3 Electronic Structure Theory for X-Ray Absorption and Photoemission Spectroscopy
75
three-dimensional crystal momentum vector k is conserved up to a reciprocal
lattice vector G.
2. Propagation of the excited wave to the surface, with damping due to inelastic
scattering.
3. Transmission through the surface by matching the Bloch wave |nk to a plane
wave exp(ik
· r). The matching conditions are dictated by conservation of energy
and the surface parallel component of k, i.e. k
|| = k || and k
2
⊥ = k
2
⊥ + 2mV 0 /
2 ,
where V 0 represents the surface potential barrier.
The three-step model is very useful for relating the photoemission data to the threedimensional band structure of the material. However, for an accurate calculation
of ARPES intensities, the one-step model should be used, where the photoelectron
final state is calculated in all space (bulk, surface and vacuum) as a single wave function with proper boundary conditions (so-called ‘time-reversed low-energy electron
diffraction’ boundary conditions). A suitable computational scheme is the layered
Korringa–Kohn–Rostoker method [16].
The hole left behind in the photoemission process is not an independent particle,
but it interacts with the electrons and the lattice, giving rise to various many-body
effects, which are conveniently described using quasiparticle theory.
3.7 Quasiparticle Theory
In a photoemission experiment, an electron is ejected from the system, which
becomes ionized. Neglecting the interaction between the photoelectron and the hole
left behind, i.e. applying the so-called sudden approximation, the photoemission
excitation is a one-electron removal process from the N -particle ground state to a
N − 1 particle excited state. In the same fashion, inverse photoemission probes the
one-electron addition process from the N -particle ground state to a N + 1-particle
excited state. The true excitations are called quasiparticles. In the limit of vanishing electron interaction, the quasiparticle wave functions are the spin-orbitals of the
ground state Slater determinant and the quasiparticle energies are the one-electron
levels. In the independent particle approximation, the quasiparticles are taken as HF
or Kohn–Sham orbitals. This neglects electron correlation and the interaction of the
electrons with the lattice vibrations. These effects change the quasiparticle energies
and wave functions. The quasiparticles are said to be renormalized or ‘dressed’ by the
interaction. In particular, due to inelastic scattering at collective excitations such as
phonons and plasmons, the one-electron quasiparticles will decay after a characteristic lifetime. As a result, compared to the delta-function-like photoemission peaks
corresponding to the independent particle approximation, the true photoemission
peaks are energy shifted and lifetime broadened. Moreover, some spectral weight
of the main peak is lost to extra (‘satellite’) peaks, corresponding to some inelastic
process.
75
three-dimensional crystal momentum vector k is conserved up to a reciprocal
lattice vector G.
2. Propagation of the excited wave to the surface, with damping due to inelastic
scattering.
3. Transmission through the surface by matching the Bloch wave |nk to a plane
wave exp(ik
· r). The matching conditions are dictated by conservation of energy
and the surface parallel component of k, i.e. k
|| = k || and k
2
⊥ = k
2
⊥ + 2mV 0 /
2 ,
where V 0 represents the surface potential barrier.
The three-step model is very useful for relating the photoemission data to the threedimensional band structure of the material. However, for an accurate calculation
of ARPES intensities, the one-step model should be used, where the photoelectron
final state is calculated in all space (bulk, surface and vacuum) as a single wave function with proper boundary conditions (so-called ‘time-reversed low-energy electron
diffraction’ boundary conditions). A suitable computational scheme is the layered
Korringa–Kohn–Rostoker method [16].
The hole left behind in the photoemission process is not an independent particle,
but it interacts with the electrons and the lattice, giving rise to various many-body
effects, which are conveniently described using quasiparticle theory.
3.7 Quasiparticle Theory
In a photoemission experiment, an electron is ejected from the system, which
becomes ionized. Neglecting the interaction between the photoelectron and the hole
left behind, i.e. applying the so-called sudden approximation, the photoemission
excitation is a one-electron removal process from the N -particle ground state to a
N − 1 particle excited state. In the same fashion, inverse photoemission probes the
one-electron addition process from the N -particle ground state to a N + 1-particle
excited state. The true excitations are called quasiparticles. In the limit of vanishing electron interaction, the quasiparticle wave functions are the spin-orbitals of the
ground state Slater determinant and the quasiparticle energies are the one-electron
levels. In the independent particle approximation, the quasiparticles are taken as HF
or Kohn–Sham orbitals. This neglects electron correlation and the interaction of the
electrons with the lattice vibrations. These effects change the quasiparticle energies
and wave functions. The quasiparticles are said to be renormalized or ‘dressed’ by the
interaction. In particular, due to inelastic scattering at collective excitations such as
phonons and plasmons, the one-electron quasiparticles will decay after a characteristic lifetime. As a result, compared to the delta-function-like photoemission peaks
corresponding to the independent particle approximation, the true photoemission
peaks are energy shifted and lifetime broadened. Moreover, some spectral weight
of the main peak is lost to extra (‘satellite’) peaks, corresponding to some inelastic
process.