Association

22 2. Theoretical background
E F =0
E kin
photoelectrons
spectrum
E
electronic structure
hν
Figure 2.13.: Photoemission from a solid.
The photoemission process translates the
electronic density of states of the specimen
into a photoemission spectrum, if a
single-electron picture is assumed. The
intensity of the photoelectrons as a function
of the binding energy, the so-called
energy distribution curve (EDC), shows
the density of occupied electronic states
in the solid. After Hüfner (2010). 80
E vac
E F =0
E B
vacuum level
Φ 0
Fermi level
valence band
core levels
N(E)
hν
I(E)
Here, E f is the final state energy relative to the Fermi level E F , E bin denotes the binding energy
of the electron, and φ 0 is the work function of the solid which equals the difference between
the Fermi level and the vacuum level (4–10 eV). The final states are free electron states in
the vacuum and generally not relevant, and the photoemission intensity is proportional to
the density of initial states of energy E i . Thus, in the one-electron picture, photoemission
spectroscopy images the occupied density of states (DOS) of the solid. The exit process of the
photoelectron and the resulting spectrum (energy distribution curve) is sketched in Fig. 2.13.
Exciting the sample involves diffraction (i. e. a change of k ⊥ ) at the potential barrier of the
crystal surface. In core-level photoemission in the XPS and HAXPES regime (hν > 1 keV),
however, the so-called XPS limit can be applied: for most materials at room temperature,
the combined effects of phonons and angular averaging in the spectrometer yield photoemission
spectra that are directly related to the matrix element-weighted density of states (MW-
DOS). 85 This means, in HAXPES, the k dependence in eq. (2.19) is negligible. Furthermore,
element-specific atomic cross-sections can be used to weight the MW-DOS for quantitative
evaluations. 86 which the simplified three-step model does not account for.
The one-step theory of photoemission
An exhaustive treatment of the photoemission process is provided in a the one-step theory.
80,87,88 It contains the Golden Rule equation with proper wave functions of the initial
and final states, and the dipole operator for the interaction of the electron with the incoming
light. Inverse LEED wave functions are used for the final state of the photoelectrons and
the wave functions are expanded as Bloch functions in the periodic crystal, as sketched in
Fig. 2.11b. This allows one to discriminate between different scenarios of matching types
between the free electron wave in the vacuum and the Bloch function in the crystal. This
matching can be (i) simply inside the crystal, (ii) in a band gap yielding an “evanescent wave”,
or (iii) in a band-to-band photoemission for small escape depths. Damping due to inelastic