologies for determination of crystal structures. (1)

Crystal structures can be determined experimentally. This kind of work began in 1911 by von

Laue and his colleagues (Freidrich et al., 1912); and then Bragg (1913) performed crucial

experiments that determined the structures of some very simple crystals. However, the

determination of such information is very time- and cost- expensive. (2) Nowadays, the

evolution of high-speed computers and the derivation of one-electron potentials, which

greatly simplify many-body interactions, make it possible to predict structures using quantum

mechanic principles, as has been demonstrated by Wolverton et al. (2001), Asta et al. (2001),

Blum and Zunger (2004) and Curtarolo et al. (2005). These modelling efforts have made

computational materials design a real possibility for many years; however, difficulties still

exist: i) As Chelikowsky (2004) wrote in a review: “Although the interactions in …

compounds are well understood, it is not an easy task to evaluate the total energy of solids…

As the energy of an isolated atom is in the order of about 106 eV, but the cohesive energy only

in the order of about 1-10 eV/atom, one must have a method that is accurate to one part on

106, or better.” ii) The number of particles involved complicates the situation of evaluating the

cohesive energy and makes the computations less accurate than experiment (Villars, 2000).

(3) With the help databases of known structures or models which have physical meanings,

certain regularities, such as laws, rules, principles, factors, tendencies or patterns, might be

found to help predict unknown structures. This method, known as structure mapping, is one

kind of the most successful method for crystal structure prediction and has been review by

2

Burdett and Rodgers (1994), Villars (2000) and Pettifor (2000). Examples of structure

mapping include Mooser-Pearson Maps (1959), Zunger Maps (1980), and Villars maps

(1983, 1984a, b), but the best known are Pettifor maps (1984, 1986). This method orders the

very large database into domains of different structure types and can be used as an initial

guide in the search for new compounds with a required structure type. Combinatorial methods

from both computational (Yu et al. 2009, Hautier et al. 2012) and experimental (Pullar

2007a, 2007b, Zhang 2007, Rossiny 2008, Scott 2008, Chen 2011a, 2011b) approach have

been used to search new materials and explore crystal-structure and materials properties

correlations. In those situations when a large amount of new materials data is involved, a

simple and fast evaluation method is needed. In this paper the application of ANN on the

parameters of tolerance factor tBV have shown a faster and accurate evaluation of the

structural stability of perovskites.

1.2 Data mining method in crystal structure prediction

The data mining (DM) method has become a powerful tool in different branches of materials

science. This technique offers the possibility of making descriptive and even quantitative

predictions in many areas where traditional approaches have had limited success. Generally,

people try to make predictions through constitutive relations, that is, derived mathematically

from basic laws of physics; however, in many cases, the problems are so complex that

constitutive relations either cannot be derived or are too approximate or intractable for

practical quantitative use. The data mining method is based on the assumption that the useful

constitutive relations exist, and these relations could be derived primarily from data rather

than from basic laws of physics (Morgan and Cerbrand, 2005).

Due to the fact that there is no generally reliable and tractable method to predict structure, and

considering that a lot of structural data exist in crystallographic databases such as ICSD

(Bergerhoff et al., 1983), Linus Pauling files (Villars et al., 1998), CRYSTMET (White et

al., 2002), and others mentioned in the review paper by Allen (1998), crystal structure

3

prediction is an area well suited for data mining. Examples include works by Woodley et al.

(1999), Curtarolo et al. (2003), Fischer et al. (2006), Kazius et al. (2006) Hautier et al. (2011)

and, in particular, Morgan et al. (2003), who developed a clustering algorithm to automate the

construction and testing of Pettifor maps based on data from a materials crystal structure

database.

1.3 Structural stability and formability of ABO3-type perovskite compounds

Due to the diverse physical/chemistry properties of perovskite and perovskite-related

materials, which can be applied in a variety of fields, it is useful to discover the regularities

that govern the formation of perovskite-type compounds, and then use them to further guide

the exploration of new materials with unknown structures (Zhang et al., 2007).

The study of ABO3 compounds has a long history. Megaw (1946) accurately determined the

structure of a number of doubled perovskites by examining high-angle lines on X-ray powder

photographs. Salinas-Sanchez et al. (1992) introduced a parameter called the global instability

index (GII), which can be used to determine the overall structural stability of perovskites.

Giaquinta and Loye (1994) later predicted the perovskite structure for a number of

compounds based on the combination of ionic radii and bond ionicities, predicting the

structure of InMO3 (M = Mn, Fe) with this method. Reaney and Ubic (1999) reviewed the

relationship between the tolerance factor (t) and the temperature coefficient of permittivity

(τε) (and therefore resonant frequency, τf) in perovskites and also discussed the effect on

of changing t on τε in the perovskite-related solid solution series Ba6-3xNd8+2xTi18O54. Lufaso

and Woodward (2001) used a bond-valence model to calculate ideal A – X and B – X bond

distances in order to calculate the bond-valence based tolerance factor (tBV) which they

proposed as a new criterion of the structural stability of ABO3-type perovskite compounds.

Ye et al. (2002) applied a pattern recognition method and found some regularities in the

formation and the lattice distortion of perovskites. Li et al. (2004) used rA – rB structure maps

to study the perovskite formability of 197 ABO3 compounds; and then, with the knowledge of

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the importance of the octahedral factor (rB/rO) and tolerance factor ( ), another

structure map was constructed to predict the perovskite formability. Jiang et al. (2006)

investigated the regularities governing lattice constant of ABX3-type compounds by a

statistical regression method and found a good correlation among lattice constant, the

estimated bond length between ion B and ion X (rB + rX) and the ionic-radii tolerance factor

tIR; however, they used ionic radii of all ions A, B and X appropriate for sixfold coordination,

which are inappropriate for A and X. Ubic (2007) later used a different approach based solely

on ionic size and derived a simpler and more accurate empirical relationship between ionic

size and lattice constant. In 2007, Zhang et al., based on the bond-valence model (BVM) and

structure-map technology, investigated the structural stability and formability of ABO3-type

perovskite compounds, established new criteria for the structural stability of such compounds,

and created new structure maps that can be used for exploring novel perovskite-related

compounds. In that work, the calculated global instability indices GIIs are compared with

bond-valence based tolerance factors tBVs for 232 ABO3-type perovskite compounds, and the

results are shown in figure 1. From this result, they found the GII values are close to 0 when

tBV values approach 1, and all GII values are less than 1.2 v.u. for ABO3-type perovskite

compounds. Also, it was found that the structural stability of ABO3-type perovskite

compounds follows the order A1+B5+O3-type > A2+B4+O3-type > A3+B3+O3-type, which agrees

with the experimental measurements. In 2008, Feng et al. used a method similar to that of Li

et al. (2004) to predict the formability of ABO3 cubic perovskite using the octahedral factor

and tolerance factor. Recently, Verma et al. (2008) developed an equation to predict lattice

constant values for cubic perovskites by using the product of ionic charges and average radii

of ions A, B and X, but the results are much less accurate than those of either Lufaso and

Woodward (2001) or Ubic (2007).

This work is based on the structural stability and formability data of ABO 3-type perovskites

used by Zhang et al. (2007) and makes use of the neural network data-mining method to 1)

5

test whether it is possible to use a backpropagation neural network to make the prediction of

GII from tBV within each ABO3-type perovskite subclass (i.e., in A1+B5+O3 type, A2+B4+O3

type, and A3+B3+O3 type), and within the whole ABO3-type perovskite; 2) test whether using a

probabilistic neural network could improve the prediction accuracy for perovskite formability

based on the ideal A-O and B-O bond distances (dA-O and dB-O) compared to the work done by

Zhang et al. (2007).

2.0 Experimental Details

2.1 Configuration of the neural networks

In this work, two different neural networks were used. One is backpropagation neural network

(BPANN), which is versatile and can be used to model mapping problems; the other is

probabilistic neural network (PNN), which is a type of radical basis network suitable for

classification problems.

2.2 BPANN

The construction of the BPANN involves the choices of i) number of neural network layers,

ii) number of neurons in the first hidden layer, iii) method for improving generalization, iv)

partitioning of the database and v) data normalization method. The details of this technique

can be found in previous work (Zhang et al., 2008, 2010a, 2010b, 2011).

2.3 Partitioning of the database for PNN

As for BPANN, the generalizing ability of PNN depends on the training database size. It is

necessary to generalize for the unseen data, so the data set used for training should be large

enough to cover the possible known variation in the problem domain. The parent database

here is partitioned into two sub-sets: training and testing. Following the method used in

construction of BPANN, as discussed previously (Zhang et al., 2008), the sets are picked as

equally spaced points throughout the original data. The whole data set is partitioned into five

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groups, four groups being used for training, and one group is used for testing. Because the

problem domain is not clear in this work, a loop program was used to redistribute the database

in order to make the training set cover the problem domain. One way that can be used to view

the results is a confusion matrix, where matrix element mij gives the number of times a sample

belonging in class Ci was assigned to Cj. As the result, the distribution was selected on the

basis of the total number of samples incorrectly placed into the class (false positives).

The ideal value of should be zero.

2.3 Choice of spread for PNN

The spread of radial basis functions in PNN need to be chosen. As mentioned in MATLAB

(Mathworks), if spread is near zero, the network acts as a nearest neighbour classifier; as

spread becomes larger, the designed network takes into account several nearby design

vectors. In this work, a looped program is used in order to find the best combination of

database distribution and value of spread.

2.4 Determination of input and output parameters

This work was composed of two different parts: 1) prediction of the global instability indices

(GII) from bond-valence based tolerance factor (tBV) using backpropagation neural network

(BPANN); 2) prediction of the perovskite formability based on the ideal A-O and B-O bond

distances (dA-O and dB-O) using probabilistic neural network (PNN).

In part 1, the bond-valence based tolerance factor (tBV) and global instability indices (GII) are

input and output of BPANN, respectively; in part 2, the inputs are A-O (dA-O) and B-O (dB-O)

bond distances, and the output are the formability of perovskites.

3.0 Results

3.1 Prediction of GII from bond tBV

7

From the three curves in figure 1, it is interesting to see whether there is a correlation between

GII and tBV, that is, can one of them, say tBV, be used to predict the other (GII in this case)?

The results of using tBV to predict GII for A1+B5+O3 type perovskites, A2+B4+O3 type

perovskites and A3+B3+O3 type perovskites are shown in figures 2(a) and 2(c) in which the

blue circles represent the training datasets and the purple stars represent the testing datasets.

The best linear fit equations and regression coefficients, R, show that they give sensible

correlations; R=0.999 for figure 2(a), R=0.997 for figure 2(b) and R=0.999 for figure 2(c).

These three results indicate that correlations exist between the tolerance factor ( tBV) and global

instability index (GII) for each subclass of perovskites (i.e., A1+B5+O3, A2+B4+O3, A3+B3+O3).

The reasons will be discussed later.

It is also interesting to know whether a uniform correlation between the tolerance factor ( tBV)

and global instability index (GII) exists for the whole class of ABO3 perovskites. In order to

address this question, a similar procedure is used to make predictions of GII from tBV. The

result is shown in figure 2(d). By comparing the best linear fit equation and regression

coefficient, R, in this figure (R=0.949) with the results shown in figure 2(a-c), it is found that

a correlation between GII and tBV for all ABO3 perovskites exists, but is not as

straightforward as those shown for the individual classes of previous separately.

There are three different underlying links between variables that can explain the observed

correlations, as shown in figure 3. The dashed line represents an observed association

between the variables x and y. Some association can be explained by a direct cause-and-effect

link between the variables. In figure 3(a), an arrow running from x to y shows “x causes y”.

When thinking about an association between two variables, lurking variables need to be

considered. figure 3(b) illustrates the common response, in which the observed association

between the variables x and y can be explained by a lurking variable z; and both x and y

change in response to changes in z. The common response creates an association even though

there may be no direct causal link between x and y. figure 3(c) shows confounding, in which

8

both the explanatory variable x and the lurking variable z may influence the response variable

y. Since the effect of x is confounded with the effect of z, the influence of x cannot be

distinguished from the influence of z; also, it cannot be said how strong the direct effect of x

on y is. It is difficult even to tell if x influences y at all.

From bond-valence model, the bond valence of the ith ion with respect to the jth ion, sij, is

calculated using equation (1)

(1)

where dij is the cation-anion bond distance; Rij is empirically determined for each cation-anion

pair based on a large number of well-determined bond distances for the cation-anion pair in

question, and has been tabulated; and B is an empirically determined universal constant with a

value of 0.37.

From the calculated bond valence sij, the atomic valences of ith ion Vi(calc.) can be calculated

from equation (2) by summing the individual bond valences (sij) about each ion:

(2)

Then, the discrepancy factor di can be calculated from equation (3)

(3)

where Vi(OX) is the formal valence for the ith ion.

The overall structural stability, referred as the global instability index (GII), is determined by

comparing the calculated atomic bond-valence Vi(calc.) and formal valence Vi(OX), and calculated

from equation (4)

(4)

9

where N is the number of atoms in the asymmetric unit. The bond-valence based tolerance

factor, tBV, shown in equation (5), is calculated like the tolerance factor ( ), but

substituting rA+rO and rB+rO with bond distances A-O (dAO) and B-O (dBO) calculated from the

bond-valence model.

(5)

From the analysis above, it can be found that both the global instability index (GII) and bond-

valence tolerance factor (tBV) are calculated fundamentally from dij and other parameters, and

so there is a common response, as indicated in figure 3(b), between GII and tBV. This can

explain the results for each ABO3 type (A1+B5+O3, A2+B4+O3, A3+B3+O3) shown in figure 2(a)-

2(c); however, for the whole range of ABO3-type perovskites, the correlation is a bit poor,

which indicates that there is no unified relationship between tBV and GII for the all types of

ABO3 perovskite; however, if each GII value in figure 1 were normalized by the valence of

A-site cations, a unified correlation would emerge. The result is shown in figure 4. Again,

ANN was used to make the prediction of GII values from tBV and the valences of A-site

cations, and the result is shown in figure 5.

3.2 Prediction of perovskite formability

In this part, a probabilistic neural network (PNN) is used to simulate the work done by Zhang

et al. (2007) to predict the likelihood zone of perovskite compounds being formed, based on

the same parameters they used, i.e., ideal A-O and B-O bond distances derived from bond-

valence model (BVM). The predictions are made for each type of perovskite (A1+B5+O3,

A2+B4+O3 and A3+B3+O3) and also for all ABO3-type perovskites collectively. The results are

shown in figure 6(a-d).

10

Figure 6(a) shows that six A1+B5+O3 structures are located in the wrong zone (six non-

perovskites locate in perovskite zone), whereas in Zhang et al.’s (2007) work, seven such

structures are located in the wrong zone. Similarly, figure 6(b) shows 11 incorrect A 2+B4+O3

perovskite assignments, whereas Zhang et al.’s (2007) have 17 incorrect predictions; and

figure 6(c) shows four incorrect A3+B3+O3 perovskite assignments where Zhang et al. (2007)

shows two.

From these three comparisons, it is found that in the first two cases, the neural network

performs a bit better in terms of predictive accuracy; however, in Zhang et al.’s work (2007),

the boundaries for separating perovskite and non-perovskite zones are regular, which helped

them to point out the conditions that determine the formability of ABO3-type perovskite

compounds. In the third case, Zhang et al. (2007) made all the predictions of non-perovskite

structures correct but made two incorrect predictions of perovskites, while the neural network

makes predictions of all perovskites correctly, but makes four false predictions of non-

perovskites. Overall, the neural network performs worse in this case.

5.0 Conclusions and Further Work

In this work, 1) A neural network modelling method is used to make predictions of global

instability index GII from bond-valence based tolerance factors tBV for 232 ABO3-type

perovskite compounds that Zhang et al. (2007) used and found that correlations exist between

these two parameters within each subclass of ABO3-type perovskite compounds; however, for

the general ABO3-type perovskite, this kind of correlation is not obeyed. It was found that the

valence of A-site cations can be treated as another parameter to induce a unified correlation.

2) Neural networks have also been used to make the prediction of formability of perovskites

by using A-O and B-O bond distances in order to find the possibility of improving prediction

accuracy. It was found that, in terms of the prediction accuracy, neural networks can yield

better performances, but it is difficult to give simple and physically meaningful explanations.

11

Neural networks have the ability to make predictions in high dimension spaces, i.e., they can

have more than two inputs to make the prediction. For the prediction of formability of ABO3-

type perovskite compounds, as mentioned by Zhang et al. (2007), besides the A-O and B-O

bond distances, other factors also contribute to the formation of ABO3-type perovskite

compounds, such as i) bond-valence based tolerance factor tBV, ii) temperature and iii)

pressure; therefore, these parameters can be incorporated into future work in order to improve

the accuracy of predictions, which can be used for practical applications.

12

Captions of Figures

Figure 1 Global instability indices (GII) versus bond-valence based tolerance

factors (tBV) for ABO3-type perovskite compounds. (Redrawn from

Zhang et al., 2007)

Figure 2 Prediction of global instability indices (GII) from bond-valence based

tolerance factors (tBV) for (a) A1+B5+O3; (b) A2+B4+O3; (c) A3+B3+O3 and (d) all

type of ABO3 perovskite compounds.

Figure 3 Explanations for an observed association. The broken lines show an

association. The arrows show a cause-and-effect link. The x and y are

observed variables, and z is a lurking variable (Redrawn from Moore

and McCabe, 1999)

Figure 4 Global instability indices (GII) normalised by the valences of A-site cations

versus bond-valence based tolerance factors (tBV) for ABO3-type perovskite

compounds.

Figure 5 Prediction for perovskite formability of all ABO3-type compounds

from tBV and valences of A ions by ANN.

Figure 6 Prediction for perovskite formability of the (a) A1+B5+O3; (b) A2+B4+O3; (c)

A3+B3+O3 and (d) all type of ABO3 by ANN.

13

-0.10

0.10.20.30.40.50.60.70.80.9

11.11.21.31.41.5

0.8 0.85 0.9 0.95 1 1.05 1.1 1.15

t BV

GII

(u.v

.)A¹⁺B⁵⁺O₃A²⁺B⁴⁺O₃A³⁺B³⁺O₃

Figure 1

14

0 0.2 0.4 0.6 0.8 1 1.20

0.2

0.4

0.6

0.8

1

1.2

Experimental GII Values (T)

Pred

icte

d fro

m N

N (A

) A = (0.996) T + (0.00335)

R = 0.999

Training Data PointsTest Data PointsBest Linear FitA = T

Figure 2(a)

15

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Experimental GII Values (T)

Pred

icte

d fro

m N

N (A

)

A = (0.986) T + (0.00734)

R = 0.997

Training Data PointsTest Data PointsBest Linear FitA = T

Figure 2(b)

16

0 0.2 0.4 0.6 0.8 1 1.2 1.40

0.2

0.4

0.6

0.8

1

1.2

1.4

Experimental GII Values (T)

Pred

icte

d fro

m N

N (A

)

A = (0.998) T + (0.000384)

R = 0.999

Training Data PointsTest Data PointsBest Linear FitA = T

Figure 2(c)

17

0 0.2 0.4 0.6 0.8 1 1.2 1.40

0.2

0.4

0.6

0.8

1

1.2

1.4

Experimental GII Values (T)

Pred

icte

d fro

m N

N (A

)

A = (0.907) T + (0.0604)

R = 0.949

Training Data PointsTest Data PointsBest Linear FitA = T

Figure 2(d)

18

19

X Y Y YX

Z Z

X

Causation(a)

Common response(b)

Confounding(c)

Figure 3

-0.10

0.10.20.30.40.50.60.70.80.9

11.11.21.31.41.5

0.8 0.85 0.9 0.95 1 1.05 1.1 1.15

t BV

GII

(u.v

.)/V

alen

ce o

f A io

nA¹⁺B⁵⁺O₃A²⁺B⁴⁺O₃A³⁺B³⁺O₃

Figure 4

20

0 0.2 0.4 0.6 0.8 1 1.2 1.40

0.2

0.4

0.6

0.8

1

1.2

1.4

Experimental GII Values (T)

Pred

icte

d fro

m N

N (A

)

A = (0.998) T + (0.000753)

R = 0.999

Training Data PointsTest Data PointsBest Linear FitA = T

Figure 5

21

Figure 6(a)

Figure 6(b)

23

Figure 6(c)

24

Figure 6(d)

25

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