noc19-mm04 Lecture 08-Semiconductor Basics-I

So, welcome to this lecture number 8 of Solar Photovoltaics course Principles Technologies and Materials. So, we will first do a bit of recap of last lecture and then move on to the contents of this lecture. So, in the last lecture we learnt about energy band diagrams, which are basically E-K diagram, E is the energy and K is the wave vector. So, these diagrams for a single electron, the curve is like this. So, this is a parabolic curve because E is equal to h cross square k square divided by 2m, when you plot it for a lattice, the situation changes for a lattice you have to plot this energy on all the reciprocal lattices space points. So, this is you can say K which is the reciprocal lattices space points and then you get a kind of many curves for different zones. So, you have curves like these and you know you can have curves like these and then we have this curve, this curve and so on and so forth. Now, here what is important is that you so basically material if these band if these lines are curves are continuous across the space and you can always jump from one state to another without having a discontinuity then it is not a insulator or semiconductor. However, if you have a gap like what you see here across the K space there is a gap, there are no energy states present within this state, then this represents what we call as a semiconductor or insulator with the energy gap E g. And if this maxima and minima of conduction band and valence and this the band on top is called as conduction band and the band at the bottom is called as valence band and if the maxima and minima of valence and conduction band coincide with each other then it is called as a direct band gap semiconductor. So, maxima and minima, maxima of valence band and minima of conduction band they coincide in K space and if the maximas and second one is indirect band gap semiconductor, if these two do not if maxima and minima do not coincide. So, for example, direct band gap semiconductor is like gallium arsenide is a direct band gap semiconductor, indirect band gap semiconductor is a material like silicon. You can read the details from other books such as solid state physics by Kittel, this is a very nice physics book. You can also read electronic and magnetic properties of materials by R. E. Hamel. These are few good books which you can consider for building a background on these topics, we have not covered them in full detail. So, as a result, you need to get better exposure if you want to learn more about these things. So, key thing is, if you have energy states which are continuous across the EK space and you can always find energy state any at any given energy level, that means, it is a conductor, but if you have a forbidden energy gap where you do not find energy state, then it is a semiconductor or insulator depending upon the magnitude of EG. So, presence of EG, presence of a forbidden gap will mean it is a semiconductor or insulator. Now, having known this now, the next thing we want to know is what is the so, in order to calculate that in order to work out the carrier concentration. So, carrier concentration, we require first thing is density of states. So, which means, how many energy states are available per unit volume, let us say and then what is the probability of occupation, the product of these 2 will give you what is the number of carriers because suppose you have 100 energy states, but probability of occupation is only 0.5, which means there are 50 carriers which are there. So, that is what we need to do, we need to find out first what is density of states and what is probability of occupation. So, let us first look at what is density of states. So, density of states in a semiconductor, so density of states is given as we write it as DE, which is proportional to e to the power half. However, for a semiconductor, we can write it a little bit more precisely. So, for semiconductor, the band structure is like this you have conduction band, you have valence band. So, this is conduction band, this is valence band and we have a forbidden energy gap. So, if you plot just D as a function of E, it is basically a parabolic sort of behavior, basically a square root relationship DE goes as E in this fashion. Now, when you plot it now, for a semiconductor, you have a conduction band and you have a valence band. So, you need to plot the density of states in both the bands. So, density of states in conduction band is given as DCE is given as we write it as m n star or m e star rather into square root of 2 m e star into E minus E c. So, this is E c. So, here it goes from E c to infinity and here it goes from E v to minus infinity or some negative energy. So, basically E c to negative energies and E v, E c to positive energies and E v to negative energies basically. So, this goes from the equation is m e star into square root of 2 m e star into E minus E c divided by pi square and h cross cube, where h cross is nothing but h divided by 2 pi, h is the Planck's constant and this is valid for energies E greater than or equal to E c. So, of course, at E c this will become equal to 0. So, you have 0 density of states at E c and higher density of states at higher energies. So, the energy, so states increase in number as you go to higher energies and then DVE is analog of it. So, this is basically m h star into square root of m h star into E v minus E divided by pi square into h cross cube, this is for energies E less than or equal to E v. So, if that is the case, then you can plot these. So, if let us say this is E, then if I make a y axis here, let us say this is a y axis, then in this direction it goes as this and in this direction it goes as this. So, these are you can say this is DVE and this is DCE. So, we have found the density of states. Now, the next thing we want to find out is, so this comes from the quantum mechanical analysis and we have not gone into details of that, but basically we want to find out. So, basically you can say that DCEDE, what would that be? That would be number of conduction band states per cc which are in the present in the range, present in the energy range. What is the energy range here? Energy range will be E and E plus DE within that energy range. Similarly, you can say DVE would be into DE would be number of valence band states per cc present in interval DE to sorry E to E plus DE. So, here we say that E is greater than EC and here we say that E is less than EV. So, basically these are, they are numbers, number of energy states per unit volume. So, we can say that numbers per centimeter square per EV. So, you can do energy distribution in that fashion. Now, what we need to know next is, what is the probability of occupation? This comes from what we call as Fermi-Dirac statistics, Fermi. So, basically here FE, so we write a function here. So, this basically the Fermi function, we can say this basically it is a function. Fermi function depicts how many of the energy states will be filled by an electron or in the case of valence band we can call it as a hole. So, what happens in semiconductors is basically, when you have, so this is conduction band, this is valence band, you have energy gap EG, these are electrons which are present. So, at 0k you will not have any electron in the valence band all of them are filled up to here because they do not have energy to cross over. But as you increase the energy some of them will cross over and they go to conduction band and then leaving empty states behind and these empty states behind are called as holes. So, these are electrons and these are holes. Holes are also carriers because they provide states for conduction of electrons. So, FE, the probability with which an electron, probability of occupation of energy state by an electron is given as 1 over 1 plus exponential of E minus EF divided by kT, where EF is a reference energy which is called as Fermi energy and k is the Boltzmann constant, T is the temperature. So, if you plot this now here, so this function as a function of temperature if you plot it for example, at 0k when you plot FE as a function of E at 0 Kelvin for temperatures for energy less than EF, when E is less than EF at 0 Kelvin, then this will be e to the power minus infinity. So, this will be equal to 0. So, probability of occupation will be 1. However, when E is greater than EF, then this will be equal to e to power infinity basically and e to power infinity is a very large number which means it will become 0. So, at energy is greater than EF. So, this is energy EF. So, at energy greater than EF, the probability of occupation is 0. So, this is 0. At energy less than EF, the probability of occupation is 1 and for of course, the function is sort of undefined at E is equal to EF because it becomes E exponential 0 by. So, you can say that every energy state up to EF is filled when. So, you can take this as sort of 1. So, but the mean value will come at half. So, this function is asymptotic around half. So, however, so what this tells you is that at 0 Kelvin for E less than EF, the FE is equal to 1. For E greater than EF, FE is equal to 0 which means all the energy levels are filled up to Fermi level and at 0 Kelvin and above EF all the energy levels are empty and as you increase the temperature, as you increase the temperature, this curve goes in this fashion. So, as you increase the temperature, the curve becomes like this. So, this is at T greater than 0 Kelvin. So, basically what happens is that this curve on this side follows the relation FE is equal to 1 minus. So, basically you can say approximately equal to 1 minus e to the power E minus EF divided by k T and at these energy levels it follows FE is approximately equal to e to the power minus E minus EF divided by k T. So, now this number becomes negative. So, the Fermi energy at these, so the probability at these energy levels reduces and probability at these energy levels rises. So, basically what it means is that the state, the carriers in this range have left and they have gone here. Now, this is what, so basically essentially you can say that this function and the function changes around half, the function changes around a value half. So, if you increase the temperature further, this will, this curve will go in this fashion. So, its node is at half. Now, if this is the case, then so at E is equal to EF, you can say the FE is half. So, at E is equal to EF, you can say FE is equal to half and then if another observation that you make is when E is greater than equal to FEF plus 3 k T, then you can write exponential of E minus EF divided by k T is much larger than 1 and FE is generally given as exponential of minus of E minus EF divided by k T and for E less than or equal to EF minus 3 k T, this exponential of E minus EF divided by k T, this becomes less than 1. As a result, you can write FE as 1 minus exponential of E minus EF divided by k T and this is what we have written there. So, if you recall, that is what we have written here. This is the behavior. So, this energy is, so at these temperatures, these temperatures are the temperature which depict the energy which is E less than EF minus 3 k T. So, this 3 k T is sort of a number beyond which you start having significant changes in the curve on the both sides and you can approximate this function because this value becomes significantly higher as at these magnitudes. So, now if you want to find out the carrier concentration now, the carrier concentration is for a given semiconductor, we say that this is our conduction band, this is our valence band, the DE goes as this. So, this is DE. So, this is CB and this is VB and our for intrinsic semiconductor, so this is for a pure semiconductor, pure or intrinsic semiconductor. For a pure or intrinsic semiconductor, the Fermi level lies right at the middle. So, this is the position of Fermi energy. This is FE. So, you can see that, so this is, so if you now plot the, let us say, this is the value of FE and this is the value of, sorry, this, hang on, this is E and this is DE in this direction. So, y axis is energy. This is FE. So, if I now plot FE, this is 1 here. So, the FE will go as something like, at a finite temperature, it will go in this fashion, right. So, this value is approximately, this is 1 and this is 0 on the y axis. So, basically the, this region and this region, they are equal in area. It is symmetric around this value, that is half. Now, what is NE? So, if you want to now plot NE, this is conduction band, this is valence band, this is Fermi energy. So, NE is now equal to FE into DE. So, when you plot these, you get these lobes, which show you, so this is energy and this is N or P. So, electron hole concentration. So, this is basically the hole concentration and this is electron concentration and in an intrinsic semiconductor, N is equal to, you can say, NE or NH, instead of N and P. So, we can say, NE is equal to NH because the Fermi function is symmetric around the Fermi energy and Fermi energy lies right in the middle of the band gap. This changes when you make a P or N type semiconductor. So, we would do that when we come back again to this. So, now what we do is that, we go to essentially, so we say that, for an intrinsic semiconductor, for an intrinsic semiconductor, NE is equal to NH, this is taken as NI and E and EG is equal to taken as the difference of EC minus EV. So, you can take mod of this basically. So, if this is your conduction band edge, this is your valence band edge, this is valence band, this is conduction band, this is EC, this is EV, then whatever reference you take, the EG is the difference between EC and EV. So, this is your band gap and this is given as, this NE is equal to NH, for an intrinsic semiconductor, this NE is given as NC into exponential of, so we can calculate using quantum mechanical calculation, same density of states. Basically, you will calculate NE, E as a function of DE into FE. So, this is what will give you the electron concentration. Similarly, you can calculate the whole concentration NPE as D, so this is DCE, this is DVE into 1 minus FE because we are looking at the other side that is 1 minus FE. So, if you do that, then we get NE is equal to NC into exponential of minus EC minus EF divided by KT and we get NH as NV into exponential of minus of EF minus EV divided by KT, where NC is essentially effective density of states, density of states in the conduction band and NV is equal to effective density of states in valence band. And you can see this from this expression, if you look at this NC is equal to nothing but NE divided by some exponential function, this is EC minus EF divided by KT. So, basically this is the probability function, you can take this upstairs, you can take this up and it becomes exponential EC minus EF divided by KT. So, this is the probability function, 1 over exponential whatever and this is nothing and this becomes the concentration. So, essentially the concentration divided by probability function is nothing but that effective density of states. So, that is why this NC is taken as effective density of states in conduction band and NV is taken as effective density of states in the valence band. Now, so we say that NE is equal to NH is equal to NI, so we can also say that NI square is equal to NE into NH, which means it will be NC into NV into exponential of minus EC minus EF divided by KT into exponential of minus of EF minus EV divided by KT. And if you put them together, it becomes NC NV into exponential of, so this becomes exponential minus of EC minus minus EV divided by KT and EC minus EV is equal to EG. So, this becomes equal to NC into NV exponential of minus EG divided by 2KT. So, NI is NC NV to the power half exponential of sorry, 2 will not be here, now the 2 will come. So, basically you can see that intrinsic carrier concentration is exponential function of temperature and band gap. So, as the band gap decreases, the NI will increase for a given temperature and as the temperature increases, the NI will increase. So, the lobes that you saw there, they will, so this is at let us say temperature T 1. So, so this lobe will be let us say a temperature T 1, if the temperature T 2 is higher than this will become bigger. So, this will be at T 2, but they will remain equal in size. So, T 2 is greater than T 1. So, you can say that NI at T 2 will be higher than NI at T 1. So, this is what will happen for intrinsic semiconductor. Now, intrinsic semiconductor is has lower conductivity and to make devices generally we dope the semiconductor with a dopant and these doped semiconductor are called as extrinsic semiconductors. And these extrinsic semiconductors have 2 types one is of N type and second is of P type. N type is the one where you have doped with ions which donate or doped with impurities which are also called as dopants which donate extra electrons and these are doped with impurities which donate extra holes. So, these electrons will go in conduction band and these holes will go in valence band. So, generally for a for a some for something like silicon if you have silicon, silicon has 4 of these electrons. And so you have multiple silicon atoms and these silicon atoms are coordinated in such a manner. So, that each silicon is coordinated by 4 silicon atoms. So, you have 1 silicon atom here, another silicon atom here, another silicon atom here, this is a simplistic design although the tetrahedral coordination is not like this. So, this is silicon, this is silicon, this is silicon, this is silicon. So, it has 4 fold coordination from each silicon. So, that you have covalent bonding and you have sp3 hybridization. Now, when you replace this silicon with phosphorus let us say. So, this is silicon, this is silicon, this is silicon, this is silicon, these have 4 extra. So, this electron, this electron, this electron, this electron they are all they all are shared, but phosphorus brings an extra electron because phosphorus is plus 5 impurities silicon is plus 4. So, this extra electron is the one which is because of phosphorus atom and this extra electron goes to conduction band and it makes it a n type semiconductor. So, this extra electron is in goes to conduction band leading to a n type behavior. So, what it leads to in terms of band representation is we have silicon whose valence band is here, we have conduction band here, this is E v, this is E c, when you dope it with phosphorus, phosphorus creates a energy level which is called as donor energy level E d and phosphorus for each phosphorus atom if you are able to ionize all of them, then these electrons cross to conduction band provided this donor ionization energy the difference between E v minus E d is a small enough of the order of k T. So, which is generally the case and phosphorus will ionize and give rise to give all its electrons extra electrons away to conduction band. So, in this case the number of carriers basically N d becomes equal to almost equal to N e. So, if you for example, dope with 10 ppm phosphorus 10 ppm will become 10 to the power minus 6 parts per million. So, 1 part per million which means 10 to the power minus 6 for every part. So, basically in a million you have 1 10 of phosphorus atoms. So, based on the concentration of phosphorus atoms you will have the carrier concentration. So, in a given lattice if the number of atoms is 10 to the power of 24 per centimeter cube and electron an intrinsic carrier concentration is about 10 to the power let us say I do not know 14 per centimeter cube at a given temperature let us say at about 298 Kelvin. So, let me get the exact numbers. So, N i is about 10 to the power 10 for silicon for silicon it is about 10 to the power 10 per centimeter cube at 298 Kelvin. Now, this number can increase to if you if you dope phosphorus with 10 to the power minus 10 ppm 10 ppm is 10 to the power minus 5. If you have 10 to the power 24 atoms per centimeter cube this becomes. So, N d will become 10 to the power 19 per centimeter cube. So, you can see that N d is much higher than N i and if all the N ds are able to contribute to electrons which means N d is almost equal to N e and this is much more larger than N i. So, for a N type semiconductor N is equal to N d. However, because of mass conservation N e into N h is always equal to N i square when you increase the when the electron concentration suddenly increases then recombination also decreases increases. So, when the recombination increases then of course, the whole concentration also tends to decrease in the valence band. So, as a result this relation has to be always followed. So, for a N type semiconductor this N h is equal to N i square divided by N e or it is equal to N i square divided by N d. So, for a N type semiconductor some in summary you have N e is large almost equal to N d and N h is small as against the intrinsic semiconductor. If you take for a P type semiconductor P type semiconductors are doped with impurities such as boron which are trivalent impurities phosphorous pentavalent these are trivalent. So, when these are trivalent then if you make the same configuration you have these silicon atoms silicon and then you have boron. So, boron comes with only 3 electrons silicon has 4 electrons to share. So, this boron atom now it basically you can say now where does it need to bring an electron from it does not it cannot bring an electron is short of electrons. So, this creates a hole in the valence band. So, in the in the in the in. So, this will be basically you can say a 3 electrons are shared and this is basically you can say there is a hole there. So, basically what happens is that in the band diagram perspective this is conduction band this is valence band this is E c this is E v the P doping will causes energy will create energy level E a accepted energy level which will take the electron from here to here to fulfill its configuration for the bonding and leaving a hole. So, it will take a electron some from somewhere here leaving a hole in the lattice somewhere. So, to make this 4 4 to make this covalent bonding it will create a hole somewhere else. So, this is the hole that is created it is taken electron away from some silicon. So, essentially it will create holes because these electrons will go here to the. So, this is acceptor energy level. So, this is again this energy level is E a minus E v is of the order of k T and that is how P type doping will work for a P type doping in the analogous manner this N h is equal to N a as a result N i square is equal to N e into N h we can write this is equal to N a into N e. So, N e is equal to N i square divided by N a. Another thing that happens in these semiconductors P and N type is the Fermi level tends to shift. So, change in Fermi level. So, the Fermi level for a intrinsic semiconductor is here this is E f i for a P and N type semiconductor. So, this is E f p sorry E f n and this will be E f p the Fermi level tends to shift up for a N type towards the conduction band this is E c this is E v and Fermi level tends to shift down for a P type semiconductor. Now, this has repercussion in terms of what we drew earlier the energy band diagram. Now, you can also you can do the similar energy band diagram picture that we did earlier for a we drew that. So, now if I draw this in this fashion let us say this is E c this is E v and for a N type semiconductor this is E f somewhere on top this is E c this is E v. If you plot density of states D e, D e will go in this fashion on this side like this on this side in this fashion. If you now plot the Fermi energy the Fermi energy would be the Fermi energy will go. So, this is. So, this is Fermi energy now we want to plot the probability of occupation probability of occupation will go as if this is 0 this is 1 this is half it will be symmetric around this point. So, essentially you can say it will go in this fashion. So, this is 1 this is 0 this is F e. So, the center point has shifted to top now. So, now if you want to plot the electron concentration. So, electron concentration I can use the same color for lines these are the blue lines here again the blue lines here. Now if I want to plot the electron concentration since there is more area here this gives a bigger lobe and this gives a smaller lobe much smaller lobe. So, this is N e and this is N h and you can see that N e is much larger than N h and this is because the Fermi function has shifted. So, it is not drawn very nicely, but it is sort of little it is like this for a p type on the other hand you can draw it in a different manner for a p type if this is if this is E c this is E v the Fermi level is now here E f p this is E f n again I want to make the same things. So, here I want to plot D e I want to plot F e and I want to plot N e or P e N e or N h e or basically carrier concentration. So, if I want to plot for this, this will remain same the same behavior in this case now the Fermi level has come here. So, Fermi level has come here now you can see that the Fermi level will sorry if this is let us say 0 this is 0 this is half this is 1. So, it will go something like this in this fashion. So, it will have more area. So, this is 1. So, it will have more area here, but it will have no area on the other side if it is in fact, I am not drawn it very well, but it would be something like this. So, there will be some area on this side let me see if I can draw it a little better basically you can say this curve will go something like this and it will go something like this on the other side touching 0 here and now if you plot electron and hole concentration the hole concentration will be bigger and the electron concentration will be smaller. So, this is N h and this is N e and N h will be much larger than N e. So, this is the difference between the P and N type semiconductors in terms of their concentration. Now, so, this semiconductor the last thing I would come to what is what is N c and N v. So, expression for N c and N v we will see in the next class and, but we will just see how the semiconductor behavior changes as a function of. So, what happens is that when. So, this is your N type this is your P type this is E v this is E c for a N type semiconductor if this is the intrinsic level for a N type semiconductor this Fermi level is somewhere here and for P type the Fermi level is somewhere here this Fermi level keeps shifting to upside as you dope more and more. So, with doping it increases up. So, N type doping right and this will shift towards E v with more P type doping and if you keep doping them they will keep shifting up or down. So, when the difference between E c and E f n is less than 3 k T or when the difference between E f p and E v is less than 3 k T then they tend to become what we call as a degenerate semiconductors they become degenerate semiconductors or more like a metallic behavior they become more and more conducting. So, you will see in certain circumstances N type is heavily doped P type is heavily doped which means their Fermi levels are very close to conduction band or valence band. So, this is what we have done today we have looked at the carrier concentration and the reason behind having higher and lower concentrations and expressions and this on this we will further dwell in the next class to look at little bit more analytical treatment of carrier concentration and Fermi energy in the next lecture. Thank you.