noc19-cy16-Lecture 58 - Metals, Insulators and Semiconductors

[Music] [Music] now I will go to the third lecture of week 12 and in this lecture I will talk about the relation between the bank structure and the density of states and the electrical properties of materials that is I'll talk about metals or conductors insulators and semiconductors okay so week 12 lecture three will be talking about metals insulators and semiconductors by metals I had mean I mainly refer to conductors okay the whole idea of electrical properties of materials is based on the band structure and the possibilities that you get from filling of the bands okay now if you recall what we said in one of the earlier lectures was that was that the the way we calculate the electronic structure of materials is to first make the band structure okay so this band structure has a e as a function of K so you first so you first calculate the band structure and I will just again again for simplicity I will show the one-dimensional picture but you can do this in higher dimensions so let us say this is a range of allowed values of K okay and you have a band structure that could be something like this and what we said is that once you have the band structure you look at all the possible states okay so these are the possible states so so so basically the states correspond to different values of K so if you look at the K axis you can tell how many states are there okay again these the number of states is given by the bond one carbon boundary conditions okay so you keep you have all these states you have all these possible states okay the end in some states human for this man then what you do is once you have these states you start filling electrons into band so you count the total number of electrons okay and you start filling them into the bands so so so depending on the number of electrons you start filling them okay I am showing only one okay so you can think of it as a spin state or you can think of this line as as two electrons okay and you start filling from the lowest energy onwards so so you keep filling okay and let's say you have you have lot of electrons you have lot of bands also okay so you keep filling them and and again again you fill you feel from the lowest energy onwards so you have to fill you have to look very carefully how you fill the fill the electrons into the bands but when you are done with this exercise then you see the you are left with some the the electrons you keep filling bands up to some energy okay this is the highest this is the energy of the highest band that is filled the highest highest state that is filled okay now now this approach that we took is a zero temperature approach in the sense that in the sense at T equal to zero okay electronic configuration is strictly ground state okay that means that means you start filling from the bottom and you fill only up to up to the highest available state and this energy is called the Fermi energy EF okay so the Fermi energy is the highest occupied State at T equal to zero that is called the Fermi energy now there are different possibilities okay now in this particular case your Fermi energy in this case okay so this is case one Fermi energy intersects several several bands okay one or more bands it doesn't matter one or more bands so so basically obey basically one or more bands are partially filled partially fills so if you see if you see this band okay it is only partially filled ready it is only filled up to here whereas this band is completely filled similarly this band is completely filled ok this band is partially filled ok so one or more bands are partially filled ok the second case ok you could have another case ok I will just show it schematically below here now in this in this case let me I'll draw a slightly different like ok let me take a fairly similar band structure but let me consider the case now where the Fermi energy is right here ok so that means all these bands are completely filled whoo well okay I want to change that a little bit because I had I don't want to dislike sorry I'll just change this a little bit so what I want to show is something like this and let me say this goes somewhere here okay so so now what you find is that these two bands the tube and said the low two bands are completely filled okay so the Fermi is so they are filled all the way all the states are filled or occupied whereas the upper two bands are completely empty so so this is the second case Fermi energy is at the top of one band okay implies some bands are completely filled okay emphasize the word completely and rest are completely empty okay now in this case in case two there is clearly a band gap okay there is a band gap in this case okay because because there is a difference in energy between the between the last filled band and the first vacant band okay and and so and so there is a band gap okay in this case there may or may not be a band gap in the first case but clearly you see that the Fermi energy at least intersect several of the Bands so now we can see that these two cases case 1 where the Fermi energy intersect several bands and case two where the Fermi energy is is at the top of a band these two will have fundamentally different electrical properties so now let's let's now look at the electrical properties so so the first case ok that corresponds to a metal one is a metal or more more precisely it is a conductor the second case can be either an insulator or a semi conductor semiconductor is so in this case you have a bandgap okay so so in this case we have already seen that there is a bandgap okay now an insulator has a large band gap whereas a semiconductor has a moderate or small bandgap okay now when we use a word large or large or small you should ask immediately what is large with respect to what okay so this were large and moderate small are with respect to two thermal energy okay now the thermal energy measure of the term thermal energy is actually the Boltzmann constant times the temperature okay and when we look at this value okay this so at at T equal to 300 Kelvin which is about room temperature okay you have K KB T equal to 1.3 8 into 10 raise to minus 23 into 300 joules okay so the unit of Boltzmann constant is joules per Kelvin okay Boltzmann constant Boltzmann's constant is 1.3 8 into 10 raise to minus 23 so KBT is so many joules okay and I can write this as into 3 okay into 10 raise to minus 21 okay / 1.6 into 10 raise to minus 19 Joule per electron volt okay so this will give me the value of Boltzmann constant in electron volts okay and this works out to about 0.026 electron volts okay so the thermal energy in electron volts is typically of the order of 0.02 6 electron volt okay so usually if your band gap is so typically insulators have band gaps much higher than the 4 or 5 okay so so insulators have large band gaps okay this is several electron volts okay semiconductors have handle of the order of one to three electron volts okay now what that means is that is that with sufficient thermal energy your semiconductor can actually conduct okay now if you see if you see this is point zero two six is still much less than one or three electron volts okay but semiconductors and use another property called doping which is used to increase the the number of charge carriers okay I won't be discussing that but just in terms of bandgap we see that insulators should have a much larger band gap but if you have a band gap of the order of one to three electron volts then you would call it a semi conductor if you have much larger than four or five electron volts then you would call it an insulator metals of course do not have a band gap metals or conductors do not have a band gap okay and so this is the basic picture of electrical conductivity of solids now let me hit now look at the density of states and what that implies for the for the case of the band gap okay if you look at the density of states and I will just I'll just go matically draw the density of states for for a metal so for a mental you will have you'll have some density of states okay again again I'll use the same I'll plot G of F G of epsilon on this scale and epsilon on this okay you will have some density of states what is what is interesting is where the Fermi level is so the Fermi level will be let me show it slightly differently just to emphasize a few points I will get I will have I could have I could have states like this okay now the Fermi energy could be will be somewhere here that is in between some set of density of states okay now this is a direct consequence of the fact that the Fermi energy intersect several bands so it'll be in one of these one of these big bands okay in the case of in the case of a of a half an insulator okay now your G of E will look like this and now the Fermi energy will be right here at the top of some set of mats and there would be a band gap I'm showing this I mean you could have again you could have several you could also have things like things like you know having lot more structure so if you look at the G of e you could have you could have things that look like this it could have multiple bands okay here I'm just showing two bands okay you could have you could have things like this and you could have you could have the Fermi energy somewhere here EF okay then what you'll say is that all these bands are all full okay these all all these all these energies okay these are States states corresponding to these energies are fully occupied States corresponding to these energies are occupied in the case of the conductor you would say something like this that all these states all these states are fully occupied now the states corresponding to these energies are actually partially filled okay in this case you'll just have all these states being occupied okay these are insulators okay so these two are either insulators or semiconductors depending on the size of the bandgap madisyn this would be conductor so a lot of information about the about the nature of the material the electrical properties can be got by looking at the band structure now notice that in addition to band structure this is a very important point so I'm just going to emphasize this that in addition to two band structure yeah we also need EF okay so you need the Fermi energy okay now at zero Kelvin okay at zero Kelvin we just do the we just find the Fermi energy by looking at States by filling States from the from the bottom states okay at finite temperature actually this you know some of the there can be some excitations so at finite temperature there is a probability that even the even the higher excise even the more excited states are also possible okay so at finite that is non zero non zero temperature this is greater than zero okay excited states are also possible are also occupied okay and you define something called a Fermi level level okay which is basically this is something this is something that is that is slightly different from the from the Fermi energy so the Fermi level okay that usually usually if the Fermi energy is in this case you have a Fermi energy here but there can be excitation so some of these higher states might also be occupied okay the Fermi energy is some sort of it is it is some level this is the Fermi level okay and this I'm not I'm not going to go into this in detail but the Fermi level is a concept that is useful at finite temperature and this this has to do with something called the Fermi Dirac distribution direct distribution okay and I'm just mentioning this okay but I won't expect you to know this and I just expect you to know the Fermi energy at t equal to 0 but the Fermi level is a is a useful concept at nonzero temperatures okay that has to do with the Fermi Dirac distribution and the Fermi level is where the Fermi Dirac distribution takes a value of half the occupation number of a state is is equal to half but again we are not going to describe that in detail but just this is just for your information that at finite temperature you can define the equivalent of a Fermi energy which is called a Fermi level okay but again for this course I just expect you to know the Fermi energy and how do you understand the difference between semiconductors between conductors and insulators so with this I will conclude this lecture the third lecture of the last week of this course and in the next lecture I will I will I will use the same band structure and band gap to talk about the optical properties of materials okay so whether how what what should be the property of the band structure in order for a material to absorb why it what should it be for it to emit light and so on thank you [Music] [Music] [Music]