Chaotic crystallography: how the physics of information reveals structural order in materials

Highlights

• Review information-theoretic and computation-theoretic measures that describe structure in disordered materials.

• Extends view of exact periodic crystalline order of group theory to include partial symmetries using semi-group theory.

• Opens up the possibility of automated search for useful physical properties in disordered materials.

We review recent progress in applying information-theoretic and computation-theoretic measures to describe material structure that transcends previous methods based on exact geometric symmetries. We discuss the necessary theoretical background for this new toolset and show how the new techniques detect and describe novel material properties. We discuss how the approach relates to well known crystallographic practice and examine how it provides novel interpretations of familiar structures. Throughout, we concentrate on disordered materials that, while important, have received less attention both theoretically and experimentally than those with either periodic or aperiodic order.

Introduction

It is difficult to exaggerate the importance and influence of crystallography over the past century. Twenty-nine Nobel prizes have been awarded for discoveries either in or related to crystallography, with at least one prize per decade [1]. Crystallography strongly influences and is influenced by other fields, such as chemistry, biology, biochemistry, physics, materials science, mathematics, and geology, making it perhaps the quintessential interdisciplinary science.¹ So ingrained in other disciplines, it is now often thought of as a service science, in the sense that the techniques and theory developed in crystallography have become standard tools for researchers in these other fields. Often among the first questions in a research problem is ‘What is the crystal structure of this material?’— or, more colloquially — ‘Where are the atoms?’

Unquestionably crystallography is a mature field. The International Tables for Crystallography consist of eight volumes (A-G, A1) and if printed out would, collectively, require nearly 6000 pages [2]. Together they coalesce and codify the combined knowledge of the worldwide crystallographic community. Additionally, there are at least a dozen major crystallographic databases, some cataloging hundreds of thousands of different solved crystal structures² with tens of thousands being added yearly.

As successful as this research program has been, there has been an inordinate interest in those material structures that possess periodic order and thus have discrete reflections in their diffraction patterns, called Bragg peaks.³ Even in the early days of X-ray crystallography, though, some materials were known to have considerable diffuse scattering between the Bragg peaks [3] or even to lack Bragg peaks altogether [4]. While an observed broadband spectrum is sometimes a result of thermal agitations or limited experimental resolution, it can be and often is a signal of disorder within the material. And this disorder can be mild, preserving the integrity of the Bragg reflections, or it can be severe, where no identifiable long-range order is present. These cases have not, however, received nearly the same attention as those with “an essentially sharp diffraction pattern” 5, 6 nor has the progress been nearly as impressive. Indeed, in some sense disordered structures have been defined to be outside the field of crystallography.⁴ Nonetheless crystallographers, defined broadly here as that community of researchers tasked with understanding and characterizing the atomic arrangement and composition of materials, have shown a persistent interest in them 4, 7, 8, 9.

Researchers are increasingly discovering that disorder has profound effects on material properties and, perhaps surprisingly, disorder can improve their technological usefulness. For example, it was recently shown that significantly disordered graphene nanosheets are excellent candidates for use in high-capacity Li ion batteries due to their unusually high reversible capacities [10]. Theoretical investigations suggest that the band gap in ZnSnP₂, a promising candidate for high-efficiency solar cells, changes considerably (0.75–1.70 eV) as the material transitions from an ordered chalcopyrite structure to a disordered sphalerite structure [11].

The growing importance of disorder in materials, then, contrasts sharply with the lack of tools available to characterize disordered materials. And, just as researchers developed new conceptual models and theoretical techniques to understand the novel organizational structure in quasicrystals [12], new approaches are needed to characterize disordered materials. Here, we detail a recent initiative that exploits information- and computation-theoretic ideas to classify the structure of materials in a new way, one that can seamlessly bridge the gap between perfectly ordered materials, those materials with some disorder, and finally those that have no discernible underlying crystal structure.