A combined multicell‐WENO solver for the Boltzmann‐Poisson system of 1D semiconductor devices

Abstract

PurposeTo present a new direct solution method for the Boltzmann‐Poisson system for simulating one‐dimensional semiconductor devices.Design/methodology/approachA combination of finite difference and finite element methods is applied to deal with the differential operators in the Boltzmann transport equation. By taking advantage of a piecewise polynomial approximation of the electron distribution function, the collision operator can be treated without further simplifications. The finite difference method is formulated as a third order WENO approach for non‐uniform grids.FindingsComparisons with other methods for a well‐investigated test case reveal that the new method allows faster simulations of devices without losing physical information. It is shown that the presented model provides a better convergence behaviour with respect to the applied grid size than the Minmod scheme of the same order.Research limitations/implicationsThe presented direct solution methods provide an easily extensible base for other simulations in 1D or 2D. By modifying the boundary conditions, the simulation of metal‐semiconductor junctions becomes possible. By applying a dimension by dimension approximation models for two‐dimensional devices can be obtained.Practical implicationsThe new model is an efficient tool to acquire transport coefficients or current‐voltage characteristics of 1D semiconductor devices due to short computation times.Originality/valueNew grounds have been broken by directly solving the Boltzmann equation based on a combination of finite difference and finite elements methods. This approach allows us to equip the model with the advantages of both methods. The finite element method assures macroscopic balance equations, while the WENO approximation is well‐suited to deal with steep gradients due to the doping profiles. Consequently, the presented model is a good choice for the fast and accurate simulation of one‐dimensional semiconductor devices.

A kinetic approach to tunnelling at Schottky contacts

“…Moreover, electrons may tunnel from the silicon to the metal and vice versa with a certain probability P T , leading to the tunnel currents j TL sm and j TL ms , respectively. The thermal current from the semiconductor to the metal can be approximated by assuming perfect absorbing boundary conditions for the outflowing electrons, as we have shown in [7], and is given by…”

“…The model equations are obtained from the BTE by applying the multicell-WENO method we have presented in [7]. This technique is a combination of the method of weighted residuals [24] and high-order shock capturing algorithms [25].…”

“…where, in general, the fluxes h and h are obtained by a WENO approximation [25,6]. In this case, we apply the third order WENO approximation for non-uniform grids presented in [7].…”

“…They were successfully applied to study bulk semiconductors [1,2] as well as diodes [3][4][5][6][7], MESFETs [8] and MOSFETs [9]. However, contact phenomena are often neglected or dealt with by very simple approximations, namely quasi-ohmic and perfectly absorbing boundary conditions.…”

“…Second, the electrons are treated as quasi-free particles with a parabolic dispersion relation. Both points can be avoided by applying kinetic transport models, as has been shown, for instance, in [7]. From the quantum-mechanical point of view, interference effects as well as tunnelling, reflections and other interface effects must be taken into account at the barrier [14].…”