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Computationally Assisted Identification of Functional Inorganic Materials

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Science  17 May 2013:
Vol. 340, Issue 6134, pp. 847-852
DOI: 10.1126/science.1226558

Modules of Desire

Using computational methods to design materials with specific properties has found some limited success. Dyer et al. (p. 847, published online 11 April) have devised a method, based on extended module materials assembly, that combines chemical intuition and ab initio calculations starting from fragments or modules of structure types that show the desired functionality. The method was tested by identifying materials suitable for a solid oxide fuel cell cathode.

Abstract

The design of complex inorganic materials is a challenge because of the diversity of their potential structures. We present a method for the computational identification of materials containing multiple atom types in multiple geometries by ranking candidate structures assembled from extended modules containing chemically realistic atomic environments. Many existing functional materials can be described in this way, and their properties are often determined by the chemistry and electronic structure of their constituent modules. To demonstrate the approach, we isolated the oxide Y2.24Ba2.28Ca3.48Fe7.44Cu0.56O21, with a largest unit cell dimension of over 60 angstroms and 148 atoms in the unit cell, by using a combination of this method and experimental work and show that it has the properties necessary to function as a solid oxide fuel-cell cathode.

The identification and synthesis of functional materials is a substantial challenge for both experiment and theory, especially for complex crystalline materials (those containing many atoms of different elements in distinct geometries). There are a vast number of arrangements of the atoms in the unit cell of a complex structure that need to be adequately sampled for structural identification and prediction (1). Different theoretical approaches to this problem have been developed; some involve an unbiased search through the different atomic arrangements (27), whereas others use existing chemical knowledge and understanding to reduce the number of arrangements considered (814).

In describing complex solid-state structures, it is customary to break them down in terms of modules or fragments and their combinations (15). Structural units—such as blocks, rods, or layers (as shown in Fig. 1, A and B, for two different structural families)—derived from archetypal structures are combined by structure-building operators. Within this approach, the combination of two or more layers from different crystal structures, a concept known as polysomatism (16), has proved powerful. The diverse families described as polysomatic series in terms of layers include heterogeneous structural series of intermetallics (17); light-element networks, such as the aluminum carbonitrides (18) (Fig. 1A); mineral families (19), such as the sapphirenes (20) and biopyriboles (16); and accretional series within the perovskite family (21), including Aurivillius and Dion-Jacobson phases, and families of hexagonal perovskites (22). It is possible to envisage polysomatic series using larger layers, including the complex ferrites (Fig. 1B), some of the largest-volume functional inorganic structures known (23). The functional behavior can be directly controlled by the constituent layers; for example, lone pair–based fluorite-like layers in the Aurivillius series can produce high-temperature ferroelectricity, whereas the Ba-containing R and T blocks (red and blue, respectively, in Fig. 1B) in the hexaferrites impose large magnetic anisotropy (24).

Fig. 1 Assembling structures with extended modules.

(A) A modular description of the aluminum carbonitride structural family. Modules of composition Al2C (blue), AlC (red), and AlN (orange) are used to describe the extended structures of Al6C3N2 and Al8C3N4 (23). Atoms are colored as follows: light blue, Al; black, C; and dark blue, N. Al-C/N tetrahedra are shaded in light blue to guide the eye. (B) A modular description of the hexaferrite structural family. Modules of composition Ba2Fe8O14 (T block, blue), (BaFe6O11)2– (R block, red), and (Fe6O8)2+ (S block, orange) are used to describe the extended structures of W- and Z-type hexaferrites (32). Atoms are colored as follows: green, Ba; brown, Fe; and red, O. Fe-O polyhedra are shaded in brown to guide the eye. (C) A schematic outlining the EMMA method of constructing and screening possible structures arising from extended modular units. In this case, four layered units are chosen, with two equivalent units. All 12 permutations of these units are built into structures, of which three are unique under translational symmetry. These three are tested, and the best structure chosen.

We use the combination of layers in the extended module materials assembly (EMMA) approach adopted here to identify new candidate structures as outlined in Fig. 1C. First, a set of extended layers (modules) must be chosen. Next, the number of times each module appears within a single repeated unit cell is selected. At this point, all possible permutations of the modules within the unit cell are created by using chosen structure-building operators. The present examples used translation along the long axis and inversion symmetry; however, other symmetry operations could be included. The number of candidate structures is reduced by removing duplicates of the same cyclical stacking sequence. Those remaining are ranked according to some suitable selection criteria to give the most promising candidates for experimental synthesis. We initially ranked a large number of structures by using energies obtained from a preliminary structural relaxation by classical force-field methods (25) (although any cost function suitable for the system under study could be used). The relative stabilities of the best candidates were evaluated quantitatively by using ab initio calculations (25). The extended modules encode bonding information directly into the candidate structures, mainly creating structures in which atoms are in chemically sensible environments where the use of force fields is valid. There is no conceptual limit to the nature or size of the modules, although we have concentrated on one-atom-thick metal oxide layers in this study. A practical limit exists on the number of modules within a single repeat, because the total number of permutations becomes very large with more than 15 to 20 modules. This limitation could be circumvented by searching through possible permutations with use of methods such as evolutionary algorithms (5, 7). Finite computational resources will also limit the overall size of the generated unit cell (a product of the number of modules and their individual sizes).

The perovskite ABO3 structure can be described by stacking AO rock-salt and BO2 square-layer modules that contain large A and smaller transition-metal B cations. There are a large number of possible superstructures derived by coupled cation-site and anion-vacancy ordering, such as YBa2Ca2Fe5O13, which functions as a solid oxide fuel-cell (SOFC) cathode (26). We select anion vacancy–containing modules to construct superstructures containing Fe3+ ions. Such units afford both electronic conductivity and oxide ion transport, which are required for functionality as an SOFC cathode. We validated the EMMA approach by reproducing the structure of this material. By generating a set of structures that use 2Y4 + 4Ba4O4 + 4Ca4O4 + 2Fe4O4 + 8Fe4O8 modules (Fig. 2A) in a cell of dimensions 2ap by 2ap by 10ap, where ap ≈ 3.9 Å is the dimension of an ABO3 unit cell, we obtained 14,190 structures with 184 atoms per unit cell. The AO1−δ and BO2−δ (0 ≤ Δ ≤ 1) modules were permuted separately. For each permuted arrangement of AO1−δ and BO2−δ modules, a candidate structure was constructed by stacking alternate AO1−δ and BO2−δ modules. During the initial relaxation (25), 7628 structures failed to converge because of poor starting geometries and were rejected. The three structures shown in Fig. 2C are significantly lower in energy than the others (Fig. 2B), with an energetic spread of only 0.02 eV per formula unit (FU, here YBa2Ca2Fe5O13) and a separation in energy of 0.15 eV per FU to the next structure; the experimental YBa2Ca2Fe5O13 structure is the second most stable. These four most stable structures were then relaxed by using density functional theory (DFT) (25) to improve the energetics. The most stable structure by 0.09 eV per FU using DFT (Fig. 2D) corresponds to the known experimental structure for YBa2Ca2Fe5O13, demonstrating that the EMMA approach can correctly identify a known structure.

Fig. 2 The 10ap structure of YBa2Ca2Fe5O13.

(A) The A- and B-site layered units used as the base structural modules throughout this work, (B) a graph showing the force-field energies of all calculated structures, (C) the three lowest-energy structures predicted from force-field calculations and their relative energies calculated by using both force-fields and DFT, (D) the lowest-energy structure from EMMA, and (E) the refined experimental structure. Atoms are colored as follows: yellow, Y; green, Ba; blue, Ca; brown, Fe; and red, O.

We experimentally explored the partial substitution of 7% Cu2+ for Fe3+ in order to increase the electronic conductivity of YBa2Ca2Fe5O13 and therefore improve its performance as an SOFC cathode. This resulted in a multiphasic sample, suggesting that other phases are accessible near this composition. One phase gave a powder x-ray diffraction pattern with Bragg reflections indicative of an 8ap repeat and an energy-dispersive x-ray spectroscopy (EDX)–derived cation composition of Y2.1(2)Ba1.8(2)Ca4.1(2)Fe7.4(2)Cu0.6(2) (the estimated standard error in units of the last quoted decimal place is shown in parentheses). We attempted to identify the structure of this phase by the EMMA method, which targeted the idealized compositions Y2Ba2Ca4Fe8O21 and Y2Ba2Ca4Fe7.5Cu0.5O21 (Fig. 3). These are candidate Fe3+/Cu2+ compositions based on eightfold multiples of the basic ABO3 perovskite formula unit with repeat dimension ap (8ap). By using these target compositions and a box of dimensions 2ap by 2ap by 8ap (where ap = 3.9 Å) that accommodates multiple orderings of the cations on the basis of the target composition, we generated 6300 structures for Y2Ba2Ca4Fe8O21 with use of three chemically sensible choices of modules, expanded from those used for YBa2Ca2Fe5O13 in which the constituent modules were already known. The module sets each included 2Ba4O4 + 4Ca4O4 + 5Fe4O8 modules and then a combination of either 2Y4 + Fe4O4 + 2Fe4O8, Y4 + Y4O4 + 2Fe4O4 + Fe4O8, or 2Y4O4 + 3Fe4O4 modules. The three sets of modules place the oxygen vacancies in different combinations of Y4 and Fe4O4 layers, allowing for flexibility in the final structures while using chemical intuition to guide the initial location of oxygen vacancies. A total of 17,640 structures for Y2Ba2Ca4Fe7.5Cu0.5O21 were constructed by using 2Ba4O4 + 4Ca4O4 + 2Y4 + 5Fe4O8 + Fe4O4 + 2Fe3CuO8 modules, with Cu atoms at different locations within the two Fe3CuO8 modules.

Fig. 3 EMMA calculations on structures with an 8ap repeat.

The energies of structures with the composition Y2Ba2Ca4Fe8O21 with converged force-field calculations are plotted in (A), and those with composition Y2Ba2Ca4Fe7.5Cu0.5O21 are plotted in (C). The most stable Y2Ba2Ca4Fe8O21 and Y2Ba2Ca4Fe7.5Cu0.5O21 structures are shown in (B) and (D), respectively. Atoms are colored as follows: yellow, Y; green, Ba; blue, Ca; brown, Fe; light blue, Cu; and red, O.

After initial relaxation of the structures (Fig. 3), one Y2Ba2Ca4Fe7.5Cu0.5O21 structure was 0.42 eV per FU more stable than all others (Fig. 3D). We relaxed the 20 lowest-energy structures at this composition by using DFT, confirming that the predicted lowest-energy structure is the same from both force-field and DFT calculations. The same lowest-energy structure was also found for Y2Ba2Ca4Fe8O21 (Fig. 3B). We note that extensive relaxation of local structure and cell dimensions away from the initial module-constrained structures was observed at the force-field stage, suggesting that considerable movement away from the starting structures is possible and reducing the bias in the prediction because of our original choice of modules (fig. S1). However, the lowest-energy structure was only found by using one of the three sets of starting modules for Y2Ba2Ca4Fe8O21, which illustrates the need to consider a variety of different starting modules to ensure identification of the optimal final structures.

In parallel with the EMMA calculations, the experimental phase diagram was investigated by synthesis at similar compositions (Fig. 4F). Synthesis of Y2Ba2Ca4Fe7.5Cu0.5O21 generated a polyphasic material containing a perovskite-derived phase with the apparent 8ap repeat (fig. S2). Competing phases observed in this system were two 10ap phases related to YBa2Ca2Fe5O13, a 3ap Sr2LaFe3O8-related structure (27), YCa4Fe5O13, and Y2O3. Isolation of the target compound in phase-pure form suitable for structure refinement and property characterization requires experimental optimization of the global composition and synthesis conditions (25) through a classical phase-diagram study (Fig. 4F) to afford compound 1, with the nominal composition Y2.24Ba2.28Ca3.48Fe7.44Cu0.56O21 [EDX-derived cation composition Y2.07(15)Ba2.49(20)Ca3.44(5)Fe7.63(18)Cu0.37(7)]. Selected area electron diffraction (SAED) (25) evaluation of this sample revealed that the longest axis in the unit cell is a 16ap repeat derived from body-centering superposition of the 8ap blocks (Fig. 4J and fig. S3). A compound with the same 16ap repeat but without any Cu in the material, 2, has been synthesized as part of a polyphasic sample at a nominal composition of Y1.95Ba2.1Ca3.95Fe8O21 (fig. S4). A second series of EMMA calculations were performed on materials with the composition Y2Ba2Ca4Fe8O21 in the enlarged 16ap repeat, adding an inversion center to generate the body centering observed by SAED on the experimental compound to reduce the number of total permutations of modules. By using the experimental SAED results to inform the EMMA procedure, we reduced over 108 possible structures with 16ap repeats to a restricted set of 47,040 candidate structures that have body centering in agreement with experiment (25).

Fig. 4 The 16ap structure.

(A) The 16ap structure as predicted by EMMA is compared with (B) the refined structure of 1. Atoms are colored as follows: yellow, Y; green, Ba; blue, Ca; brown, Fe/Cu; and red, O. (C) Overlaid EMMA (green) and refined experimental (red) structures. (D) Observed and (E) calculated HAADF-STEM images of the 16ap phase. (F) Compositional phase diagram showing the compositions investigated to isolate 1 (open blue circles) and 2 (solid brown circles). The compositions of 1 and 2 are highlighted with a star. (G) A Z-Z* plot showing cathode performance of 1 in a symmetrical cell on a GDC electrolyte at temperatures between 923 and 1073 K. (H) ASR values obtained at temperatures between 773 and 1073 K for 1 compared to literature values reported for 10ap (26) and LSCF (31). Ea, activation energy. (I) Refined x-ray diffraction data (25) showing observed (black crosses), calculated (orange line), and difference (gray line) data as a function of Q, the momentum transfer. Tabulated crystallographic data can be found in table S1 and refined neutron powder diffraction data in fig. S13. An electron diffraction pattern of 1 showing the 16ap repeat (J) and an enlarged high-Q region of the diffraction data (K) are shown inset.

The final predicted 16ap structure from EMMA (Fig. 4A) contains five elements with a 61 Å longest repeat. The 16ap structure consists of nine crystallographically distinct cation sites. There are four Fe sites in three different coordination environments: one square pyramidal (Sq), two octahedral (Oh), and one tetrahedral (Td). Y cations occupy two 8-coordinate sites (between Sq Fe layers and between Oh Fe layers), whereas the larger Ba cations occupy a 12-coordinate site (between Sq and Oh Fe layers). Ca cations occupy two chemically similar 8-coordinate sites (between Oh and Td Fe layers). The total B-site stacking sequence can be represented as SqOhTdOhOhTdOhSq. We note that the 16ap structure is related to the lowest-energy 8ap structure found in the first set of calculations by the addition of body centering; the smaller 8ap unit cell dimensions still allowed sufficient complexity for the observed local chemical environments of each ion to emerge.

The symmetrically unconstrained 16ap structure from EMMA was transformed into the experimentally (SAED) determined symmetry of Imma (fig. S5) and used as a starting point for Rietveld refinement with combined x-ray and neutron powder diffraction data collected on the experimentally synthesized phase 1 (Fig. 4I) (25). During the course of the refinement, the phase composition was changed from the idealized EMMA 16ap composition of Y2Ba2Ca4Fe8O21 to that of 1, Y2.24Ba2.28Ca3.48Fe7.44Cu0.56O21. The experimentally determined structure from Rietveld refinement closely resembles the EMMA-predicted structure (Fig. 4, B and A, respectively), with complete preservation of the predicted Fe coordination environments. The determined A-site distribution shows the same majority cation ordering as predicted by EMMA but with significant cation mixing between layers, which is both made necessary by the difference between the EMMA composition and the experimental composition and expected because of entropy considerations at the synthesis temperature. The fractional cation occupancies can be modeled by swapping the cations on the A sites in the EMMA-derived structure and using finite temperature Monte Carlo sampling (25, 28). The similarities between the predicted and refined structures are notable (Fig. 4C). The diffraction-based long-range average structure was observed in shorter length-scale high-angle annular dark field scanning transmission electron microscopy (HAADF-STEM) (25) imaging of 1 (Fig. 4D), which was in good agreement with a simulated HAADF-STEM image of the refined structure (Fig. 4E and fig. S6). Mössbauer spectroscopy (25) also confirmed that the structure contains only Fe3+ in Oh, Sq, and Td environments in a 2:1:1 ratio, in agreement with the calculated and refined experimental structures (fig. S7) and iodometric titrations (25).

The 16ap material 1 has the structural and electronic prerequisites for mixed ionic and electronic conduction because of the presence of open-shell Fe3+ and anion vacancies, which was directly encoded in the choice of EMMA modules, suggesting its application as an SOFC cathode. It showed good kinetic stability, with no detectable decomposition after heating at 950°C for 5 hours. No reactivity was observed with typical electrolytes Ce0.8Sm0.2O2−δ (SDC), Ce0.9Gd0.1O2−δ (GDC), and La0.9Sr0.1Ga0.8Mg0.2O3−δ at common intermediate temperature (IT)–SOFC operating and processing temperatures of 1023 and 1223 K, respectively (fig. S8). The 16ap material 1 shows an improved stability to the state-of-the-art barium-rich cathode material, Ba0.5Sr0.5Co0.8Fe0.2O3−δ (BSCF), which decomposes to a hexagonal phase (29) and reacts with SDC (30) under the same conditions used in the reactivity and stability tests. Symmetrical cells were fabricated by screen-printing porous cathode layers on both faces of a dense GDC electrolyte disc, and the resulting area-specific resistance (ASR) value of 0.27 ohm cm2 at 973 K compares well with the common IT-SOFC cathode La0.6Sr0.4Co0.2Fe0.8O3−δ (LSCF, 0.46 ohm cm2) (31) and the layered 10ap YBa2Ca2Fe5O13 cathode (0.88 ohm cm2) (26) (Fig. 4H). dc conductivity measurements are shown in fig. S9. The EMMA method has therefore identified a structure that has experimentally demonstrated functionality as an SOFC cathode material.

Compounds 1 and 2 are likely to be metastable at temperatures below ~1250 K, based on both experimental and computational evidence. To produce phase-pure 1 (16ap), it is necessary to quench from the synthesis temperature (fig. S10); slow cooling produces a mixed phase sample containing a significant proportion of 10ap. Calculated enthalpies of reaction at 0 K based on DFT predict that Y2Ba2Ca4Fe8O21 (16ap) is less stable than a multiphase assemblage of YFeO3 + Ca2Fe2O5 + YBa2Ca2Fe5O13 (10ap). Finite temperature effects must be included to explain the synthesis of 1 and 2 (supplementary text). In a complex compositional range of this type, reaction conditions often have a controlling influence on phase stability, leading to the conclusion that a practical target is to develop computation to assist rather than to direct the synthesis of such complex phases. Nevertheless, we have demonstrated the use of EMMA in searching a phase diagram to find new structures, by investigating the (Y, Ba, Ca, Fe3+, O) phase diagram for layered phases with 7ap long-axis repeats (supplementary text and fig. S11).

By including an initial bias toward the use of specific modules, EMMA provides a compromise between exhaustive approaches, which seek to sample all of phase space, and the need to predict complex structures. It can identify candidate structures for compositionally and structurally complex inorganic materials, given a reasonable choice of starting modules. Furthermore, the large set of ranked structures generated by EMMA contains a wealth of information, which can be analyzed further to gain more general understanding (supplementary text and fig. S12). The modular approach adopted in EMMA allows desired functional properties to be built into the predicted structures; the selection of modules containing the relevant chemical bonding known to give rise to functional behavior in other materials allows the targeting of specific functions.

Supplementary Materials

www.sciencemag.org/cgi/content/full/science.1226558/DC1

Materials and Methods

Supplementary Text

Figs. S1 to S13

Table S1

References (3353)

References and Notes

  1. Materials and methods are available as supplementary materials on Science Online.
  2. Acknowledgments: This work is funded by the European Research Council (ERC Grant agreement 227987 RLUCIM). It was carried out with the support of the Diamond Light Source (DLS) and ISIS. We thank C. Tang, J. Parker, and S. Thompson for assistance in using beamline I11 (DLS) and A. Daoud-Aladine for assistance in using the High Resolution Powder Diffractometer (HRPD, ISIS). CCDC 887926 contains the supplementary crystallographic data for this paper. These data can be obtained free of charge via www.ccdc.cam.ac.uk/data_request/cif, by e-mailing data_request@ccdc.cam.ac.uk, or by contacting the Cambridge Crystallographic Data Centre, 12, Union Road, Cambridge CB2 1EZ, UK; fax: +44 1223 336033. We also thank M.S. Islam (Department of Chemistry, University of Bath) for helpful discussion with regards to fitting force-field parameters. Via our membership of the United Kingdom’s HPC Materials Chemistry Consortium, which is funded by Engineering and Physical Sciences Research Council (EPSRC) (EP/F067496), this work made use of the facilities of HECToR, the United Kingdom’s national high-performance computing service, which is provided by UoEHPCx Limited at the University of Edinburgh, Cray Incorporated and NAG Limited, and funded by the Office of Science and Technology through EPSRC’s High End Computing Programme. We thank EPSRC for studentships for C.C. and D.H. M.J.R. is a Royal Society Research Professor.
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