 So, in the previous module, we discussed the qualitative model of a large uniformity dope bulk MOSFET. As we have remarked, first we are going to consider the DC characteristics. In this module, we will consider the equations, boundary conditions and approximations which are to be used to model the MOSFET. Now, let me remind you our device modeling procedure which is abbreviated as square bar strip. So S here stands for structure and characteristics, Q stands for qualitative model which is what we have completed so far and now what we are doing is these equations, boundary conditions and approximations. So, the learning outcome of this module is as follows. At the end of this module, you should be able to do the following for a bulk MOSFET with uniform substrate doping and large channel length L, channel width W, oxide thickness T ox under steady state conditions. So, what should you be able to do? So at the end of this module, you should be able to explain the boundary conditions and approximations related to the above distributions employed to solve for the device current from the drift diffusion transport equations. That is our goal here. So let us start with the equations. Now we have laid out these equations in an earlier module. These are basically the equations for the current density. So these two are equations for the current density Jn and Jp, electron current density and hole current density and then you have the equation related to the electric field. Now these are called electrostatic equations. How do you find out the current though the current I in the device, right? So let me sketch the device here. So this is your MOSFET, this is schematic and you are applying voltages to all these three terminals with respect to the bulk. So you apply let us say Vdb, Vgb and Vsb, b stands for bulk, d is drain, g is gate and s is source and you are getting a current Id, drain current Id. So you want to model this drain current, okay. So this drain current is the current I here. So this current is obtained by integrating the current density J which is sum of Jn and Jp. This is Jn and this is Jp. Over ds that is the contact area, okay. Now as far as electric field is concerned, how do you calculate the electric field? So you get the electric field or rather how do you use the electric field to get the potential because it is current as a function of voltage. So let me go back to this diagram here. So we want this Id as a function of Vdb, Vgb and Vsb. So we want to relate the current to the voltage. So how do you get the voltage? Let us say I want to get the voltage between the drain and the source. So how do I get that? So I will solve for the electric field E and using this equation I can get psi by integrating E as shown here. So E dot dl where dl represents the differential length along the path of integration. So for example if I am going to integrate from drain to source, this is the path, right? So the arrow here shows the path. So along this path you are integrating the electric field. That is what is shown here. That is how you get the potential difference between drain and source. So like that you can get the potential difference between any two points. I can get Vdb, for example Vdb means potential difference between this and this, Vgb means potential difference between gate and bulk. So I have to go along this path and do the integration. Further when we are solving the continuity equations, here the delta N is the excess electron concentration and that is given by N minus N0 where N0 is the equilibrium carrier concentration of electrons. Similarly delta P is given by P minus P0 where P0 is the equilibrium concentration of holes. And tau is the minority carrier lifetime, right? This tau here is the minority carrier lifetime. The divergence of E here, please ignore this rectangle. Here what you have is a dot like this, okay? Let me place the dot here. So there is a dot there. So del dot E, divergence of E is rho by epsilon s where rho is the space charge given by the sum of positive charges minus the sum of negative charges, okay? Here too you please ignore this rectangle which is an artifact coming up. All you have here is a dot showing divergence of Jn and divergence of Jp. So these are our equations starting from which V model any device. So even for MOSFET, these are the equations which are used at the starting point. Now let us go to the boundary conditions. Here is your device structure. So we are starting with the boundary condition on the potential psi and we are starting with the potential psi at the contact. Now at this point let me take you back to the equations. So here you can see that these 6 equations can be converted into essentially 3 equations. So how do you do that? So you take the equation for Jn and substitute it here. Then these 2 equations get compressed into 1 equation. Similarly you take Jp and put it here. You get 1 equation for holes and then you take this equation and put it here, okay? And then you get let me put that dot there. So you take this equation and then put it here. So when you do that, since the Jn is in terms of n, your equation here will be converted into an equation for electrons and similarly you will get equation for holes and similarly when you put this here you will get an equation for psi, okay? So to repeat from an earlier module you have 3 equations for modeling devices either n or Jn, p or Jp and e or psi, okay? So there is an r here. So either one of these 2, one of these 2 and one of these 2. So I can use for example n, p and psi, okay? As the variables. I could similarly use Jn, Jp and e. Other possible combinations are also there. Now at the contacts we use np and psi, okay? So psi at the contact is given by the sum of applied and built-in potentials experienced by a positive test charge moved from the point of reference that is 0 potential along the circuit to the contact. So this is the approach to get the potential at any contact. Now let us follow this approach and see what are the potentials at the various contacts. So we are choosing the bottom contact as the reference. So that is why here your potential is 0. Supposing I want the potential at source that is let us say I want this, how do I get this? So I move along this path from bottom contact to the source contact. Let us see what will happen when I do that. So when I do this first I will get a built-in potential between n plus and p. So this region is p type, okay? So I will get a built-in potential between p and n plus here. What will be the direction of the built-in potential? If I am moving from n plus to p the built-in electric field is from n plus to p so I will experience a fall in potential, okay? And then when I go from p to n plus I will experience a rise in potential because n plus side is positive with respect to p for the built-in potential, right? So if I now assume that the doping in this n plus source and n plus drain is the same as the doping in the poly either at the gate or at the bottom here then this built-in potential here for the source bulk junction is the same as the built-in potential for the poly n plus poly p junction. And since they are in the opposite direction when I move along this path the two will cancel each other that is why in size source you are not getting any built-in potential component and you are getting only the applied voltage that is VSB, okay? So you are applying a voltage between source and bulk that is something like this. So this is your VSB, okay? Now similarly you get the side drain as VDB you move from this contact first you will experience a fall in built-in potential then you will experience a rise in built-in potential here and then since these two potentials will cancel all you are left with is the applied voltage VDB for your benefit let me mark the built-in potential directions here. So built-in potential direction here is like this and here it is like this same thing applies here. So we are saying that this when I move along a path like this, this and that get cancelled. So if I want to show VDB then it would be like this, this is VDB that is what has been marked here. Now let us look at VGB that is this, what is the potential of the gate? Once again we move along this path here, what about the potential of the bulk, psi bulk? So again let me mark the substrate here P type. So potential of the bulk is modulus of sorry minus of modulus of phi MS. Now what is phi MS? Phi MS is nothing but the potential difference between metal and semiconductor, bulk function difference between metal and semiconductor. We have shown this in an earlier lecture right. Now M here stands for the poly region right, metal is actually the poly here, metal gate is the poly gate here. So phi MS, phi M minus phi MS is same as phi M that is the work function of this region minus work function of this region. So it is nothing but the built in potential whose polarity is like this and therefore the potential of the bulk with respect to bottom contact will be negative right according to this. Now often psi bulk is taken as 0 okay that is the bulk is taken as the reference in which case boundary potentials are given by potentials obtained assuming the bottom contact potential psi bottom is 0 plus the magnitude of phi MS. So if I want to now estimate these potentials here for gate, for source gate and drain with respect to the bulk. So I have to change the bulk to 0 then all other things will change as follows okay. So all that we need to do is to add the magnitude of phi MS to each of these because that is how you will make the bulk potential 0 right. So here you can see if bottom potential is 0 then bulk potential is minus of modulus of phi MS. If I want to make this 0 I should add here phi MS then it will become 0 plus mod phi MS if I add it becomes 0 that means I should add mod phi MS to each of these okay to get the potential. So here I am adding mod phi MS here I am adding mod phi MS okay and here so here also I have to add okay. So when I add a common potential to all of these of magnitude phi MS then this will become 0 okay so that is how you get the potentials shown in this slide when the bulk potential is 0. So we shall be using this reference once again here this is the P type substrate. Now let us look at the boundary conditions on electrons and holes at the contacts. So first we are considering the contact region of the device and then we will be considering the non-contact region right. So this device has contacted boundary and non-contacted boundary. So here this is contact, this is contact, this is contact and this is contact. All other boundaries such as this one, this one okay and so on they are non-contacted. So coming back to the boundary conditions and the contact, first we have discussed boundary condition on psi. Since there are 3 differential equations to be solved you need 3 boundary conditions. So we have completed psi now we are looking at N and P. So what are the N and P conditions at contacts? This is fairly straightforward at each of the contacts because they are ohmic you assume the electron concentration and hole concentrations to be equal to their equilibrium values this is what you find here okay. So whatever is the electron concentration at this boundary here that inside the semiconductor that is the boundary condition at the contact and same thing applies for holes. Now let us go to the non-contact. It turns out that while for the contact it is easier to talk in terms of potential and carrier concentration at non-contacts it is easier to talk in terms of the electric field and current densities okay. So we can easily identify the values of electric field and current density at the non-contact. That is why here we are choosing a different set of variables namely E, JN and JP. Now let us start with E at the non-contact. So examples are shown here. Let me take the important interface that is this non-contact interface. So what this particular equation says is that if you want to know the electric field in the y direction at the surface. So let me now remark here which is the y and which is the x direction because electric field is a vector so it has direction. So we are assuming that this direction is y and this direction is x. So horizontal direction is x vertical is y. So that is what is shown here the y direction. So E ys means a field in this direction at the surface s okay because we are talking about a boundary condition at this surface. So that field will be related to the field in the oxide in the same direction here. So field here so E ox would be this field in the oxide but at the boundary please understand this at the boundary. Now as we will see the since there is no space charge field in the oxide is constant that is why we have not put a surface s here right. So through this across the entire thickness field in the y direction will be constant okay. So E ox plus Q f the Q f is this fixed charge okay. So in other words the y directed field is contributed by both the oxide field as well as the fixed charge right this is nothing but the Gauss law and you have to take into account the dielectric constants in silicon E epsilon s and dielectric constant in oxide. Now let us look at this boundary non-contacted boundary. So here the relation is epsilon s E x is equal to epsilon A E Ax. So E x is a field that is shown here inside the semiconductor region or poly region at the boundary and E Ax is a field in the ambient in the x direction that is the field here like this in the ambient outside the device but at the boundary. You can similarly see this interface between poly and silicon dioxide. So here E y in the poly and E ox in the oxide so E y in the poly is this white arrow and E ox is this arrow taken here right. So this is a relation between E y P and E ox you can see the boundary condition here epsilon ox E x is equal to epsilon A E Ax that is similar to this okay except that here we had put silicon because we are considering the poly region which is semiconductor region here we are considering oxide region right. So this epsilon s here is replaced by ox epsilon ox okay. Finally you can look at this boundary here it is epsilon s E x this is your E x is equal to epsilon A the ambient dielectric constant into electric field in the x direction in the ambient that is that will be this field here at this point. Now whatever we have written here same thing applies at this non-contact similarly whatever we have written here this boundary condition applies at the corresponding parallel contact here and finally this boundary condition here applies at this contact okay that is how you get the boundary conditions on the electric field at the non-contact. Now let us look at boundary conditions on the current density at the non-contact J n and J p this is very straightforward as shown here J n x and J p x both are 0 at all the non-contact so no current can escape out of a non-contacted boundary current can only go out of the device through the contacted boundary okay. Current density is a vector therefore you have x and y components I want to emphasize so x is this direction horizontal direction this is x okay and this is y direction. So you can see that in y direction here at this non-contacted boundary current is current density is 0 J n y and J p y are 0. Now let me make the point here that if there is a tunneling current from gate to bulk okay or substrate then this condition will not hold it will be violated okay so you will have a non-zero current density. So here we are assuming that the tunneling current from the gate is 0 in other words there is no current from the gate to the bulk for any reason tunneling or for any other reasons. So let us now summarize all the boundary conditions at contacts and non-contacts okay now this would be the picture we have taken the bulk region potential as 0 okay so psi bulk is 0 that is our reference and that is how these potentials bottom then source gate and drain are decided okay. So I will leave this slide for a few seconds so that you look at all the contacted and non-contacted boundary conditions okay. To repeat all the contact boundary conditions are in terms of potential and carrier concentration you can see that this is a contacted boundary potential carrier concentration potential carrier concentration another contacted boundary potential carrier concentration this contacted boundary potential and carry concentration whereas all non-contacted boundaries it is electric field and current density electric field current density here electric field current density electric field current density right this non-contacted boundary that is how you are going to use the boundary conditions on your equations for solution ok. Now let me give you an assignment to check whether you have understood the concept that we have discovered. So this assignment is that compare the values of psi bulk psi bottom psi gate and psi drain that is the potentials at the various contacts and psi source in an n type MOSFET having n plus poly as a bottom contact with those in an n MOSFET having an ohmic contact at the bottom. So we have just now discussed the boundary conditions for this kind of a structure I am just replacing the n plus poly by an ohmic contact here at the bottom. If I do that how will the boundary conditions of the various contacts change? So I leave it to you to figure out. Now let us discuss the approximations because E, B, A equations boundary conditions approximations. Let me recapitulate from our previous course or rather a previous part of the course. So you have approximations related to the qualitative model and the approximations related to the quantitative part right. So therefore we can say the approximations related to the qualitative model are approximations set 1. Now these approximations decide what kind of equations you will use. So we have decided to use the drift diffusion equations ok and therefore there are a approximations related to arriving at these equations. These approximations will be used to get the drift diffusion equations and boundary conditions and you can solve these equations without any further approximations using a numerical solution. So I have shown you the equations in the previous slide right. So those equations if you want to solve them as it is without doing any further approximations you can use the boundary conditions we just now discussed. Apply it to those 6 equations and get a solution. However you cannot get an analytical solution because the situation is fairly complicated in terms of the number of terms in the equations. So you will have to only do this by using a computer right and therefore you can get a numerical solution of those equations. To get an analytical solution you have to make further approximations. So you take this drift diffusion equations and boundary conditions and approximate the terms of these equations. This is called the Aprogram capsule set 2. These are the approximations of the drift diffusion equations and boundary conditions. So you have a Aprogram probation set 1 which lead you to these drift diffusion equations and boundary conditions. And then you have a Aprogram flame set 2 which are the approximations of the equation so that you get analytical solution. Now let me just tell you what are the sets of these approximations and then we will discuss this in detail. So approximation set 1 which lead to the drift diffusion equations this is just recapitulation from our previous part of the course okay. So I am just going to state these approximations here. So if you take the electrostatic equations E equal to minus grad psi and divergence of E is equal to rho by epsilon here there should be a dot and ignore the rectangle that is coming there. Now using the electrostatic equations means in the Maxwell's equation you are neglecting the magnetic field. In other words electric field has no circulating component that is it arises from static charges only. Now you consider the current density equation J equal to some of the electron current density and hole density. When you write this equation you have made the approximation that E is quasi-static on the scale of dielectric relaxation time. Please refer to the earlier module on characteristic times and lengths where we have defined what is dielectric relaxation okay. In other words we are saying displacement current is small. Now look at the current density and continuity equations for electrons and holes. When we write these equations okay we have made the following approximations between two scattering events carriers are particles with an effective mass determined from their wave nature. Then volume averages of concentration momentum and kinetic energy of carriers are used ignoring their standard deviation. This temperature T L is quasi uniform that is thermoelectric current is small. So you see that there are no thermoelectric current terms in the current density equation for Jn and Jp. You have the diffusion and you have the drift there is no thermoelectric current here or here. I is quasi-static on the scale of momentum relaxation time. So the current I is quasi-static. The term W the term W that is energy density right is quasi-static on the scale of energy relaxation time and quasi-uniform that is the drift energy is much less than the thermal energy okay. So drift energy is because of the directed motion of carriers whereas thermal energy is because of random motion of carriers. You recall we have been saying that the transport of carriers or the movement of electrons in semiconductors is due to a directed motion superimposed on random motion okay. Now let us look at approximation set 1 which lead to the ideal boundary condition. So how are things different for ideal contact and ideal non-contact this is what is put here. So you have no restriction on J an ideal contact can take any amount of current density however an ideal non-contact no current can escape from there. Ideal contact has no contact resistance the contact resistance term is not relevant for non-contact. The surface charge at an ideal contact is zero it is also zero for an ideal non-contact. The surface potential psi s is pinned to the applied voltage and on the other hand there is no restriction on the surface potential at a non-contact like there is no restriction on current density at a contact there is no restriction on the potential at a non-contact it can be any value. Surface recombination velocity is infinite at an ideal contact that is why your carrier concentration is equal to the equilibrium value. At an ideal non-contact there is no surface recombination finally for an ideal contact and non-contact structure the ambient dielectric constant is taken as zero which means that you are assuming no electric field escapes from the device okay this is not real but this is an ideal condition that we assume to simplify things. Now the above approximations lead to the following boundary conditions at the ideal contact and ideal non-contact. At the ideal contact carrier concentration N and P are equal to the equilibrium value and the potential psi is sum of the built-in potentials and the applied voltage is repetition from our previous discussion in this module. On the other hand at an ideal non-contact the gradient of electrons and holes or gradient of carrier concentration normal to the contact surface that is why you have this perpendicular sign okay perpendicular means normal so gradient normal to the contact surface of electrons and holes they are 0 gradient of psi normal to the surface is also 0 that means the electric field is also 0 in the ambient right so no electric field is escaping that means electric field inside the semiconductor at the contact is also at a non-contact is also 0 because no electric field escapes from the device. What is of interest to us in this course greater interest is the prog mission set 2 related to the drift diffusion equations because these are the equations we are going to solve recapitulating from one of our earlier slides we know that in any device you can have a space chart region and a quasi-neutral region you can divide any structure right device structure into these parts okay so for example if you take the p-in junction this is a space chart region right and this and these are quasi-neutral regions now if you put a metal here and you consider contact then here also you will have a space chart region right at this contact also okay so these are the space chart and neutral regions in a p-in junction you can similarly think of other devices and identify the space chart and neutral regions in a MOSFET for example if this is source this is drain this p-type substrate now this is your space chart region okay all of this is your space chart region you will have some space chart region on the n plus side also whereas the remaining part here is neutral okay and so on you can similarly consider the oxide poly and sketch the spatial and neutral regions there okay so these are the basically the spatial and neutral regions in the device the spatial region could be either p-type or n-type the quasi-neutral regions also could be p or n-type right therefore you have basically these equations which have to be approximated in each of these regions so this is the framework for any general device okay so you can have an approximation related to each of these blocks so you can approximate the current density in some way for p-type space chart region you may have the same approximation or a different approximation on the n-type space chart region and so on so each of these blocks you will have an approximation in principle some of these approximations may be common okay so same approximation may be used in different blocks let us see for the MOSFET what do we do for the MOSFET we have to consider only the space chart region of the p substrate beneath the gate okay let me sketch that here this is the p substrate this drain source this is the space chart region right which may consist of inversion layer depression layer and so on but this is the region of interest to us this is what we are talking about here now in this region since current densities and electric field are vector quantities you have to consider both x and y direction so this is x direction and this is y direction okay so we have to consider things in both directions you may say why are we not considering z direction actually the MOSFET is a 3 dimensional device it just means that in the z direction or the direction perpendicular to the slide we assume all conditions to be uniform so let us start with the current density of electrons what approximation do we make so we assume that the current density of electrons is approximately in the x direction in the space chart region right though it is a function of both x and y so this i is a unit vector in the x direction this is Jnx okay and xy so if you take this if you take this point as the origin this is y this is x so at each of these points here your current density in the x direction would be different so that is why you have Jnx as a function of x and y so we are assuming there is no current density of electrons in the y direction why is it so because we are neglecting the current which may be going out of the bulk contact and similarly current going into the gate contact right so gate leakage is neglected and bulk current also is neglected we have discussed this point in the qualitative model so electrons only flow in the x direction okay then what about y direction so just now we said that the current density of electrons y direction is 0 that is what is said here so Jn y at any point x,y is 0 then another approximation we make related to the current density is that we are making the charge sheet approximation so we have said that the inversion layer will be regarded as a sheet of charge right so it has a finite thickness in the y direction we are going to assume that all the inversion layer is concentrated at the surface therefore the x directed current will flow only over a negligible thickness at the interface so in the y direction therefore your current is restricted only to the surface that is the meaning what about continuity equation we neglect generation recombination and tunneling in the space chart region because we are neglecting the bulk current okay and therefore the continuity equation reduces to divergence of Jn is equal to 0 right because in the continuity equation other terms are 0 this steady state so dou n by dou t term is 0 generation term is 0 recombination term is 0 right so divergence of Jn is equal to 0 now what does this mean this means in the x direction Inx is constant independent of x that is the current due to electrons in the x direction please note the difference between current density Jn and In current density is current per unit area In is the current over the entire area right you integrate the current density over the area then you get Inx so this is constant independent of x from source to drain so the current density what we are saying is in the x direction here is constant so you may change your location x but the current i does not change total current i because no current is flowing either in this direction or into the gate right so the current in x direction is constant bulk current is 0 that is this current this is bulk current and gate current also is 0 okay now what about whole current density so the whole current density is 0 we just do not take into account the current density of holes either in the x or in the y direction therefore the continuity equation also is not considered right so we are not really bothered about holes okay this is an approximation now as far as the equation E equal to minus grad psi is concerned we are going to use it as it is and therefore there is no approximation related to this now in the x direction when you use E equal to minus grad psi there is no approximation we are going to use this equation in the y direction it leads to what is called the surface potential equation E equal to minus grad psi in the y direction so this is your y direction this is your gate voltage this is your bottom so when I apply the E equal to minus grad psi in this direction I will get what is called the surface potential equation we will see what that is Gauss law in the x direction we are not using any approximation in the y direction we are going to use the gradual channel approximation for left hand side of the equation okay so what is what is this approximation so let us sketch draw the Gauss law equation divergence of E is equal to rho by epsilon now since we are considering both y and x directions this divergence of E is dou E x by dou x plus dou E y by dou y so you have 2 terms there so left hand side of this equation is divergence of E so we are going to use gradual channel approximation that is we are going to neglect one of these two terms now which term do we neglect so let us look at the structure so you have field in the x as well as y when you are drain voltage you have a drain to source voltage then the field you know will be drain voltage is more than source voltage your field in the x direction is like this E x and then you have a gate to bulk voltage therefore your field is also there in y direction so what we are going to assume is that the dou E y by dou y is much more than dou E x by dou x in other words the E y varies rapidly in the y direction much more rapidly than E x varies in the x direction okay now why is it called the gradual channel approximation because the x direction is along the channel we are saying that conditions along the channel in the x direction vary much more slowly than in the vertical direction in other words the conditions along the channel vary much more gradually the electric field variation along the channel is much more gradual than the electric field variation in the y direction from gate to bulk now this is what we are going to assume that this is what is called the gradual channel approximation so this is valid for the left hand side when you do use this approximation this term will go away now on the right hand side of the equation we use the charge sheet approximation and depletion of the approximation let me write the equation again now left hand side we said this is dou E y by dou y we are neglecting the x term right hand side rho so let us write rho it is q into p plus n d plus but there is no n d plus here minus n minus n a minus that is rho so charge sheet approximation means that we are going to assume that the inversion layer of electrons is close to the interface and depletion approximation means we are going to assume that it will all be depleted of holes right so when I take this space charge region in a MOSFET so here all the inversion layer electrons will be close to the interface and remaining part of the space charge region here will be depleted of all electrons and holes okay that is the depletion approximation so there are the two approximations we are going to use for the right hand side okay of the equation Gauss law in the x direction we are not using anything right now towards the end of this module let us discuss how the various MOSFET models have evolved in the context of these approximations the very first model that was proposed for the MOSFET was by Pao and Sa and this was published in SSE that is solid state electronics okay so reference is given there. Now let us see what are the approximations that were employed in this model and what were the solution features so this particular model employed the gradual channel approximation it assumed that in N channel MOSFETs you will have only Jn you will not bother about Jp in P channel MOSFETs there will be only Jp you will not bother about Jn okay so in the previous slide when we discussed the approximations they were meant for the N channel MOSFET we are going to take N channel MOSFET as an example right and discuss everything for that we have remarked this in the qualitative modeling step itself. Now it neglected the generation recombination the spatial region and assume that the mobility is a constant does not change with location of the point in the channel right so mobility is constant all along the channel constant effective mobility and as a result of the approximations what the solution what was the solution like so it was a numerical model because these approximations are not sufficient to allow you an analytical solution okay another feature was it required an estimation of a double integral there was a double integration involved okay and on the other hand however this was regarded as a standard model for long channel devices it became the gold standard so all further models which were developed always try to show that though their model is simple it is as accurate as this Pausa model. So Pausa model was a reference you want to show your model is accurate you just compare with the results of Pausa model and then you can say that yes my model is as it gives the same results as Pausa model right so this is a reference with which all other models were compared in 1978 John Bruce proposed what is called the charge sheet model this was again published in solid state electronics okay now what were the features of this it made one additional approximation as compared to the Pausa model and that was the charge sheet approximation for the inversion charge so Pausa did not make the charge sheet approximation for the inversion charge right it did Pausa did not assume that the entire inversion charge would be stuck to the interface as a thin layer of 0 thickness so this charge sheet approximation simplified the solution a lot so even though this was a numerical model it was much simpler than Pausa model it removed the double integration right so Pausa model you had to do double integration this is a previous slide. So the double integration was removed that was a simplification achieved by assuming the inversion charge to be charge sheet but its results were almost as accurate as the Pausa model so after 1978 people would compare their model with the Bruce model and say that you know their model is as accurate though it is simpler now while these physics based models were being developed so first model was in 1966 you know we have discussed the history of the MOSFET basically the MOSFET became very important after the invention of the integrated circuit around 19 early 1960s 1961 and so on so 66 was the first model that was given for the MOSFET physics based model and an approximation to this was given in 1978 by John Bruce but the MOSFET had to be used in a circuit and these physics based models were not computationally very efficient right to calculate currents and voltages and so on in for design purposes quickly okay therefore parallely there was an effort to develop simpler models so first simple model suitable for circuit simulation was proposed by Shishman and Hodges okay and this model was perplexed in the general of solid state circuits IEEE general of solid state circuits because this was suitable for circuit simulation it was published in general solid state circuits now what were the approximations and solution features let us compare with the Bruce model so additional approximations that were made were the following the depletion approximation for the depletion charge assume only the drift current neglect diffusion current assume that the drain current saturates due to pinch off that is assuming the inversion charge goes to 0 at the drain that is a pinch off condition and the depletion charge is constant from source to drain equal to the value at the source so let me depict this pictorially source drain substrate this is depletion region so when saturation happens the inversion charge here goes to 0 this is the inversion charge okay so idea saturates due to pinch off and QB constant from source to drain so actually you can see this is QB all this this is QB right now we are assuming that this is constant actually it is not constant okay it is increasing from source to drain but this Shishman and Hodges model assume it to be constant equal to the value at the source and further this current flowing along the channel was assumed to be only because of drift though you know the charge is varying concentrate charge concentration is varying from source to drain so definitely there will be diffusion current but they neglected the diffusion current so what are the consequence so the solution features so you got the first analytical model right because of a large number of approximations as we have remarked earlier you have few approximations you have a complicated equation and you want a simple equation you will have a large number of approximations then it was a regional model this means that there was a separate model for linear and saturation region so if this is your current voltage curve ID VDS let me put IDS because drain to source current we are talking about we are neglecting bulk current and so on so IDS VDS then there was one equation which would work for this so called linear region and another equation which would work for the saturation region this is what is meant by regional model different regions different equations then it neglected the drain to source current in sub threshold so for threshold conditions when you measure the current at threshold voltage you will still get a small nonzero current okay so they assumed however that at threshold the current is below threshold there is no current then they got a simple equation analytical model and it was a square law IDS VDS so current varies as a square of the drain to source voltage this was the model which introduced the concept of threshold voltage right if I see the ID versus VDS this ID versus VDS with VDS as a parameter instead I can plot ID versus VDS and it would be something like this so they said okay so let us call this as a threshold beyond which you have a current right below this you assume then no current so they introduced this concept of threshold voltage unless your VDS exceeds some value you do not get significant current right they said zero current until VDS is less than VT and beyond VT you get current now this model was valid for large devices in 1971 a simpler model for circuit simulation was proposed this was called the mayor model now what were the features of this model right so the approximations so what mayor showed was you need not do this approximation of QB constant from source to drain you can assume QB to be varying right and still you can get an analytical model so what was the result the square law model of Schismann-Hosch was replaced by a 3 by 2 power law model however there was no threshold voltage term in the drain to source equation whereas Schismann-Hosch model there was a threshold voltage term that was the result so these are so called 3 by 2 power law model we will derive this model and we will show how there is no threshold voltage term now finally came the velocity saturation model okay so what was the features of this model the approximation was that rather than pinch off idea saturates due to saturation of the drift velocity in the channel however it assumed the QB to be constant from source to drain equal to the value at this source so let me explain that so this is the inversion charge now saturation is not occurring because inversion charge is going to 0 in fact it is not going to 0 at the drain there is a region here where the drift velocity VD reaches saturation so you know the velocity field curve drift velocity versus electric field I am putting a modulus because electrons flow in opposite direction to the field so this velocity saturates at V sat when the field is high so in the qualitative model we have discussed how the y directed field that is this field I am sorry x directed field in this direction this field is low at the source goes on increasing and becomes high at the drain so once the field exceeds the critical field your velocity saturates so that is the reason for saturation of the current because of velocity saturation however the model could not accommodate a varying QB because it wanted the people who developed this model wanted it to be analytical and so they assume the depletion charge to be constant right from source to drain so what are the features of this model so what the model achieved with the help of this particular condition that idea saturates due to velocity saturation and QB constant from source to drain is that they achieved a square law ideas VDS model okay instead of the 3 by 2 power law and moment it is square law you know it becomes threshold voltage based so threshold voltage term reappeared in the equation however now this model advantage was it was valid for both long as well as short channel devices so while the long channel device use the pinch of criterion for current saturation you get the same result as you use the velocity saturation condition right so velocity saturation condition and pinch of condition give approximately the same current for long channel however for short channel the conditions are the currents are very different okay and this correctly predicted the current for short channel devices the velocity saturation model it reduces to shishman hodge model for long channel devices now with that we have come to the end of our module so let us make a quick summary of the key points at the end of this module hopefully you are now able to do the following for a bulk MOSFET with uniform substrate doping and large channel length channel width and oxide thickness under steady state conditions so you should be able to explain the boundary conditions and approximations related to the above distributions employed to solve for the device current from the drift diffusion transport equations so we considered the boundary conditions first we started with the boundary conditions at the contact and we said this boundary condition can be obtained by moving along a path from the reference point and summing up all the applied and built in voltages along that path so using the bottom contact as ground we got the potentials at source bulk and drain as given here so at source it is VSB in the bulk it is minus of 5ms where 5ms stands for the built in potential of metal to semiconductor contact that is this this but this is the same as this because this contact because you have the n plus poly here as well as there at the gate your voltage is VGB and at the drain it is VDB however it is more common to use the bulk as a reference point in other words you make psi bulk equal to 0 a consequence of this is that the modulus of 5ms gets added to all the potentials potentials at all the various contacts and these are your potentials you know you add the boundary conditions on the carrier concentration at the contacts you get this picture the carrier concentration at the contact is equal to the equilibrium value at the non-contacts it is easier to talk in terms of current density and electric field as compared to the case of contacts where you talk in terms of carrier concentration and potential so at the non-contacts you have the current density normal to the contact as 0 as far as electric field is concerned use the Gauss law right and then you write the electric field in terms of the dielectric constants on either side of the contact right that is what you find here these are all your boundary conditions at the contacts and non-contacts then we discussed the approximations used to derive the model you have two sets of approximation, approximation set 1 is that which is employed in the qualitative model and it leads to the equations that you use for solution the equations that we have decided to use are drift diffusion equations right so this approximation set 1 leads you to the drift diffusion equation and boundary conditions which if you solve without any further approximations that is if you follow this path then you can only get a numerical solution if you want an analytical solution then you have to take these equations and boundary conditions and make further approximations of these after which you will get an analytical solution now we focused on the approximation set 2 because we decided that we are going to use the drift diffusion equation and then solve for the model so these were the approximations we said that you have to consider only the spatial region of the MOSFET in the p substrate beneath the gate we are discussing the model considering the n channel MOSFET as an example and then you have to consider both the x and y directions so the MOSFET modeling problem is inherently at least a 2 dimensional problem in the z direction we are assuming conditions to remain constant so for these 6 equations we identified these set of approximations x direction and this set of approximations in the y direction in the x direction we assume that the current density is due to electrons alone you do not bother about holes at all and further this current density is in the x direction ok in the y direction the current density of electrons is 0 and we do not bother about the holes at all and we assume that all the carriers in the inversion layer are concentrated in a charge sheet we neglect the generation recombination tunneling as a result we get the continuity equation as divergence of Jn equal to 0 whose consequences are that the current of electrons in the x direction is constant independent of x and since there is no current in the vertical direction the bulk current and gate currents are 0 we are not considering the holes at all as far as equal to minus grad size concerned it leads to the surface potential equation in the y direction we have not discussed these equations we said that we will talk about these equations in the next lecture for Gauss law we make the very important approximation in the y direction left hand side the divergence of E term is replaced by just the variation of electric field in the y direction so dou Ex by dou Ex plus dou Ey by dou Y is approximated as dou Ey by dou Y you neglect dou Ex by dou X you assume that conditions in the channel along the x direction change gradually that is a gradual channel approximation right hand side of the equation space charge term which contains holes electrons and acceptor concentration you assume that the electrons are all concentrated in a charge sheet while beneath the inversion layer of this charge sheet of electrons the entire region is depleted of both electrons and holes that is you make the depletion approximation using these approximations we are going to derive the analytical model in the next lecture finally we discussed the approximations and solution features of the various models that have been developed since the time the MOSFET became an important device so the first model was the PowerSaw model published in Solicited Electronics in 1966 then was the Bruce charge sheet model published in Solicited Electronics in 1978 both these models are physics based models okay they are very accurate but they are based directly based on the physics do not make too many mathematical approximations okay so therefore you can only get a numerical solution for circuit simulation purposes models developed were Schismann-Hoss model in 1968 this was published in general solid state circuit since it is suitable for circuit simulation further simplifications of the models are improvements where the mayor model published in RCA review in 1971 and the velocity saturation model which applies to both short channel and long channel devices so today we use the velocity saturation model okay in this model you assume that the current saturates because in a region near the drain where there is a high electric field the velocity of carrier saturates this condition is used instead of the condition that the charge becomes 0 at the drain right at the saturation point okay that is called the pinch off condition so pinch off condition is no more used in MOSFET modeling because that is not a physical condition the velocity saturation condition is the more appropriate it so happens that the velocity saturation model gives you the same current as the pinch off model current when the channel length is large that is why in the beginning when the channel length of the MOSFET was large people could use an unphysical approximation like the pinch off approximation and yet get currents which were matching reasonably with the measured data.