The Monte Carlo Method for Semiconductor Device Simulation

The name of the Monte Carlo method is inspired by the gambling casinos at the city of Monte Carlo in Monaco. The mathematical techniques used by this method are in fact based on the selection of random numbers [1–4]. In its present form, the method is attributed to Fermi, Von Neumann, and Ulam, who developed it for the solution of problems related to neutron transport during the secret research at Los Alamos for the construction of the atomic bomb during world war II. There are, however, indications of previous uses of methods based on selections of random numbers. In particular the name of Lord Kelvin is mentioned for a paper of 1901 [5], and Gosset (better known with the pseudonym Student) used experimental sampling to support his well known theoretical studies of statistical distributions. Fermi himself used already Monte Carlo techniques in the 30′s in connection with neutron transport [6].

Electrons in a perfect crystal¹ can be described in terms of Bloch states, whose wave functions can be written as where n is a band index, ħ k is the quasi momentum of the electron, and u _nk (r) is a function of the space coordinate r with the periodicity of the crystal.

The Monte Carlo method, as applied to charge transport in semiconductors, consists of a simulation of the motion of one or more electrons inside the crystal, subject to the action of external forces due to applied electric and magnetic fields and of given scattering mechanisms [1–4]. The durations of the carrier free flights between two successive collisions and the scattering events involved in the simulation are selected stochastically in accordance with some given probabilities describing the microscopic processes. As a consequence, any Monte Carlo method relies on the generation of a sequence of random numbers with given distribution probabilities. Such a technique takes advantage of the fact that nowadays any computer generates sequences of random numbers evenly distributed between 0 and 1 at a sufficiently fast rate. The generation of random variables with any distribution, starting from random numbers evely distributed between 0 and 1, is discussed in Appendix B.

In an era which is dominated by an always faster and larger flow of information, microelectronics plays a major role. The building block of today’s microelectronics are semiconductor devices, which are used either as single components in a variety of applications (process controllers, antennas, sensors, radios, etc.,...) as well as in integrated circuits. Since the invention of the bipolar transistor in 1949, many new devices have been proposed and improved performances have been constantly achieved. Before this date, semiconductors were only used as thermistors, photodiodos and rectifiers. The advances in the field of semiconductor devices are the combined results of better understanding of the physical processes that underline the electrical behaviour of devices, of an improved handling of technological processes involved with the fabrication of the devices, of the mature knowledge of the chemical properties of the materials that are used, and of the combination of all these factors. In other words, electronics have been able to make big steps forward in the last few decades thanks to the progress in the physical, chemical and material sciences, as well as the development of new technological tools. The best example is given by the fact that we are currently able to put hundreds of thousands of devices onto a single chip, well into what is called very-large-scale integration (VLSI) [1].

After introducing the general methodology of the Monte Carlo algorithm and reviewing some of the essential features of semiconductor devices, we are now ready to discuss in detail the application of the Monte Carlo simulation to semiconductor devices. Although every specific device presents peculiar aspects that make its modelling not easily extendable to other devices, it is nevertheless possible to give some general guidelines and define specific blocks that are common to all simulations [1].

In the present chapter, we will present a series of applications of the Monte Carlo method to the simulation of specific devices. It will become obvious that the Monte Carlo simulation of devices has followed a very different path than the one that has governed the actual evolution of the real devices. It was shown in chapter 4 that while the first steps towards the development of semiconductor devices for microelectronics were based on covalent semiconductors (Si and Ge), GaAs and other polar materials entered the scene only at a later stage. Furthermore, a large gap existed (and still exists) between the state of the art of fabrication, which is still dominated by Si C-MOS and bipolar structures, and the realm of “research and development”, where GaAs fied-effect transistors with very sophisticated geometries and extremely reduced dimensions are currently produced. As pointed out in chapter 4, one exception is certainly found in the microwave area, where GaAs single devices are currently used.