Publication Date

Summer 2018

Degree Type


Degree Name

Master of Science (MS)




Daniel Brinkman


Finite difference method, Numerical analysis, Scharfetter-Gummel, Semiconductor, Solar cell, van Roosbroeck

Subject Areas

Applied mathematics


Since the 1950s, semiconductors have played a significant and daily role in our lives, as they are the foundation of our computers, phones, and other electronic devices. Aside from their obvious uses, the equations that govern semiconductors have peaked the interest of mathematicians and numerical analysts. In 1950, van Roosbroeck described the fundamental semiconductor device equations as a system of three nonlinear coupled partial di↵erential equations. The van Roosbroeck system poses a challenge numerically because of its strong nonlinearity and coupled equations. Its diculties lie in simultaneously solving drift-di↵usion equations for electrons and holes and using their solutions to solve the Poisson equation. To start o↵, we will numerically solve the one-dimensional drift di↵usion equation with constant velocity using unwinding techniques and illustrate the results using MATLAB for a toy model. This attempt will only complicate solving two drift-di↵usion equations and will not suce in solving the full van Roosbroeck system. Thus, we will analyze the standard finite di↵erence scheme proposed by Scharfetter and Gummel that deals nicely with the nonlinearity and coupled equations. Then we will compare and contrast the solutions by the standard finite di↵erence scheme proposed by Scharfetter and Gummel and direct discretization of the fluxes in Slotboom variables. We will conclude that the Slotboom discretization performs better than the Scharfetter-Gummel in cases of large forward bias. We will also briefly discuss solar cells and their simulations.