Surface Electronic Structure

BY S. J. GURMAN AND M. J. KELLY

1 Introduction

Dramatic improvements in the production and control of ultrahigh vacua (p <.ca. 10-9 Torr) and methods of sample preparation in the last decade have allowed experiments to be performed on atomically clean or systematically overlayered metal and semiconductor surfaces. The progress in experimental techniques has been parallelled by theoretical developments in surface science which have drawn heavily on the methods used in the study of the physics of bulk solids, suitably modified to incorporate the crystal–vacuum interface. The behaviour of electrons localized in the surface region of a crystal, and of electrons in the propagating (i.e., bulk) states near the surface is being extensively investigated at the present time, and in this brief review we wish to take the concepts and results of the solid-state-physics approach to the problem and interpret them, wherever possible, in terms of the chemical concepts typified by the bond model, particularly in terms of the dangling bond at the surface. It is impossible to be exhaustive, or even moderately complete, in a short review, and readers are referred to the Table where some major reviews of the field are listed, together with short notes as to their contents and style.

The major link between the theory and experiment is the density of states, the number of electron states in a given energy range. Near the surface, this becomes strongly position-dependent and we define the local density of states (LDOS) by equation (1) where the wavefunctions Ψn(r) and their energies En are the

[MATHEMATICAL EXPRESSION OMITTED] (1)

eigensolutions of the Schrödinger equation (2) for a potential V(r) appropriate to

[MATHEMATICAL EXPRESSION OMITTED] (2)

the crystal-vacuum system. The number of electrons in the energy range E -> E + dE and in a volume element dr about position r is given by N(E,r) dEdr. he theorist can determine the LDOS, subject to (often quite severe) approximations which are necessary to make the problem tractable either in terms of the usual methods of band-structure calculations, 1 or the more recently developed local methods. The experimentalist can sometimes measure quantities related to the density of electron states, but perturbed and modulated (often severely) by the experimental probe used. Nevertheless, the LDOS is a useful place for theory and experiment to meet, and to compare and contrast the information from different experiments.

We know from work-function measurements that there is a surface potential barrier for electrons extending some 5eV (1 eV/electron [??] 23.05 kcal mol-1) above the highest energy of the occupied part of the electron distribution in the solid (the Fermi energy), and so no electrons can escape from the crystal. If we are working in terms of electron wavefunctions, the results of elementary quantum mechanics tell us that we may require wavefunctions which decay into the crystal 3 in order to match the crystal and vacuum wavefunctions at the surface, and so obtain the eigensolutions of the solid–vacuum system. Such spatially decaying solutions, which are forbidden in the infinite solid (see the next section for details), lead both to electron states localized near the surface (surface states) and to the modification of the propagating bulk states in the region near the surface.

It is at this point that we come to the difference in methods used by the majority of physicists and chemists. Consider the (111) surface of silicon. To the solid-state physicist, silicon is a material in which the valence electrons behave as if they were nearly free, interacting only weakly with the ion cores via an effective potential known as the 'pseudopotential'. (The precise reasons for this are very subtle and details are to be found in recent texts on solid-state theory.) Their wavefunctions are moderately perturbed from a single plane wave, being generally described as a combination of plane waves whose wave-vectors differ by a vector in the Fourier transformed lattice of the ions, a reciprocal lattice vector At (r.l.v.).the surface we match these plane waves through the surface barrier to the plane wave solutions of the vacuum, and so derive the surface-state wavefunctions in terms of plane waves. The chemist sees silicon as a classic covalent material with its valence electrons paired in bonds between adjacent atoms. At the (111) surface, one orbital on each atom projects out from the free surface, and we can treat surface states as being derived from combinations of these 'dangling bonds'. In this Chapter we shall try to link up these two approaches to the problem, since they have both been used extensively to treat semiconductors. We shall also describe the energy band theory results on metals, and wherever possible translate these into the local orbital picture which is perhaps more useful in considering chemical bonding.

We shall review separately (i) the results of studies of intrinsic surface states, i.e., those states which are localized at the surfaces of clean crystals, as well as some of the relevant experimental data, and (ii) the progress made so far in understanding the behaviour of the bulk electron states in the region of the surface, together with the complementary experimental data. The distinction leading to these two sections is rather artificial since the localized or extended character of the electron states does not enter the local density-of-states expression [equation (l)]. In the final section we comment on the current directions of research efforts.

2 Intrinsic Surface States

The theory of intrinsic surface states has been almost exclusively carried through in terms of a band-structure formalism, and we must therefore use this prescription in describing the general theory. In the subsection on the theory of semiconductor surface states, we use both the bond and band formalisms, linking the two by way of the electron-density distribution in the crystal. A full description of the relationships between, and the uses of, the two systems has been given by Phillips.

General Results on...