 Let us begin our fourth lecture on surface potential and threshold based solutions of the drain to source current. First we will summarize the key results of the previous lecture. In the previous lecture we got a solution of the surface potential psi s as a function of gate to bulk voltage and channel voltage from y-dimensional analysis. The steps were as follows. To integrate the y-dimensional Poisson Boltzmann relation to get E y s that is the y directed surface electric field. So if I want to show you that this is your source and drain and this is your y direction right. So you are getting a field in the y direction. This is your depletion region, this is the so called inversion region, this is n plus n plus and this is p. Now how do you get the E y s? So you have dou square psi by dou y square. You integrate this equation once and we have discussed the steps of integration, I am not going to go through again. When you integrate it once you get dou psi by dou y. In fact that is what is the electric field in the y direction. Then after you have got E y s we have seen that right hand side of E y s is a square root term and therefore you have both plus and minus roots of E y s out of which the plus or positive E y s is used for depletion and inversion because it is a p type substrate right that we are considering and the negative E y s is used for accumulation right. So when the E y s is in this direction which is what it happens when you have a p type substrate E y s is positive. Now we approximated the E y s by removing some terms considering the practical situations and then using this approximated E y s we use the equation Q s equal to minus epsilon s E y s to get the following in inversion right and that is this Q s as a function of psi s. After getting Q s as a function of psi s we substituted this into the surface potential equation or S P E with and in this S P E we put psi ox equal to minus Q s plus Q f by C ox using the gradual channel approximation and so on right and then this surface potential equation with these substitutions led us to the surface potential equation that is given at the bottom of the slide. Now in this equation here that term psi s is equal to psi s naught surface potential is psi s naught for V equal to V s B if you substitute V s B here you will get psi s naught okay all these terms will become psi s naught and similarly if you substitute V equal to V D B you get psi s l or the surface potential at the drain. Now the important point is this psi s expression saturates with increase in channel voltage V causing saturation of I D s so this is the reason why the I D s saturates in our theory because the surface potential psi s saturates I gave you as an assignment to show that this expression for psi s saturates I am sure you would have attempted it right now in case you had difficulty in appreciating how this particular expression leads to a saturating values of psi s let me quickly give you some hint so this is psi s as a function of V now you can see here that this entire term has a negative sign right this term has a negative sign so I can push this term to the left hand side so that all the psi s dependent terms will be on the left hand side then this will come as a positive term so now on the left hand side I will have psi s and this entire term okay. Now concentrate on this particular exponential when your V is 0 this term is psi s minus twice psi f by V t okay and let us say you get some value of psi s evidently it will be non-zero okay because on the right hand side you have V G B minus V F B so let us put some non-zero value here for psi s now as you increase your V what is going to happen is this exponential term is going to go on decreasing right why is it going to decrease because those psi s is also going to change but since the V here appears against a negative sign okay this entire quantity will go on decreasing as you increase your V so if this quantity decreases your value of psi s will increase because you can see that we have moved this to the left hand side and if the exponential term is reduced in other words you have 3 terms on the left hand side dependent on psi s if any one of the terms reduces the value of psi s will increase because right hand side is remaining the same right so it goes on increasing so you get increase here now let us appreciate why it will saturate you can see that if this term right if this exponential term if the exponent becomes more and more negative the exponential term falls and you know exponential of a negative variable goes to 0 if the variable goes on increasing so this term is going to vanish this entire exponential term is going to vanish and once that happens right you are left with an equation for psi s so if I remove this term you are left with a very simple equation for psi s it is a quadratic okay and when you solve it psi s will be a constant because it will be a function of Vgb minus Vfb and gamma all of which are constants so therefore for large v it will reach a saturation okay and in fact you also know what will be the saturation value saturation value is obtained by removing the exponential term and solving the remaining quadratic equation okay so now let us put together or rather in the previous lecture we put together surface potential based model as follows so ids consists of diffusion and drift the diffusion current is given by this formula the drift current is given by a somewhat longer formula where you have the linear square law and 3 by 2 power law terms the psi s is solved by using this equation and to repeat the above psi s expression saturates with increase in V hence this model predicts the ids behavior over the entire bias range without the need for any partitions such as sub or super threshold non saturation or saturation and weak or strong inversion okay so see it is such a nice expression right it includes drift and diffusion and saturates naturally you can carefully see here no threshold voltage term or no saturation voltage term is horizon okay so let us illustrate our result pictorially you remember we had sketched this kind of the current voltage curve okay on a semi lock plot note that here current is on a lock plot whereas the VDS is on a semi lock sorry VDS is on a linear scale right the id is on a lock scale while the VDS is on a linear scale okay now what we have done is we have struck out some terms such as threshold voltage and this line which partitions between saturation and non saturation right and we are striking out this also because this current is because of leakage and we have not modeled leakage in our model okay so the surface potential based model predicts this entire variation of the current without introducing any threshold voltage or this partition though of course it does not model the leakage current so that is why we are striking it off and therefore if you use a surface potential based model and make a calculation you will get all these curves okay smooth you will get all these curves in a smooth manner right okay let us see what kind of curves would the surface potential based model give when you plot id as a function of VGS okay so here two types of plots are shown this is the plot which uses the current on the right hand side right that is why this arrow is shown here and this is a lock scale so this current versus voltage is on a semi lock plot whereas this blue curve uses the current scale which is on the left hand side that is why this arrow is shown here and this is linear scale so this is linear linear plot okay the voltage is always on a linear scale so we are plotting id as a function of VGS keeping VDS and VBS constant right now this thing is struck off in other words there is no threshold voltage concept so you will not have any partition like this in the surface potential based model that is why this is struck off and this part is because of leakage which we are not able to model so the surface potential based model would give us these curves smoothly okay so let us return to our previous slide giving all the expressions of the surface potential based model and let us make another important point this is about the channel length L, L denotes that part of the source to drain channel over which 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 okay now let us understand this part about interpretation of channel length and so called channel length modulation okay so let me sketch a diagram to illustrate this point so this is your MOSFET, this is your gate, this is your depletion layer, this is your inversion layer and this is the so called region right here at this end so the edge of the channel comes only up to this point when your VDS goes on increasing right beyond a value that I am going to tell you shortly so this distance here I am sorry not this this distance is delta L so the channel length L is up to this point now let us give the characteristics of this point this is the inversion layer so this is rounded here you are applying VDB you are applying BGB here and you are applying VSB there this is a P type substrate okay so let me sketch below this the potential variation so I am plotting the surface potential along this okay so this is your X along this axis so at the source here the surface potential that is the potential drop like this right in the depletion layer this is PsiS0 on a other and on the other hand here this is the so called PsiSL at this point okay and you are going on increasing VDB but the PsiSL has saturated right so we talked about it as you increase V the channel voltage V the PsiS saturates so at the drain it will saturate at PsiSL but you are going on increasing VDB now what is the potential at this point okay so if I put the potential here at this point at this edge this is Psi0 plus VDB okay where Psi0 is a built in potential of the PN junction right so that is somewhat like the potential drop across a reverse bias N plus P junction when you apply a voltage VDB the total potential drop here is Psi0 plus VDB so now the difference between Psi0 plus VDB that is at this point and the PsiSL which is the potential here at this point that difference falls across delta L so let us show this in this graph so this potential goes on increasing and it is something like this okay so this value is PsiS0 what we are plotting is PsiS so this axis is PsiS and when you come here it becomes PsiSL and when you reach the drain it becomes Psi0 plus VDB so when you start from VDB equal to VSB your potential PsiSL will follow VDB as you increase VDB but beyond some point so let me sketch that part if I sketch PsiSL versus VDB you will get something like this right so this is the saturation value of PsiSL because that is the nature of this PsiS versus VDB function so this is the saturation value of PsiS so when you are increasing VDB up to this point PsiSL is following your VDB but beyond this point it is saturating but you are going on increasing VDB so the difference between the potential at the drain contact and the potential at this point which is the edge of the channel length L okay that difference falls across delta L so delta L will go on increasing from 0 as you increase your VDB right and this Psi0 plus VDB minus PsiSL that is this potential difference it will fall across delta L so this edge of the channel will go on moving towards the source so this is the channel length modulation this is the interpretation of channel length okay so over this L your garage channel approximation will be valid but over this delta L it will be invalid now let us move on further again we come back to the slide of the various expressions what about the average mobility let us focus on that okay we are using this average mobility expression here or average mobility term for the diffusion current this average surface mobility for electrons is approximately equal to mu n effective given by mu n 0 by 1 plus EY effective by E0 power n okay so you see that here there is no longitudinal field effect that we need to consider we only need to consider the vertical field okay because we are in the diffusion regime okay and if you recall in the diffusion regime across the channel the voltage does not fall much so let me recall that while for drift current you have your depletion region varying like this okay where it is less at the source and more at the drain but when you are talking about diffusion it will be constant until the drain though the depletion region between the drain and bulk this will be large but the depletion region in the channel it will be uniform okay so that is the condition in the subthreshold region please recall there will be an inversion charge which is very weak so subthreshold is weak inversion region it will vary from source to drain no doubt but this will be very less compared to the depletion charge so that is why we are not considering the effect of this this field because this field from drain to source along x this field is very small you only need to consider the vertical field so that is why we are considering the mu n effective now here when you try to evaluate this now you recall that mu n 0 is the bulk value of mobility corresponding to the particular doping in the substrate when you evaluate this where E by effective is the effective electric field in the y direction and E naught is a critical field the E by effective is written in terms of QB plus 0.5 QI by epsilon s with the modulus because E y is positive right whereas the charges are negative here inversion charge and depletion charge are negative now in this expression in subthreshold QI is much less than QB so just now we have remarked that because QI is much less than QB even though QI varies QB does not change it is uniform so that being the case since QI is much less than QB we can remove QI term here and we can write the E by effective in terms of QB alone so that would be your expression for average surface mobility here how do you find out QB well you know how you can find out QB in terms of the surface potential okay using P injunction theory. So we can further write this QB in terms of the surface potential as gamma into C ox into square root of psi s naught you can write QB as this numerator term okay so this is what I said you can do in terms of P injunction theory psi s naught is a surface potential at the source now you will say but what is happening surface potential at the drain so I want to take you back again to this diagram and we have said that from source to drain here the channel depletion layer is same so the surface potential does not change much in weak inversion okay so you can take it to be equal to the value at the source. Now let us look at the drift current what is the average surface mobility for drift current now the average surface mobility for drift current you will recall the expression that we had written it is 1 by L into 0 to L into this term right where the mu n effective is given by this formula and of course the EY effective we had written in the previous so both these expressions we had written in the previous slide right the new thing is only this so let us try to understand this particular expression right so what is that expression really okay so you will recall we defined let me write here on the side mu n s average as 1 by L integral 0 to L dx by mu ns okay where now what is mu ns so mu ns is equal to the mu n effective which includes the effect of the vertical electric field divided by a term which includes the effect of the horizontal electric field that is 1 plus E x s I am putting a modulus because E x is from drain to source whereas x axis is from source to drain by V sat by mu n effective okay that is so called critical electric field and here this term is raised to beta and the whole thing is raised to 1 by beta okay this is the so called critical this thing again I will draw the axis for your convenience this is x this is y this p type substrate okay now it is this formula which is written here okay now just understand this how do you get this so this minus 1 here is this minus 1 okay and this mu ns is this so since there is this is 1 by mu ns your denominator here will go up and numerator will come down so let us write 1 by mu ns this will be equal to 1 plus this whole quantity to the power beta to the power 1 by beta upon mu n effective now I take this mu n effective inside so when I take it inside since there is a power 1 by B mu n effective will get raised to the power beta okay but with a negative sign because mu n effective is in the denominator right so mu n effective raised to the power minus beta so that is what is happening here when I take it inside similarly when I take this mu n effective inside this term here it will cancel with this mu n effective term okay and therefore you will be left with V sat alone okay so that is how you get this E xs by V sat whole power beta so now you understand how we have got this expression here now let us look at how do we integrate this so to integrate we have to make some approximations beta is equal to 1 because otherwise analytical integration of this equation is not possible and what about mu n effective it actually varies in the x direction from source to drain however we will assume mu n effective to be constant with x at its value mu n effective 0 at the source okay so these are the 2 approximations we are going to do so we will set this beta as 1 so okay this beta will set it equal to 1 and this mu n effective though it varies from source to drain according to this formula here where E y effective is given by this formula we are going to use its value at the source that means we are going to use this here we will put a 0 here and we will put a 0 here right that is what we are going to do so this will now be a constant when I am integrating with respect to x so when I do that I will get this mobility expression okay mu n s bar average mu n s is given by 1 by mu n effective 0 plus bds by L divided by V sat okay where mu n effective 0 and E y effective 0 are given here now let us look at this a little bit how do I get Vds by L in the numerator so let us go back to this slide so what is happening is this E x s okay is being written as Vds by L how does it happen so since these 2 terms are being summed up when I put beta equal to 1 okay this whole power this bracket power 1 by beta becomes just 1 here okay so this goes off you are really just summing up 2 terms okay and integrating them so let us consider this integration individually so you are getting integral 0 to L mod E x s by V sat the power beta is again power 1 so there is no beta dx right now V sat is a constant now E x s consider your mass device E x s the field at the surface in the x direction this is E x s when I integrate this in dx so this is your x direction from source that is x is equal to 0 to drain x is equal to L this is 0 this is L then what will I get I will get the difference in potential between this point and this point right and what is the difference in potential between these points I can either talk in terms of surface potential or in terms of applied voltage if I talk in terms of surface potential here the potential is psi s L and here the potential is psi s not so really when I integrate E x s from source to drain I will get psi s L minus psi s not now when I substitute psi s L as let me create some space so I substitute psi s L in terms of VDB and psi s not in terms of VSB so this quantity will be VDB plus phi not and this quantity will be VSB plus phi not okay phi not is twice phi f plus 6 VT here so this phi not phi not cancels when you take this subtraction so it becomes VDB minus VSB and VDB minus VSB is nothing but VDS so that is how the integral of E x s over DX becomes VDS okay and there is this 1 by L here outside so that you take inside so this becomes VDS by L so that is how you are getting this VDS by L now why are you not getting the L division here for mu ineffective because this has become a constant after you have made it a mu ineffective 0 okay and you are integrating that with respect to DX from 0 to L so you get an L out of this integration in numerator that cancels with the L in the denominator that is how you get this equation okay. Now mu ineffective 0 is this formula this is straight forward we know this the E y effective 0 you have to determine Q i and Q B at the source so that is why you are going to use the expression for Q i in terms of psi s and put psi s equal to psi s not and you are going to use expression for Q B and you are going to put psi s equal to psi s not in that expression now the expressions for Q i and Q B in terms of psi s are given here so Q i as a function of psi is given by minus C ox into VGB minus VFB minus psi s minus gamma into square root psi s while Q B as a function of psi s is given by minus gamma into C ox into square root of psi s so you substitute psi s 0 for psi s so you make this 0 here that is what you are supposed to do and how do you solve for psi s not and psi s l so well that we solve using this equation okay by substituting V equal to VSB here you get psi s not substitute V equal to VDB here you will get psi s l okay so that is how you can get this average mobility so you can see that the computational procedure is a little involved right it is a nice model it predicts current voltage variation smoothly over entire region of operation okay but there are lot of computation involved so that is why people have gone in for some simplification right and other types of models so further simplification of this average mobility expression so you are writing the denominator here VSAT as there is an l coming here okay so you are writing this VSAT in terms of a critical electric field right and you are moving this mu n effective out please look at this so let us show how that happens so you take this mu n effective out so you will get since there is a reciprocal there so mu n effective 0 into this you have taken out so you get 1 plus BDS upon l and this mu n effective goes up when you put it here so mu n effective 0 by VSAT or I can put this mu n effective in the denominator and write it as VSAT by mu n effective 0 now this term VSAT by mu n effective 0 you can clearly see that it has units of electric field and this is actually the critical electric field so this bracketed term is EC you can check this that this has units of critical electric field from we have discussed this mobility expression in one of our earlier modules otherwise you see dimensionally that is electric field dimensions VSAT is centimeter per second and mu n effective is centimeter square per volt second so second second cancels 1 centimeter goes in denominator you are getting volt per centimeter right that is electric field that is how here you get the critical electric field of course there is a power minus 1 over there so in other words this term comes in the denominator so that is what is actually shown here okay where EC is VSAT by mu n effective 0 so let us once again summarize our surface potential base model in the form of a plot and then use this plot for the surface potential base model to show how we are getting into the threshold base model okay so this shows IDS as a function of VGB on a semi log plot IDS is on the log scale here let us show the segments from which this behavior is obtained so you have the diffusion component given by this expression and if you plot that this part of the curve which is approximately a straight line okay which actually because of diffusion is because of this expression however when you increase your VGB as you get into the strong inversion region your current because of diffusion goes on increasing but at a slower rate than the total current because of drift and diffusion right so this is a diffusion component. Now let us put the drift component so in strong inversion your drift component dominates over diffusion component the drift component is given by this formula however for lower values of VGB your diffusion dominates over drift okay that is the way the picture is and there is some region here where both are comparable this is really the difficult region to model in very simple expressions right. Now in the background of this let us introduce the threshold base model okay the first step in the threshold base model is to partition these smooth curve okay the smooth blue curve into 3 regions. Now the partitioning voltages are the threshold voltage with respect to bulk okay the dash here indicates that this threshold voltage is calculated using the twice 5F criterion okay so we have seen that there are 2 threshold voltages one is the threshold voltage calculated based on the twice 5F criterion as we have done in the first level course and other is the threshold voltage that you get by linear extrapolation of the current voltage curve so theoretically if you want to calculate this that is the VTB that is obtained by assuming the surface potential to be twice 5F plus 6 VT. So V dash TB is the threshold voltage calculated using the twice 5F criterion while VTB is the threshold voltage calculated using the twice 5F plus 6 VT criterion okay. So below V dash TB on the VGB axis you have weak inversion between V dash TB and VHB you have the moderate inversion and beyond VHB you have strong inversion. So you are partitioning this curve this blue curve smooth curve into 3 regions weak inversion moderate and strong inversion and what you find here is in weak inversion the blue curve on a semi-lock plot is almost a straight line and we can show that similarly the red curve rather the blue curve in the strong inversion region okay is a square log has a square log behaviour okay. So let us show this explicitly the dashed lines show the exponential approximation for diffusion now we have not shown how you get a simple exponential expression okay we are right now telling you the outline how it is going to be so it will turn out now why is it exponential well that is very simple on a semi-lock plot if a curve is straight line you know it has to be a power law okay an exponential is a form of power law. So we will show that the diffusion a component in the surface potential expression will reduce to a simple exponential after some approximations okay so that is this dashed line on the other hand this other dashed line here shows the square law approximation and we will show that the drift part of the surface potential based model will lead us to this simple square law expression after some approximations. Now notice here that if we were to use the exponential approximation for diffusion even in moderate inversion you will get very high currents much higher than the actual current on the other hand if we use a square law approximation okay which is valid for drift current in the moderate inversion you will get much lower currents than the actual current okay so that is why this moderate inversion region is really very difficult to model we will be able to model using simple expressions the weak inversion and the strong inversion using exponential approximation in weak inversion and square law approximation for strong inversion and then later on we will discuss how to stitch together the square law and exponential approximations so that you get a smooth variation here without adding any separate expression for the moderate inversion region. Now with that introduction of the threshold based model let us summarize what we have achieved in this lecture so which we achieved number of things so in this lecture we first summarized all the expressions of the surface potential based model and then we gave an interpretation of the channel length we said that when you go on increasing the drain voltage from value equal to the source voltage your surface potential at the drain psi sl follows the drain voltage up to a point but beyond some drain to bold voltage the surface potential saturates naturally right as a behavior of that surface potential expression. So therefore the excess potential between the drain contact and the saturation value of psi sl falls across a channel length modulation portion delta l right quickly this is the diagram which shows you that so this is your inversion region this is delta l this is l this is the so called psi sl sat and here you are applying Vdb so at this end your potential drop is psi naught plus Vdb so this difference between psi naught plus Vdb and psi sl sat falls across delta l so this is another important point made we made in this lecture then we also discussed the average surface mobility right what is the expression for the average surface mobility how to calculate it for the differential component of the current and for the drift component of the current so we found in the differential component of the current the mobility does not depend on the drain to source voltage while in drift component of the current the mobility does depend on the drain to source voltage right it also depends on of course gate to source voltage because both diffusion and drift mobility are a function of the vertical field which is always strong so in fact without the vertical field in a MOSFET here vertical field is this you cannot really get any inversion layer right whether it is diffusion or whether it is weak inversion or strong inversion and finally we showed how when you plot the surface potential base model drain current as a function of the gate to bulk voltage on a semi lock plot you get a linear segment for the weak inversion right on a semi lock plot so this linear segment on the semi lock plot amounts to an exponential kind of behavior and in the strong inversion region thus behaviors obtained from the surface potential base model can shown to be square law we have not shown it we said it can be shown and we are going to show it in the next lecture the moderate inversion region is a region which is difficult to model so in weak inversion diffusion current dominates over drift current in strong inversion drift current dominates over diffusion moderate inversion both currents are comparable we mentioned that the exponential segment of the weak inversion and the square law segment of the strong inversion can be stitched together in a way in which you get a smooth transition from the weak to strong inversion in the moderate inversion region so how to do that we will see in subsequent lectures.