NANOSCALE DEVICES: APPLICATIONS AND MODELING - Bio-Inspired and Nanoscale Integrated Computing - page 53

Nonetheless, there are many different levels of theory that have been developed
over the years that could be applied based on the level/depth of the model
itself. These include methods ranging from semi-classical treatments of atoms to
single-electron methods to lower level methods directly relying on the solution of
the fundamental equations of quantum mechanics for all the atoms. In the next
few sections we shall briefly introduce some of the common methods in this
context.

2.5.4. The Methods
Generally speaking, the methods of modeling and simulating nanodevices could
be divided into two major categories: the so called ab initio (from the beginning) or
first-principles approaches, and methods that have more of a phenomenological or
empirical aspect to them. In first-principles methods, as the name suggests, the
idea is to start with the basic physical equations for the entire system, such as the
Schro¨ dinger and Poisson equations, and solve them to find all the desired
characteristics like electronic structure or transport properties. The Hartree-
Fock theory is one such method that provides a framework for treating the
quantum many-body problem by solving the Schro¨ dinger equation. As discussed
before, this could be a remarkably difficult task to accomplish (in terms of
computational complexity) for a reasonably large system such as a nanodevice
with hundreds of atoms. Therefore, other methods have been developed that rely
on approximations and fitting parameters that, although not as accurate as first-
principles approaches, depending on the problem at hand could still capture most
of the underlying physics and handle much larger systems in a practical manner.
Molecular dynamics and tight-binding approximations are some important
examples in this category.

2.5.4.1. Ab Initio (First-Principles) Approaches. Quantum mechanics and
relativity are the theories that describe the world around us (the best theories we
have so far, though not necessarily the absolute truth!). In everyday experiences,
quantum mechanical and relativistic effects are often masked and simpler theories,
such as Newtonian mechanics, provide an accurate enough description. At very
small scales, such as the nanoscale, quantum mechanical effects are dominant
(although relativistic effects can be negligible in many nanodevice applications).
We will begin our discussion here with a quick look at what is at the heart of
quantum mechanics.
In one common approach to quantum mechanics, the behavior of a particle
can be described by the Schro¨ dinger equation:

H C ¼ eC :

Here, H is the so called Hamiltonian operator of the system, including both
potential and kinetic energy terms and e represents the allowed energies in the
system. Particles are described in terms of waves ( C ). The amplitude of the wave