nanoHUB-U Fundamentals of Nanotransistors/Lecture 2.5: Mobile Charge: Bulk MOS
========================================
 >> 

[Slide 1] Hello everyone. This is Lecture 5 in Unit 2. This is the first of two lectures on the mobile charge in an MOS capacitor. The mobile charge is the charge that's going to carry the current from the source to drain. So it's important that we understand this charge which occurs once we bend the bands past the onset of inversion. 

[Slide 2] So you remember that we have sketched qualitatively, the charge versus surface potential in an MOS capacitor. And it was easy to understand the pile-up from majority carriers and accumulation, the depletion of majority carriers in the depletion region, and then the pile-up of mobile inversion layer electrons in inversion. So we have some quantitative understanding of the depletion charge. I simply written down that the inversion charge goes exponentially. And we want to talk a little more quantitatively about what happens in this inversion region when the bands have been bent past this critical point two psi B. And then we also want to relate this mobile charge. You know this is the charge we're focusing on now. We want to relate that mobile charge not just to the surface potential, but to the gate voltage that we can actually control and use to manipulate the surface potential. So that's what we're after. There's a little bit of heavy lifting in this lecture. This is a problem that can be solved analytically. And sometimes you need to use some numerical techniques as well. If you've had an advance MOS course, you might have done that. I am going to try to establish the important relation and do it as simply as possible. But no rocket science, but there will be a little bit of derivations and equations that we'll have to step to carefully. 

[Slide 3] So, before we dive into that, just to remind you that, you know, the current through a MOSFET is charge times velocity. We've been discussing the total charge in a semiconductor and the depletion charge but those charges are not the important ones to us. In this expression, it's the mobile charge in the inversion layer mobile electrons in an n channel MOSFET, mobile holes in a p channel MOSFET. We want to understand how those mobile charges vary with surface potential and how those mobile charges vary with gate voltage. 

[Slide 4] All right. So let's look at that. First of all, and let's ask ourselves about the mobile charge volume density chart, the charge per cubic centimeter in the semiconductor. So the way I would get that is if I could-- if I knew the density of electrons per cubic centimeter at any depth in the semiconductor and that's n of y. I take minus q because they have a negative charge and then I integrate through the depth to the semiconductor and get all of the charge in coulombs per square centimeter. That's what I'm after. So first of all, let's ask ourselves, what is the volume electron density at some arbitrary location y? So we have this simple relation that gives us the electron density in terms from the effective density states, the material parameter that we know. And then if we assume  non-degenerate statistics, it's exponentially related to the separation between the fermi level and the bottom of the conduction band. But the bottom of the conduction band varies with position because the electrostatic potential varies with position so the electron density varies with position. You can see it's higher here than it is here because the bands are closer to the fermi level. The conduction band is closer to the fermi level at the surface than it is in the bulk. OK, so the conduction band is bending because of the surface potential. So we can re-express this relation in terms of the potential in the semiconductor. And I can simply use this expression to write the second equation in this manner. And I'll recognize this first term as the electron concentration in the bulk. So it's the electron concentration in the bulk times e the potential over kT. So anywhere in the semiconductor, the bands bend down by the-- according to the potential at that location and the electron density increases exponentially with that increase in potential. So I can write this as bulk concentration, times e to the potential over kt. And remember the bulk concentration in a p-type semiconductor, we have np is equal to ni squared. So the bulk electron density is just ni squared divided by the p-type doping density and the bulk. 

[Slide 5] OK. Now let's see if we can evaluate that expression. We want to find out what the total charge per square centimeter is. So we'll do an integral with the expression that we just developed. And now, I'll do a change of variables so that I can introduce-- I can innovate with respect to potential rather than with respect to distance because that's an integral that I can do. I can do that as long as I recognize that the gradient of potential is the electric field. So, dy dpsi is actually one over the electric field, you know, actually minus, I need a minus sign there but I've got a minus sign out front. So, this is the electric field. Everything is going to happen very near the surface. All of the electrons pile up very close to the surface. So, I'm going to make an approximation, I'm going to assume that where everything is happening, the electric field is approximately constant at its value at the surface. And then I'm going to pull that electric field out of the integral and bring it out front here, E sub s. That leaves me with an integral that's easy to do, you'll work it out. This is what you'll find, we'll find that the charge density per square centimeter is minus q times the density in the bulk ni squared over doping, times e to the surface potential over kt, times something with the units of thickness. We interpret that as the thickness of this thin layer of electrons near the surface. And it's kT over q in volts divided by the electric field at the surface in volts per centimeter. All right, so we can simply then write it as q times the charge at the surface times the thickness of this layer of silicon. Thickness of the inversion layer, although there may not be very many electrons there in that inversion layer if we're below threshold, but they're still there. 

[Slide 6] OK. So, we have an expression now for the mobile charge density as a function of surface potential that is really valid above or below threshold. And it wasn't too hard to derive. And it's actually pretty accurate too. 

[Slide 7] So, let's look at this below threshold, you know, something is going-- we're going to have to be careful with something below threshold. And what we're going to have to be careful with is the electric field. Below threshold, we have an expression for the electric field in terms of the potential. We can simply use that expression in the first equation and now we have an expression that gives us the mobile charge density in terms of the electrostatic potential in the subthreshold regime because I'm using the electric field that we got from the depletion approximation which is only valid in subthreshold. So, I have some pre-exponential factors that vary slowly as 1 over the square root of surface potential. But what I really focus on is this exponential dependence of the inversion layer charge or the mobile electron charge with the surface potential. So, the important take away is that the mobile charge density increases exponentially with surface potential. 

[Slide 8] OK. Now, we're going to go above threshold, I said the same expression holds above threshold, well, not quite. We have to be a little bit careful. Now, above threshold, everything-- all of these electrons are piling up in very high densities right near the surface, right? All of the action is right near the surface because all of the charge is right near the surface. The electric field is going to vary rapidly right near the surface. It's not a very good approximation to assume that the electric field is constant at it's value that we got from the depletion approximation. So, what I'm going to do is I'm going to use-- I put a bar over that electric field. I'm going to use the average value of the electric field in that thin layer. So, it's going to be very strong at the interface at the surface of the silicon. And it's going to drop very rapidly and be maybe almost zero by comparison at the end of that layer. So, I'm going to insert a factor 2 here, and I'm going to interpret that as the average electric field inside this thin layer. OK. Now, if I do that then I can use this Gauss's Law which relates to charge density to the electric field in the first expression. And then I'll have a charge density down stairs and a charge density over here. I'll get a quadratic equation for the mobile charge density, I'll take the square root of both sides and I'll get this expression. And now I get, you know, I've waved my hands a little bit because I'm avoiding the full exact solution, but we've ended up getting to correct the answer when we're in the strong inversion layer regime and there's a high density of electrons in that layer. And now you can see because we took the square root, we get a 2kT downstairs. So, the take away message is that, above threshold the inversion layer charge varies exponentially with surface potential, but it's exponentially with surface potential divided by 2kT 

[Slide 9] over q. All right, so if we go back, remember when we sketched this qualitatively. We argued that mobile-- that the depletion charge when it's a square root of potential here, the mobile charge went as the square root of-- as e to the q surface potential over 2kT. So, now we see, we see where that 2 comes from, we see that in subthreshold, the 2 isn't there and we've got our hands on expressions that work reasonably well. You know, even below threshold, when there aren't very many electrons in that thin layer, there's a few there and those few contribute to the leakage current of the MOSFET. So, we're going to worry about those few. Above threshold, the charge is dominated by the numbers of electrons in that inversion layer. In both cases, it goes exponentially. 

[Slide 10] All right, now we want to relate these charges above threshold and below threshold to the gate voltage, because the gate voltage is what we can control. Let's do subthreshold first. And this is where our simple way of relating gate voltage to the surface potential that we discussed earlier. This simple two capacitor voltage divider network, this is where it's going to prove to be useful. So, we saw that we could write the surface potential as some fraction of the gate voltage and the fraction was determined by this parameter m, which involve the ratio of these two capacitors. We're going to just, going to take some average value for the depletion capacitance in this subthreshold regime. So, if we do that, we can take our expression for-- the mobile charge in subthreshold, and we can use these relations and we can write that in terms of gate voltage. But we have to do a little bit of work here with the electric field because that's a little bit harder to that. 

[Slide 11] And let's go through that argument carefully, just so you can see how it's done and how people get the standard expressions. So, we've got an electric field. Remember, the electric field was given by this Gauss's Law as charge divided by dielectric constant. OK. I can recognize dielectric constant divided by depletion layer thickness as the depletion layer capacitance, and then I had qNa upstairs. OK. So, I can-- Now I can recognize that my depletion layer capacitance is related to this parameter m, which turns out what we'll see later is something that we can get a handle on experimentally, easily. So, I can solve for that depletion layer capacitance in terms of this parameter m and the oxide capacitance which presumably we know. And that gives me an expression for the electric field in terms of parameters that I'm likely to know. 

[Slide 12] So, that's the expression that we're going to want to use. So, if we go back to our general expression for mobile charge in the subthreshold regime, and if we use this expression that we have just developed for the electric field. And if we use this expression where the gate voltage in term of-- for the surface potential in terms of the gate voltage here, then we have an expression for the mobile charge that is simply in terms of the gate voltage and parameters that we'll see-- we tend to know for our MOSFET. Now, one more thing I want to go through here quickly. This isn't quite where people stop. What people recognize is that this ni over Na, I've seen that somewhere before, you know, where have I seen that before? Well, where we've seen that is in this critical potential psi B, this involves a logarithm of 1 over that ratio, NA over ni. Now, you'll also remember that if I bend the bands 2 psi B, that voltage it takes to bend the bands 2 psi b is my threshold voltage. So, I can use these two relations to solve for ni squared over Na. And I'll find it's e the minus q threshold voltage over mkT. That's nice because threshold voltage is something that I can determine in a device. I can use that expression back in here. And when I do that, we get the classic expression 

[Slide 13] that you'll find in every  MOS textbook for the mobile charge below threshold. And this is it, so it took little bit of work, you know, go through it carefully, run through it again after you listen to this lecture. But you'll see that there's nothing too difficult there it's just a series of steps that we go through. Now, the important take away message here is that, this mobile charge increases in maybe small it subthreshold, but it increases exponentially as e to the q, gate voltage minus threshold voltage over mkT. That's the important part about this derivation. Below threshold, that inversion layer charge increases exponentially. That's the leakage current of the MOSFET, and that's a current 

[Slide 14] that we worry about a lot. Now, we also want to know when we turn the device on. How is that mobile charge related to the gate voltage? So, let's look at that again, this is our relation between gate voltage and surface potential and charge-- total charge. Total charge in the semiconductor. The total charge in the semiconductor is part depletion charge, and it is part mobile charge. Above threshold, we'll have a lot of mobile charge. So, if we look right at the onset, right at threshold the only charge in the semiconductor is depletion charge. We're just beginning to build up significant mobile charge. We can ignore it right at this point. So, we get this relation at threshold. Beyond threshold, we bend the bands just a little more, remember it's hard to bend because the semiconductor is becoming like a metal it's getting harder to bend the bands. We bend the bands just a little bit. We really don't change the depletion charge by very much because it varies slowly with surface potential, that's the square root of surface potential. We bent the bands a little more, so we're a little bit higher than 2 psi b but not very much, a negligible amount. We just bent it a little bit. But we've seen that the mobile charge increases exponentially with surface potential. So, just that little bit of increase in surface potential beyond 2 psi B gives us a lot more mobile charge. So, we begin to get a lot of mobile charge. This term is close to this term, this last term is close to this last term. If I subtract the two, VG minus the VT, all I'm left with is minus mobile charge divided by oxide capacitance. So, isn't that surprising? That's remarkably simple. This is the expression in unit 1, I just sort of postulated and said this is reasonable charge is capacitance times voltage, but we have to subtract out this threshold voltage because we're only interested in a certain part of the charge, the mobile charge. So, we've seen where it comes from now. The mobile charge is approximately minus the oxide capacitance, times VG minus VT. And below threshold, the mobile charge is approximately zero but maybe not negligible  these days. All right, the reason that we have an approximate sign here was because of these approximations that I was going through. So, it's not quite exact. 

[Slide 15] In fact, you know, it's something that people worry about a little bit here. And we have to talk about that. You know, this is really where we've been heading. We have an expression for the mobile charge in terms of gate voltage in subthreshold and above threshold. This one, we have to talk about a little more because it's not quite right in a way that is important these days to understand, OK? 

[Slide 16] So, I want to look at this gate capacitance a little bit more. Below threshold, we have an oxide, we have an oxide capacitance, we have a depletion layer and we have a semiconductor capacitance. We have a series combination of the two. The semiconductor capacitance is the depletion capacitance. It's just dielectric constant of the semiconductor divided by the thickness of the depletion layer. Now, I also want to point out that this depletion capacitance, this is the capacitance of the semiconductor. Capacitance is charge divided by voltage. If it's a non-linear capacitance is-- derivative of charge divided by voltage. In the depletion region, the charge is negative when the surface potential is positive. So, to get a positive capacitance I have to use a negative sign here. And it's just depletion charge. If you take our expression for depletion charge, do this differentiation, you'll just get epsilon divided by thickness of the depletion layer. OK. Now, let's look beyond threshold. We've applied a gate voltage higher than the threshold voltage. We have a very strong inversion layer. The metal-- The semiconductor is becoming almost metallic, there's lots of electrons there but it's not quite metallic. If it were an ideal metal, I would just have two metal plates an insulator between them, then all I would have is the oxide capacitance. That's what we said earlier. But, this is not an ideal metal. There is still some capacitance of the semi conductor. That capacitance, we're going to call that-- The capacitance of the semi conductor in depletion, is the depletion capacitance. We're going to call the capacitance of that semi conductor in inversion, CS inversion, it's the dielectric constant of the semi conductor, divided by the thickness of the inversion layer. The inversion layer is very thin. So, this is a very large capacitor. Again, if you want to know what is this capacitance it's just the derivative of the total charge in the semiconductor, divided by the surface potential. In inversion, most of that charge is mobile electron charge. And it's increasing exponentially. So, when we do this derivative we get a very large capacitance. 

[Slide 17] OK, so the bottom line then is that, we can use our expression for sub threshold charge. But, if we go above threshold, instead of using the oxide capacitance here, we have to replace it by a capacitance that is a little bit less. It is the series combination of the oxide capacitance, and the semi conductor capacitance, in inversion. And, it is less than the oxide capacitance enough so that people worry about it, these days. 20 years ago, we didn't worry about it. But, today, we have to. 

[Slide 18] OK. Let's do an example, you know, how big is this semiconductor capacitance. Under typical on current conditions, we might have 10 to the 13th electrons per square centimeter in the channel of the MOSFET. So, we can compute the charge by multiplying by q, negative sign because it's a negative charge. Let's assume that I have 1 nanometer thick oxide, all right. So, that's a reasonably thin oxide of the kind-- close to the kind that people might be using these days. So, we can compute the oxide capacitance, right. I can compute the semiconductor capacitance in inversion. Its just dq dv. q is the mobile charge, it goes as E to the q psi over 2kT. If I do that derivative, it's just minus mobile charge divided by 2kT over q. You put in numbers, it is almost 10 times bigger than the oxide capacitance. So, when I ask for the total capacitance on the gate, I have to add the two inverses. And when you do that, we're going to find that the total capacitance is about 90% of the oxide capacitance. It's a little bit less than the oxide capacitance. Actually, it's even worse than this, because when you throw in some things that were leaving out, fermi dirac statistics, quantum confinement, they all tend to lower the semiconductor capacitance and make the capacitance even lower. So, the fact that this capacitance is lower, then-- you would like it to be, is an issue that the device designers have to struggle with these days. 

[Slide 10] Now, because it's a little bit complicated to actually properly compute this inversion layer capacitance, you will often find people when they do write papers that they will quote something called the capacitance equivalent thickness. So, if you quoted a capacitance equivalent thickness, what it means is, take the dielectric constant of oxide, divide it by the capacitance equivalent thickness. That will give you the correct total capacitance on the gate, including all of these other affects that I've discussed and some of them that I haven't discussed but it will give you the proper total gate capacitance. 

[Slide 20] Now, just briefly, I want to go back to this question about, why is it so hard to bend the bands more than 2 psi B. You know, we've been arguing that no matter how large you make the gate voltage, you can't bend the bands more than that. So, one way to think about that is that, you know, we have this voltage divider that we think about and this parameter m that this-- that determines the fraction of the gate voltage that gets in to the surface potential. And, below threshold, the second capacitor is the depletion capacitance. This ratio is greater than 1, m is greater than 1. But the numbers worked out so it isn't-- m is not much bigger than 1, 1.2 or 1.3. So, what it means is that most of the gate voltage gets in to the semiconductor and is reflected in surface potential when I'm below threshold. But above threshold, the semiconductor capacitance in inversion can be very large. What that means is that this perimeter m is very big. What that means is that only a tiny fraction of the gate voltage gets in and bends the band. What's happening is that most of the additional charge that we're putting on the gate metal is not getting into the semiconductor. It's being imaged by this mobile charge that's piling up, and it doesn't get in to the semiconductor and is not able to significantly bend the bands very much. So, the bands bend a little bit, but they don't bend a great deal beyond 2 psi B. 

[Slide 21] So, what we've discussed in this lecture is this mobile charge versus gate voltage in a bulk MOS capacitor. We developed an expression. Wasn't too hard, took a little bit of effort. But the bottom line here, the takeaway point that you want to remember-- the takeaway point that you want to remember  is that below threshold, the mobile charge varies exponentially with gate voltage, it goes as e to the gate voltage minus threshold voltage over mkT. There's that parameter m again that keeps coming up and that's so important. Above threshold, things are even simpler. Above threshold, the charge varies linearly with gate voltage. It goes as VG minus VT. The capacitance out front is approximately roughly that oxide capacitance. It's a bit less and we call it the gate capacitance under inversion conditions. Now, you might ask, isn't there an expression that goes smoothly from subthreshold to above threshold. Well, you can do that numerically, and people are doing that frequently these days. You can also do it-- empirically. You can define an empirical function that stitches these two expressions together so that you have a smooth transition from below to above threshold. We'll see that that's useful and can be done when we revisit the virtual source model during the last lecture of this unit. 

[Slide 22] OK, so, for bulk MOSFETs, we now have an understanding of how the mobile charge varies with surface potential and with gate voltage. What I want to do in the next lecture is to talk about the same topic, mobile charge versus gate and surface potential, but do it in the context of some more modern structures. So, these days the industry is moving towards structures like, extremely thin silicon insulator, or FinFET which is sort of a vertical silicon on insulator. The structures are significantly different from the bulk structure that we've been discussing in this lecture. We need to discuss how the inversion charge versus these two voltages works out under these structures, and we will see that the bottom line is very similar to the results that we've obtained for the bulk MOS structure. So, we'll continue this discussion in a next lecture, and I'll see you there.