After taking a look at abrupt junctions, we can now move onto linear graded junctions. To solve for the equations relating positioning, doping, field, and potential, we shall be using Poisson’s Equation:

For linearly graded junctions, the graphs look like this:

Due to the symmetry of linearly graded junctions, we can equate xn = xp = xd/2. Based on this assumption, we can begin to solve for the turn on voltage, the electric field based on position x, and the potential based on x. First we will find the turn on voltage.

Once we have derived the turn on voltage, now we must relate the electric field based on what point you are in the linearly graded region.

Now with the electric field found, we must now solve for potential. Note, that potential is equal to the negative integral of the electric field

For capacitance, it is the same as abrupt junctions: