pyGACE: Combining the genetic algorithm and cluster expansion methods to predict the ground-state structure of systems containing point defects

Abstract

Searching the most stable atomic-structure of a solid with point defects (including the extrinsic alloying/doping elements), is one of the central issues in materials science. Both adequate sampling of the configuration space and the accurate energy evaluation at relatively low cost are demanding for the structure prediction. In this work, we have employed a framework combining genetic algorithm, cluster expansion (CE) method and first-principles calculations, which can effectively locate the ground-state or meta-stable states of the relatively large/complex systems. We employ this framework to search the stable structures of two distinct systems, i.e., oxygen-vacancy-containing HfO_2−x and the Nb-doped SrTi_1−xNb_xO₃, and more stable structures are found compared with the structures available in the literature. The present framework can be applied to the ground-state search of extensive alloyed/doped materials, which is particularly significant for the design of advanced engineering alloys and semiconductors.

Introduction

Point defects occur naturally in materials and affect the properties of the materials in many aspects [1], [2], including the mechanical [3], [4], [5] transport[6], [7], [8], electronic[9], [10], [11], optoelectronic [12], [13], and thermoelectric properties[14], [15], [16]. Explicit configurations of the point defects at relatively high concentration can significantly change the electronic property of the materials, as in the case of the oxides employed in resistance random access memory (RRAM) [17], [18] for information storage: clustering of the oxygen vacancy leading to the low resistance represents the signal ‘1’, while the uniform distribution of the vacancies inducing the high resistance represents the signal ’0’. Unfortunately, locating the ground state structure of the point-defects-containing materials remains a challenge, especially when the concentration of the point defects is high, due to the numerous atomic configurations that can be formed and the huge cost to evaluate their energies accurately with density functional theory (DFT).

In the past decades, extensive efforts have been put on the efficient searching of the most stable crystalline structures given the compositions of the systems, and two strategies are mainly focused so far. The first one is enhancing the sampling of the phase space and reducing the candidate structures for the density functional theory calculations, by using the genetic algorithms[19], [20], [21], [22], evolutionary algorithm [23], [24], [25], particle swarm optimization [26], random sampling [27], [28], deep learning[29], etc. These methods work very well for the unit cell design given the composition of the materials, while when the number of the atoms in the unit cell is large, the high computation cost for the DFT is inevitable. The other strategy is using empirical potential to replace DFT or to prescreen the ground-state candidates for DFT [20]. However, empirical potentials comparable to the accuracy of DFT calculations for extensive materials are still lacking, although recently the machine-learning potentials based on a quite large data set of DFT calculations shed some light on that [30].

Apart from the structure prediction for the materials from ’scratch’ which uses the composition of the materials as the only input as mentioned above, in certain circumstances, searching the stable atomic structures given a rigid lattice is also demanding. These ground-state searching methods with confined phase space are of great importance in engineering-alloy and semiconductor industry, where additional elements or defects are introduced to the parent materials to improve their performance [31], [32], [33], [34], [35]. In principle, figuring out the structures of the alloyed/doped systems with a confined supercell should be easier than structure prediction from scratch where no ‘template’ exist, while the challenges are the expensive energy calculation since for the alloyed/doped systems a larger supercell is required to avoid the periodic interactions between the defects with their images. So far, the efficient techniques designed for the ground-state searching of the relatively large defect-containing systems are seldom reported in the literature. The worth mentioning technique dealing with this issue is the cluster expansion method. Cluster expansion (CE) method, first proposed by Sanchez et al. decades ago [36], [37], describe the energy of the multicomponent systems in terms of the orthogonal discrete Chebyshev’s polynomials, in which the parameters, i.e., the coefficients for the effective cluster interaction (ECI) among clusters formed by two or three atoms, can be obtained by fitting moderate DFT energies. With reliable ECI, energies of any structures can be estimated at a meager cost; then the ground state structures can be localized after enumerating all the possible configurations. CE method coupling with DFT calculations has been integrated in a few codes [38], [39], [40], and widely used in the ground-state searching of the alloying systems (i.e., substitutional defects) [41], [42], [43], [44] or in the thermodynamic calculations in conjunction with Monte Carlo simulation [45]. The key step for the CE method is fitting reliable ECI parameters using DFT calculations. However, for a large supercell, even if its shape is constrained, the fitting process will be computational demanding. Therefore, when the large supercell is involved in the ground-state searching process, both the efficient sampling and efficient calculations are needed.

In the present work, to handle the ground-state search of a relatively large system, we have employed a theoretical framework by combing the genetic algorithm for enhanced sampling of the configuration space and the CE method coupling with DFT for efficient and accurate energy evaluation. Particularly, to obtain the more effective clusters, we use the previously found ground-state structures in GA to re-calculate a new set of clusters in CE, which will be further employed to predicate new structures. The loop will continue until the ground-state candidates do not change anymore compared to the previous step, which guarantees that one can obtain a precise cluster and reasonable ground-state configurations. It is worth mentioned that CE was once connected to GA [46], while we do not see extensive parameter exchange (as using the iteration) between the GA and CE in the previous work [46]. Here the efficiency and accuracy of our framework are validated in two examples, namely the oxygen-vacancies configuration in HfO_2−x and the substitutional-defect structures in SrTi_1−xNb_xO₃.