 I have actually checked on the website, the Amazon India has this book. This is the ad for that. If you have not purchased it so far or couldn't, this is what recent I saw yesterday. It is 718 rupees or whatever possible. And this is second edition, 1st January 2009. This is available at website www.amazon.in, silicon, vales or technology. This is written here. You can take, they say 5 days, 5 to 8 business days. So in case you have not purchased or don't want to purchase yours, but in case you want to, this you can note down or maybe you can go on Google and say Amazon India. So whichever way it is. So don't tell me that book is not available or something, something. Is that okay? Coming back to those who want to write that website, maybe you can in case you feel that that is what I am not. We were other day looking for Bruce Deel model, Bruce Deel and Andy Grove model for oxidation. The first paper published in this area in way back in 65 and this model is very popular in the literature called Deel Grove model and I just told you other day Grove was the, one of the CEO or chief of Intel from say 1992 to 97, is the emulators scientist or emulators CEO of Intel. Andy Grove also has a book on physics and technology of semiconductors or other technology of semiconductor, which is a whole book but many things remain same irrespective of the today or yesterday. So one can also look for Grose book in the library. There is another book on VLSI technology by Wolf, WOLF, Wolf which is also very good, okay, which is rather recent compared to Plummer of course older but around 2006, Wolf has also published. So any of these books are good enough, please do not say that the books are not there because of that whatever happened, okay, at least do not blame books, okay. So the assumption what he did was that initially there is a finite thin oxide and at t is equal to 0 minus, this is the oxide available and t is equal to 0 plus oxygen attacks this silicon layer, silicon dioxide layer and starts the oxidation. We did last time I am just trying to recapitulate. Then the oxidant species then reacts with diffuses through the thin oxide, reaches to the silicon-silicon dioxide interface, reacts with silicon and forms SiO2 and we already last time said there are three fluxes, F1 in the gas phase, F2 in the oxide and F3 near the interface of silicon-silicon dioxide and we also discussed other day that in steady state if the oxide oxidant is coming, diffusing, reacting the system has to be in steady state and in steady state all fluxes must be equal, okay. This we have done last time I am just trying to push it again and we say F1 is equal to F2 equal to F3 and that we call it F, okay and we will show you this process again but this is just to show you how oxidation proceeds. Inside the furnace where temperature is kept 800 to 1200, wafers are vertically start on a quartz rack, oxygen is entering here and oxidizes this silicon. So let us start with the model, this is what we did last time. So now we actually look for the model, okay. Here is the model, same thing what I wrote but there are now few more definitions. Let us say Cg is the oxidant gas or oxidant concentration in a gas stream that is not near the surface but inside the whole tube actually. This is called Cg, then corresponding to this Cg we have a oxidant concentration at oxide interface is Cs, okay or rather C0, sorry C0, forget it, Cs is the concentration here. So there is a gradient because there is no enough oxygen here or oxidant here so there is a gradient from Cg to Cs, okay. Now these are called gas phase concentrations. So equilibrium concentration at the solid also can be found and we define C star as the equilibrium oxidant concentration in solid related to this Cg, Cg is the gas phase concentration and C star is the equivalent of that in the solid phase whereas C0 is similarly is equilibrium oxidant concentration in solid related to Cs. So this relates to here and this relates to here, okay. This is then we say initial oxide thickness is Ci, initial concentration at the interface is Ci, okay and we say already Xo oxide thickness exists or Xi rather but Xo is the oxide thickness which we want to find at a given time and temperature. So let us look at the fluxes now. According to Deal-Grow model flux F1 which is in the gas phase from the ambient to the surface we know the flux will be proportional to the gradient. If Cg is the gas concentration in the bulk, Cs is the gas concentration near the surface then F1 must be proportional to Cg minus Cs. Whatever is the surface and what is available in the stream the gradient is set and that difference between the two is the flux F1 and the proportionality constant is called mass transfer coefficient Hg, mass transfer coefficient Hg. So F1 is Hg times Cg minus Cs, this is the first flux which we are looking into that is the gas phase flux. However we are more interested to know there are terms which probably you should learn from thermodynamics but there are words called pressure, total pressure and there are words called partial pressures. Partial pressure is defined as the pressure of a gas inside a stream in a given temperature in a given area okay. So if I know F1 is Hg this I want to replace this Cg and Cs which are not known to me, Cg and Cs cannot be monitored so I will try to know is there equivalence of these in terms of partial pressures and if I know the partial pressures then I will also know correspondingly to that partial pressure using Hertz equation we will find out what is the solid phase concentration for this partial pressures. So at the end I am interested to correlate Cg, Cs with C star and C0 this is what I am looking for. So the first thing I do is after I write Hg into Cg minus Cs as the first flux and where I repeat Hg is called mass transfer coefficients okay. We define Cg by ideal gas law Pv is equal to RT there is nothing great about this is concentration so Pv is equal to RT. So Cg is equal to Pg by kT, n is the number one molecular this so we Rkt is nkt can be written as n is equal to 1 therefore RT is kT. So Cg is Pg by kT, Cs is Ps by kT where Pg, Ps are the partial pressures of the oxidant at gas ambient and oxide surface okay. Again same Pg in the gas stream and Ps is the at near the interface of oxide and gas stream okay. So if I substitute Cg and Cs in the last equation by Pg and Ps term then I get F1 is equal to Hg Pg by kT minus Ps by kT or is equal to Hg by kT into Pg minus Ps okay. Please remember if I fix temperature and if I fix the gas flows total gas flows Hg is a constant which is the mass transfer coefficient proportional to the total pressures okay. We will see then in case of CVD. We invoke Henry's law for gases and fluids there is a Henry law which actually relates the partial pressure to solid state concentration this is called Henry's law. According to Henry's law the C star which is the equilibrium oxide concentration in solid state is proportional to the partial pressure in gas stream similarly C0 which is at the surface of SiO2, C0 is proportional to partial pressure at the surface which is Ps and there is a constant associated with this equilibrium this which is called Henry's constant. So C star is H time Pg, C0 is H time Ps where H is called Henry's constant okay. So now you can see I have relationship with C star and C0 in terms of Pg and Ps through Henry's constant. So I can use these 2 equations to go back into 3 and find the flux F1. What is the method? First I converted Cg and Cs into equivalent partial pressures by mass transfer coefficients. The partial pressures are related to their equivalent solid state concentrations by using Henry's law or Henry H is the Henry's constant. So now I have C star and C0 which are at this surface please remember C star and C0 are at the surface of SiO2 which are in solid phase that is the oxygen we are going to actually diffuse through okay. So F1 therefore is that okay I repeat first we relate Cg to Ps, Pg, Cs, Cg, Ph, Pg, Sa and then we wrote this equation then we say okay using invoking the Henry's law of flux C star is proportion to Pg, C0 is proportion to Ps then C star is H time H is capital H Pg and C0 is H time Ps and now I have substitute Pg, Ps from here in this equation 3. So if I do that F1 is Hg by kT C star by H minus C0 by H or F1 is Hg by capital H kT C star minus C0 and I can redefine this Hg upon H kT which is all constant at a temperature and mass total flow and pressure then H is Hg upon H kT is redefined as term H which of course is proportional to mass flow okay. So F1 is H time C star minus C0 so first flux in the gas phase is related to solid phase concentration difference H time C star minus C0 so this is the first flux we obtain. Now as I say what is our game is to find F2, find F3 and then what do we do? We write F1 is equal to F2 and then solve ultimately what I am really trying to do is I want to find dx0 by dt what is dx0 by dt? Rate of oxide growth that is what I am interested in rate of oxide growth so if I know the time I know how much is oxide I will be growing in case of oxidation so that is my purpose. So first flux F1 I got now secondly if we look at our figure again just a minute if you look at this figure F1 we have found in terms of C star C0 F2 is the flux which is diffusing inside in oxide F2 is the flux of oxidant passing through the oxide thickness and that is equal to C0 minus Ci okay Ci as I say is concentration of oxidant at silicon-silicon dioxide interface Ci so there is a gradient diffusion okay. So F2 is then proportional to C0 minus Ci by X0 this is gradient C0 minus Ci by X0 is essentially a gradient and what is the gradient constant should be since it is diffusing what should be the constant of proportion to here diffusion coefficients okay of that oxidant in the oxide please remember diffusion coefficient is for the material oxide or whatever species going in that material diffusivity is different in different material for different gas flows going in so specifically you had to earlier we had talked about impurities in silicon now we are talking of equivalent solid concentration of gases in the oxide and how do they diffuse through okay. So if I write that I write F2 is minus D effective minus is because I am subtracting Ci minus C0 by X0 where the why I did this because final concentration minus the initial concentration divided by X is actually slope okay it is essentially how we define that is going down so that is the final concentration minus the initial concentration divided by X0 is the gradient so minus D effective Ci minus C0 by this where D effective diffusion coefficient of oxidant in oxide okay this is flux F2 now this is now made available to react with silicon oxidation oxidant is now diffusing please remember what is the process we are saying the gas from the gas phase oxygen is coming equivalently it gets to the silicon oxide surface then diffuses through and then reacts with silicon to form fresh oxide is that clear that is what deal go model is suggesting gas phase through oxide and to react with silicon that is the process we have defined so we want to know what is the flux 3 which is proposed and we know flux 3 is essentially as many silicon atoms are available our concentration of silicon available Ci at that place that is the only one which it can react with oxygen cannot react more than Ci because whatever available only can react so we say the flux F3 is proportional to Ci and the proportionality constant is called reaction rate constant KS reaction KS is called reaction rate constant so F3 is KS Ci so now three fluxes F1 is from gas phase to the silicon dioxide surface from silicon dioxide surface to silicon surface diffuse and then reacts with silicon to form fresh oxide okay and as I say if I assume it is in steady state which will happen I put the vapor in a constant temperature constant flow system will go into steady state in that case the flux F is F1 equal to F2 equal to F3 okay no no no what we are saying available concentration in the solid phase from the bulk is C star of which only C0 is really going to diffuse because that is at the surface Cs is available which correspond to C0 in solid okay correspond I just now said gas concentration equivalent in solid concentration is related I just now said through the mass transfer coefficient okay and through that to partial pressure through that to Henry's constant to the actual concentrations so we are saying the gas is coming equivalently how much gas is available overall of which how much will diffuse from the surface is C0 it will go to Ci because there is a gradient there is very few oxidant here and very large oxidant here so it will diffuse and when it reaches here it will start reacting with the silicon surface and form oxides so we do little maths and there is nothing very serious maths since we are already done F1 F2 F3 are equal we say first we equate F1 equal to F2 and next time liquid F2 equal to F3 so if I write F1 equal to F2 I get SC star minus C0 minus D effective Ci minus C0 by X0 and if I write F2 equal to F3 I get minus D effective Ci minus C0 by X0 is KSCI so I equation 9 and 10 which are we I got it from equating F1 equal to F2 and F2 equal to F3 two equations to announce I am I am interested to know C0 I am interested to know Cci in terms of C star available gas phase concentration and correspondingly I derive solve these two equations 9 and 10 how do I take Ci out of this substitute here and find C0 or get C0 from here substitute here to get Ci okay this mass is a little longer so I just wrote down the final equation I repeat solve 9 and 10 to get Ci and C0 substitute one of them to the other you will get the other value or substitute from the C0 into the second equation you will get Ci first whichever way you do so I get Ci is equal to C star upon 1 plus Ks by H plus Ks X0 by D effective then C0 is 1 plus Ks X0 by D effective times C star and the denominator is same please remember denominator is same 1 plus Ks X0 by this Ks by H so I have now Ci and C0 okay I may not be very I do not need to know C0 very much but I am interested to know Ciy because that is the concentration it is going to react with silicon so I figured out both can be calculated but I am more interested to know Ci in terms of C star which I got so what deal model says now what is the oxidation rate can anyone suggest what is the oxidation rate if F is the flux and N is the number which is available for reaction so F by N is essentially equal to DX0 by DT if what is DX by DX DX by DT available flux divided by available concentration of where it can react the ratio of that is essentially DX0 by DT so that is grow deals model which says if anyone is the concentration of oxidant molecules then DX0 by DT is F by N1 and F is equal to F1 equal to F2 equal to F3 so F3 is the smallest term Ks Ci so I use F3 instead of I can write any one of them but then there will be two variables coming so I just want to remove I have got Ci value anyway so I got Ks Ci by N1 please remember this N1 for oxygen is 2.2 to 10 to power 22 N1 for H2 OH HOH molecule is 4.2 of that OH OH double the concentration whereas pure oxygen has N1 as 2.22 into 10 to power 22 this has been chemically monitored measured by many experiments by atomic spectroscopy to FTIR everything we one is monitoring that so if I DX0 by Ks Ci by this I substitute Ci from the last equation I just derived an expression for Ci so substitute Ci here so it gets Ki upon C star upon N1 1 upon 1 plus Ks pi H K sorry Ks X0 by defective so how does it look like we will do some simple looking expression for this plus you have not done this yes so I get DX0 by DT is C star by N1 1 upon 1 plus Ks pi H plus Ks X0 by D effective so this equation now I know I can modify this equation to suit some kind of good looking expressions okay what is good looking some AB constant so it looks very simple equation algebraic of course now algebraic first order differential equation is that noted this one DX0 so I have now the oxidation rate and I will modify the form that is what I am doing is it okay okay so DX0 by DT is 2 C star D effective by N1 I actually multiplied 2 D effective 2 D effective by Ks both sides norm this and here and I get this expression readjust the terms 2 D effective 1 plus Ks by 1 by Ks plus 1 upon H plus 2 X0 into this constant and now I define some terms I define term A as 2 D effective 1 upon Ks plus 1 upon H as A and I 2 D effective C star by N1 I define as B okay same expression this I defined as B and this I defined as A so I get a very nice looking simple differential equation first order DX0 by DT is B upon A plus 2 X0 please remember Ks is a function of temperature H is a function of mass transfer coefficients or mass flows in normal case H is much higher than Ks but let us see how much DX0 by DT B upon A plus 2 X0 is the simplest equation we get okay which good looking assuming that B and AR constants at a given temperature for a given mass flow okay if you change that these terms will B and A values will also correspondingly change is it okay so I say very trivial looking this but it is important why are we doing this again because at the end when I monitor the oxide I have the furnace I actually oxidize the wafers and then I may do some characterization to know the oxide thickness but if I am doing something process on a system where there is no furnace and there is nothing to this I must be able to get relative oxide thicknesses every now and then a change in oxide thickness or whatever I am doing in process which is running on a CAD tool okay that is called T CAD for technology tools are available earlier ones we used to have program from Stanford called Supreme process simulator there are now many one is Centura other is DOS there are many such program this is which is device plus process simulator since we have many process simulators now we like to know what models they use because they will also find out what is the oxide capacitance every now and then so they need to know oxide how much grown okay so you may specify temperature you may specify gas flows you may specify but then at the end software has to do something to evaluate that we need models to do that so all this effort is to see a model which is going into a T CAD tool okay so whenever you are using Centura or others you probably do not even look at the models you just it ask data you just substitute you guys are good but actually what has gone through is this and maybe tomorrow for thin oxide better models will be required so you must know how do we actually derive models okay what is the physics behind chemistry behind material science behind and your brain behind okay so I rewrite the same term ADS0 by DT okay so if I solve the if I see this equation this is a quadratic equation so we put an initial condition to solve the differential equation sorry not it is a simple differential equation I say at T is equal to 0 according to deal gross model there is a initial oxide XO is XI that is how we started with there is an initial oxide okay so XO is XI and corresponding to this time XI if I use okay so we substitute here and let us say tau is the time taken to create this XI let us say tau is the time taken equivalently actually it is existing we do not know what time we did we did not do anything so we say okay equivalently if we had to go this much oxide how much time so that time I declared as tau so I say now AX if I put it this A2X0 DX0 is BT and I integrate this then I get AX0 2X0 square BT plus initial condition I will put it at X is equal to XI tau is the time taken so I rewrite this term AX0 by 2X square BT plus B tau tau is the time taken to grow oxide thickness of XI and this is only a fictitious number why fictitious the initial oxide is already there we are we are just trying to equate it to a time frame okay all to say tau is XI square plus AXI upon B AXI XI square AXI by B is essentially this XI also I do not know in fact okay so we assume normally tau should be very small because thin oxide is there but if I somehow figure out how much was XI then I should be able to know how much time equivalently it would have been okay. Okay so this is my equation and this looks what a simple quadratic equation and therefore I can find X0 terms okay I repeat the term which I am getting is AX0 plus 2X0 square 2 2 of course will cancel X0 square is equal to BT plus tau so this is the expression you will get X0 square plus AX0 is BT plus tau for a thicker oxide grows tau can be neglected why because T will be much larger than tau tau is very small but it is existing but for a thinner oxide that may be comparable so we must figure out how much is actually thin oxide during initial time must be actually evaluated okay I will show you how. So if this quadratic equation can be have solution of minus A plus minus A square plus 4BT plus tau by 2 minus A by 2 plus minus A by 2 1 plus T plus tau upon A square by 4B and of course negative solution is neglected why but there is nothing called negative oxide growths okay so we say it is only minus A by 2 plus A by 2 terms and assumption is and time should be such that this term should because this is 1 plus so obviously this term will be larger than A by 2 and therefore positive growths are expected is that clear I repeat if this term is even if it is point something okay point 1 plus something is there which means A by 2 times this will be always larger than A by 2 and therefore positive growths are expected. So if I write only plus sign I get X0 and I put minus inside so A by 2 into 1 plus T by tau A square by 4B to the power half under root minus 1. Now we define some whatever we A and B we define we actually give some nomenclature to them and why we will see soon we call B as a parabolic rate constant B is called parabolic rate constant and B by A is called linear rate constant this is definition name wise and why we name linear and parabolic will be soon seen when we will take the two cases okay I repeat B is defined as parabolic parabolic rate constant and B by A in many models this this is given KP and this is given KL parabolic KP and KL so most of the earlier cap tools may be using or even central uses capital KP means parabolic rate constant capital KL is essentially is linear rate constant okay which is same as what I have been using this is deal go model I cannot change K is there because that is deals model if it is my model I can do any other names but this is deals model and that is how they are defined okay way back in 65 two limiting cases okay just look at this terms this term is smaller or larger depends on this T plus tau is larger than this T plus tau is smaller than this two extreme cases okay. So first we say T plus tau is much smaller than A square by 4B so we can expand it by binomial term so 1 plus X to the power half if X is less than 1 is 1 upon half X so this 1 plus half T plus tau by A square by B minus 1 1 1 cancels so X0 is B by A T plus tau for a given temperature given gas flow B and A both are constant how is the X0 related to time linear B is X0 is proportional to tau tau is T tau is very small so X0 is proportional to time what is this growth is linear it is linearly increasing okay. So now we understood why I named B by a linear rate constant because X0 is B by A times T and therefore B by A is named as linear rate constant okay so initially what will happen that what does that mean if T is smaller what does that mean initially oxide will grow linearly with time and as time increases we believe it will become parabolic and let us see how that can happen is that clear so initial growth of silicon dioxide is linear with time and then starts it reduces the rate anything why it will reduce the rate as the time increases oxide thickness will increase so the gradient will decrease is that clear to you so if the flux available is less whatever is available may react but available is less okay so the oxidation rate will go down okay so that is the essential as you go thicker oxide must smaller impurities I mean oxidant will reach there so smaller thickness relative to the time will grow okay is that point clear why this is happening so okay before we come to parabolic there are few terms to be explained B by IA is to now we substitute B and A to defective C star by N12 effective case plus this can be rewritten as C star upon C star upon N1 case H by case plus in general for a given mass flow in oxidation furnace and for the temperatures which we normally use from 800 to 1200 H is much larger than case case is e to the power minus e by kT kind minus 4 minus 5 whereas B will be H sorry H will be order of few centimeter per seconds okay which is much higher so what happens in most normal cases H is so which is it limited by H or KS smaller than one limits it larger one does not limit it okay so H is not I do not say every time in series I know it is mass limited but in this particular case we say H is much larger than KS so if I do this I can neglect H upon H it can goes so we get B by AC star by N1 into KS okay is that okay if H is larger than KS this can be neglected okay when upon H is smaller compared to one upon KS and therefore we neglect H and we get C star upon N1 into KS and as I say case how do we define case reaction rate constant what is reaction rate available oxidant reacting with silicon the rate with which that reaction takes place it is a chemical process which essentially means which is essentially temperature e to the power minus temperature dependent term so B by is also linear rate constant which also follows case dependence of temperature is that clear BS be by a is a function of case case follows temperature dependency to the power so be by will also follow temperature dependence same as case okay so we write then this is the X0 C star by N1 KST plus star this is called so initial oxide growth is related to with time as linear growth and it is also limited by available reaction rate at CI how much is available only can much please remember availability of CI I mean is the silicon atoms which can it react I may have enough oxidant but not all is possible to react okay initially enough oxidant has come but they are not enough bonds where oxygen can SI OSI bond can be formed so the growth rate is limited by available bonds so it is essentially decided by case which is temperature dependent okay typically I will show you the term but as just now want to clarify okay please remember oxidant is reaching enough amount thickness is very small much of it diffuses through whatever is available most of it will reach CI but available oxidation will be limited by the reaction there which is temperature dependent which is essentially in how many atoms can react okay this is the case one what is that case we are talking time is very low smaller times T plus tau okay however if T is much larger than tau and T is much larger than a square by B longer time oxidation is performed T will be always larger than tau is very small so if T is much larger time T plus tau is T and then we say T is larger than a square by 4B then we can neglect one there and rewrite the term X0 as 2 root TB by A into A by 2 minus A by 2 if I readjust this terms this is just substitute into the equation which we wrote earlier in quadratic solution substitute T much greater than this one can be neglected there therefore under root of that is root TB and into A by 2 minus A by 2 and if I do this it essentially comes to be equal to root BT okay we are neglecting small terms so compared to this everything is small this term is smaller than this we already said through this so we always say X0 is root of BT which means X0 square is BT what is this law parabola this X0 square is BT is a parabola so X0 breadth is now under root of BT means it reduces the oxidation rate as therefore oxide thickness as time is larger and larger initially so initially linear and then parabolic growth starts in the case of oxidation so this is the and since B is parabolic constant I mean is giving a constant of parabolicity so we call B as parabolic rate constant B is called parabolic rate constant I told you know this A square by 4B is much smaller compared to T so A is much smaller obviously this term is a is much larger than this okay so it that is negative in actual models you do not neglect anything but if it is a 100 minus 0.05 think of it whether you want to retain 0.05 if you wish fine 99.95 is the answer or 100 your choice what decimal accuracy you want 64 precision 128 bits of precision computer can do any precision okay here is some experiment was performed okay just to compare those value which I am talking about I have done an oxidation experiment means actual experiment was performed I have grown an oxide or silicon at 920 degrees centigrade and assume that tau given to me is 50 seconds okay 50 seconds now we monitor X0 at different times I keep growing 5 minute say time in hours so this is some 6 minutes and this is 18 minutes this is 24 minutes this is 30 minutes this is 60 40 minutes roughly 36 minutes so I have a different oxide thickness oxide times oxidation times in this is in hours please remember this is in hours and I have monitored by some way the oxide thickness we will see how to monitor oxide thickness and I monitored measured them for 0.11 hour it is 0.044 1 micron 0.3 0.10 0.4 0.128 0.5 0.153 0.6177 okay since you did not want to leave a I have used it to show I can take care how do you get that value also useful since X0 square is BT plus tau minus AX0 so I get AX0 is BT plus tau by X0 minus A okay what is this equation looks like this is Y is equal to a mixed plus C kind of equation linear so if I plot T plus tau by X0 this term versus X0 so somewhere at 0 that is T plus tau at tau by X0 at this point whatever is the constant is minus A so you can monitor that minus A how much is minus A so in actual I just want to make parabolic I said neglect A but in calculation I have I will take care because I need to know A okay I need to know A so by extrapolating this curve I will get minus A and the slope is B so I could get B and B A and therefore B and B by A that is KL and KPR monitored if I know actual oxide thicknesses at different growth temperatures okay different growth times okay assumption everywhere is temperature is constant and also the mass flows are constant so if you can see there that if I do experiment I will be able to measure B and B by A okay and then what do I have to do this I have to repeat for many temperatures okay to get B and B by dependence with temperature and then I figured out later maybe I do not know whether I have graph I have okay I will show you it shows that it is actually following the physics that is case and diffusivity is essentially whatever their temperature dependency same happens to B and B by A as well okay we will see this little later so is that okay so I can monitor B and B by A by actually monitoring the oxide thicknesses at different times of oxidations okay is that plain clear so let us say by typical experiment which I did or rather someone else had done of course I have calculated but this data was taken from our lab okay many years ago so B is 0.2 microns square per hour A is 0.5 micron and B by A therefore is 0.4 microns per hour 0.4 micron per hour so I I just now said if I know the data I will be able to plot time versus X0 by this and if I know this I will be able to evaluate B as well as B by A now as I said you I will repeat this experiment what do I do at different temperatures again oxide thicknesses for only thing is now I may do it for a given time okay I mean same time so that but even if you do different time graph will slow the slope so it does not really matter as long as you but preferably you do same so that you know where the slopes are moving okay just to see them why you want to do this because I want to know whether B and B by A really follows D and KS that is what we have been from the experiment from the theory we are looking that B by is following KS and B is followed D effective so can does that temperature dependence appears we will like to see that so we actually do repeated experiment at 1100 and 1200 monitor oxide thicknesses at same times and replot B and a replot X0 versus time and get B and A for all of them okay for different temperatures this is for 920 repeated for 1000 repeated for 1100 repeated for 1200 if I do this the data I get is 920 is this and 1200 is this this is B by A please remember units B is expressed as micron square per hour and B by is expressed as micron per hour linear and parabolic words square okay okay so if I plot now B versus 1 upon T okay for dry oxidation of course I have not talked about this but will come dry means only pure oxygen is passed okay so if I have a dry oxide and I plot B versus T 1 upon T I see a straight line okay and its slope is 1.23 electron if I plot B versus temperature 1 by T in fact and I actually see for a dry oxide case the slope is 1.23 electron volt if I repeat the same graphical this for B by A versus temperature I plot B by A versus 1 upon T and I get slope of 2 electron volt okay this is experimental okay this is experimental because I what I did I actually went and did oxidation monitored the thicknesses at different temperature for different times and plotted them to get B and B by A different temperatures so this is no model this is essentially what I can measure okay and now I want to prove that what I said in a model probably fits to what I monitored and therefore Brodyl model is reasonably good okay is that point clear to you okay okay so B and B by A for dry we do wet oxidations in 95 degree water vapours so I did it we did same experiment for wet oxides and I find for B the slope is 0.78 EV and B by A has a slope of 2.05 EV let us look at this term again B and B by A okay if you have noted down as I say B by is also called KL and B is also called KP in many simulators and that can be written as C1 exponential minus E1 by KT C2 exponential minus E2 by this is the model they are used okay and if you see here these are the ease E1 E2 okay this is the temperatures slope we have got it C star into case by N1 is that is what we are just derived B B by a C star case by N1 B is 2 effective by C star case C is constant N1 is constant so if this temperature dependence has to come case must have similar relations if this relation has to be followed in this the D effective must be for I mean E2 must be following the activation energy associated with D effective okay is that point clear if these are equals from the graphs if I say then the since here they are constants only D effective is temperature dependent case is temperature dependent so obviously this E1 must conform to case and E2 must conform to T effective okay and yes this experiment was further extended and once we did this we found that they did they did which essentially means what the linear rate constant B by A essentially is governed by reaction rate constant which has temperature dependence of e to the power E1 by kT where E1 is 1.23 for dry oxidation 0.78 for wet oxidation so case actually follows what the growth is this diffusivity of oxidant for dry or wet you can see from here sorry this was case this one for B we find it is 1.23.78 and we figure out D effective has the same energy of oxidant in oxide as such we know and same is reaction rates we know about in real life we actually monitored by different methods so we figured out that this process is reaction rate limited and this process is diffusion limited and which is obvious initially when the oxide thickness is small available oxidant is enough it is the reaction possible at a given temperature to convert silicon into silicon dioxide Si plus 2 O 2 Si O Si O bond has to be formed now this at a given temperature is this is the reaction so it has to this reaction will be temperature dependent that is what exactly we did experiment and we found yes it does depends on this however when I increase the time that means I have larger time by then already oxide thickness has grown so the available oxidant now at the interface of silicon silicon oxide is smaller and it is this as much as you can diffuse through is going to available for oxidant any amount coming I will oxidize but available is only what you will supply okay so it is the diffusivity which decided how much oxidant I can provide initially everything available so reaction is in next time reaction can be done any amount but available Krishna that means the reaction rate constant is dominant in linear initial times and parabolic rate constants are dominant in thicker oxide times this is called whose paper way back in 71 very famous person now we also figure out that the oxide thickness for 111 plane is different from 100 we are shown the last time that oh maybe we have to sorry we are not shown so why do you think that C1 constant is larger for compared to this the reason why 111 shows oxide thickness thicker can anyone suggest why it cannot be thicker because more atoms on that plane the question asked was exactly this Miller planes the Miller planes in how many atoms are available on that to react okay so 111 shows the maximum silicon atoms four of them in fact are four corners and three inside since they are the largest number the growth rate is highest along the plane okay this is exactly what is data has been shown taken from Karl who based on this I can again show you this same time once again and again the slow activation in case and activation energy related with the diffusion is found identical what is measured for B by B which verifies B by A is case limited and B is de-effective limited identical the energy associated with case and one upon T and de-effective one upon T their slope essentially matches with the actual data measured by B and B by A which means the great deal model to a great extent okay is valid except for the assumptions which we may have to modify as things go okay but for a thicker oxide less than say thicker oxide around 100 Armstrong's are above great deal model fits very well anything below 100 is not true for de-effective and case for whom case will be a different activation energy which is essentially what B by A we got B by A is proportion to case so if I monitor case by not by B by A method by actual reaction by thermodynamics if I evaluate thermodynamically this equation I get whatever activation is associated I figure out that is same as what B by A I got by experiment so which means B by A is case limited okay I did same thing for diffusivity experiments from the chemical point of view and whatever energies I found for both weight and this I matched it with B values and I find I got the same within errors within experimental errors attrition energy is the energy required to react okay is essentially a bind is a enthalpy either yeah by any reaction whether binding or dissociation or enthalpy formation is essentially related to energy the if A plus B has to go to C plus D then the reaction is favored forward if the Gibbs energy is plus that is enthalpy minus entropy T delta is positive if enthalpy minus T delta is negative dissociation B C plus D will go back to A plus B okay we will see in CDD this is where we adjust the growth if I want growth what should I do A plus B should be stronger in delta G must be positive so A plus B will react when I want to H what should I do I do not want reaction I want etching okay so I see to it C plus D goes back to A plus B that is exactly what we do depositions and etching are identical the reaction are favored or not favored okay is that point if something is formed I reverse it don't form okay remove it okay so that is exactly what I am trying to say so what essentially I am trying to say you that that gradial model to a great extent is a good model except for as I say very thin oxides if I see orientation as I already said B by orientation is only you know after what it is the diffusivity of in thick oxide how much is available is going to decide only thinner oxide available bonds will decide available reaction so we know B by A 1 1 0 1 1 is at least 1.7 times B by A of 1 0 0 constants are already given C 1 C 2 you can value out and this is also essentially the ratio of bonds of 1 1 1 to 1 0 0 so initial oxide will be thicker for 1 1 1 compared to 1 0 0 by 1.7 time afterwards why they it will not this because availability of oxygen is going to decide and not the rate so then 1 1 1 and 1 0 0 will have same thing as diffusion limitations but initially available bonds to react at a given temperature will be decided by which kind of planes you have okay we also have in real life polysilicon as gate and we will see that later the growth of oxide is different from poly compared to crystallines we have not done poly so far so I do not want to prion but just for the heck of it I may show you something which is of see poly crystals say let us say this is crystalline but its orientation is different at different XYZ okay they are crystallites but they are many of them poly large numbers so in this part they may be crystalline but their orientation will be different compared to orientations here and here some may be larger crystallites some will be smaller the line between two such crystallites is called grain boundary each crystallite is called grain so these are our grain boundaries that is two crystallites are meeting at that point that is called grain boundaries now grain boundaries that do not have crystals there okay so it is like a void in the system okay so if you push something it may not go through oxide crystallite but may actually go through the grain boundaries so more and more atoms may be possible to be oxidized because much of the oxidant now will not necessarily go through silicon or silicon dioxide layer on the top but through grain boundaries and react okay so do you expect thicker oxide for poly because deeper at least thicker oxide because it will oxygen can go deeper in the poly crystallines poly oxide is a many crystallite and hence many grain boundaries oxidant diffuses faster through the grain boundaries leading to enhanced oxidation rate and the model which is suggested is a e to the t to the power n where a is a fit constant n is also fit constant so we say okay this is the law poly oxides follow it doesn't follow deal group model it follows another law okay is that okay and why it is so because we are not just studied very strongly the diffusivity through grain boundaries there are many micro crystalline theories some other day if you are really working PhD for that then I will show you how even this model is not correct okay but as of now we will not discuss okay is it okay so we say poly crystallite poly when oxidize it oxidizes thicker compare faster therefore compared to normal or silicon okay and the formalite fits into is x 0 80 to the power n n is typically more than half okay which essentially is closer to parabolic and greater than half is half is parabola so it is slightly higher than parabola faster growth this is like BT okay so as root BT so essentially it is saying if you do that it is essentially parabolic growth because thicker oxide is growing but n is not 5 half but it is larger than half is that clear thicker and there is no thinner oxide because by then grain enough oxidant will be made available and it will be decided only by this available this okay the next effect for us under pressure the crystals are stress leading to enhanced effect whenever you put a wafer inside pressures and there is an experiment done for a water vapor water solution and we actually put a high pressure on that and the oxidation rate gyramans experiment very popular one this pressure effects are coming back in some other way now in in fats as well as MOSFETs what for the pressure we are talking strains okay pressure is essentially stress proportional strain so when the lattice is not matched then it is a strain which is essentially due to the pressure okay of that is stress stress is pressure so force per unit area is a pressure okay so strain is essentially now coming back once again to improve your mobility is okay but in oxide this is not crystalline it is oxide in oxide more bonds are available because under pressure lattice is heavily pressed and it breaks section it is lattice velocity is not 100 percent so more oxidation is possible so a formula was figured out in linear growth regime b by a at any pressure is b by a atmospheric pressure times p to the power n typically the pressure which we used by gyraman earlier was 1 to 3 atmospheric pressure and n was found to be 0.7 to 0.8 again fit function so all modeling people very happy they fit experiment data fit okay then you can say that I could have fitted something p should be some numbers n should be some I can fit any in such combination but I know roughly that how much pressure I can put actually before it actually breaks so I figured out up to 3 atmospheric sustains okay so I fixed as value in between and for those values how much n it can fit to the data so some physics were brought afterwards after I really see what is happening I say okay which is not true because there are n combinations I can make out of it okay but that's all models do including person like me then there is an oxidation which many a times we will have to do which is called oxidation due to our impurity is present in the oxide though our impurity present like heavily dope silicon as a different oxidation rate compared to lightly dope silicon okay so there is a difference in growth rates and typically the fitting function for doping greater than 10 to power 9 per c 19 per c c where do you think this numbers will appear which area of MOSFET source drain okay so in source drain the field oxide growths will be thicker naturally because of b by 20 times b i undoped 2 times e by f of p type beta for b is dope is twice b type undoped and b doped is 0.04 times undoped for n and p type this is a fit data again this numbers do not actually matter but close to this numbers can be fitted okay in real life it may become 0.30 0.0385 but it's okay to say 0.0554 okay the real data may not be exactly this number but this is good enough because most of the time we are not very keen we not actually just do by we actually monitor so I know how much oxide thickness I have there okay so I may not make mistake in future this but to model I cannot do very very accurate models if I had to then water every time I get experiment data for the lab I will retry to fit in this and actually get the fit models okay for this lab for this furnaces this everything I know this data which may not be universal only for this people okay this is how all people do we get spice parameters in the circuit simulator all kinds of effects are taken care all physics is introduced by fitting okay what do in the fitting I mean the physics cannot be fit but that is what it happens as long as IV ideas VDS with characteristics appears circuit performance appears who cares whether it is case to the power or not to the power this function fits that is what we want but if I had to understand what is happening then I am a is a mobility dependency is how much is correct okay is the oxide thickness is varying how much okay then I will start VTS constant or not constant how much is developed there so I will now put a physics and since it doesn't fit I will put some constants exponential constants so that it fits to the lab data okay for a given technology then I have a circuit simulator with modified model card okay no one stops me putting my model card so I will put my model card okay that will give me the accurate result okay this is how all designers do whatever actuality and they will fit to a value which fits to the data okay and then of course many of us will like to physics so force physics on it okay but that is more interesting to see before we quit last this all these models since it starts with thin oxide assumption if you are going thin oxide then what is the model okay since we are scaling down the technologies at least below 90 and even 65 the gate oxide thickness is now reducing below 40 Armstrong's and in the 28 nanometer or 22 nanometer it is less than 5 Armstrong's even close to 1 Armstrong 1 Armstrong of course you cannot grow and that was the question asked in your quiz if you scale down technologies the oxide thickness are becoming less than 5 Armstrong's and then there is one monolayer which you cannot grow and therefore you need high k so that capacitance remains constant epsilon a by t so increase one epsilon increase t the ratio remains same okay so use high k okay here we are not talking of high k we are just saying it is reducing till very late we were working on you know 65 even there is a si of two thickness gate insulators have been used below 14 four nanometers I started using nano because these days everyone must talk nano not Armstrong so I also wrote 4 nanometers okay the kinetics of thin oxide growth was not exactly as predicted by deal and grow okay so for example thicker it may still work because then it is not a thin oxide so we are not interested to know something about parabolic because by then it is already thick okay where is the worry the linear portion because there the thickness is smaller okay so we figured out that if I see the grow deal model and if I see the actual thickness it is somewhere different slope which essentially means thinner oxides do not follow exactly in the very thin oxide regime the deal grow model to reduce the oxide thickness what should I reduce one is of course time but time cannot be one second okay I mean you can push and take out you can do so we will put some minutes so what can I reduce temperature so I actually reduce the temperature I do not go oxide at 900 1200 but I do it at 800 possible you can say why not 600 there is no reaction the reaction minimum temperature for SiO bond to form is it at 800 so the thinnest of oxide can be grown around 800 degree centigrade which is around 1073 degrees Kelvin okay please convert everywhere into Kelvin's okay so then there are models which are available in market quickly will show you and so is that clear to you for a thinner oxide regime this is what we see this is what grow deal predicts one of my PhD student way back in 80s did this work which model I will show you later there is a first model which appeared was Risman's model which has some modification they did x0 at plus Tn to the power this Tn is a fit parameter okay which is essentially Xi by A A and B are constants okay is a fit parameter Xi is also fit parameter get the data and fit it into this model and we say okay Risman says if you grow this and fit into this it will fit okay so fit get AB and Xi from the actual data and use this model for your lab okay for your CAD tool okay you can see everywhere what we are doing is fit okay so someone should say is it B by A or is it what no I do not know what is it it is A and B then there was a hand and hands model which he says there are two parallel reactions are going on in this so he says B1 upon 2x0 by A1 and B2 upon 2x0 by A2 now this you have this B1 B2 A1 A2 as fit parameters and try to adjust x0 versus time using these two parallel terms okay so this doesn't fit the data you add from here some term and see it may be minus as well sin B2 may be minus depends on the fit you want and you see that it fits to a other level data okay this is called hand and hands model the most important model which except which was accepted for quite some time was from Plummer and his student Masood and Masood has the best of data as well in way back in 90s Masood published much of the experimental data for thin oxides as well as to thick oxides but and his data has been taken as the most standard data by almost all industries this was work done in what Stanford okay then Masood and Plummer has a model they say okay the second term instead of B2 by something they had added a term called e to the power exponential x0 by L again true and they expected that the L value which they will use should be less than 70m strong beyond which this is not valid thicker oxides so they added another parameter here C and L to fit the data and that is that was accepted for many years rather even now if the first attempt is to use Masood's data and Masood Plummer's model we also did some more kinetics on that for thin outside our work was published in jab 1989 Mysel Vasi and Moorid my PhD student so we suggested some equations that S is called the available site there is a term which we created called site so we say oxidant plus site may form a oxygen site combination oxygens O2S plus another site may form 2S at with a constant I mean proportionate constant K2 reaction rate constant then we say Si plus Si Si bond will react with OS to form Si O Si bond and create another site and that is how oxide will keep growing thin oxide will come we also introduced many constant in this expression P is some constant omega into silicon silicon bond times N K3 into K1 K2 these are called constants of reactions Q is K1 and R is K1 K2 this of course those who are very keen can see our paper of 89. We fitted this data for 800 degree and 900 degree centigrade with the Masood experimented data which he published for different pressures but 0.1 atmospheric pressure is the best result they claim so we fitted our model to this and by making a proper choices of K1 K2 also we derived what values we should have and based on our analysis of this we could get PQR values which fits into thin oxide so we could fit data up to 20 angstroms of oxide thickness below 20 of course our model also did not fit 20 to 40 angstroms or 60 angstroms our model fitted very well with the experimental data then known okay. So if tomorrow someone wants another model you can always try something so you should be able to find some reactions what should be the real materials going on what bonds can it create what is the binding energies available what is the space charge around so there are many things which you can think and add on to a model okay is that okay so this finishes the modeling part next time we will do oxidation techniques and characterization.