 Good morning. We start today's talk with answering a few questions that came up yesterday on chat. So the first question which came from Rangaswamy College Center 1313 and the question is, is superconducting state reversible or rather I should say is superconducting transition reversible? The answer is of course yes but let me sort of try to make a couple of comments on this. See what do you mean by a reversible phase transition? By a reversible phase transition what we mean is that if you have gone from let us say state a to b by some process and if you reverse the procedure in small step following exactly that path. Supposing I have gone like this from point a to b and I have arrived at this state. Now whatever are the processes by which I went from a to b then if I reverse it I should come back exactly by the same path and coming back to a. So look at for example the classic case of a non-reversible transition that happens when you talk about the how does the magnetization behave against an applied field in case of a ferromagnetic. See when you apply a field the initially the very increase in magnetization is linear gradually it becomes non-linear and when all the magnetic moments have saturated have been aligned it sort of becomes aligned. So this is what is known as saturation magnetization. Now supposing I reduce the field and I go this way. Now it turns out that it does not quite take that path but will come like this and even when the magnetic field has actually become 0 you will find instead of returning to the magnetization value of 0 it will still have certain amount of magnetization and this is the what is known as the spontaneous magnetization. You want to reduce the magnetization to 0 you have to actually apply what is known as a coercive field and then of course you have seen this BH curve completely. So therefore this is an example of an irreversible process. Reversible process would be one if I went like this and if I came back like that. Now let us come back to the graph that I drew for the magnetization. As I have told you I normally plot a minus M against H not for any reason but I could have plotted M against H but then my curve will be in the fourth quadrant it sort of doesn't look good other than that there is no reason. So if you look at what happens to that when you apply at a particular temperature the thing is this that supposing you have the curve is basically this. Now what we are trying to say is this that look if you now supposing you have a super conducting state and you have achieved it by reducing the temperature. So this is let me come back to that I am let us say I am plotting some physical quantity. I could be plotting resistance curve let us say resistance in the absence of magnetic field. So resistance at H equal to 0 and I am plotting it against temperature T. See the idea is something like this that as I have told you yesterday that as long as I am below Tc the curve goes as A plus Bt square and if it were not so at certain temperature Tc which we have talked about that temperature. So this is a normal state and if you have go to a superconducting transition then all this state here is a superconducting state and the resistance or resistivity suddenly drops to 0. Now let us reverse the procedure supposing I am at this temperature and magnetic field is still 0 then if I increase the temperature I will go like this that will as long as the temperature T is less than Tc the resistivity will remain 0 and then as soon as Tc increases by epsilon this I am talking about type 1 conductor type 1 superconductor. So resistivity will suddenly rise and after that if you increase the temperature it will follow the usual T square behavior. So therefore this goes to show that superconducting transition is reversible the answer is yes. The second question is a quick question this came from center number 1028 I have not picked up the name. It simply ask the question do you today have room temperature superconductors the question of what we call as high temperature superconductivity will be discussed by professor Suresh in his last lecture but the quick answer to that question is no we do not we have not achieved transition temperature which is 300 Kelvin even today. Next question is from 115 to San Joseph college Palai it is a very interesting question it says that on one hand you say superconductor is a diamagnet on the other hand you say there is a persistent current. So I have persistent current and I have diamagnetism a very very very intelligent nice question is there a contradiction between these two statements. So let me first explain to you what does he mean by a contradiction sorry I keep on using the word what does he mean I it could be a lady who has asked that question. So he or she but I will not be able to be politically correct all the time. So whoever has asked that question the point here is the following that on one hand we say a superconductor is a perfect diamagnet that it does not allow a flux to penetrate it. So the the so therefore but on the other hand we are saying that there is a persistent current which is running which is creating its own field okay the is there a dichotomy in this the answer is no and let me let me explain what the problem is the problem is something like this the type this thing that we are talking about takes place in any material which has a geometry with a hole in it. Now somebody had asked the question what happens in a solenoid the thing is not whether it is a solenoid or it is a toroid or it is a ring all the all the question answers are similar the the point actually is this that in the absence of a magnetic field when I have a superconductor then of course everything is fine nice but let us suppose that I start with a temperature T greater than this I am also simultaneously explaining the question asked by question 1100 M.E.S.Pallai right M.E.S.Pallai yeah so it says simply explain flux repulsion so I am trying to answer that also along with the point here is the following that supposing I start with a normal material but the shape is such that that the the material has a hole in it for instance it could be a tube like structure it could be a toroidal structure it could be a ring whatever you like disc now the idea is this that when you put in a magnetic field the in a at a temperature higher than Tc the flux lines enter not only the material of the thing but they will of course enter the hole also now let us so this is in the presence of B now let us let us decrease the temperature so that T reduces so that the temperature becomes it becomes superconducting now when it becomes superconducting one thing is of course the since it is a perfect diamagnet a perfect diamagnet which has a negative susceptibility of one it says that it does not allow the magnetic flux because remember essentially H plus M is your the magnetic field B so therefore what I will have in such a situation is a situation of this type then there would be field lines outside close to that because of the geometry it will bend a little bit but it will still be outside but there will be flux lines in the hole because the hole there is no material so there is nothing which is stopping you from having flux line the hole so this has expelled now once it is expelled what you do is that you switch off the magnetic field now when you switch off the magnetic field the situation that happens is the following that you had these lines which I drew outside the material now they of course simply vanish these lines these will simply vanish they in our language they will go away to infinity because infinity has the source and sync for everything forget about that but look at those flux lines which sort of came in the hole now how do these flux lines go they cannot go because in order to move to infinity they will have to pass through the material of the superconductor so this is this is the situation what happens when B is equal to 0 now of course since there is no source no sync so therefore these flux line will sort of close on themselves I am coming back to this in connection with today's lecture when we did take up this type of geometry now notice what I have what I have is a very peculiar situation I have magnetic flux lines outside the material of course but there is no source of a you know magnetic field now how can such a thing arise we know that the magnetic fields can be only created by means of a steady current now where is the steady current now like it has been pointed out that look I do not expect the material of the superconductor to have a magnetic field but you see remember that even in a conductor we made a statement that it is not that conductor cannot have charges the conductor certainly cannot have charges but only on the surface the because these charges are free electrons and inside there cannot be a magnetic field so the electric field so here what happens is the following that in a very small see here close to the edge there is a circulating current this is the persistent current we are talking about the this persistent current is will allow this magnetic fluxes to be sustained here to be sustained here and it is it is a surface effect and so therefore it stays so there is nothing like a in this if you try to do it what is the current here the answer is you so there is no contradiction here the currents will be in a very thin surface there magnetic flux exponential I have talked about but let me make the another point here I wanted to point out the fact that it is a 0 no flux is not a consequence I should say not necessarily a consequence of of zero resistivity the confusion arises because of the following that if you follow Maxwell's equation you have del cross of a equal to minus dv by dt this famous Faraday's law now I have a superconductor I have a superconductor and obviously my electric field is 0 so E is equal to 0 means curl is equal to 0 now if curl is equal to 0 it tells me well I could take the line integral of surface integral of both sides and you know that we have been talking about it for quite some time that del cross E dot ds is same as E dot dl that is equal to minus d5 by dt this is the Faraday's now if the I have del cross of E equal to 0 so therefore I expect E to be equal to 0 so this quantity I want it to be equal to 0 so phi is equal to constant now so the phi equal to constant tells me that if I have a superconductor I have a superconductor there now the flux inside it was 0 to begin with now I have now put in a external magnetic field now since this 0 resistivity demands my flux cannot change so my flux cannot enter the sample if I suddenly put in a magnetic field so this is this is not expulsion of flux this is not allowing flux to enter in this you can understand by means of the fact that this is this can be understood by means of the fact that superconductor is also a perfect conductor the expulsion of flux means that if these material already had flux at a temperature T greater than Tc now you cool it according to what I just now told you the flux should even on becoming a superconductor because it is a perfect conductor the flux should remain the same this does not happen this does not happen the flux gets expelled not understandable not understandable from the point of view of a Maxwell's equation in order to understand that you have to have other theories the most commonly accepted theory today is the BCS theory I will touch upon BCS theory but I will not be able to I will not it is not possible for in a course like this to actually derive the entire BCS theory so the expulsion of flux like somebody has asked me to explain well it is not a very easy thing to explain based on ordinary electromagnetism you need quantum mechanics okay next question is from sector 1319 which asks me the question why do you draw the you know when we talk about superconductor and we sort of said that we we have we draw the plot in terms of the magnetization versus the magnetic field why not T versus Tc I mean why not why not magnetization versus T or minus m versus T look at what would you get it is a question of your benefit what would you get if you plotted the magnetization versus T the idea is something like this that we know that supposing you start with the superconductor we know your susceptibility is minus 1 so that your minus m versus h curve is actually linear now it remains linear at all temperature below Tc which means you have to plot plot a set of curves one for each temperature and there will be no feature in that the moment it becomes Tc and more the m suddenly goes to 0 so again you do not need a curve for that so the that is why you can describe the phase transition by for example the plot that I had of rho versus T because there the for T greater than Tc I have a normal state for T less than Tc I have this so magnetization versus this will not help 1047 ask my question superconducting state contains both normal and superconducting electrons so can the equipartition principle be valid let me first make one comment there is nothing like a normal electron and a superconducting electron there only one type of electron what I said is the following that before people understood the mechanism of superconductivity there were many phenomenological theories what is meant by a phenomenological theory a phenomenological theory is one where you take certain experimental fact now these are given to you by experiment to show you cannot change it now once you have taken some experimental fact you try to say that let me make minimum number of assumptions now if with my minimum number of assumptions which may not always be rigorously justified I can explain the phenomena then all that I do is that later on search for whether that I had any basis or not so since people had no quantum basis for understanding the superconductivity what they did is to say that alright this is behaving funny so maybe there are some of those electrons which are and this came up particularly because of the discovery of type 2 superconductors the type 2 superconductors were funny because the they were partially allowing flux to enter there was there was a question of partial entry of flux as a function of the magnetic field now so people said that maybe there are two types of electrons one type of electron let us call them red electrons they are the ones which behave normally they carry entropy they can scatter against things like ions and if you want for which the equipartition principle is valid provided you are classical physics there is another type of electron which we call as the green electron let us say now notice I am using now a phenomenological thing let us suppose there is a red electron there is a green electron there is nothing like a red electron a green electron I am just giving it as an illustration so they said there is another type of electron or another fraction of electrons which do not carry entropy which in the presence of an electric field will simply go by Newton's law not and move without resistance now this is this is a phenomenological assumption so there is no question by which I can explain this the next question next question was on explain penetration depth which was not very clear I will do that because they were the last slide so therefore I will take care of it and another question which came by your forum is the energy gap is delta why do we talk about 2 delta always ok now there is a good question say basically what we are trying to say is this that super conductor has an energy gap of 2 delta now but on the other hand as you will become clear in when I talk about the cooper pairs the superconductivity arises because there is not one electron which is participating it at a time but pairs of electrons are participating it at a time so if you want to break superconductivity if you want to make it normal you have to break a pair in other words supposing a pair where here and this is this is a couple pair then when you break it the and you want to take it there each of the electrons has to be promoted to that level that is each electron must at least be excited by an energy delta and that is why since there is a pair of electron I need 2 delta for that so with that I complete the question may be some clarification on the questions themselves yeah Gyanumani college you have a clarification on the questions please go ahead the main purpose of the superconductor is that we consider only the current factors is it correct it is 2 or not see there is nothing like a purpose of a superconductor there are material which behave like that way so it is not that superconductors purpose is to talk about current factors the there are material these are physical things like copper aluminum etc are conductors the if you takes in fact it is very surprising that most of the cases most of the cases particularly when you come to the HDFC you will find materials which become superconductor at a temperature below TC in their normal state they are rarely good conductors so for example you know I mean the you notice copper that the copper has such a low transition temperature a superconducting transition temperature that copper will not become a superconductor at reasonably low temperature as well so it is not true that good conductors will become superconductor if you raise their temperature that correct statement is not true SGS college of technology you have a question yesterday you specified that conductors have positive temperature coefficient and semiconductors have negative temperature coefficient so why this could be possible as in both the cases amplitude of vibration of electron no no no do not go so fast you said yesterday I mentioned that resistivity coefficient for conductors is different from the resistivity coefficients of semiconductor yes but what is your question on that temperature coefficient of conductors is positive yes and that of semiconductor is negative yes but amplitude of vibration of electrons in both the case yes yes okay I have understood your question so let me explain see the point actually is this the following that you see you have to realize first that conductor has a lot of free electrons available so the basic problem in conductor when you increase the temperature is the following that look at what is the simplest mechanism I will I will explain that answer in two ways both with respect to our classical way of understanding what is conduction and with respect to the quantum way so the way it happens is this that the when you are talking about electrons participating in conductivity what happens really so basically the electrons when you are subjecting them to an electric field they are moving around going from one end of the wire to another that is because they are free now when they are going the because there are ions which are more or less fixed at their positions they also bump against those fixed ions now so when they bump against it they are you know there will be a collision between them typically elastic collision and their velocity direction would be changed elastic collision velocity magnitude will not be changed so but they will proceed till they meet another collision now this is what happens now if you raise the temperature now as you said that there would now be a vibration of the these ions now when these ions vibrate the probability of collision with the electron or probability that an electron collides against an ion becomes more that is the ion the electron gets more resistance to its motion imagine you are in a room with fixed chairs now these chairs are fixed so therefore you can avoid the chairs and go once in a while we will hit the chair but you can go out of the room but suppose now that some of your friends are continuously moving those chairs in random fashion now your possibility of hitting that chair as you are trying to move out of the room becomes more so you incur more resistance now so as a result as the temperature rises the vibration amplitude rises so you are absolutely right and because of that it happens now your question was that why didn't it happen in case of a semiconductor you see in semiconductor there is a dip there is another effect which is much more important and that effect is because of the fact remember in in in the conductor there is nothing like a band gap the whereas in semiconductor there is a band gap now first thing that happens when you raise the temperature now in semiconductor for example at zero temperature intrinsic semiconductor there is nothing like a conducting electron because of the valence band is full now when you raise the temperature the collision becomes a secondary effect there because the number of electrons is so few anyway therefore the first effect is there are no electrons in the conduction band there are no electrons in the valence band so there is something to conduct so the the number of electrons participating in conductivity rises so your n value rises so there are two effects there the you are right that it is it is not very different the amplitude of vibrations would remain the same but you see there was nothing to conduct there are now electrons there so therefore there will be some conductivity and conductivity will increase is it clear sir one more question yeah sir yesterday we have seen the magnetic field expulsion from superconductor yes but but it might be that at certain depth over the surface of metal magnetic flux cannot be cancelled completely so is this depending on the intensity of magnetic field or shape of sample yeah no no see the the magnetic flux expulsion say like in conductor if you apply an electric field I know that in the bulk of the conductor in main conductor there I am not talking about a current carrying conductor because current current conductor there is a battery supposing you subject it to an electric field due to static charges and all that now the inside the bulk of the conductor the electric field has to be zero but the there would be induced charges generated in a on the surface so what we are trying to say this is what I am going to anyway come back to the explanation of that that there is a skin depth distance there the it is not that the a magnetic field flux does not penetrate at all it does penetrate a little bit but it very quickly dies out that is that is the whole idea that there is a depth to which it goes something like the skin depth in case of a conductor okay let me since we have already taken up our please continue to send questions I understand that today is the last lecture on superconductivity but I will take up questions that are raised today maybe tomorrow in the part of the optic session there is no because the questions are coming in so I will answer okay so let me now go over to the main lecture so before I concluded yesterday and also during my clarification today I talked about some phenomenological theories by due to London brothers they are called two fluid model and remember that these were before Bardeen Cooper's theory of superconductivity came into the picture so we said that there are two types of electrons one type of electron are the usual electrons which are normal electrons which carry entropy contributes to conductivity suffers resistance and so that has this type of a structure another type of electron which doesn't offer any resistance moves just following Newton's law and that is that fraction is Uranus so that's the what London assumed now so we said he said that the total charge density or number density of electrons consists of a normal fraction and a superconducting fraction and the current density of which is of course obtained by multiplying with the charge is also a normal current density and a superconducting current density now it is this js that we are interested in j and we understand this is our usual dissipative thing the the superfluid they are called fluid in a two-fluid model satisfied so normal one satisfied j equal to sigma ohm's law the the second one is djs by dt now js is n times velocity number time velocity is the current you can multiply with the charge then so this becomes there is a e missing here the normal the superconducting fraction the charge which was missing here and d by dt of s now this d by dt of s is by Newton's law equal to minus ee because that's the force so therefore I get ns e square over m into e so it tells me that djs by dt is proportional to e so I can write e as equal to after bringing that m by ns e square to the left hand side I define that quantity as some capital lambda so I get electric field is given by rate of change of capital lambda js so what we are now trying to talk about is this that supposing I took this okay and set up as I did yesterday equation from the electromagnetism what I get is the following I assumed that I have a one I have a semi-infinite slab now what is meant by semi-infinite slab firstly it is in two direction it is infinite so for example supposing the x direction is from 0 to infinity now that's a semi-infinity because the minus infinity to 0 is not included y and z direction are infinite now this symmetry tells us that there cannot be a variation in the y and z direction because after all they are symmetric so only variation that is possible is in the x direction so therefore your equation for the magnetic field or whatever del square h that becomes can only vary in the x direction and I had shown you yesterday that it gives me a solution of the type that hx equal to h0 the surface value and it falls off exponentially and and typically a skin depth is always defined as the depth at which the surface value becomes reduced by a factor of 1 over e so by a factor of e so that that's 37 percent that's that has been electrical convention for years that this depth at which the magnetic field strength which has penetrated the sample becomes 1 over e of its value is known as the London penetration depth now this penetration depth for example in aluminum it's about 500 nanometer angstroms that is 50 nanometers and well this is this is roughly this order but if you take the current high temperature superconductors they are much more so that takes care of what we did yesterday let me now come back to today's talk now I also spent a bit of a time in explaining the what happens to this flux so basically I tried to tell you that supposing I take any substance which has a hole in it a solenoid with a tube which is a tube and inside there is a magnetic field established the or just take a ring have a an external magnetic field there are flux lines which are passing make it a superconductor now when you make it a superconductor the flux lines cannot be outside if the flux lines are not outside then they but they can be inside the hole the when they are inside the hole the the reason why they continue to be inside the hole is that they cannot get out if they tried to get out they have to enter the superconductor and superconductor is not going to allow it all right so then when I switch on the magnetic field the since the flux lines have no sources this is the picture that you find this is the picture that you find the flux lines must close on themselves and as we explained a little while back this requires that there must be a persistent current in a small thickness of its surface now that is understandable but what is important is that the measurement of these fluxes they found something very interesting they found that the flux is not just anything but the flux is quantized in this unit h by 2 e unit in fact this factor 2 keeps on coming back to haunt us that is because two electrons are always involved so in all words you will find phi naught the flux that is there can be either h by 2 e or 2 h by 2 e 3 h by 2 e like this but not any fraction there so the trapped flux is also quantized now let us try to understand why trapped flux is quantized I will try to explain it to you on the basis of again very basic I mean phenomenological old quantum theory if you like remember your Bohr summer field model of quantization which you had utilized in case of Bohr model now in Bohr model we had said that the angular momentum is quantized and the way Bohr derived it is this that since I have circulating current remember this Bohr's argument that since there is a circulating current the charge which is circulating moving in a circle obviously is being accelerated now an accelerating charge must radiate the accelerating charge did not radiate in Bohr model so therefore there was a condition on it they said that this is you say normally an accelerating charge would radiate and then it will lose all its energy and the electron will fall back on the nucleus now in order to stop it what Bohr had suggested is that there should be a statement in Bohr model which says that there are stable orbit in which such radiation does not take place but what the needed was a statement that integral of p.dl if you are talking about the a circulating electron in Bohr model it says integral of p.dl is a multiple of h this is there in any school textbook now we all know that when you put on a magnetic field the role of the momentum is taken by p minus I have put in not ea but I have put in e star ultimately this e star will become 2e so p minus ea well those of you who are using other units you might find in the books it says p minus ea by c but that is all right we are using si units so in other words this thing which says p.dl equal to nh has to be replaced by p plus e star a.dl is equal to nh now so therefore and this quantity remember this p is now an ordinary mv so it tells me by separating it mv.dl plus e star a.dl is equal to nh now so far as this term is concerned you are taking this over a circuit the and so therefore as it returns back to that point the line integral of v.dl over a circuit must be equal to 0 so I am left with e star a.dl is equal to nh now this a.dl I had shown is nothing but flux and you remember what we did we said that look the you convert this into del cross a.ds using the inverse of the stokes theorem see stokes theorem said that surface integral of a curl is the line integral so I am saying this line integral is equal to surface integral of del cross a but del cross a is my magnetic field surface integral of del cross a is nothing but my flux so integral a.dl is an equivalent expression for the flux so my flux is given by e star phi there is e star there it is equal to nh this is from Bohr's upper field condition which tells me that this phi is nothing but nh by e star which is nh by 2 and this is very simple explanation using old quantum theory and this I am plotting from a the picture is not very good because I have taken the picture from a journal and it tells you about the way the fluxes change as your you know the strengths etc increase so you notice these coming steps it is this value it keeps this value and then suddenly it will jump okay when you increase the magnetic field it jumps from here to here so this step picture is what you get. So let me now go over to some qualitative description of the BCS or the Bardeen Cooper Schreffer theory so Cooper was a graduate student with Bardeen when this theory was done and in fact about Bardeen I would like to tell you something that you know there are already very few people who got Nobel Prize and still fewer people who got Nobel Prize twice okay and there are just two people who managed to get a Nobel Prize in the same subject twice I mean of course there are people who got two Nobel Prizes Madame Curie got it once in physics once in chemistry Linus Pauling got it once in chemistry once in peace but I am talking about people who got Nobel Prize in the same subject Bardeen is one of them Bardeen got twice Nobel Prize once for this theory that I am talking about in physics and second time again in physics for his contribution to the discovery of transistors okay two different areas totally but that is why Bardeen is so famous all right Cooper was a student so the point was that what the problem that Cooper was doing it is going as goes as Cooper's problem that they found that it is possible to think in terms of an attractive interaction I am I know I am using words which you will find very disturbing but let me explain that to you due to an attractive interaction between a pair of electrons you know that pair of electrons normally have a repulsive coulomb interaction so what is this attractive so let me explain that how it happens it is not a direct interaction so imagine an electron moving through a lattice you have been talking about solid state already so you know that the structure of a solid it has ions in it which are arranged in a geometrical structure which we call as lattice so imagine an electron or electrons entering a region of a metal okay and with some typical velocities typically 10 to the power 5 meter per second etc now the what it will do is this that because the electron has charge the because the electron has a negative charge the ions have a positive charge now this electron in its way as it is traveling can polarize ions now what does it mean it means that because the ions are positive the presence of this negative charge as it is moving will disturb this ions a little bit from their one position mean position because the in other words close to the positions of the ions they there would be a redistribution of the positive charge now but however the electron is moving at very high speed so it polarizes the medium and goes away now but as it moves away supposing a second electron is coming in now now this second electron finds that the electron which we talked about earlier has already pulled the ions a little upward or little closer to its path so as a result the electron experiences an attractive force which is stronger than it would have had if the first electron was not there so this means that in effect there is an attractive interaction between the two electrons which happen due to or mediated due to this vibrating ion if you like which it has been able to polarize if this is also called due to phonons because the vibrating ions are basically phonons so as a result what we have got is an effective interaction between the pairs of ion which is attractive now mind you there are fairly large distances large in some scale I will talk about but the because of this fact that the first electron has polarized the ions so the that is what I mean by polarizing ions is that it has created region with more positive charges than it had so therefore the second electron feels an attraction so therefore what I got is an effective interaction between the electron pair which is attractive in nature now how do I even know it is correct now in other words is there any evidence of the fact that the ions are participating in the process of superconductivity because till now I have remained silent about ions I have never said that the ions were there anywhere we said these are electrons now if I look at the transition temperature the critical temperature of various material and their isotopic mass you understand what is an isotopic mass a substance having the same atomic number but having different neutron numbers that is its charge is the same but the mass is different now people have found that the transition temperature inversely varies as the one over square root of the mass tc there is a mistake there m to the power half okay so in other words the there is evidence of the fact that ionic mass is actually playing a role there so cooper was asked by his professor to work out on this problem cooper was asked that look can you find out now remember our idea of what is a metal now my my electron in a conductor or was filling for example they all the states up to Fermi energy or they were in what is called a Fermi scene now what we are saying is this that talk about two electrons which are inside the Fermi but instead of talking about attractive repulsive interaction imagine they have an attractive no matter how small it is as long as they are attractive now solve the quantum mechanical problem of these two electrons this is what cooper did this is what cooper did and found out that this pair which has an attractive interaction between them will become unstable against pharmacy in other words the lowest energy state would be one in which the energy of the pair happens to be lower than the Fermi energy so that would be the preferred thing now what we are talking about is this now notice the energy will obviously increase if the momentum of the pair is taken to be large or nonzero so what we do is we say that energetically the most favorable situation is that for which the pair of electrons which are interacting by a attractive indirect interaction they are their momentum is taken to be this pair is called a cooper pair now this is a picture which you will see very frequently so basically what we are saying is this that there is an electron which is shown by a negative charge which comes now imagine what the phonon is doing is to bind them by something like a spring bind the two electrons by a spring maybe they are far apart but on the other there this is a to lose picture and this is a phonon so the electron continues to exchange phonon with the other electron and though they are both negatively charged but because they are bound by a phonon they it remains attractive so this is the picture now what am I trying to say here we said that look supposing this is my Fermi surface this is the field for machine this is the field for machine and I am talking about two electrons there and these two electrons I have taken the momentum to be 0 which means if one has a web vector k the other one has a web vector minus k and and this two pair would then have an energy which is lower than the Fermi energy and what we are trying to say is this that the superconductor essentially has large number of such pairs of electrons and this pair of electron is called a Cooper pair and this is this was actually his PhD thesis so you can imagine at what young age he worked it out now the wave function of the pair I write it the usual way I have the position of one position of the other I have taken the usual you know block electron type of behavior and uk e to the power ik dot r 1 minus r 2 and this this if it is even the function could be this way if it is odd the function could be that way but the supposing the now I am talking about two electrons you know that when I talk about electrons they are fermions now since the electrons are fermions I cannot allow two electrons to sit on the same place but on the other hand if that happens if that happens that the event the odd term right means by wave function will then become 0 but only the event term will allow two electrons to sit one over the other now but total wave function is a product of the space wave function and the spin wave function if the space part is even cosine is an event function then spin part has to be I t c that is if I am talking about a pair say I must have only singlet possible combination okay now you will hear a term which is known as characteristic length in superconductor called coherence length in fact typically one talks about two characteristic lengths in superconductors one is the coherence length which is usually written as xi and this is the length over which the cooper pairs can maintain phase coherence and this can be as large as something like about 1.6 few micrometers two as low as 14 nanometers in case of now alumina was a very large coherence okay so these are these are the things that we are talking about and the lower state is separated from the normal state by of the order of a fax of a mu v this is the gap this is the gap and the presence of the gap is verified by standard tunneling experiments and the specific heat in activation energy alright so this is this is what we are talking about that that the Fermi energy of in a super conducting state is lower than that in the normal state and there is a gap between the normal state and the superconducting state and that is my delta as I told you while explaining somebody had asked why do you talk about 2 delta that is because becomes clear now because I have to break two electrons and take both of them above the gap so let me come to a little more I will not be solving equations but let me talk about some things see in the quantum mechanics course you wrote down how to write down a Schrodinger equation so that was your usually the way we write it down is ih cross d psi by dt there might be a minus and wrong here is the p square by 2 m plus the v psi of course now this term as I have been pointing out repeatedly the momentum becomes momentum minus your qa or e star a momentum operator is minus ih cross del at this phi is just a scalar potential in which these things are there so this is my wave function I will not solve this but without solving it we will be able to make some comments about it so the thing is the following let us talk about what is the ground state of a superconductor in the superconducting state I I have electrons but pairs of electrons are bound the pairs of electrons are bound together okay and this is now there is a difference between saying I have two electron and I have saying and saying that I have a pair because it can be shown that this pair which is bound it behaves like a boson not each individual electron they are fermions but this pair can be shown to behave like a bosons now if they are bosons then there can be many pairs in one state because remember it is the Pauli exclusion principle which does not allow fermions to fermions to occupy the same state but on the other hand if you have bosons there is no such restriction so therefore in my lowest state all my pairs have the same momentum I have taken them to have zero momentum that is the lowest energy state so basically I have a gas of bosons which are in the zero momentum state okay so let us look at then it becomes now a very simple job to do suppose I talk about that occupancy of the lowest state is n what it tells me is this that this is this occupancy of the lowest state is higher than any other state by remember your amplitude of the wave function is square root of the probability density so therefore if I write now I now I am writing down not the wave function of a single electron okay I am writing down the writing down wave function for the entire system of electrons or entire system of cooper pairs so I write down my wave function as square root of rho which is taken as a real and e to the power i theta and both of these I will take them to be real because any complex number can be always split like this that I can write any complex number as a e to the power i theta so I have one part which is a real part rho r and another part is e to the power i theta which is the phase part now if you now substitute this psi of r into the expression for the probability current density remember your probability current density from quantum mechanics it simply says minus ih cross del well psi should have been inside here also del psi star psi plus the complex conjugate here now in this particular case there should have been a psi here in this particular case the my probability current density expression if you simply substitute here I have to simply take a gradient and things like that I get ih cross by m del psi minus I have used q instead of e star because I am getting this star confused with this star so let us put it this way q by h cross a rho so this is what happens to the probability current density simply there is a very trivial arithmetic substitute here and write it like this I will I will go to some of the questions questions seem to be piling up but but before that let me make one or two comments so I have said that I have a wave function for the collection of the pair and that is given by root rho r which is a real thing time surface factor e to the power theta the one comment I want to make is I know a wave function is a single valued thing in other words supposing I am here and I come back by this path again back to the same place I could not have changed my wave function so this means that if I go around any circuit a loop so that the theta changes by either 2 pi 4 pi 6 pi etcetera that is a multiple of 2 pi then I must have the value of psi to be the same all right so suppose I do this I say I am well within a superconductor I am considering a superconducting ring and not I am not near the edges where I have seen there can be persistent current but I am within the material of the superconductor then my current density must be 0 because there is no current supported here and the expression for the current density was here so if this quantity is 0 then I get delta theta equal to q by h cross a right so that is what I get now notice this that so let me now calculate how much is integral of a dot dl on this say phi is integral a dot dl that is my flux and now you write down that we have said delta theta is q by h cross a so therefore that becomes a is equal to h cross by q a delta theta so that is what I have written down there and when you complete a circuit the integral of delta theta has to be a multiple of pi so I have written it as 2n pi that tells me the flux phi is n h by q and this is because h cross has a h by 2 pi in it 2 pi 2 pi cancels out and I am left with this so this is the whole story about the flux quantization. Before I go to Josephson junction let me take up a few questions yes KK Wagg Institute please go ahead. In an electric circuit yeah if anyone of conducting wire is replaced by superconducting wire then what will be the mechanism see the point is this that the you are saying that you have an electric circuit where you connect two parts of a circuit by means of an external wire which is a superconductor assuming you can do it by maintaining only that part of the circuit at the temperature lower than the critical temperature all that it means it you know I mean you do not expect anything great and I will tell you why because even today when you connect your circuit by copper wires you do not expect a drop of voltage along the copper wire because copper is fairly highly conducting so all that you are saying is now I have connected them by ideally resistance less wires. Now in all your electric circuit you also assume that the lead wires they do not contribute to any resistance. So if you ask what will happen the answer is theoretically something will go wrong be different but it will not matter at all from what when you take for example a good quality copper wire or any such thing because they are there you know I mean today the type of purity that you can get in copper is very good and copper has excellent conductivity. So yes there would be theoretically a difference because you have connected them by what is ideally lossless but if you had a copper wire in principle you have to worry about there is a voltage drop between the point of connection to the other point because there is a current passing but superconductor there is no such issue but nothing great happens. Knowledge institute yes go ahead yes madam. Question is related to superconducting phenomenology. Yeah. Suppose if you are taking the superconducting ring we are having persistent current and diamantism. Yeah. Suppose if you are taking in the shape of bar what happens to that phenomena. Shape of. Suppose the shape instead of ring if you are using a bar. Yeah. As a solid what happens to this phenomena. Nothing really you see the ring was very special because what happened is you see if you take a bar the flux is simply expelled out you see what is the way in which you understand the flux lines or field lines. See what we normally do is this take let me let me explain it by talking about supposing I have a positive charge you have seen how positive charge electric field lines due to positive charges are written because the let me take a positive charge and what will happen is that you will draw lines which start from that source and go to infinity right that is because at infinite distance I assume there is a sink also. So all the field lines are drawn like that now if you have take a bar and you subjected to you make its temperature lower now then supposing you had some flux going through it because you had subjected to a magnetic field now then you make reduce the temperature by Meissner effect the flux will be expelled out. So what will happen is close to that bar the field lines will bend okay because they cannot enter in and there would be these are the fringing effects that you have seen while drawing magnetic field lines etc even for ordinary conductors right now they so close to that bar the field lines will bend they will not enter and so there would be certain amount of edge effects that you will see but because there is no trapping of the flux the thing that happened in case of a ring or a tube will not take place here. So a bar is a much simpler thing to explain because the moment you switched off the all the flux lines vanished because they have all gone away to infinity MS midway Indian calling yes go ahead yes. Good morning sir. Good morning. So what is for me what is for me the direct distribution? See the thing is this that there are the all the fundamental particles that we have now I am talking about quantum mechanics all the fundamental particles that we have they can be classified into two groups depending upon what is the way they behave the first type that known as fermions the fermions have this property that they satisfy Pauli exclusion principle. In other words no two fermions have the identical fermions can occupy the same same quantum state. Now the bosons are the other one they are called bosons I am sure many of you know that the name boson came from S n bos who was actually an Indian physicist and so there are two classes of fundamental particles. Now if you consider the first category namely the fermions now the by statistics what we mean is that what gives you how is the probability distribution at an arbitrary temperature. Now remember when you had a classical gas you said that in classical gas if you are looking at the fraction of let us say whatever is the gas which have it an energy E. Now you go by what is called a Boltzmann factor higher the energy lower is the occupation now this is generally true because nature likes to have nature likes to have I think somebody is using a flash there anyway. So nature likes to have lower energy states occupied more than the higher energy states. Now if you have a collection of electrons for example or protons or neutrons all of them are fermions then if you now say that what is the fraction occupying an energy epsilon. Now this fraction would be given by a factor now this distribution is called a statistics the fact how are the electrons distributed energetically. So in Fermi Dirac comes in because this form was with Fermi and Dirac Fermi Dirac statistics is applicable to fermions which says that if you have and if you consider energy epsilon then the fraction of electrons which have an energy epsilon is given by a factor which is not exactly the Boltzmann factor as we talked in case of a classical gas but 1 by e to the power your energy epsilon minus a term known by as the chemical potential all occasionally called Fermi energy divided by k t plus 1 in case of Boson it will be a minus 1. So Fermi Dirac statistics if you like tells you how are the energy states occupied if you connect consider and collection of fermions. One more question sir. Yeah. Type 2 superconductors. Yeah. In the type 2 superconductors. Yeah I am hearing you yes. In between HC1 and HC2. Yeah. The Messner effect is not satisfied. Okay. That is in mixed state sir. Yeah. Why sir? Yeah. So the point actually is this that the it is not true that Meissner effect is not satisfied Meissner effect is incomplete there okay this is the property of the superconductor the Meissner effect is incomplete. What happens is that when you have exceeded HC1 now supposing you had a type 1 superconductor the moment HC is exceeded the superconductor is totally destroyed. In other words it allows the flux to penetrate completely. But the in type 2 superconductors between HC1 and HC2 there is a part entry of the flux now you are asking why the general belief is the following I cannot just tell you because there is not a course in which one can go to the quantum theories the general belief is this that what happens in type 2 superconductors in the intermediate region what is also called a mixed state is that there are regions in which the material becomes normal. Now what is found is these regions which is there appear in the forms of tubes okay that is what I mean by tubes is they are very localized distribution there they so therefore a in the mixed state what I have is a collection of such these are called vortices and it turns out that these vortices like to arrange themselves in a lattice and they are called vortex lattice the so therefore what I have is a material where the mostly it is superconducting but there are areas where which have now become normal and this normal part can behave like any other normal battle. So it is because of that the mixed state arises but complete understanding of what is a mixed state we need to go to theory of type 2 superconductors which cannot obviously be talked here alright I have very little time now so what I will do in this time is the following I will I would like to in fact in case this lecture is not complete when I come back in the later part in the afternoon to talk about optics I might take a part of the lecture on to complete this because I have only 15 minutes left and I do not think I can finish Josephson junction but Josephson junction is such an interesting quantum phenomena and phenomena connected with superconductivity that it is better to talk about it also okay according to my colleagues here what I would do is I will go to because the questions are piling up and because my superconducting session is only likely to last only one you know this lecture what I am going to do is this I have Josephson effect which explaining it properly will take half an hour which I do not have I will do it in the evening session so let me finish the remaining 10 15 minutes in simply taking more questions yeah shower Susheela yeah so go ahead please Collapur morning sir yes good morning madam my question is which property of superconductor is to be improved in order to increase critical temperature up to the room temperature means which property we need to focus more I wish I knew that questions answer in fact many scientists they they are wondering if they know the answer to that question okay her question is that what do you need to do to increase the you know critical temperature of a superconductor you see the point is this that the type of temperature we are talking about today for there are elemental superconductors or the conventional BCS type superconductors where the temperature the maximum temperature achieved is about 23 degrees Kelvin or so okay but we are not talking about that type of temperature we are not talking about temperatures which ideally I guess what you mean is can you go to 300 degrees Kelvin okay that is the room temperature now the thing is the following that what we have found is the following that there are certain materials particularly cuprates which professor Suresh will describe in detail which have this property that they remain super they are all type 2 superconductors but they remain superconducting for a much higher temperature than most of the others are okay so the idea is more a answer to your question is more a material science question than so much a physics question see what they have to do is to examine what are the type of materials which seem to be having you know they have to look at something like if you like a periodic table it is not really a periodic table but a table of materials versus their critical temperature and see that which what are the differences between them for example you take let us say a cuprate YBCO now then you knew that if I put in some amount of nitrogen what happens so what is happening today is if you like alchemy by physicists and chemists why it is alchemy is that you are just shooting in the dark that you are trying to say okay let me add this let me add this material try to see whether the temperature is increasing or not so what we are trying to do is there doesn't seem to be a very methodical or scientific answer to your question as yet I mean it probably is there but at this moment it is not there what people are trying to do is trying their luck by changing compositions in fact this is precisely what our alchemists did in the past they say you know or what our medical scientists do even today see if I ask you the question that look we all know what is the drug for a diabetic now what should you do to improve the efficacy of a diabetic drug drug now what are our scientists doing they are simply trying let us try this let us try that because there is nothing like a microscopic phenomena behind medicine so it's a it's a phenomenology and this is where phenomenology becomes very important so this is one question where I have no answer I have no answer but that is because nobody has an answer so you have to look at the material science scientists who will try out various materials try to see that if I take for example this material I applied a pressure to it TC went down so obviously that is not a good thing to do okay so sorry this question yeah are you I know you are not happy with my answer but I do you at least understand my inability to answer that question yes I have also seen variation in materials only that's why I ask this question so type 1 or type 2 is more beneficial in this respect mostly these are type 2 because because type 1 is very restrictive and these are the BCS type there one has realized that you don't stand a chance of getting anything more than 23 to 25 degrees Kelvin so we are we are talking about mostly type 2 yeah no sir yeah one question yeah why do various superconductors have a different TC well ask madam who is a material scientist the same same answer see properties of material is not something which is explained on the basis of a thoroughly microscopic theory and this is true of why conductivity of copper is more than the conductivity of aluminum okay both of them are conducting materials now what I mean the idea is this all of them have free electrons but some amount of bondings which are here and there makes these behave differently so properties of material is something which you know if you want to explain it fundamentally there is really no answer I mean at least nobody tries it that but one understands that look if the theme we are talking about material where certain conditions can be satisfied then it will become superconducting but why TC of aluminum is different from TC of mercury okay it is the property of the material itself that is a core institute yes go ahead with your question good morning good morning good morning good morning go ahead I have two questions are yes question is first question is which materials are including high temperature superconductor yeah and what are the advantage of that yeah second question you say it because I have already answered previous thing high temperature superconductivity is not being taken up by me there will be one full lecture on high temperature superconductivity I am doing conventional superconductors physics of it last lecture on superconductivity is on high temperature ask that question at that stage yeah next question you said you have two questions yes why this a ferromagnetic materials are not superconductor yes that is a good question I will be able to answer that question at least partly see the thing is you realize that in some sense particularly when we talk about the conventional superconductor the ferromagnetism and superconductivity are basically opposing effect because you remember when I wrote down the wave function for this the paired electrons I said they must be in singlet state that is their spins are opposite and that is because I need the two electrons to be in the same state and so therefore I want the space part of the wave function to be symmetric so the spin part has to be anti symmetric so the two electrons must have their spin oppositely directed you know I wrote down that wave function up down minus down up by root 2 now on the other hand the in while you are talking about a ferromagnet you essentially are asking that the spin should be aligned okay so in other words if I had just two particles that is not a ferromagnet but I am just explaining putting I just had two electron if it is to be magnetic then it should be a triplet and not a singlet because singlet has s equal to 0 the magnetic moment is 0 there so a superconducting ground state prefers the magnetic moment to remain low so that is the reason why the ferromagnetism and superconductivity do not exist together yeah waltz on this shoot please go ahead yes so so basically what we said is that the electrons are of course fermions there is no question on it but what we said is that this is a phenomena where what we find is that the electron never is alone okay one electron is always paired with a second electron and it is a combination it is a composite material that we are talking about and this composite okay until you break it it will not become a fermion and it requires an energy of 2 delta where delta is the band gap to break a pair of electrons so in other words they always go together I mean if you want to use a crude analogy it is like the two electrons are married together and they are always go together okay and this combination of composite thing their statistics have been found to be in fact you can see why you see if two electrons the mind you I am not saying if you take any two electrons together it will become boson I am just telling you why the two electrons which are you know welded to each other if you like by by a constant exchange of boson their total spin right see each electron has a spin half the if I add two electrons the spin can be either 0 or 1 right this angular momentum addition now those particles which have 0 angular momentum or 1 or all integral angular momentum their statistics is a boson so yes so that is the reason why this collection now remember do not think of it as a two electrons think of it as a cooper pair so what the statement we have made is cooper pairs are bosons two electrons are not bosons two electrons are fermions 10 electrons are also fermions but 5 cooper pairs are bosons have I made it clear yeah done okay we close this session