Numerical study of the systematic error in Monte Carlo schemes for semiconductors

Abstract

Abstract. The paper studies the convergence behavior of Monte Carlo schemes for semiconductors.A detailed analysis of the systematic error with respect to numerical parameters is performed. Different sources of systematic error are pointed out and illustrated in a spatially one-dimensional test case. The error with respect to the number of simulation particles occurs during the calculation of the internal electric field. The time step error, which is related to the splitting of transport and electric field calculations, vanishes sufficiently fast. The error due to the approximation of the trajectories of particles depends on the ODE solver used in the algorithm. It is negligible compared to the other sources of time step error, when a second order Runge-Kutta solver is used. The error related to the approximate scattering mechanism is the most significant source of error with respect to the time step.Mathematics Subject Classification. 82D37, 65C05.

An introduction to probabilistic methods with applications

“…For a more precise construction of these stochastic models, we refer the reader to [19]. The article by Muscato et al [33] also provides a rather detailed discussion on these jump type probabilistic interpretations in the context of the Boltzmann transport equation. By the law of large numbers, the law of the random states η t can be approximated by the occupation measure…”

“…We also refer the reader to [33] for an alternative discrete time approximation model based on fictitious jump events. In this article, the authors also provide a discussion on discrete time approximation of the stochastic jump integrals introduced in the definition (2.8).…”

“…The article by Muscato et al [33] studies the numerical error in Monte Carlo particle methods modeling carrier transport in semiconductors; both errors due to finite timestep and number of particles is examined. Due to the Monte Carlo nature of the algorithm, analysis of the systematic error requires removal of stochastic sources of uncertainty, which in many cases can be very resource intensive.…”

“…[33]), small-scale dilute-gas hydrodynamics (see the article [1]) and small-scale solid-state heat transfer, where their ability to efficiently treat high-dimensional problems offsets the limitations due to the statistical uncertainty associated with their results. The latter feature, namely the statistical fluctuations, can be an advantage or a disadvantage, depending on the application.…”

A hierarchy of hydrodynamic models for silicon carbide semiconductors

“…To solve the BBP equations is not an easy task also from the numerical point of view, because they form a set of partial integrodifferential equations. Particle-based solvers [1][2][3][4][5][6] of the BBP system can be proposed but with a huge computational effort. For engineering purposes, one has to introduce hydrodynamic models, which are obtained by taking the moments of the BBP equations and by using a suitable truncation procedure.…”