 Remember what we discussed yesterday? So, what materials we can use for solar cell? And we cannot use metal and insulator for very good reason. What else? We can use the combination of materials to get a base optical and electrical properties. And how to tune the tuning of the band gap? Metal tuning is we can do. Why we come to the 14 percent? All the way from 100 percent we have looked at the various losses in terms of the transmission, in terms of the higher energy photons, in terms of the recombination and some other losses that will happen at the junction and the contacts. Very nice. What else? Challenge in the PV cells. Any technology, any solar cell that you want to develop in future, you must be take care that it has smaller energy, payback period, smaller money payback period, material is abundant, there is a lot of efficiency are good enough and things like that. Yeah, sorry. The fill factor and the open circuit voltage and the short circuit current and that reminds me of the homework that I have given to you. Thank you very much. You done the homework? No? Ok fine. So, today was just introduction, you know, what are the possible issues that actually goes with the solar cell. And today what we will go is little deep into the theory. Again it is a it is not a complete thing because the whole P N junction theory takes about 8 to 10 hours of discussion. So, this is again not to be going to be sufficient, but it will get a kind of basic idea how the solar cell works and therefore, what we are discussing yesterday, you know, how to generate up, how the voltages get generated, you will make little more clear from the today's lecture. So, we look at the P N junction theory. So, what we are going to look at is this basic points, the carriers, the drift of the carriers, diffusion of the carriers, generation and recombination. So, these are the four major things that happens. Drift of the carriers, diffusion of the carriers, generation and recombination. This is how you define the device operation. In any solar cell, these four things are happening all the time always. And we need to understand this process is very well. So, that then we can actually design a new kind of solar cell or even if solar cell is not performing well or if it is performing well, we can understand which of the process is actually contributing to the good performance or not so good performance. And again, then we will try to look at the IV equation of the diode in a very qualitative manner. The quantitative description or the proof requires little bit more time, which we do not have for this workshop. But we look at the equation of the diode and then we look at the equation of the solar cell, which we are ending at the point we ended yesterday. So, the carrier drift and diffusion, these are the two forces that acts upon charge carrier or the carrier simply a particle, which results in the motion of the charge particle or just the particle. Drift is a phenomena that happens in presence of the electric field. The carriers moves because there is electric field. In and in our solar cell, the electric field does not exist everywhere. That is our only at the junction, this is electric field. And at the other part of the junction, there is very weak electric field. So, they will see how it works. The diffusion is phenomena in which carriers move because of the concentration gradient. So, that is a force. The force behind the motion of the carriers due to the diffusion is the concentration gradient. Generation, as we have seen yesterday, we will look at little more detail, what is the absorption coefficient, absorption length and how much what should be the absorption coefficient and what is the material thickness requirement and so on. And then the recombination, what are the various mechanism by which the carriers recombine. We have seen yesterday, but we will go little bit more into detail. And then qualitatively, we will try to put the equation of the diode, p-n junction diode and we will modify it for the solar cell. So, that is the agenda today. In any semiconductor, the electron is moved. So, E is electric field here. By the way, I wanted to make one correction before we go further regarding yesterday, we had actually get the approximate value of h nu. And I wrote that it is 1.124 lambda in micrometer. Well, this is wrong. The correct answer is 1.24 lambda in micrometer. From this calculation, the highest photon energy was coming to about 3 sorry, 3 electron volt and the lowest was coming about 0.28 electron volt. From this calculation, the number I think will become about 3.5 electron volt and the lowest is about 3. something electron volt. So, this is actually not 1.124, it is 1.24. That is correction that you can make. When this has come simply from putting h nu, nu equal to c or lambda, you put the value of c and put the value of h, the Planck's constant you will get the value which is 1.24 and lambda in the micrometer. Nothing great about it. So, in any semiconductor, if you are not putting any electric field applying electric field, another thing is this E is actually normally used many times for the potential energy and this involves zeta is used for electric field. So, this E here is actually electric field. So, even when this electric field is 0, the electron is moving all the time. Why it is moving? Why it moves? The electron is moving all the time. So, this is electron, it goes here, here, here, random motion, it is going all the time here and there even if there is no electric field. Why do you think it moves? It keeps on moving all the time because of the thermal energy. It is still at even room temperature, it still has enough energy to go around, but the net motion is 0. The net motion is 0. Therefore, if net motion is 0, which means what? Current is 0. No charge transfer per unit time. So, dq or dt is 0 and therefore, current is 0. So, without electric field, the net current is 0. When there is a non-zero electric field, when you apply some electric field, this electron will experience a force and it will still go through the random motion, but there is a in total there is some motion happening which is in the direction opposite to the electric field. Force in the electron is in the direction opposite to the electric field. Is that clear to everybody? If I take a big positive charge, this is the direction of the electric field. If I keep a negative electron, it will move in this direction. So, this is the direction of electric field and this is the direction of electron one. So, the motion, electron motion is in the opposite direction of electric field. So, what you see here is this electron is actually going through the further. So, it is not straight path because why it is colliding with the available the electrons in the particular semiconductor. So, it is not going to the straight path, but it assumes some certain velocity in the direction opposite to the electric field and that velocity. So, at the time it takes from going to here, that velocity is an average velocity you can say because it is going here, here, here, here. So, it assumes a certain velocity against the force which is there because of the non-zero electric field and that velocity is normally referred as a drift velocity. That velocity is referred as the drift velocity. So, you can see there is a force which is equal to in the negative direction because of negative sign electric field and the electron accelerates and de-accelerates, accelerates and de-accelerates. De-accelerates because of the collision it is going through and as a result it gets the average drift velocity. So, now there is some current flowing because there is a net displacement of a charge from one point to other one there is some current flowing. Let us say the concentration of electrons in that particular case is n. Can you write the expression for the current? Write it, what is current dq or dt? So, you need to write how much how much charge is flowing per unit time and you should get the divided by the time and you should get the current. Can you write the expression for the current? So, what you want? You want the expression for the current which should be this. So, what I said how many charge are there? You can assume a concentration of let us say n, let us say electrons are moving. So, n electrons per unit volume. Normally in a solid you give concentration n electrons per unit volume. Now, if I multiply with the charge q then I am having n q charge per unit volume right. Ideally what I need? I need charge per unit time. If I multiply this. So, if I multiply this with velocity drift velocity then what I got? So, this is per unit volume centimeter q and this is centimeter per second. So, this will cancel out, we will get 2 here and if I multiply with the cross section area which is in centimeter square. So, this will get cancelled out and this was having the unit of charge. Now, I am getting the per unit time right. This was having unit of charge coulomb and this is per second. So, what I got eventually? n q v d and this is having unit nothing other than the coulomb per second which is the current. So, if in any case if there are concentration of n particles per unit volume moving with the velocity v d across the cross section area. So, I have a also from here across the cross section area a then the current that will be flow in that condition will be n q v d a that is the current that I am. Now, this current is because of the drift. So, you can call it I drift n I assume that the particle is electron here n right. So, you can say I drift because of the electron right is it clear? Now, same thing can also happen with the force exactly same. So, just coming back here if the direction of electric field is this the motion of electron is in the other direction. So, the electrons are moving here electrons are moving here and because electrons are moving in this direction the current is moving here ok. So, direction of current is same as direction of electricity. So, there is no negative sign same right direction of current is same as the direction of electric field is that point very clear to everybody ok. Now, look at the holes put a electric field here the hole force will be there and therefore, current will also be in the moving in the same direction. So, again the direction of electric field and the direction of current is the same for the holes also ok. Now, let us write the expression for the hole let us say hole concentration is p number per unit volume. So, my expression for the drift of holes will be same p q v d a. So, the drift current is similar to the the the drift current for the electron and drift current for the hole ok and direction of the current for both electron as well as hole is in the same direction as the direction of electric field clear. Everybody, direction of the current for both electron as well as hole is in the same direction as the direction of electric field yes or no yes everybody not everybody seems to be coming as area ok. I am considering cross section here because you are consider the volume right. So, if I take a cross section here they are like this. So, this is the area this is the where the electric field is applied. So, this is the cross section area. If any doubt please ask we have lot of time no hurry if you are not asking I am going to ask you the question. So, you cannot be neutral you have to this side or this side like bush you are with me or you are against me cannot be neutral is that clear the expression for the right current is very simple you can write drift current expression for any other situation very simple you have to just say how many charges moving per unit time over a cross section area ok. So, very simple now this is expression this is in terms of current, but you can also write I drift per unit area you will get p cube V d. If you are writing this then what is this current density current density per unit area and many times we in solar cell we talk about the current density like yesterday when I told you that I see I told you current density. So, how much current is flowing per unit area. So, you can refer it to current or current density ok. These are the same expression we will not go into the detail sorry. So, these are the same expression we will not go into the details, but eventually arrived at the current density the expression for the current density is normally j. So, commonly use expression for the current density is j and I is the current I is the current j is the current density we are considering. So, I mean we have considered because because the force is also in the negative direction. So, the charge on electron is negative and the force direction negative got cancelled and therefore, the net is positive ok. There is a the force on electron it is also in the negative direction and there is a negative charge coming because of the minus cube. So, both cancel out and therefore, in simplicity what we have looked the direction of electric field is the same as the direction of current which means everything is cancelled out there are two negatives in that expression ok. One another parameter which is very very important that we need to define is how much a velocity a particle attains per unit electric field ok. What is the velocity of the particle in attains per unit electric field? Analogy very simple right if there is a runner which runs on a smooth road and if there is a runner which runs on the rough road ok. The runner same runner same strength everything same it will run faster on the smooth road right then the rough road. Is that clear? Same thing will happens to electron also when electron is moving in a crystal if the crystal is perfect without any disorder without any defects without any impurities then it will run faster or the velocity it will attain is higher as compared to the case when the crystal is defected ok. And the examples of the defected crystals are polycrystalline silicon ok example of defected material is cadmium to the right CIGS and amorphous silicon very defected ok. What does it mean for a given electric field? The velocity that a electron will attain in the defected material will be higher or lower? Lower. Just to check it will be lower right. The velocity that electron will attain in a defected material will be lower then the velocity it will attain in a perfect crystal. And it is represented by a parameter called mobility mu ok. So, mu is nothing but velocity or drift velocity per unit electric field. So, in a good crystal the mobility is high in a not good crystal the mobility is low ok. From solar cell perspective we would always like to have high mobility ok. So, this is one of the electric important electrical property right I said yesterday two properties we need to get the best one is optical other is electrical. High mobility is one of the important electrical property that we are always looking for ok. So, when people are trying to develop the materials and trying to get the and trying to get the job done. So, they are actually trying to show that. So, the what kind of velocity that you can get what kind of drift velocity you can get for a given electric field. And this is the parameter which can vary a lot. The mobility is one of the key parameter for a scientist and it can really vary a lot depending on the condition and the defect and the impurity etcetera. So, keep this in mind this is a very important parameter for solar cell right. Everybody will remember mobility what is the unit of mobility? This is centimeter per second this is volt per centimeter. So, overall centimeter square per volt second centimeter square per volt second ok. There is a drift current for the electrons and there is a drift current for the holes. What is the total drift current for both electron and hole right? What is the total drift current? Some of the electron drift current and the hole drift current. So, mobility as I said is one of the important parameter and the various numbers you can see here. The mobility of electron for silicon is 1350 what 1350 centimeter square per volt second and for holes 480 4 germanium 3900 and 1900. The total current density for the drift current density is the sum of the two current density. So, what we have done in this expression is we have used this expression ok. So, is the V D in your expression can be replaced by mu times electric field right. V D can be replaced by mu times electric field and therefore, expression for J n and J p will become q n mu n zeta n q n mu a q I am sorry this should be what is the mistake here? This should be p. So, q mu a q p mu p zeta and the subscript for the mu n and mu p is different because the mobility for electron and holes are different right. So, you can put together the sum of the two current component is this and if you actually look at this carefully this is nothing, but the ohms law J equal to sigma e zeta actually electric field is nothing equal to i is a ohms law right V equal to i r if you do that analysis you will come to the same expression. So, this expression is nothing, but the ohms law. So, what is the expression for and this sigma is what? So, this expression is nothing, but the conductivity ok. So, what is the expression for the conductivity q in the bracket n mu n p mu p ok. So, you can find out the conductivity of the material if you know the mobility if you know the concentration of electrons you can find out the conductivity q is known. If you know the conductivity you can find out the what also you can find out resistivity if you know the resistivity you can find out the resistance. So, knowing the doping knowing the number of electrons and holes and knowing their mobility you can find out lot of parameters about the materials. Knowing knowing the n n mu n p n mu p you can find out the the total current density you can find out the current if you know the area you can find out the conductivity you can find out the resistivity you can find out the resistance etcetera ok. So, far with me everybody with me very good. So, this is one mechanism by which by which electron move the mobility by the way is not a constant it depends on the doping density ok. If you dope more your mobility decreases any idea why? Very simple doping more means you are disturbing it more right. So, because there are silicon atom sitting now you are doping with some concentrations you are putting more and more boron or phosphorus which means you are disturbing your lattice more which means you are making the path of this electron rougher. And therefore, as you dope more and more the velocity it will attain the drift velocity will attain will be lower and lower and therefore, mobility lower. So, therefore, a very high doping level transfer 17, 18, 19 you can see the mobility are smaller for both electrons and all. Mobility is also function of a temperature ok. So, initially if you increase the temperature mobility increases, but after some after some increase in temperature decreases because the lattice vibration starts ok. So, the the atom starts starts starting vibrate and therefore, it decreases yeah doping density means we have p type silicon and n type silicon right and we have just silicon intrinsic silicon, but in order to make a junction we need to make a p type. How do we make a p type? We put boron into it and so the how much boron is there in silicon that is a doping density ok good. So, this is a tensor 15 boron atoms per centimeter cube in silicon. By the silicon what is the atomic density of silicon? In 1 centimeter cube of silicon how many silicon atoms are there? Very easy to calculate, but I will give the number directly. So, 5 times transfer 22 atoms of silicon there in per centimeter cube that is silicon atom density. Now, when we do doping we put boron into silicon that boron doping is normally in the range of tens for 16 atoms of boron in silicon per centimeter cube that is the doping density. So, when you say when there are tens for 16 atoms of boron in silicon we say doping density is tens of 16. What does it mean? How many atoms of boron are there in per per silicon or so what is the difference tens for 16 to 22? So, 1 million for 1 million atoms of silicon there is 1 atom of boron right. Silicon atom is tens for 22 boron is tens for 6. This is controlled while making silicon itself. When you make a silicon, silicon is actually melted ok. So, in melt. So, you are putting 100 or 1 million kilogram of silicon put 1 kilogram of boron. So, then you will get this kind of doping density ok. So, it is controlled while making silicon itself in the melt itself. Boron atom can come can deflate silicon as well as interstation. So, where is the boron going because crystalline silicon is silicon right. So, boron has to go somewhere that. So, it is first of all it is in melt right. So, when silicon is melted at very high temperature about 1800 degree centigrade at that time you also put boron and then you stir it ok. It is like it is like what making kheer mix it properly you know put sugar or whatever you put mix it properly that is it and then when when when the silicon actually comes up and it gets colder and solidifies. From liquid it goes to the solid form within that form it is a very critical state. So, that your all silicon atoms are nicely arranged you know how many atoms zillions and zillions of atoms are nicely arranged and somewhere you also get the boron atom. How many boron atom 1 in 1 million very small number is still right. If you are doing doping of tens for 16 only 1 boron atom is coming in 1 million silicon atom will have very still number. You can do it by diffusion, but you actually when you grow the silicon is grown in the form of ingot which is like meter and half long 8 inch or 6 inch the diameter. So, that is done at that time it is a different from the diffusion. In diffusion also you actually insert other kind of impurities where you are diffusing you are also doing similar thing. Not really I mean look at the number it is so small 1 in 1 million really small, but still very large. When silicon is purified you know how much it is purified not 99.9 it is purified to this much 1 in billion and then you can make your device 1 in 1 billion atom. It is similar to find 1 foreigner in the Indian population in the whole of the country that is 1 in 1 billion right we are almost 1 billion. You need to purify some of the impurity to that level then only you can make a device. Of course, solar cell is more tolerant to the more forgiving to the impurity. So, you can actually make a solar cell in a higher impurity concentration also. So, 1 in 1 million is also ok and therefore, now people are trying to make. So, this kind of silicon is called electronic grade silicon very expensive, but now people are trying to make what is called solar grade silicon because you are allowed to have more impurities it is cheaper relatively, but the science of making silicon pure silicon itself is a it is a it is a 3, 4 years back it was in hands of only few people. Only 6, 7 companies in the world was doing every silicon everything in silicon know that only 6, 7 people. Now, of course, there are M. M. C, Tokyama, R. E. C. They are the new people they are in more polycrystalline, but earlier people as I am talking about. These are the these are the Mitch Bushy. These are the main players earlier, but now there are many new people are coming particularly in China ok fine. So, drift you know lot about it now you are all expert of the drift mechanism of the charge motion right. Now, I will make a expert of the diffusion mechanism of charge transfer, but these two these are the two force which is always working in a semiconductor. So, diffusion is. So, the force behind the drift was electric field ok. The force behind the diffusion is concentration gradient ok. It is similar to like if I have only half of the people sitting here in this room in the next person comes in is not going to sit there you will sit here right where there is a space. So, it is kind of diffusion happens ok. Other very classical example of diffusion is the perfume. If I spray perfume here right now everybody will know which perfume it is right because it diffuses the concentration is more of the perfume particle here and it is less there. So, there is a motion is always from the high concentration of particles towards the low concentration of particles. For example, you have one condition where all the particles are sitting at time t equal to 0 at x equal to 0 ok. If so, what does it mean? There is a lot of concentration here and there is no concentration here. So, there is there naturally the diffusion will occur and if you allow the diffusion to occur after sometime t 1 you will see the distribution is increased or the time distance and some other time t 2 which is greater than t 1 it will further diffuse and diffuse and if you wait for infinity time what will happen? The concentration will become same everywhere concentration that is the job of the diffusion force that is the job of the concentration gradient that to make everything similar uniform everywhere over a period of time you have to wait for sufficient period of time. That is the diffusion is the force behind the diffusion is the concentration gradient. Now, we have written expression for the current and current density as a for the drift can you write the expression for the diffusion current? Current is always going to be the same right you take drift at diffusion or anything right it is always charge per unit time. Can you write the expression for the diffusion current or what we need to know if you want to write expression for the diffusion current what you need to know? Concentration gradient so, let us say the concentration gradient is d n over d x that is our concentration gradient. Now, what else you need? You need area let us say area is a and n is the number right. So, you need charge you can multiply by the q what is the unit of this expression just find out what is the unit of this expression and we will see what else needs to be multiplied do it quick quick what is the unit of q a d n over d x n is n what is the unit of n per unit volume concentration per number per unit volume. So, what is the unit Coulomb centimeter square per unit volume n centimeter. So, this will go this will become 2 and this will go. So, Coulomb per centimeter square I still need per unit time I still need per unit time. So, I need to multiply with something which has unit of centimeter square per unit time does not it. If I multiply with centimeter square per unit time this will get cancel I get Coulomb per second I will get Coulomb per second right got it. So, basically and that is true also this is some coefficient which is called d which is diffusion coefficient like mobility is related to the drift there is another coefficient called d which is called diffusion coefficient which is related to the diffusion and having a unit of centimeter square per second we can actually derive this also and this is nothing but the expression for the current I for diffusion of electron I for diffusion of electron Q d Q this is a d n over d x ok. Two important thing we are going to do here now. So, let me write fresh again. So, I diffusion for electron is d Q a d n over d x I can also write similar for the following for the holes right d Q a what should I write here? What should I write here? At the corner my favorite students are actually corner students. If the corner students can understand everybody else can understand corner that is idea what should I write here? P right I am writing a diffusion current for the holes I should write hole concentration right. Similar that mobility was different for the holes in electron diffusion coefficient is also different for the holes in electrons I should write d n and d p right. Now we have done one more important thing in the last time while writing the drift equation for the current we have seen what is the force electric field and then we have seen what is the direction of the current and we found out that the direction of current and electric field for both electron and hole is same. We should do the similar exercise here and find out what is the force that is the concentration gradient and for that concentration gradient what is the direction of current? It is important to find out that right got it the problem everybody I need to know whether my my current is in the same direction of the force or is the opposite direction if there is opposite direction I should put the negative sign right let us do it. So, suppose I plot n over x right and this is my plot of the concentration gradient n over x right if I do the d n over x I can find certain constant right fine. So, I cannot point a straight line then there is no gradient right the fact that I have not plotted a straight line which means that there is a gradient concentration is changing with respect to x fine. Now, because of the diffusion my carriers are always going to move from low concentration to high concentration good because of my diffusion carriers are always going to move from higher concentration to lower concentration which is this direction which is this direction my carriers will always move towards the positive x axis in this case ok. What is the direction what is the d n over d x where it is positive d n over d x is concentration gradient is it positive towards the plus x direction or minus x direction the concentration gradient is positive in the minus x direction. So, this is my d n over d x right. Now, where is the current flowing where is the direction current direction. So, this is the motion of electron and current is this direction ok. So, this is my I for electron. So, what you can see here the current for electron is in the same direction of gradient for the electron. So, I should not put any any sign it is perfect because both of the force the current is in the same direction as my force. So, I should not put any sign there it is perfect. So, this equation this equation here is correct got it this equation is correct following me everybody everybody. Now, let us do for the let us do yourself do it for the holes what is happening do we have to put negative sign or not just do it yourself plot the profile for the holes plot the direction of the gradient plot the direction of the motion and see the direction of the motion is in the same as the direction of gradient or not plot the profile first p versus x do not make it a straight line either horizontal or vertical just make a slope curve like we have done earlier see where is the direction of the motion where is the direction of the gradient. So, if I plot p versus x similar the gradient is positive in this direction d p o d x right holes will go from high concentration to low concentration which means this direction moment is this direction and therefore, because it is holes the current is also in the same direction right. So, I diffusion for the holes in same direction. So, you see the force and the current are in the opposite direction right the gradient is in the minus x direction the diffusion current in the plus x direction and therefore, I should put a negative sign here got it. So, very important to find out the right direction for the current motion. So, now, electron drift current, hole drift current, electron diffusion current, hole diffusion current there are four ways current can flow in a semiconductor there are four ways current can flow in the semiconductor and if I tell you any x y z direction you should be able to give the direction appropriate equation for the current with appropriate sign got it everybody very clear. This is nothing, but the expression how the diffusion will actually have the distribution over the period of time equal. So, starting suppose the particles are in the one compartment therefore, compartment at t equal to 0 over the time the particles will start redistributing themselves and if you look at the very long time the particle will be equally distributed in all that is nothing, but the diffusion mechanism and this is how we can derive the expression for the diffusion that suppose this is the this is the profile of the concentration you have n 1 concentration and 1 compartment n 2 concentration and other compartment at a given time the n 1 particle will actually move in the either side there is a good probability that it can go to this side. So, good probability that the electron which are in this compartment can go to this side or that side and we can say equal probability either side. So, after sometime t half of the n 1 particle will be here and half will be here and it is happening everywhere. So, after sometime t half of the particle in n 2 will be here half will be here it will keep on happening. So, what you will see here is the half of the particle over the time t goes to the other side and this way you can actually write the expression for the flux that is the number of particles crossing per unit area per unit time. So, there is a flux of the electron due to the diffusion you can write in the in the form of the gradient you can write in the form of the gradient n x and n x plus delta x. So, this width of the compartment is a delta x. So, you can write the same expression in this form and you actually write the expression of the flux in the derivative form d n over d x that is what we are writing and this is the flux right this is the number per unit area per unit time what we are looking for current. So, we have to still multiply by the q to get the charge per unit area per unit time and then we multiply with the area. So, you get this. So, this expression here is nothing, but we will have the unit of centimeter square per second which is our diffusion coefficient and this is the flux. So, as I said we have to multiply by the charge q to get the number charge per unit area per unit time and you multiply by the area. So, you get the charge per unit time which is current density or sorry current for the. So, this is this is being replaced by d n diffusion coefficient and this is the gradient of concentration and this is what you get the flux right flux of anything is in the number per unit area per unit time flux of anything flux of charge flux of particles. Multiply by the q you get the charge and multiply by the a you get the current ok. So, the diffusion current density right j is expression of density is per unit area right and if I multiply with the area I get i right j is the expression for the current density and i is a expression for the current. So, this is j means area is not multiplied to see here. So, expression for the electron diffusion current no positive sign here you see expression for the whole diffusion current there is a negative sign and now we know very well why there is a negative sign right right we know very well why there is a negative sign. So, there are how many components of current now 4 current electron drift and diffusion whole drift and diffusion. So, the total current in a semiconductor at any given time should have should have all this expression. So, j total is j n plus j p j n is drift current and diffusion current j p is drift current and diffusion current. So, total expression for the current is sum of all this right sum of all this electron drift electron diffusion whole drift whole diffusion there is a total current in semiconductor and under different condition various components of the current gets modified under various condition various components of current gets modified. Now, few important point to note here looking at this expression what this says is minority carrier current can be significant due to diffusion. Let me write one expression for you which we have not covered because the lack of time, but so in any semiconductor there will be electrons and there will be holes always right one is in majority other is in minority right. So, in any semiconductor the electron concentration is given n 0 equilibrium electron concentration whole equilibrium carrier concentration given p 0 and when semiconductor is not doped it is given with the intrinsic carrier concentration n i right for silicon n i is fixed because this is intrinsic case n i is about transfer 10 number per centimeter cube ok. This is fixed for a material n i is fixed for material. So, if I look at germanium n i that is intrinsic carrier concentration different if I look gallium arsenide intrinsic carrier concentration is different if I look cadmium telleride intrinsic carrier concentration is different. So, it is fixed for the material at room temperature what can be proven is that n 0 p 0 is always equal to n i square always equal to n i square. What does it and if I am doing the doping majority carrier concentration if I am doing doping as we have taken example of 10 square 16 boron atoms per centimeter cube it gives rise to 10 square 16 electrons per centimeter cube more or less same which means my equilibrium electron concentration is 10 square 16 ok. Now, if I put this number here if I put this number here. So, my p 0 is equal to n i square by n 0 n i is how much 10 for 10. So, n i square is 10 for 20 divided by n 0 which is how much 10 for 16. So, my p 0 is 10 for 4 ok because electrons are in majority your very huge number of electrons 10 for 16 what is the whole concentration 10 for 4. So, whole is in minority right whole is in minority and the number difference is huge. So, my p is. So, my n 0 is 10 for 16 and my p 0 is only 10 for 4 ok. This is what normally the situation when you start with the silicon for making solar cell your doping is of the order of 10 for 16. So, majority carrier concentrations of the order of 10 for 16 minorities of the order of 10 for 4. Now, 10 for 16 may become 15 17 yeah it is it is not boron it is phosphorus concentration we are talking about the n phosphorus concentration fine. So, coming back to the slides. So, what does it mean the minority concentration is much much smaller right several order of magnitude in this case 12 orders of magnitude smaller right keep that in mind. So, coming back here this is very important point to note with the solar cell perspective that here n if I am talking about electron current density here it is n and here it is gradient of n ok. This number will be large the drift current density will be large if n is large. So, diffusion current density will be large if gradient is large for gradient to be large n need not to be large gradient can be large. So, if my concentration varies from 10 to 10 to 4 in 1 nanometer then my gradient is very large right even my n is not large. So, therefore, for minority carrier concentration the drift current is always going to be small where there is number is very small, but the because the gradient can be large. So, diffusion current can be very very large for the minority carriers and minority carrier plays very very significant role in the operation of solar cell ok. Take a note minority carrier plays very very significant role in the operation of a solar cell and therefore, current due to minority carrier flows due to what drift or diffusion current due to the minority carrier is always because of the diffusion because then diffusion can be large otherwise the number of minority carrier is so small the drift current component of the minority carrier is always going to be very very small. So, the diffusion plays a important role and therefore, minority carrier plays an important role in the operation of solar cell. Point taken. So, we are making some progress towards the solar cell now. So, these are the forces and these are the direction of current you know very well how to how it has come right. Electric field in this direction current is also in the same direction due to the whole drift and due to the again one of them has to be no that is ok. Whole drift and diffusion electron drift and diffusion. So, this is the gradient this is electric field and you can write the direction you can actually put the direction of current right. Let me take a let me take a quiz. So, this is direction of my electric field and this is n versus x ok. What is the direction of electron drift current and electron diffusion current? How many agree with him? How many do not agree with him? You agree with him raise your hands. If you agree with him drift current is the same direction as the electric field ok. So, this is J n drift fine any objections ok. What is the direction of? Diffusion current. So, this this is my plus x direction. So, tell me with respect to this is it plus x and minus x. Diffusion current for electron is plus x or minus x direction. I I just want with respect to this this is my plus x direction with same is same is J n I do not understand tell me plus x direction or minus x direction. So, you are saying plus x who is saying plus x plus x ok. Do you agree with her everybody? No some people do not agree you do not agree ok. There is a confusion. Let us do it a this is very simple right. This is the electron will go from high concentration to low concentration. So, they will go in this direction right and the current due to electron is in this direction and we are interested in direction only. So, this is in plus x simple is not it. The electron will always go from high concentration to low concentration and because my direction of electron is this my current has to be in the opposite direction and that is what I was interested in the direction of current. So, therefore, direction of current is in the plus x direction. So, J n diffusion J n diffusion is in the plus x direction. If this is the case now things can be reversed right it can be this that whatever happens good it is very important to understand. If you understand this you understand a lot yeah, but this is my positive x direction I know it does not matter right. I can say any direction, but more important thing is this is happening. So, with respect to this direction my current is in this direction that is it. You call it positive x negative x you call it y z alpha beta gamma whatever you call it does not matter. The main thing is the profile. J n diffusion is the direction of electric field. This is this portion from here below is all belongs to the diffusion. Yeah, portion from here to here is all belong to diffusion ok. Go ahead should I go ahead ok. So, now if I come to the that P n junction diagram and tell you about which particle will go where you should be able to understand various values for the coefficient you know both all of these are very important mobility and diffusion coefficient are very important are the material property they are dependent on the material that you are using they are dependent on the process that you are doing that they are dependent on the deposition technique that you are using there are so many various ways in which this particle can be dependent. One other important factor that is we do not go in detail, but as I told you yesterday that if whenever you draw the energy band diagram if you do you are not drawing the horizontal lines which means there is an electric field ok. As long as there is horizontal line there is no electric field whenever there is no electric field there is a whenever the bands are not flat it means that there is electric field and we can prove that right. We always know that expression for the electric field is gradient of the voltage that is electric field in the negative direction and you put this expression here you can find out that whenever there is a gradient this also means the gradient of voltage also means gradient in potential energy E is energy here. So, whenever there is a gradient in the potential energy means that there is electric field and what is this axis by the way always energy right. So, the line is not flat means there is a gradient in energy what energy potential energy which is given by E right because my y axis is energy and because this lines are not flat which actually indicates there is a gradient and the gradient actually means electric field that is the message I want to give right. In any band diagram whenever you see in your life from today onwards and whenever lines are not flat it indicates electric field ok. Now, if your drawing is very bad you may have a lot of electric field in your drawings, but make sure that you draw horizontal line because it makes lot of difference it makes it is very important point and it can play a huge role in making your device work or not work or you are actually improve the performance and lot of people actually play with this. So, this is another skill one should have to draw the appropriate energy band diagram. If you can draw it properly you know exactly how your device is going to work draw the appropriate energy band diagram ok. So, key thing from here is whenever line is not flat it means there is an electric field that is it ok and why it is not there it is because of this simple. Electric field is gradient in potential this is my potential energy axis because there is a gradient it actually indicates electric field ok. Einstein has found out that the two most important parameters that we have seen one is corresponding to the drift that is mobility other is corresponding to the diffusion that is D diffusion coefficient are related to each other. Yes, they are related to each other again we will not go to in the details of the expression how we have derived, but what is more important is the D and mu are related with KT by K. The more important thing is this is only depends on the temperature it does not depend on any material parameter. So, ratio of D or mu does not depend on any other third material parameter it is only depend on temperature. So, D by mu is equal to KT by Q is called Einstein relationship K is Boltzmann constant T is temperature Q is charge on the electron if you put the values there for a room temperature you will find this expression is equal to 25.6 25.9 25.9 close to 26 milli electron volt ok, you will find that value to be 25.9 milli electron volt. No it is milli volt milli volt this is centimeters square per second this is centimeter square per volt second everything will get as a milli volt and only KT is electron volt. If you only expression KT then it is energy unit of energy KT by Q is unit of volt only KT is unit of energy. So, if you are looking only KT then it is 25.9 milli electron volt KT by Q is 25.9 milli volt that is the value of that expression. So, D and mu are related with that I think should be true from this expression also if you divide mu KT by Q D by mu KT by Q you should find the similar relationship D by mu Dn by mu N KT by Q Dp by mu Q KT by Q should be the same.