 You can follow along with this presentation using printed slides from the Nano Hub. Visit www.nanohub.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. Donors and acceptors and how the statistics of donors and acceptors when you put electrons in donors and acceptors are different from when you put an electron in a big solid and like a silicon or germanium which has lots of atoms. Remember donors and acceptors are like isolated atoms and analogy would be if you go to a foreign country the people of course in that country has a way of doing their business and getting distributed in various ways they want to but when you are a foreigner like a donor or an acceptor in that case the rules that you follow might be slightly different and that is the purpose of the discussion today. And in the process we will also be trying to calculate or getting started towards calculating the electron number when a solid has a certain number of donors or acceptors. We will quickly review the law of mass action and the intrinsic concentration. This is something we have done before. We will talk about statistics of dopant levels both donors and acceptors and then finally conclude. Now this is something I have discussed before two lectures before that if you have a solid which is not doped by any foreign material then in that case the total number of electron is a constant and when you raise the temperature of the solid let us say from 0 degree to some finite temperature the electrons get redistributed and if we discuss that some of the electrons from the valence band in that case goes to the conduction band and the one that goes to the conduction band has been shown here in red and the ones that the holes that have been left behind are shown here in blue. Now we realize without donors and acceptors then the blue area under the curve for the blue must be area under the curve for the red although the picture doesn't look like that well. Now we also discussed that N multiplied by P for any semiconductor doped or undoped is always equal to Ni squared in equilibrium. When you apply a bias that rule will not be valid it will be changing a little but N multiplied by P regardless of how much dopant you have put it in is always going to be equal to Ni squared a constant and so in that case what we said that the left hand side will become Ni squared and because N is equal to P and the right hand side the expression for Ni squared is effective density of state and the band gap on the exponential and you remember that we calculated by solving this equation the number of electrons in a intrinsic material beta remember is 1 over kT right. So always you know as an engineer we are not physicists we are engineers so it's always good to carry some numbers in our head like NC and NV on the order of maybe 10 to the power 20 remember in a given band the number of states is equal to the number of atoms so per centimeter cube 10 to the power 22 let's say but remember not all states are occupied a fraction of them are occupied let's say 100th of it so that number N sub C or N sub V on the order of 10 to the power 20 you see and the PG is on the order of EV that's what I have seen many semiconductors right and kT on the order of maybe 25 mille-electron volts or so so you can see that factor on the top can be e to the power minus 20 so that is what suppresses although the NC and NV are big numbers the gap band gap suppresses the number and it sort of makes sense you see if you have to jump one EV up while your average energy is only 25 mille-electron volts you not too many can jump over to the conduction band and that's why this number is small so we talked about that number and we also talked about how to locate the Fermi level or the intrinsic level for this case right by equating N equals P and from that we calculate everything you know NC and NV you already know EG you know beta is 1 over kT so you know all these quantities for a intrinsic semiconductor therefore you know the Fermi level the intrinsic level and the Fermi level the subscript I is the same thing for intrinsic semiconductors now in a dope semiconductors of course our life isn't so easy it's a little bit more complicated and that's why we have to discuss that's what we have to discuss little bit more today remember however this is also an approximation this formula of Ni squared when it is non degenerate if it is degenerate and the Fermi level is very close to the conduction band in that case this formula is not correct now I have already mentioned that intrinsic concentration is too small and therefore a semiconductor under normal temperature you know many of the common semiconductors is essentially an insulator the electron concentration is so small that if you apply an electric field you will not even get a femtorempere of current in a given 1 centimeter solid so we need something to do to increase the carrier concentration and we said that this is done by doping the semiconductor okay so this far we already know now one thing I want to emphasize from the very beginning that if I assume for example here the rates are donors in this picture and the dark blue is acceptors no matter what I do if I throw in a bunch of these things this whole solid as a whole is charged neutral do you understand why because yes a donor brings in an extra electron but it also carries with it an extra proton right so it's charged neutral holes yes it has one less electron compared to all its friends but it also has one less proton so therefore if you count the all the protons in the solid in this new material and all the electrons in that solid in that new material the sum when you calculate them all up this is 0 over the volume but in addition if the material is homogeneous that is that you take the solid and put the dopants uniformly everywhere not one more in one little bit more in one corner little bit less in another corner not that uniformly everywhere in that case you can make further this this additional additional statement that not only it is the integrated thing is charged neutral but if you took a microscope and sort of a very small region a micron cubed region in throughout the volume you'd still find it is charged neutral within this only if it is specially homogeneous right because of course otherwise at every point the dopant density could be different and so that wouldn't be correct in general locally however specially homogeneous this would be correct now I have written here p is number of holes n is number of electrons and this is charged so p has a plus sign and n has a minus sign I have also put a n d plus n d plus means the donors which has given away its electron if it hasn't given away its electron then it's charged neutral it wouldn't come in this equation only the fraction that has sort of donated its electron those will become charged charged and only the charged one will get into this equation and similarly n a minus is the ones the holes that have been filled by electrons because otherwise it's charged neutral when it catches a hole and fills up the hole catches a electron and fills up the hole then it becomes n a minus and the sum total 0 now actually with this statement even in a dope semiconductor I already know what the Fermi level is let's see a little bit carefully that the value for the holes p is given by n v an exponential of e f minus e v over k t you know that right because it depends on how many holes you have depends on where the Fermi level is if it's closer to the valence band then you have more if it's farther away you have less so that's the expression for holes the next one is an expression for electrons you can see that there is a e c and n sub c sitting there well that's fine now there is an expression for n d plus which is looking a little strange but it's almost looking like a Fermi Dirac statistics you see I have n d divided by one plus that exponential thing and I have a little two sitting there almost like a Fermi Dirac but not really because Fermi Dirac wouldn't carry around that two in the denominator is that right and the final one is the acceptor you see this but again like a Fermi Dirac accept that there is a force sitting there in the denominator apart from that Fermi Dirac's okay I'm going to derive these two relationships shown here in red and blue that will be sort of the most of the class today but the answer of what we are after is already here what is the only thing unknown in this expression you see you know n sub v n sub c e d is a donor level in the last class we calculated that do you remember that that was a hydrogen levels hydrogen like level m star and other things and we also know e a is a acceptor level wherever that was the whole level so the only thing that is unknown in this expression is e f Fermi level is the only thing unknown and as soon as I know by solving this equation where the Fermi level is then I will insert it in the first expression it will give me the number of holes insert it in the second expression number of electrons and so on so I can I have in fact solved my problem of calculating the Fermi level remember that was the unknown thing calculating the Fermi level in a arbitrarily doped solid so what I am going to do now is spend probably next 20 minutes thinking about this donor and acceptor level and why is why I have to use a Fermi like fact function where for the first two I did not right why do I have to do that for here also and where is this factor of two and four coming from right but this is a detour your answer is already here okay so now let us talk about it now that will be the statistics of dopant levels now I have already mentioned in the last class that when you have a dopant in a solid in a semiconductor then if the dopant brings with it an extra electron in a donor then and also it brings with it an extra proton electrons goes around and you can calculate where the level is where the level is and we will indicate that with ET the T subscript over there is for trap they call it trap but you could also call it ED if you wanted to associate it with the donor level I mean that is just a nomenclature compared to the before and I have also mentioned to you that you can have one level two levels and levels below the mid cap now I will give you a homework in which instead of having one level you will have let us say two levels and then you will have to rewrite the formula in the previous slide in the presence of two levels because over there I had just one level at ED you should see how to handle two levels and you should be in good shape now I want to physically tell you what does a dopant do yes it gives an electron but remember it's a foreigner it doesn't do everything that the locals do in an exactly the same way first of all it doesn't want to play the standard game assume that I have four silicon atoms let's say now in general if when the silicon atoms are far apart or idealize atom let's say far apart then let's say each one of them have the two levels this is what I meant about you know last time we talked about potlack that let's say each one has two dishes and you have those two levels and when you come to potlack being come close together you share now remember in each one of the levels the size square integrated size square within each rectangle that is of course one so the electron is sitting in the own atom when they are far apart now when they are close together what happens that they begin to share because the electron in the first one now can go to the second one third one and fourth one so it's spending sort of one fourth of the time in each one is own one and so this whole thing is spread out but you can see still on the bottom you have four electrons it is just these are spread out and they have more places to sit this is where the band came from right the bottom one let's say was the valence band the top one was the conduction band this is the sharing of the states that is where the bands came from and that is why when you have four atoms per band you have four levels without spin now when a donor comes in donor is a little different because the donor levels it has one less proton one more proton one more electron it is not levels are exactly not in the same place and so that is what I have shown here in green now when a donor comes then it doesn't want to donate anything it is like one of those persons who come to the port leg but gives his food to his own and in that case what will happen that the others will share but the donor wouldn't share on its own of course if you give it a little bit of temperature then from one from the green they might jump up and join the others but you will have to give it a little bit of energy it doesn't do it on its own right and so in general anytime you have a donor or an acceptor in that case the number of continuum states that you have is 2 and minus 2 here with spin because it didn't give the donors did not give the electrons now in a big solid remember you have a 10 to the power 22 so 2 times 10 to the power 22 minus 2 well you don't really care that you had a few donors here and there didn't want to share the number of states are so many anyway it doesn't matter but think about a small molecule right where you have only a few states in that case these things might become an important consideration but the main point I wanted to make was when you have a donor it doesn't give its states or electrons easily unless you give it a little bit of temperature so the green level where it is sitting compared to the bottom of the band that is what ED is or ET is right that amount of energy you have to give before it goes to the conduction band now let's talk about what rule these electrons in that donor level should follow now you have done this before so I don't really have to work very hard on explaining it do you remember that the formula we used for deriving Fermi statistics using the partition function we use three techniques one of them was partition function and in the partition function method we had on the numerator the energy level EI and the electron level N I those things you have now think about this donor levels donor levels have actually always have two levels right spin up and spin down why I cannot ignore spin here I will come in a second in a donor and an acceptor which are localized which doesn't share its electron with everybody you cannot neglect spin anymore you cannot neglect Coulomb interaction anymore right and I will explain why but you can get this idea first first you say if both of them are empty no electron over there right if you don't have any electron over there you have energy is 0 well no electron N I is 0 and if you insert EI equals 0 and N I equals 0 in the equation on the top you get the idea that it will be E to the power 0 and that's one and that's why in the first line I have 1 over Z okay what about the second one in the second one I have one spin moving one spin up you see U is for up spin D is for down spin so that's why I have U and D written there so you have one on the upside and nothing on the zero side fine one level one electron you have you have EI and then one electron so I have N I equals 1 put it in I get that expression and this I have you have seen before in the previous expression you have seen this before what about the third one well third one is again one down and nothing on the upside same expression no problem but the trick is and that's what makes a donor so different is that when you try to put one and one both up and down so close together remember donors they don't give out their electrons so close together this time they are not sitting together at all because they have strong coulomb interaction they will kick the other one out the first one will sit and anytime the second one tries to come in the coulomb repulsion will push it away so this level if you want to put two will require such a high energy that that configuration is not possible as a result I will not have any configuration associated with one and one you see now so I have three configuration that's the table the probability that the donor is empty it has given away its electron it has one more proton in the core so it's a positive charge this is what I have when it's empty it's f zero zero is what I'm after I want to know when both states are empty compared to everything else and if I insert the expressions from the top table you immediately see where this two is coming from the two on the right hand side shown in the red you can see the two comes from because you have two configuration which are now possible spin up and spin down which was not the case before and when the donor state is empty you get that expression and when the donor state is full in that case you have one minus f zero zero you can calculate the whole thing you know this is no rocket science and you will get an expression for half an expression containing half in the numerator on the right hand side so good so this is a it's not really Fermi Dirac's statistics right if you derived it it would have carried your name so in that case it's a slightly different than Fermi Dirac's statistics and this extra factor will be called a degeneracy factor and I'll explain why a little bit later this is about electrons good so I understand it now when I want to do it for holes I will require a little bit extra work by the way before I get there for the holes I want to compare it with what I told you before in terms of band electrons in the band electrons you see I did not have spin up and spin down I didn't consider spin I just allowed them to sit one at a time and from here I just calculated this you remember this table right a few slides a few classes before and from here I let me go back very quickly so I calculated this quantity and from here calculated the Fermi statistics right that has that one over one plus e f over minus e minus e f over k t so that is what I calculated now what is different between a band electron and a localized donor electron why they do have different rules for filling up the states the reason I already alluded to a little bit in a few class before is the following many times you hear this argument if you have taken a chemistry class you have heard this argument that there are a certain number of states and every state can be filled by two electrons right there is this Hohens law and other things where you can fill the electrons one spin up one spin down you can fill them up that rule is actually you see one the applies for large solid and this is not this is a fact that is seldom appreciated because you see when we say that two electrons can sit in a given level what we might have in mind the two levels are two electrons can sit very close together that is impossible only in a solid what will happen that the two electrons when they can be in the same level but physically or spatially as far away as possible only when they can do that only then two electrons will sit in a one level the two spin electrons cannot sit in one level if they are confined to small space and the argument people make that if you have for example Lx is the size of the solid then do you remember 2 pi over Lx these are the Brillouin zone and the first level the second level the third level this is 2 pi over Lx 4 pi over Lx and all those you can calculate the states but the reason they can stay together is the following in fact you could break the solid into two pieces Lx over 2 and Lx over 2 again calculate the states when people say that two electrons can sit in the same level they actually mean that they can sit in a separate places within the volume not in the same place so you can see the blue up spin one is in a very different place from the red down spin one and that is why there is no coulomb interaction and that is why band electrons can sit however in a donor they are sitting within two angstrom of each other right not 5000 angstrom apart no hope of being two being together that is why their statistics is different now let us talk about holes now the holes what is what is the hole the hole is essentially and sort of an empty space in the sea of electrons in the valence band that sort of reflects the motion of the collective electrons the other electrons it is like in the parking lot lots of electrons are lots of cars are there one empty space and if I just follow the empty space then I know where the cars are moving so that is the hole you know that already now let me make a quick argument before I go to the next slide I will call state one this is a name I will call state one when hole is present means I have a hole when I have a hole then I have one less electron right and one less proton so that will be n-1 charges when a hole is present I will call that state one when a hole is absent meaning how can a hole be absent meaning the hole was there a electron has gone in and filled up the state in that case I will call the state zero hole field means it has been filled by an electron and then how many charges do I have I have one extra electron so I have n electrons but I have n-1 protons I did not bring any proton so I have one extra charge when the hole is filled right and I will call that state zero now you remember when I have one band in the conduction band and when I localized it I got one level up out because it didn't want to donate and then I got a spin up and spin down but do you remember the whole bands are a whole lot more complicated why is that because instead of having one band there one band there I have two bands what are those bands called a heavy hole and a light hole band and I want a little bit down what is that called and split off band right it has split off a little bit so that's what I have now when let's say this donor wants to withdraw states that otherwise would have donated it will withdraw states from the light hole it doesn't want to give it to the light hole it will also withdraw its state from the heavy hole it doesn't want to give it there and since they are together when it withdraws it will withdraw four states because it gave four states before to each one of the bands two states to each band so now it will withdraw three four states from there now you realize that if this split off band was also closer if you had four bands over there that was sort of on top of each other it will withdraw how many it will withdraw eight states from there right so generally in this case it will withdraw four states so now how to fill these four states is what I am going to discuss next now this is what I mention that it will take away two states for each band and there since there are two bands it will take out four and let us see how these states are full now these four states I will be calling zero zero zero zero when the state is filled all filled zero zero zero one when you have one hole and zero zero one zero is when you have one hole but in one of the four states this is how it is distributed and then there are you can see that there is no one one because no two holes can sit on those levels localized levels because holes have a plus q charge it does not want another plus q charge to be there so no one one no one one one nothing like that all those are taken out do you see how the four in the distribution function will come because now you can see that there are this four configuration of one levels right here so let us calculate so previously we had up spin and down spin just two now I have four you see this is not rocket science this is very simple so let us say in the first state zero zero zero zero the holes are full because you have bought in an extra electron and the holes are full and that is why you have a certain number of charge you can see when you have one hole then one of the states are missing the three others are full and so on so forth you can you can easily see how the where the other states would be coming from now this I have already mentioned that the zero zero zero level is has n electrons because it is full right that is how I define zero level when it is full and therefore it has one extra charge and when you go and have zero zero zero one level then you have a hole in that case you do not have a charge and so going from if you jump from zero zero zero the four zeros to any of the other configuration then your charge has gone down by one why because previously it was full now it is at least one is empty so you have charges gone down by one and your energy correspondingly it was previously full now there is no electron over there so your energy has also reduced by ea the acceptable level so that is why you will have a minus ea now this will require you to sit down with your slide and at the reason I wrote it out clearly instead of just speaking is because I am sure that you are not probably just by listening will not follow this but follow this line by line with your printed slide and listen to the lecture one more time this will become clear because it is not an easy thing most students get out of this course probably never realizing that they really do not understand these things very well okay so I am done because I know how to fill the levels you know if I wanted to calculate p zero zero zero zero probability that the state is filled what do I have to do I have zero holes so ni is zero and I have zero energy so ai is zero so I put it over there I get that for all the other levels you can see I have put a minus one for ni because going from zero zero level to zero one level I actually reduce electron by one and the partition function is for electrons so I have a minus one you can see I also have a minus ea because the energy has gone down by ea and when I put this in I will get an expression okay that is the step three and four from the previous slide and if I wanted to calculate the probability that a hole is full right in that case I will again do the standard ratio of probabilities and do you see a factor of four appearing right over there the factor of four comes because the electron is being shared the state is being shared by light hole and a heavy hole band if you had now from this do you see if you had three bands all at the same point in a state of four how many would you have you have six right so in general depending on how many degenerate levels you have this number will be changing and that's why it's called a degeneracy level now you can also understand that why there is no 1100 because two holes will repel each other by Coulomb interaction and those states are not acceptable so in general we are interested in ND plus do you remember what was the ND plus in the first slide I wanted to calculate how many were empty and that is why the charge balance equation so ND plus means whoever whatever electrons or whatever donors has given away its electron and has become positively charged in the process and in that case I should multiply with F00 whatever number of donors I have a fraction of them will give away its charge not everybody and that's why I multiplied with F00 and therefore I will have that expression now you should realize that's why I got the expression in the first slide what about holes in the holes the ones that are charged is when a thing is the hole is charged acceptor is charged when the acceptor is full not empty full because then it has one extra electron that's why you see I write Na minus because it has taken an electron and therefore you have to correspondingly multiply it remember the meaning of 00 over there 00 on the top for the donor means it is empty 00 on the bottom means it's filled so it's not the same 00 over top and the bottom you calculate you get this expression that's why we had the four okay so the derivation the detour we took is sort of ending here now very quickly I want to point out a few things that this factor is for the degeneracy factor you know for different material for different systems this value would be different so the first thing I want to point out that this statement the distributions are physical what it means is that when you have a set of particles the way they distribute themselves in energy you say this you cannot do arbitrarily you cannot say I have a bunch of particles I will assume Gaussian distribution you know many people do that many people when they don't know anything the first thing they say well I will take Gaussian Gaussian is when you don't know anything about anything then you start by Gaussian because that's at least you can get started for Fermi Dirac when the electron is in a band only in that case I have Fermi Dirac statistics when I talk about photons or I talk about phonons I have Bose-Einstein distribution the physics of photons and phonons they gave rise to Bose-Einstein distribution when I'm talking about electrons in a localized state then I have the spatial distribution now there are distribution for quarks and there are all sorts of distribution every physical particle which follows a certain amount of rules certain rules will be given by a certain distribution so distribution is not statistics it is actually physics because physics dictates how things can get distributed in energy now I also mentioned the degeneracy factor but you see for a theorist it is easy to calculate in some way this number most of the time experimentalists don't really worry about this because if you have a new material you see in this material we already know we already know how many valence band we have how many conduction band we have and this has been calculated over last 50 years people know a lot let's say you have a new material you don't even know where the band is and how they are with respect to this exact position so in that case although the GD looks very easy to calculate once you know the band structure correctly many cases you may not have the band structure known correctly so experimental is what they often do is they say well since I do not know I might do a little fudging and that's what they do they say well I'm going to still assume that it is a Fermi Dirac statistics because I don't know GD you see and what they are going to do is write the GD in terms of GD is a constant 4 6 2 whatever they will write it as in some epsilon divided by KT you know that's a constant and pull it in so that instead of talking about ED which is the donor level E sub D the donor level physically calculated from the hydrogen level you have a E sub D prime which is a effective level because you have hidden the degeneracy in the ED E sub D prime now once you have set it up you see all the calculations will come out right because this is a property of the physical property of the material and the donor so so long you have a silicon and a phosphorus combination you exactly will have one specific value for E sub D prime now if you go to germanium and you talk about phosphorus then you will have a slightly different value of ED prime but in general you see in general the ED value would have been different anyway because of the different dielectric constant because of the different effective masses it have been different anyway so why worry about it we will just set it to a particular value and experimentally determine it so that's what's called an effective donor level so this course is full of effective things right effective mass effective density of state effective donor level because we are engineers at the end of the day we will have to calculate something theory is good but at the end we need to calculate specific things quickly so we are set and that's why I'm beginning to come to the close again is now you know the physics of donors the physics of acceptors degeneracy level and all from here you can calculate this the one reason you always have to use this full expression for donors and acceptors but not necessarily for electrons and holes is because the Fermi level wherever it is since the donor level is below the conduction band most often the donor is in the degenerate level although donors and acceptors because it's closer to the mid gap compared to the conduction and balance band as a result although you can approximate the first one first two often you cannot approximate donors and acceptors as easily sometimes you can but you shouldn't take chances too often because you can be barn okay so to conclude then we talked about why intrinsic semiconductors are do not have too many electrons right this I'm emphasizing quite a bit because there's a very little difference between and semiconductor and insulated one EV band gap is a large band gap for electrons to jump from the balance band to the conduction band and whether once you have one EV or whether you have five EV at that time at a room temperature makes very little difference so therefore you have to do something about it that's donors and acceptors now the statistics of these levels are different because these levels are localized and in localized levels electrons repel each other they cannot have it in extended band they don't that's why the statistics are different so let me end here just also this final point being that the conduction and balance band also behave differently and that's something I've also tried to explain during the lecture okay so we'll use this information in the next lecture to begin to calculate for various donor doping levels what exactly are the conduction number of conduction electron and number of conduction or balance band holes I have okay we'll end here thanks