In this video we will continue with the subject of parameter extraction for compact models. I will now give you a few parameter extraction examples. First of all, oxide thickness. This is measured through special capacitance structures of the same waver as your transistors, which you measure to determine the capacitance, and from that you infer the outside thickness for a known permittivity. Then you can perform a threshold measurement. Let us take the example of the strong inversion model. This is our simplified source reference strong inversion model. And I've extracted VDS as a common factor for in order to explain this better, but it's really the quadratic model we've been using and here the denominator is to take care of the dependence of mobility on the vertical field. Now at very low values of VGS, this term here is very small and can be neglected. And I will also assume V is P equals zero, so we can neglect this term too. Now EVG is a small, saturation is achieved at low values of EDS, so we have to be careful in order to remain in nonsaturation, we need to maintain low vds as well. So, as I said before, the denominator is approximately one, so the whole equation reduces to this one. So now you can see that the current for a low value of vds, for which you will be doing this procedure, when VGS is equal to VT plus 1/2 of VDS, the current is predicted to be zero from the strong inversional model. If you now have the measurements, as shown here, of course the measurements will deviate at very low current values, because you're not in strong inversion anymore, and the model is not valid. Also you will have deviation over here, because of mobility dependence on the gate voltage, but for low VGS values, you can assume that this equation is valid. And as I mentioned already, when Vgs is equal to VT plus half alpha Vds, in other words here, the current is predicted to be zero, so you take two points where you have the maximum slope here, and you extrapolate to this point and that gives you this quantity. Now you know you Vds, for which you're doing this, you're performing this procedure. Approximately you know your alpha, so to very good accuracy, you know your threshold. So that's then the way to measure the extrapolated threshold voltage. I want to emphasize that we do all this at the maximum slope, and because the maximum slope here, approximately this is the constant of proportionality and we already have the oxide capacitance from the oxide thickness measurement here, and we know our W over L at least approximately we can find mu zero. The constant mu zero. Now from the deviation of this, from the straight line, we know that this is because of this denominator, we can estimate theta, so we can extract theta from this behavior over here. Now let us say we repeat this measurement at other values of VSB, so we find the threshold for different values of VSP. That, of course, will give us information about the body effect. This is the equation we have been using for the dependence of threshold on VSP. This is the so called body effect equation. It has some parameters like the flat pan voltage. It has free zero which is the pin surface in the strong inversion, and it has gamma the body effect coefficient. Now if we already have Vt for different VSP values, you can fit them. You can fit this equation to those values. And after you finish your fitting, you can determine the parameters in this equation. For example, if you plugged Vt as a function of square root of Vfb plus V0 for an approximate value of V0, that should be a straight line, right, with respect to this whole quantity. It should be a straight line. If it is not, you use a different value for zero, and you try again. Of course, all of this is done automatically by optimization software, but you can do it by hand also. So once you have finished and you have fit this one satisfactorily to the measurements of VT versus Vsb, you have determined the values for phi zero, Vfb, gamma, and Na. So notice that we started with oxide thickness measurement, then we proceeded to measure the threshold and from that we got they squandered this. So you continue in this way, you find an appropriate region and the IV characteristics from which we extract parameters. Continue extracting parameters, and in the end, as I mentioned in the previous video, you do a global optimization to fine tune the parameter values. This is the same plot of I versus VGS, which deviates from straight-line at large values of Vgs because of mobility dependence. We said we take a straight line, we pass it through two points of maximum slope. Now the slope of this current versus voltage plot is called the transconductance. The first transconductance it's a small signal parameter is defined as the derivative of ID with respect to VGS, assuming that the other bios voltages are held constant. We will have a lot to say about transconductance once we talk about small signal parameters. Now this is the current equation we have been using for this procedure, so if you differentiate to get the transconductance, after some algebra you find this expression and you can plot it and you expect it to be maximum at some point here. Gm maximum is the value it attains and it corresponds to the slope of the ID versus VGS plot over here. So we will be using Gm max to illustrate how you extract a couple of other parameters now. I will show you now how to extract the parameter delta W. Let me remind you what that is. When you design a device on your computer screen you specify a mask, so called mask value for W. This is what your device looks like on the computer screen. It looks like it has a gate width of w sub m. But because of fabrication details, the final value of w that the real device ends up with differs from wm by the quantity delta w. We need to know how much delta w is in order to find this quantity and use the real W for the device in the model. How can we extract the value of delta W? Now you will recall from the previous slide that gm, the transconductance, was proportional to W over L. I will rewrite the expression for gm slightly. Here is your W, and here is your L. I wrote it in this way, so that you can see that. The entire Gm is proportional to Wm minus delta W. So that means that gm should be a straight line with Wm. And in fact, if you extrapolate, you can find the value of delta W. That would be the extrapolated value that makes this whole thing equal to zero and therefore gm equal to zero. So you plot the maximum Gm for a few different devices of different masks Ws. So this device has a large Wm, this device has a smaller Wm, and so on. You take these dots, each of which gives you Gm as a function of Wm, and you say according to this equation, you're expecting to lie on a straight line. So you pass your best fit of a straight line here, and then you extrapolate and you know that according to this equation the extrapolation is your delta w. So now you have determined the extrapolated delta w that you can use in here and find the real w for your devices. You can do something very similar to find the corresponding correction in the channel length but instead of Gm, I'm going to use one over Gm. So to remind you the real length of a device is the mask length minus the correction delta L that depends on fabrication details. One over gm is proportional to L over w. Here is my L, and here is my W. From this I know that if I plug one over Gm, versus Lm, the mask length, I should get a straight line. So I do this for a few different divisors of mask lengths, different values of the length. I extrapolate and here I get my value of delta L, according to this. So there is a significant number of such procedures that with experience people have developed and they can extract the parameters that go into a model. I think from these examples you get the picture, I cannot afford to spend too much time, there's more information in the book about this procedures. But let me talk about something else that is related. First of all when you have a plot, it can be misleading. Let us take an example of drain current versus VDS, it looks perfect, right? The model, the solid line passes between, passes through the dots, which are the measurements. But let us now take the slope of these lines. Now, the slope of these lines, theta ID Theta Vds goes like this, the slope is large over here and becomes close to zero eventually, so slope goes like that. And for different Vgs's you get the different slope curve line like this. We consider even the slope is predictable, at least for the range of values we loaded here, but It could be about for very small values of the slope over here, we do not have enough resolution to see how good a job we're doing with this slope. So rather than plotting this slope on the linear axis, let's plot it on the logarithmic axis. Now the slop looks like this. Let's take this one. This slope passes through the dots pretty well. So this model obviously does a good job with the slope. This is for a different VGS value, a different VGS value. They all look good, but let's just take one of these points, this one. This seems to be very close to the curve, right, so you would think that the model does a very good job predicting the value of the slope, which is represented as a measurement by the thick dot over here. But what is the error that the model makes? The eye is misled to think that because the dot is close to the curve, horizontally, this is a good match. But the error is the difference in the value of the slope. So it is the vertical distance between the dot and the line, in that distance is significant. It could be a factor, I don't know, let's say three or four. So plots can be misleading. You really have to calculate the error, the mean square error, we covered that in the previous video, and only then can you judge. And related to this, you have to be aware of the fact that the weak inversion, where, let's say, a model, the delta line and the measurements are solid lines seem to be almost the same. Again, the vertical distance on the log access can be a significant factor. So your model could be off by 100% and still these things seem to be matching pretty good to the eye. So don't get fooled by semi-log plots of this kind. As an example, I mentioned here that 15 milivolts of error in the flat line voltage can result in a corresponding shift of 15 mV here, that can result in a 50% error in the drain current. These are subtle details that people who do parameter extraction know very well. Binning, what is binning? Ideally, we would like a model to give you accurate predictions for one set of parameter values in it. Unfortunately, this is very difficult to achieve, so what some of those parameter extraction procedures do is they divide the entire range of W's and L's into bins. For example, if that bridge between w1 and w2. And L is written L1 and L2. In other words, we are some worry in this shaded region. Then we use one set of parameters for our model, and if we are in another bin, let's say here, we use a different set of parameters for the model. This unfortunately has to be done because the model is not perfect. Now in addition because fabrication proceeds rather rapidly, and there is not enough time for modelling people to catch up with the models, they have been forced to also use empirical expressions for some of the parameter dependencies on Ws and Ls. And such expressions take usually this form. A parameter has a nominal value plus some dependence on W, some dependence on L, and some dependence on the product of W and L, which is a term used mostly when there is an interaction between short channel effects, and narrow channel effects. Now you can see here, if I have large device, with large W and large L, all of these go practically to zero, and P is equal to zero. But as W becomes small and L becomes small, they start becoming important. And unfortunately, you need to use different quantities, P0, PW, PL, and PWL, for the different bins, and this practice, as I mentioned, is forced by the fact that the model is not perfect. One problem with this binning, is at the boundaries between the bins. Here is an example of VT0 versus length. Because of short channel effects, which we have already covered, we expect that VT0 will go up, and it does. But the weight goes up because different values, different sets of parameter values are used for L larger than 1.5 micrometers. And different sets of parameter values is used for L between .5 and 1.5, you get a rather unnatural change as you can see the slope of this solid line is not continuous at the boundaries between bins, this is called cusping. So such effects are bound to happen unless you can come up with a model that with one set of parameters, it models everything for all Ls, all Ws, all bias voltage, and that is not an easy thing to do. With these few examples, I think I gave you a brief idea of what the parameter exaction procedure is. In the next video, I will present a few representative compact models that are widely used.