 So let us go over looking of impurity, contribute to resistivity we have seen last time. This is what we are doing it. We also calculated, I just want to quickly go through what we did last time. We calculated the available substitutional impurities or interstitial impurities by putting this thermodynamics and we then said that these are available to the n e to the power minus e s by k t where n is the number of atoms available in the lattice per c c. And then we also said if it is interstitial we declared it n i 0 if it is vacancies it is n v 0 and we derived an expression which n i 0 is 27, it tends to be 27 times this n v 0. This is all that we did last time. Today we start with quickly something more about it. There is also a possibility as I said last time. Frankel defects are also available and they can also be created at given temperature. If n is the number of atoms in crystal per c c or per volume, n dash is number of available interstate sites, sorry it is sites per volume and n f is number of Frankel defects per volume and e f is the activation energy Then by similar arguments we have entropy is equal to k time, Boltzmann constant time, ln of c n f. Please remember I am now looking for vacancy interstitial so both these are we are taken into care okay. And by same argument I can show you that by similar d we can write s by into k t and t delta g delta g by same method we can calculate the available Frankel player will be n into n dash e to the power e f by 2 k t where silicon, e f in silicon is 1.1 electron Boltzmann constant time. Please remember this energy is smaller so Frankel pair creations are not smaller okay. Frankel pair creations are not smaller simply because an interstitial site vacancy can pair very easily and move together. One jumps there then the vacancy is created on the back side okay. It is like electron hold transport. So this of course is not so important so math you can do again I have not done it. I just wrote down the final answer. The method is same as we did for interstitial and vacancy. Once we say we know the defects at a given temperature we like to know how these impurities move inside a crystal and we are not looking impurities concentration in amorphous semiconductors and in polycrystalline semiconductors. They we are interested at least in the case of solar cell doping of amorphous materials and we are interested in the case of CMOS poly gates of silicon gate material devices poly diffusions. However as we say we will first look into crystalline structure in which impurities are entering okay. Now these impurities if they sit only on the substitutional sites as I said other day then only they can contribute to resistivities otherwise they will sit into interstitial site and do nothing except creating strain okay. Okay if you are written down I may move further please remember this is available on Google sites if you wish to read sometimes fine time if not very much busy with other more important activity like TV mobile and more internet look for this may be interesting. This is a diffusion process and nothing to do with electrical engineering or something it is a diffusion in anything okay so basically it can be available on chemical engineering sites, chemistry sites, material science sites, many places you can get same expressions because the thermodynamics related situation is that okay. So let us start how impurities move please remember silicon has a primitive cell which is shown here which is one silicon atom is bonded to nearest neighbor by 4 atoms and this is called primitive cell if you have seen our unit cell there was a primitive cell colored shown by the corner one the one which is just in interpose from 1 by 4 1 by 4 side and 3 other this so these are essentially is called primitive cell the minimum amount of bonding which makes silicon atoms go is silicon lattice go is this cell primitive cell. Now there are if you see primitive cell there are 5 voids they are arranged in tetrahedrally 1, 2, 3, 4 and back one side so 5 sites here are some are occupied but most of available sites for impurities not all voids are substituted by something but they are voids available okay. So the one of the possible mechanism is how interstitial diffuses through interstitial so let us say an impurity sits a position 1 it can hop through to 2 to another interstitial site it can hop to 3 another interstitial of course this is random this is only shown one method one place but can have any random number placements 3 can go to 4, 4, 5, 6 may be a further ahead so this is called interstitial diffusion impurities hop from one void to the other this is called substitution interstitial diffusion even now though it is doing this process may not contribute to resistivity but it has more important because as it releases the void strain releases and there is a possibility of silicon moves from here it can occupy another void and release a vacancy down so the whole purpose is how vacancies also can be moving with interstitial motion. Now in case of silicon this date this is of course few lines you can always see silicon diameter of interstitial void is 2.36 amstrons what is the void essentially I am saying between this the circle which touches all 4 or other backside one as well is called sphere there and the circle which is on shown here has a dia of 2.36 amstrons which means radius is 1.18 amstrong this is called tetrahedral radius 1.18 amstrong is called tetrahedral radius and of course between the two constriction between the two atoms on the constriction side the gap is 2.10 amstrong since lattice vibrates at any temperature there is a lattice vibrations okay the lattice keeps on rocking as well as stretching more physics someday and it has some frequency which is called jump frequency or frequency of oscillations or the vibration and typically it is 10 to power 13 to 10 to power 14 per second in different lattice. This is typical monitor number derived from and measured from atomic spectroscopies if interstitial impedance has to jump from one side to the other it has to overcome energy barrier you cannot just go it has to cross some barrier. Now this barrier which is 700 to 1200 descent pair thermal vibrations occur with frequency nu which is given by 4 nu 0 e to the power e m e m is the barrier energy it has to cross this much energy to come out you know it should get excited enough pass through barrier and jump to the next side this 4 of course is called degeneracy from where it comes it is a random it can go this side it can go this side it can go this side it can go this side. So it has a 4 possibility of motion so it is called degeneracy factor. So typically the jump frequency is 4 nu 0 e to the power minus e m by k t typically e m for this substitutional sites the barrier is something shown here so if atom has to go from this side to this side it must cross this much energy. This is equivalent model of energy I hope in second year or maybe earlier you might have done chronic penny model this is replication of that to some extent. So you have an atom here and it has to occupy this it must cross the barrier of energy e m to reach to the next site okay this essentially is what this expression shows. The jump rate of frequency is 1 per minute at around between 700 to 1200 day it varies little bit but it is around 1 jump per minute is what the rate with which substitutional this sorry interstitial actually jump okay. This is important because how many atoms are where at a given instant is relative to at a given temperature is somewhere 11 to know how many impurities are available where okay I introduce some impurities in silicon where they will lie so I like to know where they can at best go and will how will they go okay. So this is physics telling that there is a possibility now if this jumps impurities can come and occupy that point so if they are entering in okay. Now this jump frequency why I brought this there is something this expression this equation is to do with diffusion coefficient but maybe we will see this later okay. This relationship which I am drawing I need to have for creation of constants diffusion constants or diffusion coefficients okay is that okay everyone okay. So this is something called interstitial the most important transfer of impurities inside a material is through substitutional sites substitutional means wherever silicon atom is there and if there is a vacancy an impurity atom can occupy that vacancy and sit there but it can actually jump from cellar position 1 if it finds a vacancy here so it actually may jump here if it finds a vacancy here it may jump here and keep jumping wherever it finds another vacancy so impurity can move from one vacancy site to the other vacancy site by process called substitutional diffusion this is most important diffusion how impurity is actually travel inside silicon okay that is what I am what are why are we doing all this maybe it should be very clear to you that my interest in doing all this is not just because I want to understand physics alone or maybe I am interested X of y or may be interested but the interest part is how much resistance it finally offers because of the profile it gives because current is related to that are somewhere I am very keen to know how much is the current I can get if I apply X voltage okay. My interest in electrical engineering is only IV and CV if anyone hurts me on IV or CV I am going to look into why why why cannot I do better so I look into physics I look into material I look everything because at the end of the day circuit must function the way I thought I had designed okay and to do this I must understand everything around that which which helps me to improve okay so please do not think this technology course is only of this side this is essentially covering those areas which normally we do not cover anywhere okay normally I do not say please remember generally vacancies are fear compared to words so essentially you can say the jump rate of diffusion process will be smaller because the available sites are smaller compared to interstitials interstitials are almost everywhere vacancies are need to be created by some energy into the body is by KT so you are now two energy one is you have to first create a vacancy and then allow it to move so that will be now new energies will be will be actually energy creation for vacancy plus energy creation energy for barrier to cross so actually it will be two energy some now which will be required for vacancy motion is that point clear why is not to be created they are there but in case of first vacancy are to be created once vacancy it can jump only if it crosses a barrier okay so based on this same similar analysis one can write mu is now 4 mu 0 e to the power E n plus E s where E s is the E n and E s as a binding E s is binding energy and E n is the vacancy creation energy is that okay I just now said to create a vacancy I need E n energy to move it I need a barrier to cross which is E s now silicon silicon bond by binding energy is larger than silicon impurity bond is that word clear two silicon atoms okay maybe first you write down this part is most important to show that substitutional impurity diffusion has much lesser chance compared to wide ones because there are enough words so many many of them can actually enter which is very contrasting people believe that impurities must be sitting first vacancy no first they may likely to sit to the voids itself or interstate site and then maybe move by seeing a vacancy around okay so this very interesting part silicon is also atoms are also moving because of thermal energy some bonds are broken anyway but silicon silicon bond is very strong very very strong in case of bond strength that is Coulomb's law if you apply the bond force is very high compared to silicon impurity atoms the binding energy for silicon silicon is much higher so it is unlikely that silicon bonds will be keep breaking every now and then unless you heat it very heavily view lot of thermal energy on otherwise silicon does not break very burns doesn't may do not break very easily therefore silicon self diffusion going from one side to the other is lesser event not that will not happen but lesser chance of moving one silicon atom to the other silicon side is very smaller probability event compared to impurities getting inside the crystal because after all you are giving temperature enough thermal energy is provided every possible mechanism can happen so we want to eliminate saying that okay silicon silicon self diffusion is smallest among all of them doesn't mean 0 finite small compared to the interstitial and substitution there is one more possible may be two more possibilities are there which also are very low may be this one there is a possibility that atoms may move by the vacancy vacancies or at silicon atoms and they can actually keep moving around the other sides okay this is very interesting which is very very small probability that this atom of impurity or silicon jumps to the next side next side next side and come back this as a very very small this is called interchanged diffusion very very unlikely event but can occur one in billion or even lower probable but can occur but at given temperatures higher than it may occur as well to some extent however which is the best possible possible diffusion therefore substitution and interstitial together is very possible atom first come to interstitial side jumps to vacancy allows another impurity come to interstitial side jump to vacancy this vacancy atom may move to another vacancy it may create another void there from where interstitial move another vacancy is brought in and this is called cooperative diffusion this is the most likely diffusion in which interstitials and vacancies go together to push atoms inside okay is that okay they both can help each other to actually get more and more impurities getting diffuse inside the crystal this is most likely event and has the largest probability okay I can do some quick calculation for this as well is that okay to possibly so there are four possible mechanism in which impurities can get inside most likely is the last one but the first two actually tell you that together they will help in the fourth case okay is that okay everyone if ns and ni are concentration of available substitution interstitial sides at temperature T then the effective jump frequency can be given by available substitution side into total sides into jump frequency for substitution plus available interstitial side to the total sides into jump frequency for interstitial this is very standard average method available once to the total with a jump rate available once with ratio to the total into that jump rate this is the net possibility if it is only substitution what will happen ni will be much smaller than ns then you say this term will be negligible only substitution impurities may move if ni's are much larger than ns we say only interstitial diffusion but in a given this both are together and ns ni also concentration keep changing as numbers start getting more and more inside okay. So relative to for concentrate both if I relate to each other is effective new please remember this is most important thing which you should understand this expression is only trying to say both together can happen and these expression I have derived from the average method one among so much into this second among so much into this however it is important to note that natural random emits may not be very large these are called natural random jumps okay most of the impurities actually travel because of concentration gradient larger compared to here smaller impurities here and they try to diffuse through to make equalization so question rather than why do we all do this if we know it is only gradient dependent no some of the effects what I should call them anomalous effect I see some profile none of the standard expression fit to that then I come and see is this material at this time has some other diffusivity going on so I look into which is the other mechanism which might have added or reduced it so that the profile should have gone up but it is going down so to understand the actual profile which I will get in real diffusion I like to model it and to model it I should know from where these possibilities can occur okay. So it is not that these are very strong forces there but in case the profiles do not match as in the case of bullion it does not match password is it does not mostly it matches with arsenic in normal this but all other impurities show what is called as anomalous effects anomalous means from standard diffusion theory does not match my profile so I say why this is more important if you are actually looking into very highly dope crystals areas or very low in between the diffusion coefficient diffusivity is essentially given by gradients okay most devices are in that range but source range 10 to power 20 some kind of p i and i volts 10 to power 13 so in those areas these diffusion techniques are very very important other ones we have two laws for now we will start with this how the diffusion starts and I want to find at the end of the day maybe I will show you one figure what is my ultimate aim of doing all this why I am so much worried by some way this is my crystal maybe silicon right now and this is my surface I call it and I introduce impurities this is what I will do technique of doing will be discussing techniques how impurities are introduced actually now if they are getting since there is a large concentration of impurities at the surface compared to what silicon it has it can be p type n type smaller doping higher doping but difference so there is a gradient these impurities tend to enter the lattice and keep moving downwards because the gradient it created if I let us say this is x is equal to 0 then if I plot concentration of impurities nx versus x impurity is getting down this is x down okay then I may say there are number of profiles one possible profile is this other possible profile is sorry exponential this is Gaussian this is exponential and one more we shall that is what we will start with so this nx is what I am interested to know what is the profile that means from the surface the concentration will be normally very high but as you go down the concentration starts reducing now this profile decides some kind of resistance available to because carrier available will be proportion to x now normally we say dopants are equal to the electrons are holds depending on the what dopant if it is n type all the doping impurities actually are equal to available electrons not everyone of it but mostly we say n is equal to nd plus ionized one p is equal to any plus ionized ones they are almost equal okay so resistivity decided by n and p so that that means nd and n is okay so I but if nd and n are not constant that means there is a profile so n and p is are also of functions of x which means if I calculate the resistivity of resistance I will find it is a function of x so in many cases the way I do it I take an average of that value integral over that range and say okay average resistivity is this but in real life resistivity will higher higher up and sorry resistivity is smaller up or conductivity higher up and we will go down now this matters lot in a smaller devices now because your channel length is of the order of 14 10 nanometers your diffusion of substrate at n to power 19 kind of things very thin layer of channel is going to be created of course there are other physics effect called quantum effects but some other day now there how many real atoms are available to us how much really annex I have do I have a profile or uniform all those issues will finally decide IV of the most transistor or BJD transistor so this number is very crucial for me how many actually okay and this how many makes me actually go through all of it since I am interested to know nxpx for my electrical properties I am now looking into how are the impurities going to contribute to this nx and px profiles is that clear so that is the purpose of doing all this I am not just doing this physics because I am I like maybe I like it I keep saying I do really like it over the years you will also like it when you know there is no stress of exam you also start liking okay I mean this is natural okay okay so there are two laws in which impurities diffusion can be modeled first is called fixed first law and second is fixed second law I think what I will do is I will leave it fixed first and second law for a while okay I will first finish the impurities how because someone will say lab you want some numbers so I will show you actually how diffusion is done there okay so what we are essentially doing here is the impurity which are coming inside the silicon how they get inside and what is the profile it creates is what is our major interest of doing all of it okay so I start with something which is related to that and as I said time permitting I will come back to that this is very interesting since our diffusion is solid impurities are getting into solid material okay if it is a liquid liquid system which we have seen solid bits getting into liquid both forming liquid crystal growth okay there we can as if you know steroid and uniformly doping but that is not so in the case of diffusion in normal sense so we are interested in if I start this impurities diffusion from the top let us say on the wafer what is the maximum concentration I can get at the surface for your kind information maybe just a minute I will just say numbers and come back to this the maximum silicon concentration at a normal temperature is 5 into 10 to power 22 atoms per cc this is the maximum carrier concentration or maximum doping or maximum atoms per cc available to you based on this all density of a calculations were performed or rather from the measure density this number has been found okay for the lattice of 8 atoms in per cell okay so this is the maximum atom so how much doping we can do certainly not 5 to 10 to power 22 because if it replaces all the atoms then there is nothing silicon okay so obviously it has to be less than 5 into 10 to power 22 so this is a natural limit up to which silicon can get into impurities can get into silicon that at a given temperature is different and this term is called solid solubility solubility is essentially between liquid and solid but here is a case we are talking terming it as solid solubility is that clear atoms how many atoms per cc are inside at the surface we like to know how maximum can reach there okay at the surface that is called solid solubility here is some type of impurities you know which contribute to electrons and holes there are some query about if there is a why do not you buy put impurities from the second group or even the first group then it can create two electrons or two holes three electrons or three holes why only one kind of thing three or five it take there is the answer later when I calculate diffusion coefficients what will happen if there is a double such system appears however as of now arsenic phosphorus antimony are standard n type impurities and boron aluminium gallium are mostly p type impurities 99 percent now we use arsenic as n type dopant and boron even now is the only good dopant for p type okay yeah there are some devices in which boron plus aluminium has been tried but it is not a great success there is some new methods are being tried there some other day about this p type impurity concentrations difficult to get into the much numbers we will see this number soon so here is some graph shown here this is temperature versus solubility I can say this is my 10 to power 22 or maybe 5 into 10 to power 22 is the limit of silicon so one nothing much can go below also the graph shown here arsenic phosphorus phosphorus dip at higher temperature will explain this is anomalous at high temperature phosphorus why it comes out okay so it actually reduces the concentration boron of course at 200 does not do much now one can see from here the numbers with roughly it is 4 into 20 for arsenic a boron but arsenic can go up to 10 to power 21 4 into 10 to power 21 what does that means arsenic is much easier to get in compared to boron so what is the problem why solid solubility of arsenic is highest compared to boron so here is some numbers which a sheet which last time I forgot so maybe taken from plumber yesterday and hand written on something if you are taken down the graph these graphs are also available in plumbers so nothing great about of course they if you see their book I do not know yesterday I have not said but there will be also two such graphs one is this and the other is dotted lines for both arsenic phosphorus can you think what could be they they this will be slightly lower than all all all of them say arsenic 4 4 into 21 maybe it is a 5 into 21 but actually dotted curve will be slightly below 3 into 21 or 4 into 10 to 1 why this dots come lower than is that clear to you something not all atoms actually get ionized they may come in okay only at the substitutional sites if they said they are ionized which essentially means this is the actual number which I come in okay but the actual ionized atoms will be slightly lower than the available so actually our interest in that number not even this but for theory let us look into solubility at given temperature so please look at the books they will show you three graphs for solid solubility three graph solid solid is it activated impurities activated means ionized they are actually sitting at substitutional site in this everything which is getting into lattice without straining it is possible okay is that okay so this fact has to be understood by that number is slightly smaller and in our calculation when I will give you graph I will give you both I mean that full graph you will have to always choose the dotted curves which are activated numbers for the concentration is that point clear you must use dotted curves instead of the hard ones because dotted are activated impurities activated means these are the ones only contributing to resistivity and we are only interested in currents and voltages no more so we will say okay where I dotted okay so we will see that graph later okay whenever impurities try to get into a lattice it try to even if we say strain free it does strain the lattice okay for example when the even if interstellar side between the two atoms going there is a construction there so it has to pump in okay it has to push in okay now that means some of the earlier atoms may not retrace back to its original position but may be slightly moved okay so there is a partial strain which always exists even if we do room temperature problem is we recurrent it by some other technique that is another issue for a silicon we are actually discussed other day they are just now that there is a tetrahedral radius which is 2.36 Armstrong was dia so 1.18 Armstrong was the radius for each such atom size of phosphorus arsenic antimony I think they are given boron aluminum gallium indium gold silver all impurities we are actually found out the tetrahedral radius for their lattices okay now one of these impurities may come into silicon this is is that clear they are impurity atoms they also have a crystalline structure many of them not necessary all of them so if their tetrahedral radius is known and I know silicon tetrahedral radius now we say if the tetrahedral radius of silicon matches with tetrahedral radius of that impurity then the maximum will come because they will not strain anything okay same size if the size is bigger the smaller atoms and one is bigger sitting there so it will strain the lattice anyway if it is smaller it will stress it okay so if the size tetrahedral radius of an impurity is not identical to silicon or tetrahedral radius then it strains the lattice marginally but it does this is essentially called replaced by this is explained by term called misfit factor okay is it called is it is called misfit factor so maybe I write down in that sheet okay I will come back and this number maybe I will read out for you so that you know for silicon R0 is the tetrahedral radius 1.18 I am strong phosphorus it is 1.10 okay arsenic it is 1.18 boron it is 0.88 please note when I say please note down silicon 1.18 phosphorus 1.10 boron arsenic 1.18 boron 0.88 aluminium 1.026 gallium 1.26 aluminium 1.026 in gallium 1.26 indium 1.44 largest atom around okay but gold and this is even higher gold is 1.5 I am strong silver is 1.52 I am strong is that okay and the difference between R0 to this tetrahedral radius or other is called misfit factor epsilon is that clear to you the difference between silicon tetrahedral radius and impurity tetrahedral radius is called misfit factor it can be plus or minus if the inferior term is a larger tetrahedral it will be minus if it is smaller it is positive. So what I define you have noted these numbers you noted down what I wrote okay what I am now saying you that I have a formulation which says R is equal to R0 1 plus minus epsilon where epsilon is called misfit factor you subtract 1.18 from each impurity atoms each impurity atom tetrahedral radius and find out what is epsilon for each R0 is for silicon R is the tetrahedral radius for impurities. If epsilon is 0 when this can occur when R is R0 if R is R0 epsilon is 0 which impurities according to you has arsenic since arsenic has the least misfit factor the possibility of number of atoms of arsenic getting in silicon is the highest at solid solubility is that point clear because they do not strain the lattice compared to others. Next will be which depending on of course 0.068 is for phosphorus so next best may be phosphorus boron has 0.254 as epsilon difference okay so it will have a smaller number. So do you get the point that graph which I showed you is somewhere related to misfit factor is that point clear why those graph arsenic shows highest concentration followed by phosphorus follows by boron and if you plot for all the impurities you will correspondingly get solid solubility curse for all impurities but since we are only interested into silicon IC process I am restricting only these three but please remember in case I need it I will get those values for my other impurities as well. So this fact that misfit factor decide solid solubility should be understood that why people actually say that arsenic is the best n type dopant or best dopant in silicon but if a gallium arsenide lattice it may have different kinds of misfit factors for different impurities there okay of course there is no by the way which is the easiest doping this in the case of gallium arsenide silicon okay some other time okay of course I am not teaching gallium arsenide but my own PLD work was on gallium arsenide 35 years 40 years ago so I still enjoy that okay and as I said those days when I shifted to silicon my guide was saying that you know this is the area of future why are you shifting then I told him the future never comes so I would prefer to be in present so I will shift to silicon so my choice was not bad as far as technology goes but if I would have continued I would have published many more poor papers because silicon there were 20 lakh people working gallium arsenide so I should have been there but didn't realize that I will become teacher maybe I should have become the otherwise though I did try work in industry as well as R&D labs before I became teacher that's why I said technology was my first jobs wherever I did for 15 years so I understand more of technology compared to many not because they are smarter they are they know more knowledgeable maybe many of them have been taught by me so maybe more better than me but simply because I enjoy okay and after great fight with my head you had the sufferers for that but I forced them to give me this course for the last time because I said I want to record it once many of my old students who learned technology from me 20 years 15 years they kept on saying sir your course is not on the web okay last time I recorded now this was the old time case that students thinking have changed student attitudes have changed maybe I have also changed so things may not be as good or as bad as they were earlier but I am known as a technologist in earlier phase of my career and suddenly I became designer for no good reason okay and this is where I am working on metamaterials and antennas and something else okay okay there are two laws of diffusion one is a fixed first law and the other is fixed second law as I said we will derive them or I will leave it to the posting they if what is fixed first law says let us say there is a semiconductor bar okay it has two planes I have made A by root 3 A by root 3 are the two planes this fixed law statement I am just showing a figure this is the cross section area impurities are coming inside this area and getting inside okay this is my x direction this is my y direction and this is my z direction okay in a crystal now it can be found by not going to detail on this this will derive again I just wrote down there maybe I said a j is called flux density this plus please use this word j which is current density but here I am using at flux density because over the years I have been using it some other books may use something else what is flux density number as someone asked you what is the definition of flux density number of atoms or number of particles moving per area per second okay is is essentially called flux density so when someone asked us many years ago that why IIT Bombay and many other IITs only are electrical engineering department and not electronics communication communication instrument has some mixture of n of them so I said I have my statement was simple after all in all electrical engineering we are interested in the electron transport and nothing else maybe whole is a additional feature and it is only the flux density matters if it is very high flux density we say it is a power area very large flux a large amount of current amps 10s of m 50s of 100s of m flux density is very high actually if it is very very small we say nano okay in between if the signal you need moderately flux density required for electrons motion so all areas are covered essentially by number per cc per second so electrical engineering is only electron transport and nothing more okay so we keep working only on electron transport so this flux density is per unit area so is dn by dt this is a statement we will derive this letter 1 upon a dn by dt where n is the number which essentially we are saying a per unit volume actually impurities are coming and going from this plane n1 or 1 into 2 and there is a diffusion possibly means it some numbers can go from 2 to 1 but there will be net numbers going from 1 to 2 if there is a gradient set now this gradient is let us say if n2 is the number here and n1 is the number here per cc so dn by dx is n2 minus n1 upon if this plane distance is a root 3 so n2 minus n1 by a root 3 what is this a root 3 the distance between the plane which distance I am talking is the Miller distance we will see like next time okay you are on the planes so Miller planes we will see what is the minimum distance they have along 1 0 0 1 1 1 other planes so if I do this which I have done there again please just note down don't note down because I am going to post this I just wanted to rewrite because to show you and then we define this newer square by 6 whatever term is coming here as diffusion coefficient or diffusion constant d then I write 1 upon a dn by dt is minus d dn by dx or j which is just minus dn by dx this is fixed first law that the flux density is related to gradient proportional to gradient I repeat the flux density is proportional to gradient this is the fix first whatever this minus sign gradient down okay minus sign d is the diffusion coefficient or diffusion constant okay and we know diffusion coefficient can be rewritten as 4 new a square by 6 exponential en plus es by kt for vacancy transport so d is equal to d0 this term is called d0 exponential minus en plus es by kt so first fix law says j is equal to minus dn by dx gradient that is our first thing so if the impurity concentration is higher here and lower here impurities will move towards the lower side it is like a potential difference unless there is a potential difference energy does not move the only difference there is one can say it is not a random motion in this case the way it is as I say probability wise 50 percent chance going ahead 50 percent chance but keep going plus minus plus minus at some number if there is a gradient you will be further away from this starting point okay so this is essentially statement of fix first law that the amount of impurities per unit area per unit time at the surface of silicon or rather when they enter silicon it will be proportional to the gradient it has set in okay now that is the term we want n we want to calculate so we must first get j value some way and must get dn by dx relationship with that later and if I can calculate nx that is what all that my interest is is that okay so for you this of course you need not add it because I have written there again but just to repeat in case I do not then I said I will show you where from fix first law is coming the fix second law is essentially a statement I do not know how many of you have done your device as well but hopefully so but any others devices is one there where we talk lot of it but continuity equation has nothing to device continuity of transport of any fluid solid gas anything is continuity and according to the divergence theorem the dn by dt dj by dx that is the flux density gradient is equal to minus dn by dt this will discuss this is called continuity equations so time dependent term is related to space dependent term this is called continuity equation we will derive this next time or as I say may post it and if I use dj by dx is minus dn by dt use this j here differentiate j here so I get this equation dn by dt is d d2n by dx square this is called diffusion equation this is what we want to solve and then this is what we want to solve why for a given time impurities are coming in and also moving in is that point clear why this equation is 11 impurities are coming inside going with time but also moving in space so I am not interested in only nx but I am also interested in nxt but if t is known I know I will only get nx profile at the end of t is that clear that is what I want to do so this equation is my precursor of finding nx functions okay this equation is what we are going to solve now and once we solve this only thing catch word in this maybe I maybe I have said it okay here I assume d is a function of nothing that is constant d is independent of everything but in real life that is not so d is a function of concentration itself you can understand some way if there are larger atom the other infinities will require more effort to get in so it is it is a gradient dependent term okay so if nx are present that means d will be get affected by n itself n is larger d will be smaller you take from me okay because they will be stopped by some other people it is a crowd business okay since d is a function of nx d is a function of x so if I differentiate j d dn by dx then I must write this as a function of x and if I then differentiate I will get two terms one related to dd by dx the other related to d2n by dx square now this term many cases this equation is not linear equation okay it is a non-linear equation and therefore analytically cannot be solved easily some people can do by linearization if you are expert in maths there are certain condition in which you can linearize it if you cannot what is the easiest way go on a system and solve numerically in this equation any non-linear second order differential equation can be solved by n methods okay whichever method you prefer you can solve linearize it also by then come to pass riddle or cellular whichever method you can choose boogalov methods n methods are solving second order non-linear differential equations I will assume right now linearity for analytical purpose but in real life the models which I will substitute in software for process simulation I will use d as a function of n itself and let it take because there is a bed I it will find water and find d there why should I care for it okay but why I do I care many a times even if in a software when I write what is the criteria I normally put for writing good software time taken to solve is the major criteria in writing a good software you may have very interesting software written but if it takes ages to solve then there is no point in using that okay so as much as simplicity you can put some small model inside which may not be accurate but enough for that and partly linearize it okay so there are tricks in all modeling people they keep using some tricks okay and then so so fast it works okay it works fast because you are assumed some few things if you do not it takes hours or ages okay so please remember we will do only linear system because that is easy to solve analytically real life since this is not very strong term d by dx dd by dx so for first order this term can be neglected and you can use only first ones okay the first thing we start is now looking into profiles that is our ultimate that is what we are going to do work at okay so we start with profiles first let me say and then draw this is silicon surface and as I say this is the depth in silicon surface shown x this is the silicon surface okay this is silicon surface this is the silicon wafer of thickness T and T is very large compared to anything okay T is for a boundary condition what will I say T equal to infinite means our thickness is infinite okay so I have impurities introduced from surface side essentially wafer sit in a rack like this and source of impurities are impinging on it in any technique I assume and that is very important in time frame okay okay maybe we will come back to this later is that model clear what I am saying impurities are impinging at x is equal to 0 at this which is silicon surface and they will get inside silicon along the x axis okay the assumption is it is isotropic diffusion means y and z do not play it is not true actually I should do n x y z or delta as the term we should solve for but most cases this is good enough okay dn by dt is dd to d2 and this is our diffusion equation just now we wrote okay it is fixed second law we take of course this will come back this is the condition I am putting because I need to solve the initial condition I will create I take a Laplace transform I hope 99.999 people know Laplace transforms if not read it at least communication people if they do not know Laplace transform Fourier transform they will not be in communication next day okay of course this is a trivial maths any network person must know it okay s n x s is equal to the Laplace transform this is s n x s minus the initial condition n x t equal to 0 equal to d time d2 n x s upon dx square this is the Laplace transform of diffusion equation I can rearrange this diffusion equation slightly is it okay there is nothing wrong with this Laplace transforms is very tough without transforms is it okay rearrange the that equation again I write d2 n x s by dx square is s by d please remember Laplace transform is only for time x does not change okay so d2 n x s upon dx square is s by t n x s minus n x t 0 by d which is my initial condition term now here is my to solve this equation this is very easy to solve if I know this this equation I can solve if I know this that means I must know my initial condition so I have conditions which I impose myself and say this is my initial condition so let us assume if you have written down the formula which is trivial let us assume impurity source provides impurity at the surface which is x is equal to 0 at t is equal to 0 so that means what does that mean prior to t equal to 0 there are no impurity at t is equal to 0 source starts is that clear prior to t equal to 0 there is but once it starts in never ends certain number of atoms per cc are constantly available to me infinite times just a minute before I show this at t is equal to less than equal to 0 no impurities are impinging at t is equal to 0 the available concentration is n0 which remains constant for all times to come okay this is my initial condition which I start with and this is real life condition that is why I did it okay is that okay at t is equal to 0 no impurities we start the source at t is equal to 0 and some fixed number appears for all time to come which means nx t0 is less than equal to 0 however t equal to 0 plus what I said I have written again source of impurity that the surface thus impurity source is unit step function in time as shown okay our next assumption it was a condition like why what is the other assumption I need or other conditions I need x k condition t kato diha x kabhi chahi how many conditions you need boundary conditions second order equation need two bcs okay so let us see which are the two boundary conditions we have okay our next is our first assumption is it is a unit source t is equal to 0 then it starts I have a constant source available okay or also called infinite source why it keeps coming there is no stopping on that so either it is called infinite source diffusion or called constant source diffusion okay all the time in infinite or constantly available for all times okay the second time bc we secondly we want to see bcs so we say our next assumption is that impurity source keeps constant impurity concentration at x is equal to 0 at the surface we always get n0 whatever number all the time at x is equal to 0 this number is fixed how much n0 and how much will be n0 roughly solid solubility because at that temperature the maximum available to enter there is so much okay so n0 will be actually you pick up from solid solubility graph is that correct n0 will be picked up from solid solubility graph because we know at that temperature how much n0 can and reach at the just below surface of the silicon okay this value is defined as n0 because that is the number which we are constantly pushing so we said x is equal to 0 this number is fixed so the first boundary condition therefore say nxt0 plus onwards is n0 which is constant now once I know my initial conditions I I know the equations which I wrote has a equation analytical equation given by solution is as exponential under root of s by d into x plus bs exponential minus s by d to the power half x this is the solution of second order differential equation which I have used okay this is very simple second order differential equation simple solution here now the first boundary condition you have said here the second boundary condition boundary is what first is x is equal to 0 where is the second boundary call away x is equal to infinite okay if the infinities are coming in with a constant this why should it become 0 all infinities will go there why it should go 0 so there is an issue which this is what you are saying if infinite source is there the infinite end it will reach okay because there is no stopping it okay just goes away now the problem is if I put first x is equal to infinity in this term what does that mean this is positive s by d is constant x is positive x is infinite what does that mean this nx will become infinite but not I already told the diffusion is always gradient based so obviously in infinity concentration cannot reach from n0 to infinite that is not possible so what should be happening a s must be unequivocally 0 the first constant pre exponent must be unequivocally 0 is that point clear if x is equal to infinity the first time will go to infinite that means concentration will reach infinite which is never possible because impurity will diffuse down with the gradient which essentially means the first term must vanish which means a s must be guaranteedly 0 if that is so the second term has removed the first term itself second boundary condition then I get nxs is bs exponential minus s by t to the power half into x so this is the solution but what is still not known to me bs first as I because of this second boundary I just removed that but now I must know bs so I look into the real life situation let us say what it happens is that okay please note down this is the solution if x goes to infinity this term blows and since this term blows it is again the principle of diffusion and therefore a s must be 0 so the actual solution in this specific case is bs times exponential minus s by d to the power half is that okay everyone the first of course initial condition I already showed a step function of source I have introduced the second is I say at the surface the concentration is fixed that is what I said n0 solute limit that is the number which is available at x is equal to 0 if you are not very satisfied say x is 0 plus because at the surface we do not know outside but just below surface you can say or at the surface we say the concentration is n0 if I know this boundary condition x is nx so we can see from below that we do not know in the silicon there is no concentration x minus there is no concentration but just at x is equal to 0 it becomes n0 okay now this essentially means nx0 at all times your impurity source are coming anyway all times is n0 ut is that correct it is a step function okay so if this is my second boundary or rather first boundary condition that at x is equal to 0 concentration is n0 ut why this ut has to be done because step okay so I had to because in Laplace transform it will give something okay what will it give what is Laplace transform n0 ut n0 by s by s so we have to take care of that ut term constant by s okay ut is only giving me that constancy this condition is also the kind of boundary condition initially used is also called infinite source condition or constant source condition we have a constant source so we are nx0 s is n0 by s making Laplace transform the initial the second boundary condition rather first boundary condition so we write nx0 s is n0 by s equal to bs exponential 0 be a x is equal to 0 so exponential 0 so we get bs is equal to n0 by s is that correct second boundary condition first boundary condition removed is the other first boundary condition give me the bs is equal to n0 by s this is just substitution of x is equal to 0 in the equation okay and therefore the infinite source or a constant source case the solution of diffusion equation is nxs is n0 by s exponential under root of s by minus s by d times x this is the solution of diffusion inside silicon when starts with constant source at x is 0 constantly n0 this is the equation you get for it okay is that okay solution so now I have the diffusion equation solve solving done for profile this is my profile which this is in what this this is explained and I want to come back to time frame so what should I do take a inverse Laplace transform of this anyway it is not so easy for you and you are not seeing that function so do not try unless you have done a course somewhere or done something okay taking inverse Laplace transform nxt is n0 by minus error function of x upon 2 root dt that is very important I will explain the error function soon quickly before we leave can anyone tell me what is the unit of this d is always defined as centimeter square per second is that correct into time under root of that means it is centimeter if everything is so dt is root dt is essentially a distance is that correct root dt is essentially a distance we will see d is a function of temperature these some temperature dependence so what does that mean if dt term which is temperature dependent and time dependent so for a given temperature for a given time I have fixed root dt to do dt is that clear so now I know where the impurities are going for this time and temperature at every x is that point clear d is fixed for a given temperature t I have fixed okay I will do one hour diffusion so I know the time please remember everywhere we do seconds so one hour and a chat is so second is that okay to you so please 3600 do not mischief one there okay so if I plot this function normalize nx0 by n0 versus yy I define x by 2 root dt I define y to plot nothing this so if I say between plus y and minus y this is symmetric function it is initial value is this and as time proceeds and at where it will all go finally infinite it will go to 0 at infinite okay same way this has reached to 1 actually it should asymptotic it should reach to 1 asymptotically now this function is called error function is called error function now for those last slide for the day this is please note down this some data about error functions and since I am going to use this constant source diffusion very often I first want to give little expression for error function because those will be used directly by me in my solving the actual profile evaluations okay is that okay everyone note it down those who wish to everyone does not but those who wish to okay error function x is has a definition in maths 2 upon root pi it is an integral 0 2 upon root pi 0 to x e to the power minus alpha square alpha is any other parameter variable so you can write y you can write z any parameter so e to the power minus alpha square d alpha this is the definition of error function plots I already shown you if you see this integral which is shown bottom please see the last line 0 to z e minus y square dy if I expand them in series it will be y minus y cube by 3 into 1 factorial y 55 plus 5 into 3 factorial minus y to the power 7 7 into 3 factorial and so on and so forth so this series is essentially a e to the power minus y square dy 0 to z whichever it is or I should not say that I should not say why only okay so this expression is a series but I am I know this is the expression I need and this is 2 upon root pi 0 to x e to find this called error function error function at 0 you can take from me when it is 0 all y are 0 so what is the sum 0 so one can say error function 0 is 0 you say from the series everyone has a x term so x 0 0 0 0 0 0 0 everywhere 0 so error function x is equal to 0 is always 0 error function x equal to infinite is very important 2 upon root pi 0 to infinite e to the power minus alpha square d alpha now this integral 0 to infinity e to the power minus alpha square d alpha mathematically can be derived as equal series sum of this is root pi by 2 is root this is a slightly diverging series and difficult to sum up but 1 minus p kind of equivalence can be done and you can sum it up so it gives you root pi by 2 0 to infinite e to the power minus alpha square d alpha is root pi by 2 so if I put here this 2 by root pi into root pi by 2 means exponential error function infinite is 1 error function infinite is 1 and that graph was shown to the 1 it will go to the 1 always maximum okay there is few more terms we actually our profile which we are going to get is 1 minus error functions and that is called since 1 is the infinite part so you subtract rest is complement to that so error complement to error function erfcx this is remember error you want to say infinity minus error function x error function infinite is 1 this term so 1 minus error function x is called complementary error okay you can think like this an integral 0 to x x to infinity okay so that is essentially doing the same job if I differentiate a function it is 2 upon root pi exponential minus x this is the most important differential meaning because it is essentially which is this term coming exponential minus x square define it is a normalize x rise up now have defined it is a Gaussian profile okay so up error function say Gaussian may join it that is what exactly we are going to see this is in the next time similarly if I take second order differential then error function as x is my isca differential curve minus 4 upon pi x e to the power minus 2 x square x square these are the error function terms which you note down because I will be assuming that you know error function algebra so we will just substitute whenever any differential second order first order comes or infinite 0 comes we can just substitute there as it is is that okay so we have found from our diffusion equation before we quit that nxt is n0 a complementary error function of x upon 2 root dt this is the diffusion profile which I got for which case which is the case I have discussed today constant source or infinite source whichever book is we are using some use infinite source some use constant source this will give always complementary error function profiles.