Association

2.4. Hard X-ray photoemission spectroscopy 21
Figure 2.12.: Momentum conservation during
photoexcitation in a reduced zone scheme
inside a solid. For X-ray excitation, transitions
are mainly vertical, i.e. shifted only
by the photon energy hν. Free states must
be available at the energy E f and in the momentum
space at k f = k i + G.
vector) has to be fulfilled for photoexcited electrons within the solid,
k f = k i + k γ + G, (2.18)
where k f , k i , and k γ denote the wave vectors of the electron’s initial state, its final state,
and the photon’s wave vector, respectively. G is a reciprocal lattice vector of the crystal, as
sketched in Fig. 2.12. Concluding, we can write the photocurrent I inside the crystal, taking
into account the transition probability (2.16) and the momentum conservation (2.18), as
follows:
∑
I(hν) ∝
∣
∣ M ∣∣ 2
if ·δ(Ef − E i − hν)·δ(k f − k i − G). (2.19)
i,f ,G
2. Transfer of the electron to the surface of the solid What happens during the travel of
the excited electron through the solid to the surface? A fraction of the electrons experiences
collisions with other electrons and phonons. The material parameter describing inelastic
electron scattering of electrons in a specific solid is the inelastic mean free path (IMFP). The
IMFP is a measure how far a photoelectron can travel through a solid before an inelastic
scattering events changes its energy information. The IMFP value is strongly dependent on
the kinetic energy of the photoelectron. IMFPs for common solids in practice are estimated
by the the modified Bethe-equation (TPP-2M formula) derived by Tanuma et al. (1994), 83
λ =
E 2 p
E kin
(
), (2.20)
β ln(γE kin ) −
E C + D
kin E 2 kin
where E kin is the electron energy, E p is the free-electron plasmon energy, and β, γ, C, and
D are material-specific parameters. In this thesis, we use recalculated values for hard X-ray
excitation from Tanuma et al. (2011). 84
3. Transfer of the photoelectron through the surface into the vacuum The third step
considers the transfer of the photoelectron through the surface of the solid into a vacuum
state. Only those electrons can leave the solid by dispensing energy of the value of the work
function φ 0 , which have a sufficiently high kinetic energy. Escaping electrons, thereby, have
the kinetic energy
E kin = E f − φ 0 = hν − E bin − φ 0 . (2.21)