 In the previous lecture we have sketched the energy bands as a function of position perpendicular to the silicon-silicon dioxide interface. Then we also began sketching the quantities as a function of x that is along the silicon dioxide interface from source to drain. We have sketched the surface potential as a function of x that is along the interface from source to drain and then we have sketched the x and y components of the electric field. In this lecture we shall sketch the spatial distribution of energy bands with x and with x,y that is we shall consider the 2 dimensional energy band diagram also and then we will summarize the module. Let us begin with the plot of energy bands with x. This means we are plotting the energy bands from source to drain along the interface. The interface is nothing but y equal to 0. This is the y direction and y equal to 0 means this interface. Now we shall sketch the energy band in saturation and sub threshold conditions. This means for bias point 2 and 3, bias point 2 is shown here and bias point 3 is shown here. Let us look at the energy bands versus x at y equal to 0. This is your MOSFET. Let us begin with the flat band condition. This means VGB is equal to VFB. The depletion regions at the source and drain are shown here. Source is grounded, the drain is grounded and bulk is the reference so that is also grounded. This is your x direction. First we identify the depletion regions then we sketch the constant Fermi level. Fermi level is sketched constant here because this entire region is in equilibrium. You see for this p-n junction this n-region is grounded and p-region is grounded. For this p-n junction this n-region is grounded and this p-region is grounded. Therefore EF is drawn as a constant line throughout. Next we locate the conduction band edge in the n-region which is shown to be coinciding with the Fermi level because the region is heavily doped and we show the valence band region in the p-region. So the distance between EF and EV depends on the doping in the p-region. We have also sketched the EC in the source region n plus region here it coincides with the Fermi level. Next we sketch EC in the p-region, EV in the n plus drain and n plus source regions. So this EC and EV are located based on the energy gap. We join EC and EV as a continuous line across the depletion region. We have not shown the E0 or the vacuum level because in silicon or rather if the entire region is consisting of one material then the distance between E0 the vacuum level and EC is constant that is electron affinity. So if E0 is continuous EC also will be continuous E0 and EC are parallel. Therefore we do not need E0 if this entire region is made of the same material. We identify this barrier to be Q times psi0 where psi0 is the built in potential of this n plus p junction. So potential drop across the depletion region here along the interface is psi0. Now let us focus on the EC alone. Let us move to the bias point VGB greater than VFB but less than the threshold voltage. This corresponds to the weak inversion condition. In this case you will have a depletion region controlled by the gate and this is the edge of the depletion region shown by the red line. The potential drop in this direction is psiS. In other words psiS is the potential of the surface with respect to bulk. Now you know that because the device is in weak inversion psiS will be greater than phiF but less than twice phiF which is in turn less than psi0 which is the built in potential of n plus p junction. Now we identify the regions controlled by the source and drain. As you can see from here the consequence of raising the surface potential is that the depletion region controlled by the source has shrunk from this blue line to this red line because the surface has become positive, p type surface has become positive. This is equivalent to a forward bias. This is not exactly a forward bias because in forward bias there will be a current flow from p to n but in this device there is no current flow. So from the width of the space charge region that is controlled by the source from this point of view we can say that forward biasing of the or rather raising the potential of the p type substrate at the surface has shrunk the space charge region that is controlled by the source and similarly by the drain. Now if you sketch the new variation of EC it will look something like this as shown by the red line. Clearly the barrier has decreased from Q times psi0 because the space charge region controlled by the source and drain these two have shrunk. The EC has been lowered as compared to the flat band case by psi s because the energy band diagram shows increasing electronic energies. If in a certain region the potential becomes more positive then the conduction bandages in that region will move down. So now the new barrier from the source region to this point at the silicon-silicon dioxide interface is psi0 minus psi s and thereafter along this entire region until this red line you have the potential psi s that is what is shown by this constant line of the EC and then in this region also you have a potential drop equal to psi0 minus psi s. Now let us clear up the slide and every time we show the conditions for a new bias condition those conditions will be shown with red line. So that we can distinguish between the conditions corresponding to the present bias and the previous one. So we can then show the changes very easily. So this is the condition for weak inversion and now we are going to move to the next point when we apply a drain to bulk voltage. Progressively we are moving to the bias point conditions that we had shown earlier that is bias point number 2 that was sub threshold and bias point or rather bias point number 3 that was sub threshold and bias point number 2 that was saturation. Now because of application of VDB the depletion region at the drain has increased consequently the space charge region controlled by the drain near the interface this has also got expanded from the blue line to the red line. Now effect of that will be the potential drop across this depletion region controlled by the drain will be more. So that is what is shown here by the red line okay earlier the potential drop was shown by this blue line. Now the depletion region has expanded and this is the new conduction band H. The difference between the previous EC and the new EC corresponding to non-zero VDB is Q times VDB. So this is the increment in the potential drop across the depletion region controlled by the drain. Now let us clean up the slide show the quasi-formal level for holes. Notice carefully that the quasi-formal level for holes is located at the same place as the EF corresponding to the case VDB equal to 0 and this is drawn as a constant line throughout. Let us understand this point very carefully. Why is EFP drawn as a constant line throughout until this depletion edge okay but not inside N+, because inside N+, EFP corresponds to the Fermi level for minority carriers right. So that we will consider later. If you recall from our previous module when we are drawing the band diagram of a P-in junction we first show the quasi-formal levels for majority carriers okay and then we add the minority carrier quasi-formal levels later. Now there is however no such problem in showing the EFP in this N+, region because here the N+, region is grounded and P region is grounded therefore this particular junction is under equilibrium and therefore the Fermi level would continue to be the same throughout for this junction whereas that is not the same for N+, P junction. Here this is under applied bias, you have applied VDB and this is reverse bias across this N+, and this P junction. That is why when you come to this junction here we show the quasi-formal level for holes in the P region only. Now let us explain why this EFP is constant throughout. Now if you recall the band diagram that was drawn as a function of Y near the channel midpoint you see the EFP was shown as a constant line from the P region right until the interface okay. Now following from here we can conclude that from any point in the substrate when I move to the surface along this line the EFP will be constant. If I consider another vertical line away somewhere here also when I move along this line perpendicular interface EFP should be constant however you know that at all points in this neutral P region the Fermi level for holes should be at the same point therefore Fermi level for holes at the interface all along the interface should be constant right. So let me repeat the EFP is constant everywhere here in this region from any point here if I move vertically up to the interface EFP should be constant as we have drawn in the previous lecture and if EFP should be constant along this line and along this line it should be constant along this line also because here EFP is constant everywhere. Let us now show EFN. Now in the N plus region the EFN coincides with the conduction band edge in the neutral N plus region rather the space charge region in the N plus region is really very very narrow so we have not shown it explicitly. Now as you move into the space charge region on the P side the EFN will continue to be constant throughout this is because of the quasi-equilibrium assumption for junctions under bias. So under quasi-equilibrium the Fermi levels are constant. The split between EFP and EFN that is this distance is equal to the applied bias that is Q times VDB. Now we can join the quasi-Fermi levels across the we can join or rather we can sketch the quasi-Fermi level for minority carriers as a continuous line. So we know that far away from the junction in the N plus region EFP and EFN should coincide because this region will be under equilibrium away and therefore the EFP should be somewhere here far away and EFP until this edge is at this point so I join by a continuous line. So this is how you sketch the minority carrier whole Fermi level, whole quasi-Fermi level in the N plus region and you do the same thing for sketching the minority carrier electron quasi-Fermi level in the P region. You have come up to this point EFN and thereafter you join it as a continuous line until you reach somewhere here where this PN plus junction is under equilibrium. So EFP and EFN will coincide somewhere here close to this. Now these are your EFP and EFN now shown by black lines let us clean up the slide. Now let us change our bias we will now go beyond threshold VGB greater than VTB and if you do that for the VDB condition we will be reaching saturation. The result of that would be expansion of the depletion width from the source to drain because now once your VGB is greater than VTB you have a strong inversion region here at the interface though inversion charge varies from source to drain it decreases but still you have an inversion charge and therefore the potential will vary continuously along this region and therefore the depletion region will continuously expand. So that is what has happened here. What is the consequence of this on the energy band diagram? Now you see when VGB is greater than VTB the portion of the space charge region controlled by the drain has shrunk because the control of the gate has widened right. If you keep the VGB constant and increase VDB then the space charge region controlled by the drain will expand. If you keep VDB constant and increase VGB the space charge region controlled by the drain will shrink space charge region controlled by the gate will expand and this expansion of the gate control region has shrunk the source control space charge region also here. Now as a consequence of this we modify the quasi-framing level position here EFN remains constant until this edge of the space charge region controlled by the drain at the interface and thereafter you have show the variation. Then we show easy variation so you see that the barrier from source to the interface in the P region has reduced because the depletion region controlled by the source has shrunk. Now similarly if you come to this end the variation in the EC is less because this space charge region has shrunk. So this variation across the space charge region is smaller than the variation shown by the blue line which corresponded to sub threshold condition VGB less than VDB. Also you see that in sub threshold the EC was flat over most of the P region whereas here you see in the saturation there is a continuous variation in EC right from the source end. This is because your potential at the surface continuously changes from source to drain because now you have the inversion charge which connects the drain and the source region and that is consistent with the fact that depletion region is also continuously changing. Let us clean up the slide note that the EFP has not really changed throughout all this because it is a concentration of electrons which has undergone lot of change because of the change in the VGB above VTB. Now let us compare the energy band levels associated with the sub threshold and saturation condition. So saturation is shown by the red lines here and sub threshold that is bias point 3 here is shown by the black lines. What can we infer from this? The variations in EC and quasi-familiar for electrons with X which are emphasized here reveal the importance of diffusion current in saturation and sub threshold. Now let us see how this happens. The first point to note is that if there is an existence of gradient of EFN that is if EFN varies with X it means that there is a current Jn okay because you recall from the previous module that current is reflected in the gradient of the quasi-formal level associated with the carrier. So now we know therefore wherever EFN is varying it means that there is an electron current. The second point to note is that at point 2 that is in saturation near the source if you see here EC is parallel to EFN you can see that this EC which is solid line here and EFN which is the dashed line they are parallel near the source here. Now what does it mean? It means that Jn is due to drift please recall the energy band diagrams under various conditions which we have drawn in the energy band module. So if EC and EFN are parallel it means that the current is because of drift. Near the drain EFN diverges from EC so you can see here in saturation the EFN is going like this whereas the EC is going like this so distance between EC and EFN is going on increasing and this increase is quite rapid near the drain. Now this means Jn is due to diffusion as well as drift. There is drift no doubt because if EC is varying with X there is an electric field and there is a drift but difference in EC and EFN if it changes it means that electron concentration is changing with X and therefore there is diffusion also. Point number 3 to note is that at point 3 that is at this point sub threshold this is the band diagram shown by the black line EC is flat over most of the channel okay which means that the drift is very small if EC is flat the electric field in this region is very small and therefore drift current is very small and this amounts to saying Jn is due to diffusion rather than drift because EFN is going on varying with X though EC does not vary so distance between EC and EFN goes on varying which means electron concentration is varying in the channel from source to drain and therefore there is a diffusion though there is no drift current okay so this is how you can conclude the importance of diffusion in saturation near the drain and in sub threshold throughout the channel. We can now summarize by saying we sketch the energy bands versus X at Y equal to 0 in saturation sub threshold to highlight the following. The contributions are drift and diffusion to the current IDS the behaviors of electron and hole concentrations Jn, Jpe and Psi all in a single diagram the behaviors of Np, Jn, Jpe and Psi all in a single diagram. Now let us look at the plot of electric field and Psi with X, Y so we have now completed the one dimensional plots okay along Y and along X. Now let us look at the two dimensional picture okay so correlate the X and Y directional plots we correlate the X dimensional distributions of Psi, EX and EY over the surface Y equal to 0 to their Y dimensional distributions near the channel midpoint. So we will try to correlate it here okay in other words I will draw distribution of the function of X and then for X correspond to the middle of the channel we will draw it as the function of Y. So here is the reproduction of the diagrams we have already drawn as the function of X for Psi S the Y component of the electric field at the surface that is the field in this direction at the surface and EXS that is the field in the X direction along the interface at the surface. We have put a modulus here because the field is directed from drawing to source drain is more positive with respect to source however we are sketching only the magnitude of the field. Now we look at the middle of the channel that is somewhere here between 0 and L and then we draw the Y axis okay this is the perpendicular axis that is what is shown here. YD shows the edge of the depletion region that is controlled by the gate okay somewhere here and we are going to plot the surface potential as a function of Y and the electric fields as a function of Y. Now this is how the picture would be the surface potential at the interface at this point between 0 and L is shown by this value from this value it decreases and goes to 0 the time you reach the depletion edge. Similarly here EYS the Y direction field that is the field perpendicular to the interface it has this value at the surface which falls on this particular diagram and it decreases first it decreases rapidly because you are crossing the inversion layer and then relatively slowly when you cross the depletion layer and this field goes to 0 at YD. Please recall the EY at the surface becomes negative near the drain because we are considering the bias point VDB greater than VGB so drain potential is more positive than gate potential that is the field near the drain is from drain to gate that is why it is negative. Now what have we achieved using this? The first point is we are able to reflect the inherent 2 dimensional nature of the MOSFET. Second as we will see later this diagram will illustrate the worsening of the gradual channel approximation towards the drain. We will discuss the gradual channel approximation in detail later right now let me just quickly tell you what it is this approximation nothing but comparison between the gradient of EY as a function of Y and the gradient of EX as a function of X okay. So DOEX by DOEX and DOEY by DOEY. Now you can see here that DOEX by DOEX that is this slope near the drain is rather high okay whereas if I consider near the source DOEX by DOEX is very small DOEY by DOEY which is actually this slope EY versus Y is this graph this slope is rather high as you can see this will be high both near the source as well as near the drain however you see the slope of EX as a function of X is small near the source whereas it is high near the drain. So when gradual channel approximation is valid the slope of EX versus X is very small compared to slope of EY versus Y. Now because the slope of EX versus X is increasing near the drain both slopes that is EX versus X as well as EY versus Y become comparable and therefore we say gradual channel approximation versions. Now this approximation is used to simplify the two dimensional Gauss law to one dimensional Gauss law. Two dimensional Gauss law contains two terms DOEX by DOEX and DOEY by DOEY well you will understand this point more clearly when we actually discuss the gradual channel approximation in more detail. Right now just focus on the mathematical properties of the slopes of the field components. Now let us plot the energy bands with X,Y that is the two dimensional energy band diagram the two dimensional energy band diagram highlights the inherent two dimensional nature of the MOSFET. Here is the device source and drain L is the channel length W is the width of the channel the source and bulk are shorted also they are grounded you apply VGB to the gate with respect to bulk and VDB to the drain with respect to bulk. X direction is from source to drain along the interface Y direction is perpendicular to the silicon-silicon dioxide interface. Now we are going to plot the energy levels E as a function of Y and X. First we identify the depletion regions controlled by the source and by the drain then we sketch the quasi Fermi levels. In a two dimensional band diagram the quasi Fermi levels and other energy levels would be surfaces. When you draw the energy band diagram in one dimension these levels are lines. So what we have shown here is the intersection of the surfaces corresponding to EFP and EFN with the silicon-silicon dioxide interface. So those will be lines because two planes will meet at a line. So if I see over the silicon-silicon dioxide interface EFP and EFN would look something like this. We have sketched this already earlier and these are the variations of EC and EV even these we have sketched. So essentially what we have done is to draw the two dimensional band diagram we have first sketch the band diagram over the silicon-silicon dioxide interface as a function of X. Now slowly we will add the Y direction information. So first we will sketch the surfaces in the neutral N plus source and drain regions corresponding to the EC. So this conduction band edge surface in drain and this is the conduction band edge surface in source. Similarly we sketch the valence band edge surfaces, EV surfaces in source and drain. Then we sketch the conduction band edge energy level EC at the bottom surface that is along this surface here. So you see along this surface the barrier between N plus and P would be equal to the built-in potential because this is grounded and this is grounded. So that is the built-in potential barrier height. This height is more than the barrier here because application of the positive gate voltage reduces the barrier between source and the interface regions near the source here because electron concentration is increasing as you move from bulk to interface. When you come to this end that is between P and N plus region here along this line the reverse bias is equal to VDB. So that is what is shown here. This is the barrier which is reverse bias plus the built-in potential here. Now you can join the back and the front surfaces by appropriate lines to complete the conduction band surface in the P region. Now you can show the intersection of the whole quasi-framy level surface which is in the y direction with the silicon-silicon dioxide interface that is this line and with the conduction band surface that is drawn here that is this curved line here okay and this curved line here. So the surface intersects silicon-silicon dioxide interface, the surface corresponding to the EFP intersects the silicon-silicon dioxide interface along this line and it intersects the conduction band surface associated with the P region along this line. So that is your 2-dimensional energy band diagram. Now finally let us list the variables, constants and parameters of the model. So this is our table reproduced from earlier module. So you have dependent and independent variables, physical and empirical constants, geometrical process and other parameters. This is your device where this current IDS is being modeled as a function of drain to bulk, source to bulk and gate to bulk voltages. Therefore IDS is your dependent variable VDB, VGB and VSB are your independent variables. The physical constants that we have come across so far are the electron charge Q, dielectric constant of semiconductor S, epsilon S, dielectric constant of the oxide okay, this region that is epsilon ox, so semiconductor regions are these regions and this region. Phi MS, the potential difference, work function difference between the gate and the semiconductor materials and saturation velocity associated with the electrons in the inversion layer okay. There are no empirical constants. The geometrical parameters are W which is the width of the device in this direction perpendicular to the slide, L that is the channel length from source to drain and T ox that is the oxide thickness here. The process related parameters that is parameters we have controlled by the process are fixed charge QF, add the silicon, silicon dioxide interface, doping NA of the substrate and mobility mu of the electrons in the inversion layer that causes the current. At this point we have no other parameters to list. Now with that we have come to the end of the module. So let us summarize very briefly what have we achieved in this module. I would like to remind you that a previous course, a more fundamental course, more basic course on solid state devices by me has covered the topic of MOS junction and MOSFET in these lectures and whatever we are discussing here is more advanced version of those details. The first important point to note in this module is that while the device is biased with respect to source in practical applications for device modeling purposes it is more useful to consider the biasing arrangement with respect to the bulk. After we have derived the equations for this biasing arrangement in which we derive the expression for IDIB as a function of VDB, VSB and VGB we can then express the same model in terms of the voltages with respect to source using these equalities. We discussed the bias, we discussed the regions of operation for that purpose we start with the MOSFET in which the VGB is applied but VSB and VDB are 0. So drain and source are shorted to bulk. The regions to be identified here are the depletion region which is between the flat band point and the point at which the surface concentration of electrons is equal to Ni that is the depletion region. Then when you increase the VGB further your surface concentration of electrons at the interface becomes equal to the whole concentration in the bulk and this boundary now is the boundary of the so called weak inversion region. So between Ns equal to Ni and Ns equal to Pp is not you have the weak inversion. Now at the point when inversion charge becomes equal to the bulk charge this is the charge per unit area that is the area under the electron distribution okay whereas this is volume charge this is aerial charge when the area under the electron distribution or inversion charge becomes equal to the depletion charge per unit area that point we reach strong inversion and for VGB beyond this point your inversion charge per unit area will exceed the depletion charge this strong inversion. So between weak and strong inversion you have the moderate inversion. The threshold voltage VTB falls somewhere in the moderate inversion region. The region from VFB to VTB is referred to as sub threshold. Now if you add a VSB and keep VD equal to VDB equal to VSB that is short the drain to source now what happens is the effect of VSB is to expand the depletion region and sub threshold region as shown here by these lines. Now when you add a VDB which is greater than VSB now the device has a current drain current flowing between drain and source and may be small amounts of current between drain and bulk and drain and gate also. However we are neglecting the drain bulk and drain gate currents in our course. What is the effect of that that is shown here. So we now erect a VDB axis perpendicular to VGB axis this point corresponds to VDB equal to VSB when there is no current and then in this VDB VGB map you have regime such as non-saturation where the current increases linearly in the beginning for small values of VDB and then tends to saturate as you enter this region here as you increase the VDB and when you increase the VDB very much you enter the breakdown region. All these regions are situated for VGB greater than VTB. Now this is your VTB correspond to non-zero VSB your VTB correspond to 0 VSB is this so your non-saturation saturation and breakdown regions will be shown by the solid line if your VSB was 0 okay and this will be your VDB VGB map. So the dotted lines and the solid lines here show the change in the non-saturation saturation breakdown regions when you change your VSB then we will state the factors responsible for creation and continuity of JNJP and E. So we said that JN the electron current which is important in a N channel MOSFET is created by both drift as well as diffusion. So the electric field is created by voltages applied to the gate, source and bulk and this electric field causes several things. One is an inversion layer is created near the interface and this inversion layer then transports current. So you have variation of the inversion layer from source to drain and therefore current is because of diffusion in addition to drift which is caused by the field between the drain and the source. So that is about the creation of JNJP and E we really do not bother about JP because this current is very small in a N channel MOSFET. Now what about the continuity aspects? So you see that generation of electron hole pairs in the space chart region and within a diffusion length from the space chart region contributes to the drain to source current. So here you can see that all the electrons which are generated join up with the current that is being supplied from the source and they move to the drain. The holes however move to the bulk and these holes are responsible for the bulk current. The generation can really increase in the space chart region near the drain at the interface because of high electric fields here which causes impact ionization shown by this red lines okay. Then you can have some loss of electrons due to tunneling. So while generation can be sources of electrons tunneling could be loss of electrons right tunneling from bulk to gate. As far as electric field is concerned the continuity is maintained by the space charge here which can be because of depression as well as inversion. Now based on this qualitative understanding we explained the IDVDS characteristics how the current increases linearly at first and then tends to saturate beyond some voltage and how the saturation current increases with VGS. We also explained the IDVDS curves when ID is plotted on a log scale. This was done to emphasize the sub threshold regime of operation where the saturation voltage is constant independent of the gate voltage and it is at value approximately 3 times VT and on a log plot the current appears to increase linearly with VGS. In other words IDVDS is exponential this is in the sub threshold region. Then we explained the IDVDS curves. So when ID is plotted on a linear scale the IDVDS graph is a straight line over a significant portion near the threshold however there is a rounding off okay of this corner and for high gate source voltages there is a tapering off of this slope because of field dependent mobility effects. Now when you plot the ID on a log scale then near the threshold you can see a straight line portion for IDVDS showing the exponential variation of ID with VGS. Next we explained the IBVDS curves for a device operating in saturation near the breakdown. So what we said is for VGS less than VT there is no inversion charge and there is no IDS and therefore there is really no subset current because subset current is a consequence of multiplication of the drain to source current because of impact analysis okay. So this current depends on two factors the amount of electric field available to the impact analysis or multiplication and the amount of drain to source current that is a source that can be multiplied. Now near this end when VGS is small the inversion charge is small the source current that can be multiplied is small and therefore the IB is small. See for large values of VGS the IB is small again because the electric field in the x direction okay decreases though the current itself is large electric field is small and therefore multiplication is small. So in between you have some maximum value of IB and this IB increases as your VDS increases. Then we also explained the variation of the breakdown voltage as a function of VGS which follows from the IB VGS behaviour we discussed just now. We however postponed to a later time the explanation for why the IDVDS region there is a slope of the characteristics these are short channel effects corresponding to small geometry devices. A modern MOSFET is a small geometry device whereas we are considering large geometry device here. Then we consider the factors responsible for boundary conditions okay on these quantities the boundaries we considered where silicon-silicon dioxide interface that is boundary 1, boundary 2, silicon dioxide, poly interface, boundary 3 between the device and the ambient. Boundary 4 device and the ambient there are two parts one is between silicon and the ambient other is between silicon dioxide and the ambient that is 4 and 5 is the electrodes. At these boundaries the conditions on these quantities are decided by dielectric constant of silicon, dielectric constant of oxide, dielectric constant of the ambient, fixed charge, surface recombinant velocity, the potential applied at the electrodes and thermo ionic emission and tunneling currents which are negligible perpendicular to the interfaces for the conditions considered. Then we sketch the flow lines for Jn, Jpe and equipotential lines for psi. Now that is what is shown here so you see that the picture is two dimensional these are the equipotential lines and these are the field lines okay similarly we sketch the current flow lines also based on this we concluded that the MOSFET is inherently a two dimensional device. Then we sketch Np, rho, Jnx, Jpy, Ey and psi versus y near the channel midpoint and they had this kind of variations. The Jnx that is the current density in the x direction as a function of y was something like this showing that the current is restricted to the inversion layer. In y direction Jn and Jp are 0 that is current perpendicular to the interface that is 0. Then we emphasize the depletion approximation of the charge controlled by the gate and the charge sheet approximation of the inversion layer. Similarly you have depletion approximation of the charge controlled by the gate in the N plus poly region also. Now this was the picture of the electric field as a function of y, Ey as a function of y and from Ey as a function of y we also sketch the psi as a function of y from gate to substrate. We sketch the energy bands with y that is from gate to substrate and the band picture looks like this which shows E0, EC, EF and EV variations and then we sketch the electric field components EXS, EYS and the surface potential both with x and with y. Finally we sketch energy bands as a function of x around the interface okay for sub threshold and saturation conditions. We also sketch the 2 dimensional energy band diagram and finally we list at the variables constants and parameters of the model. So I hope that at the end of this module you should be able to do the following for a bulk MOSFET with uniform substrate doping and large LWT ox under steady state. First explain the shape of the IDVDS, IDVGS and IBVGS curves in terms of the charge and field conditions in the device. Then sketch the field lines, potential lines, current flow lines and energy bands in the device for various bias conditions. Then sketch the spatial distributions of the charge, current, density, field and potential in the device for various bias conditions. With that we come to the end of this module on qualitative theory underlying the operation of the large uniformly doped bulk MOSFET.