Diploma - Max Planck Institute for Solid State Research

14 2 Theoretical foundation
Figure 2.6: different quantum states: upper row – final states, lower row – initial states; (a)
bulk Bloch wave weakly damped, (b) strongly damped Bloch wave, (c) “surface” / gap state,
(d) bulk Bloch wave and (e) surface state inside the bulk band gap [28, p. 273]
which resembles Snell’s law (refraction of the wave vector). Choosing the reference
frame outside, the maximum angle equals ϑ out, max = 90 ◦ . Since E kin = E f − E F +
Φ, the angle ϑ in < ϑ out and thus all electrons “inside” this cone (ϑ < ϑ in,max ) will
transmit to the vacuum (cf. [28, p. 248]). Evidently, all inelastically scattered
electrons – depending on the number of scatter events – will have different escape
cones.
A reduced type of this model has been used for evaluating the experimental photoemission
spectra and transfering the results to reciprocal space.
2.2.2 One-step model
The decomposition of the PE process into several parts neglects important interference
effects between different emission channels (e.g. bulk and surface emission) and simplyfies
the transmission probability severely [28, p. 280]. Hence describing it properly as a
transition between two quantum mechanical states will respect the wave / particle duality.
Using Fermi’s Golden rule and an approximation for the interaction Hamiltonian
H int w fi = 2π
∣
∣〈 f | H int | i 〉 ∣ ∣ 2 δ (E f − E i − ω) (2.15)
with e.g. H int = 1 (A · p + p · A) (2.16)
2mc
as well as a reasonable composition of initial | i 〉 and final states 〈 f | yields generally the
“correct” spectra. For H int the interaction between an electron (momentum operator p)
and a photon (field operator A) can be assumed. In a manybody description the transition
operator (e.g. f emission: t f (ω) (fψ + + ψ + f); f + (f) is the creation (annihilation)
operator for an f electron and ψ the corresponding photon operator, t f (ω) represents the
weight for this emission channel) may be used for specific emission channels. In addition,
spectral broadening can be dealt with special representations for the δ-distribution