PN Junctions

Equilibrium Aspects

We will consider an abrupt PN junction with the following simplifications as stated in Pierret:

1. A 1D device

2. A metallurgical junction at x = 0 (i.e., the change from p-type to n-type occurs here).

3. A step junction from N_A to N_D with uniformly doped p and n regions.

4. Perfect ohmic contacts far removed from the metallurgical junction

A typical doping profile, idealized and actual valence band hole and conduction band electron concentrations plot are shown in the figure below. It is seen that the system does not stay as the idealized hole and electron concentrations would predict but there is a redistribution of carriers near the pn junction. This is caused by diffusion as a result of the nonzero concentration gradient of hole and electron concentrations across the pn junction. In fact, at first there is an abrupt discontinuity of p and n concentrations on either side of the junction producing a strong incentive for diffusion effects at this location. Diffusion will indeed occur with holes diffusing into the n-type region and leaving behind negatively charged acceptor atoms that have a fixed location in the lattice. Likewise, electrons will diffuse into the p-type region and leave behind positively charged donor atoms that also have a fixed location in the lattice. The net effect of this is a build up of negative charge on the p-type side of the junction caused by negatively charged acceptor atoms and the diffused electrons and a buildup of positive charge on the n-type side of the junction caused by the positively charged donor atoms and the diffused holes. This buildup of charge in turn creates an electric field directed from the n-type side towards the p-type side.

It is obvious that this diffusion process can not continue indefinitely and at some point the electric field cause by the charge buildup becomes large enough to offset the incentive for holes to diffuse against the field to the n-type side and electrons to diffuse against the field to the p-type side and equilibrium conditions will exist with time-independent carrier concentrations.

After equilibrium conditions exist, there will be a region near the junction where the majority carrier concentration (holes in the p-type side and electrons in the n-type side) will be reduced below their equilibrium values. This region is called the depletion region.

This electric field x that has been produced near the pn junction as a result of diffusion in turn causing band bending of the conduction and valence bands in the energy band diagram because of the relationship:

This results in the qualitative figure 2.4 of Pierret shown below.

We will now derive a very important parameter of pn junctions, namely the built-in potential across the device V_bi. From the figure above, we see that a conduction band electron or a valence band hole will experience a potential drop as it progresses across the junction. This voltage drop is caused by the electric field in the depletion region and will be equal to:

The Depletion Approximation

To do a detailed calculation of the electric field and the voltage as a function of position x is extremely difficult. Therefore, approximations are made to simplify the problem but still result in accurate solutions. The depletion approximation makes the following approximations:

1. N_A>>n_p or p_p hence r = -qN_A for –x_p <= x <= 0.
2. N_D>>n_n or p_n hence r = qN_D for 0 <= x <= x_p.
3. The charge density is zero in the bulk regions

Thus, Poisson’s equation becomes:

for

for

and x = 0 in the bulk regions outside the depletion region. We can use these equations to easily solve for x throughout the device resulting in:

for

for

Since there will not be any sheet charge at x = 0 and the permittivities are the same, then the electric field will be continuous across the junction. This produces the following result:

We can easily solve for the potential V(x) using the relationship between x and V(x), namely, , and after arbitrarily defining V(-) = 0 results in:

for –x_p <= x <= 0

for 0 <= x <= x_n

for 0 <= x <= x_n

Depletion Width

We now need to acquire information about x_n and x_p that up to this point has been undetermined. We do this by looking at the continuity of the voltage at x = 0. The voltage will be continuous across the junction because there will not be any dipole layer. Hence equating the voltage on either side of the junction produces:

and using the relation gives us the following relation for x_n and x_p:

Electrostatics of Forward and Reverse Bias

It is easy to determine that any applied bias will be applied solely to the junction region (i.e., depletion region) which also produced V_bi. Hence, any applied potential tends to reduce the junction voltage from V_bi to V_bi + V_A where V_A is the applied voltage. This produces the following equations for x_n and x_p: