Built-in potential of a p-n junction

An abrupt silicon p-n diode consists of a p-type region containing 10¹⁶ cm^-3 acceptors and an n-type region containing also 10¹⁶ cm^-3 acceptors in addition to 10¹⁷ cm^-3 donors.
a) Calculate the thermal equilibrium density of electrons and holes in the p-type region as well as both densities in the n-type region.
b) Calculate the built-in potential of the p-n diode.
c) Calculate the built-in potential of the p-n diode at 100°C.

Answer:

The hole density (electron density) in the p-type region equals p = 10¹⁶ cm^-3 (n = n_i²/p= 10²⁰/10¹⁶ = 10⁴ cm^-3)
The electron density (hole density) in the n-type region equals n = 10¹⁷ - 10¹⁶ = 9 x 10¹⁶ cm^-3 (p = n_i²/p= 10²⁰/9 x 10¹⁶ = 1.11 x 10³ cm^-3)
The built-in potential (at 300 K) equals f_i = kT/q ln(10¹⁶ x 9 x 10¹⁷/n_i²) = 0.77 V, using kT/q = 25.84 mV and n_i = 10¹⁰ cm^-3.
The built-in potential (at 100°C) equals f_i = kT/q ln(10¹⁶ x 9 x 10¹⁷/n_i²) = 0.673 V, using kT/q = 32.14 mV and n_i = 8.55 x 10¹¹ cm^-3 (from Example 20).