 We are still under the work about making silicon from SiO2, we went up to yesterday creation of polycrystalline rods using Siemens process and then we want to create now single crystal and wafers and we proceed further and the process which is possible in the case of silicon is there are many, most of the materials can be grown in two techniques, there are two standard processes which allows single crystal, highly pure silicon rod to be pulled and these two techniques are called CZ technique and FZ technique, though the names are, the full forms are different but short forms are similar CZ and FZ, one is Zokralski technique and the other is float zone technique. In the case of Zokralski technique, most integrated circuit manufacturing industries use CZ crystals and the advantages of this process, they are very large diameter wafers can be created, very, very large size, maybe as I said 16 inch wafers can be pulled, whereas in the case of the other observe we shall see they may not be easier to be pulled. Then the second is the process is relatively simpler and much more easier to control the tree doping, much more easier as I say, of course there are disadvantages, it contains small amount of mini impurities like carbon, oxygen, iron, bismuth, lithium, many of them, okay. Of course not all of them in a large amount, some of them are in parts per billion or less than part per billion, some are in parts per million and these are the ones which may actually create hell of a job to actually get rid of them. We will be surprised to see that the oxygen which is sitting here is a bivorant because the way the process is done, oxygen will always be present, so is the carbon and because of that the any silicon process which we will do will have these two standard impurities carbon and oxygen, both have some advantages and some disadvantages. So CZ essentially is the standard technique called Zokral ski pool technique and I repeat larger wafers are possible only through this technique and therefore most integrated circuit manufacturing companies use CZ crystals. Then why we are really looking for flow zone, yeah flow zone is also important of course for the flow zone you need CZ crystal initially, so you can say it is added process, it is dam costlier comparatively, it is impossible to grow large wafers in this technique. Of course it has the feature and that is why it is used often, it is very very very high degree of uniformity of impurities across the wafer and along the rod as we shall see soon and they are very highly pure silicon wafers, very low concentrations of both oxygen carbon and iron and other impurities, very highly pure material is possible in flow zone and for example ICs may not require that many that much of impurity, please remember most of the impurities act like either the donors or acceptors in a semiconductor, so if they are donors they will make it N type, if they are acceptors they will make it as P type and all our effort in the earlier ways is to dope a material to your choice but if intrinsically it is present there then it is difficult to control. However in the case of flow zone these impurities can be easily wished off, where do you need such a highly pure, so one can say purity of a wafer can be determined by this electrical property which will say the resistivity, resistivity of a sample is decided by say one upon Q mu N plus Q mu P, so if N and P's are very very small from the impurities then the rows are very large, okay resistivity of a wafer is very large, so larger the resistivity pure is silicon. In CZ at best you can do 100 ohm centimeter wafers or may be at best 150 to 200 sometimes but in flow zone I can go up to 20,000 ohm centimeter wafers and that is fantastic about it. The reason why we are interested in such wafers with high resistivity 20,000 ohm centimeter, one can see if I calculate the carrier concentration out of 20,000 ohm centimeter it will be around typically may be one can say it is 10 to power 14, 5 into 10 to power 14 or 10 to power 14 or even sometime less, the intrinsic carrier concentration of silicon is 10 to power 13 per cc and since I actually 10 to power 10 but 10 to power 13 is the smallest value which we use as called intrinsic semiconductors. Now we come very close to intrinsic value which means we have a wafer which is intrinsically doped and you need such wafer to create what we call power rectifiers, we also need fileries or rather photorextrary detectors, lasers, many of this P ion based devices actually require I type of material and that means intrinsic wafers, there is where the large amount of expenses are spent by these people to get intrinsic like avalanche diodes in the case of microwaves, they also need P ion kind of structures. So all those devices which are special devices and not IC group of devices, they are actually many times require a P ion structure and in those cases flow zone wafers are only possible. We are also trying that for solar cell improved efficiency and we will show you how I region does improve some efficiency if not great way. So please take it that flow zone is one of the feature that it is highly pure wafer, highly pure wafer. Now the problem starts how do we actually go through from the polycrystalline process where a lot of rod has been created maybe since I can create rods of polycrystalline in the rim reactor which I yesterday shown and then we actually break them into pieces and these pieces are called nuggets. Now these nuggets are used to create highly pure silicon rod which is shown here. This is a Okraski reactor shown here, the method is okay maybe one can think of it, this is the rod being pulled, this is the seed crystal, you can see it smells, there it is, you are in some kind of a crucible melt the nuggets at around 1400 and 12 to 1500 degree and then slightly cool it up below 1400. So it is a molten state not fully liquid and then you put a seed crystal from the top and touch it and starts pulling okay. As it touches the mold it picks up some mold to its because of the surface tension. As it starts pulling it becomes cooler and cooler so it becomes solidified. So crystal starts growing up as you start pulling the crystal, single crystal above and the orientation of single whatever grown by you is same as whatever starting single crystal you start with. So essentially this is just pulling up the rod from the melt and there are few things which we will discuss soon when we model it, there is a rate of pulling and also during pulling we also rotate the, many a times there are some reactors in which the whole mold was rotated but mold is very high density material so at times it breaks the crucibles. So nowadays only the rods are rotated, they have their own problems but essentially now rods are only rotated and also pulled up okay. So these are the two methods in which we control the group as well as the impurity uniformity as we will see later. Now this once as I say from the polysilicon nuggets are put here please remember this chamber is a carbon chamber and which is essentially a rather graphite chamber and which is lined with silica. Now you can see the two materials I said silica, silicon dioxide and a rather quartz liner and also carbon crucible or graphite crucible. So carbon and silicon, carbon and oxygen are only present in the mold itself even if their concentration is low because their melting point is higher but they are released in the melt and when you solidify something even part of this impurities will also go into solids. So do whatever, there will be oxygen and carbon in the silicon grown in zucrose kipul crystals. May be better figure, all these figures not necessarily colored ones are available in Plummer's book so if you please have a look at it some way. I repeat this is rewritten again, you can see a slightly shown here earlier there is to be a bottom rod on which this crucible was kept and that was rotated and it was found that many times crucible is to break. So in lower cases now this lower chuck is never rotated, it is only fixed, crucible sits on it, you heat it on arc furnace and as soon you start heating it goes up to 1417 degree centigrade and you start pulling and also rotating to control the growth. So these are all same figures differently shown to show you that how actually zucrose kipul crystals, please remember the values are something like this. If you see this tip where the single crystal was, the solid which actually picked up from the melt has a much larger diameter because of the curvature it has. So the depending on the kind of crystal size you initially start and kind of initial touch you do, the size of wafer is different, size of rod is different. So how much you allow it to grow along this and then start pulling. So there are tricks in that to increase the size, it is not very easy for outsiders but people who do it very routinely. Now the tip there is much smaller than the size of the crystal which you are pulling. So once I was just telling if you spend too much with high, you will spend it definitely then it is possible that that joint may not survive between the single crystal and the grown crystal. If you pull it also this very heavily, the melt will actually climb over. So you do not want to pull it very heavily but you want to pull with certain rate because you want to increase the size of crystal. So you will pull it but at the same time you will have to decide how much are and how much rotating speed you should have to have a good uniform single crystal coming out of this. Here is the same figures, these are the nuggets. You can see the better way of showing the figures. This is seed being pulled and rotated and this is how it will look. And you can see the tip is too small. You can see the joint is all, if you do, if you do too much, this joint will. So there are tricks in doing saving this which is not very difficult but I just want to show you how things happen. This is our picture from the real life K-sex corporation wafer company which produces 300 millimetre silicon ingot. The larger rod is called ingot and it can be as long as 1 meter to 1 and a half to 2 meters. Depending on the reactor you have, please remember larger the rod you pull, the machine require large heating because mold has to be larger. So cost is very high. However cost to cell ratio decides how much size you should do. This is another picture for the same Zocalski pull. This is the real life picture which is taken. This is of course a old plant in Germany from where this has been taken. So before we go to the float zone, we will like to show you that Zocalski pull crystals need to be doped. Right now we are not doped anything. So how do we do them? What we do is during this melt situation, we actually add per gram of mold some amount of impurities which you want to do. Like P type you add boron, boron oxide impact. And in case of phosphorus you have P2O5, arsenic, arsenic oxides. So this oxygen is anyway added by you by this, whether you like or you do not. Of course boron itself can be added because boron is a powder material. It can be added but it is very hygroscopic. What does that mean? It picks up humidity, humid very fast. So it becomes jelly. So we cannot add that so much. So we normally use powder which is boric acid which is available in market. It is not pure but it is available. So you add to this and since the melt is dissolved that and when you start pulling the crystal, some part of the impurities also go along the mold to the surface and then to the crystal lord. This is how the impurities are added in the case of CZ. Now we already said the problem with impurities in corporation is that is the model we would like to see it. How the impurities can be made uniform? That is major worry for us. How the full rod has same resistivity or same concentration along the axial axis and along the real axis. Both sides it should be uniform doping. Then only I can say when I cut the wafers as I show you later, I will have uniformly doped wafers. If I go and buy a market, the wafers, many things we actually specify. One of them of course is for example, one is doping which is that is either P type or N type. The other thing they will say crystal orientation like say 100 oriented. Then they will also specify to you resistivity. Normally resistivity is never specified one value. For example if I am buying wafer for integral circuit which has 10 to power 15, 10 to power 16 per cc as concentration. Typically I am talking of 0.1 ohm centimeter. But they will give you range 0.1 to 0.2. They themselves will not be sure of putting exact value. So some wafer may have 0.1, some may have 0.2, some in between. So you have to monitor each wafer before you actually load in. So you segregate those ones which are similar resistivity in one run and put in the others. Then they also specify what is called as dislocation count. We will see this part little later. Maybe next when diffusion we start. That is number of dislocations per centimeter square actually is called EDP. And this is very important. Larger the number of this verse is the wafer. Smaller the dislocation counts better will be the wafer. So one has to actually see how much EDP you are looking for before you buy. There is also radial resistivity variation. This is wafer to wafer. Some companies give radial resistivity variation. This is also specified. Then they specify thickness. And of course dia is you have ordered it but diameter. And that also plus minus delta D is specified. So when you order wafer, you have to specify in your indent itself everything. I know depending on that, they will give you quotations which you can then buy. Please take it. Larger the diameter of the wafer, thicker will be the wafer requirement. Why? Because larger the wafer may, larger is the mass. It has. If you take very thin wafer, silicon is extremely fragile material. Of course I do not say it but if you say it may break, it does not. But if you handle it little, not enough care, it breaks immediately. Particularly silicon crystals, wafer breaks very fast. Each crystal may cost you around $50 to $100. So think of it. If you lose one, you may be charged for $100. Therefore handling it is a very important thing in this is. So the larger the thickness, thicker will be, larger the wafer dia, thicker will be the wafer size, wafer thickness. Typically for three, maybe six inch wafer, I may require 500 microns of thickness. 12 inch, I may have 1 millimeter thickness and 16 inch may be even little more than that. Because it should give physical support. But unfortunately at part in all this you should understand. What is the economics behind? If you have larger size wafer, I have some figure later, you will get larger number of chips in same length, okay. Larger space means larger chips will come. Initially in 3 inch if you have a 10, in 12 inch you will get 250, 200, 300, 500 chips. The problem starts, any amount of thickness you create, it does not matter because the whole silicon integrated circuit processing is within first few microns, 1 or 2 microns. So if you have a 1000 micron of wafer and you only use 1 or 2 microns, now even it is less than that. Now in that case since it is a planar processing everything is on the top plane only, few microns are even less. So that means the rest of all of it is not doing anything, it is just getting wasted, okay. I will not say it is wasted, it has some thermal sink issue, good sometimes. But otherwise it is really wasting silicon. So that is the main problem that larger wafer you take, one side you may have larger number of chips but you will waste larger silicon amount. So one has to plan how much thickness, how much, so depending on the, you know a company decide which product they are on the sale, they decide which run line they should have. If they do not have that then they will actually go to a company which only manufactures. These are called foundries. So we will send all these to foundry to make for us, we will not do our self anything, okay. Because that kind of things we do not have, so we do not want to really get it done from somewhere else, okay. So that is the trick in all calculations of, you know when you set a line, you have to think too many things before actually you are productive. And at least within start of a company or line, it takes at least 2 to 3 years before you are on equal basis, send money, receive money and then profit starts. By then if the technology changes, some new products appear, this whole line may go far, okay. And there is the issue that how much investment wear decides the company's success, okay. And that is why many of the people who may not be that intelligent in processing still may actually tell only go for this, this is what is going to survive, okay. These are managers who only manage and actually rule the world for doing nothing, except telling, okay. Of course, due regards to them, I am not a manager or maybe good manager at all, neither man or machine or anything. Before we go to this, let me do something more interesting about this because I am going to dope the crystal. Since I am going to dope the crystal, as I said already in the melt itself, I will add impurities as per the doping I need, okay. Problem of modelling is very difficult for variety of reasons but very important reason. If you see the maybe next graph little clear and then I will come back to this. This is like this is a process shown here from the liquid or rather maybe you call it melt and the very close where the solid starts interface is the liquid. There is slight difference between melt and liquid, okay. Melt has slightly lower density than the liquid. So we may differentiate but some books do not so I may not also or I may or may not. CL is the concentration of impurity. This is CL, this is CS, we will define and do the modelling. So we can see I am pulling a crystal from this liquid or small and we are pulling it and also as I said rotating it. Now there is a concentration of solid, concentration of impurity is going towards solid and part remaining in the liquid area. The problem is not that difficult. If I know how much it goes, I can recalculate and I say okay how much impurity will go above. The problem is the reverse process there. There is a word which is called segregation, okay. Some impurities like to be more in one region. For example some more like to be in the solid than in the liquid or more in liquid less in solid. Normally when we have same material both side, the name given to this so-called coefficient called segregation coefficient is named distribution coefficients. If you have two different materials silicon and silicon dioxide and impurities go through and also do reverse processing, it is called segregation. For example boron tends to segregate in oxide than in silicon. Phosphorus tends to segregate more in silicon than in oxide when I put the impurities down. This segregation is a process which is distinguishing impurity passing through each material. Now in the case of solid and liquid also this process happens. Whatever you are pushing up partly it is coming back and so it segregates down, okay. Depending on of course the value of ratio of solid to liquid concentrations, okay. This segregation is the major cause. How much segregation you will get? And how much segregation coefficient as we shall define now we should have so that you have more uniform crystal available. Please remember every time some impurities are pushed down then the concentration in the melt will increase and lower then it will go lower above. So there has to be some balance that uniform amount of impurities go up so that the wafer rod which is pulled has uniform doping, okay. This is our major worry and we will like to model how many impurities can go up uniformly so that rod is uniformly doped. Though as I said rarely it is not but vertically it is mostly very good, okay. And as I said last time the P type impurities which I add are boron, aluminium, gallium, indium. N type impurities which I dope may be phosphorus, arsenic, antimony. Other impurities are omnipresent which are oxygen, carbon, bismuth, lithium and gold and I mean iron, okay. I forgot iron, iron and gold, okay. All these impurities can have segregation with solid to liquid when they are present. Please remember impurities do not differentiate when they go from liquid. They all will like to go above or come back to the liquid again. So each should be, we should know what is the segregation for each of them means how many impurities will go actually in solid has to be controlled and this is the model which will allow us to find how do I do but what rate I should pull, what spinning I should do that uniformity is achievable, okay. So segregation coefficient is the material property which is decided by the actually the band structure or equivalently saying the atomic structure of the material. In case of amorphous material for example there is no actually atomic bonding. So the crux of space charge available at any given time decides how many items can actually allow, there is some kind of a shielding occurs, a wall is occurring, so I cannot go above, so I will be pushed down. So segregation is a property of the sizes of the atoms as well as electrically if the impurities are going, they are strong also by electrical fields then they will say what kind of bonding it sees to reflect back itself, okay. So these are essentially not very clearly told, we only monitor it, okay that this is the segregation, okay. We are not putting them, we are not putting any of these impurities except gold one times we will do it probably but otherwise no impurities put, they are only present, okay. As I say in Zokraski carbon is, the crucible is graphite, oxygen line, silica quadliners, oxygen is silica is present there, I do not want to do anything but it will be there. Same is, of course the traces of bismuth lithium and iron is very small, iron sometimes is higher but other bismuth lithium is very small but they are there. So in float zone we actually see to it that they actually go away, that is exactly what float zone allows us to do, okay. We remove those impurities on the one end, the basic idea is if we segregate, we all of them should segregate to one side then the rest of the crystal is only of one kind, okay. So that is what float zone is trying to do compared to CZ, okay. But CZ nothing very much I can do, I can just control to some extent, okay. So we define the segregation coefficient or as I say some books called distribution coefficient as far as medical things are, you can also call it distribution coefficient is defined as CS divided by CL, where CS is the concentration of solute, solute means the impurities you are talking or solid silicon itself could be one, in solid divided by concentration of, here of course I am talking of impurities, concentration of solute by weight in liquid, okay. The ratio please remember if there are net amount of impurity, some weight and the crystal has some weight, so C is defined as total impurity divided by this weight is actually called concentration by weight, I will explain you little later. Concentration is defined by net impurities divided by the weight of the available impurities, total weight of the impurity. The ratio of this is essentially called concentration, the concentration is always defined as per CC, but it is not per CC, it is per weight is how the concentrations are defined, okay. So this segregation coefficients if you look at it, there are different segregation coefficients, these are of course data available in many books, journals, papers, Google, any number, anywhere. Most important thing why I show you is this, the impurities for P and M type which are our worrisome, we can see from here that someone was asking, for example Bismuth lithium has a very low segregation coefficient, so they actually segregate more on the melt side and they do not go to the solid, but which has boron for example, you can see 0.8, so it will segregate the other side, okay, more compared to Bismuth lithium. Phosphorus for example has 0.35, arsenic 0.30, antimony 0.0, antimony is just the other way, gallium has 8 into power minus 3, aluminium, please remember aluminium has, is a type 3 dopant for silicon, okay and that is very important because this word which I am using is very important because in many older ICs, when we are making aluminium as interconnects, we never did lot many effort to get what we call ohmic contacts. As soon as I changed over to copper, I have now a problem between copper material and silicon to make an ohmicity there, whereas in aluminium because it was a dopant, it just doped it heavily there and it made a contact there at least for P area, okay. So earlier technologies to copper has found that though it has created fantastic interconnect now comparatively, that one process step has increased because earlier aluminium was never sintered, it just went through, okay. So let us come to it later. So these are some kind of impurities which are possible in silicon and these are the actual N type impurities. We will also see later when we do diffusions that antimony is rarely used except some new sensors which are coming up now. There are variety of reasons why antimony, you know these are also impurities decided by where they lie in the band gap. They will give some level for their own impurity level and depending on where the level lies the holes and electrons can be created. If the energy required for a hole to create is very small from the impurity level, then we can say that point 0, bullion is point 08 for example EV. So most of the bullion atoms can be ionized and it can give holes. Phosphorus has point 017 EV. So most of the electrons can go to the conduction band. So these are also choices made from electrical properties. Here we are not looking so much their electrical properties. So the decision of which impurity to use is now fixed. Earlier I used bullion, P type 90 percent I will use bullion. Maybe bullion plus gallium these days that people are trying what to call complex. People are only in P type. We prefer to use arsenic compared to phosphorus. Earlier we use only phosphorus, N type. For N type we only use phosphorus for 30, 40 years. Of late 10 years or so we figured out that we will have to move to arsenic. Question can be raised here. Why not arsenic earlier? I mean so what so big? Arsenic the way we will go to diffusion then and show the source of arsenic whichever we create is arsine and arsine is extremely toxic gas. If many of you were not born probably when the Bhopal tragedy took place the gas there was Phosgene or called MIG, MIC. These gas has a minimum, the maximum allowed quantity was 1 ppm, 1 part per million whereas the death part for arsenic can be 1 ppb. So one can see even 1000 times less amount of arsine released in air and even hell you may not have time to tell I am no more. In phosphorus maybe we will say I am dying. That is a joke people used to do with cyanide. They say no one could test KCN simply because by the time he realized he is no more to tell. But that is only a joke, test is known. Okay. I start with two cases we discussed. One is called rapid steering. So we are actually steering the liquid very rapidly. Other is called partial steering very slow steering. The formulas which we want to create we want to figure out what is K0. What do you expect K0 to be good? The concentration near the interface should be same as concentration in the solid. So ideally I am looking for K0 to be 1. How much I get and how do I improve is all that technique is about. I normally would like to get 1 but it may not be possible and let us see how much nearest we can reach there. Okay. So we define. I think this is not given in the plumbers group. This is my own old notes. As I say I am in 45 years in this area. So some terminology is mine which may not be suited to new books or something. Maybe some old books and so please pardon me if you can go and read plumbers book, change over to whatever names they are putting. I know objection. But normally this problem is always worrying some for many of you. For example in the case of mass transistor this current is given by Mu C ox W by L and I call it beta. This number I call it beta. But beta is very popular known in bipolar as a beta of the transistors. Most of the books give K. They call this Mu C ox W. But I have born in area where we used to started calling equivalent from bipolar to mass. We say this is a gain function. So we started calling beta. Now this when I teach I teach beta and then people say what is K? So I say okay beta by 2 is K. That is true. But the way I look at it and why many new books do not produce the age difference. So unfortunately my question paper will follow what I actually teach. So please follow what terminology I use. May be correct or may be not correct. Let us say Wm is the initial weight of the melt. Cm is the concentration of solute in the melt. Solute means impurities. During crystal growth at any instant remember there is a difference between Cm and Cl. Cm is at initial time before it starts pulling. Cl is when you start actually melting and going towards interface okay. So any given instant near the interface value we call it Cl and inside the mold initially what was available was we call it Cm. Cs of course as we defined earlier is the concentration of solute in solids other side of the when you pull okay. There is an interface solid liquid okay. Is it okay? Everyone I repeat this is my terminology better you have it because I will not change for your sake though I should and current trend in the world is to change what student want. But please help me not ask me to change for you okay. It is too late for me to change okay. But this is similar nomenclature is available in Plummer's book so do not worry. Plummer is no different from me but I do not know he has used the modern names. We define as is the weight of the solute. S is defined as weight of the solute okay in the melt. Please note that at T is equal to 0 before the pull starts. I am talking of this. Cm is the concentration of solute in the melt. Wm is the weight of the melt. So we define weight of solute at T is equal to 0 as S0 which is Cm times Wm. This is how it is defined. I repeat Cm is the concentration of solute by weight in the melt. Wm is the weight of the melt by weight of course weight itself and weight of the solute at T is equal to 0 which I call it as 0 is Cm times Wm initial condition. Because when I start pulling I need initial condition because some equations may come and then I will have to put initial value okay. This is before I start I defined it. Now consider between solid and liquid there is a layer which has DW as the thickness crystal this which is between liquid and solid. You are pulling in between neither liquid nor solid something different that thickness we call it DW okay. Now how many impurities will be in that? The weight for that is concentration of solute into it that is DW in that small element. I just now defined C into W is the net value S. So solute available in that DW is the concentration CS multiplied by DW the weight whatever you are talking about. Why minus? Because this is loss from the melt side. You are actually transferring from melt towards solid. So this is minus sign is given to that. So that is lost from the actual melt. This impurities have been lost. Initial weight of the melt was Wm. Let us say currently the weight is W at a given instant. Then the concentration of the solute in the liquid is the total solute weight divided by the remainder of weight of the solute okay. Wm minus W is the remainder. Upper part W has gone out. Wm minus W is remaining. S by this value is defined as I repeat this is concentration not per CC. This is how it is defined in all metallurgical processes. So I had a Cl which is S divided by W. Please remember what is Cl concentration of solute near the liquid surface which is total available impurity. This has gone out of melt. So initially weight was S divided by the new weight is some part is lost so Wm minus W. So this is the concentration near Cl. The ratio we have already said the K0 which is the segregation coefficient is CS by Cl. However we have just now said DS is minus CS DW small change. Now all crystal goes we will go up to W okay. Is that okay? Just write down. As I said this is available in Plummer's book in some other form but available. This is my day I have done it myself so I do not know whether they follow this. They do not I know. As I said to you 83 December or rather 83 January to 84 no 83 July to December I taught this first time this course in IIT maybe first time in India okay. So some you know since EDS increases starts in U you start building on that and you say this is correct. So something may not be correct but it is okay. If I substitute this CS in terms of K0 then DS is minus K0 Cl DW I just represent K0 CS by Cl so I put it there. I also represent Cl as S upon Wm minus W just now I wrote the whatever is remainder ratio okay. Then I say DS by S is minus K0 DW by Wm minus W. How is the crystal grown from initial concentration of S0 to any instant S okay. S is changing and how much is the weight in this? This is changing in the solid. You could say in this 0 initially and it goes towards total weight of this W. So if I integrate with this condition S0 is Wm Cm I get this relation which is CS equal to this minus sign comes because of minus W here. So K0 Cm 1 minus W by DM K0 minus 1. So now I had a concentration of solute in solid related to concentration in the melt related to their weights okay initial weight as well as current weight and segregation coefficient. This is a simple integral you can solve yourself. What is interesting for me? I just now said for uniformity CS by Cl must CS by Cm or CS by Cl that should be equal to 1 that is what my ultimate aim is. So if I plot W by Wm versus CS by Cm for different value of segregation coefficients okay. K0 is 0.1, 0.2. How do I adjust K? I will see later but as of now if I plot for different value of K0's I find K0 by when it reaches around 0.9 okay which is what we are looking for. The melt, liquid and solid are almost as universal concentration all three same okay. Very close to this 1, 1, 1 for example. You can say there is a uniformity from melt to the liquid to the solid. So the doping is as good as possible you want to get. Smaller the K value you get worst will be the solid distribution. Best you can get as close you reach K to the 1 you will get very uniform dope crystals okay. Now in this process which we did just now we will like to do little more further processing on it and we find that K cannot be really controlled because K is something constant for a given material for this but I can do little trait to vary the key. This is the whole game. From this I figure out that for going on this K is known so how do I change that K? So I say okay let me see if I get this value if I reach K higher I will be able to achieve uniform doping. So I made a trick. I do not mean I, I means those who are done first maybe Joklowski and others. What we did? We said okay do not stir it with high speed do it partial stirring, slow stirring that has some advantage. If you are you can think of like this if you pull it faster and rotate it faster the amount of liquid which it can stick will be smaller isn't it. But if you pull it slowly and rotate also very more liquid can touch the solid area okay and the rate which you are pulling and rate which it segregates down is such that this layer remains uniform. This is called stagnant layer. If you have done anyone some course in chemical engineering all our analysis in transport there is based on stagnant layer models okay. Air is going through particles will move or not. How much stagnant layer you have? Every process in the life if you actually analyze is somewhere related to this stagnant layer okay. So okay the same model has been picked up from chemical people. Let us say the stagnant layer has a thickness delta okay between solid and liquid and now we say since it is a stagnant layer there are two things are happening. One is the impurities from the liquid are going to the solid by process of diffusion. What is process of diffusion? If there is a higher concentration higher and lower it will start to diffusion from looking at the gradient okay. Impurities start diffusing through stagnant layer. There is also a segregation going on which is inverse of diffusion. In equilibrium they will be because if the stagnant layer remain constant this will remain this process will balance each other for a given time. Initially it may be higher or lower but later it will adjust itself. Now we want to see this can we solve the diffusion equation and also segregation related flux back and if you solve this we somehow figure it out and we will show you this figure later again. If this is my X and this is my concentration I am plotting on the left is my solid, on the right is my liquid and this is my delta is a stagnant layer delta is my stagnant layer. CS is that solid concentration which you are actually looking for okay. Now we find from here near the solid liquid interface or rather above the solid to stagnant layer interface since the layer concentration is only possible because of segregation below but on the stagnant layer doing equilibrium the concentration there can be higher okay that is the fun part because it is sticking there so it is larger concentration here but below it can't because it has no force to go up okay. This means CL dash at the interface is much larger is that clear. However as it goes towards the process towards the liquid the concentration reduces to CL value. When we calculated in rapid case this CL and CL dash were same okay CL and CL dash were same. So K was decided by ratio of CS by CL but now this is not the value which is being used. Which value I am going to use now? This value which is much smaller than this value is exponentially decaying function okay. What does that mean? CS by CL will be higher than rapid thermal rapid pulling case. Is that clear? Please remember this if this is equal somewhere CS by CL dash or CL which was equal then K value was decided by whatever distribution available for you. In this case depending on the thickness of stagnant layer this concentration and this concentration will be different okay. Two processes diffusion process followed by segregation process. Both will be trying to adjust each other such that depending on the delta use. Now question is how do I control delta? The pull rate decides the delta faster I pull delta will be larger and we break also. Smaller and how much I rotate? How much I allow it to stick? So depending on the pull rate and the spin rate I keep I can adjust the value of delta and delta will decide the ratio of CL dash to CL. So if I adjust this delta value then I have adjustment of ratio of CS to CL differently from different delta values. Is that madam clear? As I change this the ratio will be varying so is K will be varying and if I change K I just showed in the last graph I can build the K then I will get more uniform crystal okay. Here is maths. Maths is very interesting always. Many of you may not think that this course will have maths. Please take it this course will have 30 percent of maths requirements. Look of your differential equation solutions better different kinds of equations may appear and we may have to solve them. So let us say if we pull the crystal solid liquid interface and it is pulled by the same pull rate let us say R is the growth rate of the crystal. Please remember growth rate and pull rate are same because as much you pull that must solidifies. So growth rate and pull rate are almost same almost exactly they are not but almost right. The diffusion coefficient of solute atoms in the liquid top portion of the melt is normally for most impurities is given by 10 to power minus 5 centimeter square per second. Now we have in this stagnant layer please remember impurities diffuse through stagnant layer into solid and segregation returns them back to stagnant layer and therefore to liquid. We write this is the forward reaction diffusion equation d into d2c by dc is the concentration d into d2c by dx square is the diffusion term. R into dc by dx is the segregation term it returns back the actually they are minus of that is equal return means minus. Total net value should be at equilibrium must be 0 some of this process plus this process must always be 0. So forward pushing is balanced by reverse pushing so that the net pushes therefore uniform is that clear therefore uniform okay. Is that equation clear? This is the we will prove this again in diffusion process when we look into right now you assume this is the diffusion equation which is giving a dc by dx is the concentration gradient second differential of that is essentially is when I multiply by this is the diffusion term. This is coming from which equation continuity equation okay. So if I solve this continuity equation which I did you can also solve. Then the concentration is Ae e to the power minus Rx by d plus b solution of a differential equation is simple and if I differentiate it dc by dx is minus Ar by d e to the power minus Rx by d these are the two why I did this? I have two unknowns A and B I need two boundary conditions to solve to get this value okay. I need A and B to unknowns I need two boundary conditions. These are essentially one boundary condition but the other one of course I will come back to it. At x is equal to 0 what is the concentration this figure we have just now at x is equal to 0 concentration is Cl dash where that the concentration gradient at x is equal to 0 by actually using this term is R by d Cl dash minus Cs I have not derived it but you can derive in the night I finally got tired. So I wrote this it is very easy to define from here and if I make the these two equations and substitute to get A and B I get final term which I am looking for please note down note down this these are the this is the solution these are the two boundary conditions which are used to define the solution of this A and B can be found and once I find A and B please note down because as I said I repeat whatever I am doing is possibly available in Plummer's book it is not that it is different only their names nomenclatures may be different but the equation will not change because diffusion and segregation physics cannot change chemistry cannot change maths cannot change. So if you find new segregation coefficient which we call effective segregation coefficient which is called Ke effective which is not Cs by Cl dash but it is Cs by Cl we solve those using the solution of that equation we arrive at this term K is K0 upon K0 plus 1 minus K0 e minus r delta by D so what is r I repeat rate of the growth of the crystal what is the diffusion coefficient for impurities near liquid stagnant layer interface so one can see from here I can adjust key K through which terms you are in the two terms r and D delta sorry not D D is constant r and delta how delta adjust by deciding the spin rate and r I adjust by assigning the pull rate r pull rate spin rate will decide how much is the delta for it if I adjust this r and delta I can improve Ke is that I can improve Ke if you want Ke higher one can say from here you should pull it little with higher this and spin very slow because delta is proportional to one upon spin speed so if I want larger delta which I thought will be good then one can see from here delta is one upon spins proportional to this so slowly spin little higher pull rate one can get higher Ke and if I can get higher Ke means what do I get higher uniform doping all the game is to show that mathematically also I can show that as a user tomorrow one of you let us I may tell you now there is a possibility other day there is a news report India may have first polysilicon plant growth plant which year when only God knows and maybe our prime minister knows maybe I do not know even that but there is a proposal now which is almost going to be finalized we may have around the problem with making a plant is simple if I make 200 million ton plants a year is there consumption of 200 million tons of polysilicon in India okay if it is not why do I put a plant I have to compete with the world with the companies which are 50 year old they already set up everything my initial cost would be high so I cannot sell cheap any polysilicon so I know I survive if I had to market out but if I have internal consumption yes solar cell this has come from solar cell department so I hope it may be okay it is not from IC group so solar cell people have wanted polysilicon and they are now negotiating I am told I am not very sure how good people the news released to the press and available on other places is identical but assuming it is so you may have a polysilicon plant in India and as of now we are they are thinking of 500 million tons of per year hopefully yes so many solar cells may be made hopefully okay so you may have a job later or at least say sir we have learned we know or I learn I know remember me then oh if you don't spin delta will be infinite what does that mean that the solid liquid portion is now touching there is infinite delta essentially means there is no solid and there is no liquid will just jump down because such a large this just volume will pick down because the weight of solute will be so high on the top it will just fall down so too much spin or too much sorry lower spin and too much R R is also if I pull very heavily that small portion single crystal will break okay so I have I have shown you two figures when I rotate very heavily and I pull it heavily I may actually lose the contact either side and therefore I cannot do too much so K cannot be made one any day maybe 0.8 0.7 0.9 at best okay but even then K 0.9 is good enough for us because it's giving relatively uniform crystals okay so all CZ crystals are not greatly uniform but mostly uniform and integrate circuits we don't bother too much about so much variation as long as variation is less than 10 percent it's all fine why 10 percent anyone circuit man this is circuit something to a circuit why 10 percent if you see here in particular in digital let's say in a digital circuit your noise margins are half VDD okay so you can see from here that if even if there is a variation of thresholds 10 percent is much lower than the available noise margins only analog this may be killing so there you may not be able to do this but in digital this is good enough of course in some speed it may affect you even then because the speed is depending on the essentially this value percentage VT because current is available to charge a capacitor so there is a problem variability issue but otherwise it's sustainable as long as I know what is the variation I will adjust my critical paths okay otherwise but in analog I may which way first I should prefer even float zone but then it will be so costly that no one will buy my chips if I make it on CZ I am not doing well I am sure and also there are very few analog chips are manufactured purely analog 741 keep me correct analog yeah a to d both today all that people buys on chip analog that is there is a digital block and some small 10 percent is analog mixed signal their technology is not analog it's a digital you work for the worst technology final off still want to give good result okay that's the trick that's why a lot of designs are much more interesting because you are fighting every second every nanosecond to get what you are want okay and digital you sleep next day it may work okay so that's what digital is all about those who are fighting digital I will tell you something else today I will not tell okay before I am not going to detail of float zone you can read basic idea in float zone is the following I already shown you this float zone reactor maybe you can see from here you have full ingot of single crystal available to you okay full ingot you hinge it on the top and also on the bottom and there is a coil you can see coil which is RF coil which is trialing it can move from one end to the other that's why wherever you apply RF to this coil by induction heating the internal part actually also the carbon rod there which actually receives the energy and heats this is our induction heating is done so once that zone gets heated only that portion where that coil is sitting gets melted molten okay the rest is still solid so what we do is we start from say anywhere and keep passing this zone up and down okay is that point clear we pass start passing zone from up and up and down too too much speed is also difficult because if the mold moves very fast it breaks if it is too slow it does not distribute well so there is always catch optimizer so this is called float zone why it's called because the zone of heating is floating so it keeps floating up and down so what happens everywhere wherever it goes there is a segregation you come down another segment so if number of passes you go through this because of the segregation and diffusion impurity is going through this zone part impurity is finally after say 8 to 10 passes becomes uniform since there is no quads or anything inside here there is no oxygen of course carbon is but which is internally to it so it normally does not interact directly to it and therefore this is almost impurity free so pure crystals can be grown only so initially which crystal I got CZ1 so I got a CZ crystal rod and I put it here and pass through number of zone passes and then you can see from here this is my zone this is my solid this is my melt or other side of this so what happens impurity crosses this melt to the other side wherever concentration gradient is there to keep moving this when you come back the impurity starts moving the other way if you keep going like this like this you will get almost uniform concentration of impurities across the lot that is how float zones are actually made before coming here I may tell you where the carbon segregates carbon normally segregates on the surface so we try to remove that with that process will show you by doing similar analysis which I did for the CZ1 one can see CS to CM ratio can be 1 minus 1 K similar like what we did for partially steering case okay and you can see all three impurities if I do number of passes zone passes will reach maximum K value of uniformity 8 to 10 passes of the length I push everyone gets almost uniform concentrations please remember this is the best possible crystal grown and since it has no impurities generally its resistivity is very high is that clear that is why it is called intrinsic material it has very few 10 to power 13 per cc or 10 to power 5 into 10 to power 12 per cc you can obtain using this method okay normally in mass you require 10 to power 16 per cc 10 to power 17 now maybe someday 18 or 19 as well so four zone process is only used when you want highly pure crystals with very high resistivity you can attain and you can always make doping to make a doping what do we do is I just put a for example this is good rod and this is the doping rod which is doped by me okay earlier fixed doping so I put this the rod which is to be doped with the one which is already available doping and keep giving the zone passes so the impurities from that concentration higher will keep going and finally after 10 or 12 zone passes it will be uniformed it is this is called zone leveling pushing an impurity from the source of rod to the whole remainder rod is called zone leveling so you give number of passes impurities distribute all along okay so float zone is a very costly process as I say a typical 12 inch wafer is these days are aware of course comparatively has gone down now but it is still 70 dollars whereas the float zone of course no one makes that size of wafers 6 inch wafers is available at 450 dollars so it is a huge money differential so unless you are working on something which gives you money on that side don't go for intrinsic wafers okay so this last but not the least as I said you these are the impurities which you see in CZ as I said the other impurities segregation is very small and therefore very little goes into the actual crystal but this math lithium ion hardly go into it they will be their interest but very little however carbon and oxygen have a large content you can see carbon car this oxygen concentration is 10 to power 18 per cc okay which is very high okay extremely high and carbon concentration is 10 to power 16 per cc which is also very high however it looks as if the presence of oxygen also in carbon is not very good if we thought yes a priori it seems to be nobody has some advantages also okay since the oxygen is a type anti-dopent it has a tendency to actually silicon gets oxidized very fast silicon has the highest affinity for oxygen so if the oxygen is there silicon gets oxidized it is called precipitates now these precipitates are more like a amorphous material no order of course I won't say zero order but some less much less order now when this someplace and if these contents are known where they are at the edges or surface okay then what happens that other impurities like let's say material has carbon what I should say gold or any other impurities they find there is a vacancy or possibility where they can occupy a site and stay there okay so many of if you heat the wafer now with any other impurities this precipitate actually absorb most of the other other impurities okay then you remove that area okay and you now have a wafer which is free from most of the impurities as well as the precipitates so that many people thought that oxygen precipitated or carbon precipitated may actually harm us it really doesn't as much in some areas if they are in the localized yes those wafer will go away they have very low resistivity there they actually burn at the end these are called hot spots in chips so very few areas burn but normally they don't okay so don't think that oxygen and carbon is all bad as if they are impurities and we do have a problem as such so this of course finishes the growth part okay now a few more things we will have to do for next time maybe we quickly see anything to be shown here on the figure okay this is what some things we may show before because these are only show there is nothing to learn this is the ingot and I actually grind the diameters okay a great result ingot for grinding and then you actually grind it into a wafer something like this may be better you have a wafer saw as they call the rod goes in and there is a rotating matrix of nickel which actually grinds this upper surface as I said you most dislocations are on the surface they are actually removed okay same thing better figures after crystal pulling the ball is shaped and cut into wafers by using this is how internal part of that okay this is how we will get wafer okay this is the age we see in the wafer magnified one these are how they are actually handled to clean these are agents in which wafers are handling putting in together inside okay and typical dimensions as I say is 150 millimeter to 400 and you can see weight per 25 wafers this is 13 LBS 150 millimeter as 1.5 LBS okay so less than a kilogram here it becomes 6 kilograms for 25 wafers so handling can you understand the handling larger the mass handling the wafer is a tough job and that it should not break okay because one wafer break means one heart breaks okay this is an example if you have a 200 millimeter wafer and whatever die side you have chosen which is 1.5 by 1.5 this may give me 88 dies if I have a larger wafer of 300 millimeters 12 inch then it is 232 dies so you are earned lot much good chips out of the same process so the cost of wafer to the cost of processing depends which is higher or lower will decide what size of wafers you should use okay okay they are polishing another thing maybe some other day so essentially we are now having a wafer which is polished clean and it is always mirror finish the way I polish the wafer you can actually see your face much better than any mirror okay so it is highly polished wafers which we actually going to use it is also highly hydrophobic no water can stick to silicon but silicon dioxide is highly hydrophobic hydrophilic so as soon as silicon is seen it will form an oxide immediately so that when you see a wafer you put into water water sticks which means it has an oxygen oxide on that so the first thing you have to remove that oxide because I want silicon surface so this all has to be now done in an area all etching everything in an area which we call fab lab which is where clean rooms are required okay so next time we start with clean rooms what are clean how much clean is the cleaner this room is called class trillion room this is class trillion I am looking for sub class one I am looking for a clean room which is sub one class whereas this is class trillion that means trillion particles of 0.5 micron per cubic feet floating here I am expecting less than one particle one cubic feet okay so the kind of purity we are looking for is very important and we will see next time what what goes into human thank you