A fully coupled scheme for a Boltzmann-Poisson equation solver based on a spherical harmonics expansion

Abstract

The numerical properties of a deterministic Boltzmann equation solver based on a spherical harmonics expansion of the distribution function are analyzed and improved. A fully coupled discretization scheme of the Boltzmann and Poisson equations is proposed, where stable equations are obtained based on the H-transformation. It is explicitly shown that the resultant Jacobian matrix for the zeroth order component has property M for a first order expansion, which improves the stability even of higher order expansions. The detailed dependence of the free-streaming operator and the scattering operator on the electrostatic potential is exactly considered in the Newton-Raphson scheme. Therefore, convergence enhancement is achieved compared with previous Gummel-type approaches. This scheme is readily applicable to small-signal and noise analysis. As numerical examples, simulation results are shown for a silicon n + nn + structure including a magnetic field, an SOI NMOSFET and a SiGe HBT.

Numerical Modeling of Noise and Transport in SOI Devices

“…The second very important solution algorithm for the DD TCAD model is the simultaneous Newton algorithm considering the complete coupling between all equations. This solution method has already been demonstrated for the BE-PE system [7]. For the BE-PE-SE system the consideration of the coupling between the equations is more difficult since SE is an eigenvalue problem which must be solved in order to establish the coupling between PE and BE.…”

“…Please note that a too low expansion order may introduce an error of nearly 10% even for the drain current. For direct BE solution methods the box integration method can be used for the space discretization, which comparable to the DD model case allows to make the discretized transport model charge conservative [7]. This implies that for the discretized model Kirchhoff's current law holds for the terminal currents as soon as the transport equation is solved with sufficient accuracy.…”

“…Another notoriously difficult simulation example even for larger channel lengths and body thicknesses are PD SOI MOSFETs with high [12], [13]. [7].…”

“…Finally, even to calculate a DC solution without a good initial solution is a real challenge where many DD and HD solvers fail. Figures 7 and 8 demonstrate, however, that the numerical solution algorithms of the BE-PE solver described in [7] are already so mature that not only a DC solution for a PD SOI NMOSFET with 2 · 10 17 cm −3…”

Inclusion of the Pauli principle in a deterministic Boltzmann equation solver based on a spherical harmonics expansion

“…In order to have quadratic convergence of the Newton-Raphson scheme like in [12], the dependence of g l ,m g l ,m on the electrostatic potential due to the Htransform [6] is considered in the Jacobian matrix. Significant simplification can be obtained when the transition rate does not depend on the scattering angle: The in-scattering term is given by…”