  The standard theory of thermionic emission developed for three-dimensional
semiconductors does not apply to two-dimensional materials even for making
qualitative predictions because of the vanishing out-of-plane quasiparticle
velocity. This study reveals the fundamental origin of the out-of-plane charge
carrier motion in a two-dimensional conductor due to the finite quasiparticle
lifetime and huge uncertainty of the out-of-plane momentum. The theory is
applied to a Schottky junction between graphene and a bulk semiconductor to
derive a thermionic constant, which, in contrast to the conventional Richardson
constant, is determined by the Schottky barrier height and Fermi level in
graphene.
