 Let us continue our qualitative discussion of the DC model of a large uniformly doped bulk MOSFET. In the previous lecture, we have discussed the distributions of electrons, holes, then the space charge, the electric field and the potential as a function of the y direction. That is, in terms of this diagram, the direction from the interface into the bulk. Now in this lecture, we continue this and move on to some other aspects of the spatial distributions of NP, JNJP, ESI and energy bands. Specifically let us consider the plot of JNX with y. Refer to the diagram here, JNX is the x component of JN, x component of electron density. X is the direction from source to drain as shown here and we want to plot this x component of electron density as a function of y. Now here is the section of the device rotated by 90 degrees so that the spatial axis is horizontal. So in this diagram, the x direction would be from this end to the other end, so vertically upwards. Now to plot the current density, we must look at the electron concentration and the distribution of the electron concentration is as follows, I am repeating a slide from the previous lecture. So this is the concentration of electrons in the inversion layer as a function of distance. Now the current can be because of drift or diffusion. The drift current would be proportional to the concentration of electrons shown here and the electric field in the x direction. On the other hand, the diffusion current would be proportional to the gradient of electron concentration in the x direction. This means that like you have this kind of a distribution for n as a function of y for some x, at another x you will have a similar distribution as a function of y but the values of the electron concentrations would be different. Now it is easily observed that the gradient of the electron concentration itself would also depend on the concentration shown here. So if the concentration itself goes to 0 or very small values, then the gradient of the electron concentration in the x direction that is this direction would also go to 0 by the same distance or at the same location. And therefore we can easily conclude that the Jnx which depends on the electron concentration and its gradient in the x direction would both have qualitatively the same shape as shown here for the electron distribution. And this is what is shown here that as the electron concentration falls to 0, the current density also falls to 0. The current density shown in the negative axis because the current is from drain to source whereas the x direction is from source to drain. So we conclude that the current of electrons from source to drain would be very close to the interface. At the interface the current density would be maximum and then it would decay rapidly and it would be restricted to the inversion layer. Now let us move on to the plot of energy bands with y. Now here are the various bias points where we could draw the band picture. We sketch the energy bands versus y at some x near the channel midpoint in saturation and sub threshold that is at bias points 2 and 3 that is this point and this point. I will be doing it for 2 and I will leave it as an exercise for you to do it at 3. You can also do it at 1. Let us consider the device in saturation. Now here is the structure of the section of the device that is relevant to us and here is the procedure for constructing an energy level versus y diagram. This procedure is repeated here from a previous module. Recall we have done a complete module on energy band diagrams and there we had listed out the sequence of steps in which any band diagram should be drawn. Now these are the steps. Let us look at the first step. We locate the junction, space charge and neutral regions. Now let us do that. Junctions in this device structure are the N plus gate silicon dioxide insulator junction that is shown here, the silicon dioxide insulator substrate junction that is shown here. Then you have the substrate and substrate electrode junction that is shown here. Then you have the space charge regions. This is one space charge region which is controlled by the gate and this is another space charge region because of the substrate-substrate electrode junction. YD represents the edge of the depletion layer due to the space charge layer controlled by the gate. The next step in this procedure is to draw EFN and EFP continuously everywhere from a knowledge of applied voltage across the junction V and current densities of electrons and holes. So you know that the gradient of EFN represents the current density Jn and the gradient of EFP represents the current density Jp. Now whenever we have options we should first draw the EFN and EFP in neutral region and among or between EFN and EFP we should first catch the majority carrier quasi-firmly level. Now this order is dictated by the ease in which we can draw the Fermi levels. So let us do that second step. So here we have drawn the Fermi level across the substrate-substrate electrode junction. This is easiest to do because as we have remarked earlier there is no voltage drop across this junction because the voltage drop across this junction that is any part of the applied voltage if it drops across this junction that would amount to a current flow across this junction and such a current flow is not possible because that current would have to flow through this insulator layer and this is not possible. Thus this PN plus junction is in equilibrium and therefore we can easily draw a constant EF across this junction. So EFN and EFP are both the same. Next let us extend this Fermi level over the neutral region of the substrate until the depletion edge. Now while doing so we have shown it as a dashed line because we want to first catch the majority carrier Fermi level and that is EFP. So the quasi-firmly level for holes will be shown with a dashed line. Evidently the quasi-firmly level for electrons in this region which is next to the depletion edge would not coincide with the quasi-firmly level for holes. Thus you can easily appreciate from your knowledge of a PN junction. So when you have a biased PN junction then in the region near the depletion edge in the neutral region near the depletion edge the quasi-firmly levels split. Now here this Svesta region corresponds to the inversion layer P substrate junction which can be regarded as an N plus P junction. The inversion layer is N plus and the V, the channel voltage V can be regarded as the reverse bias across the inversion layer P junction and therefore the EFN would not coincide with EFP. Now the EFP would be a constant line and it is easy to draw because the whole concentration would not be disturbed in spite of the bias. This is a reverse bias and therefore the injection level has to be low and therefore the majority carrier concentration is not disturbed. Therefore EFP remains a constant line okay which is the same as under equilibrium and that is why in this entire neutral P region the EFP would be constant. Next we sketch the EF in the gate region. Once again here we have sketched the EF in both the neutral as well as the space charge region of the gate and we have shown it as a constant line because the N plus region here is under equilibrium. No current flow can occur perpendicular to the silicon dioxide interface here and that is the reason why N plus region is in equilibrium therefore we can draw the EF as a constant line. EFN and EFP are the same. Now where do we draw this constant line? We know that there is an applied voltage VGB between this N plus region and the back N plus contact. Therefore the Fermi level in the back N plus contact and the Fermi level in this N plus region would be separated by Q times VGB. This VGB is positive and therefore the Fermi level in the gate is below the Fermi level in the substrate electrode because this is energy band diagram and any movement upwards would amount to moving to more negative energies or more negative potentials and therefore a positive potential will be realized by moving downwards. Next we extend EFP in the space charge region using the assumption of cos equilibrium according to which in the space charge region of a p-in junction the whole density, whole current density rather will be approximately equal to 0. Note that in the space charge region you have a large gradient of holes and as well as a large electric field and therefore you have large drift and diffusion currents which are however in opposition and the difference between the two currents is what results in the net current and this net current is what is represented by the gradient of the quasi Fermi level. So we are saying that individually the drift and diffusion are very high but they are in opposition and the difference between them is very small and that is what is implied by the statement that the gradient of EFP is very small or EFP is a constant line and therefore this is simply extended from the EFP outside the depletion region. Next we sketch EFN again as a constant line using the same cos equilibrium approximation for electrons. The distance between EFN and EFP is equal to the reverse bias across the junction that is Q times V so the channel voltage V is actually the reverse bias across the N plus inversion layer T substrate junction and since the inversion layer is positive with respect to substrate the quasi-familiar for electrons is below the quasi-familiar for holes. Now we connect the EFN from the depletion edge to the EFN deep inside the bulk as a continuous line. Now slope of this line represents the so called diffusion current of electrons which is flowing okay because of the reverse bias across the inversion layer P substrate junction. Now let us clean up the slide and show other aspects referring to the procedure for constructing the EY diagram. The third step is sketching of EC and EV in neutral regions from a knowledge of energy gap EFN, EFP and concentration of electrons and holes. Now that is what we do next so we first locate the EC in N plus region the distance between EC and EF is given by concentration of electrons in the N plus region. Then we sketch EV below EF in the P type substrate the difference between the Fermi level and the valence band edge in the P type substrate is given by the hole concentration in the substrate. So we are sketching the EV in the neutral region of the substrate. Then we sketch EC in the neutral region of the gate okay. So EC is a little above EF in practice for poly gates for heavily doped poly silicon gates EC and EF are assumed to be coincident however in our diagram we have shown a small difference between EC and EF both in the gate as well as here because we want to show the EC and EF clearly. Now using energy gap we place the other level we place the valence band edge in the substrate electrode and in the gate and we place the conduction band edge in the substrate in the neutral region of the substrate. Now the fourth step is to sketch a continuous E naught everywhere from a knowledge of potential and electron affinity so that is what we are doing here. So we place E naught in the neutral regions above EC at a distance given by Q times electron affinity. So we have done that in all the neutral regions okay. Now after sketching the E naught in the neutral regions we can now join these various E naught levels by continuous line as shown here. So we draw a continuous line for E naught in the space chart region joining the various ends. Now note that the slope of E naught is important and it represents the electric field therefore for example if you take E naught at the silicon silicon dioxide H in silicon dioxide region the variation of E naught has a constant slope because the electric field is constant. The gradient of E naught represents the electric field. In the substrate at the interface the electric field is different than the electric field in the oxide because of the dielectric constant differences between the oxide and the substrate and therefore you see the slope of E naught at the interface in substrate is different from the slope of E naught in the oxide. So specifically E naught in the oxide has a higher slope than the E naught at the interface in the substrate. In fact the slope difference would be a factor of 3 because of a factor of 3 difference in the dielectric constants of silicon dioxide and silicon. Now the next step is to sketch EC and EV in insulator and space chart regions based on electron affinity and energy gap. So let us begin with the insulator we place EC at a distance Q times the electron affinity of oxide below the E naught and we place EV at a distance EG below the EC. Now we place EC and EV in the space chart regions controlled by the gate. So both the space chart regions in the substrate as well as in the gate. Now this is done in a similar way as we have done for the EC and EV in the oxide. We are doing it from the knowledge of electron affinity. So this EC line here in the space chart region would be Q times chi s below the E naught line in the space chart region similarly for this EC line and the EV line here in the space chart region would be energy gap below the EC line and similarly here. Now let us repeat the same process for this space chart layer. Now we are not going to show any more levels such as a donor level or any trap level and so on. I leave that as an exercise to you. Let us look at some further aspects of this energy band diagram. Before that let us clean up and show only the energy levels. Let us show the voltage differences. Now Q times VGB is the difference between the Fermi level here and this line which represents the Fermi level in the bulk electrode. The same QVGB also appears as the difference between the E naught levels in the substrate electrode and the E naught level in the gate. As we have remarked earlier, the difference between the quasi Fermi level for holes and the quasi Fermi level for electrons in the space chart layer controlled by the gate is equal to Q times the channel voltage V, Q times the potential drop across the poly, Q times the potential drop across the oxide and Q times the potential drop across the semiconductor or Q times the surface potential. These are all represented for the E naught variation from the gate into the bulk. Just for the information of the students psi p, psi ox and psi s are the potential drops occurring in these regions of the device and this is the polarity. Q times phi MS is actually the total variation of E naught across the space chart region of the substrate electrode junction. Now that is essentially the potential drop or the built-in potential drop across the substrate electrode substrate junction. Now let us summarize what have we achieved by drawing these band diagrams. We sketch the energy band versus y at some x near the channel midpoint in saturation to highlight the following. Now this is essentially the bias point at which we sketch the band diagram. The first point we highlighted was the applied voltage VGB and the channel voltage V. So go back and see this applied voltage VGB and channel voltage V. Then we wanted to show the potential components psi s, psi ox, psi p and phi MS of the applied voltage VGB. So psi p, psi ox, psi s and phi MS. These components of the applied voltage VGB which is the difference between E naught levels. So you see the various potential drops are shown on the variation of the E naught level okay whereas the applied voltages VGB and channel voltage V are shown as differences between quasi Fermi levels. Finally we wanted to show the behaviors of electron and hole concentrations, electron and hole current densities, electric field and psi all in a single diagram. So when you look at this energy band diagram, from this diagram you can conclude okay you can derive the information about electron concentration, hole concentration, electron current density, hole current density, electric field and psi. Now let us just explain how. Now you know that the difference between EC and EFN okay gives you the electron concentration at any point. Similarly the difference between EFP and EV gives you the hole concentration. The gradient of EFP gives you the hole current and the gradient of EFN gives you the electron current density right. Now the gradient of E naught gives you the electric field and similarly gradients of EC and EV also give you the electric field and the variation of E naught is actually nothing but the variation of the potential. So this is how NP, JN, JP, E and psi all are reflected at once okay in the energy band diagram. Now here is an assignment for you sketch the energy bands versus y near the channel midpoint from gate to bulk for sub threshold conditions that is for this particular bias point shown on a log ID versus VGS graph. Now let us turn our attention to plots of NP, JN, JP, E and psi as a function of X. Now so far we were plotting quantities as a function of Y that is from the interface into the bulk. Now let us plot along the interface from source to drain that is what we are doing now. Now we are not going to plot all of these quantities we are going to plot some of these relevant quantities I will leave it as an exercise to plot those quantities which I am not going to do here on the slides. First let us sketch psi S, EXS and EYS versus X in saturation that is at this bias point. Psi S is the potential of the interface okay silicon dioxide interface or surface, EXS is the X component of the electric field that is the electric field directed along this X direction from drain to source or source to drain whichever we look at it along the interface. So S here means it is along the interface and similarly EYS is electric field in the Y direction but plotted along the interface in the X direction. Let us look at psi S as a function of X this variation would look something like this so 0 is the source and X is equal to L is the drain so from source to drain the surface potential goes on increasing progressively okay the variation is slow towards the source but rapid near the drain. Now let us explain this with the help of this particular diagram so psi S is the potential of this interface okay with respect to the bulk. Now if I take the source end that is this so this is X is equal to 0 and this is the drain end this is X is equal to L now we are assuming strong inversion so the inversion charge concentration at the interface or the electron concentrate at the interface may even be more than the doping in this heavily doped source okay so we are ignoring any small potential variations that may occur okay across this region because of the difference between the inversion layer electron concentration and electron concentration in the source. So we will assume more or less both are at the same potential. Now psi S goes on increasing from source to drain this is evident because VDB is more than VSB so the psi S should increase. Now the psi S increases slowly near the source but rapidly near the drain. Now this is evident from the fact that the inversion charge is very strong near the source but weak near the drain because of which the electric field directed in this direction now this is your X direction. Now our electric field at this here it is like this okay you also have an electric field in this direction but we are concerned with the electric field in this direction right because we are interested in the surface potential variation in this direction. Now this electric field is high near the drain and low near the source because as we have remarked earlier the current has to remain constant from source to drain approximately constant at least there may be some contributions from thermal generation and so on which are small. If the same current has to be there when the inversion charge is more as well as when the inversion charge is less wherever the inversion charge is more the electric field in this direction should be small and wherever the inversion charge is less to get the same current I should have a higher electric field. Now that explains why the slope of psi S is small near the source end but high near the drain end. The value of psi S at the source is equal to phi t plus vsb where phi t is normally given by 2 times phi f where phi f is the difference between the Fermi level and the intrinsic level in the substrate. So this approximation is normally given in most textbooks however when you do numerical calculations then the accurate value is twice phi f plus 6 times the thermal voltage. This we have explained in our discussion of the MOS capacitor at the drain end the psi S is equal to phi t plus vdb so source end is here and drain end is here. So we are saying that at this end the potential of this point with respect to bulk is vsb plus phi t and at this end it is this vdb plus phi t so this potential with respect to this. So from psi S versus X we can easily plot ExS versus X. As we have already remarked ExS is nothing but this electric field okay this is the exaggerated electric field and at the interface. So this ExS is nothing but minus dou psi S by dou X and since the ExS is directed from drain to source it is negative and that is why we are plotting the modulus here. So this curve is nothing but the slope of this curve. Let us turn our attention to the Y component of the electric field. Now here is a plot of EYS as a function of X. You see the EYS is crossing the zero line and becoming negative for X is equal to L. Now this is because you recall we were considering a bias condition in which the vdb is greater than vgb which is same as saying vds is greater than vgs. Now for this bias condition we have sketched the equipotential lines field lines and so on right in one of the earlier modules and you recall when vdb is greater than vgb near the drain end the field is directed from substrate to gate whereas over most of the channel length near the source the field is directed from gate to substrate. At some point the field reverses direction. Now let us explain the shape of the EYS versus X curve. The EYS is high near the source and it progressively goes on decreasing. Why is it so? Now look at this diagram and this is your EYS that is the EY at the surface or interface. Now you know that the applied gate to bulk voltage falls partly across the polysilicon partly across the oxide and remaining across the depletion layer in the substrate. So we can write this as vgb is equal to psi p plus psi ox plus psi s where this is psi ox and this is psi s. Now let us ignore the psi p that is the potential drop in the poly. Now vgb is constant from source to drain but what happens to psi s? So you see the psi s is going on increasing from source to drain consequently psi ox goes on decreasing from source to drain so this potential drop decreases from source to drain. Now the total charge in silicon that is inversion charge plus depletion charge should decrease from source to drain because the psi ox is decreasing from source to drain. So total charge in semiconductor if you represent it as qs which is sum of qi the inversion charge and the depletion charge that is this charge and this depletion charge. Then the magnitude of this depends on psi ox so this is equal to psi ox into c ox this is by the parallel plate capacitor law. Now since psi ox decreases from source to drain the qs decreases from source to drain. Now if qs decreases from source to drain then by Gauss law the y component of the electric field which terminates on this charge which is actually the eys that eys also should decrease from source to drain. Now note that we have to argue in terms of psi ox because if you take the inversion charge and depletion charge individually inversion charge goes on decreasing from source to drain but the depletion charge increases from source to drain. So purely from this knowledge of variation of qi and qb we cannot conclude about qs because this is decreasing from source to drain this is increasing. So we cannot talk about the sum unless we talk in terms of psi ox. Now the next point of interest is the fact that the variation in eys is slow near the source and it is rapid near the drain. Now this is because of the same reason because of which the x component of the electric field that is the field directed from drain to source increases rapidly near the drain and the variation in this field is slow near the source. So the conditions in the channel change rapidly near the drain but they vary rather slowly near the source. Now let us see what can we do with this information. So we have sketched psi s, exs, eys versus x in saturation to highlight the mobility degradation because of the following factors. So what we are saying is from this information regarding the variation of the y and x component of the electric field we conclude something very important regarding how the mobility varies along the channel. The first point is the mobility degrades due to transfers field eys near the source where eys is much greater than exs. So here you can see near the source eys is much more than exs and therefore mobility degradation near the source would be because of eys that is it is because of the vertical field or the transfers field. On the other hand the mobility degradation is due to parallel field exs near the drain. So when I come near the drain the exs is much much larger than the eys. So near the drain the mobility degradation would be due to parallel electric field, electric field parallel to the interface okay and that is what causes the velocity saturation effects. Now here is an assignment for you sketch ns versus x in saturation that is at bias point 2 that is this point and ns, exs, eys and psi s versus x in sub threshold conditions that is at bias point 3. So at bias point 2 that is in saturation we have sketched exs, eys and psi s and therefore you are asked to sketch ns alone whereas for sub threshold conditions you sketch all these quantities with that we have come to the end of the lecture and so let us make a summary of the important points. Now in this lecture first we sketch the electron current density directed from drain to source in the y direction and showed that this current density is restricted to the inversion layer thickness. In other words the current in a MOSFET is restricted to the inversion layer thickness. Next we drew the energy band diagram in y direction near the channel midpoint. Now while drawing the energy band diagram we made the important point that the material associated with the electrode connected to the substrate and the electrode connected to the gate that has to be taken into account while drawing the energy band diagram in the y direction and therefore whenever you draw energy band diagram in the y direction you must explicitly show the gate electrode and the substrate electrode. For simplicity we assumed that the gate electrode had the same material as n plus gate and the substrate electrode was of the same material as the gate electrode and therefore that was also n plus. The band diagram was used to highlight the applied gate voltage VGB, the channel voltage V and the potential drops psi p, psi ox and psi s across poly oxide and the substrate as well as the built in potential or work function difference phi ms. Another thing we achieved by drawing the energy band diagram is that from the energy band diagram we are able to show the variation of all important quantities namely np, jn, jpe and psi. Then finally we sketched the potential variation at the interface from source to drain that is psi s and the electric field components directed parallel to the interface at the interface that is exs and the electric field component directed perpendicular to the interface Ey at the interface, variation of these quantities as a function of distance from source to drain. The field components were sketched to highlight the causes of mobility degradation and we remarked that near the source the mobility degradation occurs due to transverse electric field that is electric field from gate to bulk whereas the mobility degradation in the channel near the drain occurs because of the parallel electric field that is electric field along the silicon silicon dioxide interface. We shall continue this discussion in the next class.