A quartic solution covering freeze-out and exhaustion effects in doubly doped P-type equilibrium semiconductors

The local charge neutrality condition for same-P-type doubly-doped uniform equilibrium semiconductors is shown to lead to a quartic polynomial in a Z-parameter that is directly proportional to the concentration of holes. Analysis demonstrates that the problem is solvable exactly by identifying the only physically permissible root to a quartic polynomial. The unique solution for Z leads directly to analytic expressions for temperature dependent majority carrier concentration and the Fermi level