Starting from the mid-1970s, metal oxide semiconductor (MOS) devices technology
was accompanied by an enormous pace of scaling. Transistor dimensions soon
reached a point at which initial modeling assumptions about physical effects
broke down. So far device simulation was mostly based on the classical
drift-diffusion or the hydrodynamic equations. These equations represent
macroscopic balance equations for the electron and hole concentrations and the
corresponding current densities. They can be derived from the Boltzmann
transport equation (BTE) by the method of moments [105].
In the sub
regime, these macroscopic device equations are
no longer predictive. One way to overcome this problem is to
introduce higher moments [106], another is to rely on a direct solution of
the Boltzmann equation. Due to the complexity of the Boltzmann
equation a discretization and direct solution as in the case of
device simulation is prohibitive. An alternative approach to solve the BTE is
provided by the Monte Carlo (MC) method which has been established as a
stochastic method for the solution of the Boltzmann equation [107].
Credit for inventing the Monte Carlo method often goes to Stanislaw Ulman, a
Polish born mathematician who worked for John von Neumann on the United States'
Manhattan Project during World War II. The method is named after the city in
Monaco, where the primary attractions are casinos where games of
chance are played [108,109].
Monte Carlo method, as it is understood today, encompasses any technique of statistical
sampling to approximate solutions to quantitative problems.
One common technique in the area of solid-state physics and the investigation
of materials with a regular crystal lattice as for e.g. silicon is the so
called band theory of solids. This theory deals with the widening of
discrete energy levels for electrons, as observed in the case of free atoms, to
available energy states from bands. This split occurs, if several atoms are
brought together, as it is the case for e.g. in a crystal structure, due to
the Pauli exclusion principle [110]. Such a constellation produces an
exceedingly large number of orbitals, which is proportional to the number of
atoms, and the difference in energy between them becomes very small. But there
are also gaps of energy which contain no orbitals, no matter how many atoms are
aggregated [111].
To find out the density of carriers in a semiconductor, the number of available
states at each energy has to be determined from the Schrödinger equation. The
number of electrons at each energy is then obtained by multiplying the number
of states with the probability that a state is occupied by an
electron [112].
However, due to the perception that carrier momentum is proportional to its
position in the reciprocal space, the dispersion relation between the energy
and momentum of a carrier can be described in reciprocal space. For
crystalline solids the reciprocal lattice is also periodic and the primitive
cell is called the first Brillouin zone.
For full band Monte Carlo (FBMC) simulation the discretization of the Brillouin zone has a major
impact on the accuracy of the simulation, so this chapter deals mostly
with the construction of a judicious tessellation of the whole Brillouin zone
and a particular part of the first Brillouin zone, the so-called irreducible
wedge. The following is related to relaxed silicon or pointed out otherwise
explicitely. The chapter is closed with three examples which show the impact of
different discretization schemes.