74
P. Krüger
In summary, TDDFT with linear response provides a rigorous and efficient
framework for calculating absorption spectra. Compared to the independent particle
approximation, TDDFT takes the screening of the electromagnetic field into account
by introducing the induced field ϕ ind (t) which is calculated self-consistently with
the density change δn(t). In practice, the problem is to find good approximations for
the unknown exchange–correlation kernel K XC . The Hartree part alone, i.e. putting
K XC = 0, yields the well-known random-phase approximation (RPA) [10]. Apart
from the single particle–hole excitations included in χ 0 , the RPA can describe plasmon excitations, i.e. collective oscillations of electron gas, which can be observed,
for example, as satellite peaks in core-level photoemission spectra. The RPA kernel
also gives rise to a redistribution of spectral weight between different transitions.
This may strongly change the peak intensity ratio, e.g. between the L 2 and L 3 white
lines in transition elements [10]. In adiabatic TDDFT, K XC can be obtained from
standard DFT [8], but such static approximations to K XC do not improve much over
the RPA [10, 12]. It appears that complex configuration mixing such as multiplet
excitations cannot be described by the common, adiabatic kernels. Going beyond
the adiabatic approximation is difficult, but some non-adiabatic kernels have been
proposed and applied to the X-ray absorption problem [13].
3.6 Photoemission Spectroscopy
PES is probably the most direct way of probing the electronic structure of materials.
In a PES, light is shone on a surface and the kinetic energy, and possibly exit angle
and spin, of the emitted electrons is measured. In core-level PES, electrons from
the inner atomic shells are excited. As these levels are element specific, core-level
photoemission is a powerful tool for chemical analysis.
Angle-resolved core-level photoemission from crystal surfaces is known as Xray photoelectron diffraction [14]. The photoelectron wave spreads from the core
hole site and is diffracted by the neighbouring atoms. Analysis of the diffraction
pattern gives precise information about the local structure around the atoms of a
given chemical species. X-ray photoelectron diffraction can be well modelled with
real-space single or multiple scattering theory on a finite cluster of atoms.
3.6.1 Angle-Resolved Photoemission Spectroscopy
Angle-resolved photoemission spectroscopy (ARPES) is the major method for measuring energy band dispersion (the ‘band structure’) of crystals. An intuitive picture
of ARPES is provided by the three-step model [15]. The three steps are as follows:
1. Photon absorption in the bulk of the material resulting in an inter-band transition
|mk → |nk, with (n, k) = (m, k) + ω. Here n, m are band indices and the
P. Krüger
In summary, TDDFT with linear response provides a rigorous and efficient
framework for calculating absorption spectra. Compared to the independent particle
approximation, TDDFT takes the screening of the electromagnetic field into account
by introducing the induced field ϕ ind (t) which is calculated self-consistently with
the density change δn(t). In practice, the problem is to find good approximations for
the unknown exchange–correlation kernel K XC . The Hartree part alone, i.e. putting
K XC = 0, yields the well-known random-phase approximation (RPA) [10]. Apart
from the single particle–hole excitations included in χ 0 , the RPA can describe plasmon excitations, i.e. collective oscillations of electron gas, which can be observed,
for example, as satellite peaks in core-level photoemission spectra. The RPA kernel
also gives rise to a redistribution of spectral weight between different transitions.
This may strongly change the peak intensity ratio, e.g. between the L 2 and L 3 white
lines in transition elements [10]. In adiabatic TDDFT, K XC can be obtained from
standard DFT [8], but such static approximations to K XC do not improve much over
the RPA [10, 12]. It appears that complex configuration mixing such as multiplet
excitations cannot be described by the common, adiabatic kernels. Going beyond
the adiabatic approximation is difficult, but some non-adiabatic kernels have been
proposed and applied to the X-ray absorption problem [13].
3.6 Photoemission Spectroscopy
PES is probably the most direct way of probing the electronic structure of materials.
In a PES, light is shone on a surface and the kinetic energy, and possibly exit angle
and spin, of the emitted electrons is measured. In core-level PES, electrons from
the inner atomic shells are excited. As these levels are element specific, core-level
photoemission is a powerful tool for chemical analysis.
Angle-resolved core-level photoemission from crystal surfaces is known as Xray photoelectron diffraction [14]. The photoelectron wave spreads from the core
hole site and is diffracted by the neighbouring atoms. Analysis of the diffraction
pattern gives precise information about the local structure around the atoms of a
given chemical species. X-ray photoelectron diffraction can be well modelled with
real-space single or multiple scattering theory on a finite cluster of atoms.
3.6.1 Angle-Resolved Photoemission Spectroscopy
Angle-resolved photoemission spectroscopy (ARPES) is the major method for measuring energy band dispersion (the ‘band structure’) of crystals. An intuitive picture
of ARPES is provided by the three-step model [15]. The three steps are as follows:
1. Photon absorption in the bulk of the material resulting in an inter-band transition
|mk → |nk, with (n, k) = (m, k) + ω. Here n, m are band indices and the