 We are going to talk about MOS devices, alright. So in a straw pool that we conducted in our department, we just asked which, so we admit lots of our M Tech students who come from outside colleges and there is much hair pulling saying they do not even know this, they do not even know that. So we did a straw pool on saying which course is worst taught in outside institutions but the course on devices one hands down. Actually the course on DSP on microprocessors on digital even analog are done quite well outside but the course on devices is unanimously chosen by people as being done very badly. So therefore we thought that we should include even though this is not a devices course, you might have a more detailed devices course outside of basic electronics but that varies from university to university and we thought that we will include a fair amount of devices in this course, it may or may not be taught as part of basic electronics that is one. Second of course we are doing it together we are not teaching you first principles of these devices and so on and therefore we should consider this discussion as a teaching methods class rather than a class on MOS, MOS happens to be a topic that we have taken by way of example. You have done a diodes thing in the morning with Professor John and did he cover basic semiconductors in that or did he start directly with diode characteristics? So I will begin with basic semiconductors, let us just first of all understand what happens in a semiconductor device. So in the old days George Gamow designed you know defined semiconductors as materials which are terrible conductors and worse insulators. So they were supposed to be no use at all because they could not be used as conductors, they could not be used as insulators either. So however history has seen that there are lots of uses for these. In today's class I will be talking mostly of elemental semiconductor which is silicon and germanium, I must point out that there are other semiconductors which are made of 3, 5 and 2, 6 pairs where the average valence is generally 4. So you can have 3, 5 or 2, 6 and so on when the average valence is 4 then you have covalent bonding and that means there are not that many free electrons to move about. All the electrons are part of the bonding and therefore they are not free to move about. So let us consider a material like this and essentially what happens is that most electrons are in a bound state and to go to a free state a certain amount of energy is required you have to break this bond. This leads to a band gap I am not teaching you solid state physics here but it leads to a band gap that means electrons can either have the energy corresponding to being involved in a bonding process or totally free in which they are not bound to a specific electron and there is a gap of energy between these two. So this gap leads to a band gap and the properties of a material depend on this band gap. Now if this band gap is very small almost 0 that means there is a continuum of states available to you any small amount of energy and you can accept that energy and start moving about. This is the case for metals or this band gap could be very high. In that case there are there is practically no energy available for taking a carrier to its free state almost all the carriers are bound and therefore while this material is full of electrons full of charge carriers when you apply a field there is no movement of carriers they are all bound in their position. By the way one side effect of this is that most materials which are transparent they are also bad conductors of light. There are very few exceptions to this bad conductors of electricity reason is obvious. They are transparent because the band gap is larger than the energy of light. Otherwise if the band gap is smaller then the energy of presented by light then a photon will be absorbed and this will lead to a transition from a lower energy to the upper energy state. So therefore if you are transparent that means you are unable to absorb light and if you are unable to absorb light that means the band gap is higher than that and if the band gap is higher than that then you are a bad conductor. So therefore most glass and all these transparent plastic transparent materials are therefore bad conductors. So all of this we will take as a background. So let us say that there is a material which has a this in between kind of band gap of the order of an electron bolt. Now if after all at any non-zero temperature there is a Boltzmannian distribution of energy. It does not mean that everything has energy kT. The number kT or 3 by 2 kT depending on degrees of freedom or whatever that comes essentially as the average energy. But the actual energy has a statistical distribution. So essentially you have some distribution of energy right. So this is the amount of energy and this is the number of simple harmonic oscillators so to speak which have this much energy. So essentially the average of this is given by the temperature. The average energy is represented by a temperature. But even though the average energy is here there are a few carriers through few atoms where energy higher than this is available. And this distribution of energy in fact can be described by many things. Actually the energy is somewhat like this and it is described in the Boltzmann distribution by this relation. It is a Boltzmannian distribution in that case if the energy is distributed like this. By the way much of it is as a discussion with you. It is not necessary that your students understand this. But it is important because after all it is this exponential which leads to the exponential in the diode equation. That is why the diode IV characteristics in the forward region are exponential. It comes from there. Now if there is an exponential distribution of energy that means there are quite a few carriers which have a much higher than average energy. And what will this so this when I talk of energy what do I mean? It could be the local vibration of atoms or whatever. So essentially there is this metal it is not at 0 you know this material. All atoms are connected to each other through springs and because the temperature is not 0 they are all vibrating away. And some of them are sluggish dancers oldish and some of them are teenagers and they are jiggling away to glory and there is a statistical distribution of this energy. And if somebody is you know has too much energy that energy may get transferred to an electron they let that may break a bond and this electron will then become free. So if you think of it as a chemical reaction so that means at any given time there is an electron which has become free and gone away. Now what it has left is the state which is deficient in electron therefore a hole. So therefore if you write it not like a physicist like I have described it. Suppose I write it like a chemist. So then it is like a chemical reaction in which I supplied some energy and this led to the formation of an electron and a hole. And whatever created the electron also created the hole. This part is clear. However should an electron chance upon a position where an electron is missing namely there is a hole then it will capture that electron. And what will you have recombination that means the electron is not there anymore. The hole is not there anymore and what has come this electron had high energy that therefore it was free. Now it is bound its energy is low. So what happened to the difference of energy that energy is released how is it released probably as vibrations or radiation depends on the material. But as in some form of energy that means it is exactly this reaction in the opposite direction. So if I look at it like a chemist in that case I would say that this is a reactant this is a reactant this is the product. And when electron and hole combined they give rise to some fixed amount of energy which depends on the band gap on the construction of this material. And if I have that much energy available I can create electron hole. So these two processes just like any reversible chemical reaction they remain in equilibrium. And in equilibrium there is some concentration of electrons and holes. And just like any chemical reaction is concerned it is this is the first order reaction. And therefore it is controlled by chemical kinetics which is which says that the concentration of electrons times the concentration of holes right these are the two reactants that will be proportional to the concentration of the product. But there is no product there is only two energy and we know that the total energy remains constant right it may change its form that energy may be absorbed to form an electron hole or given back. But the total energy remains constant right what it means is therefore if n is the concentration of electrons and p is the concentration of holes then the product of these will be constant in equilibrium right because after all this chemical kinetics applies only in equilibrium. That means this is somewhat not not easy to grasp for a person who is hearing for the first time. What you are saying is that if I increase the number of electrons in a material the number of holes will actually go down. What has what have holes done you are increasing the number of electrons in reality what have you done by increasing the number of electrons you have increased the probability that an electron will meet a hole. Therefore the probability of recombination has gone up right. So these holes are just waiting if only a few electrons are moving about there was some probability of this hole being destroyed because an electron fell in. If there are many more electrons this probability is high and therefore the holes will diminish which is natural which is part of this chemical kinetics. Therefore the two concentrations are not independent variables they are dependent on each other by this relation. This is very important therefore essentially what it means is that the independent degrees of freedom is not two but one in this way once the concentration of electrons is given the concentration of holes is fixed and the other way round in equilibrium. This part is clear now if you have a truly intrinsic semiconductor that is that means this semiconductor has no dopant or something which can change the balance between electrons and holes then any time an electron is generated a hole is also generated. Therefore the electron concentration and hole concentration will always be the same. Let us call that concentration n i in that case this constant will become n i square and since the material remains the same silicon or germanium or whatever even if you dope it this constant will remain the same because this has to do with the band gap what is the probability of recombination. Now as you all know I can add a dopant to it which will artificially change the number of electrons or number of holes. If I put a pentavalent material in this number of electrons will go up but you also note that correspondingly therefore the number of holes will come up. So the things which are in profusion are called the majority carrier and things which are in smaller numbers are called the minority carrier. Now I have a strong objection to this terminology because we live in India and majority and minority means nothing at all a few seats here on this day some few seats there majority becomes minority, minority becomes majority. Semiconductors are nothing like this in that the majority just smothers the minority always in most cases. So if you are majority you just kill the other kind. Let us look at this example a typical value of n i is about 1.5 into 10 to the power 10 per centimeter cube. But we are all lazy. Let us believe that n i is actually 1 into 10 to the power 10 who wants to do multiplications immediately after life. So let us say that n i is like 10 to the power 10 not very far 1.5. So that means this constant is like 10 to the power 20. Now let us say that I have some material and I replace one part per million of silicon by phosphor. So if I have a part per million there are roughly 10 to the power 22 atoms per centimeter cube advocated number light. Part per million means 10 to the power 16 dopants are there. Now originally how many electrons were there? 10 to the power 10. I have added 10 to the power 16. So that 10 to the power 10 is negligible. So the electron concentration has become how much? 10 to the power 16. So how much is p? 10 to the power 4 in a centimeter cube of this material. There are only 10,000 holes. You can practically tag them and give them individual names. So there are as many holes in this material as there are students in this institute. So that means this number is very very small. Indeed let us see what is the n to p ratio? What is n by p? 10 to the power 12. 10 to the power 16 electrons, 10 to the power 4 holes. So the n to p ratio is 10 to the power 12. Now all this scientific notation you know 10 to the power 12, 10 to the power 13 we get blasphemy about it. So I challenge you to think what is this population distribution? What is the population of the world? 6 billion? Yeah. So let us again we are lazy. So let us zap it up. Do not tell the population control people but let us zap it up to 10 billion. So that means there are 10 to the power 10 human beings on this earth. This ratio of 10 to the power 12 means that if you go around the earth meeting every man woman and child shaking hands with them and do it 100 times then you will find one marshal. This is the ratio of electrons and holes in this material. When we have replaced one part per million if we had replaced more this ratio would be even more dramatic. So what it means is that this term majority and minority the picture that it creates in our mind is completely distorted. Majority is really overpowers the minority except in a very small region when p is of the order of n. All right. This is very important to keep in mind one thing. The second thing is that the actual number of p and n it came from this exponential relationship. Because how many electrons were created was how many things had energy of that much band gap. If the band gap is high then even fewer places would have had that much energy. If the band gap is lower then quite a few will have this much energy. So this exponential relationship is important. So what we can do is we can use you know I do not see any white hair here unfortunately nobody is from my generation maybe you are not quite. So in that case you do not know slide rules. But this was one way pre-calculator days which we used to do calculations. So multiplication was reduced to addition by taking logs. So why do not we define something in terms of log so that we will express this n by p in a convenient way. So what is this n and p? What is n into p equal to constant saying in terms of log says log n plus log p is constant. That means there is some distance which is constant. If I that means there is something like this then if I represent if there are lots of n then this will represent n and p will correspondingly reduce so that the sum of these two logs. This is log n and log p. The sum of these two logs will remain constant. I may change the electron concentration. I might make the electron concentration less then this will be n this will be p but the width will always be constant. The sum of these two will always be constant. Now this is a very convenient graphical notation. Then what we say is that let me scale this distance so that this represents the band gap. Then inside the band gap is a level which logarithmically determines both the electron concentration at the whole constant. This level. If I increase n this level will go that way. If I increase p this level will come this way. So this level will essentially move between these two and the absolute value of this distance I have scaled so that it is the band gap. That means this level will move within the band gap as electrons and holes move and this one level will now rule the concentration of electrons as well as concentration of electrons. Is this part clear? This is very important and this level is the Fermi level. Very often even good students you try to explain what is Fermi level and generally the way it is done is I am myself a physicist by the way. I have no electrical engineering degree. So the tendency is to tell them like the your physics teacher told them. So you say there is this Pauli exclusion principle and you put an electron and you put one more electron and you put one more electron. They cannot be at the same energy and it fills up and at absolute zero it will fill up to that and the picture that it creates is totally out of the way. Because absolute zero you are filling this thing with the electron holes it is very unreal. This is absolutely real. This is you can see this. So this is simply a question that the product of N and P is constant. Both of them are exponential. Therefore you take logs and that logarithmic quantity represents this and if you scale it actually by KT by Q as it turns out. So then this distance is proportional to the number of electrons and this distance is proportional to the number of holes as simple as that and therefore since N times P is constant and is determined by the band gap that we know. So therefore this represents the band gap and this moving level is essentially a measure of the balance between electron and hole. So now we got the Fermi level. Rest is so now essentially I have a band gap. Where is the Fermi level when N was equal to P? Exactly in the middle. So this is the EFI, EF of the intrinsic semiconductor and if I increase electrons the distance is measured from here for electrons from the valence band is the electrons. If I increase the electrons the Fermi level will rise up. If I increase the holes the Fermi level will come down and it is as simple as that the sum of the two distances. So one distance measures the electrons the other measures the holes and this is the total statistics inside the semiconductor. So let us say that we have a semiconductor like this. Let us say it is P type and number of holes is 10 to the power 16 per centimeter. I have a thin oxide on top and I put some metal here. So this is like a capacitor except that I have taken this metal away and put a semiconductor. And this semiconductor is P type and it has a doping of 10 to the power 16. Now let us see what happens to this capacitor, this strange capacitor that we have made. First of all let us understand something that we are all familiar with. Suppose this side was metal the lower part was also metal then what is this? A parallel template capacitor. So how much is its capacitance? It is area into epsilon divided by D. Why is it D? Yeah but why is it D? This D is the thickness. What is the physical significance of this D? Why should this be D? No dielectric constant is epsilon. So why is it related to the thickness of this insulator? And the answer is what is the property of this metal? This parallel plate capacitor. If I should put some positive charge here it will induce negative charges. These negative charges are mobile. This is the metal now. So they are mobile. So where will the positive and the negative charges be found? They will attract each other. So all the positive charges will come and sit here and all the negative charges will come and sit here right at the surface both sides. And what is the distance between them? So it is the coupling between the two charges. Capacitance is a measure of the coupling between charges. So it is the coupling between these two charges which is controlled by D and therefore you have that term D there. So it is the distance. Now why do they stop there? Because it is an insulator. It cannot get into that. It cannot move into that. But they come as close as they can because they are attracted to each other due to the electrostatic force. So they come as close as they can and that brings them right at within an atomic layer of this surface and they sit close to it. So they sit there you know like Romeo and Juliet across the street in their balconies as close as possible but separated by this plate right. And this is the ordinary parallel plate capacitor. Now I take this metal away and put down that aforementioned P type semiconductor. Now we all know about semiconductor. That means there are lots of holes here and practically no electrons very few electrons right. We agree with that. However law of electrostatics are non-relenting. So you have seen those turned pictures of Maxwell and Boltzmann in text no they will not compromise. So the laws will apply even here. So when I put a certain 8 kilo charge here then I want 8 kilo charge here. There is no getting around do not say this is metal, this is semiconductor, this is insulator. I want that much charge here. So let us do some accounting of what is positively charged what is negatively charged here. This is a semiconductor silicon. It has dopants maybe both kinds of dopants. So let us say it has boron and phosphorous but much more boron very few phosphorous atoms and there are about 10 to the power 16 boron atoms here. Now a hole means this is it has given away a hole that means it is negatively charged. That means it is boron it wanted only 3 electrons but because of the crystalline structure there is this fourth guest which it did not want which is sitting there because of the bonds. Because there are 4 bonds coming here and it wants only 3 it is boron but this fourth uninvited guest is sitting in its home and there is nothing it can do about it. So it is negatively charged. It has more charge than it wants right. So what are the various things in it which have charged? First of all obviously electrons. Electrons are have carry what charge? Negative. Holes positive. Boron atoms which are ionized. So therefore they have negative. They have this because they wanted only 3 electrons but because of the crystal structure 4 bonds. So 4 electrons are sitting in their little veranda. So they are negatively charged. And what about phosphorus? They had 5 but when electron has gone away because there was no bond to retain it there. So it has only 4. So it has one less electron but the nucleus has not changed. Therefore it has one positive charge. So let us do a little bit of accounting. What is negatively charged? Electrons are negatively charged and boron is negatively charged. It has accepted. It is an acceptor. It has accepted an electron negatively charged. It has one electron too many. What is positively charged in this? A hole is positively charged and if there is a phosphorus then that is positively charged that is sorry. Yeah phosphorus has lost an electron. So that is positively charged. But there is a further distinction apart from this. The top guy is mobile. It can move about. Whereas these are ions they are fixed. The atom cannot go for a walk just because there is a field. But these free electrons you apply a field they will move. So the electrons and holes are mobile and boron and phosphorus are fixed. Now we have this scenario. Now let us say that I put some negative charge here. I put some negative charge on the top metal plate. So what are Maxwell and Boltzmann going to say? Turnly or maybe Gauss. So you put negative charge there. So it says give me positive charge. So we go around our table and say who is positively charged? Holes are positively charged. Are there holes in this material? Lots. It is P type. So lots of holes. So then what do we say? Holes love and this is negatively charged. What will it do to the holes? Pull them to the surface. Where will these holes stop? Right at the surface. So the situation is the same as it was in case of a metal. So let us plot the capacitance of this as a function of bias. I have applied a certain bias which is the amount of charge already DC charge already on this and then I jiggle it a little bit to measure its capacitance. So if this charge is negative then the charge required here is positive. That charge is mobile and all mobile charge will end up right here at the surface. Agreed? Alright? Therefore the capacitance is expected to be roughly the same as the parallel blade capacitor as long as it remains strongly negative. This part is clear, makes sense? Now suppose I put some positive charge in small amount of positive charge. What will happen? What kind of charge do you need? Negative charge. So you will say who has negative charge? You come here saying we want negative charge. Right? This is all sorts of going on in this material of phonons and photons and so on. Who has negative charge? So who has negative charge? Electrons but there are no electrons. Okay? Number is very small. 100 times the population of earth and then you will find one electron. Hardly any electrons are there. That would not be. But the negative charge has to be found. Okay? Uncle Maxwell and Uncle Boltzmann are non-relenting. You have put positive charge there. You must find negative charge. So where is the negative charge? Boron atoms. Lots of boron atoms are there. 10 to the power 16. Right? So what will happen effectively? That the holes are pushed away. That leaves exposed boron which is negatively charged. Agreed? So what is the charge distribution now? The charge is uniformly distributed in this region. Right? Because I put more positive charge there then I need more fixed atoms. Therefore I must push away holes even further so that the total charge in this region adds up to the total charge that I put on the top. This is clear? Alright? So as a result the average coupling between the freshly put charge and the new charge is further and further deeper and deeper into the semiconductor. This part is clear? As soon as I put some charge there are no electrons. So I pushed away the holes from the surface. Now I put some more charge. I must push them away from a further off point. Put some more charge even further. Right? Therefore this region which is now depleted. Right? So as a result it is as if the thickness of the insulator has become more. Right? And that means that the capacitance is reduced. Right? The coupling is reduced because what was that D? That was the distance between positive and negative charges. Right? Now the distance is higher. The amount of charge is the same but the average distance between positive and negative charges is more. The action is taking place further and further deeper into the semiconductor. You agree with this? So as a result what will the capacitance do? Come down. Right? Capacitance will reduce as if the insulator has become thicker. Okay? This makes sense. How long will this continue? Okay? So I have taken advantage of the fact that you have had a good lunch and pulled a fast one. The point is that the argument that there are practically no electrons in this is a conditional argument. That argument says that there are lots of holes. But if there are no holes then that is not true. At some point the average number of holes will become so small that now holes will become the minority carrier and electrons will become the majority carrier. Remember this diagram. So when I have pushed away the holes essentially I am fiddling with this. I am moving this to the other side. So that has an effect not only of reducing the holes but also of increasing electrons. Okay? So up to some point the net charge will be produced by pushing away holes. But once the holes have been pushed away electrons are not in such small numbers anymore. As soon as holes reduce electrons increase. And at some point electrons will be as numerous as holes were initially. Okay? And now this ratio will become exactly the other way round. That means there will be 10 to the power 12 times as many more electrons as holes. There are practically no holes left. Right? So now the action will take place with electrons. Right? So now when you want more negative charge there are willing electrons who will come jumping up. And where will they go? If there are electrons which respond to the more additional amount of positive charge. Where will they go? They are free. And there is positive charge on the top. So they will go bounding up the surface till they come to a stop here. Right? So they will all come. They will all come right here. Agreed? And therefore the capacitance will be restored to the parallel plate level. Agreed? And therefore my capacitance will suddenly bound up and become like this. Okay? So essentially what I mean is, if you look at the relative position of the Fermi level, initially this had lots of holes and very few electrons. Right? That means very few electrons means it was right here. Right? And lots of holes. Now as I push away the holes this will keep inching up. And as soon as it goes above EI then there will be fewer holes and many more electrons. Right? Now the band gap of silicon happens to be about 1.12 volts or so. Right? That means a potential change of about 1 volt or so, in fact much less than 1 volt, will cause this complete reversal of roles between electrons and holes. Okay? Whereas electrons were 10 to the power 12 times more numerous initially. Now holes will be 10 to the power 12 times more numerous compared to electrons. Okay? The entire character of this semiconductor has changed. Starting out from P type it has now become N type locally under this metal. Without too much ado, no current has flown, not much power has been consumed. Right? And we have applied some 1 volt or so and we have changed this material property. What material property? The ratio of N to P. It has gone from 10 to the power 12 to 10 to the power minus 12. It has changed by 24 orders of magnitude. Okay? So I am fond of this back of the envelope calculations. Do not do it right here. But calculate the distance between sun and earth and calculate an atomic radius and find out the ratio of these two. It is nothing compared to what we have done. By just applying 1 volt, it is a capacitor without drawing any current. Nothing doing and we have changed this particular property of the material 24 orders of magnitude. Okay? 10 to the power 12 itself was overwhelming. 100 times the population of the earth, more than 100 times the population of the earth. Just imagine what 10 to the power 24. Okay? So therefore, this is a very big thing. This is called inversion. Alright? This is called the MOS capacitor. This is the heart of the modern day MOS device. Okay? However, what I have told you is true in equilibrium. Alright? It is not true. All this N to P ratio, etc., that we calculated, that applied only in equilibrium. If there is no equilibrium, this is not true. So the next question is how long before equilibrium applies? Right? So therefore, let us say that to invert it completely to N type, I need to put, let us say 2 volts on the metal. Okay? Now if I put 2 volts on the metal, suddenly at T equal to 0, I know that the material should turn from P to N type. Okay? However, where are these electrons going to come from? These electrons were not there initially. How many were there? 10 to the power 4 per season. Okay? Now, to give you a typical number which occurs in these situations, from 10 to the power 4, I need 10 to the power 12 surface electrons. Okay? Not volume. 10 to the power 12 on the surface to equal this charge. Alright? So 10 to the power 12 electrons, where are they going to come? Suppose I need 10 to the power 12 electrons. Let us say that this metal is 1 centimeter square. Okay? It is a centimeter long, centimeter wide patch. And I have put a couple of volts there and C times V requires 10 to the power 12 electrons. Where am I going to find 10 to the power 12 electrons? In 1 centimeter deep material, how many electrons were there? 10 to the power 4. Right? So this semiconductor should be how deep? So that I find 10 to the power 12. Each centimeter will give me 10 to the power 4. Right? So how deep does it have to be? 3 centimeters. 3 centimeters will give me 3 into 10 to the power 4. 10 to the power 8 centimeters deep. Right? That is 10 to the power 6 meters deep. Right? So that means you can go from here to Delhi. Right? And then only then you will find enough electrons. Our semiconductor is not deep. It is not deep from here to Delhi. So something else has to take place. Okay? So you can say Maxwell Boltzmann be damp. There are no electrons here. Where am I going to give you this many electrons? Okay? So in equilibrium, all right. So the question is how is this equilibrium maintained? So what happens is that suppose you have put suddenly 2 volts here. So let us say in 1 centimeter you had 10 to the power 4 electrons. So you pull them away. Now the rest is completely left without the electron at all. Because it is devoid of electrons, it is away from equilibrium. In equilibrium there should be 10 to the power 4. Right? So thermally these electrons whole pairs have to be generated. Right? When there are 10 to the power 4, you drag them as well and you do it 10 to the power 8 times. That will take you time. Typically this time in case of silicon is of the order of 1 tenth of a second. Okay? It varies over a lot. I am just giving you orders of magnitude. Okay? This is very slow. While this is very promising, this is very slow. There is a problem with it. Right? So while it means that it is very promising, I can convert a p-type semiconductor to n-type semiconductor. But if I suddenly apply this bias, then it will be a tenth of a second before then of course equilibrium will prevail. Then the surface will be inverted. It will be n-type. The base will be p-type. Okay? But it will be slow and there is a time response of this which is quite slow. Alright? However, let us put that aside and come up with some bright idea. Okay? So what I am going to do is the following. This is my same old MOS structure, metal oxide semiconductor. So this is the same. This is the same material p which is 10 to the power 16. Okay? But now I put two strongly endoped materials here outside of this just outside. Okay? So that the corner overlaps. Alright? Now you have done diodes this morning. I will attach electrodes here and here and try to pass current. Alright? So let me assume that this metal is not there. Then what do I have? n plus p here and pn plus here. These are back to back diodes. Right? So therefore p made positive will make it forward bias. Right? So it is like this. So this is the structure that I have. Whatever I do no current will flow because at least one of these will be blocked and they are in series. Okay? So no current will flow. This is like a switch which is off. Now let me put that one or two volts on this. Now what happens to the surface? Becomes n type. Right? Becomes very rich in electrons. So therefore the surface now becomes n type and now there is no junction. I have n plus n, n plus. There is a current path. I apply the voltage. Lots of current will flow. Okay? Now this is a very good idea. Now I got a switch where without consuming too much energy without doing too much I can convert this off switch to an on state just by applying a voltage at the gate. This is called an MOS transistor. Okay? However there is one problem with this. The problem is that the rate of switching of this will be very slow. Right? Suppose I convert it to n type suddenly. How long will it be before it starts conducting? It appears that it should take one-tenth of a second before it becomes n plus. Right? Only then will conduction begin. So that is bad news. Our brilliant idea does not sound so brilliant anymore. Okay? But very often God comes to the help of the Sheikh Chilli. So it turns out that there is something which helps us. Why was it that it was taking so long as one-tenth of a second? Because there were no electrons. For equilibrium we wanted electrons and there were no electrons in this material. Right? You wanted material which is as deep as from here to Delhi or Pune or whatever. Right? And that was not available. Electrons were not available. Correct? But suppose I put suddenly positive charge and a cry goes up, electron lana. So this guy will say, oh yes I have electrons. It has any number of electrons you want. Right? So the electrons are available right where you want them. You want them at the surface and there is a source of electrons right there. So rather than creating them thermally from the bulk, there is a guy sitting there. Now of these two, which one will provide electrons? In order to pass current, I have applied a voltage. I have applied negative voltage to one and positive voltage to the other. So which one will supply the required electrons? The negative. Right? Because the positive guy is holding electrons like this. And the electron guy is pushing them off. So all the electrons will come from the one which is negatively charged. So that is called the source. That is the reason that terminal is called the source. It is the source of electrons. Okay? So that is called source. However, if electrons came and then equilibrium reached, then you will have a pulse of current and then nothing. But what happens is that even as currents come, conduction starts. Right? So this guy keeps on supplying electrons and the other guy keeps on taking them away. And in this process a dynamic equilibrium establishes so that the average concentration of electrons, the net concentration of electrons, these are not the same electrons. Okay? So Ramlam came and went away and Chagandas came and went away and so on. But at any given time, the population under the gate is 10 to the power 12 or whatever you want. Okay? And this establishes a certain amount of current. Okay? Higher the current, meaning higher the concentration that you require, that means higher the voltage on the gate, higher is the amount of current that will flow. And the net concentration of the electrons in this surface is then determined by the vertical. Right? And the net current is now determined by both. This part is clear? Okay? So now without doing any algebra, without doing any basic quantum physics and so on, we have established the operation of a MOS transistor. Okay? This is the basic MOS transistor. There is much more to do actually beyond this. We do not have time. So we will not do in great detail. But it turns out that if you draw the IB characteristic of this, then you get characteristics like this, very much like a bipolar transistor characteristic. That means the current does increase. This is the drain voltage and this is the drain current and this is for different gate voltage. Okay? So as you increase the drain voltage, initially the current increases. But then it becomes constant. All right? And this is the biggest mystery for students. Why does the current become constant? Okay? So we will just have a 5 minute discussion on that. That is all the time I have. All right? And if you have some questions, then later on we will discuss this in much greater detail. All right? So first thing, what is current? Definition of current. Rate of flow of charge. What is flowing here? This is an N-channel transistor. Electrons are flowing. Right? So electrons are flowing. So what does the current depend on? It depends on two things. How many electrons are there and how fast they are moving? Right? If there are more electrons, there is more current. If the same number of electrons are moving faster, then also there are more current. Okay? So these two things determine the total amount of current. Right? Who determines how many electrons are there? The gate voltage. No, source is always heavily doped. Source will provide as many electrons as the gate wants. Right? So the gate determines how many electrons are there. And who determines at what speed these will move? The horizontal field. Therefore the drain voltage. Right? So it is inherently a bi-dimensional device. The vertical dimension determines the amount of charge. The horizontal dimension determines the velocity. And therefore both of them determine overall how much current will flow. Therefore it is not very surprising that under ordinary conditions before saturation, the current is actually given by an equation somewhat like this. Okay? Now this is algebra. But you really have to see beyond algebra and see what physics hides under this. Okay? So let us take VD common here. All right? So what do you get? Now let us take a qualitative picture of this. Not just algebra. There is an integral. My notes have that. You can derive this equation. But we really have to understand physically what is going on here. So as I said, there are two things which determine how much current will flow. One is the amount of charge. Right? This second is the velocity with which it is. Therefore the field. So first of all look at the constants in this. It is a product of two constants mu and C ox. C ox is electrostatics. Mu is electrodynamics. C ox determines how much charge will arrive in response to a vertical field. Right? C ox times V will give you the charge. Right? And mu determines how fast that charge will move for a given field. Right? So therefore it is not surprising that you get the product of these. The W is quite clear. If you double W, it is like putting two transistors in parallel twice as wide. So you put two transistors in parallel, current should double. So therefore it should be proportional to W. Okay? All of this makes sense. Look at this expression. When VG is equal to VT, there is charge but none of this is mobile charge. Right? Up to a particular point, if you recall, we had got this charge by pushing holes away. Right? We had got only ionic charge. So up to the point that you reach VT, there is charge but this charge does not take part in conduction till you reach VT. So it is only the voltage beyond VT which produces mobile charge. Right? All right? So how much charge is there? Well at this end, C ox times VG minus VT. Right? The mobile charge is 0 at VG equal to VT. So any voltage in excess of this will produce mobile charge in excess of VT. So C ox times VG minus VT. Agreed? That is the vertical field here which is producing that charge. And how much charge is here? No, no, no. Don't look at algebra. How much? What is the vertical field here? No, no, no. We are looking only at the vertical. Okay? Don't go by textbooks. We are understanding this now. We are not proving algebra. What is causing charge at that point? The potential difference across that dielectric, how much is it? VG on that end and VD below, drain voltage below. Right? So the net available charge is VG minus VT minus VD. Agreed? So VG minus VT at one end, VG minus VT minus VD at the other end. What is the average charge in the channel? Average of these two? So VG minus VT minus half VD. Right? VG minus VT minus 0 here. VG minus VT minus VD here. Take the average. VG minus VT minus half VD. That is what this is. Multiplied by C ox, this is the amount of charge. Right? VD by L is the field, horizontal field. Multiplied by mu, that is the velocity. Charge times velocity and scale it by W. Right? And that is where this equation comes out. Okay? Otherwise, this is just algebra. You just derive it, etc., etc. But now you look at it, this is your friend. Now it tells you what is really happening here. Right? That is what leads to MOS conduction. However, this equation is valid only as long as this term VG minus VT minus VD doesn't become negative. Okay? As soon as it becomes negative, that means there is a portion here where the mobile charge density is 0. Right? So then what you should expect and this is what the students ask all the time, that the current should drop to 0. There is no channel at the drain end. So why does the current remain constant rather than dropping to 0? Why should the current saturate? It should actually drop to 0. There is not sufficient vertical field here to have mobile charges here. Okay? And that is where the two dimensional nature of the MOS becomes important. Electrons can arrive from two sources. They can arrive due to the vertical field and now none are arriving because of the vertical field. Once you reach VG minus VT. However, electrons are arriving horizontally. So those electrons which are flowing, they will not come to a dead stop there. They will get into this region. And the number of electrons need to be quite small to keep the current constant because the field will now become very high. Therefore, this region over which the voltage is negative will become very narrow. And across, this is called the pinch off region. And across this region, a very high field will develop and all the excess voltage over VG minus VT will drop across this pinch off region. And very few electrons will be sufficient to give you current continuity. Okay? And these electrons will not come because of the vertical field. Vertical field is not enough to draw electrons. They will come because of the horizontal flow of the electrons. Okay? So now we have the picture complete. When the as long as you increase the drain voltage initially up to VG minus VT, it will follow this equation which is initially linear, but you subtract a parabolic term from it. So it kind of stoop below the linear in a parabolic way till its tangent becomes 0 at VG minus VT. And when it reaches VG minus VT, the current becomes constant. Okay? This is the pinch off. This is the saturation of current. And that is your MOS transistor. And it is not limited by that slow speed which the undoped MOS capacitors was. Experiments have been done. The MOS capacitor is indeed slow. The MOS transistor is indeed not. Otherwise, you will not have Pentium with a clock rate of 3 Gigahertz. Okay? So the actual MOS transistor can switch in fractions of a nanosecond. And that is because you have heavily doped region which are willing to supply electrons. And they are so close to the region where it is required that there is hardly any time lost for those electrons to come within that region. And therefore, this transistor changes from conductive to off in literally picosecond. Okay? And that is how all the modern digital and analog devices are made using this MOS. Okay? I think that is enough dose for one hour lecture. We will stop here. So let us break for T. But I am around till 3.30. If you have some questions bugging you about this semiconductors or MOS and so on, we can discuss. If you require this infrastructure, we can discuss it here. Not everyone needs to wait. Only those who have questions and others, T is there. The current density concept comes into that. Current density is proportional to the field. Yeah, it is current density. So at that time, the current density will be high? Current density is, it is not really the current density which is high. Current density is the same. Current density has to be continuous. So it is not dependent on current density. Okay? And you have walked into the trap which I was hoping you will. So now I will give you the experiment. Okay? Do this thought experiment. This is my channel. Okay? The question that we did not have time to go through. The question I am asking is, how much of this channel is actually depleted? All right? So what happens is that I must have current continuity. Right? It is all in series. So I must have current continuity. So let us say that I have a drain voltage which is greater than VG minus VT. All right? So all the excess voltage above VG minus VT drops across the pinch of region. The rest appears across the channel. All right? So let us assume that the voltage VG minus VT is reached here. This is VD greater than VG minus VT. Okay? And this region is delta. All right? Now the, if you look at things to the left, this is a standard MOS transistor with drain voltage equal to VG minus VT. So it has certain amount of current. That current is arriving here. Right? Now this material, there is no vertical field. Vertical field is not producing electrons. Right? So it is only the horizontal field which will determine the current. How much is the field? VD minus VG minus VT. This is the voltage across this. Right? This end is VD. This end is VD. And this end is VG minus VT divided by delta. This is the field times mu. This is the velocity times whatever is the charge. Okay? This must be equal to I d sin. Okay? Now because this charge is very, very small. Practically 0. That means the velocity has to be very high. When will the velocity be high? When the field is high? When will the field be high? No, VD is the same that we applied. VD is not under delta is very small. Okay? So delta becomes infinitesimally small to produce a very high field. It is like how do you get current in a bipolar transistor? The collector junction is reverse biased but it sets up a high field. Any electron which happens to amble into its field is then whisked away. Exactly the same thing is happening here. Electrons have been brought up to here by the regular MOS action and then they are whisked away. Okay? And the size of this delta is determined to keep the current continuity. As a result you can see that delta should be linearly dependent on VD. Okay? If VD, because this I d sat is constant. So therefore delta should depend on VD linearly. Okay? So in short channel transistors what happens is that wherever we use, where we were using W by L. Now it should be W by L minus delta. Okay? So what you will get is W by L minus delta. So I am going to cheat. I am going to write it like this. This is fine. So this W by L is the old value. Right? And this I can write as 1 upon 1 minus delta by L. Right? So this will be 1 plus delta by L binomial theorem. Right? But delta is proportional to VD. Right? That means the current will linearly increase with VD in short channel transistors. It will not become stable but you will get transistors where the current increases linearly and that is exactly what we see. So in a practical transistor, short channel transistor you do not see full saturation like this. You see a linear increase here. Okay? And it will become. So you can break for T then.