Monte Carlo simulation

Monte Carlo simulation is a computational algorithm in which events are stochastically simulated using the process of repeated random sampling.
In the case of SEM, the Monte Carlo method is used to simulate the behaviors of diffusion of the incident electrons in a specimen. The simulations are performed on the basis of 1) Rutherford scattering formula for elastic scattering and 2) Bethe’s stopping power formula for inelastic scattering in the specimen. According to these formulas, the electron trajectories are calculated one by one where multiple parameters needed at scattering events are stochastically obtained using the random numbers, and the diffusion behaviors of the incident electrons are simulated. Here, the parameters to be used are the scattering angle and azimuth angle of elastic scattering of the incident electrons, the free path length of the electrons (the distance from one scattering to the next scattering), and the value of energy loss due to inelastic scattering.
Commercial and free software programs are available for Monte Carlo simulations. Using those programs, when one inputs the parameters, such as the constituent elements and the layer (film) thickness, the number of the incident electrons, the incident electron energy, the incidence angle onto the specimen, the software stochastically provides the scattering angle, the electron free path and the energy loss value due to inelastic scattering for a scattering event, and then electron trajectories are automatically calculated and the diffusion behaviors of the incident electrons in the specimen are simulated. In addition to the simulation of the diffusion of the incident electrons, the software programs are available which can simulate the generation and absorption of characteristic X-rays in a specimen for elemental analysis by X-ray spectroscopy.
Figure illustrates a simulation example using a software program. This Monte Carlo simulation shows the diffusion behaviors of the incident electrons normal to the surface of an aluminum specimen where the number of the electrons is 100, the accelerating voltage is 10 keV and a specimen thickness of 1μm. It is seen that the incident electrons are diffused into a depth of 0.7 to 0.8 μm from the surface, and that the backscattered electrons generated at a region distant about 0.5 μm from the incident electron beam point are emitted from the upper surface.

Fig. Example of a Monte Carlo simulation.
1) Specimen: Al 2) The number of incident electrons: 100 3) Accelerating voltage: 10 kV