Obtain the quadratic equation and the explicit analytical

Question

This and the next problem summarize the equation and methodology used by physicists to calculate N_DD, N_AA, E_D and g_D in a semiconductor from a set of N versus T data from very low temperatures to above T_i.

Using the mass action law for all the equilibrium reactions present In this problem, derive the general cubic equation for the electron concentration. In a nondegenerate semiconductor that has N_DD > N_AA ≠ 0 which is valid at all temperatures.

Obtain the quadratic equation and the explicit analytical solution for N which is valid at all temperatures below the intrinsic temperature.