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The photoemission process can be considered using the so-called three-step model of Berglund and Spicer [70], consisting of: (i) the photo-excitation of an electron; (ii) its passage to the surface of the solid; and (iii) its penetration through the surface into the vacuum. The discussion here is based upon that of H¨ufner [71].

In the first step, using Fermi’s golden rule to give the transition probability between an initial state, |ii, and final state,|fi, the energy distribution of photoex- cited electrons, N(E), can be obtained as

N(E)∝X

f,i

|Mf i|2δ(Ef −Ei−hν)δ(E−(Ef −φ))δ(ki+G−kf). (3.3)

In the presence of electromagnetic radiation, the Hamiltonian of the system [55]

H = 1 2me · p+ µ eA c ¶¸₂ +V(r) ≈ H0 + e mec A·p (3.4) whereH0 = p 2

2me+V(r),pis the momentum operator, Ais the vector potential, the

3.1. Photoemission spectroscopy 30

and termsO(A2_{) have been neglected. Consequently, the transition matrix element}

Mf i∝ hf|A·p|ii ∝ hf|(E·r)|ii (3.5)

where the final expression corresponds to the electric dipole approximation, appro- priate if the photon wavevector is small. The first delta function in Eqn. 3.3 ensures conservation of energy during the excitation of the photoelectron, while the second ensures that the kinetic energy of the photoelectron in the vacuum is that of the final state inside the crystal minus the work function [71]. The third delta function ensures conservation of momentum, up to a reciprocal lattice vector.

The second step considers the transport of the excited photoelectrons to the surface of the solid, during which time the photoelectrons can be inelastically scat- tered, predominantly by other electrons, but also by phonons or ionized impurities. Such inelastic scattering events cause the photoelectrons to be emitted from the solid with a lower energy than would be expected from their initial state, and these therefore contribute to the background of photoemission spectra.

The total intensity of photoelectrons emitted from a distance d below the surface that have not been inelastically scattered by the time that they arrive at the surface follows the Beer-Lambert law

I(E) = I0(E)e−d/λsinθ (3.6)

where I0(E) is the initial intensity of photoemitted electrons of energy E, θ is the

polar angle that the detector makes to the surface, so that d/sinθ is the effective path length to the surface, andλ is the inelastic mean free path (IMFP) of the elec- trons. Consequently, PES is a surface specific technique with, in a normal emission geometry, 65% of the photoemission signal originating from withinλ of the surface and 95% from within 3λ of the surface, with the majority of the signal originating from photoelectrons generated closest to the surface. The IMFP can be estimated for specific materials from semi-empirical means such as the TPP-2M predictive for- mula of Tanumaet al.[72]. However, Seah and Dench [73] showed that the IMFP for a very large number of elements and compounds all fit approximately on a ‘universal curve’ which shows a pronounced minimum in IMFP ofλ≈5 ˚A for electron kinetic

3.1. Photoemission spectroscopy 31

energies of ∼30−70 eV, with a marked increase towards both lower and higher kinetic energies. Consequently, PES measurements made in this energy range are the most surface sensitive.

A characteristic of the inelastic scattering events is a stepped background, with an increase in background intensity observed to the lower kinetic energy (higher binding energy) side of spectral features [74]. For non-monochromatic sources, photoemission due to Bremsstrahlung radiation can also give rise to a general background, in addition to satellite features due to lower intensity satellite lines of the photon source, although these two features do not need to be considered for monochromatic sources. The background can be accounted for using various methods. However, for most purposes, fitting a Shirley integrated iterative back- ground [75] is sufficient.

The removal of the photoelectron from the solid into the vacuum is described by the final step of the three-step model. Assuming perfect two-dimensional transla- tional symmetry in the plane of the surface, parallel momentum must be conserved in this step, up to a parallel reciprocal lattice vector,Gk, and so the momentum of the final electron in vacuum, p, satisfies

pk

~ =ki,k+Gk. (3.7)

Assuming a free electron dispersion of the final state bands, the parallel wavevector of the initial state can therefore be determined

ki,k = kf,k = ¯ ¯ ¯ ¯

¯sin(θ) (cos(φ)ˆx+ sin(φ)ˆy) r 2meEk ~2 ¯ ¯ ¯ ¯ ¯ = sin(θ) r 2meEk ~2 (3.8)

where the coordinate system is defined in Fig. 3.1(b). This ability to determine the initial state wavevector simply from the angle of the photoemitted electrons is the central concept of angle resolved photoemission spectroscopy (ARPES). However, the wavevector is not simply conserved normal to the surface. Consequently, ARPES is most effective for looking at electronic features characteristic of two-dimensional

3.1. Photoemission spectroscopy 32

solids, such as surface states on metals or semiconductors. However, by applying various approximations, such as free-electron final state bands and a step function for the surface potential, it is possible to obtain information on the perpendicular momentum, allowing, for example, bulk band mapping of semiconductors [71, 76].

Including all steps of the three-step model, the total photocurrent,I(E,pk), therefore becomes I(E,pk) ∝ X f,i |Mf i|2δ(Ef −Ei−hν)δ(E−(Ef −φ))δ(ki+G−kf) ×D(E)× |T(E,pk)|δ(ki,k+Gk−(pk/~)), (3.9)

where D(E) accounts for the attenuation of the excited photoelectrons by inelas- tic scattering (step 2) and T(E,pk) is the transmission factor for photoelectrons penetrating the surface (step 3).

Although it has proved remarkably successful in explaining photoemission features, the treatment of each step separately in the three-step model is a simpli- fication. A more accurate picture is the one-step model, where the photon induces an excitation from the initial state directly into a damped final state, which propa- gates in the vacuum, but decays away into the solid near the surface. In fact, the arguments leading to Eqns. 3.3 to 3.5 are appropriate for a one-step model, provided that the final state wavefunction is taken as the damped state propagating into the vacuum. However, theoretical treatments of this model, such as the inverse LEED formalism, are beyond the scope of this thesis, and the interested reader is referred to Refs. [71, 77].