Beyond the surface atlas: A roadmap and gazetteer for surface symmetry and structure

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Throughout the development of single-crystal surface science, interest has predominantly focussed on the high-symmetry planes of crystalline materials, which typically present simple stable structures with small primitive unit cells. This concentration of effort has rapidly and substantially advanced our understanding of fundamental surface phenomena, and provides a sound basis for detailed study of more complex planes. The intense current interest in these is partly motivated by their regular arrays of steps, kinks or other low-coordination structural features, whose properties are little understood and may mimic specific highly-reactive sites on dispersed nanoparticles. Furthermore, the lower symmetry of these planes may give rise to other equally interesting properties such as intrinsic chirality, with exciting potential applications in enantioselective heterogeneous catalysis, biosensors and surface magnetism. To aid exploration of this new territory for surface science requires a depth of understanding that goes beyond the character of individual surfaces to encompass the global relationships between all possible surfaces of a given material, both in their structure and in their symmetry. In this report we present a rigorous conceptual framework for ideal crystalline surfaces within which the symmetry and structure of all possible surface orientations are described. We illustrate the versatility of our generally-applicable approach by comparing fcc, bcc and hcp materials. The entire scheme naturally derives from the very simple basis that the fundamental distinction between symmetry and structure is paramount. Where symmetry is concerned, our approach recognises that the surface is not a two-dimensional (2D) object but actually a truncated three-dimensional (3D) one. We therefore derive a symmetry scheme specifically formulated for surfaces and naturally encompassing their chirality where necessary. Our treatment of surface structure, on the other hand, highlights elementary one-dimensional (1D) features lying parallel to the surface plane. Crucial to the utility of these concepts is that both structure and symmetry can conveniently be represented independently within a single stereographic projection; this then serves as a “roadmap” that fully embodies the essential relationships between different surfaces and facilitates navigation amongst them. Key locations on the map identify surfaces of particular structural simplicity, which are collated in a “gazetteer” with an accompanying description of their essential symmetry and structural character. Our symmetry–structure surface stereography (4S) analysis of fcc, bcc and hcp crystals reveals various new insights about the types of surface that they present. For instance, although the asymmetry of chiral fcc planes has hitherto been associated with the presence of kink sites, we show that the same is not always true of either bcc or hcp chiral surfaces. Kink-free chiral bcc planes offer particular advantages as model systems for asymmetric applications. We further reveal that the hcp crystal structure gives rise to intriguing types of surface that are not observed for either fcc or bcc materials. These include surfaces with intrinsic glide symmetry, surfaces with intrinsic racemic character and others displaying intrinsic double-chirality, analogous to the existence of diastereoisomers in molecular chemistry. We also identify several elementary surface structural categories that are specific to the hcp case. Having thus established a secure framework via ideal bulk-terminated crystalline surfaces, we subsequently demonstrate its extension to real surfaces. In considering how these differ from ideal surfaces we discuss relaxation and reconstruction within the same symmetry-resolved and structure-resolved perspective, drawing on numerous examples from the literature. Finally we illustrate the application of our scheme in one selected branch of surface science by exploring the symmetry-constraints on the surface chemistry of chiral molecules at chiral substrates.