3.2 Tunneling: a Stochastic Process

The first assumption is that electrons can not probe the past, that is they
posses no memory, and thus their tunnel rate depends only on the
momentary state of the system. This is exactly the
criterion for a Markov process [100]. If one
further assumes, that
the system evolves at random times in a jump like fashion, as is the case
with tunneling, one can describe such a system with a
ME [37]

where
*p*(**S**,*t*) is the probability density function in state space, and
denotes the transition rate from state
**S'***
to state
S. If the states are discrete the ME becomes
where
denotes the transition rate from state j to state iand P_{i}(t) is the time dependent
occupation
probability of state i.
Fig. 3.5 is a typical state transition diagram for such a
process.
*

The ME method for the simulation of SET circuits tries to solve (3.12) which is the general description of a SET circuit, numerically, as was done by L. Fonseca et al. [32]. Taking the simplification one step further and considering only a single tunnel junction, where the number of tunneled electrons gives the state of the system, the state can only change to a neighboring state, since only one electron is assumed to tunnel at a time. Processes with this property are called point or birth-death processes [52] and can be visualized with a simple state transition diagram Fig. 3.6.

Further assuming that states can only evolve in one direction, this is the case if only tunneling in one direction is considered, and that all states have an equal transition rate to the next state, which is the case when neglecting any charging effects, brings one to the so called Poisson process, Fig. 3.7.

The Poisson process starts at

This first order differential equation with constant coefficients may be solved by Laplace transforming both sides.

where is the Laplace transform of

This set of recursive equations for the state probability Laplace transform is easily solved by induction.

Taking the inverse Laplace transform, one finds that for and ,

This is the well known Poisson distribution. Building on the stochastic description of a single tunnel junction with the Poisson process, it is possible to use a MC method for the simulation of SET circuits.