1.10 Diffusion

Introduction

Let us turn our attention to what happens to the electrons and holes, once they have been injected across a forward-biasedjunction. We will concentrate just on the electrons which are injected into the p-side of the junction, but keep in mindthat 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 adiffusion process, while at the same time, some of them are disappearing because they are minority carriers (electrons inbasically p-type material) and so there are lots of holes around for them to recombine with. This is all shownschematically in .

Diffusion process quantified

It is actually fairly easy to quantify this, and come up with an expression for the electron distribution within thep-region. First we have to look a little bit at the diffusion process however. Imagine that we have a series of bins, eachwith a different number of electrons in them. In a given time, we could imagine that all of the electrons would flow out oftheir bins into the neighboring ones. Since there is no reason to expect the electrons to favor one side over the other, wewill assume that exactly half leave by each side. This is all shown in . We will keep things simple and only look at three bins. Imagine I have 4, 6, and 8electrons respectively in each of the bins. After the required "emptying time," we will have a net flux of exactly oneelectron 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 thepicture.) We find that the net flux across each boundary is now 2 electrons per emptying time, rather than one. Note thatthe gradient (slope) of the concentration in the boxes has also doubled from one per box to two per box. This leads us to arather obvious statement that the flux of carriers is proportional to the gradient of their density. This is statedformally in what is known as Fick's First Law of Diffusion :

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

What about the electrons?

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 theirconcentration is greater than what they would be if they were in a sample of p-type material at equilibrium. We willdesignate them as $n^{\prime }$ , and since they could change with both time and position we shall write them as $n^{\prime }(x, t)$ . Now there are two ways in which $n^{\prime }(x, t)$ can change with time. One would be if we were to stop injecting electrons in from the n-side of thejunction. A reasonable way to account for the decay which would occur if we were not supplying electrons would be towrite:

Using the ambipolar diffusion equation

For anything we will be interested in, we will only look at steady state solutions . This means that the time derivative on the LHS of is zero, and so we have (letting $n^{\prime }(x, t)$ become simply $n^{\prime }(x)$ since we no longer have any time variation to worry about)

Just so you can get a feel for some numbers, a typical valuefor the diffusion coefficient for electrons in silicon would be $D_e=25\frac{\mathrm{cm}^2}{\sec }$ and the minority carrier lifetime is usually around a microsecond. Thus