An Efficient Scheme for Crystal Structure Prediction Based on

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An Efficient Scheme for Crystal Structure Prediction Based on Structural Motifs Zi-Zhong Zhu, Ping Wu, Shunqing Wu, Lin Han Xu, Yi Xu Xu, Xin Zhao, Caizhuang Wang, and Kaiming Ho J. Phys. Chem. C, Just Accepted Manuscript • Publication Date (Web): 15 May 2017 Downloaded from http://pubs.acs.org on May 17, 2017

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The Journal of Physical Chemistry

An Efficient Scheme for Crystal Structure Prediction Based on Structural Motifs Zizhong Zhu1, *, Ping Wu2, Shunqing Wu1, Linhan Xu1, Yixu Xu1, Xin Zhao3, Cai-Zhuang Wang4, Kai-Ming Ho2,3, *

1

Collaborative Innovation Center for Optoelectronic Semiconductors and Efficient Devices, Department of Physics, Xiamen University, Xiamen 361005, China

2

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. 3

Department of Physics and Astronomy, Iowa State University, Ames, IA 50011, USA 4

Ames Laboratory - US DOE, Ames, IA 50011, USA

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 two fold: 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

* Corresponding authors. Email address: [email protected] (Z.Z. Zhu); [email protected] (K.M. Ho) 1

ACS Paragon Plus Environment


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material for lithium-ion batteries. Numerous new structures of LiFePO4 have been found, compared to those currently available, available, demonstrating the reliability of the present methodology and illustrating the promise of the concept of structural motifs.

1.

Introduction 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 are 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 discovering 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 understanding the experimental studies, when the experimental data are incomplete. For example, powder diffraction data may be 2


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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 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, meta-stable 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 algorithms (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 motif-based structure searching method (as programmed in a software called XMsearch). The present method shows two key advantages: firstly, the total degrees of freedom (DOF) of the 3



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system could be greatly reduced due to the use of 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 effectively reduction in the degrees of freedom of the system could then reduce exponentially the computational cost. Secondly, 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 meta-stable 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 the step 1. The “filtered” structures from step 1, which will be passed to step 3, are then selected here based on an interatomic potential which will be described in detail below; 3) the filtered structures from step 2 are now fully relaxed by the 4


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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 degrees of freedom of five, i.e., (R, θ, φ), with R is the position of the motif and (θ, φ) are the orientation of the motif. That is, no matter what kind/shape of a motif 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 the 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 cells 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 is no atomic positions at hand, the structure then consists of only motifs (noted that a single motif is represented by five random numbers) and random numbers, in this way, the 5



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structure of a crystal can be predicted from only the chemical composition, i.e., from scratch. Thus, the structures of a crystal are predicted. It should be pointed out that “good” structures can be predicted if there are inter-atomic potentials at hand for the studied material, since the predicted structures can then be relaxed based on the known inter-atomic potentials. For example, “excellent” structures have been found if Tersoff potential 13 for C or Si is adopted when predicting the polymorphs of graphene or silicene. Since the creation of a structure, as mentioned above, cost very little time, 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 first-principles method. If there are inter-atomic potentials at hand, each predicted structure can then be geometrically optimized based on the inter-atomic potentials (as mentioned above). In this way, the structures can be filtered simply based on the calculated total energies of the systems. On the other hand, if there are no inter-atomic 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 6


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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 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 corresponds 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 meta-stable 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, more possibility 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 the DOFs to be predicted, i.e., all the unknown DOFs (unknown atomic coordinates), 7



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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 based on the inter-atomic potentials, if there are such potentials at hand. 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 inter-atomic potentials are available, they could greatly improve the efficiency of the structure prediction.

Fig. 1. Flowchart of the motif-based structure prediction method.

3. Results and Discussion 8


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Our searching strategy is used to predict the stable and meta-stable 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 the LiFePO4, all 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, Fe atoms (all the O atoms are used up to form the PO4 tetrahedra). All 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 9



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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 functional, considering the effects due to the localization of

d-electrons 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 Monkhorst-Pack scheme.21 In all the 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 website MaterialsProject.org are shown in the figure 2 and Table 1. In this paper only structures with a unit cell up 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 the 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 powerfulness 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 10


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EAM potentials are not very accurate, therefore, DFT relaxations are performed in the final step, which make the structures predicted here having the accuracy of the DFT level. About 250 (less than 300) predicted structures are relaxed by using first-principles DFT method.

Fig. 2. Formation energies of LiFePO4 as a function of unit cell volumes. Those structures given in the website 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.

Based on the predicted crystal structures, the low-energy 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 three-dimensional (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 is served as the cathode material for lithium-ion batteries, the Li ions will be 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 11



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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. 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.. The Value “N” in the Table Indicates That the Corresponding Structure Is Not Found. All Structures Are Relaxed by Using VASP. Index 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

E(XM) -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

E(MP) -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

Index 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

E(XM) -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

E(MP) 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 12


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40 41 42 43 44 45 46 47

-47.266 N -47.260 -47.258 -47.254 -47.254 -47.254 -47.253

N -47.264 N N N N N N

87 88 89 90 91 92 93 94

-47.083 -47.037 -47.036 -47.027 -47.015 N -46.902 -46.122

N N N -47.026 -47.015 -46.951 N N

3.1 Structures with 3D Fe-P-O framework As illustrated in Fig. 3 and Fig. 4, differences between the structures of polymorphs come from not only the orientation of polyhedra but also the coordination number of Fe atom. PO4-tetrahedra and FeO6-octahedra in olivine LiFePO4 form a 3D framework as plotted in Fig. 3(a). Fig. 3(b), 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 Fig. 3(a), (b) and (c) show some similarities, that is, all the PO4-tetrahedra are distributed between the two FeO6-octahedral layers, but with different orientations for different polymorphs. Different from the framework in Fig. 3(a) where neighboring FeO6-octahedra in each layer are oriented in different directions, the FeO6-octahedra in each layer in Fig. 3(b) (a high pressure phase) point to the same direction. On the other hand, Fig. 3(c) presents the AB-stacked FeO6-octahedral layers, i.e. the octahedra have same orientations every alternating layer. The structure plotted in Fig. 3(d) can be regarded as a deformation of Fig. 3(a), where the FeO6-octahedra change into FeO5-square pyramid. And, the rearrangement of PO4 tetrahedra and FeO5 square pyramids in Fig. 3(d) can lead to different crystal structures, such as Fig. 3(e). Mixing PO4 tetrahedra with FeO5-square pyramids can 13



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result in another pattern, as shown in Fig. 3(f). It is believed that the larger unit cell is considered, the more polymorphs can be found theoretically.

Fig. 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.

Fig. 4(a)-(f): Examples of different structures with 3d Fe-P-O frameworks, including FeO4- tetrahedra. 14


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Actually, we found that more frameworks consisting of PO4- and FeO4-tetrahedra exist as shown in Fig. 4, where the orientations of the tetrahedra are highly diverse. The FeO6-octahedra in Fig. 3(a) can deform into FeO4-tetrahedra, leading to the formation of Fig. 4(a). In the structure of Fig. 4(a), the neighboring two tetrahedra at each line have opposite orientations, while in Fig. 4(d) and 4(e) the orientations of the neighboring tetrahedra are at random, resulting in different crystal structures. In Fig. 4(b), the orientations of the tetrahedra in each line point to the same direction. In Fig. 4(c), all the tetrahedra are toward the same direction. Although the structure in Fig. 4(f) is similar to that of Fig. 4(a), compared with the orientation of tetrahedra in Fig. 4(a), some tetrahedra in 4(f) 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 framework

Fig. 5 Examples of structures with (a) 2d and (b) 1d Fe-P-O frameworks.

15



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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 Fig. 5(a) while the 1d disconnected [FeO6-PO4] rods are shown in Fig. 5(b), 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 Fig. 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 eV/f.u. 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 eV/f.u. and 0.19 eV/f.u. lower than those of the structures with only FeO4-tetrahedra and with the mixing of FeO4-tetrahedra and FeO5-square pyramids, respectively.

4. Conclusions 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 16


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

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 Nos. 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. DE-AC02-07CH11358, including a grant of computer time at the National Energy Research Scientific Computing Center (NERSC) in Berkeley, CA.

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Table of Contents (TOC) Graphic

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An Efficient Scheme for Crystal Structure Prediction Based on

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