Notes on Semiconductors: P-N junctions: Linearly Graded Regions

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:

Poisson 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.

Relating holes with the linear graded limit
Rearranging Equation
Relating linear with electrons
rearranging eqn
Solving for Vbi

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.

Finding charge density
Setting up Poisson
Integrating Poisson
Solving for C
Finished Linear Electric Field

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

Potential relating to integral of E field

Finished Equation for Potential

Width of linear junction

For capacitance, it is the same as abrupt junctions:

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