NANOSCALE DEVICES: APPLICATIONS AND MODELING - Bio-Inspired and Nanoscale Integrated Computing - page 52

2.5.2. The Challenge of Nanoscale Modeling
If you have two planets interacting with each other through gravitational forces,
you can use Newtonian mechanics to trace their motion analytically. Add one
more planet to the system (the so called three-body problem) and a closed-form
analytical solution will generally not exist. This is also true for interacting
electrons and protons. There is no analytical solution to the interacting many-
body problem and one will need to resort to numerical methods to study such
systems. Although simple in theory, the practical difficulty arises from the fact
that with the addition of every new particle to the system, the amount of
calculations needed grows considerably.
Now let us imagine that we want to study a system that includes only a few
electrons. As discussed, we know the fundamental laws that describe the system—
that is quantum mechanics. One tends to think that it should not be too hard to
analyze such systems with a reasonably good computer. In fact, that might be the
case when there are only a few particles involved. However, if there are hundreds
or thousands of interacting atoms in the system, such as in nanoscale devices, then
the exact solution of the equations of quantum mechanics for the entire system is
absurdly out of reach for even the most advanced supercomputers of the modern
day. At the same time, hundreds or thousands of atoms do not represent a large
enough system where we could neglect the individual character of each atom and
simply describe the system in terms of statistical averages. Thus, nanoscale devices
present a regime of physical systems—in terms of their size—where we can neither
neglect the atomic nature of the structure, nor treat it exactly using numerical
methods and fast computers. It is noteworthy that this topic focuses on new
approaches to making computers with significantly more power than our tradi-
tional computers. When such efforts eventually become successful, our method of
dealing with nanoscale devices from a modeling and simulation perspective could
also change in major ways.
2.5.3. Smart Physical Modeling
So how do we deal with nanoscale devices? The solution lies in a smart approach
to modeling. Often, a hybrid, multiscale approach is necessary to treat the system:
more classical methods for aspects of the system that are less affected by the
atomic nature, and more fundamental, quantum mechanical calculations where
needed. Another important factor is that often a given property or experimental
outcome could be explained based on a small part of the overall device. The art of
modeling nanostructures consists of seeing what is essential to retain in a minimal
subsystem (that we can handle with our available computing power in reasonable
time) in order to study a certain subject relevant to the nanodevice in question.
Nanodevices constitute a very new field and there is not, at least not yet, a general
approach to modeling all nanodevices (such as is the case for larger devices like
microelectronic transistors). Therefore, every researcher working in the field will,
at some point, have to deal with modeling the specific system he/she is studying.