 It is a pleasure to start our valedictory function of the this year's orientation come selection camp in physics. We shall have the formal function later at 11.30. Before that we have talk on physics, on core physics by our chief guest today, Prasap Pratap Rajudhuri from Tata Institute of Fundamental Research, Mumbai. It is a great pleasure to have Pratap with us today. I and Pratap go back a long way actually to our school days. We both got admitted to the same school, we shifted schools in class six on the same day. So we really go back a long way and we are also colleagues during our PhD stint at TFR. So Pratap did his master's integrated MSc in physics from IIT Kharagpur and after that he did his PhD at TFR in the condensed matter group. And then after brief stint as postdoctoral fellow in the University of Birmingham and St. Andrews in the UK for a couple of years he came back to India and joined as a faculty in TFR in 2002. And since then he has been at TFR, he is a professor at TFR now and Pratap has is very prolific in his publications. He has received several awards starting from his earlier days, the Young Scientist Award and so on and so forth. Recently he has received the Bhatnagar Award last year in 2014 and in 2015 he was elected as a fellow of the Indian Academy of Science, Mangalore. So Pratap will talk more, he will himself talk about his work and we look forward to it. Pratap. Thank you for your kind introduction. It's a great pleasure to be here talking to you today in this valedictory function. So what I will be telling you today is about phenomena that we actually encountered and use in our everyday life and that phenomena is called electron tunneling. So let me first go with what the phenomena is and towards the end I will tell you why all of us should care about this phenomena though it's a phenomena that is only seen at very small length scales. All of you I guess are familiar with the game of squash. So squash is like tennis but you have both the players which are on the same side of a wall and you have one player which hits a ball to the wall, it bounces back and then it's the turn of the other player to hit the ball again. So this game of squash is of course played like this with both the players on the same side and this ball of course stays on this side of the wall whenever you are hitting it to the wall it just comes back. But suppose you were in the quantum world which is the world of very very small particles. Then if you were playing squash with such a small particle you would need of course you have a wall here you need two players here but you also need a player on the other side because there is a possibility every now and then that this ball instead of hitting this wall and bouncing back will somehow find its way on the other side of the wall and then you will need another player to take that ball and hit it back. However this of course does not happen in your everyday life the way you see it with a tennis ball this does not happen and as I have told you this ball has to be very very tiny for this to happen how tiny. So let us have a sense of scale you take a tennis ball let us say about 5 centimetre this of course is completely the macroscopic world you make it smaller you go to a grain of sand or you have human hairs. So these for example are of the order of 100 micron so 1 micron is 10 to the power minus 6 meter 1 millionth of a meter and again here you know that your hairs do not disappear all of a sudden they might disappear slowly over time but they do not tunnel quantum mechanically and disappear. You go to even smaller and let us take let us say you take a virus which is of the order of 100 nanometre 100 nanomi 1 nanometre is 10 to the power minus 9 meter even then these objects are fairly classical though they have internal process that could be quantum mechanical but these three objects as such will not undergo the phenomena that I showed you before. To see quantum mechanical phenomena you have to actually go to atomic length scale so let us say you take this grain of sand and you keep on so magnifying it eventually what you will observe is these atoms of sodium and chlorine. Now the sodium chlorine atoms they will be separated by a distance typically of few angstrom one angstrom is 10 to the power minus 10 meter and this is where quantum mechanical phenomena starts playing a role and of course then if you go to an atomic length scale which is about half an angstrom here the world is entirely quantum mechanical and you have these electrons that actually do not remain here like a particle they form a cloud and you have a quantum mechanical system. So this is the typical length scale that we will be talking about today. Now how does the quantum mechanical world differ from the everyday world that we see let us say you have a cannon here which is firing balls you have a screen here a hard screen which can take the cannon ball and not shatter and it has two holes here and here you have a brick wall so I start firing sometimes the cannon ball will pass through this is not a very accurate it is not a very good cannon so there is some spread in the direction we should and eventually you will see that you have made one hole here and one hole here which will be corresponding to the lines joining this cannon through this hole to this wall. Now instead suppose I take a bucket of water or a large container of water and now here I put a similar wall I have water all over I am not showing it here but I have assumed that I have water all over this place and here I have this wall. Now I tap here on this water I will have a wave front that will propagate when it will encounter this obstruction this wave front will break into two new wave fronts one starting from here one starting from here these two wave fronts will interfere and you will get a diffraction pattern of this kind. So now you will get sort of profusions in the wall if you keep on doing this for a long time which will form a pattern of this kind which we know as the fringes. So I am sure you have sort of done some experiment on fringes with light and grating and so on. Now electrons depending on the momentum that they have and the scale that you are looking at could either behave as particles or could behave as waves. When they behave as waves for example if you have these electrons which have a momentum p then they will be associated also with a wave with such a typical wave length and if you put here the double slit which is separated by a distance of the order of the wave length you will see that these electrons now will no longer behave like individual particles they will behave like a wave. So you will form a diffraction pattern on a screen that you see here. Such diffraction actually can be seen and for example this is one of the early experiments where you pass an electron beam through a double slit and you get this diffraction pattern. So this is an experimental demonstration that the electron behaves also as a wave. So you have essentially in the quantum world you have this continuous dichotomy whatever is the particle can also behave as a wave depending on the conditions that you are looking at, the energy scale that you are looking at and the length scale that you are looking at. Now what is how does this how to understand that is intuitively the process of tunneling. We know now that electrons can behave as waves and waves as we know when they encountered a barrier like a tennis ball they need not just bounce back they can get attenuated through this barrier but a little bit of the wave leaks on the other side. So if you have a cop shouting on the on this side of the wall this guy might feel safe that the cop cannot catch him. Nevertheless if the cop shouts loud enough he will be able to hear the diffuse sound of whatever he is shouting. Similarly if you have an electron which is incident on a barrier here from this side then this electron of course is associated with the wave like this. This wave will get attenuated in this barrier but on the other side a little bit of the wave will leak through. How do I now reconcile this wave picture with my usual understanding of electrons as particles this wave I can write it like this some amplitude and then e to the power i k x minus omega t that is a travelling wave. And the way I interpret this wave in the particle picture is that this mod square of the amplitude is the probability of finding a particle here. At any point if I calculate this mod square of the probability that is the sorry mod square of the amplitude that is the probability of finding a particle at that position. So since I have an electron which is incident from this direction I have a large probability of course of finding the electron here. And of course most of the electron will get reflected back most of the wave will get reflected back that means even after reflection I have a large probability of finding the electron on this side of the wall. But there is a finite but small possibility of finding the electron on the other side of the wall. And this is what is called as this is what is called as electron tunneling and this is intuitively the process this is intuitively the way to understand this process. Now let us try to translate this tunneling into particle language we have been sort of looking as the wave which gets attenuated. But suppose now I start with particles itself then what this equation tells me is that t 1 to 2 is the tunneling to the particle probability of an electron from this side to this side. So which means that it is a probability that an electron which is on this side of the wall will be found on the other side of the wall. Now there are these expression is reasonably easy to understand. If my barrier is very thick the probability of this tunneling will be very low. So I have here this exponential kappa d where d is this thickness kappa also incorporates the height of this barrier. So therefore if the barrier is very high then again the probability of the particle tunneling through the barrier is low. There is another condition that you have to meet. First is you need to have particles there. So you need to have states on this side of the wall that are populated and you need to have states on this side where the particle will tunnel through. So you need to have empty states here. So if the states for example here were already occupied you could not have tunneled here because that is already there is no place to tunnel to. So therefore tunneling probability will also depend on this product of n 1 and n 2 which is n 1 is the number of field states the number of occupied initial states and n 2 is the number of unoccupied final states. Interrupt me at any point if you feel lost because I am not very familiar of the background from which all of you have come. So if there is at any point you feel lost just tell me. And there is yeah suppose you have no states here what would be your tunneling probability yeah I can hear you it is fine. But is it not intuitively 0 if I have no state to tunnel to how will my particle go on the other side. Oh these are occupied states these are empty states here you can see these are occupied these are empty is it clear. There is a third condition and this is the condition which makes for example which is sort of which is a characteristic of tunneling. For example if I take a tennis ball and if I hit really very hard on a wall and the if the wall is flimsy it can break the wall and go through. But that is not tunneling because in the process of breaking the wall the tennis ball will lose a lot of its energy and even if I have thrown it very hard it will possibly not go very far after breaking the wall on the other side. Tunneling is a process where energy is conserved. So if I start with something which has a energy E on this side on the other side the energy will be the same. So therefore you have now this delta function here which means that this occupied states and this empty states need to be at the same energy. If they are a different energy then there will be no tunneling. So this is the governing equation for my tunneling process. Now before I go to showing you where tunneling actually happened let me first tell you that what I will be telling you today is tunneling of electrons. But electrons are not the only thing that tunnel. In fact tunneling the process of tunneling was first formulated by Gamo to understand alpha decay in radioactive elements. How many of you are familiar with alpha decay? Good. What does alpha decay produce? Helium. Yes. And then this helium nucleus gets electrons from outside and it becomes helium atom. How do you know? Any indirect evidence? Do you know where do you get helium from? Do you get helium in the atmosphere? That is yeah. There may be some cosmic ray. Yes, precisely. So almost all of the helium on earth has come from deposits of uranium and thorium where they have decayed into where they have undergone this alpha decay and produced helium. So that is stored under the crust and typically when mining oil it comes out with natural gas. So often natural gas contains a lot of helium. So that is why for example the helium that we normally buy in India we do not have much of reserve of helium. Most of the helium that we buy is from Gulf countries. When they take out the oil the helium comes out any way from inside. They collect that and sell it. So this process of tunneling happens in many, many different nuclear reactions inside the sun and so on. But these are sort of natural phenomena that are happening and we understand them in terms of tunneling. What I will be telling you today is how to controllably use this tunneling and manipulate it towards an end. So it was essentially Max Born who realized that tunneling is not a phenomena that is restricted just to these nuclear reactions. It is something that is a fundamental aspect of quantum mechanics and you could realize that in any quantum mechanical system provided your conditions are right and the conditions are the ones that I have shown you in the tunneling equation. So for example if I have a solid you know if I have a metal then I have these positive ions in the metals and I have this sea of electrons which are sort of mobile. If I apply a voltage you know that this electrons will move and they will move possibly in this kind of random work fashion and that is because you have impurities in the solid and they will get scattered at impurities. That is what will give you typically Ohm's law. Now if I take a solid and I cut it into two pieces and in between I put an insulating barrier. Classically I would think that there should be no conduction across this barrier. However quantum mechanics tells us that there can still be a small current flowing and that current is by an electron tunneling from this side of the barrier to the other side. Conceptually this is fine but does this actually happen. 1973 was a great year for condensed water physics and quantum mechanics both. In this year these three gentlemen Leo Esaki, Ibar Gyabard and Brian Josephson got the Nobel Prize. They shared the Nobel Prize. Essentially all of them for demonstrating the phenomena of quantum mechanical tunneling of electrons in solids. Out of them Esaki and Gyabard were experimentalists. Esaki essentially showed that tunneling does happen in a solid. Gyabard showed how this tunneling can be used for particular purposes and Brian Josephson was an experiment was a theorist who sort of demonstrated a very unique form of tunneling called the Josephson effect but that is a little bit more intricate. So we will not go into Josephson work now. Let me briefly review what Esaki and Gyabard did for winning this Nobel Prize. Now there is a story here particularly with Gyabard. At this time when Gyabard did this his experiments, tunneling was not yet established as a process for electrons in solids and in fact when Gyabard who was an electrical engineer with very little background of physics, he consulted experts in physics at that time. He was in Bell lab and he consulted them and asked them whether he can experimentally observe tunneling. Almost all of them told them various reasons all of them having some degree of validity on why tunneling can never be observed in a solid. But being an experimentalist he decided to try it anyway and we will see what he got. So first let us look at the work of Leo Esaki before we come to Gyabard. All of you are familiar with semiconductors I guess. A semiconductor has a field valence band like this and an empty conduction band. What is this kind of semiconductor called intrinsic semiconductor? Are you familiar with the concept of Fermi energy? That is possibly you are not. So Fermi energy is the energy level that is in between this conduction at the middle of this conduction band and valence band. The way you define this Fermi energy is what is the energy required to take out an electron from a semiconductor or to add an electron to the semiconductor. If you take out an electron to the semiconductor you have to take out from this energy level but if you add an electron you have to add it here. So therefore the average of these two is the Fermi energy which comes here. Now you know that a semiconductor you can dope it either with electrons or with holes and they are called N and P type semiconductors. If you dope it with electrons then suppose this is silicon and in silicon you are putting in phosphorous. So then you will have some electrons which are here because you cannot put more electrons here. So those electrons have gone here and when you have more electrons here then this Fermi energy actually will go up because now when taking out the electron you can take it out from here also but when adding an electron you cannot put it here. You have to always put it here. Earlier you were taking out from here and adding here so it was exactly at the middle. Now you can take out from here or here but you can add only here so it has gone up. On the other end if you put in holes in the semiconductor then similarly you have some states here which are empty and your Fermi energy comes down. Now this is the usual semiconductor that you know. If you take this N and P type semiconductor and join them together form a junction that is a P N diode that you have studied possibly. Have you studied? However suppose you dope them very strongly and this is what we know as degenerate semiconductors. So if you dope them very strongly now I have put a lot of electrons here. So the Fermi energy actually goes completely here because I take out an electron from here I add an electron here as well. Here again I have doped them very strongly with holes so I take out an electron from here and put them here the Fermi energy has come here. So these are essentially these have almost become metals but metals with very low carrier density because I have not filled too much of the conduction band or I have not emptied too much of the valence band. No Fermi energy is not an actual means there is no state there at all in the case of a semiconductor in the case of when it is degenerate yes you have a state here but Fermi energy actually is the energy required to take out a particle from the solid or the energy required to add a particle in the solid. The reason why you have to study the Fermi energy is that if you put this two material into contact you have a band gap here you have a band gap here. Now when you have put them into contact they will have to align themselves with respect to each other. How will it align itself? The rule and this now take it as a rule because you have not studied statistical mechanics here is that the Fermi energy on both sides will have to be equal let me show you in a moment. So for example I take a heavily endoped semiconductor I take which one is that yes I take this one and I take this one and I put them in contact. Now what will happen is that the Fermi energies have to be same on both sides. So that is how you know how these two bands the two band structure on both sides are arranged relatively to each other. No it is equal because if they are not equal you would have flow from of electrons from one side to the other side till they become equal that is why you need this quantity the energy required to take out an electron or to put an electron that determines the flow and when the flow equalizes it is exactly when the flow equalizes the Fermi energies are at the same point. So now you have put these two into contact so you have made a diode with two degenerate semiconductor as opposed to regular semiconductor. So you will have these bands which will align like this as I explained to you just now the Fermi energies will equate on both sides and now however these band bottom corresponding to here these has to connect in some way to the band bottom here because both at the conduction band bottom and the valence band top connects to the valence band top here. So therefore effectively for the electrons which are here they observe a barrier when they try to go from the p to the n. So you have a barrier when a electron tries to go from this side to that side or that side to this side this if this barrier is thin enough then you will have electron tunneling through the barrier. However here when you have not applied any voltage there will be equal number of tunneling from this side to this side and this side to that side. So there is nothing much that happens. If I apply a reverse bias like I have here then I will have some tunneling that will happen from here to here. So in reverse bias I will see some small current again this is not very spectacular. The spectacular thing about the Esaki diode comes when you have when you apply forward bias. Let us see what will happen in forward bias. So you know that when I have no bias at all I have equal number of tunneling from this side to this side this side to that side there is no current. Let me apply a small bias. Now you see here I have occupied states and here I have vacant states. So at the same energy so I can have a current that flows but if I apply a little bit more of bias then I have no tunneling possible because here I have occupied states but at this same energy this falls within the band gap of the semiconductor. There is no vacant state available. Then I apply even more bias and now I can run the tunnel into the conduction band. Again I will have current. So if I do this at a finite temperature for example I will see something like this. Initially I apply voltage and the current grows. Then the current then I reach this band gap region and the current decreases and then when it can tunnel again in the conduction band the current increases. This is not 0. You can see why it is not 0. First of all this is not 0 because one reason it is not 0 is because you are at finite temperature. So these are measurements done at typically room temperature where you have sufficient number of thermally excited electrons for which it is not 0 and then there is another reason why it need not be 0 is I am showing here only the tunneling of the top electrons but you can have tunneling from the entire band. So there can be some tunneling that happens from the bottom of the band when this level is slightly let us say below the top of this level. But which area this area here? This is not an area that you are looking at. This is a band structure that I am putting. So this is a region where you have essentially the barrier which comes a little lower that is all. It is the way I have shaded it. If I had shaded it from here it would look different. That is also true. So we had this tunneling equation. This is your governing equation and you have this delta function here. Tunneling is a process that happened. It is a energy conserving process. Why means we have not derived this so this I will not be able to explain to you without doing quantum mechanics which you have not done yet. But take this right now as a fact that tunneling is a process that does not change energy and that is how it differs from as I told you just a cannon ball breaking a wall and going on the other side where it loses most of its energy in breaking the wall. So now this is the current voltage characteristic and this is extremely strange because you are increasing voltage and this region the current is decreasing. Which means that if you look at the differential resistance which is dvd First of all this breaks Ohm's law. It is quite obvious. I is no longer proportional to v. You cannot define the resistance in the usual way. The second thing is that here your slope is negative. So this region if you are looking at let us say small within this voltage to this voltage this element behaves like a negative resistor because your dvdi is less than 0. Now Esaki diode this part this is what essentially demonstrated for the first time convincingly that tunneling is a phenomena that happens with electrons in solids but Esaki diode this was not just the only importance. It had a great application when it was discovered and this application was for oscillators. Now you all have studied LC oscillators. If you have a L if you put a L and a C and you make a circuit does it oscillate forever? I am telling you about a real L and a C. You have to understand that I am an experimentalist. There is nothing ideal in what I do. What happens? It will stop after sometime and that is because you always have a resistor. You cannot you have a L which has a resistance and you have a capacitor there the capacitor can be more ideal and eventually the resistance will cause a decay. Now in the circuit element where you have this L and a C and suppose you have some resistance you put an Esaki diode and you bias the Esaki diode at a voltage where the resistance effectively is negative. So what will happen is that it will compensate the resistance of your circuit and you will have effectively a zero resistance. So you will have LC and no resistance no effective resistance at all. So this will oscillate forever. Then what will happen is it will sort of instead of decaying it will amplify and at one point it will reach a region where you can see that this negative resistance does not extend over all voltages. So it will extend over that maximum value and stop there because then it will become like a normal resistor. So you will have an oscillating voltage with an amplitude which will be limited by the negative voltage region of the Esaki diode. This graph you cannot modify it the way you like. It cannot asymptotically go to zero because you have a conduction band here. If you did not have a conduction band then of course it would asymptotically go to zero but that is not reality. Any semiconductor will have a conduction band. So eventually it will shoot up. Tell me again. Yeah that is right. Actually an insulator and a semiconductor are the same thing. The gap is different in magnitude but there is a practical difficulty of using an insulator. If you have to apply so how much you are shifting the Fermi level with respect to each other depends on the voltage you are applying. For an insulator you may have to apply huge voltage which means huge electric field and at some point you will have a material breakdown. Your atoms will not remain in place. That is what you have for example a burnt capacitor that you might have seen in the lab is essentially that electrical breakdown. So insulators are not practical because you cannot sort of the material itself cannot sustain that high electric field. So this Esaki's work was the demonstration of tunneling. Now Geerber's work as I told you established how to use tunneling as a spectroscopic probe. Now what is spectroscopy? Do you know what is spectroscopy? Spectroscopy let me just define it in very simple words for you. Spectroscopy is when I am getting information about a system as a function of energy. You could also have more complicated spectroscopy where you have as a function of momentum or as a function of spin but normally as a function of energy. So if you see let us say free electrons then for free electrons you know that you have the energy and this is the number of available states at various energy. It looks like this and since these electrons are fermions you cannot feel one energy state with more than two electrons one with up spin and one with down spin. So you keep on feeling them with electron and you reach the top of the and when you have no more electrons then you reach the top of the band which is the fermionergy of a metal now. So this is for free electrons in a real metal you can have many bands. So for example this will be less like you saw in a semiconductor you are looking just at the top two bands the valence band and the conduction band alright a real band structure would look like this let us not bother about this. Now suppose I take a metal on this side I take a metal on this side and I put a insulating barrier in the middle. So and now I ground this metal I connect it to ground and on this one I can apply whatever voltage I want to apply. When I have no voltage at all I have equal tunneling on both sides and therefore I have no current flowing. When I apply a negative voltage I will have this which will go up there will be electrons which will tunnel into these empty states here. When I apply a positive voltage I will have tunneling of these electrons and these states here. That is all captured in this equation let me not get into the equation it will get too complicated. Now when Giaver was trying to look at tunneling the fashionable thing at that time the hot topic was superconductivity. Now what is superconductivity? In 1911 Camerleones was measuring the mercury the he was measuring the electrical resistance of mercury. Mercury because it is thought to be exotic it is a liquid metal of course by the time you cool it it becomes a solid. I do not remember exactly the solidification point but now it is not too low. So then when he cooled it further and further and eventually went to about 4 Kelvin above absolute 0. He realized suddenly the electrical resistance disappears. So later on there were more and more precise experiments done to see whether the electrical resistance disappears or whether it is a small value and it was realized that it is not a small value the electrical resistance identically is 0. This state of matter is called a superconductor. So a superconductor can carry electricity with 0 resistance there is no dissipation there is no joule heating. The other property of a superconductor is that it is a perfect diamagnet which means that if you have a metal like this and you apply a magnetic field the magnetic field passes through the metal. But if you are you have a superconductor then it expels all the magnetic field. So the magnetic induction B inside the superconductor is 0. Now a very rough understanding of the superconductor is this that you have these electrons which essentially you have one up electron, up spin electron and one down spin electrons which essentially form a bound state and this bound state have a typical size of 5 to 50 let us say nanometer. And these because they form this bound state they go slightly below the Fermi energy now. So earlier you had everything occupied up to the Fermi energy. Now they have gone they formed a bound state and since there is a binding energy which is negative the energy has come slightly lower. So earlier everything was filled up to this outer circle in terms of k this is in the momentum space and now they have all come down a little bit. So which means this if you want to look at the your more familiar band picture which you are familiar from super from semiconductors it looks like this. In a metal you have all these states which are filled and then they are filled up to this highest occupied level which is the Fermi energy. In a superconductor now electrons which are these close to the Fermi energy they have formed this bound state and they have come down slightly lower in energy. So therefore they have come down slightly lower by an amount which we call delta and because of that you form a gap here a gap that looks in a in a way like a semiconducting gap. So you in this region what you have is this bound state of 2 electrons and you have to apply an energy which is twice of the delta to break this and have individual electrons again. So that means the following that if I now look at the density of states which is the number of available states at a given energy this is typically flat with energy. There is no energy dependence in a metal for small energy range which is in the range of milli electron volt. However when I get into a superconductor what happens is that these states from here disappear because these states the electrons have come slightly lower in energy and here I no longer have any single electron states I have only these bound states of 2 electrons. So now what I get is I have a situation where I have this these states which are filled up to here then I have this gap where I have no single electron states available and then I have these states which are again single electron states but they are empty. So it look it is the band structure is somewhat like a semiconductor and now you where did these states which disappear where did they go they actually go here and because of that you start getting a large increase in the number of available states just at the bottom of the gap and the top of the gap. So Gever took one of the superconductors and he wanted to see whether through tunneling he can measure this superconducting gap. So superconducting gap is this energy range where there is there are no states available. So what he did is he took a superconductor and in his it could be aluminum, lead, niobium and you may he makes a strip like this then in a controlled oxygen atmosphere you oxidize a little bit so that the surface gets oxidized. Now oxides are typically insulators. So now you have an insulating barrier and now you put a gold electrode on the other side ok. You normally make several gold electrodes so that you have several tunnel barrier. So you have a tunnel barrier between a superconductor and insulator and then over that gold. If I now pass current like this let us say I pass current I here and I measure the voltage here essentially I am measuring the voltage across this tunnel barrier formed by a superconductor and insulator and then a normal metal ok. So now what do you see? This I have told you is the density of states now is the band structure for this superconductor. So I have a normal metal here I do the same thing I ground this normal metal and first I apply a voltage a positive voltage on the superconductor. I can have tunneling of electrons from this metal from this field state to this empty state. So I have a finite conductance then I reduce my voltage and when I have reduced my voltage at some point I cannot have any more tunneling because here if these electrons have to tunnel here they have to tunnel within the gap and there are no states single electron states available in the gap. So my conductance would come to 0 and then if I go if I apply a negative voltage then I can have tunneling from field this field states to this empty states. So again my conductance is finite this is what I expect to see and this is typically what I see when I do an experiment and this is the way you can actually find the superconducting energy gap. So this process which with gearver used not only establish that the process of tunneling can be used to measure something but it also established tunneling as a spectroscopic pro now how is that? Now let me give you one example you know that you can these are for example molecules let say I take water molecule I have oxygen here and the hydrogen here now this water molecule can vibrate. So this hydrogen can vibrate and they can have various modes this is like mass spring system is like three mass attached with springs. So they can have stretching mode bending modes and so on each with characteristic frequency. Now what I will do is I will make a tunnel junction of the kind that I showed you before but after this step here I will put a little bit of let say some molecule and I will let them settle on this barrier then I will put my counter electrode of gold. So here I do not need to really start with a superconductor I can start with a metal have this oxide barrier then put a little bit of these molecules that have some specific vibrational spectrum that I want to study and then I put a counter electrode. Now what will happen is that now what will happen is that this electron when it will tunnel it will first get trapped in the molecule then it will lose part of its energy to excite this molecular vibration and then go to the other electrode. In the process if you measure the conductance of the junction you will get specific signatures at those particular energies. So for example be here you take acetic acid and you put few acetic acid molecules in the tunnel junction then when you take let say the double derivative of the I V curve you see this typical peaks here and each of these peaks which occur at specific voltages corresponding to one of these vibrational spectra. You have the oxygen hydrogen bending modes the carbon hydrogen bending modes. Bending is typically lower in energy because you have a dumbbell kind of structure stretching these is more energy cost more of energy and therefore you have the stretching modes here which you can see at higher voltages. Now this was nice but tunneling essentially attained its full glory a few years later in the 80s and by these two gentlemen called Binning and Roder and that is through the invention of what is called the scanning tunneling microscope. So these two gentlemen of course won the Nobel prize in 1986 and this was the Nobel prize which was given not for discovering a new physics phenomena but because for using a known physics phenomena to make a spectacular new instrument and in this case one of the most powerful microscope available in the world. So what do you do here? In this system what the principle is very simple you have let us say a sample like this you take a very sharp tip of another metal. So suppose I have copper here I take another sharp tip of platinum and I bring it very very close to this surface so close that I can have electrical tunneling from this copper to this tip. Now of course this tunneling is happening as you can see here I have to bring it within let us say 1 nanometer to have significant tunneling this tunneling now is happening through vacuum or through air if you are if you do not have vacuum around and but this tunneling is happening very locally. Now what I will do is I can move this tip around and see how the tunneling current is varying. Now first of all how do I bring do this how do I bring a tip so close to a surface without actually crashing on the surface what I use is something called a piezo ceramic tube. So piezo material let so are materials where if you apply electric field they expand or contract depending on the polarity of the electric field. So you take a tube like this and you have you covered the inside of the tube. So this tube is made of a piezo ceramic material the inside has a electrode made of gold and outside you have let us say 4 quadrant electrodes like this like the one I have shown here there is a y electrode here one on the back side that you cannot see and then 2 x electrodes here. Now I apply let us say I ground the inside and I apply a positive voltage on all these 4 electrodes then it will expand the tube will expand and therefore the tube will get slightly longer. So this length change is tiny this length change typically would happen of the order of 1 micron which is 1 for let us say 100 volt or so which is typically a millionth of a meter. If I apply a negative voltage it will contract but I can do more than that I can also bend it sideways by applying a positive voltage let us say on this x electrode here and a negative voltage on this x electrode here what it will have what will happen is that this side will contract this side will expand. So the tube will bend a little bit but since the length of the tube is huge compared to this displacement the length the length of the tube is this much it is a let us say 2 centimeter tube and this movement is of the order of a micron it looks like a linear translation. So I can also so if I now attach a tip on the top of this tube that tip can move in the z direction and can also move in the x y direction. So this is a very precise positioner and what you do now is this I have this tip here suppose I have a surface like this I move the tip like this and I can measure the tunneling current wherever I have a sort of hill the tunneling current will be more wherever valley the tunneling current will be less. So this tunneling current will give me the surface topography this is normally not what you do because if you do this and you have a very big hill the tip will go and crash on it. So instead what you do is that you try to keep the current constant and you take the tip up and down and you track how much up and down you are taking the tip. Once you track this if you keep on doing this on the surface you can map out the surface roughness. So this for example is a is a particular material called niobium nitrate and you can see that the surface has sort of this kind of hillox which is tens of nanometer in size. But that is yet not very impressive I told you this is the most powerful microscope that you can make in the world. So how much zoom can you do? So if you zoom further and further eventually you can see individual atoms. So now this is for example selenium atoms that you see on an avium diselineite single crystal. So the process of tunneling has so it can be used to get this kind of a microscope where you can see individual atom on variety of surfaces. For example there is a 7 by 7 reconstruction on a silicon surface this is graphite where you see carbon atoms this is another material and so on. But is that all you can do? No, you can do more than this. Now I can actually convince you directly that electron is on wave. Now I told you electron is on wave and all that I gave you was this diffraction pattern that was taken with double slit. But if you take a bucket of water and you tap that you start forming ripples and you can see them. So you do not need to do a double slit experiment to convince yourself that there are waves on water. Can you see similarly the waves formed by electrons and with a scanning tunneling microscope indeed you can. This was a spectacular experiment done in 93 and by now this has become fairly routine. What you have what one has done here is you take a copper surface and you build what is called this quantum coral which is nothing but you put with the tip itself you can position cobalt atoms and form this kind of a ring. Now you see this is like a barrier for this electrons on copper and this electrons inside will form this wave pattern similar to the wave pattern that you have here. With the tip now you measure the tunneling current as a function of distance and you can see this wave directly on the surface of the copper. So what is the wavelength of this wave and so on will be determined by this barrier that you are creating with cobalt. If you have many disturbances on a pool of water then you have many such wave fronts and they will interfere. So you will get a pattern like this. Similarly if you have many defects on a solid you will have many disturbances here you have just created one circle that is why you had something which looks like circular wave front propagating but here you have many such circular wave front and you see a difference between them. So I am out of time is not it? So I will show you a few more interesting stuff without not going too much into details. So for example these images that I showed you you can do more with them. One of the thing that you can show is this. So this for example is a gold surface and gold surface if you look at it it has a reconstruction on the surface which is called a herringbone reconstruction and this herringbone reconstruction are these lines that you see here. So these I am just looking at the surface and I am measuring the tunneling current. Now instead of measuring the tunneling current if I measure the derivative of the tunneling current which is d i d v the conductance what I see is this herringbone patterns again but on this herringbone pattern I see these waves here. These waves are the same electron waves. What is the wavelength of this wave? What will be the wavelength determined by that is correct? So they will depend on the momentum or the energy of the electron. The wavelength of the wave depends on the energy of the electron that we know from de Broglie's principle. So now what you do is that you do this experiment and then you can take a Fourier transform of this image you will get a ring like this and this radius of the ring will give me the wave vector. The wave vector is momentum divided by h bar. Now you know that for a classical particle the energy is given by p square by 2 m which is h cross square k square by 2 m. So if I keep on measuring this wave vector as a function of energy and I plot on this axis the energy on this axis I plot the wave vector you see this nice parabolic relation which is nothing but a demonstration that you have this dispersion relation E equal to h cross square k square by 2 m. This is for gold so you have this very simple dispersion relation you can do it for more complicated materials as well. Now this I think I will so this is the last example that I will give you. We have been talking about superconductor you ever did this experiment on superconductors. So let me give you one more example where you can play a nice game with superconductors. I told you that superconductor expels magnetic field like this so the magnetic induction B which is h plus m is 0. So if you measure m which is the magnetization as a function of h you will get magnetization which is equal to minus h. So it will be minus m versus h is a straight line. This will continue up to some field which is called the critical field and here above this field superconductor it is destroyed. Now these kind of superconductors I will call type 1 superconductors and these are most of the elements that we know but most compounds that we know are a different kind of superconductor called type 2 superconductor. Here the same thing happens at low field you have the flux which is expelled but above a certain field the magnetic field can partially enter in the superconductor in the forms of this lines of flux so you have this flux tubes that enter. Just like you know that the quantum of charge is e the charge of the electron this flux tube each of them has one quantum of magnetic flux that is called the flux quantum and in a superconductor that is given by h by 2 e. Now this flux quantum you can actually image them with an STM because wherever you have this flux quantum the density of the superconducting electron is a little less. So therefore the Karneling current will have the contrast and therefore if you now take a superconductor let us say this is a superconductor on niobium disolinide. If I apply a magnetic field then I so the magnetic field will out of the plane of this screen and then you can see all these vortices. So these are called vortices or you can see all these flux tubes. If I join the nearest neighbor here you can see that each of them has six nearest neighbor. So now this six nearest neighbor is easy to understand these are flux tubes. So these are like bar magnets they are repelling each other and since they are repelling each other this interaction arranges them into the close packed structure for circular objects and which becomes a hexagonal pattern. This hexagonal pattern now effectively is a solid. So you have a solid like any solid like sodium chloride but which is not formed of atoms it is a solid which is formed out of this flux tubes which have entered. So your particles which are forming the solid are this flux tubes. However nevertheless they are solids and if they are solids you know that if I increase temperature then at some point the solid will melt through a first order phase transition and if you have a solid like ice I can put pressure on it and I can also melt ice. So what will happen if I take this solid and increase the pressure on it. Now here increasing the pressure is actually increasing the magnetic field because the more magnetic field I put the more of this flux tubes will get in. So density of flux tubes will increase and that is equivalent to increasing pressure in a solid where you increase the density of atoms. So I first start with a low field I increase field and you see that when I have increased field a little bit I start forming these defects. Now what are these defects? These defects are these red points instead of having six nearest neighbor have five nearest neighbor and these white points instead of having six they have seven nearest neighbors and you see that here every red it has an adjacent white. So this is a defect called a dislocation. So I have these dislocations which proliferate as I am applying pressure and then at some point the melting happens and this is when I start having another kind of defect which is called disclamation. Here this disclamation at this red or white points which do not have a adjacent nearest neighbor. So in a solid typically it is very difficult you can see melting of course the way ice melts but you cannot easily see how the atoms are rearranging themselves during the melting. This provides you an avenue to study melting at the atomistic level. So and then if you go to high field you start seeing that you actually have a molten state where you do not see individual flux tubes anymore but since they are moving you see these lines of flux tubes. There are plenty of other things that you can do with tunneling but I think I will end here and let me just end with one slide. The STM that I showed you you should go back and the scanning tunneling microscope give a search on Google and you will find plenty of sites on that and you will find actually these kind of sites where you have school children, where you have college children who have tried to make their own STM with very degree of success. So some of them have attained a level of success where you can actually image atoms. Some of them have been able to image let us say larger scale structure but not atoms and some of them have been able to find tunneling current but not actually been able to image very successfully. Nevertheless for those of you who are experimentally inclined may be worth a try and if you get an opportunity then try it out yourself. Thanks uncertainty will only come in the form of your line broadening. So any energy level that you have in a real system is not exactly one energy it has a broadening that broadening is comes from the time energy uncertainty principle. So that is the only uncertainty that comes in the energy. But there was a delta function. Yeah, yeah. So for all practical purposes it will be a delta function. So that is an uncertainty that you do not really need to consider. If you want to convince yourself to what extent it is a delta function then you so most of the tunneling that I have shown you where tunneling which happened from continuous energy levels and not discrete individual energy level but I think I will have something showing that yes. So since you ask this see most of the tunneling that I showed you so far was from systems which had continuous energy levels and not individual energy levels specifically. However there is one situation if you take a solid and you start making it smaller and smaller then at some point instead of having this continuous electronic energy levels like this you will start getting discrete energy levels like this. Now if you do a tunneling experiment into the small particle which is embedded in between these two aluminum electrodes then you will start you will see that you have this tunneling current which shows this very sharp peaks and this sharp peaks essentially is the tunnel is when this Fermi energy of this big aluminum is getting coincident with one of the discrete energy levels. So this is actually a proof for you that the tunneling has this is an energy conserving process. Of course all that we have shown essentially is based on that but if you want to see it for discrete energy levels for which the at equation was formulated this is the example. Oh yeah that is right, that is right not well done. It was in those years when general electrics also supported this kind of research which is very different from what it is today. Would like tunneling be a disadvantage sometimes would tunneling be a disadvantage sometimes everything is a disadvantage sometimes life is a zero sum game you cannot win all the time yes. So of course tunneling is a disadvantage for example you cannot make a very very thin insulating barrier which is really insulating that is a disadvantage in some devices you always. So in miniaturization of devices you want to make everything smaller but for example when you are making a field effect transistor a field effect transistor is where you have let us say electrode here then an insulating barrier and then your active device on the top and you apply a voltage between these two to create a large electric field that modifies the property of the material on the top. Now here you do not want any current to flow and if you want to have a large electric field you want to make this barrier as small as you want but you cannot go below a certain level because below a certain level you will start having tunneling current that is going and that is not what you want. So yes there is a there are situations where tunneling can be a disadvantage. Sir you said that we can change the distance between the tip and the surface. Yes. So we change actually but why do we do that I cannot say if there is some undulation in front or the other it may touch then also after changing distance. No no no I am I have so when I am moving the tip I am always maintaining a constant distance from the local surface. So I will be never touching the surface. And what I track in my computer I keep recording how much I have to go back how much I have to go forward in order to keep this current constant. So in principle this will never crash the tip unless of course you have mechanical vibrations that come in. Now what I did not tell you here is that the main challenge in a scanning tunneling microscope is actually to remove all kinds of mechanical vibration from your system. So you have to put your scanning tunneling microscope on vibration isolated table then take special precaution that you do not have heavy people like me walking around and so on. So you are trying to keep a tip within a nanometer from a surface and in between you have no barrier. So it is just vacuum. Any kind of vibration can make your tip crash but if you have taken care of all of these then of course if you maintain the current constant you are never going to crash on the surface. Sir when you said that we are applying a negative resistance to the LC circuit. Yes. The part like the wire which is the still the positive resistance that will still dissipate heat. Yes. So you work with a bias that is why you cannot so if you had real negative resistance you would sort of instead of having your oscillation which is damping your oscillation will be exponentially increasing. Here your oscillation will increase initially exponentially but will reach a limiting value and the limiting value is restricted to the voltage range where your differential resistance is negative. When it will try to increase further it will act as a negative feedback and constrain it to that value. My question was something different. I said that like the there are like in the wire there are some part which has positive resistance and some part which has negative resistance right. Some part means. Like the semi-conductorial pudding has the negative resistance. Some part means you are talking of physical part or you are talking of voltage range. No. Yeah. Yeah. Oh yeah that is right. Oh yeah yeah yeah there will be dissipation there. So then what will happen is that that dissipation here you have to realize that these only works as a negative resistance system under a bias. So therefore it will provide energy to the system whatever is dissipated there these resistors through this bias voltage essentially your power supply will draw some current and will compensate for that dissipated dissipated energy. So there is no energy conservation that is violated anywhere. So this is an active device which is actually pumping energy in the system. We can use what? We can use forced oscillation using battery instead of using semi-conductor that also gives constant. Yeah yeah you can that you can do. You can of course keep on kicking the system with a battery and that will also give you oscillation that will persist. These have the advantage that these gave a extremely stable and tunable oscillator. So this used to be used in all clocks and circuits where you need a timing. Of course now this is no longer in use. Tunnel diodes have become obsolete and that is because quartz crystals have replaced tunnel diode as very good oscillator material. So all your watches they have quartz crystal inside quartz oscillators instead of tunnel diode. But tunnel diode when it was discovered became the standard for any oscillator where you needed very good clocking sort of clocking timer. Really the last question. Okay you mentioned that when you bring two metals together whose Fermi energies are different that eventually their Fermi energies will align themselves. Yeah that's right. So there is going to be some transient response before that happens. Yes. Has somebody calculated the transient response? Are there any papers on that? Ask Professor Singh. Okay I will. Thank you. I think there is but it is a very transient phenomena. It must have calculated. People have calculated everything. Whether they have measured is I think possibly. So whether they have measured the answer is yes not when you are physically contacting but for example you take a diode and then people have for example applied pulses and reverse this or change this Fermi level alignment optical pulses and then look at transient response how it comes back to its equilibrium configuration. So yes it has been measured in various form. Okay. Thanks Pratap for a wonderful lecture. Let's thank the speaker once more. We are running significantly late. So I propose that we go out for a tea and come back at 11.45 and we start the next part of the program at 11.45.