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- mu m channel and at 77 K for a 1.0- mu m channel. >