Elsevier

Surface Science Reports

Volume 62, Issue 10, 31 October 2007, Pages 373-429
Surface Science Reports

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

https://doi.org/10.1016/j.surfrep.2007.06.001Get rights and content

Abstract

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.

Introduction

Some forty-two years ago, a seminal book entitled “An Atlas of Models of Crystal Surfaces” presented the first systematic attempt to catalogue the infinitely variable range of crystalline surface structures [1]. True to the metaphor implicit in its title, each page displayed a bird’s eye view of the atomic arrangement obtained by truncation of the bulk material at a particular facet; the atlas thus provides an indispensable field-guide to the basic “geology” of the crystalline surface. Furthermore, as with any true geological atlas, the preamble to this classic volume was given over to a discussion of the methods employed in preparing the individual “topographical plans”. A further “Atlas of Surface Structures” was published almost thirty years later [2], respecting a similar rationale but also containing details, obtained by the crystallographic techniques developed over the intervening period, of relaxed and/or reconstructed surfaces with or without adsorbed molecular and/or atomic species; updates of this latter reference source are available continually in the form of the NIST Surface Structure Database [3].

What both of these atlases lack, however, is a real sense of the ways in which different surfaces relate to one another. They perform an admirable function in showing what the various surfaces “look” like, but discerning the key similarities and differences between them is left to the eye of the reader. Certain high-symmetry low-index surfaces, such as the {111}, {110} and {100} surfaces of cubic materials, or the {0001}, {101¯0} and {112¯0} surfaces of hexagonal materials, have become familiar and well-understood within the surface science community, but low-symmetry high-index surfaces remain practically terra incognita to this day: relatively few have ventured there, and reports from those who have are fascinating and disturbing in equal measure.1

Pursuing the geographic metaphor to its limit, therefore, we see that what is needed is not merely an atlas, but rather a roadmap and gazetteer.2 We need to see the network that connects different surfaces–a network of structural similarities, that is–and to learn which of these surfaces are major nexus points and which relative backwaters. With an atlas showing only geological information, for instance, the task of navigating between Newcastle and Cambridge might prove tricky indeed; our aim is to provide a roadmap of the main connections between any two centres, and a gazetteer of the key intersections one would pass along the way. In doing so, we will make use of the same stereographic projection employed in the original surface atlas, and indeed commonly used to represent the symmetry of bulk materials in a crystallographic context [4]. In the present work, however, we argue that the symmetry of a crystal surface must be decoupled from, and subsequently reconciled with, its atomic structure, if a proper description of complex surfaces is to be achieved. We map surface symmetry and structural elements independently, therefore, using the stereographic projection as a suitable reference frame in which to present both data sets. Lines on the map will group together surfaces with shared structural features, while their intersections pinpoint the most important members of each structural group.

Our approach should, therefore, provide an organising principle which could be of immense value in the study of high-index surfaces. We note that the literature presently contains works on a disparate range of such surfaces, and progress is hampered by the fact that it is not clear to what extent results from one facet may be related to those obtained on another. By providing a framework within which the structural similarities of different facets become clear, we hope that cross-referencing between existing data, both experimental and theoretical, will become more meaningful. Furthermore, by proposing a hierarchical arrangement of high-index surfaces, we hope that future research effort will be concentrated towards a smaller set of facets chosen according to a unified logical pattern. Finally, by providing a guide to the high-index surfaces of bcc and hcp structures, we hope to stimulate work in areas that have been almost entirely neglected to the present time.

The separation of symmetry from structure is, we believe, a crucial feature of our approach to surfaces, and its necessity is nowhere illustrated quite so clearly as in the consideration of surface chirality.3 For some time, it has been known that cutting even a highly symmetric bulk material on a low-symmetry plane can result in a chiral arrangement of atoms at the surface. To our knowledge, the first detailed discussion of this effect was provided by Gellman et al. [5], who highlighted the intrinsic chirality of fcc surfaces displaying kinked steps with portions of unequal length on either side of the kink. Clearly such surfaces are of great potential interest as possible catalysts for heterogeneous asymmetric synthesis and/or separation. Indeed Attard et al. [6], [7] have demonstrated the electro-oxidation of glucose over various intrinsically chiral single-crystal Pt electrodes to be highly enantioselective. Furthermore, they have pointed out that a kink site at the junction of two equal length steps is also chiral if the microstructure at the kink is asymmetric. They have proposed an analogue of the Cahn–Ingold–Prelog rule, by which to assign the chirality of fcc surfaces: for an R surface, the sequence of microfacets in order of decreasing atomic density {111}>{100}>{110} runs clockwise about the kink atom, while on an S surface it runs anticlockwise [6]. In this picture, the chirality of the surface is inextricably linked with the presence of a kinked step. In contrast, our approach allows us to determine that this is emphatically not the case at bcc and hcp surfaces. Indeed, the kink-free chiral surfaces of bcc materials may prove rather more stable candidates for chiral catalysis than any of the fcc surfaces hitherto considered.

In an earlier letter [8], we have already presented some of the most basic conclusions to be drawn from our approach in the cases of fcc and bcc materials. In the present work, we extend our discussion to include rather more subtle aspects of the surface structure of these crystals, but also to address the more complex structure of hcp surfaces. The inclusion of an additional atom within the primitive bulk unit cell leads to surfaces that may display intrinsic glide symmetries, properties akin to diastereoisomerism, and even racemic qualities. To introduce these concepts in a systematic manner, we will begin in Section 2 with a general discussion of the nature of chirality, before progressing in Section 3 to a consideration of the differing surface symmetries of the fcc, bcc and hcp crystal structures. Sections 4 Primary structure, 5 Secondary structure, 6 Tertiary structure will then classify what we describe as the primary, secondary and tertiary structural characteristics of these materials, identifying the networks of shared structural features and relating the locations of the intersections to the surface symmetry. Some general results that aid understanding of these sections are included as Appendix A Assignment of termination labels, Appendix B 1D structural motifs for, Appendix C On the symmetry and size of 2D surface lattices, Appendix D Identifying convenient surface-specific reference cartesian axes, and a comprehensive Glossary is provided in Appendix E.

A summary of our Roadmap approach, emphasising the nature and strength of the relationships between similar surfaces, is outlined in Section 7. An accompanying Gazetteer of surfaces is presented in Appendix F. Issues relating to the possible reconstruction, roughening and facetting of high-index surfaces will be discussed in Section 8, and the implications of our findings for heterogeneous asymmetric catalysis will be considered in Section 9. We emphasise, however, that this latter discussion represents merely one example of the ways in which our approach may prove useful. In the most general terms, we anticipate that the categorisation of surface structure and symmetry proposed here will be fully applicable in a wide range of situations.

Section snippets

A vectorial analysis of chirality

Before discussing the symmetry and structure of specific crystal surfaces, it is essential to establish some general concepts regarding chirality and diastereoisomerism. This will be most easily achieved through a self-contained discussion of finite systems (e.g. molecules), before adapting the same arguments for their application to extended periodic bulk and surface systems. For each of these three cases our aim is to derive concise vectorial representations of both chirality and

Symmetry

The symmetry of an object may be defined, in the very broadest sense, as the set of operations that leave the object apparently unchanged. In order to discuss crystalline surfaces in this context it will be beneficial to first consider the various types of symmetry that an object of this nature may present. We will introduce and develop them referring entirely to bulk-terminated surfaces in this section. Having laid the foundations of surface symmetry in the limit of bulk-termination on a

Primary structure

The fcc, bcc and hcp crystal structures of metals have little in common; in fact the only structural feature that they all share is the extended close-packed chain of atoms. Note also that the formation of an atomic chain is the structurally simplest means for a metal to attain metallicity, since it allows the formation of a continuous band of electronic states with non-zero density at and around the Fermi level, albeit in one dimension. We thus consider close-packed chains to lie along

Secondary structure

The primary structural features discussed in the preceding section provide an appealing way in which to classify surfaces as flat, interrupted-flat, stepped, geminal, meandering row and kinked. Looking beyond this initial categorisation, however, might it not be possible to further refine our scheme by the introduction of additional important interatomic vectors and their zones? Anticipating the discussion below, we assert that such a refinement is indeed possible, and that the zone

Tertiary structure

Having introduced the idea of secondary structure in the preceding section, the extension to tertiary structure and beyond is conceptually straightforward. We begin by defining “tertiary structural vectors” (symbolically t) formed as the sum of three primary structural vectors, again subject to the constraints that the result be an interatomic vector of the crystal and that it not be collinear with any of the primary or secondary structural vectors. The consequences for the three structures

Reading the roadmap

Revisiting the geographic metaphor we introduced in Section 1, it is now clear that the network of primary, secondary and tertiary structural zones for each crystal structure strongly resembles a roadmap (Fig. 19, Fig. 20, Fig. 21). The zones and their corresponding structural motifs are hierarchical in importance, just as roads range from major highways to minor byways; the surfaces at their intersections are also hierarchical, in the same way that settlements range from major cities to

Beyond ideal bulk termination

The preceding sections outline a general framework within which the structure and symmetry of ideal surfaces are described on an equal footing. It will not have escaped the reader’s attention, however, that the discussion thus far makes no reference to any deviation in atomic positions due to the existence of a surface. In reality, of course, atoms within the selvedge adopt non-ideal positions, but in many cases the symmetry and the structural features categorised above are essentially

Exploring the symmetry properties of chiral surfaces

The separation of symmetry and structure has led us to classify surfaces on the basis of these two fundamental attributes. We have identified the resulting classes of surface for fcc, bcc and hcp materials, and highlighted the most important members within each class. Our consideration of real surfaces brought relaxation and reconstruction into the same conceptual framework; the most important observation regarding real surfaces, however, is their preservation of truncation symmetry relative to

Conclusions

The underlying philosophy of this entire work has been to disentangle the symmetry properties of surfaces from their detailed structural characteristics. These two fundamental properties are undoubtedly related, but their conceptual separation has proved highly advantageous, yielding a secure and general approach for the rationalisation of all single-crystal surfaces. Indeed we have shown that our symmetry–structure surface stereography (4S) analysis is not only applicable for primitive

Acknowledgments

We are grateful to The Royal Society for a University Research Fellowship (SJJ) and to EPSRC for Post-Doctoral funding (SJP). The valuable comments of Dr Simon Titmuss and Dr Pedro de Andres on an early draft of the manuscript were very helpful and much appreciated. We warmly thank Dr Stephen Driver for his various helpful and important contributions throughout the duration of this project. Sincere thanks are also due to Professor Sir David King, whose positive comments and encouraging response

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