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# 3 Simulation of Single Electron Devices

The computer is the primary research instrument of the sciences.
Heinz Pagels

...in real life mistakes are likely to be irrevocable. Computer simulation, however, makes it economically practical to make mistakes on purpose. If you are astute, therefore, you can learn much more than they cost. Furthermore, if you are at all discreet, no one but you need ever know you made a mistake.
John McLeod and John Osborn

The real danger is not that computers will begin to think like men, but that men will begin to think like computers.
S. J. Harris

In order to understand the behavior of single-electron devices and circuits one has to solve the many body problem of carriers tunneling and interacting with each other. The dependence of this carrier transport and interaction on applied bias voltages gives the characteristics with which one can understand the circuit behavior and assess possible applications and limitations.

Analytical solutions of the carrier transport in single-electron circuits are only possible for circuits with few junctions such as the double junction or SET transistor [53], or symmetric devices such as a one-dimensional array of tunnel junctions [13] [22] [80]. For more complex circuits and for the consequences of co-tunneling the obvious thing to do is numerical simulation. A SET circuit may be described with a Master Equation (ME) (see Section 3.2) which is a conservation law for all probabilities of states a circuit can occupy. A ME is a standard means of describing stochastic processes. Basically there are two distinct ways for a numerical solution of a ME. Either one tries to solve the ME directly, that is to solve the equation for the probability density function, or one simulates the stochastic process by letting one or more particles jump from state to state according to the transition probabilities. From a sufficient set of samples, the desired statistical properties and quantities can be calculated. We refer to the former as the ME approach and to the latter as the Monte Carlo (MC) method.

Only few publications deal with the simulation of single-electron devices. N. Bakhvalov et al. [13] was the first to follow a MC approach, and E. Ben-Jacob et al. [16] suggested a ME method, as an appropriate technique applicable to single-electron devices. From time to time simulation results were published [4] [18] [66], but implementation details, limitations and assumptions of employed simulators were not always obvious. S. Roy explained in more detail his MC analysis tools [97] which were developed from the study of linear arrays of tunnel junctions. He briefly explains an interesting semi-automatic approach to determine stable and instable regions of zero temperature operation. It is based on the critical voltage method from Section 3.1.5. L. Fonseca et al. developed a ME simulator called SENECA which is explained in [32]. They focused especially on co-tunneling and how to simulate it correctly and efficiently. M. Kirihara et al. [61] describe briefly the MC simulation method and give simulation results of basic logic circuits. H. Fukui et al. [34] and S. Amakawa et al. [3] also focused on logic circuits and studied in particular single-electron inverters. They developed similar tools to the one S. Roy describes, namely the calculation of stable and instable operation regions for zero temperature and a MC simulator. R. Chen developed the single-electron MC simulator MOSES which is described briefly in [17].

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Christoph Wasshuber