Dirac Fermions in Graphene and Graphite—a view from angle ...

information about the lifetime of electrons and many body interaction 17 . The rapid improvements in the
past few decades have made ARPES a powerful technique to reveal important information on the electronic
structure and many body interaction in various materials 18 .
Figure 2.1. (a) Schematics of ARPRES experiments (courtesy of J. Fink) and (b) typical ARPES
data from graphite.
The basic principle behind ARPES is the energy and momentum conservation. The kinetic energy and
momentum of the photoelectrons are related to the binding energy E B and crystal momentum k inside the
solid by the conservation laws.
E kin = hν − φ − |E B | (2.2)
p ‖ = k ‖ = √ 2mE kin · sin θ (2.3)
Here the in-plane momentum k ‖ is conserved because of the translational symmetry of the single crystal
and the negligible wave vector of the photons. For a two dimensional solid where there is no dispersion
along z-axis (sample normal direction), the electronic dispersion is completely determined by k ‖ . For a
three dimensional sample, the electronic structure also depends on k z and the full momentum information
including the out-of-plane momentum k z is important. The extraction of k z , however, is more complicate
because of the lack of translational symmetry at the interface between the sample and the vacuum. Thus k z
cannot be extracted directly without a priori assumption. However, k z can be extracted by modeling the final
state of photoelectrons, e.g. the ‘free-electron model’ or a calculated band structure 19,20 . The simplest and
most frequently used assumption is the free-electron approximation, where the final state is approximated as
a free electron state. This is only an approximation as the photoemission process takes place in the presence
of a crystal potential. The higher the photon energy is, the better this approximation is, because the effect
of the crystal potential becomes weaker for higher kinetic energy. Though the free-electron approximation
is a simple model, it has been a useful and accurate method to extract the k z values.
In the free-electron model, the dispersion of the final state is assumed to be that of a free electron in a
potential V in :
E k =
2
2m ∗ (k2 ‖ + k2 z) − V in , (2.4)
where m ∗ is the effective mass, which is usually taken as the free-electron mass m. V in (so called ‘inner
potential’) is a parameter that can be determined from the periodicity and the symmetry in the measured
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