 So, let us begin a new module, so this is on the DC model of a large uniformly doped bulk MOSVET, we are now going to talk about the solution to the equations boundary conditions and approximations right which we have discussed in the previous course. So, we will consider two types of solutions surface potential and threshold based solutions for the drain to source current in terms of the gate to bulk voltage the drain to bulk voltage and the source to bulk voltage. At the end of this module you should be able to explain the surface potential and threshold based solutions of the IDS as a function of the 3 bias voltages of a MOSVET. You should be able to explain where and how these two solutions differ and finally you should be able to estimate the critical quantities associated with the carrier concentration of electrons and holes N and P, the current densities of electrons and holes JN and JP, electric field and the potential side then energy bands within the device and finally the current voltage characteristics. This is a recap of the approximations that we have made in the previous module, so you can see here that you have approximations related to the electron current density given here, the approximation related to the continuity equation which are given here okay, the approximations related to the current density and continuity equations of the hole current density. So in fact we have said that we are not going to consider hole current density at all okay, then we have the approximations related to the Gauss law and there is no approximation related to the equation equal to minus grad psi okay, so these are the approximations which we have discussed in detail in the previous module, we will have opportunity to discuss about these approximations further in this module okay, so that is why I am not taking up each of these and then repeating the explanations. So let us give an outline of the solution first before discussing the details of the derivation, so these are our equations what we said is we are not going to consider the hole current density at all, so those 2 equations go out and as a result in the equation for the current density J you have only the electron current here, then in the continuity equation we said that we are going to neglect generation recombination right, so in this continuity equation here we are going to neglect these terms and since we are considering steady state this term also goes to 0, so really these equations are also not required okay and the equation reduces to divergence of J n equal to 0 which we will show can be cast in the form integral of J n x dou y from the surface to bulk is constant independent of x, this is nothing but the drain to source current, so we will show that the continuity equation will reduce to this equation that drain to source current is constant independent of x if you neglect generation recombination and consider steady state conditions, so this amounts to neglecting the gate current and the bulk current. Then about the Gauss law what we said is on the left hand side of the Gauss law we are going to make a gradual channel approximation, so we are going to consider only the variation of the electric field in the y direction, so I want to emphasize or remind you that in the device the x direction is along the horizontal and y direction is along the vertical, so the left hand side divergence of E has been approximated by the term in the y direction and on the right hand side we said that we are going to use the charge sheet approximation for electrons, so rho you have the electrons only considered here you do not have the holes okay and you have the ionized acceptors to take into account the charge beneath the inversion layer okay the depletion charge, so let me repeat that if you consider the previous slide what is happening is that divergence of E is being replaced by dou E y by dou y and the right hand side rho the P is being removed there is no donor concentration, so this term really does not arise there and so you are left with only the electron concentration and the ionized acceptors, so whatever approximations we have made in the table we are using those right and simplifying these equations. So this is what is your drift diffusion model after approximations, so further solution will be done using these equations, now let us set out the milestones in the derivation, so our approach will be that we will first show you the milestones and then we will talk about how to get to the milestones, so the overall solution outline for the MOSFET is as follows, the derivation of the drain current as a function of the various biases is a 2 dimensional problem, so you can see here the device is shown conditions are varying in this direction that is x direction as well as in the y direction okay, we have said that we are going to assume the z direction conditions in the z direction to be constant okay the z direction is perpendicular to the slide, so that is why it is a 2 dimensional problem for us, so what is the 2 dimensional problem, so you have flow of electrons from source to drain, please note that the conventional current will be from drain to source that is why this arrow is shown inward into the drain contact but the flow of electrons is from source to drain and this flow of electrons is controlled by a y directional electric field EY that is really the MOSFET, the key MOSFET phenomenon or that is called the field effect control of flow of carriers by a transverse electric field, so what is this 2 dimensional problem, the x directed current IDS created by VDS and VSB, now what we are going to do is we are going to approximate this as 2 coupled 1 dimensional problems, so this is the simplification okay, so you have a 2D problem, so this complex situation is split into 2 1 dimensional situations coupled to each other and what are these 1 dimensional problems, so solution of the x dimensional current equation together with the solution of the y dimensional voltage equation or surface potential equation, so in the x direction you have an equation for the current and in the y direction you have an equation for the potential drops okay in the various parts of the device, what are these equations, so right now we are writing the equations we are going to discuss the derivation of this equation later, the x dimensional current equation is the drain to source current is approximately equal to the negative of the electron current okay, which in general can vary with x in general meaning in a device where you have the gate leakage and you have the bulk leakage also okay, in that case the IN or the electron current from source to drain will vary right because part of this current is going to gate and part of this current is going to the bulk though as we will as we have remarked we are going to neglect the gate and bulk leakages, so this is equal to the width of the MOSFET W into the mobility of electrons at the surface, now this also varies with x, again let me show you the x direction this is x, so in this direction your mobility is varying at the surface, so this is the surface, so your mobility is varying right, you have the inversion layer there and the mobility of electrons is what we are talking about, now this is multiplied by within brackets negative of the inversion charge which also varies with x, so the inversion charge here is something like this, it varies with x okay and this is your depletion charge into the derivative of psi s with respect to x psi s is the potential at the surface here plus the thermal voltage into the derivative of the inversion charge with respect to x, so that is the equation, now before considering the detailed derivation let me quickly show you that how this equation relates to the equation for current density of electrons, if I go back to the previous slide showing the current density you can see here that there are two terms one is a diffusion term and another is a drift term the drift term you have multiplication between the carrier concentration and the electric field diffusion term you have the gradient of the carrier concentration, so you can see these two terms, so this is the term which is related to the gradient of the carrier concentration okay and this is the term which is related to the carrier concentration multiplied by the electric field okay, electric field is nothing but the derivative of the potential, so this is the so called drift term and this is the so called diffusion term, you are getting a thermal voltage because you are writing the diffusion coefficient as product of the thermal voltage and the mobility, so this is just to make you comfortable with the equation or we have not yet derived the equation we will do a detailed derivation shortly. Now so you need the surface potential, this potential as a function of X you can also show that the same surface potential can be written in terms of the channel voltage V and the gate to bulk voltage VGB and similarly you need the QI as a function of X right, so that QI as a function of X can also be written as QI as a function of psi S, you know the psi S and V vary with X okay, so these functions psi S and QI will be provided by the solution of the Y dimensional voltage equation or surface potential equation, so what is that equation? That equation is a simple equation which says that the gate to bulk voltage is divided into or rather part of this falls across the polysilicon, part of this falls across the oxide and remaining parts, remaining falls across the silicon part and then you also have the built in voltage phi MS okay coming there, let us relate these voltages to the structure, so we are talking about this is gate to bulk voltage, voltage between these two points, so part of it falls here in the poly then part across the oxide then part across the silicon and then as we have said the bottom contact here you will have a contact and for simplicity we said we will assume this contact to be of the same material as the N plus poly okay, so here you have a built in potential that is a phi MS, so that is why the gate to bulk potential can be divided into psi P, psi ox, psi S and here you have phi MS okay, so you sum them up taking the polarity appropriately, so this equation we will have to solve to get these two quantities, we will show how we can do it, so this is going to be our approach for solving the MOSFET current equation in terms of the voltages, now let me make some comments how can you solve for psi S from this equation, phi MS is a constant, so if you express psi ox and psi P in terms of psi S then this equation will become only a function of psi S and you can solve for it, now reason we have shown a cross here for psi P okay, we have cross this out because in our analysis we are going to neglect this psi P okay, so nevertheless if you wanted to include psi P you can express that and psi ox in terms of psi S and the channel voltage V, we will show that how to do it, another comment is the coupling of the two one dimensional problems via psi S makes MOSFET modeling essentially a surface potential based problem with the threshold based and other formulations as derivatives, so fundamentally if you want to give a name to this solution of the MOSFET problem you can call it as a surface potential based solution because as we have seen here the key here is to get the surface potential okay and this will also lead to QI as we will see because QI also is required in terms of surface potential, so various terms required in the current equation are expressed in terms of the surface potential, so finding the surface potential in a MOSFET at various points within the channel is actually the key problem, so we have reduced the MOSFET current voltage characteristic derivation to finding out the surface potential in the channel, so other models such as those which use threshold voltage and things like that can be shown to be derivable from this fundamental surface potential based solution, now let us go to the derivation of the X dimensional current equation and bi-dimensional voltage equation okay, let us take the current equation, we will now show that these conditions which result from the approximations that we have made we had listed these in the table of approximations okay, so the relevant approximations which are used to derive the current equation are all put down here, these approximations will lead us to this particular equation for the current flow, now let us do this exercise okay, so this is your x direction and this is your y direction, this is the origin and here this is channel length L, now what is IDS? Let us start from what is IDS, so IDS is the current between drain and source right, ID is the current that is entering here, since we are neglecting the gate and bulk current ID is same as IDS, so for us this is the IDS, now so what is IDS, we can write it as the integral of the current density X at X equal to L okay, that is the current density here as a function of y, do y because we have to integrate in this direction and multiply the current density by the area here, so that area is you have W of the MOSFET in this direction right that is the W, so W and this is integrated from 0 to bulk and since IDS is positive we just consider the magnitude of this right, so JX is the current density in the X direction, you are integrating it over the area of the current flow and that is W into the length in this direction but instead of the length since JX is varying with Y we are integrating over the Y okay, so if I show this it is something like this, so this is W your JX is like this and this is the Y direction, so you are integrating over this direction because the current density is not uniform, it is more here at the surface, it is more at the surface and less at in the bulk, inversion layer is actually restricted to a very thin layer there, so that is what we are saying here. Now we have said that we are going to consider only electron current, so this JX is nothing but JNX at X is equal to L, now we have one more result and that is divergence of JN is 0 continuity equation, this approximation, now we can show that this result will allow us to take this JNX not just at X is equal to L but at any X okay, you can do this integration at any X, to show that let us use this particular result, let us see how we can use this, so divergence of JN equal to 0 means dou JNX by dou X plus dou JNY by dou Y is equal to 0, now what does this mean, this means that dou by dou X of JNX into dou Y which I am taking from here plus dou of JNY is equal to 0, so you can write this in this form which I can now integrate as follows, so I can say dou by dou X of JNX dou Y integrated from 0 to bulk plus integral dou of JNY integrated from 0 to bulk is equal to 0, now as we have shown in our table of approximations okay, this JNY that is the current in the Y direction, current density in the Y direction, it is 0 at the surface, so Y equal to 0 is the surface here okay this is Y equal to 0, so JNY is 0 at the surface and it is 0 even in the bulk because JN is 0 in this direction therefore this quantity is equal to 0 and therefore our result is dou by dou X of JNX dou Y integrated from 0 to bulk is 0 okay, since this is 0 this quantity also has to be 0 and this therefore leads you to the condition that integral JNX dou Y from 0 to bulk is equal to constant with X because of the derivative of this quantity if the derivative of this quantity with respect to X is 0 it means this quantity is constant with X okay, therefore now you can use this result okay here and therefore instead of integrating at X is equal to L that is at X is equal that is at the contact I can integrate at any X in the Y direction, so I can write the same thing as this is same as modulus of JNX which is the function of Y dou Y okay, so this is what you get IDS is given by this particular result, next now let us substitute for JNX, so what is this JNX you can write this as the diffusion term and the drift term okay, so let me write the drift term first so QN mu NE X plus Q DN dou by dou X of N because we are considering things in the X direction you are taking all derivatives also in the X direction, now this is your JNX, now what we will do is we will use the charge sheet approximation okay, how do we use that, so here this JNX this N mu N EX DN all these are functions of X okay, so they are varying I am sorry they are functions of Y each of this is a each of this term JNX N mu N EX and DN are functions of Y, so if you move in this direction your carrier concentration is going to change your current density is going to change and since your field is changing in this direction your mobility and the diffusion coefficient depends on mobility based on the Einstein relation therefore that also changes in the Y direction, so let us write that JNX N mu N EX and DN are functions of Y, however if you make the charge sheet approximation what it means is that all the electrons are actually concentrated as a sheet at the surface therefore we can ignore the variation in Y direction and take all these quantities at the surface, so charge sheet approximation implies that variation of above quantities with Y can be ignored and we can take the quantities at the surface alone because all the electrons are concentrated at the surface, so therefore we can write that we can replace the mu N by mu N SDN by DNS and we can when we are writing EX we can write EX in terms of surface potential psi S okay, so we can write this as dou by dou X of psi S with a negative sign, so let us do that then what do we get, so result is that we can write this IDS, so this is what we will be going to substitute, so we are going to substitute this result into JNX here with mobility taken at the surface, division coefficient taken at the surface and the EX also taken at the surface, so our result will be the following, so IDS is equal to or maybe let me just clean up this board here, so I can write IDS is equal to W into I will take this mu N S out and I will write DNS in terms of mu N S, so let me write this as mu N S which is a function of X into this EX also I can take it at the surface right, this is what I am saying here, so I can write this as minus dou psi S by dou X and then I have to do the integration, so then I will put the integral from 0 to bulk and this N I will put here and you have to integrate with respect to Y, so N is a function of X and Y I am integrating with respect to Y, then let me take the next term here, so plus W has already been taken out mu N S has been taken out, so here I should have the thermal voltage because I am using DNS is equal to mu N S into thermal voltage, so I will get the thermal voltage term here and incidentally this Q also I have to take so that I am putting here and this quantity I can write as dou by dou X right of 0 to bulk N X comma Y dou Y okay and let me close this bracket, so bracket is opening here and then closing here, now we can identify this quantity as nothing and there is a Q there, this Q, this quantity as minus of Q i okay because when you integrate N with respect to Y, you are integrating the electron constant with respect to Y you get entire inversion charge per unit area, N has units of per unit volume multiplied by length it becomes per unit area, so this is Q i the inversion charge, sheet charge as a function of X and same thing applies here, so this quantity is Q i X with a negative sign okay because Q i is negative it is charge due to electrons whereas here this whole thing is positive, now therefore my equation becomes I DS is equal to W into mu N S, now I will avoid this writing as a function of X and so on to avoid to keep things simple and this quantity is bracket open Q i and that is dou by dou X of psi S this negative and this negative cancels okay plus V T into dou by dou X of Q i but there is a negative sign here, so I have to replace this by negative sign okay, so this is what I get and this is actually the current density equation, there is one thing here and that I have to take the modulus, so when I have to take the modulus I should take into account the fact that the polarity of this may be negative and I should then take the positive of this, now let us look at the terms, now one thing you can see is that algebraically the drift and diffusion current should act to each other, why? Because you can see that electrons are flowing from left to right and the flow is because of in the same direction because of diffusion as well as drift, let us see this point, so why is the flow because of diffusion from left to right because carrier concentration is more near the source than near the drain okay, so when you draw the inversion charge you know that we charge draw it like this, it is more near the source than in the drain, this is your depletion charge, so evidently the diffusion of electrons will be from left to right, then the electric field is from right to left because V dB is more than VSB and that is going to again cause a drift current from left to right, so both these should be added algebraically, now what is the sign of this, so you can see here that the electric field doh psi S by doh X, now psi S is surface potential is less here and more here okay, this is evident because the field has to be from higher potential to lower potential, so therefore if I take this doh psi by doh X this quantity will be positive but inversion charge is negative, so this quantity is negative, now similarly here if I take the doh by doh X of QI okay, I will find that this quantity though QI is negative doh by doh X of QI will be positive, why because the inversion charge is decreasing from source to drain in the X direction okay, now since this is therefore positive and there is a negative sign here, so this quantity is negative, this is also negative, the overall thing is negative, so when I take the modulus as shown here what I should do is I should take the negative of this because that will be positive, so I put positive sign here and negative sign here okay, so that is my current equation okay, now let us go back to the slide and compare this with what we have got, so you can see here that is equation written here okay, so INX as a function of X as we have seen is constant and it is equal to the terminal current IDS, W into mu NS which is a function of X minus QI into doh psi S by doh X plus VT into doh by doh X of QI okay, now let us look at the Y dimensional voltage equation, now this is obtained by integrating electric field in the vertical direction okay, now this is straight forward and as I have shown in one of my earlier slides that psi p, psi ox, psi S okay, so you can it is easy to see that how VGB is split up in this, so I am not going to do this integration of electric field in the Y direction analytically, you know that when you integrate this electric field you get this potentials right and they are all in series, so you are summing them up, so really the derivation of this Y dimensional voltage equation is straight forward, let us go to the next step, now solution of the above equation how do we solve, now let us go to the solution of this equation, so what is suggested here is the following, so you move the mu NS by DX term from the right hand side to the left hand side, now where do you get this DX term from, you can see that you have here doh by doh X of psi S X and you have here doh by doh X of QI, so there is a DX term in the denominator here, so that is what is being clubbed along with this term and this mu NS by DX is being shifted to the left and then you perform an integration and then you rearrange, so now let us go ahead and do this exercise okay, so let us perform what was suggested, so doh psi S by doh X, so here we will keep the doh psi and we are pushing the doh X here and therefore from here also we are removing the, this is nothing but doh QI by doh X, so that doh X has been put here and we are saying this we should move to the left hand side, so what does that mean, that means we are going to write this as doh X by mu NS and we have to remove the thing from here, now we have to integrate okay, so you have to integrate in the X direction evidently, so let us put that integral, so you are integrating from 0 that is X is equal to 0 here the source to the end of the channel L, so here this is L, so this is also from 0 to L, now you see this mu NS is not a constant in the X direction, so this is a function of X, the mobility, surface mobility is a function of X, why because the vertical field EY is changing with X and horizontal field EX also is changing with X and mobility is a function of both the longitudinal field as well as the vertical field right, transverse field and therefore this is varying with X, IDS however is a constant, so when you are doing the integration I can move the IDS out, so I can put the IDS here and I can remove it from here okay, so now this term I move back to the right hand side, how do I move it back to the right hand side it will come in the denominator, so what I will get on the right hand side when I move that is the following, so I will get this term inverse of this, so how do I interpret this term, now if I do a simple thing namely I multiply both sides by 1 by L or divide by L both sides, so I will do I will put 1 by L here and I will also put 1 by L here okay and then I will look at this term, so which means when I push this here what I am getting is this reciprocal right, that is what is coming here I can show that this quantity can be shown to be an average value of the surface mobility okay, now this is a very interesting kind of average, so what you are doing is you are taking the reciprocal of the mobility integrating that reciprocal over X and dividing that integral by L, so what happens is your length dimension gets cancelled, so you can see here the dimension of this whole quantity will be mobility because here mobility in the denominator and you are taking reciprocal ultimately and the length dimension is getting cancelled, so this is really the reciprocal of the average of the reciprocal of the mobility over the channel length, please look at this, this quantity here is the reciprocal of the average of the reciprocal of the mobility okay and therefore this is a interesting kind of average of the surface mobility okay, now at this point I just want to mention that normally what happens is since we use the arithmetic average very often we think the term average means arithmetic average right all our marks and so on are calculated by averaging, the marks obtained in various subjects arithmetic averaging, now we must remember that arithmetic average is not the only average, you know for example there is geometric average or geometric mean, harmonic average or harmonic mean and so on, so the averaging operation depends on the situation, so in this situation the average mobility should be calculated according to this formula okay as the reciprocal of the average of the reciprocal, let me write it here reciprocal of the average of the reciprocal, this is surface mobility over and averaging is over channel length that is what this is, so now what I am going to do is I am simply going to remove this term from here and put it here as mu ns average that is what is this, so this is what we end up as our equation for the drain current okay, this is the solution of the x dimensional current equation, now here when you do the integration over L okay you can see that the qi has to be integrated with respect to psi s, qi is a function of x, psi s is also a function of x but you have to do integration of qi with respect to psi s okay, so what we could do is we could write this as follows, so I can write this as W by L into mu ns average, so the first integral I will write as this into minus of qi should be expressed in terms of psi s, do psi s from 0 to L, so at 0 your surface potential is psi s0, surface potential here is psi s0, surface potential is this potential right and here this is psi sL, so this is psi s0 and this is psi sL, so at L you have psi sL plus VTE integrating this from 0 to L is a straight forward thing doh qi, integrate from 0 to L is nothing but qi at L minus qi at x is equal to 0 okay that is qi 0, so this is your equation that is what is in fact shown here okay, this is the equation shown here with the so called average surface mobility and this is the diffusion component of the current and this is going to be the drift component, so now you can see I have to get qi as a function of psi s, so now let us summarize what we have achieved in this lecture so far, so in this lecture we said that the MOSFET problem is basically a 2 dimensional problem, so derivation of the MOSFET drain to source current as a function of gate, drain and source bias is a 2 dimensional problem consisting of the x-directed current IDS created by VDB and VSB modulated by y-directed field created by VGB and the approach is to approximate this as 2 coupled 1 dimensional problems in which one 1 dimensional problem is the solution of the x-dimensional current equation and the other 1 dimensional problem is solution of the y-dimensional voltage equation, the y-dimensional voltage equation gives you the solution of the y-dimensional voltage equation gives you psi s and qi as a function of psi s and V which is fed into this extensional current equation to get the current. So we can express the terms psi ox and psi p in terms of psi s and that is how the surface potential equation or y dimensional voltage equation can be converted into an equation for psi s. Now in our course we are going to neglect the potential drop in the poly. The coupling of the 2 one dimensional problems via psi s makes MOSFET modeling essentially a surface potential based problem with the VT based and other formulations as derivatives. So the fundamental solution the basic solution of the MOSFET problem is a surface potential based solution. Threshold voltage based models and so on can be shown to be derived or derivable by approximations of the surface potential based equation. Then we looked at how the extramational current equation can be derived based on the approximations namely the current density is due to extracurricated current alone and that to exerted current of electrons alone. So we do not consider the whole current at all. And then we neglect the generation recombination and tunneling so that the continuity equation for electrons reduces to divergence of Jn equal to 0. And this in turn results in the result that the current Inx that is the current of electrons as a function of the position in the channel is a constant ok. This is given by W into the integral of Jnx over the y direction from 0 to bulk. And then we use the char sheet approximation right combining all these elements we can get this extramational current equation. The surface potential equation or y dimensional voltage equation is obtained by integrating the equation equal to minus grad psi from get to bulk. And finally we showed that the extramational current equation can be solved by moving the surface mobility term together with the dx term from the right hand side to the left hand side integrating it integrating the result and rearranging the same to get it in this form where the crucial thing is that the average surface mobility is given by this formula which is the reciprocal of the average of the reciprocal over the channel length.