CHAPTER 1
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 Surface State Existence. — We have defined the local density of states above in terms of the one-electron wavefunctions Ψn(r), which are the solutions of equation (2). In a perfectly periodic system, such as an infinite crystal, these solutions are constrained (by Bloch's theorem; see ref. 1) to be of the form (3)
Ψk[r] = eik•ruk (r) (3)
where uk(r) has the periodicity of the crystal lattice. In the infinite crystal the eigenvalue k which is fixed by the energy of the eigenstate (see ref. 1, which shows that the k's are quantum numbers for the periodic crystal)] must be real, since only then can we keep Ψk(r) finite throughout the crystal. However, in the proof of Bloch's theorem (see ref. 1, in particular Mott and Jones), there is no limitation to real values of k, and in fact we can show that for certain energies in a given crystal, complex k's are needed for the solution of equation (2). It is this requirement of complex k's, together with the forbidding of such solutions by the boundary conditions (Ψ finite as r -> ∞) of an infinite crystal that leads to energy gaps in the band structure.
We repeat, the problem with complex k values is that if we write equations (4)
k = k[parallel] + kz (4)
kz = kr + iq
the wavefunction behaves as e-qz which diverges in the infinite crystal. However, as was pointed out by Tamm, such solutions can be allowed in a finite crystal as long as the Ψ is bounded within the finite crystal and can be matched across the boundaries. If these solutions lie below the vacuum level, electrons in them will be localized near the surface, since the wavefunction amplitude will decay both into the vacuum and into the crystal. These solutions lie within the infinite crystal energy gaps, since they have complex k's, and are known as surface states. They are intrinsic to the clean crystal.
Since 1932 many calculations of surface-state energies have been performed, both for simple models and, more recently, on systems which accurately reflect the real crystals. They are described in the reviews of Jones and of Davison and Levine (see Table). Virtually all of these calculations, and all of our discussion, consider a semi-infinite crystal with its surface lying in the x–y plane. The semi-infinite crystal has translational symmetry parallel to the surface (i.e., we consider only a clean surface or one with an ordered overlayer) and the component of 'crystal momentum' parallel to the surface (k)[parallel] is a good quantum number and is conserved to within a two-dimensional r.l.v. of the surface (see ref. 1). Energy is also a conserved quantity. Since the crystal is infinite in the surface plane, k]parallel] must be real, and we need study only those k[parallel] in the first surface Brillouin zone. (The surface Brillouin zone is derived from the two-dimensional surface net, and it is this zone (not the two-dimensional projection of the more familiar three-dimensional bulk Brillouin zone) which gives the k[parallel] value.) Momentum normal to the surface, K., is not a good quantum number and the surface mixes together states with the same energy and k[parallel], but different Kz. Our surface states in general do not have a well-defined Kz, since they are composed of a sum of different Kz states, and so decay lengths quoted for surface states refer to the decay of the least localized component.
The results of all these calculations show that a small number of surface states, usually one or zero, exists in each energy gap of a given crystal-band structure, and that all types of material can exhibit these states. Some of these results are described below and fuller surveys appear in some of the reviews listed in the Table. Before we describe these in any detail however, we first consider the general problem of the existence of surface states, to see whether it is possible to predict that a surface state will exist in a given gap without performing a full and detailed calculation. In the general rules that follow, it is to be understood that the statements refer to a gap at a given value of k: in order to obtain the total numbers of surface states we must sum over all k[parallel] points in the surface Brillouin zone. Shockley gave a simple rule for the specific case of the nearly-free-electron model: under certain limiting conditions about the surface potential barrier, a gap at the Brillouin zone boundary (Kz = 0, ½Gz where Gz is the z-component of a three-dimensional reciprocal lattice vector) will have one and only one surface state, provided the gap is 'inverted', i.e., if the wavefunctions of p-like character lie at the bottom of the gap with the s-like functions at the top (see ref. 1, especially Harrison and Kittel). This situation corresponds to having the matrix element responsible for the gap negative. In the lowest approximation, this matrix element is just V(Gz), the Fourier component of the potential V(r) which corresponds to the r.l.v. Gz This result has been extended within the same model by Forstmann who showed that a gap within the zone (i.e., Kz ≠ 0, ½Gz), caused by the hybridization of two plane waves, always contains a surface state, irrespective of the details of the crystal or the potential at the surface. The most recent generalization of these rules has been given by Pendry and Gurman, and includes all types of crystal, with the restriction only that the crystal must have a mirror plane parallel to the surface and a centre of symmetry. Their conclusion is that a narrow gap within the zone can contain zero, one, or two states only while a narrow gap at a zone boundary can contain zero or one state only. The problem of whether or not a particular gap contains a surface state depends on a single matrix element of the potential (cf. the situation above concerning the inverted gap). In certain cases it is still possible to say definitely how many states a gap contains without recourse to calculation at all. For a gap at the zone boundary, for each value of k[parallel], there can be zero or one state depending on the sign of the matrix element of the potential responsible for the gap: one sign gives a state, the other does not. For gaps inside the zone, at each value of k[parallel] there will be one, and only one, surface state in all the following cases: (i) the bands are nearly-free-electron like, or are of the simple nearest-neighbour-interaction tight-binding (LCAO to the chemists) form; (ii) the surface potential barrier can be approximated by an infinitely high barrier located on a mirror plane; and (iii) the interaction responsible for opening up the gap is not present in the barrier. As an example of (iii) we have the gap opened up by a spin-orbit interaction (e.g., in tungsten): this will always contain a surface state.
From these general theories we would expect surface states to be a common occurrence on clean surfaces. Calculations on individual materials support this belief, and we now proceed to describe the results of a selection of such calculations.
Surface States on Semiconductors. — Although a few early papers considered the effects of the surface upon the electrons of a finite crystal, interest in surface states was really aroused by the early work of Bardeen and Shockley on semi-conductors in the late 1940's. These workers showed that many of the 'anomalous' properties of these materials could be explained by postulating the existence of states localized near the surface in the energy gaps between the valence and conduction bands. The Fermi level at the surface was fixed within this band of states by the requirement of charge neutrality near the surface, and this leads to strong modification of the surface properties: for example, since this band is half-filled, the surfaces of a semiconductor act as conductors (in ideal cases) for currents parallel to the surfaces. Since these early experiments, the main theoretical and experimental interest has centred on the elemental semiconductors, silicon and germanium, which are the most widely used in practice. We will consider as an example the (111) surface of silicon, since this has been the most popular subject for investigation. The properties of the electrons in bulk Si can be well described in the nearly-free-electron or the tight-binding models, and also by special methods (related to the tight-binding model) based on the bond picture of Si. All give similar results and silicon therefore provides a good example for comparing and linking the various approaches.
The (111) surface consists of a plane of atoms arranged in a hexagonal array [see Figure 1(a)], each linked by three bonds to its neighbours in the layer below. In a simple bond picture, the fourth bond would stick out normally from the surface, giving a 'dangling bond' surface state. The bonds are sp3 hybrid orbitals, derived from the atomic orbitals so as to satisfy the bulk bonding symmetry. This simple picture is complicated somewhat by relaxation of the outermost layer; it is energetically favourable for the surface layer to move backwards towards the second layer, and to reconstruct. The relaxation changes the bond hybridization towards sp2,characteristic of the two-dimensional layer symmetry, plus a p-like dangling bond.
Several investigations have been performed with use of plane-wave expansions for the crystal wavefunction, these being matched across the surface potential barrier to vacuum wavefunctions to obtain the allowed wavefunctions of the semi-infinite crystal. These have been completed for both the unrelaxed and relaxed surfaces and have recently been extended to achieve self-consistency in the electronic potential set up by the electrons in their allowed wavefunctions, and from these results the charge density and surface potential have been calculated (see the next section for a further discussion). In the case of the unrelaxed surface where the interatomic spacings of the bulk were used right up to the surface, the calculations give a single band of surface states lying in the gap between the valence and conduction bands. This band is ca. 0.7 eV wide and lies in the upper half of the gap. If we allow the surface layer to relax, and solve the problem self-consistently, we find three bands of surface states: one in the valence-conduction band gap, this band now being 0.8 eV wide and crossing the indirect gap, and two within the valence bands. The inclusion of 2 × 1 surface reconstruction splits each of these bands.
Calculations made by using the tight-binding model with a basis of atomic orbitals whose interaction parameters are fitted to more elaborate band-structure calculations, give very similar results. We may also use a basis of bonding orbitals as the symmetric sum of oppositely directed sp3 hybrid orbitals along the common bond. Together with the antibonding orbitals, such a basis is equivalent to the atomic ones that have been used.
