Abstract

CHAPTER 1. OVERVIEW 3
obtain. Thus, physically measuring how a normal quantum system is functioning
would require an account of the effects of the observer on the reported data. To avoid
this issue, engineers and physicists researching these quantum devices derive and use
accurate models of these quantum systems from first-principle physics.
In this work, we develop and analyze numerical methods for solving the Wigner-
Poisson equations. These equations describe the statistical mechanics of particles
under the influence of quantum mechanics. A classical analogue to the Wigner equa-
tion is the Boltzmann equation [11] for describing the statistical mechanics of gas
particles or electrons on the macroscale. The Wigner-Poisson equations consist of
two equations: the Wigner distribution equation, a nonlinear integro-partial differ-
ential equation which describes electron transport on the quantum level, coupled to
Poisson’s equation, which determines the potential created by the electrons. In the
past two decades, the Wigner-Poisson equations have been used to predict the be-
havior of nanoscale semiconductor devices [14],[22]. One particular nanostructure we
are interested in is the resonant-tunneling diode (RTD).
Recently, resonant tunneling diodes have been considerably researched in the field
of semiconductor technology [35], [36], [8]. Figure 1.1 shows a diagram of a RTD and
the electric potential within the RTD.
A RTD is created by the joining together two different semiconductors, material I
and material II semiconductors. In a semiconductor material, the state of an electron
is determined by its energy. If an electron has enough energy, then it is able to move