## The role of the Van Hove singularity in the time evolution of electronic states in a low-dimensional superlattice semiconductor

##### Abstract

In this dissertation we will study a wide range of phenomena from
atomic, molecular, and optical to solid-state physics. We will find a common
theme in problems from these different branches of physics in that they can
all be modeled by some variation of a simple bi-linear Hamiltonian. Each of
these models will also share a key feature in that they all contain one or more
singularities (called a Van Hove singularity in the context of solid-state) in the
density of allowed states associated with a branch point that results near the
edge of a continuous energy spectrum. In addition, the fact that each of these
models is one-dimensional will maximize the effect of the singularity on the
system. We will show that when a discrete state is coupled with the continuum
that in the vicinity of the singularity Fermi’s golden rule breaks down; the
golden rule normally predicts that the de-excitation rate of the discrete state
should be proportional to g
2 where g is the dimensionless coupling constant
between the discrete state and the continuum. Relying on a non-perturbative
approach, we will show that the de-excitation rate is actually proportional to
g
4/3
in the vicinity of the singularity. This results in a dramatic amplification
of the decay rate.
In the main topic of the dissertation, we will consider a nano-scale
semiconductor superlattice with either a single impurity site or multiple impurities
(which behave as electron donors or acceptors) in which there are two
Van Hove singularities in the density of electron states which occur at the two
edges of the conduction band. These singularities result in the non-analytic
g
4/3 amplification of the charge transfer rate from the discrete impurity site
into the electronic conduction band where g is the coupling constant between
the impurity state and the conduction band.
We will demonstrate other results including an asymmetry in the optical
absorption profile for monochromatic light incident on a core electron state
in the single impurity system and bound states in continuum (BIC) for the
superlattice system with two impurities.