Let us turn our attention to what happens to the electrons and holes, once they have been injected across a forward-biased junction. We will concentrate just on the electrons which are injected into the p-side of the junction, but keep in mind that similar things are also happening to the holes which enter the n-side.

As we saw a while back, when electrons are injected across a junction, they move away from the junction region by a diffusion process, while at the same time, some of them are disappearing because they are minority carriers (electrons in basically p-type material) and so there are lots of holes around for them to recombine with. This is all shown schematically in Figure 1.

It is actually fairly easy to quantify this, and come up with an expression for the electron distribution within the p-region. First we have to look a little bit at the diffusion process however. Imagine that we have a series of bins, each with a different number of electrons in them. In a given time, we could imagine that all of the electrons would flow out of their bins into the neighboring ones. Since there is no reason to expect the electrons to favor one side over the other, we will assume that exactly half leave by each side. This is all shown in Figure 2. We will keep things simple and only look at three bins. Imagine I have 4, 6, and 8 electrons respectively in each of the bins. After the required "emptying time," we will have a net flux of exactly one electron across each boundary as shown.

Now let's raise the number of electrons to 8, 12 and 16 respectively (the electrons may overlap some now in the picture.) We find that the net flux across each boundary is now 2 electrons per emptying time, rather than one. Note that the gradient (slope) of the concentration in the boxes has also doubled from one per box to two per box. This leads us to a rather obvious statement that the flux of carriers is proportional to the gradient of their density. This is stated formally in what is known as Fick's First Law of Diffusion:

Where De De is simply a proportionality constant called the diffusion coefficient. Since we are talking about the motion of electrons, this diffusion flux must give rise to a current density Je diff Je diff . Since an electron has a charge −q q associated with it,

Now we have to invoke something called the continuity equation. Imagine we have a volume V V which is filled with some charge QQ. It is fairly obvious that if we add up all of the current density which is flowing out of the volume that it must be equal to the time rate of decrease of the charge within that volume. This ideas is expressed in the formula below which uses a closed-surface integral, along with the all the other integrals to follow:

We can write QQ as where we are doing a volume integral of the charge density ρρ over the volume VV. Now we can use Gauss' theorem which says we can replace a surface integral of a quantity with a volume integral of its divergence: So, combining Equation 3, Equation 4 and Equation 5, we have (note we are still dealing with surface and volume integrals): Finally, we let the volume VV shrink down to a point, which means the quantities inside the integral must be equal, and we have the differential form of the continuity equation (in one dimension)

Now let's go back to the electrons in the diode. The electrons which have been injected across the junction are called excess minority carriers, because they are electrons in a p-region (hence minority) but their concentration is greater than what they would be if they were in a sample of p-type material at equilibrium. We will designate them as n′ n′ , and since they could change with both time and position we shall write them as n ′ ⁢xt n ′ x t . Now there are two ways in which n ′ ⁢xt n ′ x t can change with time. One would be if we were to stop injecting electrons in from the n-side of the junction. A reasonable way to account for the decay which would occur if we were not supplying electrons would be to write:

Where τr τr called the minority carrier recombination lifetime. It is pretty easy to show that if we start out with an excess minority carrier concentration no ′ no ′ at t=0 t 0 , then n ′ ⁢xt n ′ x t will goes as But, the electron concentration can also change because of electrons flowing into or out of the region xx. The electron concentration n ′ ⁢xt n ′ x t is just ρ⁢xtq ρ x t q . Thus, due to electron flow we have: But, we can get an expression for J⁢xt J x t from Equation 2. Reducing the divergence in Equation 10 to one dimension (we just have a ∂J∂ x x J ) we finally end up with Combining Equation 11 and Equation 8 (electrons will, after all, suffer from both recombination and diffusion) and we end up with: This is a somewhat specialized form of an equation called the ambipolar diffusion equation. It seems kind of complicated but we can get some nice results from it if we make some simply boundary condition assumptions. Let's see what we can do with this.