next up previous contents
Next: 6.1 Geometric Properties of Up: Dissertation Wilfried Wessner Previous: 5.3 Example


6. Mesh Refinement for Full Band Monte Carlo Simulation

Starting from the mid-1970s, metal oxide semiconductor (MOS) devices technology was accompanied by an enormous pace of scaling. Transistor dimensions soon reached a point at which initial modeling assumptions about physical effects broke down. So far device simulation was mostly based on the classical drift-diffusion or the hydrodynamic equations. These equations represent macroscopic balance equations for the electron and hole concentrations and the corresponding current densities. They can be derived from the Boltzmann transport equation (BTE) by the method of moments [105].

In the sub $ 0.1  \MR{\mu m}$ regime, these macroscopic device equations are no longer predictive. One way to overcome this problem is to introduce higher moments [106], another is to rely on a direct solution of the Boltzmann equation. Due to the complexity of the Boltzmann equation a discretization and direct solution as in the case of device simulation is prohibitive. An alternative approach to solve the BTE is provided by the Monte Carlo (MC) method which has been established as a stochastic method for the solution of the Boltzmann equation [107].
Credit for inventing the Monte Carlo method often goes to Stanislaw Ulman, a Polish born mathematician who worked for John von Neumann on the United States' Manhattan Project during World War II. The method is named after the city in Monaco, where the primary attractions are casinos where games of chance are played [108,109]. Monte Carlo method, as it is understood today, encompasses any technique of statistical sampling to approximate solutions to quantitative problems.

One common technique in the area of solid-state physics and the investigation of materials with a regular crystal lattice as for e.g. silicon is the so called band theory of solids. This theory deals with the widening of discrete energy levels for electrons, as observed in the case of free atoms, to available energy states from bands. This split occurs, if several atoms are brought together, as it is the case for e.g. in a crystal structure, due to the Pauli exclusion principle [110]. Such a constellation produces an exceedingly large number of orbitals, which is proportional to the number of atoms, and the difference in energy between them becomes very small. But there are also gaps of energy which contain no orbitals, no matter how many atoms are aggregated [111].

To find out the density of carriers in a semiconductor, the number of available states at each energy has to be determined from the Schrödinger equation. The number of electrons at each energy is then obtained by multiplying the number of states with the probability that a state is occupied by an electron [112].

However, due to the perception that carrier momentum is proportional to its position in the reciprocal space, the dispersion relation between the energy and momentum of a carrier can be described in reciprocal space. For crystalline solids the reciprocal lattice is also periodic and the primitive cell is called the first Brillouin zone.


For full band Monte Carlo (FBMC) simulation the discretization of the Brillouin zone has a major impact on the accuracy of the simulation, so this chapter deals mostly with the construction of a judicious tessellation of the whole Brillouin zone and a particular part of the first Brillouin zone, the so-called irreducible wedge. The following is related to relaxed silicon or pointed out otherwise explicitely. The chapter is closed with three examples which show the impact of different discretization schemes.




Subsections
next up previous contents
Next: 6.1 Geometric Properties of Up: Dissertation Wilfried Wessner Previous: 5.3 Example

Wilfried Wessner: Mesh Refinement Techniques for TCAD Tools