 In the previous lecture, we have derived a simple formula for the drain to source current in weak inversion and we have shown that this current is due to diffusion. The formula that we derived was as follows, Ids is given by W by L into average electron mobility at the surface multiplied by thermal voltage square into the depletion capacitance multiplied by exponential of VGS minus V dash T by NVT into 1 minus exponential of minus VDS by VT where the average surface mobility of electrons was given by this formula, okay. Let me show the structure and biasing arrangement so that we get the meanings of all the terms here, source, drain, this is the gate, you are applying a drain to bulk voltage, source to bulk voltage, gate to bulk voltage, this is the bulk which is grounded. In weak inversion regime, your depletion width is constant from source to drain. So, this term CD is the depletion capacitance where the potential drop is psi s naught and depletion width is XD. So, this CD is equal to epsilon s by XD and this is the potential psi s naught, okay, here across this. Now, the voltage VGS is the difference between VGB and VSB. The voltage VDS is the difference between VDB and VSB, okay and this is your inversion layer which is very weak, it is concentration of electrons inversion varies from source to drain, okay and the mobility of these electrons is this average surface, mobility of these electrons here is this average surface mobility, okay. In this lecture now let us derive the model for strong inversion that is VGB greater than VHB, okay. So, this is your IDS versus VGB in a semi-lock plot and this is VHB, okay. So, we are talking about this is the strong inversion region that is where we want to derive this current, this is on a lock scale. So, the step one of this derivation is as follows, approximate the surface potential base 3 by 2 power law model, given here as a surface potential based square law model. So, you are going to approximate the 3 by 2 power law model by a square law model. This is the milestone we want to achieve, we will discuss how to get this. What you find here is that this VFB has been replaced by VTB, this term remains as it is and these 2 terms have been reduced to this square law term and in addition of course some thing from here has gone into this brackets and therefore this VFB has become VTB. In step 2, so we are first laying out the milestones we will then do the detailed derivation. What we will do is we will replace psi S0 by the saturation value VSB plus phi T of the surface potential where phi T is twice 5 plus 6 VT and we will replace psi SL by VDB plus phi T for VDB less than VDB sat and by VDB sat plus phi T for VDB greater than VDB sat. So, that would be our square law model. Let us see how do we go about doing it. Now, the origin of the 3 by 2 power terms in this expression, this is the basic expression for the drift current in the surface potential based model. It is a square root term gamma square root psi S in this expression for QI. So, this formula, this inversion charge here has the following formula in the surface potential based model where you have a gamma square root psi S term. When you integrate this term in this expression, then you get the 3 by 2 power term. Hence, to get the square law model approximate the square root term by a linear function. This is our methodology. Let us look at that linear approximation. So, here we have sketched gamma into square root psi S as a function of psi S. The shape is nonlinear as shown here. Now, what is interesting however to note that if I take a segment of this nonlinear function, I could approximate this by a straight line. Now, in practice in a MOSFET, what would be the segment of interest? In MOSFET, at the source your surface potential is psi S0, at the drain your surface potential is psi SL. So, this is the regime of interest. Corresponding to psi S0, I have gamma square root psi S as gamma square root psi S0. Corresponding psi SL, I have gamma square root psi S as gamma square root psi SL. So, let us see what is the linear approximation we can do in this regime. So, the linear approximation we can write as follows. Gamma square root psi S minus gamma square root psi S0 by psi S minus psi S0 is given by this formula. Let us explain this formula. So, you take any point, any general point here. Then this is gamma square root psi S. Now, compare the value with this value here. This is gamma square root psi S0. So, we take this difference, that is this difference and divide it by psi S minus psi S0. This is psi S. So, psi S minus psi S0 is this difference. So, ratio. Now, this we can write as alpha minus 1. Like we defined a slope n or rather 1 by n for the psi S versus VGB plot in the linear portion. We are now defining a slope for gamma square root psi S versus psi S plot in the approximation of the linear function. So, this slope is same as the slope obtained using this triangle, using similar triangles. That is what is written here. Gamma square root psi S L minus gamma square root psi S0. That is this difference and denominator psi S L minus psi S0. That is this difference. So, we are defining it as alpha minus 1. Now, why do we call it alpha minus 1 and not alpha? We will find that if you call it alpha minus 1, the final expression will turn out to be simple. So, we can use this and write gamma square root psi S as shown here. So, you take this part equate it to alpha minus 1. What you get is this. Now, why have we given this? So, this tells you how you can actually evaluate alpha minus 1, okay, based on values of psi S L psi S0 and so on. So, that part we will see later. Now, if you substitute this expression for gamma square root psi S in the Q i as a function of psi S expression in the surface potential based model, that is this expression, you are replacing this gamma square root psi S by this entire quantity, okay. Then you will end up getting Q i psi S as minus of C ox into V GB minus V TB minus alpha into psi S minus psi S0. Let us see how we get that. So, this part gamma square root psi S is gamma square root psi S0. Now, what we will do is we will expand this alpha minus 1 into psi S minus psi S0. So, alpha minus 1 can be written as alpha into psi S minus alpha into psi S0 minus psi S plus psi S0, okay. Now, you have a minus psi S term here. So, when I subtract this quantity, okay, from this, so this minus psi S term and this minus psi S term will cancel. So, this goes off. Alpha psi S term will however remain and that is why you are getting this alpha psi S here. Similarly, alpha psi S0 term will also remain and that is what you are getting here. What you are left with is psi S0 and this gamma square root psi S0 here, okay. You recognize that V FB plus psi S0 plus gamma square root psi S0 is nothing but this threshold voltage with respect to bulk, okay, V TB. So, that is how you are getting this expression. V TB is V FB plus phi T plus V SB plus gamma into square root phi T plus V SB. So, this is actually the so called psi S0. This is also the psi S0. So, let us put that expression here and create some space. You have to substitute this QI into this expression here and do the integration. Clearly now, we have only linear terms in psi S. So, it is a straightforward integration and you will end up getting V GB minus V TB into psi S L minus psi S0 because this is a constant. When you integrate with respect to psi S, you will get a psi S term and you take the limits, you will get this. Then take this alpha into psi S term, you will get a psi square by 2. When you take the limits, you will end up getting minus alpha by 2 into psi S L minus psi S0 whole square, okay. This is also straightforward, just integration of this. Let us put that expression here on the top and create some more space for further manipulation. As we have said in step 2, we replace psi S0 by its saturation value, V SB plus phi T. Psi S L by its saturation value is given here. I want to remind you about those saturation values. So, this is psi S versus V, this is saturation value. When I put V DB here, I get psi S L. Now, the exact saturation value do not know but very close to that, this value is actually V DB plus phi T. And if I replace this V DB by V SB, all that I have to do is replace this by V SB and this will be psi S0, okay. So, similarly you can talk about the other saturation values. Now, so when you do that, you are going to end, when you do that, you are going to end up with this expression for the square law model. What has happened is the V DB minus V SB has been put here as V DS, okay. And this term here is same as this term. Now, for V DS greater than V DS sat, you have to use the same expression. All that you do is replace V DS by V DS sat. That is what is written here. Now, let us create some further space so that we can manipulate this. What about the average value of electron surface mobility? That is given by this formula, which we have derived already in the context of surface potential based model. The one thing that you should do as a part of assignment is that show that this EY effective 0 for N plus polysilicon gate, okay, can be simplified to V GS plus V T by 6 T ox. So, this quantity if you replace by various expressions terms and then simplify, you can show that the inversion charge and depletion charge you put in terms of the biases. It will reduce to the simple formula. To give you some support or help, you should use this fact that for N plus poly gate, flat band voltage plus 5 T is much less than V T, okay. Flat band voltage plus 5 T is much less than V T. So, when you replace this by expression in terms of bias, you will get this term V F B plus 5 T and you will also have V T. So, you will neglect this term compared to V T. Now, why is V F B plus 5 T much less than V T? You will have to draw the energy band diagram, okay. There for N plus poly and P substrate. So, if this is your vacuum level, this is your gate and this is your substrate, P type substrate. So, your Fermi level in the P type substrate will be here and N plus poly gate, your Fermi level coincide with EC, sorry E F is equal to EC. This is your EC, this is EV and you will have to use this fact. So, this is your phi M S. This is the intrinsic level and this is phi F. So, you will have to use this band diagram, okay and then you can write the expression for V F B. So, your V F B is equal to phi M S minus Q F minus Q F by C ox. Now, in most MOSFETs, modern MOSFETs, Q F does not matter. So, you can take V F B as approximately phi M S. So, this is your phi M S and you will have to then show that if you take phi T, phi T is twice phi F plus 6 V T. So, these two quantities will be of opposite polarity. V F B is this because you are neglecting the Q F but it has a negative polarity because Fermi level in the metal is or poly is above the Fermi level in the semiconductor. This is your phi M, Q times phi M and this is your Q times phi S. So, phi M S is negative whereas phi T will be positive and almost of the same value as V F B. So, that is how this whole quantity will be much less than V T and you will also need to show that the V DS sat that is being used here. It will be obtained using this formula. It is a value of V DS when D I DS by D V DS goes to 0. So, differentiate this with respect to V DS and when you do that please remember that a V DS term comes in the average mobility expression also. So, not only that you have V DS here you will also have V DS in the denominator of this particular expression that is this. So, when you differentiate you take that into account. This is also an assignment you have to show this expression for V DS sat. Now, let us summarize the threshold voltage based model both for the diffusion component as well as the drift component. So, the diffusion component of the threshold based model is given by this formula where this V dash T is threshold voltage based on the twice phi F criterion. The drift model is however valid for V G S more than V H. The diffusion expression is valid for V G S less than V dash T. Now, note that V dash T and V H are different values ok. They are not the same values. So, there is a regime between V dash T and V H where you do not have any model here really ok. Now, I DS for drift is given by this formula which we just derived the square law formula. So, we have put that in the slide that this model leaves a gap in the moderate inversion range between V dash T and V H. Now, although the expression for V G S greater than V H works right down to V G S equal to V T its estimates are accurate only for V G S greater than V H. Now, this is important because you find that there is a V T here which is different from V dash T ok. So, we have said this expression is used for V G S greater than V H ok. Somebody may say, but here you have V T. So, actually you can use this V G S until V G S equal to V T and V T is less than V H. Let me show you those values or those partitions this V G S, this is V H, this is V dash T and this is V T ok. This is the strong inversion ok. This is weak inversion and this is moderate inversion. So, your V T lies here. So, what we are saying is that when you take your ID versus V G S graph. So, it goes like this and it goes like that. Now, when you take the square law segment here it will go like this. This we showed with a dash line. So, what we are saying here? What this statement is saying is this dash line exists for V G S greater than V T here. So, it will give you some value. This expression will give you some value even if you go below V H, but this current will be inaccurate. This is the current you will get instead of this current. To complete the model the V D S sat is given by this formula where this EC is the critical electric field for velocity saturation. For diffusion current here the average electron mobility for the surface is given by this formula and for the drift current it is given by this formula ok. So, this L into EC is same as this. Now, before proceeding further though we have not derived this expression for V D S sat let me give you a feel for this expression ok, because it is somewhat lengthy expression and it is very different from what you would have used in a first level course with the square law model. So, you might recall that you use the saturation drain to source voltage value as V G S minus V T ok. So, as compared to that what are the changes here? So, one thing is the V G S minus V T has been divided by alpha ok. And what is alpha? Alpha is the straight line approximation for the square root function right. So, this function goes like this and you are going to use it only between psi S naught and psi SL. So, here we will approximate it by straight line and the slope of the straight line is alpha minus 1. So, that is how you get this ok. So, you can find out the slope expression by taking value of psi SL psi S naught right and then taking the slope of this triangle ok. In addition to this division by alpha you have a factor 2 here and something coming in the denominator. Why is this occurring? Because we are using a velocity saturation model ok and in the velocity saturation expression you have the critical electric field for breakdown. So, let us try to check this expression for limiting cases. You know we have said whenever we get a complicated expression we should verify whether all the terms have been taken care of by using a simple test ok. See whether this expression reduces to the simpler expression that you are familiar with for some limiting cases. So, this factor 2 is coming because in the denominator you have this one and this other thing. Now, let us try to reduce this expression to the expression V G S minus V T for V D S sat you have under what condition this will reduce. So, first thing is you must remove the velocity saturation effect. How do you remove that? If I make my E C infinite this is your velocity versus field curve where this is your E C. So, if I make this E C very large then my curve will go on increasing linearly then there will be no velocity saturation. Now, you will recall that the formula V D S sat equal to V G S minus V T is obtained using the pinch off condition there was no velocity saturation there. So, if you remove the velocity saturation we should get that expression. So, the way to remove it is to put E C infinity when I put E C as infinity this whole quantity becomes 0 and you are left with just one here. So, this 1 plus 1 is 2 and this coming the denominator this 2 will cancel with this 2 you will get V G S minus V T by alpha. Now how do you remove alpha? So, what is alpha doing really? So, if you carefully see this the linear approximation of square root psi S versus psi S is nothing but physically it amounts to getting an approximate expression for the varying depletion width from source to drain. So, you are replacing this varying depletion width by a constant depletion width which is some sort of an average value from source to drain. So, how do you get that average value? So, this average value is more than the value at the source, but less than the value at the drain. So, this averaging is obtained by using this factor alpha. So, if your depletion width is a little larger than the value at the source then your saturation voltage will be a little lower. So, if you use the depletion width to be constant equal to the value at the source then you get the V G S minus V T relation for V D S sat. You will recall that that square law model assume the depletion width to be constant from source to drain equal to the value at the source. So, if your depletion width is a little larger which is actually the correct thing to do right because the depletion width is increasing from source to drain then your value of the threshold voltage will reduce a little bit that is what this factor alpha is doing alpha is more than 1. So, if you want to set if you want to set the depletion width equal to the value at the source then you have to set alpha equal to 1 and then that is how you will get your first level course approximation for V D S sat. Now finally, L denotes that part of the source to drain channel over which the channel voltage V varies from VSB to V D B for V D B less than V D B sat and to V D B sat for V D B greater than V D B sat. Remaining part of the channel delta L near the drain accommodates the voltage variation from V D B sat to V D B. So, this interpretation of channel length is also very important is analogous to what we discussed in the surface potential based model. So, this is your picture in the MOSFET. So, what we are saying is that the channel voltage is VSB here and it would be V D B if your V D B is less than V D B sat at this point but if your V D B exceeds V D B sat then your V D B sat will come here and this point will be V D B. So, the difference between V D B and V D B sat will fall across this delta L and this is your L that is what we are saying here interpretation of channel length. So, with that we have completed our module on surface potential based and threshold based model expressions. So, let us do a summary of this module. So, we will summarize the module as follows. First we will list out the learning outcomes which we laid out at the beginning of this module. So, at the end of this module you should be able to explain the surface potential and threshold based solutions of the drain to source current as a function of gate bias, drain bias and source bias with respect to bulk. Explain where and how do the above solutions differ and estimate critical quantities associated with carrier concentration N and P, current densities J N and J P, electric field and potential. Then estimate quantities related to energy bands and finally current voltage characteristics. Let us compare the equations of the surface potential and threshold based models. Surface potential based models the key features are as follows. The IDS expression consists of an average mobility integral evaluated numerically. Please recall the average mobility integral okay. So, what was that? So, if you recall we said the average mobility at the surface is average of the reciprocal. So, you are taking 1 by mu NS and taking the average over the channel length 0 to L and then taking the reciprocal of this. So, this was possible, this integration was possible only numerically because mu NS depends on both vertical field as well as horizontal field. You have closed form function for psi S0, psi SL and VGB rather you have a closed form function. The next point is the drain current expression is a closed form function of surface potential at the source, surface potential at the drain and gate to bulk voltage which have square root linear and 3 by 2 power and square log terms of psi S0 and psi SL which are obtained by integrating the X dimensional current equation. So, you have the term square root psi S, you have the term psi S, you have the term psi S power 3 by 2 and you have psi S square that is what is meant here. Another feature of the model is that the IDS expression consists of implicit relations for surface potential at the source in terms of VGB and VSB and surface potential at the drain in terms of VGB and VDB. Giving a saturating behavior for psi S0 and psi SL as a function of VGB or VSB or VDB obtained from the bi-dimensional surface potential equation. Now, it is this saturating behavior of the psi S as a function of bias which gives you the saturating behavior for current as a function of voltage. So, what was the implicit relation? For psi S, let me write it here. So, psi S is VGB minus BFB minus gamma square root psi S plus VT exponential of psi S minus V plus twice phi F by VT. So, this is the implicit relation because psi S comes here on the left hand side, it comes on the right hand side at these two places. When you replace V by VDB, you get psi SL. When you replace V by VSB, you get psi S0, okay. Now, this has a saturating behavior. Finally, the channel end modulation region where GCF fails is represented by delta L. So, the IDS expression consists of a channel end modulation region, right. So, interpretation of the channel length is that is the region over which the surface potential varies up to the saturation value so that anything excess of that that is applied to the drain falls across delta L. The channel end region is the region where gradual channel approximation is varied. Now, compare these features with those of a threshold based model. One-to-one correspondence letter C, right. So, this expression consists of closed form average mobility expression. For the case of the surface potential based model, you had an average mobility expression which was evaluated numerically. Here you have a closed form expression. Then, your expression for IDS as a function of psi SO, psi SL and VGB, okay. That is split up into two parts. Exponential approximation for VGB less than V dash TB and square law approximation for VGB greater than VHB. So, one of the issues with this threshold based model is that there are several partitioning voltages, right. So, you can see here the V dash TB is one partitioning voltage, VHB is another partitioning voltage and so on. Compare this with the corresponding feature of the surface potential based model. That is this closed form function. So, IDS consists of closed form function of this thing but which consists of root, that is square root, linear 3 by 2 power and square law terms, okay. Whereas, here you have exponential approximation for VGB less than V dash TB and square law approximation for VGB greater than VHB. Now, as far as the surface potential expression is concerned, in surface potential based model, this was an implicit expression which you have to solve numerically. Whereas, here we have expressed psi SO and psi SL as linear function of VGB for VGB less than V dash TB and the psi SO is saturating to phi T plus VSB and psi SL saturates to phi T plus VDB when VDB is less than VDB SAT and it saturates to phi T plus VDB SAT for VDB greater than VDB SAT for VGB greater than VHB, okay. So, let me show this. So, psi S versus VGB. This is your flat band voltage. This is your behavior. So, what we are saying is we are approximating this behavior by a straight line here and a constant, okay. So, these constant values are psi S naught equal to phi T plus VSB if you are at the source and phi T plus VDB if you are at the drain and this is the linear function, okay for VGB less than V dash TB. So, here you are using a linear function. But beyond this, this VHB or VHB rather you are using the so called saturating function. So, this is a 2 piece model. In between however you have a problem because this is a moderate inversion region. Then your channel length modulation region is where the channel voltage exceeds VDB SAT. What is the difference between the two models in as far as the interpretation of channel length is concerned? In the surface potential base model, L denotes that part of the source to drain channel over which the gradual channel approximation is valid. Remaining part of the channel delta L near the drain accommodates the potential variation from psi SL to psi naught plus VDB where psi naught is the built in potential of the drain to bulk voltage. So, let us say this is your MOSFET. This is delta L. What we are saying here is variation of psi SL if you sketch here. So, it will be like this. So, this delta L portion. So, your psi S varies from psi S naught to psi SL over this channel. And from psi SL to this value is psi naught plus VDB when you apply VDB here. This falls across delta L. The corresponding picture in the kind of threshold base model is that L denotes that part of the source to drain channel over which channel voltage or quasi-formal level difference varies from VSB to VDB. So, in surface potential base model, the validity of gradual channel approximation is a criterion for the channel length. Whereas in the threshold base model, your channel voltage varies from VSB to VDB for VDB less than VDB SAT and when it is in saturation, it varies from VSB to VDB SAT for VDB greater than VDB SAT. So, all that will happen is that this psi S naught will be replaced by VSB, psi SL will be replaced by VDB SAT in the threshold base model. Remaining part of the channel delta L near the drain accommodates the voltage variation from VDB SAT to VDB. So, you will not have this psi naught term because in threshold base model, you are not considering voltages with respect to bulk but voltages with respect to source. Now, this lays the foundation for improving the model because we have said that this threshold base model does not have an expression for moderate inversion region whereas what do we want in practice when you want to simulate a circuit? A model that is suitable for circuit simulation is called compact model. We have explained these parts, we have explained this definition of the compact model clearly in one of the earlier modules. For most analog applications, the device is typically biased just above threshold that is for VGS greater than VT and less than VT plus 0.6 volts where the VT is approximately 0.4 volts. So, this is where you are biasing the model. Hence, the use of regional models is no longer justified. So, we cannot use the threshold base models that we discussed for this case. Let me explain this. If I take the ID S versus VGS on a linear scale, so what we are saying is this. Suppose this is your 0.4 volt threshold and you are going up to VT plus 0.6. So, 0.4 plus 0.6 may be here. This is 0.4 plus 0.6. It is about 1 volt. So, how will your ID VGS curve look like? So, it would look something like this. So, what we are saying is that there is a significant non-linear portion that you will have to model properly if you want to simulate your circuit accurately for this bias range. As against this, in olden days the same curve would look like this. Your bias VGS bias would have gone up to much higher voltages like 5 volts. This is VGS and your threshold voltage is one-fifth of this, say about 1 volt. This is 2.5 and you divide this into, so this is about 1 volt. So, in earlier days your curve IDS versus VGS curve would be something like this. So, you have a significant portion which is not non-linear and this non-linear portion here is reduced a lot as compared to the overall biasing range in this overall biasing range whereas because biases have reduced but threshold has not scaled down as much. So, here this maximum value has reduced from 5 to 1, but threshold voltage has not gone from 1 to 0.2. It has not reduced by a factor of 5. So, it has come down only to 0.4. So, that is why this non-linear regime is now significant. That is why we cannot use regional models. So, if I use a model such as saying IDS equal to 0 here and IDS varies in a straight line fraction, it would not work. You will miss this part. Whereas, olden days I could say this one segment and this another segment and still most of this regime I would be working the model would work very well. So, in the light of this the modeling goal is the following in compact modeling. You need a single expression description of ID from sub-threshold to strong inversion as well as from linear to saturation regions where which is C infinity continuous for all VGS, VDS and VBS bias conditions. And we need a similar expression for IB and IG. Finally, let us summarize at this end of this module the role of each of the approximations in the solution. So, so far we have considered what is the difference between the surface potential based model and threshold based model. We summarized all the features of the two models and then showed the differences. Since approximations are very important in device modeling, let us summarize the various approximations we have made in this MOSFET modeling and what were their roles. These approximations can be seen in the background of the basic equations for device analysis, the drift diffusion equations. So, you know that there are 6 equations, 2 current density equations, 2 continuity equations and 2 electrostatic equations. So, all the approximations we are discussing will pertain to those equations. Let us start with approximations related to current density. We assume J P Y and J N Y are 0. So, the Y direction is the vertical direction. This is Y and this is X. In the Y direction, whole and electron currents were 0. What did this approximations allow us to do? Now, first why did we make these approximations? Because we said we neglect the gate current and bulk current and there is an insulator. So, between gate and bulk, we assume no current can flow. Similarly, there is no bulk current also. We neglect leakage and all that. Now, these approximations allow us to express carrier concentrations P and N in the space charge expression of the Poisson's equation in terms of psi using the Boltzmann relations and they therefore helped us to get the Poisson Boltzmann equation. So, these exponential terms here are because of these approximations. The consequence of these approximations because this is where we are using the Boltzmann relation. I just want to point out here that when you want to derive this factor K, it is better to plot the N and P on a linear scale. So, K times P0 and N0 in exponential minus V by VTR values at the edge of the space charge layer. The above equation is written under gradual channel approximation and its integration gives you the total silicon charge Qs as a function of psi s and channel voltage. Very important approximation was pertinent to the continuity equation divergence of Jn is equal to 0. In other words, we assume the generation decombination and tunneling currents to be absent. Now, what do we achieve using this approximation? This leads to the simplification of the current Inx as a function of x. Now, what is Inx as a function of x? That is the electron current in the MOSFET in the x direction that is what this suffix x stands for and this current in general varies with x. However, because of divergence of Jn equal to 0 approximation, we could assume that this is constant along the channel equal to its terminal value that is we could use this expression. The negative sign being due to the flow of ideas in the negative x direction. While ideas is the terminal current which is positive, Inx flows from drain to source whereas your x axis is from source to drain. So, that is why that negative sign is there. Now, this approximation plus the charge sheet approximation leads you to the current equation given here. Then another important approximation we made is the gradual channel approximation. What did we achieve out of this? So, first let us state what this approximation is. So, it says dou square psi by dou x square is much less than dou square psi by dou y square. The variation of the potential along the channel is much more gradual than that in the perpendicular direction. So, x is along the channel is x and y is this from gate to bulk. So, we are saying that variations along the channel are much slower than variation in the y direction. Now, this is understandable because you see how much is the channel length. So, we have considered a structure of the MOSFET right in one of the earlier modules. So, you would recall channel length was of the order of about 30, 40 nanometers in our structure that we considered right. Modern MOSFETs have even shorter channel lengths. Anyway, let us look at that structure and get a feel for why the gradual channel approximation works. So, this distance from drain to source is about let us say 40 or 30 nanometers okay. Now, how much is oxide thickness? So, you will recall it was much smaller just a few nanometers right. In fact, today we are going down to as low as 1 nanometer right. So, in the vertical direction the distances are very small as compared to the horizontal directions, horizontal direction. But the applied voltage is of the same order between drain and source and between gate and bulk. So, which means the field in the vertical direction is going to be several times more than field in the horizontal direction because this drain to source distance is more than the gate to bulk distance okay. Now, this is the reason why variations in the horizontal direction are much slower than variations in the vertical direction because of this distance issue. Now, the gradual channel approximation can also be stated in terms of the electric field in which case dou Ex by dou X is much less than dou Ey by dou Y. The variation of the field along the channel is much more gradual than that in the perpendicular direction. What did we achieve out of this? The simplifies the two-dimensional Poisson's equation as dou square psi is replaced by del square psi is replaced by dou square psi by dou Y square only one term okay that is what we have achieved or it simplifies the Gauss law where divergence of is replaced by dou Ey by dou Y. Next directed variations are removed. Now it yields the silicon charge Qs as a function of psi s and channel voltage by integrating the Poisson Boltzmann equation that is where its utility lies the utility of the gradual channel approximation. Some more advantages of gradual channel approximation this is used extensively it allows us to write psi ox equal to minus of Qs x plus Qf by C ox okay using the one-dimensional capacitor formula. In the surface potential equation Vgb equal to psi ox plus psi s plus phi ms we neglected the poly potential drop even when the quantities in the equation vary with x. So the MOSFET is two-dimensional we are writing the oxide potential drop in terms of the one-dimensional capacitor formula which relates the potential to charge even though you have variation in the x direction or along the channel okay. So this is what is the advantage of gradual channel approximation and ultimately this yields Qs in terms of psi s in this formula which is then used in the current equation. Then let us look at the charge sheet approximation. The inversion charge Qi due to the distribution of electrons is compressed into a sheet or delta function at Y equal to 0 that is what this approximation means that is you are writing N of xy as Qix by Q into delta y where Qi of x is the integral of the distribution of electrons. So what this means is the following in your MOSFET your electron concentration in the y direction goes like this in the inversion layer. We are taking the area under this and putting this entire charge as a delta function at the interface right charge sheet that is this charge sheet approximation. The consequence of this is that various layers of Qi distribution over y experience the same values of mu n, E x and psi. This is the great simplification that results if your entire inversion charge is compressed in a sheet okay. Then the mobility, the exaggerated electric field and the potential psi right that is experienced by all the electrons would be the same whereas if you have a distribution something like this evidently your E x value at the interface and E x value little bit inside will not be the same. So different layers of inversion charge will experience different values of E x. Similarly different values of mobility also okay if the electron distribution is assumed to be you know over a distance. Now consequently the charge sheet approximation allows conversion of the equation for current density J N x which is in terms of the volume charge Q times N x okay as in this form. So this is your actual current density expression. So this gets converted into an equation for I N x in terms of the sheet charge. This is current density in terms of the volume distribution of charge. This is a more complicated situation. It is simplified to the current in terms of the sheet charge Q i okay that is the advantage of the charge sheet approximation and the same mobility value will be valid for all the inversion charge right whereas here you can see the mobility varies in x and y direction right and so different layers of electrons experience different mobilities. Another consequence of charge sheet approximation is that no potential drop across Q i because inversion charges compressed into a sheet and that is the entire potential size drops across the depletion charge. So the charge sheet approximation when combined with depletion approximation and gradual channel approximation allow us to write the formula for Q B which is given by this simply as shown here even when the quantities in the equation vary with x that is even when the quantities vary along the channel you are using the one-dimensional capacitor formula in the vertical direction to write the expression for Q B. This Q B formula allows us to derive a closed form relation for Q i in terms of psi s in the form shown here then we come to the depletion approximation. The space charge due to reduction in whole concentration that is the term Q times P minus N A on the right hand side of the Poisson equation or Goss law can be approximated as follows. This is the statement of the depletion approximation. What is the advantage of this approximation? This approximation plus the charge sheet approximation and the gradual channel approximation allow us to write the depletion charge in this way where you can see that we are assuming that the entire psi s is falling across the depletion charge though part of psi s in practice does fall across the inversion layer because inversion layer is compressed into a sheet so that is the charge sheet approximation. Therefore you are putting the entire psi s in the across Q B and that Q B you are not considering the mobile charge at all. You are only considering the ionized acceptor charge so that you can write it in this form. This Q B formula allows us to derive a closed form relation for Q I in terms of psi s. This was one of the major steps in MOS modeling right that is because we cannot evaluate Q I from the basic definition of integral of the distribution of electrons. We cannot get an analytical expression for Q I based on that basic definition of area under the electron distribution. Therefore we are writing that inversion charge as difference between the total silicon charge and the depletion charge which can individually be expressed in terms of simple expression of psi s. Now with that we have come to the end of this module.