Книги2 / 1993 Dutton , Yu -Technology CAD_Computer Simulation

interface. We will see that this does not occur. In fact, the bands are continuous across the interface as is shown in Figure 4.4 (b). One can realize that this bending of the bands implies two things. First, the concentration of holes and electrons varies across the region of transition from n- to p-type materials, as evidenced by the variation of Eo - Ep. Second, band bending implies that fields are present. The direction and value of the field are such as to oppose the diffusion tendencies of carriers. Consider Eqs. (4.46-4.47). An electric field is necessary to maintain the concentration gradient.

The electric field in the transition region (and in any region for that matter) is related to the gradient of the potential energy. That is, the force on an electron is -q£(x) and this force must equal the negative of the gradient of potential energy for electrons (because of E(x) = -fJ1f;(x)/fJx). The potential energy for free electrons can be represented by energy Eo for homogeneous material neglecting the bandgap narrowing effect. Hence,

One may also observe that the intrinsic Fermi level follows Eo under the same condition as specified above. Thus the gradients of Eo and Ei are the same, and it follows that

Notice from Figure 4.4 (b), that E(x) is positive in the +x direction in this case.

For no current flow across the transition region:

(4.54)

If we use Eqs. (4.24) and (4.53) in the above expression (realizing that the concept of Ep is valid for this equilibrium condition) we obtain

From the above equation and using the Einstein relationship (Eq. (4.48», the terms with fJEdfJx cancel and it follows that

Because P and j.tp can never be zero, so EF must be a constant independent upon the position. This derivation confirms our earlier assertion that EF is flat across the transition region.

In addition to showing that EF is flat across the transition region, we can use Eq. (4.54) to quantitatively determine the nature of the bandbending in the transition region. To maintain Jp ( x) = 0, the required field £(x) can be determined from Eq. (4.54) to be

Again using the Einstein relationship and expressing £: in terms of potential 'IjJ,

It is interesting to show from the above equation that even without knowing p(x), it is possible to determine the relative potential of one side of the transition region relative to the other. By integrating Eq. (4.58) over the distance between two points where p is known, one can find

That is, the electrostatic potential in the neutral p-region (where p = ppo) is less than that in the neutral n-region (where p = PnO) by an amount 'ljJo, where

This quantity 'ljJo is the built-in potential at the junction under equilibrium conditions.

150 CHAPTER 4. PN JUNCTIONS

Under certain circumstances, even at nonequilibrium conditions we can still obtain a relationship similar to Eq. (4.60). Specifica.l1y, assume that we apply to a pn junction an external bias, V, of a polarity so

as to oppose (or decrease) tPo, the built-in potential. Physica.l1y this corresponds to making the p-region positive in charge with respect to the n-region. Under these conditions, we assume that the current which flows externally is sma.l1 compared to the individual counterbalancing flows in the transition region, and we define this condition as quasi-equilibrium in the transition region. The concept of quasiequilibrium is that it allows us to use Eq. (4.58) to solve for the relation between potential and concentrations at the edges of the transition region. The assumption that a.l1 the applied bias appears across the transition region is reasonable as suggested by the discussion in Chapter 3 since the abundance of majority carriers outside the transition region on each side typica.l1y precludes significant resistive drops in these regions. Hence the new potential difference across the transition region is given by tPo - V. The hole concentrations at the edges of the transition region are now given by Pn, which is not equal to PnO, and Ppo is assumed not to be altered under low level conditions. Thus for the nonequilibrium case

Using the relationship for Ppo,tPo, and PnO given in Eq. (4.61) one can rewrite the above equation as

The essence of deriving Eqs. (4.64-4.65) is the assumption that quasiequilibrium exists across the transition region. The applied bias is assumed to alter the energy bands just across the transition region. Also the net current flow is assumed sma.l1 compared with the individual drift and diffusion components. The conditions for equilibrium and quasiequilibrium (for an applied bias potential of +V from P to n) are shown

Figure 4.5: The relationships of potential and Fermi-level to (a) the equilibrium condition, (b) a +V applied to the p-side (relative to the n).

in Figures 4.5 (a) and (b), respectively. Note that for quasi-equilibrium conditions the quasi-Fermi levels have been added. Their indicated flatness ({JEFn/{JX ::::::i 0, {JEFp/{JX::::::i 0) in the transition region reflects the assumption that· the total current in that region is small compared to either term in Eq. (4.54). The larger dots at the edges of the transition region indicate the points where Eqs. (4.64-4.65) are assumed as the "boundary equations". Although we have considered the conditions of the quasi-equilibrium at the edges of the transition region only, an interesting result is that everywhere in the transition region

(4.66)

Tha.t is, the quasi-Fermi levels are separated exactly by qV (again, a.s depicted in Figure 4.5 (b)). Outside the transition region, as the carriers diffuse and recombine during the process of current flow, conditions of equilibrium are restored, as can be evidenced by the return to a single Fermi level. A final observation regarding the assumption that

For lp > x> 0 the charge density is now given by p = -qNA. For £(x) to equal zero for x > lp it must be true that "+" and "-" charges are equal from the requirement of the overall charge neutrality. Hence

(4.73)

Using Eq. (4.72) and the new expression for p, one can find the expression for the electric field in the range of lp > x > 0 as follows.

(4.74)

If we define the total depletion layer width as

(4.75)

and integrate the electric field through the entire depletion region ([-in, lp)) knowing that the potential difference across this region if 'ljJo - V, one should be able to obtain the following expressions for alII's using known parameters.

Having obtained expressions for Ip and in, it is useful to consider how the "space-charge" contained within these limits varies with applied voltage. Define Q(V) as the total negative space-charge (on the p-side):

(4.79)