Numerical Constraints for the Wigner and the Sigma Equation

Numerical simulation (Technology computer-aided design, TCAD) is used in development and design of semiconductor devices in order to minimize the number of cost and time intensive experiments and to reduce the time to market. Semiconductor devices in modern integrated circuits have feature sizes in the nanometer regime. At this length scale quantum mechanical effects strongly influence and even determine the behavior of the devices.

Eugene Paul Wigner (1902-1995) was an Austro-Hungarian born physicist who developed a mathematical formulation of quantum mechanics which is formally close to a classical phase space description. This formulation allows one to define mixed quantum-classical models which are very attractive for the simulation of modern semiconductor devices.

In this project numerical methods for the solution of the stationary Wigner equation will be developed. This equation can be used, for instance, to calculate current-voltage characteristics of semiconductor devices. The Wigner equation is often solved using a method developed by William Frensley more than 30 years ago. The method has been criticized for the results being strongly dependent on the discretization parameters, especially the mesh spacing. It is known that the method can even give unphysical results.

The stationary Wigner equation has a singularity at zero momentum (k=0). We believe that the breakdown of Frensley's method is due to the improper treatment of that singularity.

In this project we suggest a revision of Frensley's method. We explicitly include k = 0 in the discrete mesh and derive two equations for that point. The first equation is an algebraic constraint which ensures that the solution of the Wigner equation has no singularity at k = 0. The second equation is a transport equation for k = 0. The resulting system which we refer to as the constrained Wigner equation is overdetermined. Such an equation system can only be solved approximately. The main goal of the project is the devolopment of such approximate solution methods. We will first systematically study both the constrained Wigner equation and the sigma equation in a single spatial dimension and develop numerical solution methods. In the second phase we will explore the extension of the method to two spatial dimensions. Solving the Wigner equation in two spatial dimensions is computationally expensive and requires parallelization of the algorithms developed.