 Yesterday we discussed something about the diffusion using constant source and we did derive some expression for complementary error function profiles. Today we will continue with that. In real life the most of the time the profile which you get may be Gaussian and how it comes. So let us see there is second case. The first case we discussed was infinite source or also called constant source diffusion. Now let us say we have a limited source diffusion to be performed. Since it is a limited source diffusion what we are assuming right now is at the surface there is a sheet charge of impurities. Sheet charge has a delta function which is shown here in the figure and it contains an impurities which is which has a concentration which corresponding to whatever number but since it is a delta function we say it contains a charge density which is number per centimeter square and if I divide by also delta then it will per cc okay. So this charge density which is sitting here that is the amount of charge per unit area sitting at the surface is let us say is known to me okay this is known to me. How do I get it is the process we will learn in the figure and it contains an impurities which is which has a concentration which corresponding to whatever number but since it is a delta function we say it contains a charge density which is number per centimeter square and if I divide by also delta then it will per cc okay. So this charge density which is sitting here that is the amount of charge per unit area sitting at the surface is let us say is known to me okay this is known to me. How do I get it is the process we will learn how do I get this sheet charge that is fixed charge which are fixed impurities have been put at the surface which is very thin layer of certain amount of impurities per unit area sitting at the surface okay. If we come back to the diffusion equation once again and we say the d to n x by dx square minus Sn x by d equal to the initial condition by d and in this case we can show that if the sheet charge is this and q into delta is the volume per this this then we could say it is q the Laplace transform of this will come qsd maybe we can think little later how but since this equation we want to solve and we call this q as the word which will use most often in implants which is called dose okay. The dose is essentially number per unit area or flux density into area into time is actually the dose okay. Now this dose as is fixed in our case because this is a limited source diffusion okay and so delta function at x is equal to 0 it is called sheet charge approximation it is called sheet charge approximation. How do I get sheet charge is another simple process we will see that if it is not exactly like sheet charge how close it is and if it is still not then can I still solve an equation of second order okay there are all possibilities are available. However in most cases we will assume that sheet charge approximation is sufficiently valid the reason why it is valid or not is the case because anyway actually diffuse it in a silicon any impurity I will actually monitor its profile and by at the end of the day I know my physics I know my numerical analysis I will fit it anyway so in worst case if I cannot get exact physics I will fit that physics to the curve I actually monitor okay and then we will say okay there are other effects is that and we will justify myself that I am right that is what all as I keep saying is most important procedure. One can see there the solution of this right now as I say please look into book I will just go into final version the solution of diffusion equation was BS exponential minus SD by 2 with linear this condition giving initial condition Laplace transform taken I get this you just write down this equation the one which is relevant the rest you read in the books NXS is Q upon root SD exponential minus S by D to the power half X essentially this is my BS term Q by root SD okay and if I take the inverse Laplace transform then NXT is Q upon root pi DT exponential minus X square by 4 DT I repeat this is the Laplace transform solution I take inverse Laplace transform and I get a relation which is Q upon root pi DT exponential minus X square by 4 DT if you look at this expression exponential minus X square by DT into some pre-explanant this is Gaussian in nature this is standard Gaussian function please remember this is constant because Q has been fixed constant I had done some diffusion at temperature T for a given time T so this is constant one can also see quickly from there if I make X is equal to 0 what is the quantity remaining Q upon root pi DT if X is 0 the pre-explanant only remains which means at X is equal to 0 what is the concentration we define the surface concentration at the wafer after diffusion has been performed is such that it is only Q upon root pi DT and that is called the surface concentration in a Gaussian profile okay now this is essentially as I say Gaussian and therefore if I plot these values these equation solution please note down you note down only these two this and this and that is most important for us please remember if I have a fixed charge charge sheet charge sitting the solution will be always Gaussian this is something most important for us because in real life there are few things somewhere we may do error function profiling and somewhere we may do Gaussian or in some cases I did not want Gaussian but will happen Gaussian okay this is what is some interesting part in this is maybe I have a graph to show you both profiles together so you will learn soon so if I plot nxt versus X and since remember the sheet charge is fixed is that correct so if I start actually giving a heating cycle at a given temperature T1 T2 T3 T1 is smallest sorry T3 is higher T1 is the smallest so initially after T is equal to 0 some impurities will diffuse okay and they will follow which profile Gaussian profile where T1 in that formula which I wrote in that case q upon pi root D1 D1 is the temperature at which T1 into T1 exponential X square upon 4 D1 T1 so this is the profile I get I repeat what I say if diffusion is performed at temperature T1 then D for that T1 if we call it D1 T1 is the time for which diffusion was performed it is 4 D1 T1 so this is the profile I get at time T1 so please remember since the charge is fixed if you push inside something what will happen area under the curve must remain same as q because that is what the assumption has total amount of impurity you pushed are constant so obviously they will get redistributed. So at first T1 they go inside to some depth okay and this concentration has decreased please initial may be very high but it was first NS1 has appeared okay NS1 is what q upon root pi D1 T1 is the NS1 so the first time I do driving for time T1 I see surface concentration has gone down I keep hitting it for another time total time of T2 same amount of impurity had to be there now this surface concentration will go down and it will go deeper but still area under this curve should be same as the initial value so I have a next profile which is at T2 I did another another cycle of temperature time cycle for time T3 it will go NS will further go down please remember but still this profiles will be always Gaussian in nature they will always remain Gaussian in nature so as long as your charge is fixed initially the profile will be always Gaussian and it will start flattening okay it will start flattening now one question which I have not said it very clearly here if you have a silicon surface maybe paper itself if there is a sheet charge sitting here there is a silicon here but on this side there is air okay so if I heat it the impurities have no such choice of direction you know it may get inside silicon partly may go out of this this is called out diffusion so the impurities then the next time what impurities you are looking for was not q there is that because part of the impurities are already lost in the air however that never happens in real device the reason is the during the driving this cycle of time temperature cycle we actually do this process we shall see technically later in oxygen ambient okay this time cycle is always done in oxygen ambient so what happens that the silicon gets oxidized and we will see later of silicon dioxide is an excellent mass for impurities okay it does not pass impurities unless it is very thin most of the impurities cannot come out of the oxide so the impurities which are getting in the outer surface gets oxidized and do not allow impurities to out diffuse they all go into silicon so now one thing you have to understand that all your processing should be such that the impurities are always in silicon and not outside of course this can be debated how some things may happen but that is some other day so interesting points to note again which I said earlier since impurity those were provided only once it remains constant however subsequent 10 cycle at any A temperature spreads the impurity profile and giving this 0 to x dash nx dx always remains q because whatever you are started that many impurities will always per unit area remain there integral of nx dx is area per unit area so that remains constant so this is something different from error function profiles or complement error function profiles there the surface concentration was held to what value solid solubility value because source was constant all the time so impurities can reach to the peak of solid solubility at a given temperature so there the surface concentration remains constant and the as you start pushing impurity the more and more deeper they may go but the ns there remains constant to the solid solubility limit and there is this ns is not saying it keeps on decreasing as you go for time cycles one after the other is that this is the difference between Gaussian profile and so if you are looking for surface concentration very high your profile should be what kind complement error function profile because at the surface I want very large concentrations this is a necessity in cases of like emitter diffusion in pnp or nbn transistors or in a mass transistor when I am doing a source diffusion or drain diffusion I want source to be highly doped so is drain and the upper surface should be very highest dope possibility the reason there we are looking for highest doping age from electrical properties we know if any other metal sits on the making a contact to this source on drain metal semiconductor junction is a diode it is called short key diode but we know if metal n plus plus or metal p plus plus kind of diode is made highly dope p semiconductor on if the metal sits on it the iv characteristics of such a diode is more like an omics very low resistance omic contact okay what is the problem in circuits if you see maybe we will this is what I keep saying that I have all the game for my device I have shown you earlier figure but I will show you again let us say this is my n plus or p plus whichever it is and I am making contact to source drain this is metal this is semiconductor now if this surface concentration is very high then normal short key diode with n this is more like of course this is not as good also but maybe slightly bad slope there whereas this is short key diode any metal semiconductor contact has a barrier and it always shows a characteristic like a diode whereas what kind of contact we are looking for here we are looking for a contact which roughly gives ideally this but at least this of course this I may be slope may be slightly more or less depending on the process you do but I want an omic contact it is called omic contact r is constant across minus v or plus v and it should be very low typically less than a ohm preferably or 10 ohms or less than 10 ohms or at least a one ohm if possible now that is not possible because metal semiconductor always unless the doping there is extremely high this is called contact resistance or specific contact resistance since doping is very high the resistance here may be smaller but this contact resistance may not be smaller this Rn plus and R contact R in series and R contact may overrule the Rn plus why Rn plus is even smaller because the area there is very high source drain areas are much higher than the contact area so in some sense if real by area increases the resistance goes down so source resistance is smaller compared to the contact resistance same thing true here so essentially you are having now parasitic resistance then there is a channel and then another parasitic resistance or source dash or D dash if I apply v voltage plus minus if Rn channel then I see the current is flowing from here to here or electrons go from source to drain so if there is a resistive drop here other than the channel there will be drop in source regions and drain regions so what may what does mean the actual Vds is not same as what you actually applied and what is the implication the current is proportional to Vds and therefore one sees that at a given gate voltage the current actually has reduced okay V by R if R it increases the current decreases now this issue looks to be trivial when we look from technology side so much but when I go to circuit I suddenly find my charging time has suddenly increased or discharge time has increased because Rs are not as small as I expected of course you cannot make it 0 any day but whatever small possible so all technology details are actually taken care from the circuit point of view so this issue I am trying to again and again impose on you that do not think we do technology for the heck of it yeah many of us do like the heck of it as well as I say 20 years I spent in lab and I enjoy doing many things so when we first were making a thyristor it never fired but after two three turns we realized that where is the why father blocking is not possible so we change our mask and we suddenly got firing that was a day when I say okay I made a thyristor it was a very small power transistor we are just blocking some 50 volts but even then that was the thyristor reverse blocking of 100 volts making a device is much more interesting than talking okay so do and go and make devices okay this is a life as well creation is most important part in life and therefore you should learn this talking of course is easier so I always challenge all my friends even when I was in TFR that if they can make a diode of a given specification on a wafer let us say it gives a leakage current of say 10 mic 10 nano ampere or 200 now whatever fix I do blocking voltage of 100 forward resistance of 0.2 ohm at 100 milliamps repeat this n times and see all them actually give these characteristics this is challenge because if I do one device out of 100 and in runs of 100 I get 10 devices no one is going to pay money for okay so the problem is reliability and problem we have process standardization so making a device in a lab is very interesting and you jump but in real life you cannot jump because one only is working okay so much money is wasted on that okay so please take it that in real life making things are very well learned because you have to keep practicing to know what is being done actually so there the theory comes to your help oh if this has happened probably this is the reason behind so I modify that term okay so that in next process run I do come closer to what I was looking and if I still do not get it then I go to another friend I say I try both what do you say he may have a third idea and try again so there is a hit and trial but based on some thinking what can go wrong if you do not get your device results as you thought okay so device design and process design go hand in hand and one has to really work hard to get a process going which which is what the characteristics device characteristics you are looking for and therefore the circuit characteristics you are looking for okay so final version of Gaussian profile is NXT Q upon root by dt exponential minus X upon 2 this is X square by 4 dt has been written in square form minus X upon 2 root dt square which is same as X square upon 4 dt so obviously one can see from here the profile has a direct relationship with root dt term okay through exponential and through pre-exponent dt term is appearing both ways and therefore dt will decide the surface concentration and it will also decide the Gaussian slopes or the depth up to which this please yesterday I talked about d has a unit of centimeter square per second dt has a units of centimeter square so root dt has this unit of length or centimeters so essentially now I am saying for a given temperature where I fix d which are given d I am now and if I know what is Q then I would say I will be able to figure out how many impurities are at the surface after a time cycle of t1 and also up to how deep they will actually move in larger the time they will start moving deeper and deeper whereas by doing deeper diffusions the penalty I will pay is reduction in surface concentration because profile is flattening is pressing as if okay next we look error function profile once again let us see why why we were so enamored by only Gaussian so let us come back and say okay let us look error function next we look into error function complement function profile again what we say if I introduce at a given temperature time cycle using constant source diffusion some impurity which is error function so I want to know how many impurities are actually per unit area gone in after that time cycle that is the dose because as many impurities per unit area into that distance if per unit integral of that is area is the dose so I like to know what is the dose if I do only error function profiles okay in the case of error function profile nxt is n0 error function x upon 2 root dt remember this term I brought 2 root dt because I wanted to have similar term both sides so that I can write y is equal to x upon 2 root dt anywhere okay to make integral solution easier okay okay we define y is x upon 2 root dt and therefore divided by dx one of the assumption I am making all through in this that d is constant at a temperature and we last time said d is a function of n and therefore x our assumption is d is constant in real life one may have to do more complicated analysis to get d dependence okay from fixed first law we know the j is the dose there we define this is not j is the flux per unit area per unit time so jxt is minus dn by dx this is fixed first law okay that is what we started with fixed first law is that okay I started with Gaussian already I said okay this profile I can get but I want to can you think now something I want to see this can this q be created out of the error function profile that is my ultimate aim I am looking for can I create q out of that so in error function how many impurities per unit area I am can push let us see at given time what how many they go in okay at a given temperature because what is the temperature is going to decide dt product is that clear and it also going to decide n0 value this is most important there the surface concentration is also decided by this but here also the pre-exponent is temperature dependent so for a given temperature n0 may be constant but t may vary and therefore impurities per unit area can be varied by just pushing either at higher temperature or longer time which will you prefer I want a sheet charge higher temperature lower times because then it will be very shallow this diffusion but highest of concentrations ahead on the top so we say okay here is sheet charge okay that is what we are trying to do okay so from the fixed first law jxt is minus d dn by dx at the surface the flux density j0 t is minus dn dn by h at x is equal to 0 at x is equal to 0 term we can evaluate dn by dx is x is equal to 0 and then say okay this is the flux available at x is equal to 0 flux density so using the nx profile for complement error function I write dn by dx at x is equal to 0 n0 d by dx of error function profile I already made the y term as x upon 2 root dt so I can write n0 d by dy into error function why this term is appearing 1 upon 2 root dt is additional term product term appeared what for y is y is equal to x upon 2 root dt dy by dx is 1 upon 2 root dt so if you make d by dx then I can write d by dy into dy by dx is that correct anything I can write d by dx can be written as d by dy into dy by dx okay. So since I have simple algebra here calculus part here this is the dy by dx this is your d by dy error function y this is n0 2 root dt d by dy 1 minus complement error function is 1 minus error function y so I write d by dy 1 minus error function y and well constant differential of a 1 is 0 and differential of this is still so minus sign appears minus n0 upon 2 root dt d by dy of error function y is that clear the idea is to get a term which is algebraically I can find error function y if I differentiate yesterday I gave you table for all of them so differential of error function y is 2 upon root by exponential minus y square now this term is dy dy of this error function y this n0 by 2 root dt was pre part of that so this is now dn by dx at x is equal to 0 why we are interested in because I want to know the flux density at x is equal to 0 how much is impinging okay I repeat I am just trying to find the flux density at x is equal to 0 I use fixed law I figure out if I take dn by dx x is equal to 0 multiplied by diffusion coefficient at given temperatures then I can differentiate I use n is equal to n0 e to the m complement error function x by root dt try to rewrite that dn by dx by differentiating this replace x by 2 root dt to y and therefore dy by dx term appears and when I expand a complementary to 1 minus error function I differentiate that so one differential goes 0 but the other one is minus d by dy error function y but I know the differential of error function y is 2 upon root by exponential minus y square this is from the error function theory since I know this I know this so I know dn by dx and since I know my dn by dx if I multiply by minus d what do I get j0 t so x is equal to 0 same as y is equal to 0 is that okay y is equal to x by root dt means when x is 0 y is also 0 so at j0 t at y is equal to 0 which means x is equal to 0 if I take the value of this the so-called flux density at x is equal to 0 is n0 d upon pi t under root of d by pi t and this is very important this is j0 t please remember this is only j0 t how do I why are interested in I will now say over the time this impurities are getting in but n0 remaining constant so I am now going to integrate for any time and figure out after a time t1 how many impurities have actually per unit area has reached by t1 okay which is called area under the curve okay so I say the dose is now 0 to t1 for which diffusion was performed j0 t dt is that okay once I get surface flux density I integrate that for a given time t1 and find the actual dose in a given time t1 which silicon has received okay so if I integrate j0 t I substitute this term here please remember the convention which I am going to use at any temperature t1 the d at that t1 I will call it d1 at any other d temperature t2 I will call it d2 third temperature I may call it d3 and so on so forth so I do not have to write dt1 dt t every time okay so then the q at a given temperature for a given time t1 you integrate this and if you integrate this this is very simple t to the power minus half plus half upon half so that gets 2 and 0 d1 d1 t1 upon root pi so now I know if I do a error function profile that means I do impurities entering for a given time t1 at temperature t1 then so many impurities per unit area can be deposited close to the surface of course there is a depth there this is not this is the total area under the curve is what we have received okay now this fact that this charge is something to do with t1 so larger the time I do impurities will get inside more and more so the area under the curve will keep on increasing higher the temperature I do d1 will increase so again but both are under root functions so root functions so they do not go proportionately so if I just increase time root of that time only the corresponding charge density increases is that point here this is relevant because suddenly you made 10 minutes to 20 minutes do not say it will double it will not double it is a root 2 times it will go okay so this fact is also used in our analysis or in our actual process okay if I double it how much I will get okay so I know roughly how much I should be able to get okay so this is a very important term which I will use it again this I will use it in actual junction formations please remember the charge density in a complementary error function profile let us look at a complementary error function profile this is n0 this is nxt and this is at a given time t1 okay now if I do much smaller time say t1 I can make even shallower okay so if I do smaller time I can make shallower or I can also make both small temperature higher but this small so I sorry it is not coming well so I can also make slightly higher n0 and sharply falling because time cycle is even reduced in both cases I may maintain Q okay I increase the temperature and decrease time proportionately so the dT does not change so I may have same charges but I may have no closer to sheet charge is that clear so to create a sheet charge is much easier if I do my first diffusion as complementary error function use this as the available charge density in the case of available density in the case of Gaussian there that Q term was appearing is that clear in Gaussian profile this Q you can use now is that clear to you this Q we can use there because first profile we did with complementary and then start actually hitting cycles without source okay so that will become Gaussian profiles with initial charge of this much okay so this is the trick we will come back to this little later so point I was trying I know how many impurities per unit area are just at the close to the surface after my so called error function diffusions okay the given time and temperature of my choice I can decide what profile I will get and therefore what Q I will get I can always evaluate this all this analysis has been done for what purpose we want to see junctions or devices made the first important thing for us is the junction formation for which all this diffusion process has been introduced you can have a starting substrate either N kind or P kind okay and we call the concentration as the base concentration or background concentration either it is called base concentrations or it is also called background concentration so a wafer can be which is a background to you can be either N type or P type. All that I will do is if it is a N type wafer to make a junction P N junction I must put impurities of P type if it is a sub background is P type you will do N type impurity diffusion to make P N junction is that point clear if I do P N it is a P N junction if I do N and P it is a N P junction but it is still a diode okay now or maybe you can see from here separately shown if it is N wafer I have a P area on the top if I have P wafer so impurities do go inside of the other kind and they form a junction okay we will see how that junction so if you see a profile normally N B will be opposite that of the impurities you are introduced to make a junction so if this is one may say plus and minus sign so if this N is positive the background is the opposite minus of that you can also show this way which is how most books show so if I have a background concentration of minus in B or whatever it is and I introduce impurities inside and they start getting inside the wafer for a given temperature cycle okay for at a given given temperatures and given time so D1 D1 something I fixed so the impurity profile is something like this as the impurities start from the surface their concentration starts reducing let us say the background concentration was some number which is smaller than this surface concentration we start with so as this concentration start decreasing at certain point the net acceptor impurities are equal to net donor impurities assuming all are ionized which may or may not be true every time or electrons and whole concentrations are equal or they neutralize recombine themselves is called compensations is that called is called compensation so a P type impurity will compensate the N type impurity and when that compensation becomes zero or concentration both sides are we call that say junction at the at so at junction what is the net concentration 0 at junction the net concentration N minus P or NC this profile minus in B is always so for a junction NXT minus NB at junction or NXT equal to NB is that clear alternatively saying whenever the profile concentration goes to the background concentration junction actually is formed at that point you can see I intentionally made this at this point whenever they will become this the actual concentration will be 0 and we say there is a junction sitting there now can you tell me how do I find an similarly many books are many simplest way don't look at for the signs just put NB on the same scale as that of NX and wherever this NX profile intersects with NB that is your junction is that clear whenever NX profile in actually intersects with NB you say you have a junction in reality this is happening but you can also represent in this fashion it is equally because you are anyway not showing you what is internal to device what we are saying calculations so for this this method is also equally good whatever I said I may repeat again by writing if you have drawn already please may I NXT profile intersecting with the background concentration you have the junction and XJ word was used sorry I forgot that this distance from X is equal to 0 or this distance from X is equal to 0 is called junction depth so in this case what is junction depth whatever the impurities p type impurities or entire impurity touches the background at that point is the junction depth so if I want to make larger junction depth what should I do I should put more time for impurities to get in so that the junction depth is larger or if you want smaller I should not put lot of temperature time cycles so that junction depths are smaller in the case of contact resistance maybe we will come back to that figure so this is what one of the values for a junction I want to calculate is XJ and what is the N0 we actually know in complementary function in the Gaussian what will happen this value will be NS N0 upon pi whatever function I wrote that will be the N0 for the case of Gaussian but just a minute that figure which I had drawn for you you can see from here the junction depth in a mass transistor for source and drain is how much impurities were coming from top source and going down so this is my XJ and if I calculate this is that three-dimensional area actually if you see it so if I see a semiconductor bar there of N plus diffusion N plus area larger the XJ larger is the cross sectional area area is this so larger the junction depth resistance is smaller but when I scale the device now I am looking for 5 micron down to 14 nanometer 11 nanometer 10 nanometers I do not know 0 nanometers this source drain areas will become smaller and smaller so what increases resistance contact resistance is increasing because area is decreasing the resistance of source drain increasing so what is the implication the actual speed goes down all the currents are your voltage scaling down so you are not allowing larger current to flow because of excess voltage you are reducing voltage you are decreasing junction depth scale down so one of the major hits is speed okay major hits is speed of course how do we recover from other day but this is one possible so scaling looks to be very good in every sense everyone said it here is a problem the first time we hit is the source drain resistance and their contact resistance should also we will also start increasing as scaling technology goes down as 19 nanometer to safer 14 16 will be even difficult and 9 11 7 may be even more difficult but I told you once which is my friend statement we all understand physics or at least think we understand physics but silicon does not so it behaves as it wants and we keep we are left to think why it is thinking that way okay that is the fun part in that okay let us take an example of diffusion of N type impurities P type substrate N type impurities N type P electron concentration increases near the surface and reduces as profile shows at X is equal to X J N type dopants is equal to P type doping and this is called compensation if they do not compensate there is no junction okay if they do not compensate there is no point anywhere this will that will happen P and N will actually merge everywhere and there will be uniform N I but it does compensate and therefore you do get junctions now here is we already said if I do pre constant source diffusion this is the profile D T 1 is D 1 T is equal to T 1 is time at the fusion and the impurities which I pushed at the surface because the depth is very small let us say 2 N O D 1 T 1 by Pi okay please remember once again at junction the net concentration is 0 net means background the doping profile minus background is always 0 at junction that is what the why the definition is of the junction okay so someone said I do I write again now so whatever I said please note down this is relevant because I am going to create Q out of error function profiles okay so again I said at junction net concentration is 0 hence we say incoming impurity concentrations is exactly same as N B and therefore we say N 0 error from complemented function X J upon root D 1 T 1 must be equal to N B at what point X is equal to X J now this is known function N B is known to me what is it what is known to me N B is known to me starting substrate N 0 is known to me why I fix the temperature so D 1 D 1 is known to me is that clear everything is known what is not known is X J so if I expand this X J is 2 root D 1 T 1 complementary error function inverse N B by N 0 okay X J is 2 root D 1 T 1 complementary function inverse of N B by N 0 so given so since error function is a function I need the value which I will get it from its table and I will show you soon what those tables are okay so knowing N B knowing N 0 because I know T 1 I know D 1 T 1 so I can evaluate the junction depth and I can therefore you can now quickly see larger the temperature this term will increase this is complementary 1 minus something so even if this increases this complementary will also increase please remember complementary is 1 minus something it did not increases 1 minus that term actually enhances actually so X J is larger larger the time you diffuse or larger the at larger temperature you diffuse this is how controls are there but this is only one control that is I have one fixed temperature and one fixed time but I want little more number of variables for me okay maybe one example before we start the other part let us take I am doing a diffusion of phosphorus in a P type wafer which is dope to 5 into 10 to power 15 per CC these numbers are typical numbers not necessarily for every device most devices mass transistors starts with 5 into 10 to power 15 when we are working below above the 100 nanometers now this concentration may be 10 to power 18 of 2 into power 18 as the scaling down is happening why you are actually increasing has anyone taught you this professor was he must have given you some idea as you scale down to keep that scale factor everything has to be readjusted by scale factors so concentration in the substrate has to be increased to get VT of what you are looking for. So initially let us give some example 5 into power 15 per CC substrate I am diffusing the impurities of phosphorus at 1000 degree and let us say I do 60 minutes of diffusion I do 60 minutes of diffusion please take it diffusion coefficient is centimeter square per second so time must be declared in seconds so it is 3600 seconds so please take in when you do calculation do not make mistake do not put 60 put 3600 60 times things may change of course it is mostly under root so at least 8 times or 7 and R 8 times it will vary so which may be a large variation in actual values. Now we can I will show you graph soon so at T is equal to T1 and 0 is 3 into 10 to power 20 for phosphorus this is data is available to us at T is equal to T1 which is 1000 degree and 0 is 3 into 20 per CC for the phosphorus at that temperature of 1000 degree the diffusion coefficient of phosphorus in silicon is 3 into 10 to power minus 14 centimeter square per second this data is made available to you either by me or by book or by the modeling in the software. Since I know D1 I know T1 I can calculate please note down this is the problem I am solving I want to know Hj for this case I have a substrate concentration 500 power 15 at 1000 degree for 60 minutes I introduce n type phosphorus impurities and then try to see what is the junction depth up to which phosphorus has gone in okay to form a junction that is my ultimate aim so I chose these values of randomly you can in real life I may give some other values and you will have to solve for that okay is that okay so I calculate you know I since I solve myself I write all steps you need not okay right now you may yourself do anyway that is much easier I normally do not use calculators so I just do some quick analysis hopefully they are okay maybe you know most times I can assure you my second decibel may not be same but otherwise I am very accurate in numerical calculations otherwise okay of course here it was so trivial but otherwise also so I get error function inverse NB by N0 is this so which is 1.66 into 10 to power minus 5 so we recall back complementary error function X is 1 minus error function X is that clear okay let us define error function X let us say this is X so what will happen complementary error function X is this value if I define complementary error function inverse 1.6 into 10 to power power is X then I say complementary function of X is this value is that okay yes inverse okay so I bring inverse on this side so error function complementary function X is essentially 1.6 into 10 to power minus 5 therefore error function X is 1 minus 1.6 okay now from where this I will show you 0.99984 this is the value after subtract please remember this minus 5 should not be thought smaller and neglected this number is very crucial 0.99984 by subtraction therefore X is now this is error function X so therefore X is error function inverse is that correct I want X to be multiplied with this for that term so X I am now say is how much the reason why I did this because the table which I have available is only error function table for a given X the value of error function X is known to me so what do I do how do I get X then I start looking from the error function table this value 0.99999 and to the left of it what is the X value I actually find out from there is that point clear so I have a table which he says X and error function X so I have so many X point 0.1 2 3 4 10 4 up at least we will give and these are so I says this value somewhere here and corresponding to this I get value from the table which is my X once I know my X I know the pre-exponent is 2 NO 1 by D 1 T 1 that multiplied by this term I will get junction depth so X j is root 2 2 2 D 1 T 1 minus into 3.06 X j is multiply of this which is 0.636 microns so if I have a substrate of 500 power 15 per cc of boron or boron doped p type and I introduce phosphorus impurities at 1000 degree centigrade for 60 minutes the impurities will go as deep as 0.6 microns or 0.63 microns or 0.64 micro whichever number you feel correct this value is some way measure how deep impurities have gone in please remember this earlier we used to say less than a micron is a shallow junction okay now 0.6 is the deep junction the shallow junctions will be of the order of 100 nanometers okay so it is something numbers have changed so earlier books myself would keep saying oh this is very shallow actually this is extremely deep junction okay compared to the earlier times to this scale down technologies all terminologies also have changed the micro word has suddenly become nano okay if you really see the nano technology only one dimension is going into nano okay if I reduce width and lengths too small length I mean but width I small then you take from me there is no current okay W by L I is proportional W by L so if I do not put large widths no currents possible so either it is a very small it is all that we are done is since it did not fit to the actual values so this way polynomially a 0 plus a 1 x plus a 2 x square plus a 3 a 99 x to the power 99 okay in practice therefore as I already now said that I can create by a complementary function fixed impurities surface charge Q and that step so we do two step process for any diffusion so one is called pre deposition one is called pre deposition and the other is drive in now there is joke many years ago some students kept on telling me driving I say but still he kept on saying the word driving so I said no it is drive in but that is what I am saying driving okay yum is yum and yum so if you ask any Tamilian do regards if say m say say yum I say it is m yeah that is yum yum so that is the way life is interesting part so if I do pre depositions basically pre deposition is a complementary error function diffusion with constant source or infinite source or whatever you want to use I do it at temperature T1 and time T1 so I know D1 I know NO1 I know therefore NO1 times complementary function X upon root D1 this is the profile I got and for this I have calculated the dose which is to NO1 why I put one because this is first step so I said one next maybe two okay NO2 because Gaussian will have another value so here I call it NO1 D1 T1 by pi under root is essentially the charge per unit area at the surface okay so the first step in diffusion is creating the sheet equivalence of sheet charge which is called pre depositions okay so I fix those this is all that we did just now I did I am just rewriting that I just said at a given pre repulsion is essentially a constant source diffusion for a given time T1 at temperature T1 for which NO is known this and therefore I know the surface concentration after the pre repulsion step the second step is since the source of impurities are now stopped so what I do it I put a refer inside furnace will show the technology and for a while I allow impurities to get in okay for a given temperature and time cycle is that clear once I have that impurity deposited I will take the refer out and we will see that it has the glass has to be etched and then that means certain amount of impurities on the silicon have been deposited by pre deposition step okay take the refer out clean it again and that process safe will come back and then put again in the furnace at any other temperature it can be same or any other temperature let us say T2 which for which D2 is known as such we know that so we do this second cycle is called drive in why word was used in your certain amount of impurities which you are driving drive in inside they are pushed in okay so they are driven inside the wafer and therefore the cycle was called drive in cycle okay this can be a time T2 at temperature capital T2 please remember you can still have T1 D1 T1 T1 same as this but that is not done we will show you why okay so we have a temperature T2 D2 is the diffusion coefficient DT is D2 T2 in driving no no new impurities are introduced whatever earlier only will keep inside go inside as time elapses but available once pre depression are redistributed this is the case of limited source diffusion and therefore we write the new profile now what was the profile Gaussian profile q upon root pi DT exponential minus x square upon 4 DT that was the Gaussian profile in this case what is the time temperature cycle we have D2 T2 for the drive in cycle D2 T2 however this q is coming from where from the first pre depression cycle which is essentially what we just now derived which was the number which we get 2 NO 1 root D1 T1 by pi so if I substitute this q here so I get 2 NO 1 by pi under root of D1 T1 by D2 T2 exponential minus x square upon 4 D2 T2 so how many variables now I have I can still get some xj by this process I have second temperature I have second time I have first temperature my first time so now I have two temperatures and two time cycles to adjust my xj and what else I have to adjust this surface concentration so two are not identical and therefore I can adjust please the point did you get a did you get my point I want larger surface concentrations and I want a given junction depth okay if there are too much junction depth will be larger but surface will go down so adjust your D1 T1 such that literally some number is available at the surface drive D2 T2 in such a way junction depth is achieved by you is that clear so now you have two temperature cycles at two different times and therefore much more control and final junction depth much easier compared to only one of them is that clear only one of them so this all diffusion cycles are always two state diffusion and here is a of course I am going to scan this some sheets now soon and put it on the this or maybe one or two I will give it to your class representative you take your own Xerox there are too many process graphs which you will use in your exams okay this is a graph okay which shows diffusion coefficient for all impurities possible in silicon and even in partially in some gallium arsenide by gallium phosphide devices please remember this is plotted again 1000 by T T is in Kelvin's is that clear so T has to be say 1000 degree means 1273 degree Kelvin so take a ratio of 1000 by 1273 to get this number 0.6 0.7 0.8 kind of thing then whichever impurity you are looking let us say phosphorus boron and phosphorus are same diffusion coefficient so let us say it is 0.7 so you go on this graph okay and somewhere here it hits and measure the surface our diffusion coefficient for this now one feature which I do not want to say it but over the years I am perturbed and maybe I hope this year it I will not be you should learn to read graphs log graphs okay which is very funny to say 30% of the students cannot read the log graphs correctly so most of the time their values goes by order absurd okay and because that order is changing so many things change okay now this fact has to be understood that this is increasing function of scale minus 14 10 to the power minus 14 minus 13 minus 12 11 10 the first one please remember this is 0.2 0.3 0.4 0.5 is very close more than half because say log scale 0.6 0.7 0.8 0.9 this one thing is this let us say your x by I mean 1000 by T becomes more than 0.8 you are doing temperature which is more than such that it becomes so this graph has to be extrapolated down is that clear extrapolated down like this okay however and this scale the values are 10 to power minus 15 into this and not minus 14 into this I repeat if you have gone down the next scale here will be minus 15 into 10 will make it 10 to power minus 14 so this value down here essentially is order of 10 to power minus 15 into 2 or 4 or 8 whatever the if the scale is here it may be 8 here it may be 6 4 2 into 10 to power minus 15 so as soon as you extrapolate roughly calculate from the scale where you are this is a log scale this is 0.9 0.8 point this is 0.2 this is roughly 0.5 so you roughly see from your extrapolation how much distance in this scale and this and evaluate the new diffusion coefficient in case for a case I give which I might not have thought properly and you suddenly find 1000 by T is more than 0.8 8 to or 8 then you will have to extrapolate and get correct value of diffusion coefficients is that clear so reading a log graph similarly between these to this value this is 2 into minus 13 4 5 into minus 13 10 into minus 13 is 10 to power minus 12 so it is increasing function in a log and therefore accurately monitored from the graph values this does not happen root DT product goes wrong then the junction depth goes wrong and then say sir we did everything right but graph read not right which is even worse okay. Okay so this is the first curve I will show you the second is this solubility curve second is the solubility curve you can see this dotted ones are what are those dotted ones activated ones these are the atomic ones so if you want to calculate some numbers you should use activated ones okay then real life atoms may go more but activation may be smaller so use solid solubility this is temperature against solid solubility and you should be this is already written solubility limits and this is electrical active so use solubility limit and zero calculations should be performed from these graphs okay so if you are doing pre-deposition at 1000 degree for go on so go it from 1000 to somewhere here and this is 2 into 10 to power 20 is the concentration for N0 at 1000 degree centigrade is that clear so any diffusivity you are calculating first take diffusion constant from the earlier graph take solubility limit from this graph so these values will not be specified as I just now give you values they will be actually given temperature this and this sheets will be provided is that okay this is second graph which I will provide you of course this is slightly different we will come back to it this graph because we are not done oxidation so far during oxidation what happens to diffusion okay so there is an issue there so these are different graphs will show you okay then there is also a case that impurity concentrations which is essentially is how bad they are how far away from intensive carrier concentration this is very important if n is equal to ni so is p will be equal to ni so what does that mean that the material is not doped is that clear to you if n or p is equal to ni it is intrinsic so there is no junction possible okay if everywhere ni nothing is possible so at a given temperature this what is the ni value is called ni as such that you must know so you cannot have impurities which are lower than that value at that temperature otherwise they will not actually diffuse so you have to verify of course that you do not write now but maybe I will show you how it hurts you many times there is some term called field dependent diffusion and that helps now these are the two profiles Gaussian and error function the lower one is complementary error function and the upper one is assuming right now same surface concentrations this is Gaussian profile okay is that okay so this error function profile and complementary function profiles looks similar but slightly different because complementary function never changes its shape it may keep on at different time temperature it may increase further and further but the profile will remain in case of Gaussian it starts flattening in complementary this does never this will never change in the Gaussian because net impurities are fixed so it will go down and down and down okay surface constant keep on reducing okay and depth will be larger and larger okay so this is the second this graph of course next time I will show you this more importantly there is a graph between surface concentration versus sheet resistance now we have not done it but will what I can monitor is sheet resistance and junction depth monitor and also in the problem what I say I want a junction at 0.6 micron I gave you value so X j will be given to you r s you can evaluate or I will provide I will show you how to do that so given r s for given substrate concentration that is surface concentration you can actually directly monitor how much is this is that clear because these are all temperature time cycle r s X j products okay so given the r s X j product I will be able to know for a given base concentration varies because X j mean junction is fixed so what is the surface concentration can be directly rate from here okay you can calculate every time but this graphs allow you to read this so what I can say this is the surface concentration and this is the sheet resistance monitored so essentially I am saying junction depth is given to you okay either way the problems which I ask are something like this is called convoluted situations something is given in some other format but it is available okay so you have to look into oh this I can get from this graph okay so I know of this value is that okay similarly this has been done for both what I should say p type impurity n type comp four graphs are there Gaussian n type p type and both error function graphs are available of course then there is oxidation graph we will see later then there will be oxidation for 100 a for 111 then there are graphs for implants this is called predicted range versus energy then there is a graph for struggle versus energy struggle is the variation we will see that delta struggle versus energy for all impurities of choice then there will be transfer struggle lateral struggles is this this is transfer struggle and this is most important this you will know before we quit okay this is a table for error function just just look at it please because this is something which we just now did so I would like to say it the table which is given to you I do not know whether it is so very visible it is a Z versus error functions there so Z is 0 and the table which I have used is around 4 3.99 which is the maximum value you will get any day no more than this okay now you can see from here at 0 of course error function 0 is 0 so that it starts with 0 so 0.1 it is 0.112 but if you start looking for 1. something 1.17 onwards it is 0.9 and as you increase this Z it becomes 0.99 larger 99 means higher Z value so for example error function of 3.99 is 69 or rather 7983 is that word clear 7983 9999999983 okay 7983 so what does that mean so if I am just now I calculated so the profile is normally complementary error function so what do I do I take 1 minus this that is x u anyway 1 so okay I put 1 minus 1 minus this value I subtract I have 1 minus 0.0 whatever value I subtract that will give me 0.999 some value okay 1 minus that you got look for this 0.99 in this graph wherever it is in this figure okay let us say I get 0.49 so start looking for 49 first okay here is 49 so okay from here so 2.27 is the smallest value for 49 values okay so once you start getting that value from there come on this table figure out what is the value of error function Z 0.999 correspondingly you get the Z value so you will have to go back and forth many times just to get exact Z value not exactly do not interpolate where exponential function is not linear I mean error functions are not linear therefore do not extrapolate either take the lower one or upper one and say I took this it is your choice okay in between 2.325 do not try any values okay you will have to really do a calculations or you can do on computer but do not try anything on this the value is nearest to what value you have choose from there the reason is all these calculations are only calculations in real life silicon will behave differently so why are so worried about okay okay so these data sheets as I called will be provided on system by the way this website is on there are 2 handouts have been put there 2 PPTs have been put there so please look at it and this will be also available to you download yourself one puppy maybe I will give it to your this maybe you can use that for Xeroxing but as your choice.