The table below lists the recombination mechanism and the corresponding receiver of the energy.

The rates of recombination for each mechanism are,

Let’s focus on the SRH recombination first. The equation looks pretty complicated (there are already several assumptions built into the derivation of this equation, including low-level injection and that the traps are very near the middle of the bandgap), but for most situations, the equation is easily simplified. Assume we are dealing with an n-type semiconductor with a typical doping concentration, and that we are in low injection—whoops! What is low injection?

Before we talk about low injection we first we need to define the excess carrier concentrations, Dn and Dp,

Dn and Dp, in other words, represent how much the respective carrier concentrations have changed from equilibrium—equilibrium becomes our reference point for non-equilibrium situations. Dn and Dp can both be negative as they are changes in carrier concentrations, and the change can be negative if you have extracted/removed carriers.

Back to low injection. In words low injection means the semiconductor is out of equilibrium, but that the majority carrier concentration has not been perturbed (changed). In equations we write it as for an n-type semiconductor and for a p-type semiconductor.

Comparing these conditions to the definitions above means that the majority carrier concentration is not changed significantly in low injection. More on low injection will appear in the next lesson.

Now back to trying to simplify the equation for the SRH recombination rate.

Let’s assume we are talking about an n-type semiconductor in low injection again.

Replace all the n’s with N_D because n = n_o = N_D — because of low injection and because it is n-type, respectively, and replace p = p_o + D_p . The equation becomes,
.

Note that p_o * N_D –n_i² = 0 because that is essentially p_on_o = n_i². Also note that N_D >> n_i. Now we have,
.

One last assumption is that _n ≈ _p. Not so much that they have nearly the same value, but that they are of similar orders of magnitude. If that is the case, then _pND will be much larger than any other term in the denominator. Further simplification results in,
.

What a nice and simple equation! The SRH recombination rate in an n-type semiconductor is dependent on the number of excess holes and on the time constant, _p , which is called the minority carrier lifetime. Let’s look at this from another perspective. Recall the figure above illustrating SRH recombination; it shows it as a two-step process, and we associate _n with the first step in the process, the capturing of a free electron, and τ_p with the second step of the process, the annihilation of a hole. Now let’s add a Fermi level indicating an n-type semiconductor. This is what it would look like.

.

Note the Fermi level above E_i. Recall that we have a rule about the Fermi level that says that any states below the Fermi level are filled—that is why the figure shows the trap state as being filled. At equilibrium that trap state is filled. The first step of the two-step SRH recombination process has already been completed before any non-equilibrium situation is created. This is what the equation above is suggesting—that the SRH recombination rate has nothing to do with _n because the first step is already completed.

The same conclusions can be drawn about a p-type semiconductor. In that situation the trap is empty so it easily captures an electron. It will release that electron to the valence band quickly because there are so many empty valence states, and we don’t care how long it takes because we are usually looking at minority carriers anyway.

The radiative recombination rate can be recast/simplified as well. Let’s assume another n-type semiconductor in low injection, and it is easily shown that,
which simplifies to, which looks like if

B is a constant known as the Einstein coefficient for radiative recombination. The key here is that we can make the simplified radiative recombination rate equation look like the simplified SRH recombination rate equation.

The Auger recombination rate can be simplified to if .

As all three mechanisms are present in all semiconductors, we can deduce that , indicating that the several recombination mechanism rates add together like resistors in parallel—this make sense because they are all paths of varying “resistance” (or conductance!) for electrons to travel from conduction band to valence band.

The equation you will see the most to represent the recombination rate in a semiconductor is for a(n) n-type, p-type semiconductor. Note that what we have done here shows that the lifetime can concurrently represent all recombination mechanisms—still, most texts, including Pierret’s, will have in mind the SRH mechanism when that equation is written because the SRH mechanism is the dominant mechanism in silicon and silicon is the dominant semiconductor.

Created on 24 Apr 2010, Last modified on 24 Apr 2010