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Equation 3.20 and its solution, equation 3.22, assume that the excess carriers Ap(t) have no space-dependence. However, as we shall see in the case of the pn junction in this chapter, the injection of minority carriers occurs at a given direction and our current assumption of no space dependence for Ap is not valid. In this case, when the excess carriers have a non- uniform distribution in space, we encounter the phenomenon of carrier diffusion.

Figure 3.7 shows an arbitrary density of holes that is x dependent (1-D model). At each point x there will be a hole current called a diffusion current, which will be sent in the direction of decreasing p(x) and which will be given by A variation of a hole density in space gives rise to a space dependent diffusion current. The arrows indicate only the direction of flow

FIGURE 3.7 A variation of a hole density in space gives rise to a space dependent diffusion current. The arrows indicate only the direction of flow.

Geometry for the calculation of the diffusion current

FIGURE 3.8 Geometry for the calculation of the diffusion current.

where Dp is a constant called the diffusion coefficient for holes. The arrows in figure 3.7 along the tangent of the p(x) curve indicate simply the direction of the current density Jd/f, left or right. This is called Fick’s first law of diffusion and it can be proved as follows.

Imagine a plane at x = x0) vertical to the direction x, of finite area A and two other planes, also vertical to x and of area A at a distance l of either side of the plane at x0, see figure 3.8. Now the distance / is not arbitrary, but it is specifically chosen to be the distance the carriers travel without being scattered. It is called the mean-free path. Hence it is related to the relaxation time x of holes that we have encountered in our elementary theory of transport by the formula

where vF is the velocity of holes near the Fermi level. Note that we have dropped the subscript p on x. Therefore during the time x, half of the holes enclosed between the planes at x0 and at x0 - / will move to the right and half of the holes between x0 and x0 + / will move to the left. This is due to the carrier’s random (brownian) motion. The mean free path l is usually a few nanometers so that the concentration p can be considered constant between x0 and x0 -1, say equal to p, and between x0 and x0 + /, equal to p2.

Then the current crossing the plane at x0 is

Now p2p = (dp / dx)l and hence

where in 3.28 we have reinstated the subscript p on both l and x and

A corresponding set of equations for electrons leads to the diffusion current due to a gradient in the concentration of electrons


with l„ and x„ the mean free path and relaxation time of an electron respectively. The absence of the minus sign in 3.30, when compared to 3.28, is due to the negative charge of the electron as opposed to that of a hole.

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