A finite element formulation for the hydrodynamic semiconductor device equations

A new formulation employing the Galerkin/least-squares finite element method is presented for the simulation of the hydrodynamic model of semiconductor devices. Numerical simulations are performed on the coupled Poisson and hydrodynamic equations for one carrier devices. The hydrodynamic equations for a single carrier, i.e. for the electrons or holes, resemble the compressible Navier-Stokes equations with the addition of highly nonlinear source terms and without the viscous terms. The governing equations are nondimensionalized to improve the conditioning on the resulting system of equations and the efficiency of the numerical algorithms. Furthermore, to establish the stability of the discrete solution, the system of hydrodynamic equations is symmetrized by considering generalized entropy functions. A staggered solution strategy is employed to treat the coupled hydrodynamic and Poisson equations. Numerical results are presented for one-dimensional and two-dimensional one-carrier n⁺-n-n⁺ devices. The presence of velocity overshoot has been observed and it is recognized that the heat flux term plays an important role in the simulation of semiconductor devices employing the hydrodynamic model.