Steady State Carrier Injection and Diffusion Length
In the steady state case the diffusion equations become
where is called the electron diffusion length and Lp is the diffusion length for holes. We no longer need partial derivatives, since the time
variation is zero for steady state.
The physical significance of the diffusion length can be understood best by an example. Let us assume that excess holes are somehow injected into a semi-infinite semiconductor bar at x = 0, and the steady state hole injection maintains a constant excess hole concentration at the injection point ∂p(x = 0) = Δp. The injected holes diffuse along the bar, recombining with a characteristic lifetime tp. In steady state we expect the distribution of excess holes to decay to zero for large values of x, because of the recombination that is shown in figure. For this problem we use the steady state diffusion equation for holes. The solution to this equation has the form
We can evaluate C1 and C2 from the boundary conditions. Since recombination must reduce ∂p(x) to zero for large values of x, ∂p = 0 at x = and therefore
C1 = 0. Similarly, the condition ∂p = Δp at x = 0 gives C2 = Δp, and the solution is
The injected excess hole concentration dies out exponentially in x due to recombination, and the diffusion length Lp represents the distance at which the excess hole distribution is reduced to lie of its value at the point of injection. We can show that Lp is the average distance a hole diffuses before recombining.
To calculate an average diffusion length, we must obtain an expression for the probability that an injected hole recombines in a particular interval dx. The probability that a hole injected at x = 0 survives to x without recombination is ∂p(x)/Δp = exp(—x/Lp), the ratio of the steady state concentrations at x and 0. On the other hand, the probability that a hole at x will recombine in the subsequent interval dx is
Thus the total probability that a hole injected at x = 0 will recombine in a given dx is the product of the two probabilities:
Then, using the usual averaging techniques, the average distance a hole diffuses before recombining is
The steady state distribution of excess holes causes diffusion, and therefore a hole current, in the direction of decreasing concentration
Since p(x) = p0 ∂p(x), the space derivative involves only the excess concentration.