NANOSCALE DEVICES: APPLICATIONS AND MODELING - Bio-Inspired and Nanoscale Integrated Computing - page 58

computers. There have also been many valuable attempts at modeling the
interatomic interactions for use in molecular dynamics simulations. One good
example is the Brenner-Tersoff potentials that are well established for describing
hydrocarbon systems [83, 84]. For a detailed review of molecular dynamics the
reader is referred to [85].
T IGHT - BINDING . One way to approach a problem involving multiple atoms is
to try to solve the Schro¨ dinger equation by finding an expanded form for the
expected solution. It is customary to write the solution in the form of a linear
combination of atomic orbitals (LCAO), that is, as a weighted sum of wave
functions that have been found for individual, isolated atoms. The parameters
involved could be derived from first-principles methods. In many cases, however,
these can be found from elsewhere, such as using experimental data. For instance,
the core of an atom could be represented by a pseudopotential, instead of
explicitly including all the individual electron contributions, just like in molecular
dynamics. Thus, as opposed to exact, real orbitals, the atom will be represented by
parameters derived by fitting to other available data. In such cases, especially
when interactions with close neighbors are considered only, the LCAO method
would typically be referred to as the tight-binding (TB) approximation. TB has the
advantage of being computationally much less expensive than ab initio methods
and therefore capable of providing, at least, a qualitative description of the
behavior of a relatively large system quickly. For more on LCAO methods and
TB, [86-88] can be studied.
M ONTE C ARLO . Monte Carlo methods are in place in many disciplines. The
basic idea is to start from a known point and trace the behavior of a system by
following a random-walk process. In each state, the system has the possibility of
going to several of subsequent states with different probabilities. The choice of
which one of those states the system will go into at each step is decided by a
random process that is weighted by those different probabilities. For instance,
consider the movement of an atom sitting on a crystalline surface through a loose
bond. This atom moves around due to its thermal, kinetic energy. At each point,
there exists several sites around it for it to move to. There would be a potential
barrier in front of the atom to cross in order to make the transition to any of these
sites, but the barriers have different heights and widths for the different sites,
which correspond to different transition probabilities. The random walk of this
atom could be traced using the Monte Carlo approach. Both semiclassical and
quantum Monte Carlo methods are widely in use in nanoscale modeling [89, 90].
2.5.4.3. Use of Green's Functions. Green's functions are powerful tools in
solving differential equations. Since the goal in nanosale modeling is to find the
solution to the Schro¨ dinger equation, which is a differential equation, Green's
functions often come in very handy. In particular, they provide a useful way of
dealing with problems involving open boundary systems. Consider, for example, a
QD or SET connected to two large metal contacts on the sides. Although the