Minority charge carriers in extrinsic semiconductors

we've seen in previous videos that if you take an intrinsic or a pure semiconductor and you add some impurities like say phosphorus which is group 15 it has five valence electrons one more than silicon and as a result it will end up donating its electrons and so now you'll have a lot more electrons a lot more negatively charged conduction electrons compared to holes and so we call it as an n-type semiconductor and similarly if you were to dope with if you had to add a group thirteen element like boron which has only three valence electrons then it can accept electrons now and as a result you have a lot of holes so more positive type charge carriers so we call it as p-type semiconductor and we also saw their band structure and we understood that the whole thing is possible because of the donor level being very close to conduction band the energy level of that extra electron being very close as a result electrons can easily jump and over here the acceptor level being very close to the valence band and so we spoke on all about this in previous videos so if you need a refresher it would be a great idea to go back and watch those videos and come back over here in this video we're not gonna learn more but we'll go deeper into this impure extrinsic conductors we'll explore some fantastic and very subtle questions and eventually we'll also see the numbers over here I mean we know over here like there are more electrons and and less holes but how much more how much more are these is it 10 times 400 times more the answer is going to be mind-boggling right so let's do it here's one very subtle question we can ask remember that the band structure really depends on which material we're dealing with silicon has one band structure and germanium has another band structure well notice this extrinsic semiconductor is no longer silicon its silicon and phosphorus together so wouldn't the band structure of that material change why did we still use the same band structure as that of silicones and another question we could ask is now that we have so many phosphorous atoms aren't the atom or orbitals of the phosphorus atoms overlapping and should we also consider the band structure for phosphorus and we look over here we said the donor level I mean the same question we can ask here let's just concentrate on one the donor level over here we just took one single discrete level shouldn't that all be a band how do we justify these and the reason we can do all of this we can justify all of this is because of one simple reason we keep our doping concentration very very low the doping is so low that the impurities that we add is so low that it turns out that about usually about one phosphorus atom for example is surrounded by about a million silicon atoms so if you could actually see that then usually we would draw something like this but if we were to zoom out then you would see that one phosphorus atom is surrounded by lots and lots of silicon atoms something like this and so you really have to travel a lot a lot of atoms to actually find the next phosphorus atom it's all silicon it might be somewhere over here here's another one here's another one so this should answer the question if you look at this crystal this is pretty much silicon and that's what we can justify and say that hardly anything has changed by adding these phosphorus atoms and so we could say yeah the band structure is going to remain the same this also helps us understand why we don't have to consider the band structure of phosphorus for example the reason is two phosphorous atoms are so far apart their atomic orbitals are hardly going to overlap and as a result we can totally justify that we are using a single discrete energy level and not the band structure of phosphorus because they're so far apart it wouldn't matter and the same thing can be explained thing will happen over here as well the immediate fall of question we could ask is since each phosphorus atom is key contributing to one and the conduction electrons so one extra electron per phosphorus atom and since the number of phosphorous atoms is so low that's just so that's what we saw right now then is the number of electrons here considerably increasing I mean what how much is it more than the holes is it a lot does it really matter and the answer is yes it turns out that even if you're using a very low amount of impurity atoms their effect is dramatically high incredibly high and to understand why that works out to be that way we have to look into the math behind it and don't worry we're gonna keep the math very simple the first thing we'll do is write down the doping concentration doping level we just saw that if you take phosphorus field thick phosphorous at example one phosphorus atom is pretty much centered by about a million silicon atom so let's write that down one phosphorus atom is surrounded by surrounded by about ten to the power six a million silicon atoms and usually people call it as 1 ppm one part per million but that's what it means okay and if we go back to our intrinsic semiconductor intrinsic semiconductor semiconductor without any doping then we've seen some numbers before and let me just write that down one more time we've seen that if you go to say room temperature if you look at room temperature and if you take a tiny box of silicon I'd say one centimeter cube box of silicon and if you look inside that okay this is one centimeter cube let's say then you will find roughly these numbers can be worked out we don't have to worry too much about them the numbers can be the numbers will be a roughly about number of electrons and the number of holes are equal and they're about 10 to the power 10 and the number of silicon atoms themselves the total number of silicon atoms is roughly about roughly about 10 to the 20 10 to the 22 I just happen to remember these numbers you don't have to remember this okay definitely not needed we're just gonna work out some things with this okay now having said this given this we want to figure out how many phosphorus atoms will be there in the n-type semiconductor alright given this if you look at an n-type semiconductor again take one centimeter cube box how many phosphorus atoms will you find and this is math okay so I want you to try and pause the video and see if you can figure this out look at this number look at this number and see if you can do it alright so we saw that 10 one phosphorus atom is surrounded by a million silicon atoms so we could just ask if you take 10 to the 22 silicon atoms how many false positives will be there so let's just do it this is that we select ten to the power six silicon atoms you will find one phosphorus atom so if you have about 10 to the 22 because that's what you find in one centimeter cube if there are 10 to the 22 silicon atoms how many phosphorus atoms will you find well if you cross multiplication you get 10 to the 22 divided by 10 to the 6 that is if you do that 10 to the 16 phosphorus atoms and now get this we we saw that one phosphorus atoms one phosphorus atom donates one extra electron which means that by adding this impurity the number of electrons that were getting is we already had this much plus 10 to the 16 plus 10 to the 16 and if you add them that's not 10 to the 26 because it's an exponent but just think about 1 and 10 zeros 1 and 16 zeros this number is so huge you can totally neglect this and you could say that the number of electrons in the n-type is roughly 10 to the 16 look at that that is an incredible amount of increase and if you think about it that's a million times higher than what we had before but that's not it now it's time for the climax what do you think happens to the number of holes pause the video and think about this okay now if we didn't think too much here here's the way we could we could answer this we could say that you see with since you are adding phosphorus and phosphorus only giving us electrons only electrons would be affected and the holes shouldn't be affected so we could say that the number of holes should remain 10 to the power 10 right guess what that's wrong not just wrong it's very wrong to understand why we need to recall two important processes that happen in semiconductors one process is a generation remove remember thermal generation that's a process by which electron hole pairs are continuously being created due to thermal energy that's happening all the time and we've seen in previous videos that that rate rate at which thermal generation happens that is a function of temperature it's only some function we don't care what that is but some function of temperature and we saw another process called the recombination where electron hole pairs meet up and destroy each other and we saw that that number depends on the product it's even proportional to the product of the number of electrons and holes and so that number would be we could say it's some constant K times this product and the product would be about 10 to the 20 and at thermal equilibrium these two must be exactly the same otherwise the total number of electrons and the holes will keep on changing he'll keep on increasing or decreasing arbitrarily this is what we saw for intrinsic now what can we write for extrinsic this these two process are continuously happening even in extrinsic semiconductor so if we go up a little bit all right so if you write down the generation rate over here well the generation rate must be exactly the same as before because the generation rate only depends on the temperature we are using the same temperature so the generation rate over here for the n-type must still be the same number it should still be K times K times 10 to the 20 but what about the recombination rate the recombination rate is the product of these two right so if the number of holes was 10 to the 10 like look what happens to the recombination rate this number would be a whopping 10 to the 26 and that's not equal to each other you now see that the recombination rate would be much higher than the generation rate and that's not thermal equilibrium so because the recombination rate has skyrocketed what's going to happen now is that a lot of electron holes will recombine with each other and as a reason a lot of holes will be destroyed I mean of course the electrons will also please try it but the electron number is so huge we can neglect that and so now we can ask the question how many holes are left well to do that we know that the recombination rate is the number of number of electrons and the number of holes we have to make sure that the rate is exactly same as 10 to the 20 so to do that let's say we don't know what that is it has to be equal to K times 10 to the 20 and so now if you look carefully you see that the number of holes he's not ten to the ten but it is 10 to the 20 divided by 10 to the 16 that is 10 to the power 4 that is 10 to the power of 4 that means it has decreased a million times Wow so can you see the subtle effect by adding phosphorus not only have you increased the electrons a million times more but if you look carefully you've also decreased the number of holes a million times smaller that is incredible and now if you look at the ratio of the number of electrons and the number of holes that's a million million that's or is that I don't even know what to call it anymore but the fascinating thing is that we have just done such a low level of doping and yet the effect is so high and that is all possible because in intrinsic in the pure semiconductor we had this much amount of charge carriers so guess what this is only possible in semiconductors and that's the main reason why we can we can change the properties of semiconductors and we can't do that for conductor it's too high nothing will work and 4 in swear is too low so it only works for semiconductors isn't is just awesome