An abrupt silicon \(\left(n_{i}=10^{10} \mathrm{~cm}^{-3}ight) \mathrm{p}-\mathrm{n}\) junction consists of a p-type region containing \(2 \times 10^{16} \mathrm{~cm}^{-3}\) acceptors and an \(\mathrm{n}\)-type region containing also \(10^{16} \mathrm{~cm}^{-3}\) acceptors in addition to \(10^{17} \mathrm{~cm}^{-3}\) 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 \(\mathrm{p}-\mathrm{n}\) junction.

(c) Calculate the built-in potential of the \(\mathrm{p}-\mathrm{n}\) junction at \(400 \mathrm{~K}\). at \(400 \mathrm{~K}\), we have \(n_{i}=4.52 \times 10^{12} \mathrm{~cm}^{-3}\).