Approximate graphical method of solving Fermi level and majority carrier density of semiconductors with multiple donors and multiple acceptors

We present a generic approximate graphical method for determining the equilibrium Fermi level and majority carrier density of a semiconductor with multiple donors and multiple acceptors compensating each other. Simple and easy-to-follow procedures of the graphical method are described. By graphically plotting two wrapping step functions facing each other, one for the positive hole-ionized donor and one for the negative electron-ionized acceptor, we have the crossing point that renders the Fermi level and majority carrier density. Using the graphical method, new equations are derived, such as the carrier compensation proportional to N_A/N_D, not the widely quoted N_A - N_D. Visual insight is offered to view not only the result of graphic determination of Fermi level and majority carrier density but also the dominant and critical pair of donors and acceptors in compensation. The graphical method presented in this work will help to guide the design, adjustment, and improvement of the multiply doped semiconductors. Comparison of this approximate graphical method with previous work on compensation, and with some experimental results, is made. Future work in the field is proposed.