In this video, we will discuss Non-ideal current. So, in the ideal diode analysis, we basically ignored the depletion region. In other words, we assume that nothing happens within the depletion region. Carriers are simply injected across the depletion region without any loss. But in reality, the depletion region contains some impurities and other defects that can act as a recombination and generation center. So, these guides that can then capture some of the carriers being injected, and also, on the reverse bias, they can also generate carriers contributing to the reverse bias current. So, for this analysis, we will consider the recombination current under forward bias, primarily, and we will use the Shockley-Hall-Read theory. For simplicity, we will assume that the recombination centers have the same hole capture cross-section and the electron capture cross-section. With the assumption, then the net recombination rate U is given by this, and the numerator here is np minus ni squared. But np product under bias is given by this equation here. So, in other words, np product is increased or decreased by this exponential factor contain your voltage. So, under forward bias, you will increase the np product because you're injecting carriers. On the reverse bias, you're depopulating carriers. So, np product goes below the equilibrium value. The numerator has all these applied voltage dependence here. When U is positive, you have net recombination, because this exponential factor becomes greater than one on the forward bias, then you have net recombination on the forward bias. On the reverse bias, Va is negative, so your exponential factor becomes less than one. This quantity has a negative sign that indicates net generation. Now, the total current is given by simply integrating your net recombination rate across the depletion region. Now, let's consider the simple case where U becomes maximum. The net recombination rate reaches maximum when n+p reaches minimum. This n+p is a term that shows up in the denominator. When that becomes minimum, your U, net recombination rate, will reach minimum. Given a constant np product because np product once again is fixed for a given bias. So, for a constant np product (p+n) reaches minimum when they are equal to each other, NE is equal to P. So, in that case, your nn is equal to P, is equal to ni times exponential the qVa over 2kBT. Further assume that your trap energy level, defect energy level, is located right at the middle of the band gap intrinsic formula level. Then, it simplifies the equation for you, and so, you can approximate your recombination current as the U max, the maximum value corresponding to this condition p equals n times the effective depletion region with X prime. From the equations for you, you can derive this. If you ignore this minus one and plus one, you can approximately get a exponential dependence which contains these voltage dependence qVa over 2kBT. Now, the tau naught, the lifetime and X prime, which is the effective width of the depletion region where you have the recombination. These guides are not really known. So, you typically approximate this X prime with the total depletion region width. If you calculate with that assumption, you calculate the ideal diode current Jt divided by Jr, recombination current, you get an exponential factor which has contains the voltage dependence as qVa over 2kBT. So, if your Va is large, then this exponential term becomes very large. Meaning that, your ideal diode current dominates over the recombination current. However, if your forward bias is small, then this ratio, the exponential term can be quite small, and the recombination current will become important relative to your ideal diode current. So, if you plot the current as a function of voltage, and it's a semi-log plot, so these straight line represents exponential function. So, here is your the recombination current which has a slope of one over two. This one here is the ideal diode current which has a slope of one. When your forward bias is large, then you can see that the ideal diode current becomes quite large compared to the recombination current and you can ignore it. However, when your voltage is small, then your recombination current can actually exceed that of the ideal diode current can dominate. So, this bowing of the current at a very small forward bias is the typical current voltage characteristic that you see in diode, and it is a result of these non-ideal recombination current due to the recombination going on within the depletion region.
