 So, now let me start with my lecture today. I will go little bit more into detail about the semiconductors and the charge carrier in the semiconductor. And why we need to go in detail because in the whole operation of a solar cell is determined by the motion of the charge carriers. So, charge carriers includes electron and hole. And therefore, we should be knowing how many electrons are there, how many holes are there, how the number of electron and hole is changed by the doping, how the number of electron and hole pair is changed by the light falling on your solar cell, how this carriers, how the electrons is moving in a semiconductor. And what is the implication of the motion of this carriers on the performance of the solar cell or when a solar cell or we all know that you know solar cell is a p-enjunction device. So, when a p-enjunction device becomes a solar cell. So, there are various things and again do not worry about if there are some complicated equation do not worry about it. Just try to understand the basic concept behind all this and the book that you have got all these details are there in the book. So, you can read more details. So, just do not worry about it, just try to understand the basic concept regarding the charge carriers and their motion and the generation recombination. And this all determines the physics behind a solar cell. So, last lecture we have learned about the electronic arrangement of atoms formation of the bands, we have learned about the direct and indirect band gap semiconductors. So, now in this lecture we are going to look about the charge carrier and their numbers. So, we look about p-type solar p-type semiconductor, n-type semiconductor, intrinsic, intrinsic, extrinsic, majority carriers, minority carriers and how many numbers. The most important thing is how many numbers of carriers are there because eventually when the light falls current should flow and the flow of current is given by flow of current is nothing but rate of flow of charge. So, rate of flow of charge means we should know the number of charge carriers. So, this is what we are going to discuss in this lecture. This you already know that the many semiconductors that can be used for solar cell application and crystalline silicon is one of the most common semiconductor and 80 percent or more of the photovoltaic modules produced in the world is actually manufactured in silicon. But the combination of cadmium and toluoride for example, this is referred as a 2 6, 2 6 material zinc sulfide, zinc sulfide. These are the semiconductor that is used, 2 6 material. So, there is many combinations of semiconductors also used. You can combine 1 semiconductor, 2 semiconductor, 3 and 4 in order to get the best electrical and optical properties so that you can make good solar cell. Yesterday we discussed about the formation of the bands. Why the bands form? So, when many atoms come together energy levels split because it follows the powerless explosion principle because no one energy level cannot occupy more than 2 electrons. So, they split little bit energy and this when many atoms come together trillions and trillions of atoms, the split the splitted energy level becomes almost continuous and they form what is called energy bands. So, if you look at the energy versus space. So, remember what is y axis of this y axis is space and what is the x axis sorry y axis is energy and x axis is space. So, when we look at the band gap there is a some space in energy there is some energy levels where electron cannot occupy the position that is called the energy gap and you have the valence band which is normally filled at 0 Kelvin all the energy levels all the houses available are filled by the electrons at 0 Kelvin. There is a conduction band where there are a lot of energy levels are available, but they are all empty it is available and any electron which has a higher energy can go from low energy to high energy and occupy those levels. So, this is the situation of a semiconductor at 0 Kelvin conduction band is completely empty and valence band is completely filled. Now, when you increase your temperature more than 0 Kelvin what will happen some of the electrons in the valence band will get enough energy to get excite themselves and go to the conduction band. Some of the electrons will get enough energy go to the conduction band. In this way what is possible now that the when the there is a partially empty conduction band and partially empty valence band is available a charge transport can happen, but in the field band it cannot happen and why I will show you. So, for example, if you have the conduction band valence band in the valence band there are many electrons there are many electrons setting it and I said that in the when it is a field at 0 Kelvin all energy levels are occupied right. So, this electron suppose it moves here this moment is only possible only possible if this electron moves here right otherwise there is no place what does it mean and if this electron moves here this is possible only when this comes here what does it mean the net flow of charge is 0 net flow of charge is 0 means current conduction is not possible and in conduction band there are no carriers there are no carriers no moment is happening here. So, current conduction in due to the conduction band is 0 current conduction due to the valence band is 0 and therefore, net current is 0 ok. So, therefore, what is the conclusion the conclusion is that the field conduction band and completely field valence band and completely empty conduction band cannot conduct current ok. So, field band cannot conduct current or completely empty band cannot conduct current. So, what can conduct current partially field band partially field band why it is possible the partially field band can conduct the current I will show you again on the on the sheet suppose again you have this scenario now your temperature is higher than 0 Kelvin as a result of that there are electrons here. But let us say this electron has excited and gone to the conduction right and now there is a hole here absence of electron is a hole and you have the other condition exist this is the condition what about this electron it can move right it because there are many houses available. So, it can go from here to here it can go here to here and so on. So, this electron can move what about this electron can this move. So, yes this electron can go from here to here and this is available house. So, no electron has to come back to the reverse direction and therefore, now this is a net flow of electron taking place. So, partially empty or partially field bands can conduct current very important and what is current flow of charge rate of flow of charge is current rate of flow of charge is current. So, partially field and partially empty bands are required the one very good thing about the semiconductor is this control about the creating partially empty and partially field band is with us and this control is that and we can actually change the number of electrons and number of holes in a semiconductor by a process called doping. So, that control is with us and that is one of the most important control that we have in the in the in the science and technology without this control without this changing the game or the changing the number of electrons and holes in semiconductor the operation of a any integrated circuit any chip any mobile any solar cell is not possible. So, we have this control we learn more about that. So, there is some terminology. So, when electron when electron goes from a valence band to conduction band it results in a generation of a electron and generation of a hole. So, if this hole is called the EHP electron hole pair. Note this point that silicon atom density is 5 times tends for 22 atoms per centimeter cube in a silicon which is the commonly used semiconductor for solar cell there are 5 times tends for 22 atoms per centimeter many atoms at room temperature because of the thermal energy of the room temperature 25 degree centigrade. Some of the electron goes to the conduction band and creates electron hole pair normally in silicon the room temperature the number of electron hole pairs are about tends for 10 EHP per centimeter cube. What does it mean? It means that there are tends for 10 electrons and there are tends for 10 holes also because it is a pair right it is a electron hole pair. So, at room temperature in silicon at there will be tends power 10 electrons per centimeter cube and there will be tends for 10 holes per centimeter cube. So, this is the concentration of electrons in silicon at room temperature and this concentration is referred as a NI intrinsic carrier concentration and this hole concentration is referred as PI intrinsic hole carrier concentration. One very simple thing we you can answer now very easily that the number of this carriers this carriers and this carrier at room temperature will depend on the band gap right will depend on the band gap. If I another semiconductor which has the higher band gap ok. So, I have same valence band level, but very high conduction band level ok. So, this is my conduction band age and this is valence band age this was a silicon, but this is let us say gallium arsenide ok. We know that gallium arsenide has higher band gap your temperature is same. So, the, but your band gap is higher now. So, energy required to go for electron from a valence band to a conduction band will be higher. What will happen to the electron in hole concentration at room temperature in in gallium arsenide what will happen? It will be higher or lower because now the more energy is required to in a gallium arsenide more energy is required for electron to grow form valence band to conduction band. There is a less probability that at room temperature the electron will get excited. And therefore, in the higher band gap material the electron in hole concentration at room temperature or what is referred as a intrinsic carrier concentration Ni n Pi will be lower. And if the band gap is lower like germanium, germanium has a lower band gap as compared to silicon. In germanium the electron and hole concentration at room temperature will be higher ok. One other point that you can note from here because it is a pair because it is a pair. So, which means Ni has to be equal to Pi ok. A electron is generated and then only hole is created. And therefore, the electron concentration and hole concentration in intrinsic semiconductor has to be equal. So, hole concentration is equal to the electron concentration. This is what I am saying that Ni is equal to Pi and this in silicon is about 10 electron hole pair per centimeter cube. This we will discuss later the purity of silicon ok. Now, what I am showing here is the recombination rate is equal to generation rate. So, this transfer of electron this transfer of electron because of the thermal energy right. Electron from here has gone to the conduction band because of what energy it got the thermal energy. And because of this thermal energy it is called the generation process ok. But electron can also come down and occupy a hole right. So, this process is called generation and this process reverse process is called recombination ok. So, when electron gets excited because of the thermal energy it goes from valence band to conduction band it is called generation of electron. When electron comes break comes break from the conduction band to the valence band by losing its energy it called the recombination. And typically a generation rate will be equal to recombination ok. We are talking about intrinsic semiconductor where we have not made an input it is a pure silicon. Intrinsic means pure silicon no other impurities are there ok. It is pure silicon that is the meaning of intrinsic perfectly pure silicon ok. But what we want? We want control over the conductivity. We want control over the conductivity. We want to change the number of electrons. We want to change the number of holes. So, that conductivity of the material can be changed and therefore, we get the control over the performance. So, as Professor Farnadis was telling you there are diodes which are either uses a diode is this uses MOSFET some is uses a BJT and IGFT and what not. How the same semiconductor can give you different performance? Because there are different kind of junctions and because there is a different carrier density there is different electron and hole density. Therefore, the control over the number of carriers in a semiconductor is very important. And I am repeating again that we are we are lucky to have this control in semiconductor and that is one of the most important thing that human kind has is the control over the charge carriers in semiconductor and because of that operation of every possible circuit is there ok. So, we want we want to increase the number of electrons in some cases we want to we want to increase the number of holes and that will bring me to the two concept in semiconductor. So, we have the three if you look at the category of semiconductor we have ok. So, we have one category is called intrinsic semiconductor SC ok. What does it mean? Highly pure no doping no impurities no desired impurities. Second we have P type semiconductor and we have N type semiconductor fine. P type semiconductor will have lot of positive charges ok. We will have holes many holes ok. N type semiconductor N represent for negative P represent for positive N type semiconductor will have many electrons. What is the situation intrinsic semiconductor? Electrons is equal to hole the number I am talking about number here ok. So, neither electrons are more nor holes are more intrinsic semiconductor both electron are equal to hole concentration. In P type semiconductor there are many holes more holes as compared to electrons. In N type semiconductor there are more electrons as compared to hole. The important thing that we should learn here is how many is the number ok. The 5, 10, 1 million, 1 billion, 1000 billion what is that number? We want to know that number because once we know the number we can control the conductivity we can control the performance of the device. We want to know the number and that is what we will try. So, first of all how to change that number? How to make a semiconductor or a crystalline silicon or intrinsic semiconductor majority of holes? How to make the holes? How to create the holes? And for that we have to look at this diagram ok. So, semiconductor is represented in two ways one we have looked the energy band diagram. What is the energy band diagram? You represent that there are conduction band, valence band, y axis is energy and x axis is distance. Another way to represent semiconductor is by the by the it is a bonds ok. So, here silicon has this kind of arrangement if you do not know the details do not worry about it. So, this red circles are actually silicon atoms ok. So, each silicon atom is and I have shown the example here. So, this is typically a intrinsic silicon. So, each silicon atom has you know you remember silicon has 8 14 electrons how they are distributed 2 8 and 4. So, in the valence there are 4 electrons ok. So, each silicon has a 4 electron, but in order to have a stable orbit you need 8 electron. So, what silicon do? Each silicon atoms partner with the neighbor ok. In order to make a stable a silicon atom has 4 valence electron, but you want to make a stable material. So, therefore, each silicon atom should have 8 valence electron. How to get that valence? It makes a partner with the neighboring atoms. So, what they do? They share electron. Similarly, it makes another partner with another neighbor it is the share electron. The share electron in the share electron with the neighboring ok. In this way now there are 8 electrons and how this electrons are there they are shared between the 2 atoms. This is called the covalent bonds that forms in silicon covalent bond. Do not worry about the bonds also the important thing is that now this is a stable material. And this kind of order will continue if it is a monochrist line this order will continue infinite that is what we discussed right. Same order continues infinite. So, this is another way to show the material this order will continue infinitely. What is this line showing here? What is this line showing here? It is showing a electron ok. So, when this line is absent what does it mean? The electron is absent ok. So, suppose if I show here if I show only 1 line 2 line here 2 line here 2 line here this means that 1 electron here is free moving ok. So, it is available here. So, it is not bonded with this item and that is what is generation of electron hold here right. And there is 1 possible electron that can come here means there is a hole here possible electron that can come here that is a hole. So, this electron is a free electron. When we want to create more holes what we do is we take a material like aluminum. Remember aluminum comes from the group 3 group 3 means valence 3 ok. I will go back to the same slide again here. So, when we have group 4 means all these items carbon silicon germanium has 4 valence electron all these items boron aluminum gallium indium has 3 valence electron all these items zinc and cadmium has 2 valence electrons. If you go to the other side phosphorus arsenic antimony has 5 valence electrons 5 electrons in the outer orbit and sulfur selenium tellurium has 6 valence electron. So, what will happen now if I want to if I replace 1 silicon atom with the aluminum atom what will happen because silicon because aluminum has only 3 outside electron 3 valence electron it has 1 shortage of electron to make a pair it has 1 shortage of electron and shortage of electron means what absence of electron means what hole ok. So, therefore, if you put 1 aluminum atom in silicon you get 1 hole ok. If you put 10 aluminum atoms in silicon you get 10 holes. If you put 1 million atoms of aluminum in silicon you get 1 million holes ok. So, by putting aluminum atom in silicon what you are doing you are creating as many number of holes as the number of aluminum atom. Are we changing electron concentration? No. So, therefore, impurities which are from column 3 boron aluminum indium gallium this all can become a what is called the acceptor impurities ok this because it is having a shortage of a electron here. So, it can accept electron and therefore, it is called the acceptor impurities. So, basically when you put aluminum in silicon you create hole right. So, now if you and this process is called doping putting other impurities and these are the desired impurities ok. Aluminum is desired why it is desired because we want to control the number aluminum is desired it is called desired impurity. So, when you put 1 aluminum atom it gives me 1 hole if you put 1000 aluminum atom it gives me 1000 holes ok. If I put 10 this for 15 huge number aluminum atoms per centimeter cube it gives me 10 this for 15 holes per centimeter cube right. So, in this way by putting aluminum in intrinsic silicon we are doing the doping and this doping because of this doping we are creating holes many holes and therefore, this semiconductor becomes p type semiconductor. Is this number is a large number hence for 15 hole is this a large number is this a large number then the question should be with respect to what. So, the large number with respect to how many atoms of silicon are there ok. So, the atomic density of silicon you know I told you earlier atomic density of silicon how much is this is 10s forward 22 atoms per centimeter cube. So, as come 4 10s forward 22 atoms 10s or 15 is a very very very very small number 1 in 1 billion silicon atom there is only 1 atom of aluminum. So, this is not a very large number just to tell you that most of the silicon still remains silicon only very small percentage of aluminum can actually create that many hole concentrations that here ok. Now, because of this aluminum what you what happens is aluminum creates another energy level remember every atom has its own energy level. So, when you put aluminum silicon it creates its own energy level for for for aluminum the energy level is created very close to the valence range ok. Now, because this energy level are so close that at room temperature may all the electrons actually will go and occupy this energy level and therefore, you are creating holes. So, this is how you can understand that by putting aluminum you are creating large many holes. How many holes 10s or 15 per centimeter cube aluminum atom means 10s or 15 holes per centimeter cube and the rest is simple ok. If we need to generate 10s for 16 holes per centimeter cube in silicon what should be the number of aluminum atoms per million atoms of silicon. So, all that question let me tell you that aluminum is normally not used for the making p-tap silicon the boron B ok boron is also coming from the third column. So, boron is normally used for the making p-tap silicon ok fine. So, with this analogy what what are the things you understand now by making the the atoms from the column 3 in periodic table like boron aluminum you can create holes. The number of holes is the same as the number of doping atoms they are putting ok and this energy level because of this doping of a p-type impurities like boron aluminum is created near the valence band right. So, this is p-type semiconductor and more number of holes. Now, similar analogy you can extend it for the n-type semiconductor where you want to create more number of electrons ok. So, if I ask you question what are the impurities you should put for the creating more number of electrons in a silicon then answer should be that I should have extra electron to give I should have extra electron to give. So, that we have told it now we have silicon here if I have some other impurity I have another silicon another silicon another silicon each silicon has is one atom one electron which is shearing with the neighboring silicon suppose I put phosphorus here. Now, phosphorus is a pentavalent it is 5 electrons in the valence it will share one electron with this atom it will share one electron with this atom it will share another electron with this atom and it is left with one extra electron. So, this one extra electron will be easily available. So, now, putting one phosphorus atom you got one electron same analogy you can continue one phosphorus atom will give one electron right 1000 phosphorus atom will give 1000 electron and 10 power let us say 17 phosphorus atoms per centimeter cube will give you transfer 17 electrons per centimeter cube same analogy as we did in the p-type n type also now you put more pentavalent impurities like phosphorus arsenic and you can create more electrons. So, this kind of semiconductor will because there are more electrons it is called n type. In the previous case in the previous case this was because we are putting the aluminum or as I told you boron then it this becomes a p-type. Now, in this case it is called n type fine the number is clear that you can put more numbers by putting the n-type impurities. Here I have given example of antimony and not spend time with the slide now phosphorus arsenic antimony because they can donate one electron they be they called donor impurities and in the n-type semiconductor majority carriers are electrons in the p-type semiconductor majority carriers are holes. Similar to the in in acceptor impurity or p-type the energy level was created close to the valence band. In this case the donor impurities like phosphorus arsenic will have their own energy level and this energy levels are close to the conduction band and because this energy difference between this donor energy level and the conduction band energy level is small. So, electrons are easily donated and now this electrons are in a partially empty conduction band and they can conduct the current. I will not go. So, let let me come to this directly that that is that is what I told you n intrinsic carrier intrinsic semiconductor the electron concentration which is ni is equal to hole concentration is about transfer 10 numbers per centimeter cube. If donor density is Nd and acceptor density is Na, Nd is the concentration of doping atoms. If donor density is transfer 16 atoms per centimeter cube what is N0 or electron concentration? N0 is transfer 16. If accepted impurities which is boron aluminum is transfer 16 what is the what is the hole concentration transfer 16 that is I am telling doping number of doping is equal to the number of either electron created or hole created. In reality things are little different, but do not worry about that. This is good enough to understand right now that if you doping we transfer 16 phosphorus atoms you create transfer 16 electrons. Notice one thing here that now the electron concentration is called N0 and the hole concentration is called P0. Here electron concentration and hole concentration was Ni Pi intrinsic i stands for intrinsic. Here 0 stands for the equilibrium condition means this is the condition where there is no temperature no external force no electric field no bias no magnetic field is applied that is what 0 means it is meaning equilibrium. So, I will put here for you again summary on the whiteboard that Ni is electron concentration Pi is a hole concentration and this refers to the intrinsic situation and we also talk about N0 and P0 these are the equilibrium equilibrium concentrations equilibrium electron and all Ni and Pi N0 and P0. There is a very interesting relationship between N0 P0 and Ni which is equal to Pi. We will come back to that. Now, one thing is fine we have so many electrons and holes we have so many electrons and holes we have this is our conduction band this is valence band H this is conduction band H conduction band is above this valence band is below this and this is this is the energy gap. So, suppose I am making N type semiconductor suppose I have N type semiconductor and I am doping with the phosphorous atoms such that there are 10s for 16 atoms per centimeter cube and therefore, there are 10s for 16 electron per centimeter cube where this 10s of 16 electron will be this 10s for 16 electron will be where it will be in the conduction band. When I talk about holes they are in the valence band and when I talk about electrons they are in the conduction band this is the energy axis you know this is the space axis fine. So, there are lot of empty steps available. So, where are these electrons whether they are here they are here they are here because this whole energy band right you know that this is a band conduction band. So, electron electron can be here, it can be here, it can be here, it can be here, it can be anywhere. That is what is given by what is called the Fermi Dirac distribution ok. Just try to understand that Fermi Dirac distribution is actually one kind of statistical model that explains the possibility of finding electron at a given location ok. So, let me read here, it defines Fermi Dirac statistics defines the distribution of carriers over available energy states following the Pauli's exclusion principle. Distribution electron over the range of allowed energy level at thermal equilibrium, remember thermal equilibrium means room temperature is given by like this ok. So, this is probability density function ok. F e is the probability of a finding electron at energy level e and the probability is given by 1 over 1 plus bracket e minus E f is any energy level and E f is the Fermi energy level and I will come back to that what is Fermi energy. K is the Boltzmann constant which is given here the values are given here 1.38 and stands for minus 23 joule per Kelvin and T is the temperature in Kelvin ok. So, this is this gives the distribution and let me go to the whiteboard here. So, basically this function this Fermi function f e Fermi function f e will tell me what is the probability that electron is here, what is the probability that electron is here, what is the probability that electron is here here. This is a probability density function. Similarly, we can write a probability density function for the whole ok. If the probability to find a electron is f e. So, the probability to find a hole what is hole absence of electron. So, probability to find a hole is what 1 minus Fe ok. If the probability is to find a electron is Fe the probability to not find a electron which is whole is 1 minus Fe. So, this is the probability to find a electron and probability to not find a electrons 1 minus Fe. I just show you how this graph this probability density function is plotted. First of all note one thing E f is the Fermi level ok. Now, when I put energy E equal to E f what will happen here? When you put E equal to E f this will become 0. E raise power 0 is 1. So, we have 1 over 1 plus 1. So, 1 over 1 by 2. So, at Fermi level the probability of finding a electron is 1 by 2 that is how the Fermi level is defined. It has otherwise no physical significance it is from the statistical point of view we say that Fermi level is energy where Fermi level is energy level at which the probability of finding electron is half 0.5. So, that is how you can draw the statistical that probability of finding electron is 1 above the Fermi level and probability of finding electron is 0 in the 0 Kelvin, but actual at actual higher temperature or the room temperature the curve is not between 1 and 0 it has a smooth transition ok. So, that is what I told you the probability f is the probability to find electron that a electronic state is occupied and 1 minus f is the probability that state is not occupied it is empty ok. So, this is the this is how we can see that in this case where the intrinsic silicon the lot of electron all the energy levels in valence band is actually filled and therefore, probability is 1 probability is 1. Here lot of electron and available energy states or houses are empty and therefore, probability is 0 ok. So, this is my probability that electron is available or not available, but what I am interested I am interested in finding the number of electrons ok. So, look here what will happen in the n type silicon because now we are doping and we are increasing the number of electrons in the in the conduction band and therefore, now your probability becomes non-zero greater than 0 right ok. Similarly, here probability of finding electron is 1 because all the states are filled everything is filled. So, there is probability of finding a hole is 0, but when you make a p-type semiconductor there are holes available here when you make a p-type semiconductor holes available here and therefore, probability of finding a electron becomes less than 1 which means probability of finding a holes become more than 0. So, this is how your probability density function gives an idea about where is the electron and how many are there. What we are interested is you are interested in finding the number of electron in terms of the problem ok. So, look at this equation interestingly it defines that if I have certain probability of finding a electron at a energy level and if I if I know how many such energy levels are there, let me repeat. If I know the probability of finding a electron at a given energy level is let us say x and n is the number of energy level is y then if I multiply x into y I get the number of electrons ok, I get the number of electrons ok. So, my number of electron will go from EC conduction band H ok, this is my conduction band H and the above ok. So, my probability density function is this the I am sorry this is the number of states the density of states the number of energy level and this is my probability and the product of the 2 and E n is the number of energy levels available and Fe is the probability that that particular energy level is occupied. If I multiply by this 2 I will get the carrier concentration. So, typically I will get this kind of arrangement ok. So, that lot of carriers are there in conduction band close to the conduction band H ok. Look at the conduction band H close to the conduction band H and similarly intrinsic carrier lot of holes are there and number of electrons are equal to number of holes the area of this green patch here is the number of holes. Let me explain you this point again. So, I know the number of energy levels available to me is n e the number of and the number of and the possibility that those energy levels are occupied is Fe and if I multiply this 2 if I multiply this 2 I will get the number ok, but number of electrons, but this number is at energy level E right, but if I draw the valence band and conduction band this is the edge of the conduction band this is the edge of the valence band and there are many energy levels above this and there are many energy levels below this ok. So, if I want to find out the electrons in all energy levels I must integrate this from conduction band H to infinity. So, that is why if I integrate this conduction band H to infinity I get the number of electrons right. Now, this is for finding a electron suppose here I am interested in finding holes hole is the absence of electron ok. So, if the probability that electron is occupied at energy level is Fe the probability that electron is absent is 1 minus Fe and if I multiply with the number of energy levels available it is n e and again if I integrate this from where to where now I have to go to lower energy level I have to integrate it from E v to minus infinity this side then I get the number of holes that is what we are talking about ok. So, this is the mathematical way of representing and this is the graphical way of representing this is how you can represent graphically. So, this was the case what happens for the intrinsic this is the case what happens to the n 0 now n 0 you see that there are more number of electrons and less number of holes or number of electrons less number of holes and your Fermi level is closer to the conduction band. In the p type silicon your Fermi level is closer to the valence band and you have more number of holes as compared to electron. This is what we have learned already that if you put more aluminum atoms or boron atoms in your silicon you get more number of holes if you put more phosphorus atoms you get more number of electrons ok. So, this is simple to understand that how we are creating p type semiconductor how we are creating n type semiconductor and more importantly we can find out the number of electrons in n type and we can find out the number of holes in p type. This is further elaboration of that mathematical thing that we have presented here we have stated here. So, do not worry about it this is extra information those who are interested otherwise if you do not understand no problem ok. Eventually I will come back to the direct expression because of the shortage of time we do not have to go we do not have time to go into detail, but I will come directly that n 0 that is electron concentration can be given here n c is the effective density of states. Similarly, p 0 is given by this expression where n v is the effective density of states in valence band n c is the effective density of states in conduction band. Most important point that I want to make here that if e c becomes closer to e f or e f becomes closer to e e c your n 0 increases and let me show you by the drawing. If I draw the energy band diagram how does it look like e c e v and in case of intrinsic level it is middle if I. So, this is intrinsic case if I draw the energy band diagram for the n type this is for the n type my Fermi level is closer to the conduction band. What I am saying is the distance between these tells me how many electrons are there. So, if Fermi level is closer if Fermi level is here then it is more electron concentration more doping same thing if it is conduction band and this is my valence band and I have p type semiconductor my Fermi level is close to the valence band Fermi level is close to the valence band and if the Fermi level is closer to the band then I have more and more doping this is expressed by this equation. If my e c is close to e f. So, that the negative coefficient is 0. So, that my n 0 is higher if my e f is close to e v valence band then my p 0 is higher. So, in this way I can actually find out the electron concentration and hole concentration. Once simple way to find out electron concentration of hole concentration is what doping level right. If your doping level is 10 for 15 8 times per centimeter then your electron is 10 for 15. If your boron doping is 10 for 16 then your hole density is 10 for 16. But these are what I am representing here is the mathematical way of doing that mathematical way of representing the electron and hole concentration in your semiconductor fine. So, now your in your in case of your the Fermi level is here in case of intrinsic which is equal to e i in case of intrinsic semiconductor your Fermi level is middle of the band gap between the two. So, then if I replace e f equal to e i in the previous expression which is this expression here I am replacing e f which is the Fermi level in the dope semiconductor which is equal to e i in the intrinsic. So, I get this expression p i and similarly they should be n i and p i ok. What is n i I told you earlier n i is the p 0 right I told you right n i and p i is the intrinsic carrier concentration of electron and hole n 0 and p 0 is the equilibrium carrier concentration of electrons and hole ok. So, so that is what now in this equation this drawing this slide you have both the intrinsic carrier concentration of n i and p i and extrinsic or the dope carrier concentration n 0 and p 0 at equilibrium ok. So, now you have both of this and this is represent in the 4 energy level e c conduction band H e i intrinsic level e v valence band H and e f Fermi level. So, this is represented. So, basically the summary of all the discussion we have done in the last is this 4 numbers n i p i n 0 p 0 do it as a homework, but if I multiply n 0 with p 0 you get and with n i and p i the expression is equal ok. What does it mean n 0 p 0 is equal to n i into p i ok very important and we will use a lot and you should know this and I will I will just put some numbers here again. So, and this is your homework prove that which is equal to p i square right prove that n 0 p 0 is equal to n i square which is equal to p i because n i is equal to p i wonderful. What does it tell me? It tells me many many things ok this expression we will use again and again and again and again. So, make sure that you know this you know all the symbols here very carefully n 0 represent equilibrium carrier concentration for electron p 0 represent equilibrium carrier concentration for whole equilibrium means no external bias no temperature no magnetic field nothing as it is n i is intrinsic carrier concentration electron and p i is intrinsic carrier concentration for whole ok very nice. This number is fixed for a given material n i is equal to p i and it is fixed for room temperature and for silicon how much is this? I told you this number already for silicon how much is this? 10 for 10 right per centimeter cube 10 for 10 electrons per centimeter cube or 10 for 10 holes per centimeter cube and I told you higher the band gap smaller is the n i in p i lower is the band gap higher is the number ok. What does it tell you? This number is fixed n i square is fixed. So, if you are if you are making a semiconductor n type if you are making a semiconductor n type you can put you can change the number of electrons you can make the 10s for 15 electrons you can make 10s for 16 electrons you can make 10 for 17 or you can make 10s for 18 any number. You buy doing different doping ok these are the various doping level. If you do the different doping you can actually create more and these are all n 0 these are all n 0 number. What does it mean? If your right hand side is fixed and if you are increasing your n 0 what will happen to p 0 for n type semiconductor? What will what will happen to p 0? It must decrease it must decrease it is not it. So, if this is 10s for 15 what should be your p 0 10s for 5 right. So, that the product of n 0 p 0 equals n i square n i square is 10s for 20 because this is 10s for 10 n i square is 10s for 20 right this is 10s for 20 for silicon actually little more than 10s for 20, but a good approximation is this ok this is actually 10s for 20 ok. So, now if this is if your doping level is 10s for 15 you are creating 10s for 15 electron your whole concentration will become 10s for 5 ok if your doping levels 10s for 16 your whole concentration would be 10s for 4. If your doping level is 10s for 17 your electron concentration is 10s for 17 your whole concentrations 10s for 3. If your electron concentration is 10s for 18 your whole concentration is 10s for 2 ok interesting. So, at the end what we are saying n 0 p 0 is equal to n i square or p i square whatever it is. Why it happens? So, basically when you are making a semiconductor n type you are increasing electron concentration, but you are decreasing whole concentration right that is what is mean. If you are increasing electron you are increasing and whole I am talking about concentration you are decreasing and this is I am talking about all about n types semiconductor. Then thing will be true for the p-tap semiconductor why it happens more important why it happens right. So, you have let me draw the scenario that you have n type I am talking about because of doping you have created many many many holes because of I am sorry you have created many many electrons, but you have some holes also and there are many electrons here also. So, what is happening because now you are doping you are increasing the electron and this electrons are very far large in number let us say tens for 16 and this holes are small in number and intensive case tens for 10. What happens as soon as the electron is created and population more this electron will come down and recombine this electron will also come down and recombine. So, more number of electrons will reduce the whole population more number of electrons will reduce the whole population and therefore, in n type semiconductor if you are having tens for 16 electrons you have only tens for 4 holes such that such that what n 0 p 0 equal to n i square same thing will apply to p type by the way and let me end with this same thing will apply to p type semiconductor that n 0 p 0 is n i square or p i square in n p type semiconductor which number will be higher p 0 will be higher hole concentration will be tens for 14 15 16 17 therefore, your electron concentration will be tens for 4532 whatever is the case, but your p type semiconductor this must must satisfy fine. Is it clear I think enough. So, this is the the main crux of this lecture is this the and I will give some idea about this later in the next lecture or let me finish this here let me this. So, intrinsic carrier concentration is I am ready to take the take the questions now let me summarize this what we have discussed in this lecture that it is important that we alter the concentration of electrons and holes in semiconductor. So, that we can change the behavior of semiconductor so that we can get the desired functioning. So, we can use semiconductor for making a diode we can use semiconductor for making a solar cell or IGBT or VJT one thing. Second thing we have discussed how to change the number of electrons hole pair by doping by creating more impurities. So, by putting impurities like boron we can make it a p type or hole majority by putting phosphorus we can make it n type or electron majority. If I put tens for 15 boron atoms per centimeter cube I create tens for 15 holes per centimeter cube and corresponding number of electrons reduces. If I put tens for 16 electron phosphorus atoms per centimeter cube I create tens for 16 electrons per centimeter cube and product of N0 and P0 electron and hole is always equal to Ni square which is equal to equilibrium carrier concentration. Ni is equal to tens for 10 per centimeter cube for silicon and it this number can be different for different semiconductor. So, that is the summary let me take a questions now and I think Calicut. We are defining a space. We are only considering energy level as far as the distribution of electrons is considered. Is the definition of space going to be significant in this? My question is basically how are we defining this space? How do I define? Because for a particular space coordinate if you define it suppose three dimension. Suppose if we take that three dimensional space the location of electrons or holes at a particular space coordinate that diagram gives. In the space for example, if I take a semiconductor which is a 10 centimeter thick right. So, one coordinate is a space right. So, I can start from the surface of semiconductor then I can go deeper in the semiconductor. So, that is my space coordinate. So, as compared to the space coordinate how do I define my energy levels right. So, this energy levels of a carriers in a semiconductor is a function of the interaction between the atoms right. Many atoms come together and when they come together their energy level of a different atoms interact with each other with interact with each other and therefore, they create energy bands. Now, this electron at any given space at a surface or in the middle of the semiconductor or the 10 centimeter deep in the semiconductor can have various energy level and that will depend on its excitation it will depend on whether there is it is getting thermal energy or not. It depends on whether it is an excited state in a conduction band or it is in the ground level or the valence band. So, that energy level within the thickness of the semiconductor of electron can be varying at a different location. So, it is very much possible that when electron travels from one side of the semiconductor in physical space to the other side it energy levels it goes through the various energy levels depending on its interaction with the light with the thermal energy and with the bias. ESG college Coimbatore for a good silicon device. Good question. So, optimum carrier concentration is defined by what the functioning we need. For example, for solar cell it is not good to have very low concentration because once you have very low concentration your conductivity will be lower or if you have very high concentration your conductivity will be higher and we will see that expression. So, how much should be the concentration determined by the function of the device that we have. And let me tell you that for solar cell the optimum concentration is about 10s for 16 boron atoms per centimeter cube. Typically solar cell is fabricated in a p-type semiconductor and in that p-type semiconductor the optimum concentration is about 10s for 16 boron atoms per centimeter cube. Thank you. Okay, NIT Agartala. Finding electron. Okay, now what we have seen in the diagram, energy diagram, energy diagram that is always in between the balance band and conduction band that means in the forbidden region. How it is possible to get the electron in the forbidden region? Yeah, I got it. So, it is a good question that because there are no energy levels in the bandgap or how it is possible that you talk about any energy level in the bandgap. Okay, so let me tell you again that Fermi level is just a concept. Okay, it only helps us to understand how the probability of electron to occupy a given state depends and changes. So, therefore, definitely the energy levels are only in the conduction band and the balance band. There are no energy levels in the forbidden gap, but this is just for the understanding we define a Fermi level and it comes like that the probability of finding energy level at is about half or 1.5 at Fermi level and this is just a concept in reality there is no Fermi level. Okay, it is just a concept for us to understand the distribution of electron at various energy levels. Okay, NIT Kurukshetra. When you have the compound semiconductor like cadmium telluride or zinc sulfide or any other how do you define the carrier concentration? Whether the same equation of probability of electron density in the whole concentration and whether the same N0 P0 equal to Ni square will be true or not. So, yes it will be true even in a compound semiconductor you can define intrinsic and extrinsic carrier. So, even compound semiconductor also needs to be doped. Okay, so even in compound semiconductor doping and when you do do doping this you can define the electron concentration and the product of the 2N0 P0 will always be equal to Ni square only under equilibrium condition. Okay, Jalgaon. Sir, my question is that while we were talking about mounting of the solar cell. So, my question is that whether with the temperature the efficiency it increases or decreases with temperature as the temperature increases its efficiency increases or decreases. The efficiency of a solar cell with temperature decreases. Okay, the solar cell does not like the temperature and when we progress and we study more this course you will understand why it happens so. Okay, for the time being yes efficiency of the solar cell decreases with increase in temperature. Sir, if it is so if the temperature with the temperature the efficiency is decreasing then why do we mount it by considering the noon time. Well, because there were radiation intensity so high at the noon time as compared to the morning time that the net effect will be positive. So, the net gain will be there in the afternoon because your intensity the light falling itself is very high. So, definitely you are going to lose little bit because of the temperature, but definitely you get more energy output at the afternoon. Okay, I will take one last question from Kakinada. What are the limitations for doping level sir? How will be the Fermi level changes? Okay, what is the limitation for the doping level doping level can be as it is as you know 10 for 10, 10 for 11, 10 for 12 or it can be close to the silicon atomic density. What is the silicon atomic density 10 for 22 atoms per centimeter cube? Okay, now you cannot actually dope any atom by 10 for 22 right it will not be silicon anymore. So, the limit to the doping level is about 10 for 19 or maximum 10 for 20. There is what something called solid solubility of no material and no other material can go beyond that limit. So, the higher side limit is about 10 for 19, 10 for 20. Is there any relation between Fermi level and doping level? Yes, whatever the expression that I have shown you is actually gives you the relationship between the Fermi level and the doping level. Okay, the N0 is equal to ND right. Let me tell you that I will come back to that. So, let me go to the slides right this N0 in this slide look at the slide N0 is equal to what? N0 is equal to ND that is the donor concentration doping concentration and this is a Fermi level. So, there is a relationship between the doping level and the look at here also N0 is related to the Fermi level P0 that is the acceptor concentration or the doping level is related to the Fermi level. So, yes these are related. Okay, last question I will answer from the SGS ITS how efficiency depends on the material structure. Okay, as we discussed that material structure can be you know very bad like amorphous or you can have you can have monochrist line or you can have polychrist line. So, when you have very good material structure like monochrist line then the defects in the material are low and because the defects are low the recombination is low and recombination low means your efficiency is higher right. But when you go for material like amorphous where the structure is very bad in fact there is no structure in amorphous silicon and therefore there is a lot of recombination that takes place in amorphous silicon there because of that the efficiency is higher. Okay, so as you go from very high quality structure to a low quality structure the efficiency of the solar cell goes from high efficiency to the low efficiency. Okay, so let me stop here.