An abrupt silicon \(\left(n_{i}=10^{10} \mathrm{~cm}^{-3}ight) \mathrm{p}-\mathrm{n}\) junction consists of a p-type region containing \(10^{16} \mathrm{~cm}^{-3}\) acceptors and an n-type region containing \(5 \times 10^{10} \mathrm{~cm}^{-3}\) donors. Assume \(T=300 \mathrm{~K}\).

(a) Calculate the barrier potential of this \(\mathrm{p}-\mathrm{n}\) junction.

(b) Calculate the total width of the depletion region if the applied voltage \(V_{\text {app }}=0,0.5\), and -2.5 .

(c) Calculate maximum electric field in the depletion region at \(0,0.5\), and -2.5 .

(d) Calculate the potential across the depletion region in the n-type semiconductor at \(0,0.5\), and -2.5 . The permittivity of abrupt silicon, \(\varepsilon_{s}\), is equal to \(1.054 \times\) pico-farads per centimeter \((\mathrm{pF} / \mathrm{cm})[35]\).