 In the previous lecture, we have begun a discussion of the qualitative theory underlying the DC model of a large uniformly doped bulk MOSFET. We have mentioned that qualitative theory for any characteristics is expressed in 5 different ways. These are first you identify the factors responsible for creation and conservation of current flow and electric field. Then you identify the factors responsible for imposing boundary conditions on carrier concentration, current distribution, electric field and potential. The third way to express a qualitative theory is to draw flow lines for current flow and electric field and equipotential lines. Then the fourth way is to sketch the distributions of carrier concentration, current density, field potential and energy bands as a function of space in the device. And finally you identify the variables, constants and parameters of the model. Now this is what we want to do for MOSFET okay which is biased with drain to source voltage, gate to source voltage and at times bulk to source voltage. In a previous course titled solid state devices in lectures 33 to 41, we have already discussed a first level model which includes a qualitative theory for the MOSFET. Towards the beginning of this module, we have been summarizing the key features from those lectures so that we can proceed further and develop a more advanced model of the MOSFET. Now so far we have discussed the conditions in a MOSFET with VGB applied but source to bulk and drain to bulk are shorted. We identified the following regimes of operation for this device. So this is a VGB axis. The important voltages on this axis are the flat band voltage, the threshold voltage and then you have some other important points which are based on the surface concentration of minority carriers or electrons in a N channel device. So you have depletion region, weak inversion region, moderate inversion region and strong inversion region. The strong inversion region is that region where the inversion charge dominates over the depletion charge that is what is indicated here. Now the region from VGB equal to VFB to VGB equal to VTB is referred to as the sub threshold region. In this lecture, we are going to discuss the following points. First we shall complete the regimes of MOSFET operation. So we will introduce a bulk to source bias and later on drain to bulk bias. Then we shall list out the factors responsible for creation and continuity of JN, JP and E. We shall explain the shape of the IDVDS, IDVGS and IBVGS curves based on the charge conditions in the device. According to the various regimes of operation of a MOSFET for drain to bulk and source to bulk voltages equal to 0, let us see how these regimes get affected when we introduce a bias between source and bulk. So first we clean up the slide and then here we introduce the bias. We are tying up the drain and source so that VDB is always equal to VSB. There is no current flow from drain to source. So conditions in the channel from source to drain will be uniform. Here we are ignoring the effects of the depletion region near the source and drain as we have remarked already because it is a large device wherein the source to drain distance is much more than the depletion widths near source and drain. Now what we find here is the effect of VSB is to widen the depletion and sub-threshold regions. Your threshold voltage of the N channel device shifts to the right. Similarly the point where Ns equal to Ni occurs also shifts to the right. Let us see why. So this is your MOS, this is your N plus, this is oxide and this is P. You have applied a VGB. Let us assume that this VGB is much more than the threshold voltage of the MOSFET so that you have an inversion layer here, a depletion layer after the inversion layer. Now let us connect an N plus region here. This is a schematic, an idealized situation where we are using the N plus only to make a contact to the inversion layer. The actual structure of the MOSFET is as shown here, there are two N plus regions drain and source. However, we can consider the N plus region only on one side because the potential of this N plus region and this N plus region is the same. Therefore the conditions in the MOSFET will not change whether I have just one N plus region or I have two N plus regions on either side. However we are looking at the picture is since this device is symmetric, I can just take one half of this device and then discuss the conditions therein. Now I am just considering a structure like this, right wherein I have brought an N plus region and contacted it from side. So in practice as I had shown on the slide, the P region would come here also, right. There will be P region here and this N plus region will be embedded inside. However, to understand this operation, this schematic is very useful and simpler. So you see I have neglected any depletion region that occurs between N plus and P here because my N plus is so small that it is only trying to make a contact to the inversion layer. And now I apply a bias. So this is VSB. What will be the effect? Note that this VSB is reverse biasing the N plus P junction. Now as we have remarked earlier that the effect of gate to bulk voltage, a high gate to bulk voltage over and above the threshold voltage is to induce an N plus P junction. So this is N plus region inversion layer. This is an N plus P junction. You can regard it like that. When you apply a VSB here, this VSB not only reverse biases this N plus P junction but reverse biases the entire inversion layer P substrate junction, okay, all along. So when you reverse bias a junction, what happens? The depletion region expands. So the effect of this reverse bias would be that this depletion region would expand. Now what would happen to the inversion charge itself? Note that we are not changing VGB. Now if you separate the VGB into the parts of the device, let us neglect the part of VGB that falls across this N plus region, assuming this to be heavily doped. Then this VGB is falling across the oxide and remaining across this silicon. Let us call this as psi S. So VGB is equal to psi ox plus psi S. The effect of VSB is to widen the depletion layer which means it is increasing the psi S. Now VGB equal to psi ox plus psi S wherein as VSB is increased, psi S increases. This upward arrow means increase. So we can probably write it like this. VSB increase results in psi S increase. But VGB is constant. This is being maintained constant. So if psi X increases but this is constant psi ox decreases. If psi ox decreases then from the law of capacitor you know that the total charge on the plate of the capacitor should decrease. This means inversion charge plus depletion charge together should decrease. So QI plus QB the magnitude should decrease. However when psi S increases, QB increases. This component increases. As you can see here, depletion region has widened. So psi S increase implies QB increases in magnitude. If you combine these two, QI plus QB should decrease but QB increases. This means QI should decrease. So these two imply QI decreases. So inversion charge is going to decrease as you increase VSB. What does this mean? This means that effectively the threshold voltage of the device has increased because the inversion charge is nothing but the oxide capacitance into VGB minus VTB. So if the inversion charge has decreased, if I write this expression, QI is equal to minus of C ox VGB minus VTB because VGB over and above VTB causes QI. So if QI has decreased in magnitude then evidently because VGB is being maintained constant, VTB has increased. Therefore this is how you can conclude that the VTB here is going to increase. Now just as the inversion charge reduces, in general we can say that the electron concentration at the surface here, the electron concentration at the surface reduces when VSB is applied. Now this means that a certain condition on electrons such as Ns equal to Ni would appear at a higher value of VGB. Now that is what is shown here. So this is how we can conclude that the depletion region widens and the substitution region widens because of application of VSB. Threshold voltage also increases when you apply VSB. Now let us proceed further and apply a VDB which is greater than VSB so that a current flow is set up between drain and source that is what is shown here. So VDB is greater than VSB. Note that I cannot say VDB can be less than VSB because moment this voltage becomes less than this voltage, this will become the source and this will become the drain. So since I am calling this terminal as the drain, this should always be more positive than the source. Now this is your VDB axis. The lowest value of VDB possible here is equal to VSB. Now what is going to happen as I increase VDB? Let us look at that. So what I have to do now is to bring another N plus contact on this side and apply a voltage to that that is more than this and then see what happens to this charge conditions. So VGB is greater than VTB because only then the inversion charge will be there here. Now please note that is the condition we are considering here. So we are drawing this VDB line starting from a value of VGB which is more than VTB. So if your VSB is 0 then your VTB would be this. If this VSB is 0, however since we are applying a VSB our VTB would be slightly towards the right of this. So this VGB that we are now considering is more than the VTB including the effect of VSB. So this is your source in an idealized structure. This is VSB and you put another contact at this end and this is your VDB. Now since there is an inversion layer present because VGB is more than VTB. The voltage between drain and source is going to drop uniformly along this. This means at any point here let us say if I take the inversion layer here at this point this point the voltage of this interface compared to the bulk will be higher than the voltage of this point with respect to the bulk. Now if you talk in terms of this voltage psi s okay we can talk in terms of psi s and psi ox because the VGB is falling partly across oxide partly across the semiconductor across this. We are ignoring the voltage drop in the poly. Then we can say that psi s goes on increasing from source to drain because the potential along this is going on increasing VDB is more than VSB. As a result we can draw the following picture the depletion region will go on expanding as you go from source to drain and the inversion charge will go on reducing as you go from source to drain. So at any point here this is your psi s, psi s increases from source to drain. Therefore applying the same explanation okay it is incidentally this effect of VSB on inversion charge and so on is referred to as body effect. Because it is a consequence of reverse biasing the body or the bulk with respect to the source. So we can say that when you apply a VDB greater than VSB the result is a change in body effect along the interface from source to drain. So body effect goes on increasing psi s increases and therefore we can say qi the magnitude of it falls from source to drain and the magnitude of qb increases from source to drain. So this is the picture. Therefore what will happen to the current? This is id. Now since the current is flowing through this inversion layer which is called the channel. We can use a loose analogy of this situation to the current flow in a resistor. As we will see actually the current flow here in this inversion layer is because of two factors drift as well as diffusion. Drift because there is an electric field directed like this from drain to source. There is a drift current in response to this electrons move in response to this electric field. But as you can see inversion layer charge also goes on decreasing from source to drain. Therefore there is a tendency for diffusion of electrons from source to drain. So there are two tendencies drift as well as diffusion. Nevertheless to understand what is happening in the device we can just concentrate on the drift part of the current and say that this is like current flow through a resistor whose area of cross section is going on reducing as you move from source to drain. Let us separate that out and show it here. So I am showing only the inversion layer and I am slightly expanding this right. So I am making it a little more thick so that I can show things clearly. So this is your at this end you have Vdb and at this end you have a Vsb applied voltage. Now clearly if I were to plot the effect of this on a graph where this axis is Vdb and this axis is id starting point here is Vsb because Vdb always has to be more than Vsb. Then as I apply a Vdb which is slightly higher than Vsb the variation of the inversion charge from source to drain then will be small. So I could assume an approximately uniform inversion charge therefore for small values of Vdb above Vsb my id will rise like it rises in a resistor in a linear fashion okay. As you go on increasing your Vdb keeping Vsb constant the inversion charge will reduce more and more towards the drain. So if this is the picture for one value of Vdb for a higher value of Vdb if I were to sketch the picture it would be like this. I can show it in fact here itself for a higher value of Vdb the inversion charge picture would be something like this. Note that things are not changing at the source end because Vsb is not being changed only Vdb is being varied so you are varying this. This means what now for a higher value of Vdb your effective area of cross section of the resistor has reduced that means the resistance itself has increased. So when you increase your Vdb your resistance is more which means the increment in the id for any given increment in Vdb will be less therefore you can see that for equal increments in Vdb your increment in id will progressively reduce. So here this is the increment for this much increment in current for this much increment in Vdb for equal increment in Vdb here I have a somewhat smaller increment in id and so on. So I will get a current which tends to taper off the rise tends to taper off like this so that explains the shape. So this is the reason why because the inversion charge goes on reducing from source to drain you are getting this effect and reduction at the drain is more and more as you increase your Vdb. Now what about the depletion charge depletion charge does not affect the current flow because current flows only through the inversion charge. So I am not showing the depletion layer you want you can show the depletion layer I will show it by dotted line here diagram is not to scale. So this is depletion layer for 1 Vdb and for higher Vdb this is for higher Vdb right. So depletion layer for higher Vdb will be more inversion layer is less but depletion charge will be more here. So it will be something like this so it will move in this direction whereas this edge moves in this direction as you increase your Vdb. For some value of Vdb the inversion charge here will become really very very small okay as we will see in advanced theory the inversion charge never becomes 0 at the drain end it saturates at a value where the velocity of the electrons becomes equal to the saturation velocity okay. So the velocity of the electrons will go on increasing from source to drain because your electric field is increasing when we plot the electric field as a function of distance from source to drain we will see that this electric field this directed electric field goes on increasing this is evident because my current flow should remain almost the same and if my inversion charge is less to get the same current I should have more velocity of electrons right. So velocity of electrons goes on increasing however you know from an earlier module that velocity of electron ultimately saturates beyond some critical field. So once the velocity saturates then there cannot be a change in the concentration so once that happens your current will also saturate okay beyond some point. In the first level course the saturation was explained based on what is called pinch off. Pinch off means that the inversion charge is assumed to go to 0. Now once the inversion charge is gone to 0 then there cannot be any change in the current okay almost no change in the current there will be a small change there will be a small slope here right that happens because actually this point of saturation starts moving inside towards the source we will just mention this point and deal on this point in detail later. So what we are saying is for some high value of VDB your picture would be something like this the inversion charge is small but constant over some distance and then this is what it is you apply even more higher VDB this uniform inversion charge region small inversion charge region or saturated velocity region as it is called because here the velocity would have saturated it goes on expanding. So if I denote the region of velocity saturation as delta L then delta L goes on expanding as you increase your VDB beyond a point called the saturation voltage. Now this is how we explain the ID VDB shape if you increase your VGB then your inversion charge will be more evidently your current will be more. So for a higher VGB if I want to sketch this I have to start with a higher slope here. So it will go on like that now evidently if the inversion charge here is more you will have to apply a higher value of VDB to make the inversion charge here go to the same small value at which the velocity saturation occurs and therefore this point beyond which saturation occurs will shift to the right. So this will be your locus of the saturation point. So the saturation voltage goes on increasing you can call this voltage as VDB set. So this goes on increasing VDB set goes on increasing as you increase your VGB. Now that is what is shown here you increase your VDB maintaining VGB constant you move along this line you encounter saturation. What happens if you increase your VDB further much further then you will encounter what is called the breakdown point. So if I increase my VDB further beyond some value of VDB I have to stretch this VDB here I will find the current increases rapidly. So this is the so-called breakdown point why does this happen because you can think about it in terms of this picture here there is a depletion region from the drain and the reverse bias between drain and bulk increases so much that a breakdown happens okay just like the Avalanche breakdown of any junction. In other words the carriers will ionize and there will be impact ionization note however the difference between a simple reverse bias junction and MOSFET is that there is a large amount of current flow through this depletion region. So here we have not shown the depletion region from the drain if you want to explain the breakdown we will have to show that region let me just draw a simple schematic showing that this is your n plus drain this is n plus source this is VGB this gate. So if I plot the depletion region near breakdown it will be like this. So you have a wide depletion region here a small depletion region here your inversion charge is something like this this is the so-called delta L okay over which velocity saturation happens that is the same as this delta L okay so this is the there is this current ID so electrons are flowing in this direction this is a conventional current. Now these electrons are encountering a high field here okay because there is a high reverse bias now because of this high field these electrons will multiply they will be impact ionization okay and multiplication. So that is what causes the breakdown and increase in the current. Supposing I see the picture for a higher value of VGB as I increase my VDB from VSB onwards I will encounter saturation and this saturation will occur at a higher value of VDB as compared to this case which was for a lower value of VGB. So this is what we explained here that the saturation point moves to the right okay as your VGB increases then for some high value of VDB you encounter breakdown here the breakdown voltage is shown to increase with VGB more about this behavior will be explained later because we will have to consider the various current components in the device in detail to understand why as you increase VGB in some range the breakdown voltage increases but in some other range increase in VGB would reduce the breakdown voltage so these details will be discussed later. Now if I were to join all these points and try to sketch the locus of the saturation point then I will find that would be like a straight line in other words as you increase your VGB the saturation voltage increases in a linear fashion almost okay the shape of this locus will become clear when you write equations you start from the point where VGB is VDB at that point the saturation voltage will be 0 and for the breakdown point locus is like this as I mentioned earlier this shape will be discussed later when we consider the various current components in the MOSFET. So we clean up the slide and then show the various regimes completely so this is your VDB axis and this point corresponds to VGB equal to VTB corresponding to the value of VSB and on the y axis the same point corresponds to VDB equal to VSB so if I were to show this on this graph here this region is the non-saturation region this region here is a saturation region and here you have the breakdown if I were to show the locus something like this that is this locus so non-saturation saturation breakdown right that is the same thing that is shown here non-saturation saturation breakdown here VDB is along this direction here VDB was in the horizontal direction right therefore non-saturation saturation breakdown please note that this line is straight here but this line is curved so you should not confuse this VDB VGB map with this ID VDB map right so although this VDB axis is common in both cases this VDB axis is horizontal here and the VDB axis on the slide is vertical but the other axis is VGB whereas here it is ID now suppose I fix my VDB at some small value and where is the VGB what will be the behavior this is the so-called transfer curve so let us explain now the shape of the transfer curve ID versus VGB now you are wearing VGB keeping VDB at some small value VDB and VSB are both kept constant and VDB is small say about 50 millivolts right what will be the shape of your ID VGB curve from this graph here you find that for large values of VGB you will be in non-saturation region and for some value of VGB some small value of VGB you will enter the saturation region as you decrease your VGB okay until the VGB equal to VTB point now what will be the shape of the curve for ID as a function of VGB that can again be understood from this picture here the conditions in the device so we are maintaining VDB at some small value which when I say small I think I must when I say VDB is small actually this is the correct way to put it is as follows VDB is minus VSB is small because the VDB should always be more than VSB so I am maintaining as VDB which is a little above the VSB and therefore the inversion conditions here inversion conditions will be more or less uniform in that situation if I go on increasing VGB that is this this voltage what is going to happen the inversion charge here is going to increase progressively and therefore the ID will increase progressively now when inversion charge is approximately uniform from source to drain you know that the inversion charge increases with VGB according to this formula okay the VTB will be uniform from source to drain along this because the inversion charge is uniform because your VDB is only slightly different from VSB we can assume more or less uniform conditions so inversion charge increases linearly with VGB and therefore the drain current increases linearly with VGB so the picture here would be something like this now the current will go to 0 for VGB equal to VTB here for VGB equal to VTB inversion charge is 0 now this straight line portion is applicable for values of VGB somewhat higher than VDB near VTB the simple linear law does not hold so you have a slight smoothing out of the curve here similarly for high values of VGB this does not remain linear and it tapers little bit the reason for that is when your field increases in this direction as you increase VGB this field increases then the scattering of the electrons close to the interface increases we have discussed the mobility as a function of electric field we have said that the mobility of carriers at any point is affected both by the longitudinal electric field and the transverse electric field so once the transverse electric field is large then you know that the mobility falls now that is the reason why your current is decreasing for high values of VGB now this is your ID VGB shape for some range above VTB here the device would be that is here the device would be in saturation region and in the remaining region here beyond this it will be non saturation okay that is what this shows for a small range of VGB beyond VTB you have saturation and then beyond that you have non saturation now you might wonder we just now said that the inversion charge is more or less uniform from source to drain then how can we have a how can we have for some values of VGB above VTB saturation because that is not the condition for saturation and saturation in the current is saturating here you have inversion charge at a very small value at the drain and inversion charge at the source is at a higher value okay that is the condition during saturation now the point is when we said that the inversion charge is uniform from source to drain that uniform condition applies when the inversion charge is really large VGB well above VTB when you come to very close to VTB and the entire inversion charge becomes small then the small VDB above VSB right even though this difference is small that small difference can create a variation in the inversion charge as shown here right. So as shown here when the inversion charge itself itself is small then at this point the inversion charge can become even smaller and you can have saturation this will however happen only for VGB very close to VTB so normally one does not bother about this region very much so long as your VDB minus VSB is small. So let us complete the regimes for a different value of VSB so if your VSB is 0 this is your VTB and therefore your VDB access will start from here so this is how your various regions saturation breakdown and non saturation they change when you change your VSB okay let us put our charge conditions on the slide here for a VDB which is more than VSB inversion charge is decreasing from source to drain depletion charge is increasing from source to drain we are assuming VGB greater than VTB correspond to this VSB so that you actually have some inversion charge here at the interface now let us put down the factors responsible for creation and conservation of electron current density whole current density and electric field now we shall do that for VDB greater than VDB set because you have more mechanisms of current flow okay when you go close to the breakdown point that is why we are trying to consider VDB greater than VDB set and we will go close to the breakdown point so that we can show all the various components of current so let us list out the flows in this MOSFET now there is a dominant flow of current from source to drain dominant flow of electrons that is shown here by this thick arrow then you have electrons provided by thermal generation these are the electrons this thick arrow shows electrons provided by source now you have thermal generation throughout the depletion layer okay as it happens in any p-n junction now this thermal generate electron hole pairs they behave as follows the electrons will move to the interface because there is a field directed from gate to bulk in this direction and they will also move towards the drain because there is a field directed from drain to source so you have two fields drain to source and gate to bulk so then net effect of that is electrons move towards the interface and then towards the drain that is why you see this arrow at an angle so all the electrons which are generated here will try to move to the interface and then if VDB is more than VSB they will try to move towards the drain holes will move out because the direction of the electric field is gate to bulk the holes will move out and all the holes will get collected by the substrate so these holes are actually causing the substrate current now note that there is generation not only within the depletion layer but also just outside the depletion layer within a diffusion length from the depletion layer so even the electrons which are generated outside the depletion layer within a diffusion length will contribute to current inside here okay and the holes will move out and contribute to the bulk current now even this generation is similar to that happens in a p-enjunction reverse bias p-enjunction since we are considering the picture near breakdown the electric field near the drain is high and therefore all the electrons which are moving through this high electric field region will tend to multiply because of impact ionization so these red crosses here show impact ionization the black crosses show thermal generation points now the electrons which are generated out of this impact ionization some of them move to the drain and some which have high energy because the electric field is high these are called hot electrons they can even cross the insulator barrier the insulator semiconductor barrier and get injected into the gate and they will cause the gate current so they are one of the reasons for the gate current this hot electrons which are generated near the drain because of impact ionization finally you can also have tunneling electrons you have a high concentration of electrons here and you have a high field from gate to bulk if the field is really high and the insulator is very thin you can have tunneling of electrons from substrate into the gate so these will also contribute to the gate current the thick arrows here show dominant current and thin arrows show leakage current so the thick arrow here is the current from source to drain that is a dominant current and there are contributions from impact ionization tunneling and so on but these currents are small therefore they are shown by thin arrows so that is why here gate current is small and bulk current will also be small whereas the drain current will be large the generated electrons move to the drain source and if hot some to the gate we are summarizing our observations so here you can see that there are some electrons are moving to the source right some are in the gate if they are hot the division between source and drain current is equal for VDB equal to VSB if I maintain this VDB equal to VSB then evidently all the current that is generated will get equally divided between source and drain most electrons move to the drain for VDB greater than VSB if your VDB is more than VSB then electrons will prefer to go here rather than going here to the source because electrons will go to the most positive terminal the generated holes move to the most negative of the accessible terminals namely the bulk so you can see all the holes here are moving to the bulk terminal and this is what causes the bulk current now we can explain the shape of IB VGS curve based on this picture let me sketch the IB VGS curve now here we will use VGB because our voltages are applied with respect to bulk so essentially IB VGB curve holes will increase when the device is nearing breakdown because there are many electron hole pairs generated because of impact ionization that is why we talk about the behavior of the bulk current near the breakdown point so in other words our VDB is near breakdown because that is where the current will be significant now you can see that if my VGB is small there is no inversion layer if there is no inversion layer there is no current from source to drain so this dominant current flow is cut off if this dominant current flow is cut off the electrons which can multiply their number is very small and therefore the current is very small therefore for small values of VGB you will not have much IB now once you increase your VGB and inversion layer forms then your IDS increases and therefore chances of multiplication increase therefore your IB increases now your this current is increasing therefore the source of multiplication is increasing therefore the IB is increasing so more electron hole pair generated here and these holes are moving to the bulk what happens for very high VGB when the VGB is high inversion charge is large then the voltage from drain to source will fall more or less uniformly okay that means the electric field here gets reduced note that when the inversion charge is small then the potential drop from drain to source will occur mostly near the drain whereas when the inversion charge is large then the potential variation becomes more or less homogenized okay and the electric field here at the drain decreases therefore for high VGB your IB will tend to decrease something like this so there will be a peak in between in other words for small values of VGB the source of electrons is small because ID is small therefore multiplication is not much therefore IB is small for very high values of VGB inversion charge is large that decreases the electric field near the drain and again multiplication cannot be much because current may be large but unless electric field is large you don't have multiplication okay that is why for low VGB and high VGB your current is small evidently if you increase your VDB then your current will increase so it will be something like this shape will be something like this so it will increase now finally we will make this comment that we neglect generation and tunneling currents that is both IB and IG and concentrate on IDS alone so we are going to model only IDS that is this thick arrow all other currents we are going to neglect for modeling which means our picture will be like this note that if only this current is considered then this current can be referred to as I suffix DS because it is flowing between drain and source we have neglected other components such as IDB and IDG so gate leakage and bulk currents have been neglected then this current can be called IDS okay then other current components are present then this current has to be referred to as ID as we did in the previous slide here this ID because it has three components IDS IDG and IDB with that we have come to the end of the lecture so let us make a summary of the important points now in this lecture we discussed the various regimes of operation namely non-saturation saturation and breakdown and then we explained the shapes of ID VDB then ID VGB and IB VGB curves okay in terms of the charge field and current flow conditions in the device