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.

Numerical simulation (Technology Computer-Aided Design, TCAD ) is used in the development of semiconductor devices to minimize the number of costly experiments and reduce time-to-market. Modern integrated circuits are based on tiny semiconductor structures with characteristic dimensions in the nanometer range. At this scale, quantum effects must be taken into account, as they have a decisive influence on the behavior of the components. Eugene Paul Wigner (1902-1995) developed a mathematical formulation of quantum mechanics which is formally very similar to a classical phase space description. For this reason, this formalism is particularly suitable for describing modern semiconductor components. The most frequently used numerical method for solving the Wigner equation goes back to William Frensley. The disadvantage of this method is that the solution strongly depends on the parameters of the discretization, in particular on the fineness of the computational grid used. In the worst case, this method can even deliver completely incorrect results. In this project, the reason for the failure of this solution method has been identified and two different, stable methods were developed. The von Neumann equation in a rotated coordinate system, referred to as the sigma equation, which is equivalent to the Wigner equation, has also been used. The latter is particularly advantageous for numerical calculations because the system matrix is only sparsely populated. The special feature of these equations is that they are over-determined: In the case of the Wigner equation, two equations are obtained for the zero velocity instead of just one; in the case of the sigma equation, two boundary conditions are obtained, although the equation only allows one boundary condition. The first approach developed here uses so-called inflow boundary conditions instead of the original Dirichlet boundary conditions. A second approach solves the system of equations approximately using the method of least squares, whereby the boundary conditions in the form of constraints are fulfilled exactly. A third method, which was developed independently by another group recently, uses the concept of absorbing layers to fulfill the boundary conditions and was also investigated in detail. The coupled system of sigma equation and Poisson equation was numerically analyzed. Bifurcation points were found, which are the cause of ambiguous solutions. After the new methods were first examined for systems in one spatial dimension, they were extended to two spatial dimensions. In this case the numerical solution of the equations is computationally intensive, which required strategies for parallelization to be developed and implemented.