The Symmetry Principles And Their Applications In Crystal Structure Predictions And Analyses Of Band Structures

Symmetry is one of the conceptual cores in modern physics and the fundamental laws of nature.It plays an important role in many fields of physics,such as astrophysics,high-energy physics,condensed matter physics,etc.The field of condensed matter physics mainly focus on studying chemical composition,microstructure,physical properties,and their relations among materials at micro level,for example,determining the crystal structure at atomic level and analyzing electronic band structures at electronic level.Symmetry,which refers to the transformations that leave materials unchanged in condensed matter physics,is an important tool for solving these basic problems.Crystals can be classified completely into 230 space group types according to their symmetry,which can be further distributed into 73 symmorphic space groups and 157 non-symmorphic space groups by the existence of non-symmorphic symmetry operations.In this paper,we developed an enumerating algorithm for high symmetry derivative structures based on group–subgroup relations between space groups and discovered a novel magnetically controllable band splittings by extending the method of symmetry analyses to magnetic system including spin-orbit interaction.Many crystal structure databases have been established,such as Crystallography Open Database（COD）,Inorganic Crystal Structure Database（ICSD）,and Cambridge Structural Database（CSD）,which lay a foundation for the application of data mining in crystal structure prediction methods.The approach of occupating elements on topological prototype structures,which are derived by ignoring the information of elements in prototype structures,is an efficient way to use crystal information in the database and overcomes the problem of scant structures in the method of replacing prototype structures directly.Every occupational style of elements on the atomic positions in topological prototype structures corresponds to a crystal structure,which is called a derivative structure.The applications of this approach in crystal structure prediction methods are confronted with two problems.One is the generation of high symmetry derivative structures because random occupation may result in many derivative structures with low symmetry while nature tends to form crystals with high symmetry.Another one is the identification and removement of equivalent derivative structures.Different occupational styles correspond to different derivative structures,however there could be equivalent relations between derivative structures.The generations of low symmetry or equivalent derivative structure led to useless first-principles calculations and hampers the efficience of crystal structure prediction.For solving these problems,we developed a systemic enumerating method for high symmetry derivative structures.Derivative structures have definite symmetries,which must be subgroups of the symmetry of topological prototype structures.This paper first classifies the derivative structures by subgroup symmetries and then introduces the constraints of subgroup symmetry to generate unequivalent derivative structures with specific subgroups symmetries.We further shows there could not be an equivalent relation between derivative structures with symmetries that are not conjugate,and the lists of derivative structures are completely equivalent which are labeled by conjugate subgroups.Therefore,enumerating conjugate classes of subgroups and choosing one subgroup in each class for generating unequivalent derivative structures,could guarantee the generation of high symmetry and unequivalent derivative structure.The conjugate classes of subgroups are calculated,using chains of maximum subgroups based on the group–subgroup relations between space groups.We wrote codes to realize the above enumerating method for high symmetry derivative structures and integrated it in the CALYPSO package.Transition metal trifluorides,a series of materials with formula MF₃,have long drawn considerable attention due to their versatile applications in negative thermal expansion materials,battery materials and hydrogen storage materials.As a fundamental thermodynamic variable,pressure can alter bonding patterns and drive phase transitions,leading to the creation of new high-pressure phases with exotic properties that are inaccessible at ambient pressure.We performed a high-pressure structure search on Ti F₃ system,using the developed method for generating high symmetry derivative structures,and found a pressure-induced structural phase transition from R-3c phase to Pnma phase,accompanied by the destruction of Ti F₆octahedra and formation of Ti F₈ square antiprismatic units.High-pressure X-ray diffraction experiments confirmed our theoretical predictions.We further show that the structural phase transition from R-3c phase to Pnma phase at high pressure is a common phenomenon in transition metal trifluorides.The physical properties of crystalline materials,such as electrical conductivity,thermal conductivity,and optical dielectric function,are closely related to their electronic band structures.The symmetry of crystal structures could apply restrictions on their electronic band structures.By selecting a crystal with a specific symmetry,the energy band structure with certain characteristics can be designed to obtain the desired physical properties.We expanded the methods of symmetry analyses into to magnetic system including spin-orbit interaction and applied it on the exploration of novel band splittings.The conventional controllable splitting patterns in electronic band structures can be mainly categorized into three types,including Rashba-type,Dresselhaus-type,and Zeeman-type spin splittings,which originate from either a spin-orbit interaction or magnetic exchange interaction respectively.In ferromagnetic materials with nonsymmorphic symmetry elements,orbital degree of freedom（DOF）,spin DOF,and ferromagnetic states（i.e.,alignments of magnetic moments）likely interact with each other,implying more complicated couplings as compared with Rashba-type,Dresselhaus-type,and Zeeman-type splittings.Following the strategy,by first-principles calculations and symmetry analyses,we identified the existence of a band splitting pattern in Pnma ferromagnetic YTi O₃,which is controllable wich a magnetic field.In sharp contrast to the conventional band splitting patterns,the mechanism of our pattern is ascribed to the electrons’orbital degree of freedom and can be generalized to various ferromagnets adopting the Pnma symmetry.