 You can follow along with this presentation using printed slides from the Nano Hub. Visit www.nanohub.org and download the PDF file containing the slides for this presentation. Print them out and turn each page when you hear the following sound. Enjoy the show. So in the last lecture we talked about spin and what I'd like to do is continue to touch on a slightly different aspect. Now with spin the point I was trying to get across is that there is a very different issues involved when you get to non-colinear spins and collinear spins. Because collinear spins you can think just as red and blue. And so what I'll be talking about now in this lecture, you could more or less I mean basically will assume collinear spins. So in that sense the extra complication due to non-colinear, those issues are not involved. But the way it is a little more complicated is it kind of involves the two topics together namely heat flow and spin flow. What I mean by that in the last lecture yesterday I talked about heat flow and then today we talked about spin and here I kind of consider the two things together and some of the very important issues that one should be aware of. And that's why I kind of wrote here again for your reference. The equations we had for heat flow if you remember. The way we talked about this was that well if you have these electrons and for this discussion let's say some level here and as we discussed whenever an electron goes through and this elastic resistor model you can calculate the current from this equation. We have written this conductance in various forms one of those is this one. So number of modes mean free path etc. And if you wanted to write heat current it almost follows from here only thing is instead of the Q you have to put in this E minus mu. That was the idea. And the idea being that when an electron comes in from here and if you think of an electron going through an energy channel at some energy E then the amount of what the electron that actually comes into the contact has an energy mu one what is here is E and it essentially picks up that much energy from the contact. So that's how you look at the heat current. And then what it dumps there. And if you looked at the overall heat current the conservation of energy would be something like this. This is the heat you take from the way I'm defining my heat currents is that if it's positive it means it's coming from the contacts. So this is what you get from the contacts. This is what you get from the IQ2. And this is the part that's coming from the battery. I think this is what we have discussed. Now we'll come back to this. Now getting back then to the spin valves I want to use it as an example to show that there are problems where even in a nanoscale device you may not be able to write the current this way as F1 minus F2 times something. You know this is kind of a central thing that no matter what you write it as F1 minus F2 times something. And that's something then we can talk about what it looks like in different conditions, right? Now, so the example we are considering is the following. So just the spin valve we discussed before. But now let's assume that the channel has lots of these paramagnetic impurities. You know, manganese impurities. So what do these impurities do? Well, they give you lots of spin flip basically. Why? Because there are all these impurities which can be either in an upstate or in a downstate. So left to themselves, you know, there's one electron in there. There's an upstate or a downstate. But you cannot put two electrons in it because of this kind of Coulomb blockade type situation that this takes too much charging energy to have two in there. But it's like, but it could be either this or that. And so as long as the chemical potential is in a certain regime this is the state it would be to either be this or that. And so there would always be an unbalanced spin. It would be, so these are what are called paramagnetic impurities. And there are lots of defects in different semiconductors and in gallium arsenide. The experiments we have done on this those involve manganese impurities. So there are different things that would give you this kind of a situation. And that's what I've indicated there as those red and blue dots. And I say red and blue because you see half of them are up spin half of them are down spin. Left to themselves, you know, there's no energy difference. It could be neither place or left to themselves. It'll be 50-50. So that's what I've shown there. Now how would you model that? Well, in our resistor style model that would appear as a major like a spin flip conductance here. That basically kind of shorts the two channels together. In a sense if you have a lot more ups it will send a spin flip current and do it. And of course normally if you're building a spin valve I mean no one in his right mind would do this. I mean they go to great lengths to avoid this. Now you're trying to get rid of all magnetic impurities. But here I'm really trying to make a conceptual point which is what I'm saying let's do. So let's say you got lots of these impurities. And the way you then model it often is you think that well the impurities are half and half, half red half blue. And so this is how I model it. That's the resistor. And the way you often calculate, how do you calculate the current in this case? Well for this discussion let's assume we have very good contacts. So in a way as if this is just open. So very good spin tronic contacts. So only these reds come in. Blues get out there. That's it. And these are anti-parallel of course. I'm taking the anti-parallel configuration. For this discussion. And so what would happen then is, you see if you didn't have this spin flip thing there would be no current at all. In such a circle. Because this is the anti-parallel thing. This would be like having infinite magneto resistance. Parallel great conduction. Anti-parallel doesn't work at all. But then as soon as you allow some spin flips here then current can flow like this. So the question is how would you calculate that current? Because here this approach isn't quite getting you there because that is based on conduction through up channel conduction through down channel. But here it is different. Up channel by itself doesn't conduct too well. Down channel by itself doesn't conduct too well. And ordinarily you get nothing. But then with enough spin flip that you should get currents through it. So if you were to write down that current it would look something like this. I is equal to a bunch of constants that one can talk about. But otherwise at different energies and we have the spin flip things that take you the electrons that flip from here to there and so you go. There is one process where it is f up times 1 minus f down which means an up spin electron flips into a down spin and goes down. And there will be a process where it will be f down times 1 minus f up. And there will be constants here which will be things like density of states in the up channel, density of states in the down channel, etc. Now the other thing is that should enter here is of course this process is facilitated. This is where a up electron or rather a let's say a red electron becomes blue. But in order for a red electron to turn blue you must have a blue impurity in the first place because the way this spin scattering works is a red electron and a blue impurity and they flip. So electron becomes blue but the impurity turns red because overall it's conserved. These spin-spin interactions that's how they work these exchange interactions. So in order to facilitate an electron to get from red to blue what you need to start with is an impurity that is blue. So that's what I'll put here as a capital F but then it needs to be blue so down. And in order to facilitate the other process of course it will depend on so this tells me what fraction of impurities happens to be blue. Down impurities. And I use capital F for the impurities small f for my electrons. And then I have the capital F. So it's like I said integral dE in a bunch of stuff that you don't write and then times this. Two terms this minus this. That's it. This will be the current. Now question is would this look like f up minus f down? Because if it is then the nice thing is then after you linearize it it would look like mu up minus mu down. You know we have done this Taylor series many times and then it will look like a conductance essentially. Why are those capital F's both down? Thank you sir, thanks for finding out. No, this one is up because you need an up impurity to turn down this electron to up. Thanks. So now as I said if those impurities are at equilibrium, yes please. What's the meaning of impurity for a speed or a magnetism? This is paramagnetic impurities. Say something like manganese or I think there are these certain centers in silicon dioxide interfaces which also act as paramagnetic impurities or any paramagnetic impurity which left to itself is either up or down. Gives you an either up spin or a down spin. And at equilibrium left to itself it would be half and half. That's what I've tried to show there. Half reds, half blues. Half and half. And the point is that if those two things are equal then you say this minus this you'll notice this FU FD cancels out. So when you take this minus this you just get FU minus FD. It looks a little bit asymmetric. Impurities have just one perfect FD for example in the upper line FD times FU times this parenthesis 1 minus FD. Don't do the same. Minus FD. Because with the impurities these are like a so while we are counting so you have these electrons that the picture is the electrons are delocalized everywhere. So the electron and the and you have certain impurities here. And if you have a red impurity then what it will do is it will allow red electrons to turn. I think this difference is partly because of this fact it's delocalized and also this thing that the impurities are kind of like this coulomb blockaded situation where it is up or down. So if I have an up impurity it can always flip to down. It's not like I have to find a down impurity to go to. What I mean by that is when you are thinking of the electrons it's like it is in an up state and has to find a down state to go to. But the impurity is like it's right here it just turns over. If you are this that must be empty because there's no way you can be in the doubly occupied one. So that's why there's this symmetry in what I'm saying. Now, so the point I was making is that if those two are equal FD and FU are equal then yeah this would just become F up minus F down and you could visualize it as this additional conductance there essentially, spin flip conductance. Capital is just numbers. Just numbers, fractional I suppose between 0 and 1. So F up plus F down should be equal to 1. So maybe you should have that clear. Yes please, go ahead. In any momentum state of a particular energy so if this thing comes in this way and we said it's better than the current one then what is the change of momentum? For low bias usually just the density of states is good enough but I'm not sure in detail to a higher order that may be important I'm not sure. What do you mean by that? Those averages I am assuming are hopefully included in M lambda etc. Those kinds of things. Whatever you see in normal transport those issues and these being localized impurities probably would be isotropic in scattering usually. So can an impurity happen in a vector more than one electron at one time? Yeah I suppose so. So you have one impurity here so the electron wave function is spread out and all these impurities sort of facilitate this But the basic interaction is still one spin, one impurity one electron, one impurity basic interaction so lot of them means those are all individual processes in parallel but individually when an electron flips some spin must flip somewhere so that overall if you look from outside you tend to see the same spin One to an effect and the assumption is that the models that people use for spin scattering is that the basic interaction is one to one but then there would be lots of them This is where I say that once you get to the in principle if things are non-colinear and all that all kinds of other issues could come in and here I was trying to keep clear of that let's assume everything we are talking about is you just have magnets which are red red and blue whatever you have injected in that case that should be enough to understand the basic thing but you are right that in general one has to worry about you could have of course to start with your injection is such that it involves non-colinear things and that would then you could ask whether the impurity or not so other issues could come in which I am kind of trying to keep clear of here because there are other conceptual issues I want to focus on Now the point is this that when you usually calculate these things you always assume that the impurity spin is 50-50 but then every time of course it flips an electron an impurity spin is going from red or blue to red but what is always assumed is that there is something else nowhere in your Hamiltonian that you haven't written that always manages to restore them back it's kind of like the contacts you know these contacts as I said if this was all an electron would go from there to there you would pick up some negative charge here pick up some positive charge here and you would be done and then it would be like a capacitor a resistor is the fact that something not in your Hamiltonian something not quite specified anywhere continually takes these out and put them back in that is a very important part of this whole conduction process similarly here those red and blue impurities you can think of it as a conductance but what makes it so is the fact that there is something again something else that continually restores it back to half red half blue otherwise what would happen is let's say there was no mechanism at all for doing that you know these impurities there is nothing else they can do then what would have happened is you see you got lots of ups and lots of downs so lots of ups are continually trying to become down hardly any downs trying to become up so continually what you're doing is you're taking your impurities and turning them red because this is a red electron it looks around for a blue impurity and turns him red but nobody is turning reds back to blue hardly any you see if everything were in equilibrium then of course there'd be as many taking red to blue as blue to red it would be kept in equilibrium by itself no problem but here because there's a lot more red electrons in there they're continually doing that and if it was isolated what would have happened is after some time that left picture would have become like this right here all reds that's it and at that point there would be no further scattering either and then you wouldn't have that spin flip conductance anymore either this would be gone and this is the essence of course of something called the overhouser effect you know overhouser is a professor at Purdue actually in the physics department and this was I guess almost 50 years old now and what a theme of course this impurity is it wasn't quite it didn't involve transport there the whole idea was you did drive the electron spins out of equilibrium using microwaves I mean something else you don't use it do it with contacts in those days of course I mean no one had things like this for doing things with current flow and then the impurities they were considering were the nuclear spins and the nuclear spins because the electron spin interacts with the nuclear spins and the nuclear spins have this property of being extremely isolated from everything else and so once you turn them they have hardly any way of getting back and so the effect can be very striking and the thing was it was when he first proposed it you know hardly everyone thought something had to be wrong with that but it was experimentally demonstrated and it has lots of practical applications you see you'll see if you do a Google search you'll see the overhouser MRI where in MRI instruments they use this effect and so on so it has lots of applications and all that and in the context of semiconductors also I think people have seen some things like this with nuclear spins but exactly the same thing you expect with anything really any impurities as long as they don't have a good way of relaxing through the surroundings up to a point you expect these effects and my collaborator at Michigan he did these experiments on gallium arsenide with manganese impurities and he sees this effect clearly what he sees is when you start you turn on a current and you get a lot of current because there is this conductance and then within a microsecond the current goes down goes down because a little later this is gone it has become a may not be as clean as what I've drawn but because of course it's with real contacts with other issues involved but the current goes down that's what it sees very clearly so the important point here though is the point I was trying to make is this essential distinction between a resistor and a capacitor that what really turns the capacitor into a resistor is forces not specified that continually restore things that's really in a way something that kind of makes transport generally harder because these are the issues that are often not all that clear and this is also a good example the thing on the left is like a conductance as long as somebody is continually fixing things continually restoring it back if you have nothing restoring it back then it's like a capacitor after sometime it sort kind of charges up in a way if you could say that there was no spin in the beginning after you're done you have a lot of spin all lined up so there's big spins as if you kind of charged up a spin capacitor in a way almost charged it up now once you have charged it up though you have got lots of upspins and this is now zero you have charged it up lots of reds now the current equation looks like this and if you now turn off your voltage you'll actually still get a current the current will keep flowing you'll get a current out of it why is that? well basically what would happen is suppose one could this way that there is all these electrons the red electrons coming in here which are trying to get become blue so I guess let me draw that previous picture where I had an energy level E and you have say mu1 is here mu2 was here and there were red electrons coming this way and of course there's also a reverse process and of course usually which process dominates depends on Fermi functions that is how many you have more what has happened is this process is facilitated by blue impurities because a red electron comes in has to become blue in order to get out while this process is facilitated by red impurities and now that you have lots and lots of reds what happens is it is as if you have just cut this off this is going the other way and so at this point even if you take off the voltage it is like the current will be flowing in this direction kind of against your potential drop so in principle once you have turned it red all red you could take this and then I guess drive a load of some kind with it from that red up to a point and if you think about it at first you say well that's not very puzzling after all if I have charged up a capacitor I can always discharge the capacitor and get some energy out of it and after some time it will discharge go back to the left sure but the interesting point here though is that although it is like a capacitor but there is no energy in that capacitor because red and blue were perfectly degenerate things they had exactly the same energy when you flip from red to blue no energy is involved so in that sense we still have an elastic thing although it is not elastic resistor anymore but I guess it is still elastic no energy exchange is involved so where exactly did this energy come from what I mean is once I have turned it red I said I will take out the battery and now drive my load and sure I will be able to drive it for a while which is kind of like discharging the spin capacitor but where did that energy come from it did not come from the spin capacitor because as I said what is on the left and what is on the right have exactly the same energy that is not where anything came from three energies different because one was less than this is what I am getting at exactly so here then now if you look at the energy though energy still must be conserved overall so whatever you are getting here must have come from somewhere if you look at it what you find then is that an electron comes in it takes that much energy from this contact goes out here dumps this much into that contact but it has more coming in pulls out more coming in than dumping and so overall it is taking energy from the contacts and lighting up that light bulb your spin capacitor isn't doing anything apart from stopping the reverse process is just facilitating one process or stopping the other one and in the process you are actually just taking energy from your contacts and turning it into useful work so this is still true in a way but the sum of these what you are getting from the contacts this is now a positive number what I mean is usually whenever we have talked about this you may take this much from one contact dump more into that contact so overall the heat you got from the contact is negative in the sense you dump more than you take normally but in this case actually you are taking more than you dump overall so you actually made this guy negative I am sorry they make this guy positive you are taking more my definition is positive means you are taking it and so this must be negative which basically implies that rather than take energy from my battery I am actually driving something so energy is conserved no problem with that one so the question is is there a problem with this and this is where I say that well this energy conservation is called the first law of thermodynamics which everybody understands I mean you tell anybody that energy is conserved you won't get any arguments the one that raises a lot of discussion is the second law of thermodynamics and what the second law says is that Iq1 divided by T1 plus Iq2 divided by T2 must be less than 0 because and because meaning this is what I want to discuss more where it comes from what it involves that this is the amount of heat you are taking from the source from one contact this is the amount of heat you are taking from the other contact and divided by T that is like the entropy flow whenever you take a certain amount of energy from a contact that is what you divided by T is the entropy and this is what I will discuss more it is not meant to be clear now if everything is at the same temperature then what it tells you is that Iq1 plus Iq2 is less than 0 so if everything is at the same temperature then you should be giving up more to the surroundings than taking it because remember these guys when they are positive it means I am taking it in so it has to be less than 0 means I have to give up more than I take in that is the basic law and this is something you feel instinctively and it is something that is again embodied in this second law but it is subtle enough that it raises lots of discussion lots of literature on this ever since ever since the 19th century this has always been discussed a lot energy conservation shared that we understand but then there is always a lot of energy all around you there is all these molecules running around there is all this energy but you cannot build a car that just takes energy from the surroundings and converts it into mechanical motion for example that would not violate the conservation of energy first law would be fine you are just taking energy from there but it would violate the second law because the second law says somehow there is a difference between these two types of energy the energy of random motion the energy of randomness all around and a mechanical directed motion like a car or whatever you are doing with it and so when you go from left to right that is something where you are going from something random to something very ordered so that will never happen spontaneously but if you are on the right hand side then it will happen spontaneously from that you can come back to the left hand side so it is a one way process unlike most mechanical things unlike mechanical processes which are all reversible there is no one wayness to it if a planet could go around this way it could have gone around the other way just as well that is fine mechanical things could go either way it is this part that we are trying to make earlier also that what makes nanoscale devices somewhat easier to understand is that processes of this type which actually drive a lot of phenomena in real life most things are actually driven more by this increasing entropy than by mechanical things why heat flows from high temperature to low temperature it is entropy driven everything else is really entropy driven and the biggest problem in all of this transport theory how to combine the mechanical part with this entropic part because they are all kind of intertwined usually and Boltzmann showed how to do it in a classical context NEGF does that in the quantum context in the sense it takes the mechanics from Schrodinger and then brings in the irreversible entropic parts through the sigmas in the NEGF and Boltzmann showed how to do this in the classical context and this elastic resistor or this nano device context what I feel the reason it clarifies a lot of these issues helps to see it all clearly is because the reversible is here and the irreversible is there it is kind of all clearly separated you know exactly what happened ok and so this is the part then that what you could show is that sure you can light up a light bulb here but then after some time of course the reds will again become blue and if you look at what is the maximum energy you could extract from it you would find that that number would be this NKT log 2 this takes a little bit of algebra if you like we can go over it in the discussion session but you can actually show it from here you know I mean without invoking second law or anything I could just take this look at the current see what is the maximum I could get out of it and I'll find its NKT log 2 and the justification then would be so how did we manage to get around the second law in other words basically what we are doing here when you have this right hand side is of course actually taking heat from the contacts and lighting up a light bulb which is almost like as a setting taking energy from your surroundings and running your car so how are we managing to do that well because the entropy of that impurity system is changing because what the second law really says is that the total change in entropy the total entropy current so this is the IS that is associated with the impurities that whole thing must be less than zero and what is happening is as far as the impurities is concerned from reds to half red half blue and in the process of course the entropy is increasing a lot and so this could still be somewhat positive I guess what we did here in our problem this could still be somewhat positive as long as it overall it's negative because the entropy flow there that part is negative that is how you justify it no problem with second law really it is still following the second law but this is how it's reconciled here and this are some similarity I often say is if you have read the literature or if you do a google search you'll often see this Maxwell's demon again one of the things where lots of literature on it and what Maxwell had proposed or pointed out back in I think 1850s around that time is that that's the picture below that supposing you have a box which has uniform temperature and so you have all these molecules running around but then some of them are fast let's say the red ones some of them are slow let's say the blue ones and if you had this Maxwell's demon and again no negative connotations or anything just somebody who knows about all these velocities that has a way of knowing this and what he does is he opens and closes more in between at just the right times so whenever he sees a blue one coming along it lets it through to the left red ones it lets them through to the right and so after sometime what will happen is all the red ones will be on the right and all the blue ones will be on the left and so the left hand side will be colder and the right hand side will be hotter and of course once you have a temperature difference you could then use that as a thermoelectric to drive other things you could do things out of it so as long as he can be opening and closing that door without spending any energy the argument was look you could get you could violate the second law essentially that was the argument and so question is what's wrong with it and this is what people have argued a lot and in a way what's up here you could think of as Maxwell's demon who lets electrons flow in one direction and not the other that's basically what it did you know with the red you cut off one group of things that's it and that's how it's supposed to be working and basically of course the resolution of all this is that the Maxwell's demon sure if he started out energetic and ready to do things he could do this but after some time just the process of doing it would basically drain it whatever it basically if he started out like all reds eventually it would become that that's the essential point usually now of course in the modern context there are all kinds of other issues now that people are worrying about things like entanglement and things like what quantum mechanics adds to this picture et cetera but those are separate issues I'm not going into just classically I'd said this gives you kind of an electronic Maxwell's demon now this to me is kind of brings out the basic problem here and that's what I want to talk about that when you try to include inelastic processes into any description the problem you have is let's say just something simple like a hydrogen atom you've got two levels so 1s 2p level and what we know is that if you put an electron here to come down if you have an electron here it won't go up by itself and so the implication always is when we always say this given any system left to itself it always goes to its lowest energy state and doesn't usually raise much questions especially by the time you're a graduate student you know that if you question that you get people annoyed so you usually don't bring that up but with beginning students I do still often have that question why does it always go to the lowest energy state and as soon as you try to build a transport model of course you run into this because especially quantum transport model because anything you put in you know these Hamiltonians are all Hermitian if there's a HMN there's a HNM so whatever you do anything that takes you down will also take you up that's a bit all mechanical things as I said they tend to be reversible no matter how you do it you usually have that and so you can't even describe this basic fact namely left to itself it will go to its lowest energy state and this is where the I guess you know again already discussed many places here's the clearest thing I've seen is I think in Feynman's statistical mechanics I think he almost in the first couple of pages he explains this and what he says he amounts to something like this he says that look in an isolated system if I put an electron there it would have just stayed there by itself the reason it is coming down at all is because it is interacting with the surroundings and the surroundings you could view as if this is energy there's a density of states of the surroundings and when I talk of density of states is then this entire complicated object that I call reservoir which could involve millions of electrons question is what in how many ways can that state that reservoir take energy from your surroundings and he argues that well if you looked at the density of states all common reservoirs would have an increasing density of states so low energy is here higher energy is somewhere up there and so let us say an electron is here and the reservoir is there when the electron comes down energy is conserved so electron is now here and the reservoir is now over there so if this energy exchange involved is epsilon then the reservoir if it was initially E0 it will now be E0 plus epsilon and the argument then is why is it easier to go down than go up because when you are going down you are going this way the reservoir is going this way and all normal reservoirs will have a lot more states up here than here so you might have say 20 states here and you might have 20,000 states there so although you are looking at it and saying well you have got one state here and one state I am going up and down it is almost as if you see this one corresponds to 20,000 states and this one corresponds to 20 states so even if the basic rates are equal the point is it is much more likely to be going down that is the basic argument here and so any normal reservoir then will have this and it is kind of good to think about these things because as you go to nano scale things there is no guarantee that if you are creative you might be able to get around some of these things so it is actually all worth thinking about clearly now but usually with all normal reservoirs this is the case and so what you can write then is so the net result of all this is with any reservoir like I mentioned there it is easier to give up energy than to extract energy from it because whenever you give up energy you are kind of increasing the number of states that is like going to the half red half blue thing and this is like the full red thing so whenever you give up energy you always have a lot more states to go into and that is why there is always this ratio that I think what I wrote there let me write it so so the probability that what the energy takes the probability of giving up an amount of energy epsilon divided by the probability of taking an energy epsilon so that plus epsilon I mean taking when I write minus epsilon it means giving up that is what I have done there and that will be equal to the density of states I wrote it as W there same thing I mean density of states W of E plus epsilon 0 E0 plus epsilon and then this famous Boltzmann's relation this is this S equals K log W that was the definition of actually Boltzmann's constant is KT the K that you see everywhere and S is equal to K log W so W is this density of states how many states you have and logarithm of that and so W is equal to E to the power S over K so that becomes E to the power S E0 plus epsilon minus S E0 over K so what I have done is instead of W I have written exponential S over K and then again big things I can use this Taylor series expansion again so you write it as E to the power dSdE times epsilon and then over K and the thermodynamic definition of temperature is this derivative of the entropy with respect to energy, I mean inverse of that so usually this definition is so that's how this becomes epsilon over K so this is the basic point that these entropic forces will always be such that whenever you are looking at a reservoir in equilibrium a contact in equilibrium d giving energy to it will always be easier than taking energy from it and by that ratio, exponential epsilon over Kt and this is what drives the flow of heat for example so the argument you usually have is let's say this is E this is entropy so if you have something at high temperature that is this is very high then dSdE will be almost zero so a high temperature thing will have where you can change E a lot S won't change much whereas if you have this is high temperature and if it's low temperature it will be like this and supposing you want to have transfer some amount of energy from one to the other so from the high let's say we take out some energy so it goes from here to there that much energy we take out and so the low one goes that way so you see the high one hasn't changed entropy much but low one has increased a lot and so overall entropy has increased so that's the way heat will flow on the other hand if you try to go the other way then what would have happened is you'd have lost a lot of entropy but gained very little etc so the point is these are things you always see in thermodynamics text and the point I'm trying to remind you is that lots of things in real life are not driven by mechanical forces that you can put into a Hamiltonian but driven by entropic forces and if you have pure Hamiltonian all you have is a capacitor finally things of this type are involved to turn the capacitor into a resistor really and they need to be included somehow in this whole discussion in any transport theory you see and in general then what happens is if you think of a process let's say like one little thing I should what I want to prove next is what I can say one of the questions that you could have is the following that ok this second law which I guess I erased it so let me write it up again here the second law which is iq1 over t1 iq2 over t2 plus any other entropy issues all that always has to be less than 0 because the way I defined it this is the amount of heat you are taking from a reservoir and whenever you take from a reservoir you lower its entropy so so this has to be negative because overall you must be increasing the entropy that was the argument and what I said is that when it comes to that spin system that spin system the entropy is of course very clearly increasing when you go from right to left but here so the way usually for a reservoir the entropy is calculated is that if you have a certain amount of energy to it the entropy changes by that energy divided by the temperature e over t so that's what I said this because ds de that's like 1 over t but that is true only of things that equilibrium so what I mean is you couldn't use that idea to figure out how much the entropy changed from right to left because that right thing is out of equilibrium all these ups and downs all have the same energy to have them all filled that's a way out of equilibrium situation really at equilibrium they would have all been 50-50 but this is a case where you can where you can get the entropy easily by just going to the basic definition of entropy namely s equals k log w so the whole idea is if you have n impurities how many ways can they all be read well there's only one way how many ways can they be half red and half blue or how many ways there can be anything else that's where approximately it would be 2 to the power n because the idea being each one can be red or blue and so overall there is 2 to the power n states so you have to subtract out the red from that but then this is a big number I don't worry about that and then this becomes like n k log 2 but that's how you calculate the entropy associated with that and in general when you have non-equilibrium problems of course they exactly what how you should define entropy and all that people argue about are still discussed it's not necessarily all that clear whereas when it's equilibrium things then it's clear what exactly you mean by entropy how much the entropy changes for a given energy etc that's all settled then then where does the second law come from or what I mean by that is when I'm building a model for a device and I'm trying to calculate currents and all that how do I make sure that I won't be calculating anything that would be violating the second law necessarily because everything we have done so far if you are using this model and you're doing thermoelectrics thermoelectrics where you're taking a certain amount of heat from here dumping it over there because there's a temperature difference and all that if you use the formulas discussed there will be no problem with second law but the question is how can you be sure and the point I want to make is that that is always ensured as long as this is correct what I mean by that as long as whatever contacts you have have the property that that when you take energy from it and when you give energy to it the ratio of those two happening is e to the power epsilon over kT as long as this is true you won't have any problems with second law now this is the one where if you're doing a Boltzmann equation for example when you write scattering rates you always try to make sure that s12 and s21 have that ratio between them for example the corresponding thing in NEGF is a little more subtle because there also phases are involved in the sense that electron kind of becomes like this correlation matrix etc but NEGF again if you follow the prescription this would be ensured so you automatically have this second law conserved but then if you make approximations or things where you're not necessarily following the exact prescription then you have to be careful because otherwise you may be doing things that are really not physical now why is this related to that well it's kind of like this supposing we have this simple elastic channel here which is operating with multiple reservoirs so you have one reservoir here with some mu1 T1 this is some reservoir here mu2 T2 etc and if you take a certain amount of energy from here say E1 and let's say consider one process where you take an amount of energy and here an amount of energy E2 from there and the probability of that happening is this FE1 times FE2 that is a process like that will have a rate that will be determined by this F that tells you how easy it is to take energy from this or take energy from that now the point is corresponding to that there will always be a reverse process where you give up energies to that and those would be like F of minus E1 and F of minus E2 and the point I'm trying to make is in order for this process to be the dominant thing the one that is actually happening because you could overall say that the rate at which this will happen is this minus the opposite process this ratio must be greater than 1 41 and what that means is FE1 that is like E to the power minus E1 over KT E to the power minus E2 over KT all that is greater than 1 sorry T1 and so E1 over T1 T2 over T2 should be less than 0 in order for that process to be happening and this is if you are just taking some energy now if actually an electron is transferred then of course while electron is transferred another one comes in here because this is an open system with a certain chemical potential so the actual energy you take from that contact is like E1 minus Mu1 so if you had done it that way then you would have come up with E1 minus Mu1 plus E2 minus Mu2 etc and to some extent that is the second law but all I am trying to show is that as long as your model the model that you are using includes this in how you define your probabilities you would be in keeping with the second law and then you need to think about separately and all this is kind of important again because as you know in the future firstly lots of concepts may need to be revisited as we go along that is this whole idea of entropy of reservoirs because when you have small reservoirs what happens because normally you know one would have said that a spin system is like a reservoir you could treat it as a reservoir it is always maintained half and half not really as I said nuclear spins deviate significantly and then there can be all kinds of other things one can think about for example and then there is this general question of energy conversion waste heat conversion that is how do you take how do you build devices that can just take energy from the surroundings and convert it into useful work or something that can take energy from your body and do useful work for example and all these involve this general basic principles again all that to be clear all because these are all concepts that took hundreds of years essentially to clarify and lots of discussion and even now you see lots of discussions about it and usually of course the discussions are lot more convoluted because again the two types of processes namely the mechanical ones and the entropic ones are kind of mixed up usually usually it is not this clean it is not like this is mechanical it is not like this is entropic it is like everything is happening all mixed up in a proper way and then it is so anytime you do a transport theory when you are trying to put in so this elastic resistor model that I described to you what we did was we said all the entropic stuff is here and I don't even get into don't have to worry too much about the details of this at all how do I take care of it I say well you know this is a big contact with the equilibrium with F2 and those are of course the functions from equilibrium statistical mechanics which satisfy these principles so when you are trying to write down the rate at which the electron can get out there or come back in they obey the correct ratio the ratio needed for second law and all that so there is no problem it is automatically taken care of and in the middle it is all just pure mechanical motion Newton's law, whatever you like see and that's how it separates out cleanly and makes sure that overall it will be consistent anytime you want to put in the inelastic processes in here then Boltzmann gives you a general prescription for doing it and of course all scattering rates would obey that property and then in any GF again corresponding thing but quantum that is what to usually give you now to finish up then the points I just to go back this was the outline of the things we went through now in these lectures what I tried to stress more was the concepts rather than the actual as I said you know calculating and understanding are often different and those have to be done together but these are some different things now if you are looking for specific problems so you are trying to learn how to calculate certain things then I would say I would recommend for the semi classical the simplest thing I usually recommend is what I call the point channel model that is a device where you treat the entire channel as a single point with a certain potential and you how you do the calculation self consistently and when you do that you can calculate the current voltage characteristics that is reasonably close to real calculations on MOS devices actually but it gives you a very good feeling for how electrostatics controls things how different parameters here would change the current voltage characteristics now if you are trying to learn quantum transport then as I mentioned those are the basic equations but the two examples I usually recommend then if I have it here so the two examples I usually recommend are this localization problem which means 1D wire with impurities and then the 2D problem which is the quantum conductance of 2D wire and quantum Hall effect so those examples usually give you a pretty good picture of things so I would say if you are trying to learn the methods for calculating those would be the two I recommend and these are things where I have if you send me an email I can easily send you the MATLAB codes I have these are things that I use for homework problems in the courses we have here we have an undergraduate course and the graduate course on the subject of quantum mechanics so if you are interested if you send me an email I would be very glad to send it to you but what I wanted to stress more was these conceptual aspects of this and that is where I said that there are all kinds because whenever we discuss these things people say that well you know we need to learn more quantum mechanics and this is kind of true but I think what is much more important is the statistical mechanics quantum mechanics I am not downplaying it at all but as you can see there are lots of things here that you can understand within a semi-classical picture itself and that is because a lot of these subtle interference effects tend to get washed away especially if you are doing it at room temperature one of the things that however is observed at room temperature these days and persists is the spin because spin is one thing where again up to a point you can think classically just as red and blue but not quite because whenever you have non-colonial spins you need the quantum aspect of it as I tried to point out those off-diagonal terms kind of matter and you have to have a way of thinking about it and this is this part of it you could say involves quantum transport in a way otherwise I would say the quantum models well if tunneling is involved then I think quantum models are needed usually so but overall though I feel that in transport what is much more important is to appreciate the statistical mechanics part of it and there of course what we need is non-equilibrium statistical mechanics but that as I said is much less understood and relatively few courses or books on compared to equilibrium statistical mechanics so what you can of course take standard courses on and learn very well is equilibrium statistical mechanics and again that is a very profound subject that took hundreds of years to really clarify and when it comes to non-equilibrium as I said there is all these issues that it's not all kinds of models all kinds of things that people are looking at and as I said with small devices I think I feel it clarifies the concepts it can actually understand things much better than the non-equilibrium statistical mechanics of big systems that's relatively much more complicated really so we have this discussion session today at 3.30 so if you please pass on your questions ahead of time if you can or raise them at this point and I'd like to answer any questions that you may have on I guess everything that we have discussed here okay and thank you