A High-Throughput Framework for Materials Research and Space Group Determination Algorithm
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A High-Throughput Framework for Materials Research and Space Group Determination Algorithm by Richard Taylor Department of Mechanical Engineering and Materials Science Duke University Date: Approved: Stefano Curtarolo, Supervisor Teh Tan Nico Hotz Thesis submitted in partial fulfillment of the requirements for the degree of Master of Science in the Department of Mechanical Engineering and Materials Science in the Graduate School of Duke University 2012 Abstract A High-Throughput Framework for Materials Research and Space Group Determination Algorithm by Richard Taylor Department of Mechanical Engineering and Materials Science Duke University Date: Approved: Stefano Curtarolo, Supervisor Teh Tan Nico Hotz An abstract of a thesis submitted in partial fulfillment of the requirements for the degree of Master of Science in the Department of Mechanical Engineering and Materials Science in the Graduate School of Duke University 2012 Copyright c 2012 by Richard Taylor All rights reserved except the rights granted by the Creative Commons Attribution-Noncommercial License Abstract Effective computational materials search, categorization, and design necessitates a high-throughput (HT) approach. System by system analyses lack the scope and speed needed to uncover large portions of the materials landscape. By perform- ing broad searches over structural or chemical classes of materials and guided by fundamental physical principles, materials with specific desired properties can be systematically found. Furthermore, the HT approach is an effective general tool for materials classification. Depending on the application, various properties can be computed leading to powerful classification schemes. To implement HT materials studies, however, a versatile and robust framework must first be developed. In this paper, the HT framework AFLOW that has been developed and used successfully over the last decade is presented. Specifically, attention is given to an origin-specific symmetry algorithm. The algorithm is designed to determine the relevant symmetry properties of an arbitrary crystal structure (e.g., point group, space group, etc.). iv Contents Abstract iv List of Tables vii List of Figures viii List of Abbreviations and Symbols xi Acknowledgements xii 1 Introduction1 2 Symmetry in Crystallography: Crystal System, Point Group, and Space Group5 2.1 Crystal class or point group.......................5 2.2 Lattice system and crystal system....................6 2.2.1 Lattice system...........................6 2.2.2 Crystal system..........................7 2.3 Space group................................7 3 AFLOW: Structure analysis tools and library 13 3.0.1 POSCAR file template...................... 15 3.0.2 WYCCAR file template..................... 16 3.0.3 ABCCAR file template...................... 16 3.1 AFLOWLIB.ORG............................ 17 v 4 Space group determination algorithm and related functions 21 4.1 Overview of the algorithm........................ 21 4.2 Origin-referenced symmetry operations................. 22 4.3 Unit cell conventionalizing........................ 24 4.3.1 Cubic............................... 28 4.3.2 Hexagonal............................. 28 4.3.3 Rhombohedral........................... 28 4.3.4 Tetragonal............................. 29 4.3.5 Orthorhombic........................... 30 4.3.6 Monoclinic............................. 30 4.3.7 Triclinic and the reduced cell.................. 31 4.4 Crystal point group and the standard crystallographic origin..... 33 4.5 Symmetry classes and comparison routine............... 39 4.5.1 Improper rotations and the RotoInversion class....... 40 4.5.2 Glide class............................. 42 4.5.3 Screw class............................ 44 4.5.4 Operator overloading in the symmetry classes......... 44 A Stereographic projections of the 32 point groups including crystal system and the Bravais lattices 46 Bibliography 58 vi List of Tables 2.1 Conditions on the symmetry of crystals in the crystal systems.....8 2.2 The 230 space groups with point group and crystal system.......9 4.1 The standard origins of the 230 space groups with point group and crystal system............................... 38 vii List of Figures 2.1 The seven lattice systems in 3 dimensions with unfolded faces..... 11 2.2 Classification of crystal structures. Based on Fig. 8.2.1.1 in ITC (Hahn(2002))............................... 12 3.1 A typical view of the online aconvasp interface. Note the input file is in the POSCAR format described in 3.0.1............... 14 3.2 An example of a parent lattice (a) and a derived superstructure (b). The superstructure is constructed by a relabeling of a subset of the lattice points. The lattice vectors of the derived structure become multiples of the parent lattice....................... 18 3.3 The opening view of the online binary materials database interface. Lattice-based structures such as fcc and bcc can be selected to refine the search.................................. 19 3.4 Using hundreds of first-principles calculations, the T 0 K phase di- agram can be constructed, and the low temperature stability of the binary system can be investigated. The diagram represented here is the phase diagram of the Au-Mg (gold-magnesium) system...... 20 4.1 Flow chart describing the basic design of the space group algorithm in four thematically distinct parts.................... 23 4.2 The geometric objects related to the crystallographic symmetry opera- tions in three dimensions: equilateral triangle, isosceles right triangle, 120-30-30 triangle, plane, and line. Small solid points indicate the intersection of an axis of symmetry with the plane defined by three points.................................... 24 viii 4.3 Cartoon diagram illustrating the method used to find point group operations in an origin-specific manner. a) shows a mirror reflection requiring selection of two atoms. The plane normal is defined by the line between the two points and intersects at the midpoint. b) shows a three-fold rotation about an axes. The axes is uniquely determined by the normal to the plane intersecting at the center of mass. This is extended to other crystallographic point operations (n 1; 2; 3; 4; 6) by reduction of co-planer atoms...................... 25 4.4 Three choices of reduced basis for a square lattice. Each has the same, minimal volume.............................. 26 4.5 Ambiguity in the choice of unit cell is shown. A rhombohedral unit cell (solid lattice vectors) or the larger hexagonal cell (dashed lattice vectors) generate the same lattice..................... 27 4.6 The Bravais lattices for the cubic lattice system............. 28 4.7 The Bravais lattices for the hexagonal (hP) and rhombohedral (hR) lattice systems............................... 29 4.8 The Bravais lattices for the tetragonal lattice system.......... 29 4.9 The Bravais lattices for the orthorhombic lattice system........ 30 4.10 The Bravais lattices for the monoclinic lattice system.......... 31 4.11 The Bravais lattice for the triclinic lattice system............ 33 4.12 Threefold, fourfold, and sixfold rotoinversions are illustrated. The geometric shapes defined by stereographic projections along possible axes are used to identify possible rotoinversion axes........... 36 4.13 Additional geometric objects comprising the full set of symmetry op- erations. (a) Screw axes (nf ) create triangles with non-simple (but determinable) angle and side relations. The possible geometries can be derived using the projected object (dashed triangle) geometry and the screw displacement (h and 2h). Because the screw axes are paral- lel to lattice directions, a finite set of plane normal-screw axis angles α exists. (b) Glide planes exist for a finite set of translations (n,d,a,b,c) defined in terms of the lattice vectors. Applying these translations and checking if an atom exists along the normal direction reveals the glide planes................................. 37 4.14 Cartoon illustrating a rotation-reflection/rotation-inversion operation. 42 ix 4.15 Cartoon illustrating a glide operation................... 43 4.16 Cartoon illustrating a screw operation.................. 44 x List of Abbreviations and Symbols Symbols used in the stereographic projections Stereograms for the 32 crystallographic point groups are listed in AppendixA. The symbols used are defined below. Twofold rotation axis 2 Threefold rotation axis 3 Fourfold rotation axis 4 Sixfold rotation axis 6 _ Inversion center 1 _ Threefold rotoinversion 3 _ Fourfold rotoinversion 4 _ Sixfold rotoinversion 6 Twofold rotation with center 2/m Fourfold rotation with center 4/m Sixfold rotation with center 6/m xi Acknowledgements The author thanks Stefano Curtarolo for his mentoring and Shidong Wang, Kesong Yang, and Junkai Xue for fruitful discussions. This work was supported by ONR (N00014-11-1-0136, N00014-10-1-0436, N00014-09-1-0921), NSF (DMR-0639822, DMR- 0908753), and the Department of Homeland Security Domes tic Nuclear Detection Ofce. xii 1 Introduction A number of theoretical approaches in materials science have been developed into powerful computational tools. From the standpoint of materials engineering, these tempt an intriguing vision of the future of materials design. Computational materials scientist often speak of \materials by design." This is a compelling phrase, but the tools used by materials scientists have until recently lacked the unifying framework needed to facilitate the realization of their full potential. A system-specific view is often taken by computational