Abrupt PN Junctions in the Depletion Approximation and Graph

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In an abrupt junction, the doping changes abruptly from p to n. It is common to solve for the band bending, the local electric field, the carrier concentration profiles, and the local conductivity in the depletion approximation. In this approximation it is assumed that there is a depletion width W around the transition from p to n where the charge carrier densities are negligible. Outside of the depletion width, the charge carrier densities are equal to the doping concentrations. So, the semiconductor is electrically neutral outside of the depletion width. Using this depletion approximation, it is possible to calculate important properties of junctions. The charge density distribution for an abrupt junction is,

The charge density can be integrated to determine the electric field ,

The electric field can be integrated to determine the electrostatic potential ,

The voltage across the junction is the difference in the electrostatic potential ,

The depletion width can now be calculated using the charge neutrality and the expression for ,

The depletion width is,