An Efficient Scheme for Crystal Structure Prediction Based on

Article pubs.acs.org/JPCC

An Efficient Scheme for Crystal Structure Prediction Based on Structural Motifs Zizhong Zhu,*,† Ping Wu,‡ Shunqing Wu,† Linhan Xu,† Yixu Xu,† Xin Zhao,§ Cai-Zhuang Wang,∥ and Kai-Ming Ho*,‡,§ †

Collaborative Innovation Center for Optoelectronic Semiconductors and Efficient Devices, Department of Physics, Xiamen University, Xiamen 361005, China ‡ International Center for Quantum Design of Functional Materials (ICQD), and Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei, Anhui 230026, China § Department of Physics and Astronomy, Iowa State University, Ames, Iowa 50011, United States ∥ Ames Laboratory - US DOE, Ames, Iowa 50011, United States ABSTRACT: An efficient scheme based on structural motifs is proposed for the crystal structure prediction of materials. The key advantage of the present method comes in twofold: first, the degrees of freedom of the system are greatly reduced, since each structural motif, regardless of its size, can always be described by a set of parameters (R, θ, φ) with five degrees of freedom; second, the motifs could always appear in the predicted structures when the energies of the structures are relatively low. Both features make the present scheme a very efficient method for predicting desired materials. The method has been applied to the case of LiFePO4, an important cathode material for lithium-ion batteries. Numerous new structures of LiFePO4 have been found, compared to those currently available, demonstrating the reliability of the present methodology and illustrating the promise of the concept of structural motifs.

1. INTRODUCTION

new materials by computational predictions, which are then synthesized experimentally and have useful applications. Many challenges exist in the theoretical prediction of crystal structures. The number of minima in the potential energy surface (PES) of a large assembly of atoms increase exponentially with the number of atoms, leading to a very difficult task to find the most stable structure of a large system. Actually, metastable crystal structures of many materials will appear when the materials grow, work as a device, or experience high temperature or pressure. For example, for the important cathode material for lithium-ion batteries, LiFePO4, which we will study in this paper, the structure may not be in the ground state during the electrochemical cycles. Although the structure prediction remains a very difficult problem, steady progress has been made over the past years. The progress in the searching methodologies has led to many successful structure predictions. These methods include adaptive genetic algorithm (AGA)1 and genetic algorithm (GA),4−7 simulated annealing,8−10 basin hopping,11,12 ab initio random structure search (AIRSS),2 and particle swarm optimization (CALYPSO) methods.3 Here we describe our simple, elegant, and powerful new approach to search for structures with density functional theory (DFT), which we call the motif-based structure searching

Crystal structure is one of the most fundamental factors in many areas of physics, chemistry, and materials science, since the unique properties of a material are intimately tied to its geometrical structure. Predicting crystal structures based on pure theoretical efforts is important for a number of reasons: (1) Computational searching can be much cheaper and easier than experiments, since a lot of systems can be quickly searched by theoretical calculations, which often obtain interesting results and sometimes discover promising new materials.1−6 (2) Computational searches can be employed to investigate materials under special conditions which might not be accessed experimentally, for example, materials under extremely high pressures (like deep interiors of the massive planets). (3) Apart from the global lowest energy minimum, low-energy metastable minima are also interesting, since they can be accessed at finite temperatures, or under pressure. During growth or processing, the materials may also be trapped in metastable minima structures. Theoretical prediction of metastable minima is, generally speaking, easier than experiments. (4) Computational searches can help in understanding the experimental studies, when the experimental data are incomplete. For example, powder diffraction data may be insufficient for a complete structural determination but may provide substantial help in the determination of the unit cell and likely the space group.2 Finally, the most exciting possibility might be the discovery of © 2017 American Chemical Society

Received: March 16, 2017 Revised: May 6, 2017 Published: May 15, 2017 11891

DOI: 10.1021/acs.jpcc.7b02486 J. Phys. Chem. C 2017, 121, 11891−11896

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The Journal of Physical Chemistry C

It should be pointed out that “good” structures can be predicted if there are interatomic potentials at hand for the studied material, since the predicted structures can then be relaxed on the basis of the known interatomic potentials. For example, “excellent” structures have been found if the Tersoff potential13 for C or Si is adopted when predicting the polymorphs of graphene or silicene. Since the creation of a structure, as mentioned above, costs very little time, a huge number of random structures can be generated. These structures need to be “filtered”, in order to have a moderate-size structure pool to be calculated by the firstprinciples method. If there are interatomic potentials at hand, each predicted structure can then be geometrically optimized on the basis of the interatomic potentials (as mentioned above). In this way, the structures can be filtered simply on the basis of the calculated total energies of the systems. On the other hand, if there are no interatomic potentials involved, the method of filtration on the created structures can be diversified. In this paper, we adopted our structural filtration method based on the embedded atom method (EAM) potential, which is generated by our adaptive genetic algorithm.1 A good EAM potential for the system is important for a high efficiency of structure prediction. However, it is very difficult or even impossible to fit a single EAM potential able to accurately describe a system under various bonding environments. Instead, we adopt several different EAM potentials obtained from AGA searches1 for LiFePO4 to sample structures located in different basins of the energy landscape. Therefore, several (three in this study) EAM potentials are employed in our search of structures. Since a very large number of random numbers can be created, a very large number of predicted structures can be filtered. Finally, the filtered structures in the final pool, as mentioned in the last step, are evaluated by the first-principles calculations based on density functional theory (DFT). The atomic relaxations are performed for all of the atoms in the unit cell, including atoms in the motifs. Generally, the shape of the motifs can be maintained after the first-principles structural relaxations (slight distortion of the starting motifs could happen). Thus, the predicted structure corresponding to the lowest energy should be the ground state or the one closest to the ground state. The structures with higher energies than the lowest one are of course the predicted metastable structures. Obviously, in the present way, there is no guarantee that the global minimum structure could be found, especially if the number of random structures calculated by DFT is not large. However, it is reasonable to assume that, as more random structures are calculated, there is more possibility that the global minimum could be reached. Therefore, when the present scheme is used to search for the ground state of a large system, it is recommended that a large number of random structures should be created and computed. The flowchart of the present scheme is illustrated in Figure 1. As shown in the figure, the known atomic coordinates (if any) are first read in. The random numbers are then created. The motifs are then constructed, with the positions and orientations of the motifs set up either by the random numbers or by the values at hand. Then, all of the DOFs to be predicted, i.e., all of the unknown DOFs (unknown atomic coordinates), are replaced by random numbers. Thus, a structure sample with motifs and random numbers is created. The atomic positions of the random structures could now be relaxed on the basis of the interatomic potentials, if there are such potentials at hand.

method (as programmed in a software called XMsearch). The present method shows two key advantages: first, the total degrees of freedom (DOF) of the system could be greatly reduced due to the use of the concept of motifs, since each motif can always be described by five degrees of freedom. Since the number of minima in the potential energy surface of a large system increases exponentially with the number of atoms,2 the computational cost also increases exponentially with the number of atoms. The effective reduction in the degrees of freedom of the system could then exponentially reduce the computational cost. Second, the structural motifs could always appear in the predicted structures when the energies are relatively low. These two characters of the present method lead to a great reduction of the potential energy surface to be explored and to the high efficiency of the present scheme. Our method requires only chemical compositions for a given compound to predict stable or metastable structures for a given structural motif, with the motif as the input. The success in the prediction of new structures of a complex material LiFePO4 demonstrates the power of this methodology and illustrates the promise of the concept of structural motif in highly efficient structural searches.

2. METHODS The present scheme consists of three steps: (1) The structure generation. Predicted structures based on structural motifs and random numbers are generated in this part (they are called random structures hereafter). (2) The filtration of structures generated in step 1. The “filtered” structures from step 1, which will be passed to step 3, are then selected here on the basis of an interatomic potential which will be described in detail below. (3) The filtered structures from step 2 are now fully relaxed by the first-principles method based on density functional theory. The central point of the present method is to make the best use of the concept of structural motifs of the materials. In the present approach, each selected structural motif of a crystal will be treated as a whole unit during the structural generation. A motif can always be described by a degree of freedom of five, i.e., (R, θ, φ), with R being the position of the motif and (θ, φ) being the orientation of the motif. That is, no matter what kind/shape of a motif it is, the degree of freedom of a motif is always five. For example, when PO4 tetrahedra in the LiFePO4, SiO6 octahedra in the Li2FeSiO4, or C60 clusters in the C60 cluster-assembled solids are considered as structural motifs in the corresponding materials, the DOFs of a PO4 tetrahedron, a SiO6 octahedron, or a C60 cluster are all only five; i.e., they are all described by (R, θ, φ). In this way, the DOFs of a PO4 tetrahedron, a SiO6 octahedron, or a C60 cluster can be reduced from 15, 21, and 180 to 5 (for all cases), respectively. It is then clear that DOF could be largely reduced in the present scheme, employing the concept of motifs. The structural motifs for a definite material can be determined according to the structural characteristics of the material observed from experiment or from preliminary small cell theoretical searches. A general structure in the present method consists of three parts, i.e., the known atomic coordinates (if any), the positions and orientations of the motifs, and all the rest of the unknown coordinates which are replaced by random numbers. If there are no atomic positions at hand, the structure then consists of only motifs (note that a single motif is represented by five random numbers) and random numbers; in this way, the structure of a crystal can be predicted from only the chemical composition, i.e., from scratch. Thus, the structures of a crystal are predicted. 11892

DOI: 10.1021/acs.jpcc.7b02486 J. Phys. Chem. C 2017, 121, 11891−11896

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functional, considering the effects due to the localization of delectrons of Fe ions, is treated within the generalized gradient approximation with a Hubbard-like correction (GGA+U).20 The effective on-site Coulomb term Ueff = U − J for Fe is 5.3 eV. Brillouin zone integrations are approximated by using the special k-point sampling of the Monkhorst−Pack scheme.21 In all cases, the crystallographic cell parameters and internal atomic coordinates are fully relaxed until the force on each atom is less than 0.01 eV/Å. Experimental data indicated that the ground state of LiFePO4 has the olivine structure.22,23 We have found this ground state structure with a moderate number of random structures. Comparisons of our predicted structures with those currently given in the Web site MaterialsProject.org are shown in Figure 2 and Table 1. In this paper, only structures with a unit cell up Figure 1. Flowchart of the motif-based structure prediction method.

Thus, a filtration process on the random structure generated is done. The process is repeated with new random numbers to generate more structures, as shown in Figure 1. Finally, the “filtered” structures will be passed to the final step, that is, the first-principles calculations. In this scheme, the “accuracy and efficiency” are very much related to the filtration of the generated random structures. If reliable classical interatomic potentials are available, they could greatly improve the efficiency of the structure prediction.

3. RESULTS AND DISCUSSION Our searching strategy is used to predict the stable and metastable crystal structures of LiFePO4, which is a very important material in the field of lithium-ion batteries.14 Theoretical studies show that LiFePO4 exhibits a rich polymorphism.15−17 In LiFePO4, all of the P atoms form tight tetrahedral units with the four neighboring O atoms. P is at the center of the tetrahedron and has a coordination number of four. In this material, Fe and Li atoms are at the central sites of the octahedra. The PO4 tetrahedra are the most compact ones among these polyhedra, showing the strongest bonding between P−O atoms as compared to Li−O and Fe−O ones. Taking advantage of this structural feature, the PO4 tetrahedra are considered as motifs in the present searching scheme. A structure of LiFePO4 consists of PO4 motifs and Li and Fe atoms (all of the O atoms are used up to form the PO4 tetrahedra). All of the positions and orientations of the PO4 tetrahedra together with the Li and Fe atomic positions are generated with random numbers. Considering PO4 tetrahedra as structural motifs, the total DOF of the system has been largely reduced. For example, for a unit cell of LiFePO4 with 4 formula units (f.u.), i.e., Li4Fe4P4O16, the total DOF of the system is then 4 (Li) × 3 + 4 (Fe) × 3 + 4 (PO4 tetrahedra) × 5 = 44 (three lengths and three angles of the unit cell are not included). On the other hand, the total DOF of the system is 28 (total number of atoms) × 3 = 84 if no motifs are considered. It is obvious that the total DOF has been reduced significantly with the use of the concept of motifs. As mentioned above, the predicted structures after structural filtration are calculated using the first-principles DFT method, i.e., employing the Vienna Ab initio Simulation Package (VASP).18,19 In our VASP calculations, all parameters are consistent with those given in http://materialsproject.org/. That is, wave functions are expanded by the plane waves up to a kinetic energy cutoff of 520 eV. The exchange-correlation

Figure 2. Formation energies of LiFePO4 as a function of unit cell volumes. Those structures given in the Web site materialsproject.org are marked by MP. Those predicted by the present scheme are indicated by a suffix XM. Volume unit in Å3. FU means formula unit.

to 8 formula units (f.u.), i.e., 56 atoms/cell, are presented. From Figure 2 and Table 1, we see that the lowest energy structure we found is the same as the experimentally observed one and the one presented in http://MaterialsProject.org. We can also see that our method has been successful in producing many new structures not previously predicted. The successful prediction of these new structures demonstrates the power of the present methodology. Our results here can also provide a more comprehensive database for the structures of LiFePO4, which should assist the further studies on the process of lithiation/delithiation in the LiFePO4 system. Structures predicted from the present EAM potentials are not very accurate; therefore, DFT relaxations are performed in the final step, which make the structures predicted here have the accuracy of the DFT level. About 250 (less than 300) predicted structures are relaxed by using the first-principles DFT method. On the basis of the predicted crystal structures, the lowenergy structures are classified into different types according to the frameworks built up by Fe, P, and O atoms. It is found that Fe, P, and O atoms bonding together can form threedimensional (3D) frameworks, two-dimensional (2D) disconnected Fe−P−O layers, or one-dimensional (1D) disconnected Fe−P−O rods. The reason for making such a classification is because that, when LiFePO4 serves as the cathode material for lithium-ion batteries, the Li ions will be 11893

DOI: 10.1021/acs.jpcc.7b02486 J. Phys. Chem. C 2017, 121, 11891−11896

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The Journal of Physical Chemistry C Table 1. Energies of the XMsearch Structures, E(XM) in eV/ f.u., Compared with Energies of MP Structures, E(MP) in eV/f.u.a index

E(XM)

E(MP)

index

E(XM)

E(MP)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

−47.616 −47.541 N N N N −47.497 −47.488 −47.479 −47.468 −47.464 −47.453 −47.435 −47.426 −47.425 N −47.404 −47.388 −47.384 −47.372 −47.372 −47.372 −47.372 −47.371 −47.362 N −47.348 −47.346 −47.342 −47.311 −47.311 −47.298 −47.297 −47.295 N −47.288 −47.278 −47.267 −47.267 −47.266 N −47.260 −47.258 −47.254 −47.254 −47.254 −47.253

−47.616 −47.542 −47.539 −47.520 −47.514 −47.507 N N N N N N N N N −47.423 −47.404 N N N N N N N N −47.354 N N N N N −47.299 N −47.296 −47.293 N N N N N −47.264 N N N N N N

48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94

−47.253 −47.253 −47.250 −47.250 −47.250 −47.250 −47.248 −47.248 −47.247 −47.246 −47.246 −47.246 −47.245 −47.237 −47.232 −47.230 −47.226 −47.223 −47.223 −47.216 −47.216 −47.216 −47.208 N −47.195 −47.182 −47.180 N −47.172 N −47.151 −47.144 −47.133 −47.127 N −47.111 −47.108 −47.104 −47.090 −47.083 −47.037 −47.036 −47.027 −47.015 N −46.902 −46.122

N −47.252 N N N −47.249 −47.249 −47.248 N N N N N N N N −47.226 N N N N N −47.208 −47.200 N N N −47.177 N −47.160 N N N N −47.114 N N N −47.089 N N N −47.026 −47.015 −46.951 N N

structures of polymorphs come from not only the orientation of polyhedra but also the coordination number of Fe atom. PO4tetrahedra and FeO6-octahedra in olivine LiFePO4 form a 3D framework as plotted in Figure 3a. Figure 3b, as a high pressure phase, has been successfully synthesized at ambient pressure, and is discovered to irreversibly transform into the olivine structure with heat treatments.15,17,24 Structurally speaking, the frameworks of Figure 3a−c show some similarities, that is, all of the PO4-tetrahedra are distributed between the two FeO6octahedral layers, but with different orientations for different polymorphs. Different from the framework in Figure 3a where neighboring FeO6-octahedra in each layer are oriented in different directions, the FeO6-octahedra in each layer in Figure 3b (a high pressure phase) point to the same direction. On the other hand, Figure 3c presents the AB-stacked FeO6-octahedral layers; i.e., the octahedra have the same orientations every alternating layer. The structure plotted in Figure 3d can be regarded as a deformation of Figure 3a, where the FeO6octahedra change into an FeO5-square pyramid. Also, the rearrangement of PO4 tetrahedra and FeO5 square pyramids in Figure 3d can lead to different crystal structures, such as Figure 3e. Mixing PO4 tetrahedra with FeO5-square pyramids can result in another pattern, as shown in Figure 3f. It is believed that the larger the unit cell considered, the more polymorphs can be found theoretically. Actually, we found that more frameworks consisting of PO4and FeO4-tetrahedra exist, as shown in Figure 4, where the orientations of the tetrahedra are highly diverse. The FeO6octahedra in Figure 3a can deform into FeO4-tetrahedra, leading to the formation of Figure 4a. In the structure of Figure 4a, the neighboring two tetrahedra at each line have opposite orientations, while, in Figure 4d and e, the orientations of the neighboring tetrahedra are at random, resulting in different crystal structures. In Figure 4b, the orientations of the tetrahedra in each line point to the same direction. In Figure 4c, all of the tetrahedra are toward the same direction. Although the structure in Figure 4f is similar to that of Figure 4a, compared with the orientation of tetrahedra in Figure 4a, some tetrahedra in Figure 4f are rotated by about 180°. As the unit cell size increases, it is expected that various tetrahedra with different orientations could connect with each other to form more crystal structures. 3.2. Structures with 2D and 1D Fe−P−O Frameworks. In our predicted LiFePO4 polymorphs, 2D and 1D structures of Fe−P−O frameworks are also found. The 2D disconnected [FeO6−PO4] layers are displayed in Figure 5a, while the 1D disconnected [FeO6−PO4] rods are shown in Figure 5b, similar to the frameworks in A2MSiO4 polymorphs (A = Li, Na; M = Fe, Co, Mn).25 Up to now, the structures with 2D and 1D frameworks have not been observed experimentally. From the theoretical point of view, such frameworks as shown in Figure 5 should appear under special experimental conditions. We have also analyzed the cohesive energies of the most stable structures in 3D, 2D, and 1D Fe−P−O frameworks. It is found that the 3D framework is preferred energetically, which is 0.39 and 0.53 eV/f.u. lower than that of the 2D and 1D frameworks, respectively. In the 3D frameworks, due to the different coordination numbers of Fe, there are three kinds of characteristic structures, i.e., one with only FeO6-octahedra, one with only FeO4-tetrahedra, and the other one with the mixing of FeO4-tetrahedra and FeO5-square pyramids. Considering those most stable structures, it is found that the cohesive energy of the crystal structure with only FeO6-octahedra is 0.1 and 0.19

The value “N” in the table indicates that the corresponding structure is not found. All structures are relaxed by using VASP.

a

extracted from or inserted into the lattice built up by Fe, P, and O atoms during the charge or discharge processes (it is like a “lithium liquid” flows through the lattice formed by Fe, P, and O atoms). Therefore, the Fe−P−O frameworks are in charge of the structural stability of the cathode material LiFePO4 in the working processes. 3.1. Structures with a 3D Fe−P−O Framework. As illustrated in Figure 3 and Figure 4, differences between the 11894

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Figure 3. Examples of the structures with 3D Fe−P−O frameworks, including (a−c) FeO6-octahedron, (d, e) FeO5-square pyramid, and (f) mixing of FeO5-square pyramid and FeO4-tetrahedron.

Figure 4. Examples of different structures with 3D Fe−P−O frameworks, including FeO4-tetrahedra.

Figure 5. Examples of structures with (a) 2D and (b) 1D Fe−P−O frameworks.

success demonstrates the utility of the motif concept for structural exploration of complex materials with motif.

eV/f.u. lower than those of the structures with only FeO4tetrahedra and with the mixing of FeO4-tetrahedra and FeO5square pyramids, respectively.

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4. CONCLUSIONS

AUTHOR INFORMATION

Corresponding Authors

*E-mail: [email protected]. *E-mail: [email protected].

In conclusion, we have presented a simple and very efficient new scheme for the structure prediction based on structural motifs. Our scheme greatly reduces the total degree of freedom of the system and can easily predict the structures with the desired motifs. The new scheme has been applied to the structure prediction of LiFePO4. Many new structures not found in the current literature have been predicted. This

ORCID

Zizhong Zhu: 0000-0001-5353-4418 Notes

The authors declare no competing financial interest. 11895

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ACKNOWLEDGMENTS This work is supported by the National Key R&D Program of China under Grant No. 2016YFA0202601, the National Natural Science Foundation of China under Grant No. 21233004, and USTC Qian-Ren B (1000-Talents Program B) fund. Work at Ames Laboratory was supported by the US Department of Energy, Basic Energy Sciences, Division of Materials Science and Engineering, under Contract No. DEAC02-07CH11358, including a grant of computer time at the National Energy Research Scientific Computing Center (NERSC) in Berkeley, CA.

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DOI: 10.1021/acs.jpcc.7b02486 J. Phys. Chem. C 2017, 121, 11891−11896