Numerical simulation of a steady-state electron shock wave in a submicrometer semiconductor device

Appropriate numerical methods for steady-state simulations (including shock waves) when the electron flow is both subsonic and supersonic are addressed. The one-dimensional steady-state hydrodynamic equations will then be elliptic in the subsonic regions and hyperbolic/elliptic in the supersonic regions. A second upwind method is used for both elliptic and hyperbolic/elliptic regions. In the elliptic regions, the second upwind method is related to the Scharfetter-Gummell exponential fitting method. The hydrodynamic model consists of a set of nonlinear conservation laws for particle number, momentum, and energy, coupled to Poisson´s equation for the electric potential. The nonlinear conservation laws are just the Euler equations of gas dynamics for a gas of charged particles in an electric field, with the addition of a heat conduction term. Thus the hydrodynamic model partial differential equations (PDEs) have hyperbolic, parabolic, and elliptic modes. The nonlinear hyperbolic modes support shock waves. The first numerical simulations of a steady-state electron shock wave in a semiconductor device are presented, using the hydrodynamic model. For the ballistic diode (which models the channel of a MOSFET), the shock wave is fully developed in Si (with 1-V bias) at 300 K for a 0.1-μm channel and at 77 K for a 1.0-μm channel