Hierarchical structures and magnetism of Co clusters: a perspective from integration of deep learning and a hybrid differential evolution algorithm

Theoretically determining the lowest-energy structure of a cluster has been a persistent challenge due to the inherent difficulty in accurate description of its potential energy surface (PES) and the exponentially increasing number of local minima on the PES with the cluster size. In this work, density-functional theory (DFT) calculations of Co clusters were performed to construct a dataset for training deep neural networks to deduce a deep potential (DP) model with near-DFT accuracy while significantly reducing computational consumption comparable to classic empirical potentials. Leveraging the DP model, a high-efficiency hybrid differential evolution (HDE) algorithm was employed to search for the lowest-energy structures of Co (N = 11-50) clusters. Our results revealed 38 of these clusters superior to those recorded in the Cambridge Cluster Database and identified diverse architectures of the clusters, evolving from layered structures for N = 11-27 to Marks decahedron-like structures for N = 28-42 and to icosahedron-like structures for N = 43-50. Subsequent analyses of the atomic arrangement, structural similarity, and growth pattern further verified their hierarchical structures. Meanwhile, several highly stable clusters, i.e., Co, Co, Co, Co, and Co, were discovered by the energetic analyses. Furthermore, the magnetic stability of the clusters was verified, and a competition between the coordination number and bond length in affecting the magnetic moment was observed. Our study provides high-accuracy and high-efficiency prediction of the optimal structures of clusters and sheds light on the growth trend of Co clusters containing tens of atoms, contributing to advancing the global optimization algorithms for effective determination of cluster structures.