Desktop version

Home arrow Mathematics

  • Increase font
  • Decrease font


<<   CONTENTS   >>

THE IDEAL PN JUNCTION UNDER BIAS

What mainly happens when a voltage V is applied to the PN junction is that the built-in voltage is changed. Given that the depletion layer is almost empty of mobile charge carriers, and hence has a very high resistivity, we consider that the applied voltage V is dropped

a Definition of the forward bias in a PN junction

FIGURE 3.15a Definition of the forward bias in a PN junction.

only across the space charge-depletion layer and not over the entire device length which includes the neutral P and N layers. This statement is going to be slightly modified later. We can view it at the moment as a very good approximation. Before proceeding we need a convention; we will consider the applied voltage to be positive when the positive electrode of the voltage generator is connected to the P-side of the junction, see figure 3.15a. As we can see from figure 3.15b, when V > 0 the built-in voltage is reduced to Vbi - V and when V < 0 it is increased to Vbi +|V|. Hence, in all cases we can write that the internal barrier becomes = Vbi — V. To see how this happens we recall from section 3.1 that when a voltage is applied to a specimen, the Fermi level is no longer unique but the Fermi levels at the two ends of the material differ by the applied voltage V. We emphasize that the term “Fermi level” here should be understood within the concept of a quasi-Fermi level discussed in section 3.1. Consider first that V > 0. Then the N-side is electrostatically lower than the P-side, which means that the Fermi level at this end of the junction is higher than the P-side (remember always to multiply by [ —c ] to go from voltages to electronic levels). Then the whole set of electronic levels at the end of the N-side is lifted compared to the neutral part of the P-side, as shown in figure 3.15b, and consequently the barrier for the electrons of the N-type semiconductor to jump to the P-side has been decreased to VbiV. A close examination of figure 3.15b also shows that the barrier for holes has also been reduced

c Band structure of a PN junction under reverse bias

FIGURE 3.15c Band structure of a PN junction under reverse bias.

accordingly to V&,- — V. Exactly parallel arguments lead to the band diagram of the PN junction in reverse bias as shown in figure 3.15c.

Once the barrier to jumping across to the other side has been reduced (for electrons in the N-side and holes in the P-side), the diffusion currents and have increased exponentially with the applied voltage because of an exponentially increased injection, as we will show, of the two types of majority carriers to their opposite side. Note that as the electrons in the N-side are injected into the P-side and the holes in the P-side into the N-side, they change character and become minority carriers, whereas they were majority carriers before the injection. The concentrations np and pn are shown in figure 3.16. These are the key quantities for the understanding of the working of the PN junction. We therefore have to obtain quantitative expressions for them. The problem of the distribution of the injected holes from the P-side into the neutral part of the N-side of the junction is mathematically the same as the example we solved in section 3.6. To see this, note that we have a steady state, it is a 1-D problem and the drift current is zero. The concentration of the holes in the N-side (minority carriers) will therefore be of the form of equation 3.38

Since the injection occurs at x„ (and not at x = 0), we have changed accordingly the argument of the exponential. A similar equation will hold for electrons in the P-side of the junction. Therefore, we need only evaluate the quantities Apn(x„) = p„(x„)-pn0 and Апр^-Хр^ = Пр^-Хр^-пр0. From figure 3.14b we can see that the bottom of the conduction band Ec in the N and P sides in equilibrium differ in energy by eVbl. Hence, assuming that Boltzmann statistics can be used instead of Fermi-Dirac statistics, we can write for the concentration of electrons at the conduction band minimum at the respective sides,

and similarly

where we have used the subscript 0 to denote equilibrium values. Rearranging 3.51a we have

Using the relation n,]0p„0 = nf that we have proved in chapter 2 (equation 2.61), we get

We have therefore obtained the built-in voltage Vbl in terms of the doping levels NA, ND and the equilibrium concentration и,.

When the voltage V is applied to the junction and the barrier reduces to Vj, - V, we can follow exactly the same argument as above and we then for electrons facing the reduced barrier get

But as we have repeatedly noted, the majority carrier concentration n„ is hardly changed by the injection and n„0 = n„. So using 3.51 we get

or equivalently

This constitutes the exponential increase we have referred to in the previous paragraph. Similarly, we also get

assuming likewise pp = ppo.

Therefore, we have obtained the preexponential factor in the injected hole carrier concentration, equation 3.50, and we get

and similarly for

We therefore have

and

The injected current density into the neutral P region and into the neutral N region can now be evaluated by taking the values of the above at x = —xp and x = x„ respectively. We get using 3.53a and 3.53b

and

The total current will be equal to the sum of the above two components. We will have

Electron, hole, and total current densities across an ideal PN junction

FIGURE 3.17 Electron, hole, and total current densities across an ideal PN junction.

where

The variation of Jp(x) and J„(x) in the entire range of я: is shown in figure 3.17, while the total current density as a function of voltage is shown in figure 3.18. An examination of figure 3.17 shows that apart from the minority carrier currents, there are also majority carrier currents, as indicated by the presence of Jn in the N-type region and Jp in the P-type region. Where do these currents come from? The total current density J must be constant and independent of x and these currents actually guarantee the constancy of J over the distance x. Equations 3.55a and 3.55b are valid only when they describe minority carrier movement, i.e. in the regions x > x„ and x < -xp respectively (when they describe an exponential decrease with x). When minority carriers are injected, let us say a concentration Ap(x) of holes in the N-side, an equal amount of electrons A n(x) is generated to preserve the neutrality of the region and this Дя(х) in the N region creates an extra current /„(x) of majority carriers. However, this current does not obey equations like 3.55a and 3.55b because these equations were derived from the continuity equations in the specific form valid only for

minority carriers. Actually, the majority carrier currents contain a significant amount of drift because our initial assumption of the applied voltage dropping exclusively in the depletion layer was only an approximation and a small electric field exists in the neutral regions. However, by adding the diffusion currents at the respective edges of the depletion layer we have taken care of all currents.

 
<<   CONTENTS   >>

Related topics