1.2 Diffusion  (Page 2/2)

We can write $Q$ as, [link] , where we are doing a volume integral of the charge density ( $\rho$ ) over the volume ( $V$ ). Now we can use Gauss' theorem which says we can replace a surface integral of aquantity with a volume integral of its divergence, [link] .

So, combining [link] , [link] and [link] , we have, [link] .

Finally, we let the volume $V$ shrink down to a point, which means the quantities inside the integral must be equal, and we have the differential form ofthe continuity equation (in one dimension), [link] .

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' , and since they could change with both time and position we shall write them as n' ( x , t ). Now there are two ways in which n' ( 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:

Where ${\tau }_r$ called the minority carrier recombination lifetime. It is pretty easy to show thatif we start out with an excess minority carrier concentration n ₀ ' at t = 0, then n' ( x , t ) will go as, [link] . But, the electron concentration can also change because of electrons flowing into or out of the region $x$ . The electron concentration n' ( x , t ) is just $\frac{\rho (x, t)}{q}$ . Thus, due to electron flow we have, [link] .

But, we can get an expression for $J(x, t)$ from [link] . Reducing the divergence in [link] to one dimension (we just have a $\frac{\partial^{1}J}{\partial x}$ ) we finally end up with, [link] .

Combining [link] and [link] (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 seemskind of complicated but we can get some nice results from it if we make some simply boundary condition assumptions.

Using the ambipolar diffusion equation

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

Picking the not unreasonable boundary conditions that $n^{\prime }(0)=n_0$ (the concentration of excess electrons just at the start of the diffusion region) and $n^{\prime }(x)\to 0$ as $x\to$∞ (the excess carriers go to zero when we get far from the junction) then:

The expression in the radical $\sqrt{D_e{\tau }_r}$ is called the electron diffusion length, $L_e$ , and gives us some idea as to how far away from the junction the excess electrons will existbefore they have more or less all recombined. This will be important for us when we move on to bipolar transistors.A typical value for the diffusion coefficient for electrons in silicon would be D _e = 25 cm ² /sec and the minority carrier lifetime is usually around a microsecond. As shown in [link] this is not very far at all.