 you can follow along with this presentation using printed slides from the nano hub visit www.nano hub.org and download the PDF file containing the slides for this presentation print them out and turn each page when you hear the following sound enjoy the show good morning so let's get started today we'll be talking about Fermi-deras statistics that is lecture 9 and what statistics has to do with the discussion that we had previously is the following this is the analogy and sort of where you are in the course you remember that the main purpose that we are going through all sorts of complicated things calculations and differential equations Schrodinger equation and all sorts of things is because we want to calculate the number of places the electron can sit all electrons although they are the same in vacuum inside the solid they have to stay in different bands different bands have different defective masses right density of state and therefore they move with various properties with different properties and therefore they are not all the same inside the solid so far that's what we have calculated where the electrons can sit if it were a high-rise apartment building in analogy you'd say we have calculated how many apartments there are where the occupants can stay but the tenants can stay but how many tenants do I have or where they actually sit you see that is the discussion for today's lecture now one thing you could immediately understand that there is a puzzle here because assume that you have 250 electrons for example and you have 10 bands now in principle if you didn't know anything else then you could say well 25 of them would sit in each of the band you see 10 bands 2250 electrons and in that case all the bands are partially empty or partially full as you say it and therefore all 10 bands should contribute in energy or somebody else could say well no no just the first two bands takes care of you know the 250 electrons and the remaining 8 are empty how do you know where they sit and that is the purpose specifying that distribution is the purpose of today's lecture now what I'll be doing is explaining that there are some rules for filling these electronic states by electrons now of course that rules tells you what distribution is most probable and we'll do that the next one will be a Fermi Dirac statistics we'll try to derive it but we'll derive it in three techniques the first one is based on what you have in the textbook just for the sake of continuity the other two will be simpler derivation that we will be able to use it in more complex situation that will come down the road but all three are exactly that he has the equivalent information ease of use is slightly different and finally and finally by using this Fermi Dirac statistics and the density of state you know solution of the Schrodinger equation we'll be able to calculate the number of electrons in a intrinsic semiconductor and that is at least finally we are getting to get go get that carrier density remember that's what we are after so that we can apply a magnetic electric field and see how much current flows so that's where we are going now if you remember the ek diagram perhaps you remember that this is a diagram from one-dimensional solid a set of n atoms and if the spacing is a then the Brillouin zone is pi over a to minus pi over a electrons sit in individual bands and these bands each has n state n number of atoms n state twice n state if you include the spin so each band actually has always even number of states right 25 atoms per band you always will have 25 states if you include spin 50 states each one of this bands will have that now we said that we don't want to carry around all this complicated information but rather we want to encapsulate them into simple things like density of states because electrons actually might sit on the bottom or top of each band so we looked at the density of state and the red one you can see associated with the red band the blue one is like a valence electron like band concave downward and let's focus on one because if you know how to take care of one we know how to take care of all of them do you remember that in the last class I derived this density of state this formula for density of state how many per unit energy and per unit length or per unit volume right this is within a unit box of material how many states I have now very quickly you notice that there is a discrepancy because on the right hand side the formula I have written is for three dimensional solid do you remember square root of e for 3d 1d it was 1 over square root of e and in 2d what was it it's a constant so here you see I haven't drawn very carefully on the left hand side I have done the one dimensional version on the right hand side I have shown the three dimensional version but you get the idea okay now you remember in the from the in the bottom in at least in this plot in the bottom I have few states e1 could be zero let's say because it's the bottom and as I'm going up in energy e2 and e3 I have few more states now it need not be like this anyway for one dimensional you remember the e1 will have more states e2 will have a fewer e3 will have even fewer states now these are the apartments in which the electrons have to leave so let's see how the electrons gets distributed among themselves among these states now there are three rules that these electrons need to follow the first one is only one electron per state now including spin but we will not talk about spin explicitly for the time being will ignore spin and we'll say one electron per state so you cannot put two electrons in a box like that that's first rule and this is a rule anytime something has a spin of half half integer spin any particle of course including electrons that has this rule that no two electrons can sit here the next one is the total number of electrons are conserved that no matter how you do it you know you have 250 electrons in the beginning you do a rearrangement you cannot have 252 electrons at the end now this may sound obvious but for many particles this is not the case for example for phonons two phonons can essentially join into one so the phonon number is not conserved so there are many particles where the number is not conserved and at least for the velocity that we are talking about the energy that we are talking about electron number will be conserved the finally that total energy of the system is conserved again in non-trivial statement but we'll assume this because it's always possible for two electrons to come in and collide at a such a high speed that a photon comes out so two electrons come seen two electrons goes out but also a photon is coming out so therefore the two electron the energy of the resulting electrons is not necessarily conserved but for this distribution we will assume that they are okay so let's see how it how just based on these three rules how we can derive Fermi Dirac statistics now by the way one thing when a quick historical note many times we assume we see Fermi Dirac statistics in solid state books so prominently we always assume that this was derived for the solid but remember Fermi and Dirac both independently derived it and they derived it before the band theory of solid that there is a band gap that there are these quantized states all these things are derived around 1935 or so after that Wilson's paper that we'll see posted on the website in our website so this is 1924 1925 so therefore at that time there is no notion of a solid state and band diagram and others so first few applications are really in the astrophysics area it was later on transferred to solid state devices and there is a very nice article at Wikipedia that you can take a look historically this is a very interesting interesting development so let's take one example and go through slowly because this is an important example in the bottom equal zero on the left hand side I have two states at the second level I have five and the four and the energy equals four equals four now this I don't show the units it could be milli electron volts for example some arbitrary unit here we are we are talking about and this has a certain number of states there and then total number of electrons you can see counting the rate and the magenta one or five and do you see the total energy is 12 because three multiplied by four is 12 on the top side the magenta one nothing on the level two and the rate yes I have two but they don't have any energy so I have a total of 12 now I always have to no matter how I arrange I have to make sure that those number equals five and energy equals 12 that are not violated right those two I have to make sure that these are correct but first of all how many ways electrons given this configuration how many ways can the electron arrange themselves without changing levels staying in the same level do you agree with their statement what I have shown W W is a number of configuration 203 which is subscript says first level has two second level has zero third level has three so take the first one I have two states factorial two divided by factorial one and factorial two two states two electrons so I actually have just one way of arranging it do you see that that's comes out the first factor the second one is empty so factorial five but then denominator has factorial of zero and also factorial of five do you remember this formula right this is simple arrangement of things within boxes and again I have certain number of states no electrons well I have just one way of arranging it I cannot arrange vacuum in multiple ways so I again have one but fortunately for four the third level I have drawn it incorrectly I should have drawn another box because I write factorial seven third level should have seven seven boxes that's a mistake so factorial seven and factorial three and four you get the idea right and that's 35 states so that means within this you see what might happen that the three magenta one all three could be on the pushed on the left side that will be one configuration the one shown here is another configuration 35 in total for this case what about slightly different way of arranging things remember we are trying to say how the electrons are getting distributed in bands that's the goal and we are trying to see we are allowing them every opportunity to distribute without violating the rules the second one do you agree that this again has 12 you should because on the top the third level I have eight magenta one four multiplied by two the blue one two two multiplied by two is four and then the third one is zero so I have energy of 12 I still have five electrons in here so I'm good I'm good with these rules and you see no two boxes a no box has two electrons in it I'm good with the first rule also again you can quickly check one two two the configuration again very quickly you can see 420 levels in this case it sort of makes sense do you see because in the blue state there are some configurations that you can have and every configuration you have of the blue state the other configurations get multiplied so therefore will be much more see is it the only one the not really there's another one that you can check this one also has four blue each having energy two gives you eight one magenta gives you a four twelve five electrons still the bottom level is unoccupied no rate again how many states again we can calculate 35 now the first one and third one being 35 is just accidental you can see in general this need not be the case now if you didn't know any other physics which is the most probable configuration that if you came in and did measurements at different times most likely you are going to get the one two two configuration right because out her out of 420 plus 70 out of 510 measurements that you can do in 420 cases you'll find the electrons distributed according to two most probable distribution for midirac statistics is most probable distribution this is not the only distribution this is the most probable distribution you see even in three states the one level is most probable is 420 about a factor of 10 or more higher probable than the other one if you had a few more states you can very quickly see the most probable states will be orders of magnitude higher than anything else you can have right so that is what this middle one will result in for midirac statistics see so let's plot it that's the point I wanted to make so the most probable state is one two two and that is where we'll have our for midirac statistics now in general I have here three I could quickly calculate which one is the most probable but if you had 50 of these states how would you calculate the most probable well the way to do that you will take W and take a derivative which with respect to the configuration you see going from 202 to 122 and at the point where it's maximum the derivative will be zero you see I change the particle a little bit number of configuration a little bit cause number of configurations goes down quite a bit then I'm I know that I'm at the maximum point so that is something we are going to think about in a use it in a few seconds but for now I already know that answer what is the maximum one two two and we'll let's go with that one this is the most probable distribution and from this most probable distribution I can calculate what is the most probable occupation per energy for example you see in the most probable distribution I had one in the red out of two states so my probability at that energy that something is occupied that I find an electron is half right one divided by two in the second one you see two divided by five two little blue electrons five states so on the average I will find two over five and at the third one two over seven you get the idea now do you also get the idea that as you are going up in energy the probability of occupation is actually going down you see that is for going down and f of e this f is the Fermi distribution because it tells me how electrons are distributed as a function of energy so the red one here on the top will be half two fifth is the blue one and the magenta is two seventh and that gives me let me quickly and that distribution essentially will tell me how they are distributed in energy now I'm now going to go to big solids but before I do let me quickly point out you know that transistors are getting smaller right many people say that there will be a molecular electronics by that they mean that the device would be a so small that a single molecule or a collection of molecules able to will be able to turn the transistors on and off in that case you see the derivation I am going to do now that is less applicable than the type of derivation I just did because this counts phenite number of states very precisely without making any assumption about large number of states and large number of electrons it doesn't make those assumption this is an exact calculation so sometimes you may have to be able to do that especially for the devices that you might handle during your research career but to go back to the book is a historical derivation where they had a lot of electrons remember 1 centimeter cube how many atoms do you have you have on the order of maybe 10 to the power 22 10 to the power 23 atoms huge number of electrons huge number of states so in that case we can take some more approximation so for n states generally what you will do is calculate it if s i is the number of state at energy e i stands for e here because i is various levels energy levels i stands for e and do you see a why it should be s factorial divided by s minus n factorial n is the number of particles it is exactly this formula you see when the c w sub c the c means configuration and the configuration his two zero three is a configuration and just like I multiplied here various configuration I am doing exactly the same you can see the multiplication symbol over i so exactly the same state now of course when the numbers are very large then I can do some extra things I can take a log on both sides I am missing a c below subscript on w but you get the idea and you also get the idea that when multiplication under log it becomes a sum do you see a sum and then you can see how the logarithmic has been spread out now at this point it is logarithmic of a factorial and we know the starling formula is given by this s i ln s i minus s i you can see just the first term gets expanded like this this is actually pretty good formula in terms of approximation any time you have s larger than 10 this is a very good approximation so if you have that now do you see all the linear term in s i this is that will disappear see the red s i and with a minus sign and towards in the middle you have a plus s i but that's a red with a red one also so that will cancel similarly the blue one will cancel and that will eventually going to give me a simple formula like this now what do you say the first term is actually a constant right you see first term is constant the last term second and the third term depends on number of electrons so you can immediately see when I take a derivative of w I want to know the most probable one remember then the first term will drop out immediately you can see okay let's try that so this is what I'm doing you see I'm taking a derivative why because I want the red one and I have many states so I cannot just go and look at each one of them individually so I look at those separately now this one just from that previous step if you take a derivative you will get the first term within the bracket now I'll ask you to check it out you know this one step what you will not get is the second and the third step third term with alpha and beta there but you can see what alpha and beta does it says that if I change the total number of electron redistribute them a little bit among different levels right what should be the sum over delta I it should be 0 why should it be 0 because sometimes I take two electrons from one level go bring it to a different different level so from one level's perspective this is minus 2 from a others level perspective it is plus 2 so when I sum them right therefore all changes because electrons cannot be created or destroyed so therefore in this case sum of delta ni that should be 0 and when you want to introduce that constraint this is something called a Lagrange multiplier you introduce it in this particular way I'll post a very nice note in which you will see how the Lagrange multiplier is a simple differential equation concept and not differential differential calculus concepts so you will see how it goes through and the beta similarly says that the total energy must be conserved right while while you are rearranging the configuration you cannot take any configuration only the subset that make keeps the total number of electrons the same and the total number of energy total amount of energy the same so only those are those are allowed and once you do that you can see there is a delta ni in every one of them and so you pull them inside and again that's the most probable configuration now the whole thing has to be at the maximum point the whole thing has to be 0 now if the whole thing has to be 0 regardless of the ni then the term inside the square bracket must have to be 0 so if that is 0 do you see that you can immediately transfer it down to the next equation how do I do that you can see alpha and beta I can take it to the right hand side it's a log I can make it an exponential and remember in the first equation I have si over ni and I define f as the number of electrons in a number of states right and so therefore I have flipped it so you can see how the 1 over exponential term might come along okay so now from here you can see that I can calculate this number what are the if you if I knew alpha and beta somehow if somebody told me alpha and beta I'm said how do I calculate alpha and beta the first is I don't I'll assume something the first one what I will assume again this is that Fermi energy EF again it will come later on the physical meaning of it but for the time being I will say that if my there is a state the Fermi level then the probability of occupation of that level is half I'm just defining it I could say probability of occupation is 3 4 whatever I want so I am defining it Fermi level to be that energy in which it is occupied a level of half so by the way do you see that regardless of what you do with energy E you cannot make it more than 1 in this particular case so 1 over the exponential so regardless of this the maximum point here for if you will always be equal to 1 now at F equals EF probability of occupation is half and so therefore do you see that if it has to be half then the top of the exponential alpha plus beta E must be 0 do you see that because e to the power 0 is 1 and 1 divided by 1 plus 1 that's half so this must be the condition and that relates alpha and beta the second condition is that when so you can get an expression for alpha equals minus beta F and insert it in there the second condition is that at very high energy you know 20 e V dump the probability that an any electron will be occupied is 0 because that high an energy nobody has as a result at E equals infinity infinity means maybe 3 e V 4 e V whatever and in that case it must be given by the Boltzmann equation a Boltzmann relationship and by comparing this to do you see what the value of beta should be when e equals infinity you know then then e equals plus infinity that means is much much greater than EF so beta e will go to will become a large number compared to 1 so I can drop 1 and so it becomes e to the power minus beta e right and e to the power beta e so it's it can be compared with Boltzmann distribution and as a result beta is 1 over kt so beta is related to the temperature does it make sense that as you raise temperature in fact this distribution will get broadened out higher levels will be occupied so therefore you have 1 over beta kt now I know this is a little complicated derivation you have to do it once so that you can torture your students with this type of derivation when you become a professor so but for the time being let's stay with that by the way do you remember from the discussion before where we had just three levels so the first level has probability of half right do you remember e equals 0 half to 5th to 7th so that level was actually the Fermi level in that case because that had a probability of half okay that's it so I know this is the distribution for electrons in any given energy e if I know somehow if somebody tells me e f and the temperature I'm also temperature I know and my computer is working in this room whatever is the room temperature that's my t but somebody has to tell me what e f is that will come later this is one derivation two more to go the other derivation is something like this so let me step back you see how did the electrons or let me step back how did the electrons I said three configurations remember in the in that one but how did they go from one to another actually what is happening inside let's say the magenta one scatters with the blue scatters with the blue and as a result the magenta goes down I'm sorry magenta scatters with the red red has energy zero and magenta has energy four they scatter with each other and then they go both go to the blue states because the bottom one got an energy of two the second one got an energy of minus two so the energy is conserved right and similarly electron number is conserved actually just from this statement you can calculate the fund me drag statistics in one sentence one slide because this process if the electrons the blue and red and magenta can create blue electrons similarly to bill a blue electrons could scatter and they could go to the magenta and the red states right they can go back and forth this is called detailed balance we'll use it a little bit later that anytime there is a scattering that allows electrons to go to some states the reverse process must also be allowed and must have the same rate so do you see that f not at e1 e1 being let's say equal zero when it scatters with an electron with an energy e2 e2 being the top level and f3 and f4 where the electrons are f3 and f sorry e4 and e3 are the places where the electrons eventually go so those will be the blue states the blue states must be empty remember that a state cannot be filled by two electrons so if they are not empty you cannot go there so the first level says that how the magenta scatters with the red and goes to the blue and the blue must be empty therefore I have one minus f not two factors of one minus f not on the first time now remember the reverse process must also be permitted so therefore the e3 and e4 those are the two blue electrons if they scatter they must be able to go to the magenta states with one minus f not at e1 and one minus f not at e2 okay that's it my derivation is done because if I further require that e1 plus e2 must be equal to e3 plus e4 because energy has to be conserved you see then the only solution to this equation is this that's it no complicated derivatives and others actually for midirac statistics is this powerful and this simple this is the only way configuration of electrons within states are possible you see now this will be using it quite a bit towards the as a semester progresses okay now let's do the third one that will be also using quite a bit now something called derivation by partition function here we'll assume something and then I'll give you an example to show how it works out so this is called a partition function where I got it I opened my statistical mechanics book if you take a course statistical mechanics you will get a formula like this that probability that a site energy of level I is occupied is given by this complicated formula I'll show you an example to show how it works will not derive it do you see that E I and N I floating around in the numerator so that's the energy of the electron at that level N I is the number of electrons in the level I'll show you an example how it works and then denominator instead of carrying it around because it's same for everybody I will just call it Z and not write it over and over again okay let's say I just take one stage did you see it in very nice animation I see here so it went down I'm just focusing on one state and this one the PI tells me any state I how it's filled so how can it be filled either it can be full or it can be empty you see one level one electron it can either be full or it can be empty all the energy conservation particle conservation all built into the partition function I don't have to worry about that anymore so I'll just have to worry about the configurations okay so if this is the case then say that there are two possibilities if the level is empty which is the top line we'll call that state zero no electrons in that box if there's no electron how much energy do I have well nothing you can see here is zero also no electrons well ni is zero and if I insert it in my partition function you will see I'll put ei equals zero ni equals zero so the numerator is e to the power zero and that's one so the formula PI for state being zero is one over z you see that now what about states being occupied that's the second line state being occupied I call that one and I have ei I have one electron so I have ei equals ei you know yes that's the energy ni well one electron ni is one I put it in and that's my formula okay I'm done actually because probability that state is filled is sometimes it's full sometimes it is not what is our probability probability is this sometimes if it is full then it's P1 sometimes it's not it's P not put that formula in you know from the from the table and that's my Fermi drag statistics you see that that's it you see this will be necessary a little bit later again because without doing some tricks like this doing more complicated derivation is very difficult I can skip all those to make your life easy but you need to know these days devices are getting so small you actually need to know this okay so you have seen three ways of deriving Fermi drag statistics so you see general property is that when energy is very small then the maximum is one when it's very large it goes to zero and somewhere in between at Fermi level of EF there's a half and many times I'll be drawing it this way the same equation rotated 90 degrees if we plotted in the x direction and energy E the y direction and EF is somewhere in between okay very good so few quick comments so it as I said it applies to all spin half particles the information about spin is not explicit many times we'll just do all the calculation and multiply the end result by a factor of 2 but this is not true for many magnetic materials people talk about spin-tronic transistors and other things you may have heard and you have to be more careful on those how many would distribute particles and the most interesting part is and this I want to emphasize that the coulomb interaction among the particles is neglected I pushed four electrons over there sitting next to each other now the electrons don't like each other right strong coulomb interaction how is it I allowed them just sit in the neighboring boxes and nobody is complaining this is because this theory actually just applies to solids of long solids you cannot really apply it to small molecules because then there is something called a coulomb blockade that's a different thing but this should have bothered you a long time I'm sure you have seen this derivation many times why is it that electrons don't really repel each other here and why didn't I include that factor in this is why because if you take and solid of length L you know 5 million atoms then these are the energy levels and the dotted start to spin up and they say the solid spin down you have a bunch of states the reason it works because the electrons although they are at the same energy they are actually far apart they don't see each other because if they did there would have been coulomb interaction so it is like you have one electron then you have an atom with its own electron another atom with one electron a million of them then another electron sitting on the other side so therefore they don't see each other therefore I didn't include the coulomb interaction term here see it's very important so therefore when you take a molecule somebody your professor asks you to do a calculation you're happy for me direct statistics put the numbers in your result will be wrong so be careful why you use it it's a powerful but dangerous tool okay I'll take 10 more minutes to get started finally on something useful but before that let's think about what we have learned density of state that density of state only applies for 3d but you get the idea and I'm just drawing the conduction valence band always remember that there are lots of bands underneath and above this this is for silicon and germanium 3s and 3p states that's what I'm sort of drawing I just learned about Fermi Dirac statistics for the electrons at with a certain given EF now you can see if I move the EF up and down the whole car will move up and down okay fine I'll see how to fix that one and do you see that if I multiply the density of state how many apartments for me dirac statistics probability and apartment being occupied multiply them to then energy by energy I can get the number of electrons that's shown here on the red on the right hand side in the middle I have nothing because although Fermi Dirac probability of that site being occupied is not zero right but there's no states and if there's no states I even a multiply I do not get any electrons or holes on the right hand side now that's about conduction band electrons what about the number of holes okay so if I wanted to calculate the total number I will start with EC on the bottom and eat up whatever is the end of that band remember band all bands are finite whatever is the top of the band multiply them energy by energy sum it up I get my electron number what about number of holes well holes is anytime there's no electron so if the probability of occupation of a electric state is f what is the probability that I will find a hole 1 minus f and 1 minus f is a blue line and again if I wanted to calculate how many holes I will multiply the blue density of state with a blue Fermi Dirac 1 minus the Fermi Dirac and that will give me the number of holes and the number of holes again I can easily calculate integrate it over I get total number of holes right this is it general prescription for calculating the any any electron distribution now you see only thing I'm known in here is a EF because I know the density of state as soon as I know a material I know the effective masses right I know the effective masses the band gap where EC and EV are I know all those only thing unknown somehow if I could get EF I'm all set I can calculate M I can calculate T and I can calculate conduction so let me do that I'm again doing it for three dimensional solid in exams sometimes I give one same derivation but I may ask you to do it for two dimensional solid in that case you see the GC G sub C we will change it with a different formula right but other than that you will do the same derivation by the way there is a two multiplication explicit because so far I have been talking about density of state without spin if you wanted to include spin then I put that two multiplication just to take care of the spin now the E top I'm setting it to infinity do you see why because the Fermi Dirac statistics actually decays very fast and within a few kt kt is 25 milli electron volts in room temperature band width is several EV so milli electron volts and several EV on an exponential so therefore 4 EV is as if it's infinity so I replace the top integral with infinity if the band is very narrow don't do this because remember the some of the bands towards the bottom were very narrow in that case the width of the band maybe actually equal to few kt if you do this you'll get completely wrong result so be careful now all I have done going from these two states is have rewritten the Fermi function with using a variable E minus EC but apart from that and call that eta C is that the right or need okay and therefore I can calculate this whole thing and I can just rewrite it in two form one is N sub C now you see capital N sub C and another is this Fermi Dirac integral of order half this is the capital F why it's half because you can see that there is a square root sign here so therefore that is why it's called half if it had a three half sign then it will be called three half order three half for the time being let's say that this is my derivation what does it mean what does it mean I just write it this way but what does it mean this is what it means so before I go there what it means I just want to point out that if you plotted the function the exact result with Fermi Dirac integral of half or an approximation which is NCE to the power eta C if you did that that is the plot on the right hand side that's the plot on the right hand side and you can see for the most electrons and I'll show that later on with examples I'll show that later on that if the energy level the Fermi level is few kt below the conduction band remember the conduction band and the Fermi level is somewhere in between if it is few kt below then the simpler expression a simpler expression eta C N C over e to the power eta C that's sufficient I don't have to do that complicated expression so this is what it means forget about all the derivation this is what the what it actually means you see my electrons are actually distributed in energy right you see that the red one and the holes are distributed in energy instead of thinking about these things as being distributed I can do the following I say I don't have like apartment buildings going from here all the way to the top I will compress all the apartments into one level at EC compress them all now that is what will be called an effective density of state because I had it all spread out energy I've squished it all at energy level EC and that's what will be called and it's in sum over all the states and that's called an effective density of state NC effective density of state for the valence band NV and I will just think that I don't have this distribution but just I have one level and that level occupied by the pharma factor that's it I will not have to do this integral over and over again because I have effectively compressed the information into that level and just one level how it is occupied now this is very important this distinction is very important to understand because effective density of state is not density of state density of state is energy resolved effective density of state is an integral so therefore this have to be you have to understand this okay so let's talk about electron and hole I have maybe two slides so let's say I have this n equals that expression NC e to the power beta EC minus EF right so NC is that effective density of state and you can see that exponential factor is the probability that that state is occupied right that's the Fermi-Dirac approximate Fermi-Dirac factor and correspondingly for the holes P I have NV effective density of state sitting in the valence band and again the probability that is hole is found you can see the change in sign one case a minus another case a plus because one place it says occupied by electrons another case it says not occupied by the electrons okay now if I multiply that gives me a very strange relationship because when I multiply n with P you can immediately see that the EF disappears you see that because in one case I have plus EF another case I have minus EF disappears this product do you see is independent of anything of the material in so long I know the material I know NC I know NV and do I know the band gap of course chronic penny model I solve the Schrodinger equation I know it and beta well beta is 1 over kT as I know temperature so the amazing thing is regardless of what you do for the solid however the electrons move around this doesn't care this product is one relationship you will use throughout the semester and this will give you a lot of information so finally let me end with this that if I have a solid in which it is intrinsic means a certain number of let me see whether I have okay I may have it later on but let me so let's say I have a solid I raise the temperature it was at zero raise the temperature some of the electrons which are in the valence band now have jumped to the conduction band because more energy so the number of electrons that I have is equal to the number of holes I have because where will the holes come from they are coming from because some of them have gone elsewhere on the conduction so for intrinsic semiconductor I have n equals p and I'll call that n I I for intrinsic now do you remember in the last slide n multiplied by p is always that relationship and CNV to the power you know that right so here n is equal to p so you have ni squared so you see now from beginning to calculate the number of electrons that can take part in the conduction because that's the number of free electrons in a partially field band ni and that's also the number of holes in the partially empty band and that's also ni so I can easily calculate it where is the Fermi level well if n is equal to p is equal to ni I can immediately solve for it I know the band gap EGA I know NC and NV I know beta and if NC is equal to NV then where is the Fermi level NC is equal to NV therefore that's one log of 1 is 0 and therefore right in the middle if NC is equal to NV right in the middle if NV is significantly larger than NC then what will happen NV is large I have lots of density of states on the bottom side and a few apartments on the top side fuse right NC is small then EI will be above this do you know why you understand why it is because unless the Fermi level is close to the conduction band they have only a few states so their occupation has to be much higher in order that the number of states multiplied by their occupation becomes a number comparable to the valence band number and as a result this value of EI you should convince yourself that this formula is physically correct otherwise in the exam what you will do many times this formula will get flipped around depending on what you remember from your memory but rather it's good to do it physically okay so let me conclude here so we have discussed Fermi Dirac statistics defined by the states occupation statistics defined by the states from the solution of the Schrodinger equation from now we will not talk about Schrodinger equation anymore because all the information about Schrodinger equation everything is hiding in the effective mass right you see for every band the effective mass is different that is where all the EK relationship the curvature the bottom of the band quantum mechanics is hiding in the effective mass we will not think about that anymore that's why it goes into the density of state so density of state hides the quantum mechanics all the chronic penny model everything will not also talk about Fermi Dirac statistics explicitly anymore all information about Fermi Dirac statistics is hiding in EF and the temperature T because you know you've seen the relationship we'll multiply we have found a new relationship n multiplied by P independent of how you feel the states how many states you have right NC and V to the power of the band gap so these three things we have learned in this in this class and then rest of the semester we'll be just applying this over and over again and you'll see all devices your MOSFET your bipolar transistor this laser pointer the physics of all of those will gradually just flow out of this very simple concepts right thanks