Explain the operation of the $n p$ junction. Your account should include a discussion of the following topics:

(a) the rôle of doping and the fermi-energy;

(b) the rôle of majority and minority carriers;

(c) the contact potential;

(d) the relationship $I(V)$ between the current $I$ flowing through the junction and the external voltage $V$ applied across the junction;

(e) the property of rectification.