Markus Kampl was born in Vienna, Austria, in 1988. He studied electrical engineering at the Technische Universität Wien, where he joined the Institute for Microelectronics in October 2012. In November 2015 he received the degree of Diplomingenieur. Since that time he is working on his doctoral degree.
Numerical Algorithms and Physical Models for Monte Carlo Device Simulation
To verify the already implemented methods and algorithms in the full-band part of the Vienna Monte Carlo simulator, we made an effort to implement all these methods and algorithms also in the analytic-band part of the simulator. This includes compatibility with the GTS Framework and the capability to simulate 1D and 2D devices. Additionally, the following features were implemented: A novel averaging method, whereby the averages are calculated every time the particle enters a new grid cell; current estimators for the forward and backward Monte Carlo methods; and the possibility to use MPI.
To compare the results of the analytic-band with our full-band simulations, we developed an electron-electron scattering model, suited for analytic bands and showed that this model works for backward Monte Carlo as well as for combined backward-forward Monte Carlo simulations. Testing of the electron-electron scattering model was continued. Comparison of full-band and analytical-band structures, as well as forward and backward Monte Carlo simulations, in some cases showed a weak enhancement of the high-energy tail when compared to the Maxwellian tail. Further tests are planned to identify the nature of this enhancement.
The backward Monte Carlo method offers the possibility to inject an ensemble of particles, created at an equilibrium with the temperature T0. We observed that the estimators of the backward Monte Carlo method drifted slightly as a function of this injection temperature. This effect could be attributed to the discretization error when evaluating the partition function by integration over the Brillouin zone. The problem could be resolved by a Monte Carlo estimation of the partition function from the injected particle ensemble.
We are currently working to improve the backward Monte Carlo method in order to reduce the statistical error of the estimators via symmetrical particle injection.
Fig. 1: Comparison of the relative errors of the current estimators using Maxwellian, velocity-weighted Maxwellian or symmetrical injection.