In this video, we will continue the discussion of the advance operation of MOSFET. The next effect we want to discuss is a bulk charge effect. So in the derivation of the long-channel MOSFET equation, we assumed that the depletion charge Q sub D in the channel region due to the ionized acceptors of the substrate is not a function of y, it does not change as a function of the position along the channel. However, we do know that the channel voltage changes from source to drain and therefore that must accompany the change in depletion charge. Depletion region width along the channel which should lead to changing depletion charge. So go back to the equation that we discussed in the MOS device section. And the depletion charge is given by this. The electronic charge times the acceptor density, times the depletion region width and that is given by this equation here. So now we have addedd this y dependent term here. [COUGH] The variation of the bulk charge means the threshold voltage changes as a function of position as well. And that also leads to the changes in the inversion layer carrier density, charge density as a function of y. So, because the inversion layer charge density was given by this equation here by substituting the y dependent, position dependent depletion charge, we get a new equation for the inversion layer charge density. There is an added dependence of y, the position along the channel here due to a y dependent depletion charge. And using this, we go back to the drift current equation that we used to derive the long channel equation. Now we use new Q sub n of y which includes the y dependent depletion charge then you can integrate this equation along the channel, same way as we did to derive the long channel equation, you get this equation here. And in the simpler case of no body bias the equation becomes this. So in order to simplify the description and a little bit more intuitive description, we can write the drain current in this form, it varies the same form as long channel device equation. Except that we use an alpha here to describe the bulk charge effect. Now this alpha, when alpha is equal to 1 then it becomes a standard long channel MOSFET equation. With the bulk charge effect, alpha value typically becomes about 1.5. That describes the effect of a bulk charge, which generally reduces the drain current. The saturation drain voltage of VD saturation, the drain current at which your MOSFET goes into enters the saturation regime is also changed. It's VG minus VT in the simple long channel equation but now wit the bulk charge effect. it's VG minus VT divided by alpha and that changes your saturation drain current as well. It is reduced by a factor of alpha. Now I want to discuss sub-threshold conduction. We assume that the at threshold voltage. When your gate voltage is reduced to threshold voltage then the inversion layer charge concentration becomes zero and therefore your drain current becomes zero. But it is really not the case. You do not have an abrupt change in carrier concentration to 0. And therefore, there is some lingering carrier concentration even at gate voltages below the threshold voltage. So that leads to sub-threshold conduction. And in the sub-threshold region, if you look at the energy band diagram near the surface, it looks like this, okay? So here's the source and here's the drain, here's the channel. Now, in order to understand where this energy band diagram come from, we go once again, look at this three dimensional band diagram we considered when we discussed the buddy bias effect. So this once again is the MOSFET on the side, gate in the front, substrate in the back and source and drain on left and right. And the 3D band diagram looks like this. And here now we're looking at the surface region only, so only this region here, this band diagram here, okay? So if you just take that then we get the energy band diagram looking like this. Now, as you increase the drain voltage, this site energy goes down, drain site energy goes down. So what that means is that you now have a back-to-back pn junction, and the source channel junction is forward biased, channel drain junction is reverse bias. And this is very much like the bipolar junction transistor. And the current through this system is determined primarily by the forward biased junction on this side. And so the current is given by this exponential equation that describes a diode current basically. In the analogy with bipolar junction transistor is not complete, because the energy band diagram that we are showing here not really a bulk energy band diagram, but a surface energy band diagram. So it's all highly localized near the surface, so the analogy is not perfect. But if you are familiar with the operation of bipolar junction transistor, is very easy to understand. Now we define a quantity eta, efficiency factor, which is a derivative of the surface potential with respect to your gate voltage. And it's related to the ratio of the depletion capacitance, junction capacitance, and the outside capacitance. And from the equation that I showed you all or the exponential equation, you will see that this factor eta is equal to 1 over n, the ideality factor that goes into your exponential equation. And so sub threshold current increases exponentially with your eta factor increases. So if you plot the drain current as a function of your gate voltage, in a semilog plot, you see this straight line region at voltages below the threshold, this represents the exponential tail [COUGH] representing the sub-threshold conduction. And the slope here is denoted as S parameter. And from the exponential equation that I showed you earlier, it is given by this. And typical value for S parameter is anywhere between 70 to 120 millivolt per decade. And this S parameter is an important parameter for design purpose, because it sets the limit on how low your gate voltage has to be in order to really turn your device off.