noc19-mm04 Lecture 09-Semiconductor Basics-II

[Music] [Music] so welcome to lecture nine of the scores on solar photovoltaics principles technologies and materials so let us just recap what we did in the last couple of lectures so we started our discussion on band structure very briefly I mean without going into any details and basically what happens is that when you plot energy versus wave vector there is a continuous gap and the ek diagram which is basically the band gap and met semiconductors and insulators have a finite band gap and if it is less than generally two to two-and-a-half volt electron volt then it is semiconductor and for band gap higher than 2.5 EB its we define the material normally as insulator now there was also something called as direct and indirect band gap semiconductor this director indirect bandgap semiconductors I important know at least we cannot get into details of this but basically when the conduction band minima and the valence band minima in the ek space when they coincides with when they coincide with each other then it is a direct bandgap semiconductor and well then when they do not coincide with each other when they when they shift so if you have a Maxima here minima here Maxima there so there is a shift in in the k space then it is indirect band gaps in here so this is direct and this is indirect and this has influence on the way carriers are generated and carriers are recombined in semiconductors we will get back to this later on and then we moved on to what semiconducting materials are so we defined intrinsic semiconductors and we define extrinsic semiconductors so moving forward on this discussion intrinsic semiconductors are the ones are the ones for for which n is equal to P and this is mainly those semiconductor which are where carrier generation is because of thermal excitation so you have conduction band you have valence band and electrons from this are excited to to the conduction band so this is conduction band this is valence band leaving rise to so these are now holes giving behind the hole so these are electrons and these are holes so these are mainly because of thermal excitation and as a result you have n is equal to P which is also called as ni some books also write as ne is equal to NH is equal to NI okay now the concentration is given as n E is equal to NC into exponential of sorry into exponential of minus of EC minus EF divided by kt and NH is NV into exponential minus of EF minus EB divided by K T where NC and NV are effective density of states in conduction band and this is effective density of states in in valence band now we were talking about where the Fermi levels are Fermi levels will is it the middle of the bandgap in case of intrinsic semiconductor whereas it is in the it is closer to it moves closer to conduction band edge in case of n type semiconductor any K it moves to valence band edge in case of P type semiconductor so let us move ahead with that discussion that we did last time so the way these carrier concentrations are come about so the carrier concentrations are basically calculated as so from eg to in EC to infinity sorry so this is EC to infinity for conduction band so this is for conduction band so this would be f e de into de so this is basically you can say probability of occupation de is density of states and this is over an interval de so interval and we have integrated it from EC to infinity to calculate the carrier concentration in the whole of conduction band for holes so this is for n e this is for NH for valence band it would be minus infinity to evey okay and in this case since we are looking at holes it would be 1 minus Fe de into small de again it is probability of occupation of holes density of states and this is the interval de okay so and we know what Fermi Dirac statistics is and G's and the density of states to be precise is given as for conduction band at least is given as 1 over 2 pi square to M star which is effective mass of electron we haven't gone into effective mass of theory and all that but I will give you references from which you can read about effective mass e minus EC to the power half and that for let me give you also the expression for holes just bear with me for a minute and density of states four valence band is 1 over 2 pi square 2 MH star divided by H cross square to the power 3 by 2 into e V minus e to the power half so this is valid for e greater than or equal to EC and this is for e less than or equal to e V so now when you make the appropriate substitutions you may you will and if you do the integral and of course you know the expression for Fermi Dirac statistics you will get n as NC into exponential of EF minus EC divided by kt + n ne and NH as NV into exponential exponential e V minus EF divided by kt alternatively you can write as that exponential of minus of EC minus EF divided by kt and this would be NV exponential minus of e for EF minus e V divided by kt and when you multiply these two together and E into NH it becomes NC into NV exponential of minus of EC minus EB divided by kt and EC minus EB is equal to eg divided by 2 so we get n e into NH which is equal to NI square which is equal to NC into NV into exponential of minus off so this is sorry EC minus EB is equal to eg this is minus eg divided by K T and hence and I becomes equal to and C + V to the power half exponential of minus eg divided by 2 KT okay well what is now n CN v so this n CN V is basically n CN v is n C is 2 into ma star KT divided by 2 pi H cross square to the power 3 by 2 and NV is equal to 2m H star KT divided by 2 pi H cross square to the power 3 by 2 these are the expressions for n CN and V which we didn't look at last time these are the expressions for basically this is the expression for intrinsic carrier concentration in a intrinsic semiconductor that is how you derive it now for extrinsic semiconductors we defined last time [Music] extrinsic semiconductors we said there are two types one is n-type and second is p-type n-type is doped by so if it is if it is in the context of silicon doped by phosphorus and silicon for example silicon is the base material phosphorus is the dopant and this is doped by things like arsenic in silicon so this is plus five this is plus three silicon is plus four as a result they create extra electrons and extra holes so basically since you have in these semiconductor n-type we said you have n e which is much larger than n H and p-type you have n H which is much larger than n E so so if you look at the band diagram of n o P type semiconductor okay so for n type semiconductor there are high electron density and very few holes so this is n-type and the Fermi level lies so this is let us say that EF I the Fermi level of n-type semiconductor lies little bit ahead little bit above the efi and the donor level lies very close to so this is e C this is donor level in this case what happens is that there are more holes and fewer electrons okay so here this is Evie again the mid gap position is this but the Fermi level lies here so this is EF P this is EF I EF I is equal to eg by 2 but EF p is little bit lower we will see the expressions of those in a little while this is EC okay and in this case the acceptor level lies here so we said last time that the donor and acceptor level are very close to EC and evey so that they are completely so that the atom sitting in them are completely donor acceptor and donor atoms are completely ionized so ionization energies of these so for donors we have impurities like antimony phosphorus which have if you put it in silicon for example and the antimony has a value of 0.0 0.3 9 sorry am i writing it 0.0039 evey and 4 for phosphorous this value is zero point zero four five evey for in this case in case of antimony in case of P type doping let me correct it it should be boron or aluminum not arsenic arsenic is plus five so it should be boron or phosphorus we're on our aluminum phosphorous typically antimony phosphorous antimony is used for n-type doping and boron or aluminum or gallium or indium they are used for p-type doping so let me correct myself there so this is p-type so this would be four for us dope silicon or antimony doped silicon in this case it would be aluminum or boron doped silicon and if you look at acceptors in case of acceptors the aluminum common impurities are boron and aluminum for boron the ionization energy is 0.045 electron volt and for aluminum it is 0.05 7 evey so you can see that these are very small energies as a result near complete ionization of these is obtained and since these energies are very small all the extra electrons which they bring are able to get into the conduction of aliens into the conduction band all the extra holes which are created they go they are created in the valence band so as a result since their energies are very small they are completely ionized we can write that n e in n type is equal equal to n D which is the donor concentration and n H in P type is equal to n a which is the acceptor concentration so this is for and type and this is for P type so when we say that but at the same time we also make sure mass conservation ensures that ND and NH is equal to NI square okay which means for n type and E is equal to ND hence NH is equal to ND ni square divided by ND for P type in H is equal to n a as a result and E is equal to NI square divided by n a so these are the concentration of carriers that we have okay all right so we can also write these expressions little bit differently so for example we said that ni is equal to we said that n E is equal to NC into exponential of minus EC minus EF by K T and NH is equal to NV into exponential of minus of EF - Evi divided by kt okay we also know that this any this is equal to NI for intrinsic semiconductors right okay so if you just turn the equation upside down and for intrinsic semiconductor EF is equal to EF I then we can write expression for NC as ni exponential of EC minus EF I divided by kt and NV as ni into exponential EF i - e v divided by kt okay so now for intrinsic semiconductor since n is equal to p is equal to ni so which means that NC into exponential of minus of EC minus EF / KT is equal to NV into exponential of minus of EF minus EB divided by kt okay if that is true then we can get the expression for and if EF is defined as EF I then we can write an expression for e fi EF I is equal to EC plus AV divided by 2 plus KT by 2 into Ln NV by and C and this will turn out to be C plus AV divided by 2 plus 3 KT by 4 Ln of M H star divided by M star effective mass of hole electron so if the effective mass of hole and electron were equal then EF I would be right at the mid gap position however if they were different then EF will be different okay and you can also see there is a strong there is a temperature dependence of EF as well so as the temperature changes the Fermi level also goes it's increases okay now another thing that you need to worry about a solid is since forever now if you create electron and hole which means you have positive or negative charges if you dope it with n-type impurity you have n-type impurities ionized atoms if you have p-type doping you have P type impurities however as a solid as a solid solid must remain charged neutral so which means no matter what happens is that a solid must always be be charged neutral which means that for a given semiconductor Q into n e so - any is the negative charge plus holes NH plus nd plus which is the donor atoms - the acceptor atoms which are negatively charged - na - that overall charge must be equal to zero and we know that NH is equal to NI square divided by n e so if you substitute in the above equation you can get n e is equal to nd minus n a divided by 2 plus nd minus n a divided by 2 square plus ni square to the power half and NH you can obtain as ng na - ND divided by 2 plus and a minus nd to the power 2 to the power to the power 2 + + I square to the power half now if you if we make an approximation that for n type [Music] if nd is equal to is is a lot greater than n a which is true for n type semiconductor donor impurities are much larger than the acceptor impurities remember you will always have some donor and acceptor impurities present in every semiconductor even intrinsic one but the effect is almost effect almost cancels out each other because you cannot completely purify a semiconductor to 0 impurity nothing is completely pure on the based on the laws of thermodynamics so as a result you always have some impurity some donor impurity some acceptor impurities now if you intentionally Topa semiconductor to become n-type which means ND is a lot greater than n a which means it's predominantly p-type and if ND is a lot greater than na then if you plug this in the above equation we get P is equal to + I square divided by nd that is what we said previously also right and for P type and a is a lot greater than nd which leads to so sorry let me just not write as P let's write it as NH okay and in this case and a will be equal to + I square divided by n a okay so and of course n II will be equal to nd and in this case n H will be equal to n a okay so these can also be derived now let us look at quantitatively the change in EF upon doping so we said that you have a conduction band you have a valence band [Music] okay so this is conduction band this is valence band alright and this is your band gap eg your intrinsic Fermi level is somewhere here and the Fermi level moves up for n type moves down for p-type okay and this is at eg by 2 for an intrinsic semiconductor alright and for n type semiconductor you will have this as e d and for p-type you will have this as ei this is conduction band edge and this is valence band edge okay now for a Fermi level of n-type semiconductor we can define e F n is equal to e F I plus KT into Ln of n D divided by n I and this is for n D being a lot greater than n a as well as ni ok and for a p-type semiconductor the EFP on the other hand is equal to EF I minus KT into Ln na divided by an i for in a being greater than n D so it's getting a small a and and it should be greater than and I okay so basically you can see from this expression that as you increase the donor concentration in the first equation as we increase the donor concentration the term on the right increases right and it's multiplied by K T so when your impurity concentration donor impurity concentration increases it when it's much larger than ni then EF tends to move up so you add something on EF I okay for a p-type semiconductor on the other hand since you have a minus term here as you add more and more to acceptor impurities the Fermi level tends to decrease or go lower with respect to the Fermi level of an intrinsic semiconductor okay so this explained so if you want if you have a PTM type semiconductor your formula will is likely to lie here so this is your EF and and if you have a p-type semiconductor the Fermi level is likely to lie here so this is EF P so this is what we have done so far we have just in this lecture we have looked at the carrier statistics and effect of doping on Fermi level and some things about the about P and n-type semiconductors what we will do in the next class is we will so this is let me just go through it so we looked at the carrier statistics in and intrinsic semiconductors so basically simple expression without going through complete derivation so this is how you get this ni is equal to NC n V to the power half exponential minus eg by 2 KT and then we looked at extrinsic semiconductors and the band diagram the ionization energies of donor and acceptor impurities the relation between the carrier concentration and so if I just draw these this is for n-type and this is for p-type okay and then we looked at some other relations with respect to carrier concentration the Fermi position and the Fermi position charge internally aspect which gives which ships ties to again carrier concentration expressions and the change in Fermi energy upon doping so we will continue with this carrier statistics and carrier mobility carrier motion and semiconductors in the next couple of lectures thank you [Music] [Music] [Music]