THE NON-IDEAL, REAL PN JUNCTION
One of the assumptions we have made in deriving the ideal PN junction current equation is that there is no recombination in the depletion layer when the majority carriers are injected into the neutral P and N regions. This is not actually true. The semiconducting materials always suffer from native metal impurities or even defects produced during the doping process which give rise to mid-gap states which in turn produce generation and recombination processes in the depletion region. These in turn give extra currents that we have not taken into account so far.
We have discussed recombination through mid-gap states in section 3.4. Equation 3.24 gives this rate as being equal to
The maximum recombination rate will develop when the denominator is a minimum. This in turn implies that n~ p. Let us now look at the value of np at the two ends of the depletion layer. We have from equations 3.53a-c
and by a similar argument
Although the equality of the product pn has been established only at the end points of the depletion layer, numerical calculations do in fact show that the product pn remains constant in the depletion region.
We then obtain assuming that approximately т„ = xp = C,
The recombination R, above denotes processes per unit time per unit volume so that an integration over the volume of the depletion layer is required to get the recombination current IR. Hence if A denotes the cross-section of the diode and W, as usual, the length of the depletion region
In addition to the recombination current there is also a generation current Ig and the observed current is the difference between the two. We note that the generation current Ig is depends only on temperature. At V = 0 the two must balance. Therefore
This current must be added to the diffusion current we calculated previously, so we get
The sum of the above two currents in equation 3.63 can be written more compactly in an empirical form
where 1 ^ m ^ 2. At low values of the applied voltage V, the current I is dominated by the recombination term and m ~ 2 whereas at higher V the current I is essentially composed of the diffusion current and m ~ 1. The constant m is usually called the “ideality factor” of the diode. A schematic graph of the total current I versus the voltage V is shown in figure 3.19.
From figure 3.19 it can be seen that at very high voltages the rate of increase of I falls below the rate dictated by m = 1. This is due to the fact that at such voltages, all of the applied V does not drop nearly exclusively in the space charge/depletion layer and there is some part of V - let’s call it V - which drops inside the neutral P and N regions. Then
FIGURE 3.19 I-V characteristic of a real PN junction.
the barrier inside the depletion region is not [Vw - Vj but it is equal to — (V —1/JJ. This would lead to the diffusion current being equal to
and the current increases at a slower rate as equation 3.65 indicates. There are also temperature effects in PN junctions as both diffusion and generation-recombination are temperature dependent processes which are well documented having been explored since the 1950s. We will not deal with these as the purpose of this book is to explore the transition from the mechanisms of conduction which treat the electrons as particles to those that need to treat electrons as waves, as is necessary in modern devices.