Okay, so if you recall in the last class we had discussed about E-K diagram, okay. And I told you that in a real semiconductor crystal the electrons will face this periodic potential of the atoms. That is why the E-K diagram becomes distorted.
And this E-K diagram has a huge impact on how we understand devices because the effective mass of electrons as it moves inside a crystal is dictated by the curvature of the E-K diagram. So it is very important. Hope you have, you know, recalled all those things. I told you in the last class while ending that I will introduce the concept of holes because that is as important as electrons, okay. So today we shall introduce the concept of holes and also the statistics that dictate how electrons and holes are populating these energy bands, okay.
And it is a very crucial concept. You will see electrons and holes are equally important in semiconductor research, okay. So let us come back to the whiteboard here.
I told you that the conduction band here I will call CB, conduction band. And then there is a valence band here VB this is completely filled with electrons okay this is completely filled with electrons okay and because it is completely filled with electrons electrons do not have any empty states to move so it cannot carry current but this is nearly empty so you can actually excite an electron somewhere here somehow you excite either by giving heat or by light or something okay you excite some electron there then this electron is actually free to move and that is how they carry current. And you can tune how many electrons you can excite or how many electrons you can have here and that is the beauty of semiconductor I keep telling you because in metals or in insulators is very difficult or you cannot do that okay. So now we will keep that in mind that there is something called EK diagram we will not forget that okay because that is very important. Now when I take an electron from the valence band to conduction band it leaves behind a vacancy here no.
If I take one electron away from here. To there it leaves behind a vacancy here, that vacancy is called a hole. Actually hole is an absence of electron, right, hole is an absence of electron, you agree right, an electron just move there, it left behind an empty space, it is a sort of a vacancy, so that vacancy is called hole and hole is the absence of electron, but in semiconductor devices it is much more convenient. and it makes sense if you define the whole sort of a particle only although it is the absence of electron okay although it is the absence of electron you can treat it as a particle and what will happen now suppose this is the valence band this is your valence band it is filled with electrons right I will give electrons like this okay it is filled with electrons.
I will fill throughout but I will only take so one of this for example this has gone to conduction band okay it has gone to conduction band it has gone to conduction band that is okay it has come here okay that is fine. Now there is a vacancy here there is a vacancy here so if you apply a field electric field in this direction if you apply electric field in this direction which means electrons will try to move in this direction right if a field is in this direction electrons will try to move in this direction. what will happen now? See if this was filled no if this was filled initially then the electrons cannot move because there is no empty state where will they move they cannot move so there is no current.
But now I told you that one electron has gone from here and it has left behind an empty space it is like a vacancy it is a hole what will happen now? If I apply a field in this direction x direction electrons will move in negative x direction so this electron will be able to come here now. right.
And then here it will be a empty space okay because electron has then nearby electron has come here it is an empty space. Now this electron can come here of course so because it is an empty space now it is a vacancy so it will come here and this will be blank. Of course the other one can come and so on and so forth so essentially what will happen is that you will have electrons like that and the hole empty it will keep moving. The hole was initially here, the hole was initially there, right but the hole is moving in this direction. The hole is actually moving in this direction in a way the vacancy is moving in this direction because the electrons are jumping jumping right so coming here right so the hole is moving in this direction so I can say that hole is moving in the direction which is in the same direction as electric field which means it has a positive charge you agree electrons are moving to the left electrons are moving to the left because they will move opposite to the field but because they are moving to the left the hole is seems like moving to the right.
Agree, because the hole is moving to the right it looks like the hole is moving to the right initially the hole is here. So, the hole has a positive charge because only then it is moving in the direction of the field. So, hole essentially is an absence of electron which can be modelled as a particular positive charge that moves in the same direction as the electric field.
That moves in the same direction of electric field and initially that if everything was packed here no hole you know then all the electrons are there. They cannot be movement the total momentum is 0, but now the total momentum is not 0 because there is an empty particle sort of a you know it is absence it is moving. So now this is actually not a total momentum is 0. So that negative of the total momentum you know that is the whole momentum you can call the whole actually is a is the negative you know is the opposite of electron, but its mass is not the same by the way its mass is not the same because this whole can now be modeled.
There is also a particle, it is called a quasi particle because it is not realistically a particle but you can model it as a particle and that makes life easier. Instead of explaining and understanding how electrons are moving here in this way, it is much easier to talk about a vacancy moving to the right, a hole moving to the right that makes it simpler. Of course in conduction when you talk about electron moving because there is all empty you know you talk about electron but here you talk about holes. So the beautiful thing is that holes.
Also have their own energy band first of all they have their own energy band and their energy band will correspond to the valence band right because they are moving in the valence band electrons are moving in the conduction band holes are moving in the valence band so holes will correspond to valence band. Solids also have their own effective masses because you can draw the EK diagram. for holes also and that will be for valence band right.
So for example this is E this is K and I told you the electrons have a E K diagram like that. The same way the holes also can be drawn this is conduction band by the way the hole because the hole energy is opposite of electron you know the hole will look like this, this is the hole. This is the whole E k diagram energy, this is the whole energy and this is k equal to An electron energy you know this is E k diagram for electron as you go up the electron energy increases right. Here as you go down whole energy increases, as you keep going down the whole energy increases. So at this point at k equal to 0 point the whole energy is the lowest okay as go down it increases.
So this is the this actually is the valence band okay. And their holes are here you can say holes are here okay and this is the conduction band so essentially we are talking about E k diagram for both electrons and holes okay we are talking about electron E k diagram of both electrons and holes so essentially you have this E k of course this is the hole energy here so this is electron this is hole okay this gap okay. Sorry, this gap is the energy band gap again, because this energy band gap is called EG as I told you, this energy band gap is the same as this conduction band, empty conduction band and a fully filled valence band.
fully filled some electrons will go here leaving behind holes here and that is electrons will be here right. So this valence band corresponds to this, this conduction band corresponds to this, this band gap of EG corresponds to this energy this is called this whole thing okay this whole thing is called sorry this whole thing is called band structure because it is EK diagram. Okay this is called band structure E k diagram and this is called band diagram just the band diagram because it is in real space okay but this is the one to one correspondence between the same. So valence band essentially you talk about hole transport in valence band it is filled with electrons by the way but holes.
are moving in the valence band and electrons are moving in the conduction band right that is what is happening. Holes are moving in valence band electrons are moving in conduction band holes are essentially quasi particles their absence of electrons that you model as quasi particles they have their own effective mass because this E k diagram that you are drawing for hole same thing that comes for electrons it also has a curvature no it also has a curvature so that will also give the effective mass of hole. The inverse of this curvature h square h bar square by the inverse of this curvature will also give you the effective mass of hole and so holes have their own effective mass, electrons have their own effective mass and this point okay this point is called the lowest energy point of the conduction band okay and this is the top most point of the valence band the distance between this energy gap between them is the energy band gap E g okay energy band gap E g. this is k remember this is k so this is k equal to 0 this is k equal to 0 there are many important things here number one thing is that unlike in electron holes are more complicated because they are not technically particle-particle right because they are quasi particle they are model the vacancies are model.
So there is a little thing here a hole might not only have this one branch this is one band it might actually have two three sort of it might have two three sort of a we can call them branches. of E-K diagram. Electrons technically do not have them but they may but typically holes will definitely have this you know mostly have this separate branches. You might see that the curvature of the top most is the lowest the curvature of the top most is the lowest then curvature increases.
If the curvature becomes higher then effective mass becomes lower. So this is called light hole branch the top one is called heavy hole branch why is it called heavy hole? because the curvature is small okay let me draw it again maybe freshly right I am only talking about the ek diagram for holes. So this is e sorry this is k this is there are three branches typically one is this one is the red one and then one is the blue one for example okay this red one actually goes to almost k equal to zero like this.
touches and come back here but the blue one has an offset here, there is an offset here okay the blue and the red has an offset. Okay so the black one has the lowest curvature you see so it has got the highest effective mass so it is called a heavy hole branch okay. The middle one has a higher curvature so lower effective mass it is called light hole branch so they are different branches of hole actually and the last one is actually S-O spin orbit coupling we do not have to worry about it it is basically some coupling between spin of the holes and other things. So it has three branches but that is fine we can talk about the top branch here and this is how holes actually the energy split looks like and of course the electron will have only one branch typically so electron is fine there and this gives the energy gap that is what we need to know and this is k right this is k.
You know it is almost like talking about a water bubble you know air bubble in a water so you have water filled and gravity is pulling you this side right gravity is pulling you this side right. But there is a bubble of air the bubble will move upward against gravity the bubble will move upward against gravity right. So how do you explain that it is easier to explain if you say that the bubble has a negative mass so that it is acting against gravity is moving up.
So in a filled water you are talking about an empty bubble same thing you can talk about a hole very crude analogy you can talk about a hole in a valence band that is moving against you know it is actually moving. in the direction of the field against in the opposite of the direction of the electrons that is how we can talk about it. So that is one thing and because I am in the E-K diagram I am talking about the holes and electrons already I told you that electrons and holes have their own effective masses they have their own E-K diagram and the separation between them is the energy band gap there is a one to one correspondence between the E-K diagram and the energy diagram here.
What is more important here to understand is that when I have a E-K diagram like this right and the energy the lowest point is at k equal to 0 it is not necessary that it is not necessary that x again I will just put it here k this is E this is k I told you that you know the hole has energy like that typically we will talk about that k equal to 0 point at this k equal to 0 point the hole will have this you know the. the top most point of the EK diagram in whole will correspond to here the bottom most point of the EK diagram in the electron will correspond to here but it is not necessary that all semiconductors will follow this rule in some semiconductors in some semiconductors you know that in some semiconductors your The bottom most point of conduction band need not be at k equal to 0, this is a k equal to 0 point right, it need not be k equal to 0 like this. In this case it is fine, this point the bottom most point of conduction band, the top most point of valence band they are the same k point, the energy difference is band gap, but it need not be same that is why I am saying, what do I mean?
I mean that there can be semiconductors where you have E, you have k. We will always take the whole for whole valence band will always have the top most point at k equal to 0 okay but for electrons you know the conduction band may have a point here but this is not here exactly and then there is somewhere here comes very low and goes at some other k okay at some other k. So you see this is the lowest point this point here. this point is lower than this point agree this point is at a much higher point no right this point is at lower point than this but this is k equal to 0 and this is k not equal to 0 this is some finite k we do not know what k is this right.
But energy band gap EG energy band gap is defined between the highest point in the valence band here and the lowest point in the conduction band which means This and this, this is the gap you are talking about and this is your energy band gap, this is your energy band gap you cannot call this as the energy band gap because this is much larger gap than this no, this gap is much larger than this, we have to take the smallest gap. The top most point in the valence band and the bottom most point in the conduction band their separation in energy is the energy band gap not this one and this one. What happens here is that the bottom most point of the conduction band is at not k equal to 0 because it is not k equal to 0. So, an electron and a hole if they have to do a transition or recombine they have to go this way you know it is very difficult or not probable why because this change is a change in k. And K is momentum so essentially for an electron and a hole to basically make a transition some way to excite or you know absorb light or emit light or so on electron and a hole has to essentially make this transition across this gap this is a momentum change you know delta K and this change in momentum is pretty huge you cannot electrons and holes do not have that kind of a you know a light particle for example a photon cannot give that kind of a change of momentum. to cross from here to there.
So transition of electron and hole this way is you know it is a process which needs the assistance of some other things like atomic vibrations because atoms when they vibrate they can give energy and change the momentum that is how they come go from here to there. This is a very inefficient process so this kind of semiconductor where the bottom of the conduction band and the top of the valence band are not exactly at the same k these are called indirect band gap semiconductors. semiconductor like silicon for example or germanium for example okay and those materials where the top of the valence band like I showed you in the previous slide here where the top of the the top the bottom of the conduction band and the top of the valence band are in the same k.
So if you want to excite an electron from here to there you do not have to change k no change in k which means no change in momentum. right, no change in momentum. So essentially if you want to excite an electron from here to there or get an electron hole recombine here, the K is only 0 here right.
So electrons and holes essentially do not have to change the momentum, it is a very efficient process, these are called direct band gap semiconductors and examples are like gallium arsenide right, gallium nitride and so on okay, these are direct band gap semiconductor and the semiconductors where EK diagrams are such that. the bottom of the conduction band top of the valence band are not in the same K they are called indirect band gap semiconductor like silicon. So transitioning from one point to another point becomes very difficult or inefficient you need to take the help of phonons or which are phonons are actually you do not have to use the word phonons so much these are actually atoms vibrations ok atoms keep vibrating about the mean position you can quantify them or you can call them as phonons ok and those atomic vibrations essentially give energy to the. Electrons or holes to change their momentum is inefficient process. So, things like indirect band gap semiconductor like silicon or germanium they cannot emit light.
They cannot emit light. Why? Because for emitting light you will see the electrons that are here have to come down and recombine with holes that are here and the energy that is there actually is released as photon. We will see that in light optical processes.
Electrons that are here and the holes that are here they will recombine. directly and the energy that is lost, this energy you know, that is emitted as photon that is emitted as photon and that energy that is emitted will be roughly equal to the band gap of course, whatever the band gap is there that is how you emit the light because they will recombine. It needs no change of momentum so it is a efficient process that is how materials like gallium nitride and gallium nitride can be used to make LED because they can emit light. But material like silicon for example you cannot emit light because an electron that is here and a hole that is here.
For them to recombine you need the help of atomic vibration to change the momentum so that energy lost actually is energy that is given to the atomic vibration okay and hence you cannot emit light it is a very inefficient process. Similarly absorption of light in materials like gallium nitride, gallium arsenide is very high okay the absorption is very high here you know a photon comes here for example So, an electron in the conduction valence band can be excited to the conduction band and that energy is used up the photon energy is used up it is a very efficient process so this also a very high absorption coefficient. Now silicon indirect band gain material like silicon or germanium also can absorb light although the absorption coefficient is much lower but it can still be absorbed that is why they are used to also make silicon solar cell for example when the light comes that light energy the photon energy will be used to excite the electrons from here to there. not a very efficient process because part of that energy has to also be given up to the you know the photons will not have enough change energy to change the momentum. So you have to take help from the atomic vibration anyways right.
So you can excite the electrons of course with taking the help of atomic vibration it is an inefficient process. So the absorption is not very the absorption coefficient is not very high unlike in direct band gap material like gallium nitride or gallium arsenide where the E-K diagram is such that the conduction band valence band and the same K so the absorption is very efficient there right. but this material like silicon cannot emit light that is one thing and these are called indirect band gap material and this is a very important concept that we should know. But both indirect and direct band gap material their energy diagram in the real space will look like this only you know you have a conduction band here, variance band here whether it is a direct or indirect you do not know here. here because it is a x axis it is a x axis in centimeter and this is energy this is called energy band diagram and from the energy band diagram conduction band valence band you actually cannot tell if it is a direct or indirect band gap semiconductor but from this diagram you can tell if it is a direct or indirect band gap semiconductor from EK diagram and that EK diagram actually tells you that is why it is very important whether a material can emit light or not can be told very easily very nicely performed EK diagram okay.
So, That is the concept about indirect and direct bandgap material okay. So now what are the things you have learnt let us take a recap and we go as we go to the next concept. So one thing that we have learnt is so let us see from where we have started this lecture.
I told you the concept of holes we have introduced okay so we have introduced the concept of hole. So, I told you that electrons can go from the conduction valence band to conduction band they can leave behind a hole a vacancy and you see if there is a vacancy here then when you apply an electric field electrons move in the opposite direction to the electric field and it seems like the holes are moving to the right because electrons keep moving to the left. So, we can treat the hole as a quasi particle with a positive charge that moves in the direction of the electric field.
Hole also has a Its own E k diagram I told you, holes also have their own E k diagram and the E k diagram of holes actually represents the valence band of the semiconductor, valence band of semiconductor and the movement of holes is always associated with the movement of holes in the valence band, movement of electrons is associated with movement of electrons in the conduction band and the curvature of this also will give you the effective mass of holes. So, holes of their also of their own effective mass, I told you that holes might actually to have three different branches of E-K diagram, the light hole, heavy hole and a spin orbit and a separate bandage here, band hole here, but we typically consider only light hole sometimes may be sorry heavy hole and sometimes may be light hole. and they have different curvature, this separation between the highest point of the valence band and the lowest point of the conduction band, it is basically defined as the band gap, if they happen to be at the same k and k equal to 0, it is called the direct band gap semiconductor and if they do not happen to be in the same position k, then they are called indirect band gap semiconductor like silicon for example, that is an indirect band gap semiconductor, so it is a very poor, you know it cannot emit light.
also it is not a very good observer of light although you can still make solar cells by making thick silicon layers and to make a transition of electron hole if in this kind of material you need help of atomic vibrations called also phonons. is how you basically make the transition between electron and hole, it is an inefficient process and that is why it cannot emit light. Now next immediate concept that we will learn here in today's class that is very important is actually how this is we have learned right, how electrons and holes are occupying the energy you know this is the conduction band for example and this is the valence band for example, I am talking about the direct band gap semiconductor here, this is E-K diagram right, this is E-K diagram.
How are electrons populating the conduction band? How are holes populating the valence band? All these concepts we need to understand by using statistics.
Why? Because electrons and holes are very large in number. You cannot individually pick up an electron and hole to do that.
You have to talk about an ensemble like a collection and that is why statistics comes handy and we have to use Fermi Dirac statistics here. Fermi Dirac statistics. The Fermi Dirac statistics nothing very fancy I mean there is two people Dirac and then Fermi of course they both got Nobel prizes when they are very young.
This Fermi Dirac statistics will tell you how electrons or holes will basically you know populate the energy bands and when we study Fermi Dirac statistics we do not have to think of only electrons it can be applied to any particle which has certain properties you know like indistinguishability and other things. So when we study Fermi Dirac statistics it is not exclusively with respect to semiconductor but it is. With respect to many type of particles you know it can be applied to many kind of particles which satisfy this we call them fermions we do not have to worry about that so much now.
But how do we actually come up with this Fermi-Dirac statistics to define the statistics of electrons and holes that are occupying this you know for that we will take a very simple case into account let me do one thing. So suppose I have many energy states. These are all at an energy level E1, so these are all at an energy level E1 and there are G1 states. There are G1 states they are all at energy E1 and the number of electrons that will occupy this is N1. N1 electrons will suppose occupy G1 energy states that are at an energy E1.
Then similarly, There are G2 energy states at energy E2 and N2 electrons have to occupy them. So similarly there are many right. This is suppose Gi energy states at energy Ei and Ni has to be there and eventually you have a large number of thing eventually there is Nn you know Nn for example total number here okay.
So first let us look about this in how many ways Can you in how many ways I repeat in how many ways can you fill this G1 states by N1 electrons. It is like saying I have for example 7 boxes and 5 say marbles. I need to fill up 7 boxes with 5 marbles in how many ways can I do that I can do that by 7 C5. which is 7 factorial by 5 factorial 2 factorial if you remember from high school physics high school maths right.
So the number of ways in which you can fill up G1 energy states with N1 electrons is actually G1 factorial by N1 factorial G1 minus N1 factorial okay. That is the number of ways in which you can fill up Now of course for the second level you can do G2, N2 and so on for ith level it will be GI by NI, GI minus NI right it will be like that. So the total number of ways in which you can fill up all these things will not be the sum but it will be the product. It will be the product of G1 factorial by G1 minus N1 factorial N1 factorial that is the number of ways in which you can fill up all these things. energy level can be filled up the second energy level can be filled up by G2 factorial G2 minus N2 factorial N2 factorial and so on if the product this like the summation is given by sigma the product is given by pi okay the product is given by pi.
So if I give sigma it is summation right if you remember so instead of summation we have to use product, product is given by big pi like that so this is basically I for example and I goes from 1 to a large number of N. So, the way is to minimize this. And so there are certain derivatives and some approximations that have to be used called Stirling approximation because it is a factorial and then you have to minimize this you know probability eventually.
So what you get essentially is something like you have to get something like say n i by g i the number of electrons that you are filling. in the number of available states that you have. It basically gives you the probability that a state is occupied. It is like you have five marbles. And 7 empty boxes the probability that a box is occupied is 5 by 7 you agree.
So, essentially you get this probability this is the probability that an energy state this is the probability the probability that an energy state is occupied by electron is occupied by an electron ok it is occupied by an electron that is the probability ok and this probability can be obtained by basically minimizing this function and doing some approximation and simple maths. We will skip that but eventually this expression this n i by g i it is actually called and this is at E i level by the way. So, this is the probability that electron is occupying that and it is corresponding to i.
So, it is called F the probability is called F and this F of E i because this is the corresponding to an energy state of E i that these are the number of states these are the number of electrons but this is the energy by the way. And this is actually given by 1 by 1 by 1 plus this is 1 1 plus exponential of E I minus E F there is something called E F by k T this k is not the reciprocal space momentum this k is Boltzmann constant k and T is temperature. So, let us write down it again.
Let us write it down again, F of E I the probability that this energy state is occupied is 1 by 1 plus exponential of E I minus E F by k T this is Boltzmann constant k B T, this is the probability that an energy state is occupied at E I right by an electron and E F here is called actually the Fermi level. You call it Fermi energy when it is temperature equal to T equal to 0 otherwise you call it Fermi level and it is some way it is also called chemical potential by many people same thing actually okay is the Fermi level and what does this Fermi level mean it is actually a statistical construct it does not exist in reality but the difference of Fermi level can be actually a realistic quantity by the way it will be different it is a statistical construct it is a statistical construct or a statistical concept that helps in understanding many things for example. If your E I the energy that you are talking about is E F then if you look here what will happen F of E I equal to E F you put here E I equal to E F that will become 0. So, e to the power 0 is 1, 1 by 1 plus 1 is 1 by 2 and this is independent of temperature does not matter what T you have this is 50 percent. So, you can say the Fermi level is 1. Is a statistical concept of course. It is the energy level, it is the energy level where you have a probability of 50% the finding the electron, of finding the electron, right, of finding the electron, an electron, okay.
The probability, the energy level at which there is a 50% chance. that there will be an electron that energy level is called Fermi level and it is independent of temperature like this probability of 50 percent is independent of temperature at any temperature, this is at any temperature, at any temperature the probability of finding the electron is 50 percent at Fermi level that is your statistical construct or a concept of Fermi level. It helps in understanding so many things and the Fermi level as you will see will become the most important thing that will go along with you in the semiconductor device analysis.
None of the device that we will talk about in this course can be understood without Fermi level. Everything will be related to Fermi level. So Fermi level is a statistical construct that tells you the probability of finding the electron. It is the level at which the probability of finding the electron is exactly 50 percent, right. So what I can do here is that I can plot the probability this F of E right versus E at say T equal to 0 Kelvin.
So at T equal to 0 Kelvin if you plot the function I just had drawn now it will look like this it will look like this okay this is EF. the Fermi level all the electrons you know it is a 100% probability this is 1 this is 0 above Fermi level there is 0% probability that electrons are there and below Fermi level there is 100% probability that electrons are there at 0 Kelvin. If I increase the temperature say you know if I increase the temperature say if I increase the temperature to say 100 Kelvin then this Fermi distribution this is Fermi function by the way that in case I had not told you.
This actually this is a Fermi function it is a Fermi function okay and it is you can call this is the Fermi Dirac probability also okay it is also called Fermi Dirac probability the probability distribution of electrons in a way and you can find out the probability at any energy level E I told you that at EF at Fermi level it is 50% but any energy level EI you can find out because you know EI minus EF that quantity you have to know right and temperature of course you have to know. So what do I do? So this is at 100 Kelvin this will become how much how do you know this will become slightly this and this will slightly become this. This point of course is half this is 50 percent. So at Fermi level you have always 50 percent probability of finding the electron but now there is a slight probability of finding the electron above the Fermi level.
also, right because you know the overall area has remain same, but this has become slightly deviated and then you have a say T equal to 200 Kelvin, I am raising the temperature what will happen then, then to you will have even more like this. So, now you have even more probability that there is a electron above Fermi level too much far away energy level you know this is also this is increase. So, you have a finite probability of finding the electron above Fermi level also what it means as temperature you know as your temperature increases as your temperature increases probability also increases that you Probability also increases that you will of finding electron above Fermi level.
So finding the electron above Fermi level also increases the probability of finding the electron above Fermi level also increases when your temperature increases that is what it means right. With higher temperature if you take for example T equal to 300 Kelvin then of course it will be when. like that even more but this at any temperature at Fermi level this is EF. At EF you will always have 50% probability of finding the electron that is what it means. So basically Fermi deduct statistics will tell you the distribution of electrons and it also you know if I want to tell you again if F of E gives you the probability of finding the electron at energy level E then 1-F of actually E is the probability of not finding the electron so this is the probability of finding holes by the way and this is the probability of finding electrons by the way right.
So the electrons and holes and how they are distributed and how we study them the devices and everything depends on the probability distribution okay. This probability it is very important that is why we introduce the concept of Fermi function. Now the another important concept that we probably will introduce in the next class of course and that will be very important to understand many of the things I will just briefly mention what that is. here and that is called actually density of states. I will not introduce it in this class, the next class will come here, okay. Density of states, this is something we will discuss in the next class.
okay and together with Fermi function I told you the Fermi function and the Fermi probability here right. So with Fermi function and density of states that I will introduce in the next class with doing combining these two concept you can find out the actual number of electrons or holes okay actual number of electrons or holes can be found out by doing these two things. So already I told you Fermi function is a probability function and density of states something that we will take up in the next class.
Density of states essentially tells you how many energy states are available per unit energy per unit volume, per unit energy per unit volume, how many empty states are there, how many energy states are there. Once you know that density of states and once you know the probability that you can occupy those empty states, the product of these two in a way will give you the actual number of electrons or holes that are there in the semiconductor. So all the devices will depend on how many electrons and how many holes you have in the device and for that you need density of states and then you need the probability. So today we learnt about probability, the probability that an energy is occupied by an electron that comes from Fermi deduct probability. And Fermi data probability says that at exactly Fermi level which is a statistical concept you have 50% probability of finding the electron but as you increase the temperature there is also probability that above the Fermi level you will find some electron.
At 0 Kelvin you have absolutely no chance that you will find electron above the Fermi level okay. So next class we will introduce the concept of density of states from there we will take it forward okay. So thank you and we will meet again in the next class with density of states okay. Thank you.