We have seen in the previous video that when you apply a gate body voltage equal to the flat-bond voltage, the energy bands are flat, and there is no potential drop as you go from the top of a semiconductor near the interface with the oxide deep into the semiconductor. We will now see what happens when you apply values of gate body voltage which are different from the flat voltage. In particular, we will look into what the conditions are near the semiconductor-oxide interface, or semiconductor surface as we call it. Let us assume that we have an externally applied voltage without restricting its value for now. The voltage between gate and body is VGB, and it creates a voltage drop across the oxide, C-ox, and the voltage drop across the semiconductor from the surface as we call it, the interface between the oxide and the semiconductor will be called the surface. All the way to the bulk outside the region where charges may appear. These charges will be called QC. The interface charges would be called Q0 and the gate charges would be called QG. Now, the sum of these three charges, QG plus Q0 plus QC, must be equal to zero for the structure to be overall neutral. So we will call this the charge balance equation. This potential from the surface all the way to the bulk outside the space charge region is called the surface potential, and is denoted by C sub S. But at any other point if the vertical dimension is Y, at any other point Y, the chart, the potential from a point at Y to a point, again, outside the space chart's region will be called C and will be a function of Y. So let us now plug the various potentials as a function of Y. We see that between the gate and the surface, we have a potential drop, which is the oxide drop. So this drop here is equal to the oxide potential here. Then we have a drop from the surface to the back which is CS, and there's this potential drop over here. And at any point, why we have a potential drop, which is this much here, C over Y. Now the sum of the oxide potential plus the surface potential plus any contact potentials that are in the loop as we know must be equal to the total externally applied voltage, VGB. So we can write VGB equals to psi ox plus psi s plus phi ms. If this were a circuit, you would call this the Kirchhoff's voltage law. Here, we will call it the potential balance. Now, a couple of important observations. We are interested not only in charges and potentials, but also in changes of charges and changes in potentials. So let us assume that VGB changes by a small amount delta VGB, we want to find how will the corresponding potentials here change and how will the corresponding charges change? If you change the gate by the voltage the gate charge will change by an amount delta QG. The space charge QC over here will change, by an amount delta QC, but the interface charge, which we assume is fixed, will not change. So therefore from this equation, to maintain charge balance it must be that the changes in gate and space charge regional charges cancel each other out. So this will be the chart's balance equation for chart changes, if you like. What will happen over here? You change delta VGB, the oxide potential will change. The surface potential will change. But VMS, because it only depends on contact potentials that are soon fixed, will not change. And therefore this equation will give you this one. This would be then the potential balance for potential changes. We will find both of these equations very useful. They represent very simple principles that we will use again and again. Now let's talk about the effect of VGB on the condition right at the surface of the semiconductor. First I would show you the condition we call accumulation. Here I'm applying a VGB which is smaller than VFB. I say smaller because as you see, the plus and the minus are this way, and I have added the little voltage here in the opposite direction so that VGB is less than VFB. Now, when you only have VGB go to VFB, you had the flat band condition where there were no charges here. But if you make this voltage less positive, or more negative than before, you will tend to accumulate more negative charges on the gate, which will require extra positive charges to accumulate at the surface. This being a p type material, these positive charges are in plenty of supply and they're called holes. So, now you have accumulation of holes over and above their concentration that you would have here in the box. This is what we call accumulation. So, in accumulation, first of all, VGB is less than the flat voltage. The charge in the semiconductor is positive, because it consists of holes, and the surface potential, CS, I remind you that is the potential of the surface with regards to the bulk outside the space charge region. That is negative. That is, after all, why positive charges go to the surface. Let us continue with another condition which we will call depletion. Now instead of making VGB smaller as I had done up here, I will make VGB larger than VFB by adding a voltage to VFB. So now VGB is larger than VFB and I assume that this voltage VGBs large enough to place positive charges on the gate now. These charges are positive, how will they be balance if the structure remains overall neutral? The positive charges repel the holes that are near the surface, away from the surface, because those holes are also positively charged. And once the holes move they leave behind depleted acceptor atoms that I remind you are immobile these are charges that can't move and that's why they are in a circle, now, for this charge to appear there must be some potential drop across this region. This is the surface potential that we talked about before. So, we have VGB is larger than VFB, in order to get depletion. We have a space charge that is negative and consists of acceptor ionized atoms. The surface potential is positive now, the opposite of this case over here where it was negative. And at the surface I have a negligible amount of electrons, so I will ignore any electrons that might be present near the surface. So now let's go to the next case which is inversion. This is by far the region that we're mostly interested in this course. Now, I'll make VGB even larger than before. And what happens then is, you place even more positive charges on the gate, which demand even more negative charges in the substrate in order to balance things out for overall charge neutrality. Now, and therefore the depletion region deepens in order to reveal more charges and consequently the surface potential increases. Finally, the surface potential becomes so large that right at the surface, the potential is so positive that it becomes attractive for electrons to pile up there and when this happens, you have inversion. Why do you call it inversion? Because this is a p type material, you expect to see a lot of holes, yet you see a lot of electrons at the surface. That's why you call it inversion. Now, where do these electrons come from? In the present case, they can pile up slowly. You can have electron-hole pair generation because of thermal agitation of the of the lattice. And the electrons generated find their way towards the surface, the holes find their way towards the bulk. Now, this process is slow and this electrons will not be generated fast enough to follow high speed variations of this voltage. Later on, when we add the source and the drain, you will see that those can supply the extra electrons and this mechanism of electrons piling up can be quite fast. So now in inversion what do we have? We have VGB larger than VFB, but that we also have indepletion. We have QC negative, discharge is overall negative. We also have that indepletion but now QC consists of two types of charges, right? And we have a positive charge for potential. Like indepletion the difference is that the surface electron concentration is significant. But how significant, we will say, very precisely, later on. Let's now calculate the surface electron concentration. We call it n surface. So here, we have n surface in the bulk, we have the equilibrium electron concentration, a small concentration because it is a p type material, which we're calling zero. And in our background review, we derive this equation. The ratio of two concentrations of electrons depends on the potential difference between the two points where you take these concentrations. CS in this case. And it is given by this, the exponential of CS, divided by the thermal voltage, kt over q. So what is in zero? When we discussed p type semiconductors. We found that then zero is approximately given by the intrinsic concentration squared divided by the acceptor concentration. And you may recall that we had also defined the thermic potential, and the thermic potential was given by this equation. Now, if I solve this equation for NA, plug in the result into this NA and plug in the result into here. And solve this equation, we find this result. That the surface electron concentration is given by the acceptor concentration in the bug times the exponential of the difference between the surface potential and two times the Fermi potential divided by the thermal voltage. This equation can be plotted and it looks like this. So this is the surface concentration of electrons versus surface potential. As we expect from this exponential, it goes like that. And you can show, and you will find the details in the book. That when CS is equal to 5F. The surface concentration is equal to the surface concentration of holes, and both of them are equal to the intrinsic concentration. And when you reach 2 phi F, then the surface concentration of electrons becomes equal to the concentration of holes in the bug which was approximately equal to NA. So phi f and 2 phi f are special points, so to speak. But notice that nothing drastic happens at phi f or 2 phi f. And in fact, you can continually about phi f. And this continues to be a smooth curve. I say this because we will need to discuss a fine point having to do with this fact. So what is the energy band picture? For simplicity here I will assume that phi MS is 0, and the effective interface charge is also 0. At flood bland, you will recall, the bands were flat. Now, in accumulation, we have this situation. In the bulk, okay, this is the gate, this is the oxide, and this is the summary conductor. In the bulk, the bunks are still flat because there, we have neutrality and potential doesn't change. And because we are talking about a P type substrate, the Fermi potential is below e sub i, as we expect in accumulation, near the surface holst by lap and in the sense we have even more pronounced pitibefects than we do at the balks. So the distance between the I and the F increases. So the bands bend like this. The Fermi potential is still the same. And it is flat throughout because we assume here equilibrium. We don't assume that there is any current through the oxide. Therefore there is no current in this structure here. Now let's go to depletion. In depletion the bands bend in the opposite direction. Why? Because, if you recall, the surface potential here is the opposite from here. Here the surface potential goes negative, that's why the holes accumulated at the surface. But here the surface potential is positive, and this is why electrons start accumulating at the surface. You will see below that an inversal, here, the bands bent even more. And now, locally here. You have a level above the intrinsic level, which is what you would expect in a semi-conductor. Although in the bulk, things look like an semi-conductor. So here, it looks like n, but in the bulk, it's d. This is again why we call this inversion. In this video we have discussed the various regions of operation at the surface of the semi conductor. We have talked about accumulation, depletion, and inversion. In the next video I will show you a general analysis that can handle all of these cases using a single equation.
