9. Dopant Diffusion - Numerical Techniques in Diffusion, E Field Effects

Key Takeaways

all right we're beginning the uh lecture on handout number 15 this is um the second lecture on uh dopent diffusion and profile measurement and we're uh working on chapter seven of your text so last time we introduced some of the basics of dop and diffusion we talked about uh basic definition of sheet resistance and the scaling requires PR L and we pointed out there's a conflict between the need for smaller serious resistance and the need to decrease function depths to improve control on the channel charge we also talked about short Channel effects and how BT decreases as L uh goes down we mentioned the use of predeposition uh either in the gas phase or byon implantation and then the use of subsequent Drive-In of dopents to create doped regions we spoke about macroscopic diffusion from the point of fixed first and second laws and gave a few cases there were only about three or four cases where there are simp analytic solutions to the diffusion equation today I want to just briefly review those analytic solution examples and then um talk about uh a brief introduction to numerical solutions to diffusion equation then we'll talk about uh design of diffused layers based upon certain device requirements such as the sheet resistance and uh our first deviation from fii and diffusion we'll we'll discuss electric field effects on dop and diffusion so let's go on on to slide number two and just review from last time uh this pictorial um view of fixed Second Law what I'm showing here is a volume element um that is uh shown by the blue sort of rectangular uh element and there is a flux F Sub in going in the left hand face and a flux F sub out uh going out in the right hand face remember again our basic definition of flux is it's a number of atoms or particles per unit time per unit area so looking at this uh in this Cube we can uh calculate the net flow of atoms into this volume uh into that small volume as shown in the upper left uh corner is equal to the area a of one of those faces times the difference between F out minus FN so that's the net flow of atoms and we can write that as a times and just looking at uh sort of the geometry it's Delta x times the the uh change in concentration Delta C divided Delta time uh delta T so we're can just rearrange just a little bit the A's go out and what we see is that the first derivative of the flux Delta F over Delta X is equal to the time change Delta C over delta T um so mathematically if we want to write this uh not in terms of Deltas or differences but in terms of um differentials what we see is a Time rate of change of the of the concentration in the Box partial C partial T is equal to the first derivative of the flux partial partial X and then we can substitute in the equation on the center of the page for the flux we know from fixed first law that f is equal to D partial C partial X so that is indeed um a way of deriving uh just sort of bookkeep of deriving fixed Second Law now uh the equation in the center of the page can be simplified somewhat if uh we have a special case where the diffusivity is a constant where it doesn't vary in space I can pull the D out of the partial deriva with respect to X and we just get um partial C partial T so the time rate of change uh of the concentration is equal to a constant diffusivity it's a number times the second derivative of the concentration with respect to X so that is uh at the bottom of the page would be then the special case of fixed second law when the diffusivity is a constant so let's go on to slide number three uh and these are um a couple of cases we solved last time where we we wrote down the solutions last time for two cases which are commonly encountered and which for which analytic Solutions are available on the left hand side and reviewing the case of a predeposition and again by sort of by definition in this case we are assuming that there is a supply of do of of dopent or atoms that are diffusing that holds the surface concentration at a constant value C subs and number is fixed and we said in this case that the shape of the profile c as a function of depth X and time is given by this surface concentration C Subs times the complimentary error function of X over 2un DT so that is the actual analytic solution to fix second law for a constant diffusivity and uh we also know that the integral of that uh that function uh integrated over over space or throughout the the Silicon uh has a value Q that's the do so the area under the curve and that can be solved for and it turns out to be 2 CS overun of Pi * DT um so this is a particular case and we in fact showing a plot um on the vertical axis of C over c s as a function of depth for uh several different uh times there are three curves here shown where the square of DT or two times the square of DT is being varied from 0.1 Micron all the way up to one micron you get a feel for how the shape of the comp complimentary airor function evolves with time um in contrast on the right hand side we're showing the case of a drive-in that would typically be done after prep and driving takes place uh usually under the assumption of a constant dose Q so the con area under the curve and what's happening as a function of time then um is the the shape of the curve is changing and it it's um we know the solution is a gausin uh the solution to fix Second Law um and in fact if you look at this equation as a function of time you'll see that the the concentration of the surface is dropping as a function of time according to 1 over the of DT and uh the P profile is it's broadening it's width or standard deviation is in inreasing according to DT and U there are a couple different cases shown on the light um so these Solutions are valid under this particular case where the concentration of a diffusing dopen is less than inabi and that that implies uh in that case that the diffusivity is a constant it doesn't vary in space we'll talk in subsequent lectures about the case when that uh the concentration is quite high or higher than nly so let's go on to slide number four and uh this reviews from last time what we showed is an arous plot of the intrinsic diffusion coefficients of some common delins and silicon and what we're showing on the left axis uh on a log scale is a diffusivity in units of C centimet squar per second and the x-axis is uh 1,000 over temperature in Kelvin the lower x-axis the upper x-axis actually has the temperatures uh uh actually indicated so um we are intrinsic diffusivities so they apply when in the carrier concent the um concentration of the dope and is less than ni uh at the diffusion temperature one thing you can notice right off the bat is there there are some fast diffusers in Silicon such as Bond fos vering Indian they the upper three curves and the slower diffusers are Arsenic and anamor silic uh it turns out just for practical reasons because of their High electrical solubility that Boron is often the um ptype doener Choice it pretty much is and arsenic uh in N type dopers you can choose arsenic or phosis but arsenic is usually chosen uh because of its higher electrical solubility and uh also sometimes it's lower diffusion coefficient it's easier to control profiles so what typically we're going to have issues with controlling diffusion in the pp regions because Bond uh is diffusing at any given temperature almost in order of magnitude uh higher diffusivity than the case of arsenic all right let's go on to slide number five and um here what we are trying to talk about is uh the effect of successive diffusions in say a moss process flow um uh on the on the doping profiles you remember from the first lecture of this course we talked uh we kind of walked through uh an example of NR circuit bation process and you know that they involve a large number of different steps at various temperatures we don't just simply do a pre-ip and then a drive-in and then it's done that could be oxidations it could be uh other uh high temperature steps in which uh the doens are going to move so if you have a whole series of steps what's the question is what is the uh actual final profile and how can we approximated well it turns out in the case of gine diffusion that uh we can write the total effective D2 product uh as being a measure of a thermal budget that's used in that process and for ging diffusion we can actually sum up the products of the diffusivity in any given time step times the length of that time step D * T we can sum up those DT products add them up and uh end up with an effective DT for the entire process again assuming that in each case the the diffusion satisfies the the gaussing assumptions what we'll typically find as you go through a whole process is that the some of the processing steps may be negligible and that's because the diffusion coefficient in this formula is exponentially activated uh or it varies with temperature exponential so the highest temperature steps in the process typically dominate um it's not always the case for example when we talk about trans enhanced diffusion uh we see that's not the case but for normal diffusion you expect the highest temperature steps to dominate this equation and so you can usually zoom in on those as being the ones that that primarily determine the DT product and therefore the shape of the profile one thing you have we have to be aware of is that this uh gaussian solution we've been talking about only holds if the Delta function approximation for the initial profile is reasonable we talked about this last time so it holds in the case where the width of the profile um initially when you start is small compared to the the final width or compared to the total um DT if that uh assumption is is not valid then and we don't have a simple uh uh ging solution necessarily okay let's go on to slide number six and uh here we're going to go through um some principles and examples of the design of diffuse layers so what we want to do is as as an engineer uh process engineer or working with Device engineers we want to uh typically design a diffusion process um or a prep in diffusion in a driving process so that we end up with a certain sheet resistance that's required electrically um by the uh by the device so that's that's sort of the uh the goal of uh of this section and I'm showing on the le- hand side just the example of um a cube of silicon where the top region is colored in Orange and that is the little sheet uh through which we're going to be passing uh a current and measuring the voltage and uh the uh this little uh she has a particular geometry what we're going to consider is the case where the uh length of this resistor um is actually equal to its its width so it's uh it's a square and it has Dimension W by W um and so we're going to define the sheet resistance uh of this function last time we talked about it um as being the resistivity which is an intrinsic property of the material depending on how it's doped in BRS divided by XJ The Junction depth you can see XJ and Mark on the left left hand side and the units of that sheet resistance then are um given it to be ohms per square so so the sheet resistance is indeed the resistance r that you would get in a resistor made of one square uh of this material and we you can easily write it as uh Roman R subs or uh row Greek the Greek letter row SS and uh uh the S indicating is for Sheet the resistivity row remember is given by one over Q electronic charge times the car concentration n times Mobility which is in general a function of N and the resistivity has units of on centimeter XJ has units of centimeter they cancel out with a sheet resistance in ohms per square um that's all reasonably uh intuitive in assuming you have a constantly doped a constant doped um layer for if a layer is non-uniformly doped you need to integrate uh the uh the doping uh profile times the the mobility in order to calculate the sheet resistance so the equation shown in near the bottom of the slide say sheet resistance uh can be obtained by one over the integral Q * the integral from zero to the junction depth of a doping of the car concentration profile n of X or the doping profile minus NB or NB is the background concentration that tells you where uh when you cut off the electron concentration that quantity uh times the mobility which is in general function of of of n uh DX integrated um so that's what you would do in the case of a non-uniformly doped layer with the dope and concentrations varying in thickness uh now it turns out this equation uh you can solve it yourself but this equation has already been numerically integrated uh by Irwin for um a couple of different analytical profiles in particular the case of the ging and the complimentary error function profiles and those uh the solutions to those integrals are given in your text uh either in chapter 7even or there's also an appendix where the solutions are given so let's go on to uh slide number seven and at the top of the slide we're showing an example of our wind curves for the particular case of the pype uh Gan profiles in an end type background and what you see uh in these curves on the uh Y axis is a Surface concentration so that would be C subs and adoms P centimeter the xaxis now is effective conductivity and it has units of uh ohm centimeter the minus one um so that's the FX axis if you just go to the bottom of the slide uh thinking about it that is the effective average conductivity the inverse of the average resistivity so we can write this as a sigma bar and it's equal to a one over the product of row sub s times uh XJ so it's one over the sheet the product the sheet resistance times the junction deck um so there are kind of three key parameters we need to identify and using these curves there's a surface concentration which is the y- axis uh in the profile there is again this is for a particular type this for a gum profile there is the sheet resistance um which is uh coming into this effective conductivity that's really part of the x-axis and there's the junction depth so the x-axis has sheet resistance and Junction depth um built into it the uh and now the metallurgical Junction is the place the point in depth where the Cal concentration in of the uh in the diffused region of the diffused dopin equals the background concentration CB and we'll do an example shortly after this um where you can see how these uh curves are used but each curve different color is for a different background uh concentration uh so this basically tells you that you can take these three key parameters and uh uh they have a unique relationship among themselves for the particular case uh where the shape of the profile is known in this case it's gum so in fact let's go on to slide number eight we're going to walk through the best way to uh understand these ear WIS curves is go do an example and what we're asked to do in this example is to calculate the drive-in conditions which means the temperature uh and the time to produce a seamoss uh P well so it's a p type a region it's called a p well of a certain thickness and and um uh what we want to do we're told that the constraint is the KET resistance of that P well shouldn't be 900 ohms per square okay and it has to have a certain Junction depth which is shown schematically in the figure XJ to be three microns uh and that XJ again is the point where the brwn concentration then will reach the background concentration which is um uh given by uh 10 15 cubic cm here so again we're informing a PO by ion implanting Bond um into this uh into the region onto the wafer and the region on the left is blocked so we're just being in one region and typically you would implant um Boron at a certain dose say 10 of the 13th and energy and then we want to drive that in and we're interested in the driving conditions what temperature in time will we need to get it to go that far and to have a particular um sheet resistance uh and we're going to use Irwin's curves to do your so let's look on slide number uh nine um so these are the constraints again repeated the sheet resistance is 900 ohms per square The Junction depth has to be three microns and the background concentration either NB or uh substrate concentration maybe WR NBC is 10 15th so given the requirements that we have been uh told I can actually immediately find the x-axis on the urban curve with the average connectivity layer it's just one over the shapeless distance times XJ so the average connectivity calculates out to the 3.7 ohm cim minus one um we can now from Irwin's curve we can obtain assuming this the galy uh that's assumption we we need to make at this point which we'll show is reasonable later we can then obtain the surface concentration so if in order for me knowing the shape of the Gan basically IR intervals tells us that the service concentration has to be about 4 * 10 17th per cubic cm in order to achieve an average conductivity that has been specified of 3.7 um uh assuming the junction depth is three microns so that's and if you can you want to see that how we actually do that you can go on to slide number 10 and this actually shows how the curve we used um again this is for a ptype gussian profile and all we've done is is uh on the x-axis picked out it's a little hard because this is a log long plot on the x-axis picked out the average conductivity to be 3.7 and it's a little tricky to find that but uh you can and then you see you read off the y- AIS as shown by that uh horizontal line surface concentration about four * 10 the 17th and again we have multiple curves we can choose I choose to use the orange curve uh which is for the case of a background concentration in this example of 10 the 15th uh if the background concentration say had been uh a little bit higher say 10 the 16th or um 10 the 17th what you see would have happened um is that the uh the surface concentration would have actually moved down a little bit uh on the y- AIS it would have be intersect a slightly different curve okay so that's how we come up with the um the surface concentration so let's go on to uh slide 11 and thinking about that surface concentration C Subs we know it's well below the solent solubility of B and silicon it's only 410 and the 17th and it's probably likely below the value of n subi at the diffusion temperature again at a th000 degrees which is probably somewhere near or a minimum temperature you'll be using to drive this in Andi is about mid 10 to the 18th and so mid10 to the 17th is well below that so it's it's probably valid then to say that the profile is galxy it's probably valid to say that uh it is a constant diffusivity d uh and therefore we can assume this a gan uh type of solution because it's a driving so we so we can write down sort of by inspection the equation in the center of the the slide which is the Gin equation um uh and then uh we know that uh at the point where xal XJ the uh car concentration or the doping concentration Cen of the Boron reaches that of the background that's a definitional exj so if you look in the middle of this slide we're saying uh the background open Con background concentration CBC is equal to the surface concentration C sub s which we just located which is fam times exponential of the minus XJ S 4 DT that's the definition of XJ so we know XJ in this equation we just found c s we know the background open concentration the only thing we don't know is DT so we can invert that equation and solve for DT plug in all the numbers and you get um a DT product uh in order to obtain this scalan of 3.7 * 10us 9 cm s that's the DT product the so-call if you would like to use the term thermal budget for the drive-in process um so that gives us an idea and we don't know what D is or what T is but we're going to design uh a certain temperature and time so we constrain to the fact that the D product will be this number so let's go on slide number uh number 12 and again I'm repeating the DT product from the top 3.9 * m minus 9 3.7 uh let's just assume a temperature and see what happen see what time is associated with that if it's a reasonable amount of time so say uh we know we're diffusing pretty deep down to three microns so let's just say we assume we drive in at 1100 cenade and then you can look up uh from your plots or you can calculate diffusivity of B 1100 it's about 1.5 * 10- 13 7 Square per second and so then you can solve for time the drive in time calculate that out you get about 6.7 hours it's plenty long that tells you probably from a practical point of view you would not want to do this driving any lower temperature because it would just take two them off okay so um now one thing we'd like to to calculate we we have um figured out the surface concentration 's curves and we have the DK product it will be interesting to calculate the initial dose or the dose the area and the curve for the Gan profile we have everything we need again we know the dose Q is the surface concentration C sub stimes theun of Pi DT and we have all those quantities we can plug them in and we get Q = to 4.3 * 10 13 that's the end on the gasi uh it's a relatively low dose compared to uh implant range so it it's um they can be implanted so this dose can easily be implanted into a very narrow layer close to the surface which justifies implicitly that narrow layer might only be a th ACS deep it justifies our implicit assumption in using a gin Pro profile that the initial distribution would be a Delta function maybe only again few th000 ANS deep and you're going to a final Junction depth that is uh three microns and um certainly um uh that that scene is Justified using the Gan profile okay let's go on to uh slide number 13 and now that that was the case we just talked aboutum we had implanted the dose let's just take the case where if we used a gas um phase uh uh prep uh step as an example to get the uh born initially uh into the wafer and say that was done at 950 you can then use the charts in your book or in the handouts to look up the Sol solubility of born at 950 and you'll see it's pretty high it's 2.51 20 at that temperature then we can also look up the deity it's about 41 Theus 15 so you can solve for the dose in the complimentary error function profile q and you have everything you need in that um equation 2 C Subs overun of Pi of KT and so the time that you were required to deposit that dose into the wafer T of the prep can be solved and you find about 5.5 seconds so you now see that the uh DT product of the predi which you can calculate it's 5.5 seconds times the diffusivity at 950 which we see from regardless 4 10us so that DT prep is about 2 * 10us 14th uh and comparing that to the DT of the drivein which was solved for earlier which is about 1 and a half time 10us 13 so you know order of magnitude uh difference there again this tells that the Assumption of gine diffusions is Justified whe even if we done a solid phase prep okay so that's uh the end of slide 13 that sort of gives you an idea of how these herban curves can be used to design a process the alternative you could use them is to uh to read off uh if you go back to just momentarily uh slide 10 you might the problem might be specified differently you might be given um you know uh some kind of surface concentration and uh background concentration have to back out um uh what the effective connectivity of the layer might be so the different ways that get in with curbs so let's go on now to um slide number 14 and I want to shift gears a little bit because um we have more or less covered the a few cases for which the um analytic Solutions can be obtain and those are important cases but primarily when the dope in concentration is low or uh satisfying certain special conditions like during preps it turn it'll turn out there's most cases we cannot solve analytically so these Solutions have to be obtained by using computers using numerical methods and um so I wanted to introduce you in the next few slides to uh some of the simplest tight CH numerical methods so you have an idea or a feel for how you yourself could Implement these numerical techniques if you need to do it and there are uh textbooks on this topic um in particular if you want to look at more detailed information on numerical Solutions of the Fusion equation there's a uh a book by Jay crank called the mathematics and diffusion has a chapter on numerical methods uh for this particular equation the nice thing about numerical methods they can be used to solve for arbitrary initial starting profiles so you don't have to make any detailed assumptions arbitrary boundary conditions and in the case where the diffusivity varies in space so if you have a concentration dependent diffusivity which will be the case uh we'll talk about in in later lectures when the uh car when the dopin concentration is quite high and you can use numerical methods so what I'm picturing here on slide number 14 is a very schematic picture of uh atoms moving around between planes in a lattice and so uh let's use this picture to get some physical insight into the diffusion process so each plane here is sort of indicated by a vertical bar and 0 1 2 3 4 and the distance between these planes is given by the distance Delta X as units of length and uh in subi here so n0 N1 and two is the planer Atomic density uh of the I plant so it's the number of atoms per square centimeter um in in a given plane and um cabi uh has volume concentration you it's it's the average volume concentration at any point so you can calculate you can get between the two c is just n subi over Delta X or n subi is just C subi * Delta X okay so let's go on to slide 15 or at the top I just reproduced that picture soly smaller version of it um and just reminding ourselves the the relationship between CI and ni now as we're looking at this lattice we know that the atoms are relatively fixed in the lattice but they do vibrate about their average position in a plane where average position and they're vibrating at some frequency V subd which is called the uh Debby frequency and that's that number is typically on the a of about 10 to 13th uh per second so it's pretty high frequency they're vibrating about their average blatus positions at a finite temperature uh now sometimes in during this vibration an atom will actually be able to surmount the energy barrier uh that exists in going from one last position to the next or one plane to the next so and it will actually be able to hop to an adjacent plane and there's an energy barrier which we'll call an Isa B uh to get from one plane to the next and uh this hopping frequency we call um BNB uh which is just a duy frequency VD that's the vibration frequency times a boltman factor exponential the minus uh e that's the energy barrier over KT so this V subb small V subb gives us an idea of the frequency of hopping from atoms from one plane to the next and what we assume in this derivation is that there's equal probability of jumping right or left because this is a random process um so if we look at any given plane we can say um the number of atoms jumping to the right of that plane per unit time is just VB over two times the number of atoms in that plane and half going go right number jumping left per unit time is just BB over two times the number in PL n so those two are equal um now if we do the bookkeeping and say look at the number of atoms crossing a plane uh the particular plane at I equals 2 in the above diagram so the plane labeled two uh number of atoms Crossing that plane per unit time well we can just uh write that as a flux and F an equation the bottom well that's just going to be uh the number of atoms um basically per unit time uh jumping to the right minus those uh jumping to the left uh so you write that as V minus VB over 2 N2 minus N1 again assuming equal probability going uh either way um so uh and then you can convert that the difference between uh Atomic uh density and uh numbered adoms per cubic centimeter just by multiplying using the definition of n is equal to Delta x * concentration you have then expression for this flux um Crossing plane IAL 2 and then you multiply and divide by Delta X and you end up with this equation that this flux is equal to looking the right hand side minus bb/ 2 Delta x^2 times the the uh Delta C over Delta X something that's starting to look like uh a slope so we immediately see some kind of fix law fix law here where a flux is related to a slope times something out in front that that is a constant so that's at the bottom of slide 15 let's go on to slide 16 and I've just repeated that equation that we just had but I added one more uh thing on the right uh one more equality and we set that flux equal to diffusivity a number D times the slope Delta C over Delta X where we Define the diusivity in this particular equation to be equal to uh the VB over 2 * Delta x^2 so there is fixed fixed first law the flux is equal to minus a diffusivity a constant number times Delta C over Delta X and the diffusivity has some atonic scale um mechanisms view to it in fact it's uh proportional to the jump frequency uh V subd uh and therefore proportional being related to the energy bar to hop between adjacent positions or adjacent clients uh so gives you some idea of what at the atomic scale what goes into to determining the diffusivity view okay in addition to giving us an atomic scale view we can actually use this type of formalism to go ahead and derive uh a numerical solution to uh fix equation so let's go on to to do that we'll go on to slide number 17 which um a slightly different sketch but the the exact same ideas of what we're going to be going through uh and what I'm showing uh on the left- hand side in the bottom is a plot of concentration as a function of X so distance and let's say has some shape this dark uh dark decreasing uh black line and it's decreasing uh from left to to right and what we're going to do is we're going to discretize this let me look at discrete uh positions in Space X and the concentration C 0 C1 and C2 and the spacing between those planes is going to be a constant just like we used before uh Delta X and so what we're going to look at uh is in the time interval delta T a certain time interval Delta 2 uh what is the number of atoms crossing a certain plane per unit area and we call that to the plane R which we had labeled R above Q subr it's just the flux F subr time delta T that's the number of atoms crossing the unit area in a given uh time interval that just basically is comes from the definition of what flux is and um we then can write in for the flux uh the flux is just the diffusivity of D times the concentration difference across that plane across plane R well the concentration difference across plane R is C1 minus C / Delta X so we can uh approximate that way the flux Crossing plane R what's the flux Crossing plane s well we do the exact same thing or the number of atoms Crossing plane s it's just the diffusivity times Delta P times this time we subtract C2 minus C1 over Delta X so we can then do some bookkeeping and calculate the knit number of atoms accumulated in that shaded region between plain R and plain s in the K in delta T it's just um uh QR minus Qs and we just subtract these two expressions and you see what you end up with is is equal to a diff time the time interval delta T times in parentheses the C - 2 C1 + C2 that whole same quantity over Delta X now if we divide that those that net gain in the number of atoms divided by Delta X is shown below that is the net gain in the concentration in that shaded region that's Delta C and that's just QR minus Qs that quantity divided by Delta x uh so I can uh substituting in QR minus Qs and you end up with the equation near the bottom which just says the net gain in the concentration Delta C in this shaded region is just diffusivity D time delta T times this quantity and parentheses which is related to the concentrations in those three regions divided by Delta x^2 so what it says is at a given point C1 in this Cur if I know the concentrations of the neighboring points so I know its neighbors I know the concentration at uh c0 and it's C2 you can calculate a new concentration after a certain time step delta T at C1 um just by knowing its neighbors um at this time step the concentration of its neighbors um so uh if we go on to slide number 18 we can then use that information to come up with something called an explicit finite difference formula that's what the equation being shown near the top of of the slide 18 uh what you're saying is that the concentration at any given point I in in one of these slabs at a new time t plus delta T so we're going to evolve this profile over time so C plus on the leftand side uh is equal to the concentration at time T so one interval below before that whatever it was before plus we add in this uh this difference this term on the right that gives a change in the concentration in the I slab after a Time step delta T and that term is just D the diffusivity delta T / Delta f^ 2times the quantity parentheses where again the quantity parentheses in order evaluated at any point C all I need to know is the concentration at that point and the concentration it's two nearest Neighbors in in whatever way that we've discretized we've discretized this continuous function into uh little intervals Delta X so that is a an explicit formula you can use to evolve a a concentration profile um in shape over time and if you look at it for a few minutes you can notice the similarity of that difference formula to the differential equation for diffusion which uh if you take the the partial uh partial G on this equation shown in the center and lo it over the right hand side you can see that partial C is equal to some constant D times a second derivative of concentration with respect to X and in fact this partial um second derivative of C with respect to X it can be numerically evaluated as the term in the first equation on on on slide 18 on the right hand side that is a numerical solution to that differential equation and um I won't derive it here if you go back to crank's book or any of the books on uh uh numerical methods and you find that uh the method is numerically stable so it works from a numerical point of view it gives reasonable answers for a particular condition we have when we're doing the solution we have a condition on a quantity R where this quantity R is is is what mult the plus uh the concentration change in a in in a slab at certain time step delta T that D delta T over Delta x^2 that multiplier has to be less than or equal to 1 12 if that multiplier gets too large the meth the whole method breaks down become unstable and you get nonsense Absolut um totally unmeaningful um Solutions so you need to keep need to adjust if you're doing this numerical make sure you adjust your time steps delta T to be small enough enough so that this holds um and uh your Delta X um to be be of appropriate size in delta T um so that you don't get yourself out of the um out of the physically um out of the numerically stable uh solution it's a very powerful technique it's very simple that's one of the nice things about it um and it's certainly going to be required to use some method numerical method uh when the diffusivity is not not a constant um um if you were to sit down you could write your own simple simulator using this very uh relatably uh easy to use a finite difference one um it is not the particular type of numerical method that's been that is being used in Supreme 4 Supreme 4 has a more sophisticated numerical solution method for diffusion equation um this is much more simplified this particular method it can be slow it has all of its own problems but the beauty of it is the Simplicity and just to give you an idea a of how one would would might use numerical Solutions uh to obtain num numerical methods to obtain Solutions of diffusion equos okay let's go on now to um uh slide number 19 what what I want to talk about at this point are from now on and this the rest of this lecture and the next couple of lectures will be the uh so-called mod modifications of pixel laws fix laws are introduced in a lot of basic courses um Material Science or or or physics um and uh when we are diffusing uh dolance in semiconductors in Practical cases there are a few a number of cases where uh fixed laws need to be modified to take into account uh the things that are really going on in semor processing and the first thing that we'll talk about here on slide 19 is so-called electric field effect and this occurs in the case when the doping concentration uh is uh is higher than insabi so uh insabi is intrinsic carrier concentration at a diffusion temperature again of a th inbi is close to mid10 to the 18th or so um if the doe concentration gets higher than that you can start to have um uh a non-uniformity in the um in the charge distribution in the semiconductor and electric field effects can become important and this is a a schematic demonstration of how electric Fields can arise when dope and are diffusing and silic and there's a plot of concentration schematically versus dep and the um the red uh the red line is supposed to represent the profile at any given point in time for say arsenic that's diffusing and arsenic is a donor if has one extra electron floating around it um and but the atom itself uh in the absence this electron this extra gance electron um is the donor itself is positively charged as an ion this arsic diffusing and uh there are the electrons um associated with this heavily dope Arion can diffuse ahead of the dopin and so this blue line is meant to represent the distribution or the diffusion profile of electrons let's say so the electrons uh diffuse and holes diffuse more rapidly than their Associated donors and acceptors because of that more rapid diffusion um they they'll diffuse sort the profile for electrons will be a little bit deeper diffuse s of ahead um being do that they're associated with until they reach some steady state condition where um the the drift flux uh from the internal electric field uh it's going to balance the diffusion uh the diffusion flux and and so what we look at here is there if you just look at this uh this schematic picture you'll see there is indeed from your Elementary um Electronics or solid state physics there is an electric field induced by this the fact that there's a net positive charge on the left region that is if you subtract all the Arsenic doners you subract the electron concentration there there's a net positive charge on the left there's a little bit of net negative charge on the right that produces an electric field pointing from left to right um so um that electric field then tends to cause electrons to drift from right to left right because they drift um against the field um so they're going to be drifting from right to left while they're diffusing from left to right because there's a concentration gradient so the electric field will build up to the point where there's steady state where that drift uh flux just balances the diffusion flux uh the electrons and again the physical orig this electric field is AFF that well electrons are are charged and holes and they have somewhat higher Mobility they can move around the wse much more readily than the dopen ions um themselves okay and again this is only going to happen in the particular case where the concentration of the doping is higher than in otherwise the uh the uh the background electron concentration is set by the temperature sure it's just a constant through uh throughout space so the electric field would not would not develop right at the edge of this diffusion profile let's go on to slide number 20 and so what we do is we modify pick's first law and we modify in a way that's that's very similar and and pretty much almost identical to what we do in our uh uh electronics or Sol State because classes we say that the total impurity atom flux and the presence of both a concentration gradient and an electric field can be written down as the sum of two fluxes one is f which is the normal fickin diffusion flux which is given by minus D partial C partial X Plus F Prime where F Prime is now the drift flux due to the electric field and for the drift flux I wrote down Z which is the the charge uh the charge number Z would be one uh you know for electron depends on the number of electronic charges associated with whatever is uh drifting times the mobility mu times the concentration c times the electric field that is uh the flux to the drift again that's not a diffusion process that's that's the drift process it's different physical process we going to add these two together now right then is the total flux so we have a non-fan term on the right um but because of the Einstein relation we can actually when we're talking about the diffusion of carriers like electrons and holes we know that their Mobility is directly proportional to the diffusivity and the proportionality constant is Q over KT so mu is equal to um D * Q over KT so I we can substitute in for the mobility in the upper right hand or equation with the substitute in uh Q over KT * D I can also substitute in the fact electric field e is degrading to the potential where the potential s uh can be related to the car concentration n uh as minus k T times the L of n over n i so basically by substituting in uh the expression proportional diffusivity for mu and substituting in for the electric field how it relates to the potential and therefore to n/ n i we get the equation um near the bottom for the flux F that's just the usual F minus D * partial C partial x * minus D time the concentration C time now we've come up with this um this s uh term partial partial x uh of me potential um then n Over N um so now this is a new um flux uh flux equation and um we can now we use for the car concentration n you have to be careful it's not just equal to ND because ni can be a order ND and so we need to make sure we use the full expression for the carer concentration and it's 1/2 this uh at the bottom equation here it's 1/2 times the quantity squ OT of nd2 + 4 n i s plus N D um that comes from charge neutrality um given that we there's there therefore there's a relationship between n the car concentration and um the concentration of the dopent C because again uh the the concentration C it's just NB minus na um so it's the net concentration so we have a relationship between n and C we use that to make this flux equation F to be written only in terms of the concentration C so with L of the next page slide 21 what we find is that uh we can do this compression of this equation and we find that the total flux now f is written by as something called minus H * the diffusivity of a dop d sub a partial C partial X so we've been able to actually write down an equation looks just like um fixed first law but there's an h factor multiplying it and this h factor turns out to be equal to 1 plus uh C / the quantity squ of c^2 + 4 and i^ 2 um so basically uh it looks this in the in the presence of an electric field it looks just like we have an enhancement in the diusivity of the dopin by a factor called H you can you can we just work that out so uh interestingly we see that it electric field effect uh on a dopant that is creating the field on that buin itself it enhances theity and it turns out H has an upper bound of two so H is not going to be any larger than two and that means that the electric field enhancement term can enhance the diffusivity the doping that causes the field by as much as or a maximum Lim factor of two and this happens when the doping concentration term uh or C is much much greater than uh ni and you can see that by substituting in um and calculating out what H is in certain conditions uh when in the special case where um you know C is NI then uh H goes back to being to so uh how do we get to this equation where the simplified age Factor well if you go on uh onto slide number 22 actually I'm not going to go through all the algebra but basically uh what we did was we um used a particular fact which is that on the top of slide 22 partial partial X on the logarithm of X is equal to 1 /x and uh uh what we needed to find in that earlier defusion um and that earlier drift term there was a term that went partial partial X when um of hando the n and so just by going to the algebra shown on on uh we can substit on slide 22 we can substitute for the car concentration n and we can find uh an expression for partial partial X of Len of n over n i and you see that in the middle of the of the slide and then we substitute in the dopin concentration in D is equal to C and we're able to see that the flux the total flux is the usual fickian term D minus D * partial C partial x minus this other term which ends up being able to be written um just like partial C partial X in terms of partial C partial x times the h factor where H was uh is equal to as it sh on the bottom 1 plus C over the square c^2 + 4 and I S so that's just justifies the derivation of each factor uh on slide 22 okay so just go back to SL 21 for one second and just want to remind ourselves that the uh total flux F now in the presence of a High car concentration can be written just like the fing diffusion but the H there's a multiplier H from the bity of the Dober atom a let's skip on to slide 23 now uh however so that's when we're talking about um the amount of enhancement you might get in the high concentration species um but you may have two different species diffusing and in the case where the species are at different concentrations the electric field term can actually cause an even larger change in the diffusivity of the low concentration building so let's look at a particular example uh shown on slide 23 and uh what we're we're showing here is concentration is a function of depth and the red curve in high concentration let's say a high concentration arsenic profile um the initial arsenic profile looks it was I a plet or something it looks like this as shown in the solid red curve um the background building is initially all constant it's p typ so this is boron is shown by the solid um blue line now we're going to simulate uh including electric field effects at a th000 degrees the diffusion of the arsic and the Boron U well it turns out the Arsenic does in diffuse as shown by the dash line but that the electric field that it produces right near its um the slope right near the edge of its profile actually that electric field can cause a flux of of boron ions ions which can cause the Bon to uh itself to move so the final Boron profile uh now is is shown by the blue dash line has this little um hump in it and then a dip and that that little dip occurs right near where the the junction is right where is a blonde the arch and the blonde cross and there's no way by ficking in diffusion he would ever predict that a constant profile of Bon would end up being looking the way it does uh in this particular plot again that's because of the second term the electric field term that non-fickian term so and when you want to think about that how that occurred look at the electric field Arrow drawn from left to right uh again drawing from positive net positive to net negative charge that direction of e pulls the the born again remember the born ions as they diffuse their B minus these acceptors andit negative charge so they're going to be pulled in they're going to move to the left the by the electric field they'll be pulled into the npl region and deplete some of the blond uh in the region past the junction and you end up with this sort of hump and dip uh which is a non- ficking type of of motion but induced by uh the electric field created by the higher concentration arsenic uh species so this can have quite a bit uh of an effect on the the diffusion profiles uh near a junction and we see this all the time in our devices in fact you go on to slide number 23 showing uh a simulation example of an in mosfet this is these are two supreme simulations both 5,000 de uh for mosfets you'll recognize the poly silic gate here in the magenta um and the source and drains uh regions are Arsenic and they are shown in black um if you look first at the left in the case of no electric field um you see uh these these different Contours the green yellow uh sort of light orange and then dark orange or red um we see the Boron sort of uniformly doped and um its Contours are shown by the different colors uh and now what happens on the right in the presence of electric field instead of these sort of uniform lateral doping Contours full of L you see uh they there's a um electric field that sort of Acts on the blor and and L changes uh the dop and distribution in its shape in fact they Dom it dominates the shape of the blonde distribution particularly near the source drain Junctions of a mosfet device the blonde basically gets pulled in to underneath the source and drains and there's quite a bit of loss of blond from the channel into the source drains uh that do electric field effect it's not predicted on the on the left hand side so even though H the h factor the enhancement in the Arsenic diffusivity is limited to uh at most a factor to so there's not too much effect on the Junction in the ARs this in the black Junctions uh RS don't change that significantly but it dominates the diffusion of the Boron underneath or at a lower concentration so let's go on to slide number 25 I just want to summarize uh this second lecture on diffusion we reviewed uh a few cases of simple constant diusivity um these generally apply well uh to uh the case of low doping concentrations uh in a seamless process flow that usually lean to diffusion of the seamless Wells which are fairly lightly doped uh say below mid 10 the 17th or below 10 to the 18th they usually diffused in at very high temperatures and subi is large and they satisfy the fact that the the dop and conentration is less than an I so you have constant diffusivities but beyond that case there uh turns out most diffusion problems in uh in devices need to be solved numerically and we introduced a very simple but uh powerful method for a numerical solution with the fusion equation and it actually was a finite difference formula formula it's not it's not that sophisticated and it's not actually the formulation that use is using Supreme for but gives you an idea if you needed to write your own simple uh method um the uh we talked about the first type of modification we need to make to fix uh first law and that was for electric field effects which can enhance the diffusivity of a high concentration species uh up to a factor two but can dramatically impact the motion of species at lower concentrations and what we're going to talk about next time are further um sort of Corrections to the simple uh uh piing diffusion that we've been talking about and more realistic doping profiles such as the case uh when we have concentration dependent dependent defic okay that's all I have for today and thanks very much