A fast semi-numerical technique for the solution of the poisson-boltzmann equation in a cylindrical nanowire

Silicon nanowire (SiNW) based devices have aroused great interest since they exhibit high carrier mobilities and sub-threshold slopes close to 69 mV/decade due to good charge control. The charge in the nanowire channel is obtained by solving the Poisson equation self-consistently with the Schroedinger equation. However this is a very computationally expensive process and it is usually preferable to solve for the potential to the first order using only the Poisson equation, and then to make a quantum correction for the charge based on the potential gradient at the surface of the wire. It is the former task, that of solving the non-linear Poisson's equation within the nanowire, that we address here. The built-in non-linear ODE solver in MATLAB does not converge in most cases because the equation is highly non-linear. Commercial numerical solvers like DESSIS are very accurate but slow. Depending on the grid size, they may take several hours to run.