Theory of angle-resolved photoemission from localised orbitals at solid surfaces

The theory of angle resolved photoemission from localised orbitals is reviewed and is cast in a form requiring the calculation of the purely outgoing wave emanating from an emitting atom, that describes the final state of the photoelectron, rather than using the more usual approach based on time reversed scattering states. An explicit expression is written down for the superposition of partial waves that results from emission from an atomic orbital, and it is pointed out that emission from more complex initial states such as localised bonds or Bloch and surface states, can be described by coherently combining such sets of partial waves. The effect of the crystal surface environment in damping and scattering the waves is described briefly. Model calculations are performed to investigate the major influences on the angular distribution of the photoelectrons. The profound effect of varying the polarisation direction of the incident light relative to the surface is discussed, with examples from the literature, showing how it can be used to determine the type of initial orbital. Emission from directed orbitals is studied and it is shown that scattering by the emitter potential can be an important effect, so that the radial wave function of the outgoing electron, which determines the amplitudes and phases of the outgoing waves, must be calculated with care. Different choices of these quantities lead in the model calculations to very different angular profiles, that sometimes bear little relation to the shape of the initial orbital. The consideration of emission from localised bonds shows that provided that bonds are not too strongly polarised, interference between waves from different centres is always significant at higher energies, and can also be important at energies of a few eV relative to the vacuum. Scattering by the ion cores of the surface region can strongly distort the angular distribution, or may have little effect. But it is generally difficult to decide a priori which influences are dominant for a particular case, so that the interpretation of angular profiles must be based on careful calculations including all these effects. The optimum energy range for the interpretation of experimental data is that from about 30 to 100 eV.