 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. That's the outline of this series of lectures that I was giving. So we're now like on number 5. So what I'll be talking about is spin. I mean that's something we haven't really, hasn't really come up so far in our discussion. But this basic model that I've tried to set up in the last few lectures about this elastic resistor. The idea that as long as you think of it like electrons go through this channel elastically and lose energy in the contacts, you can write the current in this form. And this conductance itself, we have obtained various expressions for that conductance that will help you relate to different things. And in this context today what I'll do, it will be useful to just go back to that first form that I had written down. Namely, what's the conductance? Well, it's this density of states divided by the transfer time. See, and then we said, well transfer time for diffusive transport was something, ballistic transport was something else. Now in this context actually we'll go to what we discussed last day which you might call interface-limited transport. So let's say you have a device in which getting through this part is relatively fast. The important thing is getting in and getting out. Now in practice of course this is also important in a real device. But as you'll see I'll explain why that that part of it actually makes the device not work as well. Ideally what you really want is for it to be interface-limited. And this is the basic device that I'll start with, that's this spin valve. And then we'll go on to things. So the spin valve then if you're trying to understand it, the first point is what I've mentioned before that usually all energy levels, these energy channels inside your device, they come in pairs. There's an up spin and a down spin. And they're usually exactly the same which is why we often don't pay much attention to it. It's like you remember to multiply by 2 at some point. And the only confusion is when you multiply certain things by 2 and don't etc. Otherwise it's just a matter of putting in the 2. But what I'll be describing today though, there is a distinction between them and I'll explain how you do that. And that's what I've tried to color code as red and blue. So when I draw this density of states, this diagram I've always been drawing, energy density of states this way. Now on this side I've drawn a blue thing here. There's a blue part, there's a red part. Meaning up spin, up spins red, down spins blue. And when I draw it this way I don't mean the density of states is negative or anything. It's just for clarity I've drawn it on the other side. That's all. You see they're really on top of each other. You know that axis is a density of states and of course it's not negative, it's positive. So that's this blue and red. And because it's usually degenerate, this channel there is nothing special about it. In a spin valve usually nothing special about the channel. It's just like any other semiconductor or metal, non-magnetic metal. And so red and blue are exactly the same. Now the distinction comes from the contacts. And those contacts are now magnetic. So those are magnets actually. So you've got these two magnets and what's known in magnets is that here the red and blue states are not degenerate. In fact these materials that are used, it's the way I've drawn it as you can see. One of them is somewhere here, one of them is here. And this offset can be significant. It's like one electron volt. I mean these are metals where the chemical potential is way above the bottom of the band. That's like three, four volts away. And this part is one or two volts. And the same on this side. So now the question is how do you calculate the current through something like this? Now the distinction comes then in this transfer time that we are trying to calculate. And as I explained in the interface limited transport, the important thing is how long it takes to get through this and how long it takes to get through that. And the point I'm going to make is that in a situation like this, the red ones actually can get in much easier and get out much easier. While the blue ones have a much harder time. Why is that? Because well supposing you have a red electron sitting here and it's trying to get out. And we said that well that time yesterday when we did quantum transport we put in a gamma or in general we're talking about the rate at which it can get out. How do you actually calculate it? Well usually you would say use something like Fermi's Golden Rule. You'd say okay this is a matrix element square times the density of states out there. So if you look here you'd say well the red has a big density of states. Blue has just a little. And of course ideally what you'd have liked is half metallic contacts. You know once where only the red can get out and blue has no chance of getting out. That's what you'd really like. But we're not quite there. You know people have been working on it is getting better and better every year in terms of how much difference you can get between this and this. Because if you have a big difference that means red will get out really well, blues won't get out and that's when the spin valve would work the best. I mean that's the basic. So what we have here then is I could say a red channel and a blue channel. I have colored chalk but Joe tells me that those are kind of hard to erase. So I think with R and B we know what you're talking about. So the red one then can get out easily. So I'll call that say T1 and the blue one has a tough time getting out. All that T2. And the thing to remember is T2 is longer than T1 because if you have a tough time getting out means you have to sit around for a second before you get out whereas easy means you just get out in a picosecond. So T1 is short, T2 is long. So how do I calculate the conductance? Well basically it's two channels in parallel. So no particular problem. So you could say well I have a red channel. So when I try to write conductance I'd say this Q squared D over 2. I don't need to worry too much about that's the constant part here because remember that D has nothing to do with the density of states in the context. It's the density of states in the channel. Because here you are really counting states in the channel. This is where the contact comes in because it is contact limited transport because what matters is not how fast you get through inside, it's outside. So I'd say that for the red channel then this would be 1 divided by T1 plus T1. So it's like 2 T1. Why? Because well T1 to get in, T1 to get out, total time. Like the model we used yesterday if you remember it comes out as some of those two things. And then there is the other channel which is 1 over T2 plus T2. So this then comes to 2 T1 T2 over T1 plus T2. That's what it would be. So this is what's called the parallel combination. Parallel meaning both magnets are in the same direction. Now, yes please. Say this again. Spin flow backwards. Like any electron. So it basically is the electron that's flowing. That's all. Why do you ask? Because yesterday we also considered the spectrum from the channel back to the contact line. So do you incorporate that into the time? Right. So the model we used that yesterday where you are feeding in things and how it is going back. After you have done all that then you have this expression for the T. That's all in the model. So now we don't need to worry about all that anymore. If you remember yesterday we had this. I was using the inverse of time as the escape rate, the nu that I wrote as per second. And there was a nu1 nu2 over nu1 plus nu2. And that's basically what you're doing here also. Anyway. So this is what's called a parallel combination as I said. So I'll write this as GP. That's a parallel case. Now the spin valve operation of course rests on the fact that the conductance is different when it's parallel and when it's anti-parallel. So you get a different conductance if the magnets are anti-parallel. And this is what has found a widespread use in reading magnetic information. So if the information is stored in a magnetic disk in the form of the direction of the magnetization and the point is the resistance you measure will depend on. So there's a reference one and if you're in the same direction you'll have one resistance. If you're turned around you'll have another. And that allows you to read the information. That's the major importance of this effect from a practical point of view. Now if I were to write this. So question is what do I expect for the anti-parallel combination? So again same formula. Now you see what will happen is these two channels previously the red one could get at the same time on both sides. Now of course what happens is on this side it's still a short time but this time it's the long one here. Why? Because now it's on this side you have turned the magnet around. So now it is like this is the blue side and that's the red side now. So it's a blue that has a much easier time getting up on that side. And so this picture now changes to instead of T1 here I actually have T2. And here this is T2 but then that's T1. Those things reversed. And so when I try to write the conductance you see now you notice both channels conduct equally well. One has a tough time getting in the other has a tough time getting out. But between them conduct equally well. And so you basically and the time is just T1 plus T2. But then there's two of them and so you put a 2 there. Now if you look at the ratio of the two you calculate Gp over Gap. What you get is I think I did this. I am dividing this by this. That's like multiplying with T1 plus T2 over 2 and so you get that. Usually the magneto resistance is often defined as this thing minus 1. In the sense that if those two are equal of course you really don't have anything to talk about then that will be 0. So you look at how much better it is you put a minus 1. That amount comes to T1 minus T2 whole squared divided by 4 T1 T2. Is there a problem of accumulation of the R-type and depletion of the D-type because it's difficult for R to get out and it's easy for B to get out. So once you start looking at how the chemical potential varies inside the electrochemical potential you see those differences inside. That is true. But that's all taken into account in your model for how to calculate the current. So one of the things I'll get into right after this we'll talk about is what does the electrochemical potential look like inside. But the point here is that you'll notice that this magneto resistance then it's positive. This is always bigger than that because as you can see as long as T1 is different from T2. Of course if they are equal there's no effect. This whole thing hinges on that difference of course. But any difference then you'll get something. And of course ideally what you'd have liked is one of them to be infinite and then you'd have essentially infinite magneto resistance. And this is what I guess over the years has kept improving in the beginning it's more like 1%, 5%, 10%. Now people talk of 200, 300%. But those involve having an oxide here. GMR devices which have 300 or 400%, actually it's an insulator here that you're going through. But the basic idea then is this is how you could visualize it. And one of the important things as far as injecting carriers into semiconductors, by the way the GMR devices originally the channels were copper. I mean it is usually some metallic channels. But one of the problems people had in trying to do this with semiconductors is that they never used to get much of an effect in a semiconductor. And the reason why something very simple it's this. That in a semiconductor usually there is very few modes here and of course in the beginning people were trying to make very good contacts to those things. And one of the things we discussed is when you have a ballistic channel with very good contacts, then the conductance is always you know 2Q squared or rather Q squared over H times the number of modes in the channel as long as you have very good contacts. And the point is usually with a semiconductor what would happen is this and this would look equally good contact wise. Oh sure you know you have got a few less, that's okay but then the semiconductor has only 10 modes. Let's say this one has a hundred and that one has a thousand, well big deal I think, makes no difference. As long as you're just trying to feed 10 modes 100 and 1000 are equally good. And one of the important realizations about 10 years back which I think led to much significant improvement in terms of injection and semiconductors was the idea that don't try to make very good contacts. Deliberately make it have a barrier in there so that you're actually injecting through a short key barrier so that, so you make it hard for this guy as well as for this guy but then he can overcome it, he can't because he has fewer density of states and then you start seeing, then one day finally you want interface limited transport. And so the way you can visualize this is almost like in terms of a resistor model often people draw this. You have a resistance here and a resistance here and I guess you have a, I'll call this a small resistance and if it was in the parallel combination you'd have a small resistance and a small resistance and here you'd have a big resistance. I'm writing R for big and you have a big resistance. So it could sort of take this Q squared D over 2 T1 and kind of associate that with what a resistance like an interface resistance. It's almost like two resistances in series essentially. And so the parallel combination looks like small r, small r, big r, big r whereas the other combination, the anti-parallel one looks something like this. And that's the resistor model that's widely used in terms of understanding modeling this kind of thing and there the way people visualize it is that you want these resistances the interface resistances to control what you're measuring and what happens is in a metal this part is almost a short anyway but in a semiconductor this resistance tends to be much quite comparable. That's how people usually say this. Yes. So does the contact matter where you are putting the contact? Is it, I mean, if it's a particle contact and if it's a difference in first-year engine if it's particularly underage it needs to transfer horizontally. So does it matter? Yeah, good point. This is something I'm not getting into so the question was that often the way this is done if you had a lateral transport you might have a contact like this question is, is there a difference between this and that? That's okay, right? Is there a difference between this one and that one? And the answer would be if this were really short then there would be a significant difference of course because what you can show is when an electron is coming in here if it is really short then you may not even collect it it could actually still be going on, right? And so usually these things take a certain length to get collected and this is what's called this RG length usually so you often visualize it as if this part of it is like a bunch of series resistors and resistors that take you upwards and if this resistance is relatively low then within a short distance you get sucked up so these are standard models for contacts for lateral contacts that people use and those same ideas could be used here because in a way you notice that whatever I'm doing so far it's not too different from just the standard propagation of carriers of how you think about electron flow everything it's just that you've got two species to take care of that's an up and a down in fact as you'll see the equations that people use for this usually the spin diffusion equation or Wallifert equation it is a lot like this the up spins diffusing and this down spins diffusing and then you can put in spin flip which takes one type of carrier into another this is spin diffusion equation and so when you think about contacts and things like that all the usual understanding of this that I discussed about just mentioned about how it gets sucked up how much distance it takes those are standard descriptions of contacts nothing special about spin yes please the amount of megatransits of it is parallel and unparalleled case or it's long to run and off the states that is I guess it will depend on the scheme you are using and usually of course spin diffusion is more in the form of voltages and charges but I think one of the things we have been proposing is you could probably instead of looking at the charge on the capacitor as your information look at the magnet itself as the information for example but those are all open questions now you know how you would use it what might be the best way to do things you had a question we just talked about electrons moving with the spin the electrons are moving right so in that sense nothing so it's like we figure out the rate at which electrons go and usually you multiply by q to figure out the charge here you would multiply by spin to figure out the spin current and I had some questions often from people what's the spin current very mysterious but I'd say just replace the q by whatever spin it carries that's all the electron is moving I mean nothing more profound involved here nothing collective nothing else yes so this is spin injection better if you have early background if you have so if you have dual contact you put in something which is a very good insulator bring your spin injection into there is an effect so what people have found is that I guess experimentally now have kind of figured out the best insulator thickness they're comfortable with you know that's what gives them and it's not like by putting an insulator the current gets any better no current of course gets worse no question it's just that the one spin gets far worse than the other and so you actually have a spin discrimination so if you actually try like a whole range of different how many values one is that why can't you just use something which is like the worst yeah it is just that over the last over the last 10 years I guess people have tried out different things and what I've noticed that different groups tend to have different thicknesses they're comfortable with and you can see what they're using from the contact conductance that they quote that is some groups the papers you'll see is like 10 to the 10th ohms square meter I think it's the units right or in some cases you even see 7th or 8th it is like whatever they're comfortable with they feel and it's values with material system and so on I'm not sure if it is all quantitatively very clearly understood but what is established is a bit of a barrier there helps injection significantly and this was of course a major change in mindset in the from the beginning because 10 years ago people are almost saying it's impossible to inject into this you're almost coming to that conclusion and of course everybody at that no one was thinking of putting a barrier because you know you're trying to make a good contact you're trying to make an only contact that's your instinct before you think too much anyway so what you can show is then so you have these two resistors and often what people describe as the polarization of a contact how good it is in terms of the resistor model they often write it as p equals r minus r divided by r plus r that that would then be a measure of how good your contact is polarization wise and I think you can show that this magneto resistance that I wrote down here if you just play around with this a little bit I think you'll get p squared over 1 minus p squared so in other words if p is small usually most contacts you're dealing with 10% or something then the denominator isn't terribly important then basically the magneto resistance you get is like the square of the polarization so if you get 10% 10% would give you 1% but then when p gets close to 1 then you have to include the denominator okay so this is the basic idea of the spin valve okay so the one question you could ask then is what does the chemical potential inside look like electrochemical potential and as I said the basic equations people use thinking about this are just like the diffusion equations that I mentioned before so you know we had i equals say something like the conductivity d mu dx I think I guess I was using z this was the kind of thing I had written before probably to be consistent I should put a sigma over q etc but now all you do is for up you put a up there and you would have a similar equation for the down but then you also have to so this would be i up this would be i down and usually when you solve a diffusion equation you say that d dz of this is equal to 0 that's what you normally say that d i u dz I mean if you are just one species you would have just set that to 0 and that's how you usually solve these equations now in this case though that won't be equal because of spin flip scattering if there is something inside that converts up into down then you would have to worry about that on the other hand if there is no spin flip inside your device and of course again ideally when you are trying to make the spin valves you try to make your channel so there is no spin flip I mean as low as possible in that case this is it but if there is spin flip then what happens is the total current stays the same so whatever happens you lose from up you pick up with the down so d i up plus d dz should be equal to negative of d i d dz because up plus down should stay the same and instead of 0 you then have something that would be the difference will depend on mu up minus mu down will be proportional to that and with some constancy so we won't bother about exactly what it is but that would be or I guess I should have a minus sign there so this at what rate is up getting converted down to down well because they have a different mu they have slightly different chemical potentials and so any spin flip processes kind of trying to equalize them that's how usually people think about it now for this discussion for the moment let's say we are ignoring spin flip processes so this is all 0 there is a mu up minus mu down but there is no spin flip current let's say so in that case basically this is it so whatever I wrote here kind of amounts to saying that spin flip processes give you some kind of a conductance between the two channels that's basically what it amounts to if you look at those equations like this you could visualize as sort of a distributed resistor thing you could map it on to that kind of a picture actually okay so let's do the anti-parallel case then so anti-parallel means small r bigger bigger small r and no spin flips here let's say so now if I look at the potential if I look at the up so this is the up or the red channel for that I'd see something like this so there's a potential let's say I'll call this 0 at this end and I'm trying to draw it for the up so little drop here big drop at this end why because low resistance big resistance and when you look at the down it's reversed that's the down so inside there would be a big difference in the chemical potentials of up and down lot more of one than the other okay interestingly you'll note that if I had drawn the if I had looked at the parallel one there wouldn't be such a major difference why because in the parallel case it's like small r small r big r big r so here the potential is about half here also the potential is about half okay there's a big difference in the currents that are being carried but but let's stick to the anti-parallel and the other thing is that I'm kind of assuming that the middle is has a very low resistance because it's interface limited otherwise there would be a slope associated in the middle and a lot of the when I analyze real devices you often have to take that into account that could be a major thing but the best case would be when everything's interface limited not much is dropping inside okay now the next point I want to talk about that's this non-local signal and so what I drew there that's like what's down here see see that red and the blue now the problem I want to talk about a little bit is so supposing this channel were extended out in some direction which is what I've drawn that way so let's say we consider a structure looking like this you know this channel this is like I'm looking down from top this is the conductor overall but I've put my contacts here these are the contacts so that's the magnets that we're talking about and of course current flows right around here maybe a little bit fringes around there but that's it that's where the current flows but this channel let's say extends out and ordinarily you know you wouldn't worry too much about it that's okay it extends out fine but what I want to show is that as far as spin flow is concerned this could have a major effect if you leave something hanging out there this is what I want to explain why so it goes something like this in this structure if you we just calculated the chemical potentials so mu up and mu down around here are separated so it's like mu up somewhere here mu down somewhere here now question is what does it look like if I look out here then what will happen these things are not in equilibrium they have two different chemical potentials they'll want to kind of go out so if I plot the mu and now I'm sort of plotting the mu with that as an axis so the up is somewhere up here down is somewhere here and as you go out in this direction they'll kind of try to approach each other because they will try to come back in equilibrium and to try to get to somewhere in the middle out here if you go more than what you call the spin coherence length the spin flip length whatever length it takes for spins to come together out there you'll gradually find it going down somewhere so you see something curious here though that you say remember what I'm plotting is this is the mu this axis is x and this axis was z I mean we know what it looks like in this direction now what I'm trying to figure out is what it looks like in that direction two dimensions and the point is here they are well separated eventually they'll come together now if you look at the current again no new principles or anything nothing strange about spin or anything we just know that when chemical potentials die out like this it drives up spins this way but then you know this side of course there can be no net current there's no net current upwards of course but if you look at the down you see it's coming back so if you actually just use the diffusion equations as written down again no rocket science whatever we had you just looked at that and you calculated the current you'd find that there was a lots of up spins going out lots of down spins coming back that's it and of course if you calculate the total current it would be zero as you expect nowhere to get out no current should be flowing that way no question but point is there's a spin current though why because you see when we calculate total currents we take the up current the rate at which electrons are flowing in the up direction and you multiply it by q and then you have the down and you multiply it by q same q and so total charge current is like the sum of the two things but when it comes to spin you see one carries positive spin the other carries negative spin you know this red and blue so whatever units you use for spin the point is you'll put a plus one there and a minus one there and so when you look at the total spin current it's like the difference between the two i up minus i down and that there is a significant amount of it right there because you've got ups going out spin downs coming back so there's a lot of spin current going this way so when you look at the spin currents you'll find yeah a lot of it you're losing it and whether you have this thing hanging out or not can make a big difference to how well this thing is working really for example is it possible to have a device with zero charge current and a lot of spin current here at this interface you definitely have that right if you look at what's going out there is no charge current but there is a significant spin current that's definitely true now overall whether you could engineer it so that there's no charge current anywhere that I'm not sure because of course eventually everything is driven by charge voltages we don't quite have a spin battery I mean from outside right finally you usually put a charge voltage but if there's a clever way of doing that yeah that would make it that's true because for a lot of the spin devices what you need is a spin current and it would be good if you could get it straight without a lot of other i squared r losses in this part of it right of course one thing I mentioned is that yeah when you calculate a vi you would probably yeah the way I am thinking here though is because we are using the elastic resistor model there is no dissipation as such if you think of it that way on the other hand if you're thinking of it as a diffusion equation in the way normally people think and you say that resistance is associated with v times i then I think you would say sure there's a v up times i up plus v down times i down right so this would be and that would be what you would call then the dissipation out there and you can always take something like this and write it this way v up plus v down over 2 times v up minus v down I'm sorry i up minus i down plus i up plus i down over 2 times v up minus v down think this is just an algebraic thing you can always take something like this and write it this way and sorry I should have written plus both and minus both and then you can say it's like the charge voltage voltage times charge current and spin voltage times spin current you could say that and then you could say well here you have a spin voltage and you have a spin current so that's the that would give you a dissipation based on that but if you're using the elastic resistor model then as I've said dissipation is finally all in contacts so that's a separate issue yeah the transport of electrons why would that affect that why would this spin or spin propagation affect the transport of electrons how it would affect that right right no no so in this model then what would happen is because you have this spin current going away you effectively put a g here because it is taking up currents and turning it into down currents so as you try to cross this region you're losing some option so in that model that's all the total power dissipation that's something here right here outside oh yeah this is if you looked inside somewhere here so if you're looking from here then all you have is a charge voltage and then you have a charge current so if I apply this view out here then one way to do it is just charge voltage times charge current plus spin voltage times spin current but then there's no spin voltage here but there's lots of spin current of course internally if you do some place then probably you got some of that power internal is dissipated in the spin part of it probably right but not about all this carefully but I think that's what so what use do we use for spin and we also have yeah this is where people do it differently I prefer to do it amperes because then you can compare etcetera lots of times it defined as you multiplied by h bar over 2 what I mean is the idea being that spin is this angular momentum and it's half h bar so the equivalent of q here would be but in that case your spin current wouldn't have amperes as your dimensions it would be more like h is joule second per second so the unit of spin current then would be just energy so you would say you have a spin current of so many electron volts and then you don't quite know how to compare fine or I prefer to just keep it as how many per second yeah so is it a real time I mean why are we not breaking kc as long and then the highest no I would say yeah it's like basically it's electrons that are flowing and then the question is what charge they carry and what spin they carry and it's just that some have positive spin some have negative spin and to some extent it's a lot like the heat current we discussed also if you remember with heat current there was a mu and anybody with energy above it carried e minus mu anybody below it carried negative so it is a little bit like that when you calculate heat currents you find that some of them contribute positively some of them contribute negatively and it's almost like the two spins in that sense right what's above and what's below it little guys yes please how is this spin current when you are trying to measure the spin current is there a way to say this thing between spin and spin down or you just have a value okay let me come to that usually what yeah so what the next topic I'll get into is this measurement issue and there yeah usually you are measuring charge things so yeah so let's I'll talk about that in a minute okay so this is the picture I think I had up there it shows how this thing varies now the next point I wanted to raise is what he said is that can you measure this because this is something that has actually been known for quite some time I mean I think the first experiments may have been like late 80s right Mark Johnson famous paper in those days I think showed this that you can measure a non-local voltage non-local meaning it's a voltage that is outside your current path this idea that your current is flowing one place but you could measure a voltage by putting down a probe somewhere here and the point is question is what do you measure now instinctively and I'll try to do this a little better in a minute instinctively the way you think is what a probe measures is again this electrochemical potential because it tries to it's a high impedance thing doesn't allow any current to flow so it basically goes to whatever electrochemical potential it connects to so when they are equal that's the voltage you'll have so if you had put in just an ordinary contact what it would have measured is the average of the two and as I said I'll make this more quantitative in a minute but if you put a red contact red meaning the one that only sees ups let's say you had a perfect red contact what I mean by that is it doesn't even care about the blues then what it will do is it will measure this potential now if you put a blue contact then of course it would measure that potential because it would come to equilibrium with the blue things and I'll make this quantitative in a minute you'll see no but the contact is voltage probe what that means is when you measure it you put a high impedance voltmeter the way you measure voltage that would be it and if you put a non-magnetic thing another which looks at both red and blue then it will look at the average so people actually put in contacts there and they can flip it and they see indeed the voltage changes and how much it changes that is a very good measure of how much of a spin voltage you have so one way to measure spin voltage is measure the voltage with a red contact measure it with a blue contact and then look at the difference whatever it is now to make this quantitative the way you can think is this that I've got a blue up and I've got a mu down and I'm putting here a contact which sort of connects to it through say some conductance I'll call it G up and connects through some conductance G down and of course this is a voltage probe so I have a high impedance voltmeter doesn't let any current flow so finally it will float to some mu which I'll call the probe it will go to some mu such that there is no net current basically and so the idea is whatever flows here and whatever flows here they should cancel out and that will tell you what the probe will measure and of course if these two G's were the same it would have just measured the average that you can see but you can make this quantitative by saying that well let's write it this way mu up minus mu probe yes I'm wondering if the P may be confusing because I've used P for parallel and all I'll just drop the subscript so by mu I mean whatever that floats to so this times G up because this is like voltage times conductance that's current and then I can write the current in this arm minus mu times G down and say that that must be equal to 0 when I put a probe down so what potential will the probe float to well it's a weighted average it's like G up over G up plus G down times mu up plus G down over mu up plus mu down and then mu down so you basically take a weighted average of the two and the weighting is determined by how well you couple to it which is what is reflected in those conductances so if you are doing actually modeling with any GF or things like that those would be reflected in the gammas you would put in sort of or here we are just talking conductances just thinking in terms of conductor thank you that's it so it's this weighted thing weighted average is it a red magnet or a blue magnet and if it's a normal non-magnetic thing those are equal and then you measure the average so this is what you would expect from common common sense you would have expected this should be half but what this shows is that of course real magnets are never quite ideally you would like say G down to be 0 that would be a perfect red magnet then of course you would have just measured that and if it's a perfect blue magnet you would have just measured that in practice you don't quite have that so what you measure is like the spin voltage times the polarization of your magnet so in that sense if you measure one microvolt the actual spin voltage inside could be 10 microvolts because your polarization could be 10 percent because you are losing a little losing the signal there that's important what is the real goal of the channel in this case it looks like everything is taken by contact itself because in this thing the channel has no particular magnetic properties at all it's just an inert background in that sense and everything it just provides the highway that's all and everything is about how you feed it and look at it it's really all about it yeah so this is the non-local signal that people have measured actually for a long time non-local because as I said current path is here but you measure the voltage out there and often people measure it with one probe around here and another one very far away so that it basically goes to the average anyway now so far then everything I've said it makes it sound like well nothing very profound involved it's like there's an up and a down that's about it and you have to keep track of reds and blues but there's a little more to it and that is what is the part that is kind of mysterious and different about spin that takes getting used to so I'm just wondering how does the non-local signal compare to the normal case when you don't have the magnets when you put the same probe without any magnets there would never be any spin voltage though what I mean by that is right but what I mean by that is if you had no magnets if you didn't have any spin voltage in the first place then no matter what you put here I mean red magnet blue magnet you know I mean red magnet blue magnet they would all measure the same thing no I'm probably not getting the question without any magnets for the source and drain right just normal normal voltages yeah the bigger part I'm not sure because absolute magnitude may not be but that's why when you look at spin voltage it is not enough to just measure one thing it's important to look at a difference right so for example look at this difference or take a red and a blue and look at the difference because as you know voltages can exist outside a current pack anyway I mean you know this van der Pohe measurements you put a current here measure a voltage somewhere out there so right spin voltages do tend to be small so roughly speaking you see what happens is the voltage you get is like p square what I mean by that is the source and drain that created your signal let's say have a polarization of p then the spin voltage inside tends to be about a factor of p down and then when you try to measure it you lose another factor of p so basically that's what and that's why typically you know we get say 10 micro volt for more like millivolts of input you had a question right so this is very different for different channels so semiconductors they say that the great advantage is the long spin coherence length in silicon in an apple bomb in his experiments he claims you know tens of microns is what he has shown in many experiments actually on the other hand in copper I think it would be more like a tenth of a micron or less 0.1 micron or less in copper but metals of course has the other advantages that I mentioned that injection is better demonstrated I would have thought that if there is no magnetic scattering mechanism there would be spin or decoherence but that would be both silicon and copper why is there a decoherence? yeah this part I think that lot of this spin decoherence usually is finally about spin orbit couplings residual spin orbit coupling but in copper probably it is bigger just because there are some magnetic impurities probably in semiconductors you can get it cleaner that's my impression whereas copper you tend to have residual magnetic so the mu up and down they don't change linearly so that means that around the vertical direction the current is changing it's not constant so the spin that we have the spin the spin changes that just has any it's like if you looked at up plus down it is zero everywhere but individually up keeps going down I mean up is getting reduced as you go away in the x direction and down is also reduced both of them and one is just the negative of the other so if you look right at the lower end close to the channel you might find say one milliamp of up spin going out one milliamp coming back you go far away you might find one microamp going out one microamp coming back but overall just cancel no net current of course and the spin current is going down why is there so much focus in silicon because why can't you use another material what are the trade-offs when you use anything again maybe a 3.5 material is there something other than just 10 in the beginning lot of the focus was more on the 3.5s actually I think more of the early experiments well the silicon experiments as far as spin chronicles is more recent and of course the metals copper and all that's what it kind of started the whole thing of GMR all of that was in copper now the relative advantages there are many issues one can talk about which may not be completely clear which ones would be best because remains to be demonstrated so these are things people are right now arguing about which would be the best material and so on and the issues are one is this coherence length how far can spins go and the other is injection efficiency how big can P be that is how you want to inject reds preferentially to blues how well can you do that and that's at different levels in different materials how to explain spin-off and coupling how does the skin use to operate how maybe I'll get into it a little bit and we can talk about yes please I was thinking if you don't really want this so you have a magnet up and down can you still see that then there is no current in the usual situation there is nothing but then there could be there is this equilibrium spin currents under certain conditions that people talk about but which are somewhat restricted and you have to be careful about there of course shouldn't be any dissipation I mean if there is no voltages applied but I don't have a simple one line answer not that yes or no so since there are so many questions people should write down that's true we can please do write it down and here we can continue the discussions actually at 3.30 today let me mention one thing here and that is that when you put down a probe like this I say that the way it works is it makes the net current equal to zero and that's how you figure out what voltage it should float to but you'll notice that the spin current is not zero the charge current is zero so there is actually as far as this magnet is concerned this contact is concerned there is a spin current because you know up spins are going out down spins are going out going this way and there is a spin current and this magnet to actually then turn those spins around and provide the torque to you know get rid of all this and this is where of course if it's a big magnet it does that effortlessly nothing much happens but one of the very important developments in the last 5, 10 years is the spin torque and that is what people found is that if this is a small magnet small magnet meaning something that's like you know a few nanometers thick like a couple of nanometers so not a very thick magnet something relatively thin then with enough spin current the magnet itself can flip you see so it's like the magnet is sitting this way and you're continually injecting spins this way and of course the magnet is having to take them all and turn them around but of course it feels a torque as a result and if it's a small enough thing with big spin currents it will give up it'll say ok you know it's going to be this way essentially and this is something that again can have a major effect and actually is having a major effect because before I said that the magnet stores the information and from GMR you measure you use it to read that information but with this effect you can write the information in the sense previously it was like when you want to write it you bring in magnetic fields big magnetic fields and that's how you write it but when it comes to small magnets it is much better it's much more efficient to be writing it with spin currents because when you put magnetic fields they're kind of like all over the place so if you have another magnet here it is very hard to turn this one without turning that one when you're especially when you're trying to get things small spin currents can be very directed so it's almost like the difference between microwave oven and a conventional oven it's like when you have a big thing to heat up microwaves are not very good but if you have just a cup of coffee to heat up this is perfect hit it right where it is so my belief is as you go to small magnets in general you'll see more and more of situations where you turn magnets using spins really so that's the spin talk that I won't really be talking much more about but it's a very important thing in this field now in the remaining time what I wanted to get across is that spin is a little more than this what I mean by that is it's not just red and blue because if it's just red and blue then as I said no new principles really you just have to keep track of two species that's all and then you know calculate things carefully that's all but the part that and this is okay because all your magnets are in the same direction but you could easily imagine things where you see I inject spins this way but then when I measure it I use a magnet that's not parallel or anti-parallel but somewhere in between what will I get now that requires getting in this thing much deeper you have to understand spin better than that better than what I have just described in other words supposing the same measurement discussed how you measure the red what the red magnet would measure and what the blue magnet would measure but what if you had a magnet that was sitting somewhere in between it is not quite red not quite blue you see what would it be and this is where say so what we have seen so far is if you look at the voltage signal what you measure probe then one case is the parallel parallel meaning that the magnet you are using is in the same direction as the majority spin you have injected so you have lots of red spins and you are measuring with a red magnet and if you measure with a blue magnet you will get a lower chemical you will get a lower voltage reading so if you have a red measuring majority red spins you will get bigger signal so this is the red magnet measuring things this is the blue magnet measuring things and you get something else and the question is so this axis I could label as theta and antiparallel means like 180 degrees and what we have seen is that here you have a maximum here you have a minimum and the thing is in between it goes smoothly basically so if you are somewhere here 90 degrees you measure half now question is yeah when you think about it the nearest analogy I know of is to the polarization of light electrons is like polarization of light but there is this very important difference the thing you learn somewhere in first year physics is that if you had a polarizer so photons are polarized in this direction and you have an analyzer they are parallel you get lots of light getting through and you can block it by putting it at 90 degrees that's it and you can describe the signal you will get what I have written there for photons that's cosine squared theta the angle is theta is cosine squared theta electrons also is similar but it's not cosine squared theta it's cosine squared half theta and that is related to this half integer spin etc but the point is you realize think of it as a vector because with vectors you would have you know orthogonal would have orthogonal would have meant 90 degrees it's sometimes people say it's like the square root of a vector but the thing is there is this whole algebra of spinors that you have to get used to to kind of see follow the literature on those lines you know where but the bottom line is though that it is cosine squared theta over 2 and this is a basic difference see as I said if it were really like vectors then two perpendicular magnets should have given you a minimum but the point is it's the it has to be 180 degrees to give you a minimum the antiparallel thing you see and so when you are trying to understand this then how do you include this in your models now again if you are using any gf it's actually as I mentioned before the great advantage is how little you have to understand and as I said as far as you are trying to calculate this with any gf the calculations are fairly straightforward in the sense you don't need any new equations now whatever I showed you yesterday is all those same equations it's just that those matrices are now twice as big that's about it so what that means is if you had say though let's say you had a device with four points in it those four black dots then your matrices the Hamiltonian matrix the self energy matrix everything you write down those things are would be 4 by 4 well when you include spin it will be 8 by 8 because every point will now have an up component and a down component with the up component and a down component that's the red and the blue so instead of having matrices that are n by n your matrices would be 2n by 2n and any gf as I said if you just do that know what you are doing you should be able to calculate things but as I think Mark said I think the quote he put up was that it's nice to know that any gf understands spin but of course we would like to understand it too so when you look in there in the it's just that when we are thinking about it of course it is convenient always when you think of the spin we think of it as a factor of pointing in some direction and it takes some time and I guess in these lectures I won't be able to go into it but I have other lectures on nano hub where we talk about the spin how you deal with spinors and things like that but here the main point I wanted to make is I think yesterday also when I was talking about the negf I had mentioned that as you know the wave function has these two components but in negf what you look at is psi psi dagger that is dagger means this conjugate transpose and as I mentioned that it's like this thing multiplied by that so this is like up and down when you up and down star and down star so when you take it together it's like up up star down down star and then you have up down star then down up star so g n if you just had two points would look something like this and you know one of the questions that had come up then was like what's these off diagonal terms do I really need them and of course what is here tells you the number of up spins what's here tells you the number of down spins and these things kind of have information about the other components this is the right way to think about it is g n it's like what I've written on top the top corner is the number of up spins which you could view as the total number of electrons plus the z component let me explain what I mean by that that you know you have let's say you have n up and n down then the total number of electrons is the sum of the two and the z component of the spin is like n up minus n down so when you add the two it's n and so if you have a matrix where this is n up and this is n down then of course this one is like n plus s and this one is like n minus s you know with a divided by two other than that so in that sense when you look at the top component you could say well that's like the total number of electrons plus this s c and the other one you could think of it as the difference or you could say well that's the up that's the down but it's the off diagonal components that tell you about the x component of the spin and the y component what you have in the other directions that's the information in there and do I really need that information well for example if when trying to measure it I put a probe in the you know as long as I have a red magnet and a blue magnet measuring things I really need only the diagonal information but if I want to figure out what will be measured by the middle one then actually I'll need the off diagonal information to do anything and broadly this is true not just of spin because spin is a nice example because it's just two by two you know this whole thing in general about quantum transport what is the use of all those off diagonal terms is just that with different types of probes you could actually be measuring things that depend on those off diagonal things actually and lots of times it doesn't make a difference because of all the phase breaking and that we discussed earlier but interference effects all of that is often in those off diagonal things relatively so in the spin case you see immediately the meaning of that now the way any GF works is that you know previously when I said what what do you mean by the chemical potential that does the probe measure and we found that there was something here like G up times mu up and then there was G down times mu down and when you use and divided by G up plus G down and what will happen in a quantum transport model usually is these things would become matrices and instead of this you would have something like trace of gamma dn something like this instead of this which of course if these are diagonal would amount to that but if these are not diagonal then you would have all this information in it but this is where of course the way the quantum transport models work then the information is in matrices of that type you see we have got a 2 by 2 matrix and the other hand when I am visualizing it I usually prefer to think of it almost as just of 4 numbers what I mean by that is at any point you know I am used to thinking of electron density ordinarily in the case of spin transport I would say at any point what I would like to know is what is the electron density what is the spin density in the z direction x and y so basically I need to know 4 things I need to know the electron density and I need to know what direction the spin points and how much it is that is it and if you had like normal devices you know up and down are all equal amounts and all that in that case these would all be 0 there is no spin density at all normal devices is just only electron density and then how much of this you have depends on how polarized what is the polarization of your contacts what you have been able to inject and things like that and then what you measure as I said it is like taking a trace of these 2 things but in terms of this vector the way you can think is you see you know I said that for photons it is cosine square theta for electrons is cosine square half theta the cosine square half theta you could write as half of 1 plus cosine theta and so you could think of it as half of 1 plus the angle between what you are measuring say the p the polarizer or whatever you are trying to measure and the dot product with the analyzer you could write it that way and then you see when these 2 are antiparallel it is 1 minus 1 0 and these are parallel and so the way I think of it is yes I have got this and then what I measure depends on of course there is a charge voltage and then there is a spin voltage the spin part and the spin part of the analyzer that is how you can think about it but as I said as far as any GF is concerned this is how it works and you don't really need this as long as you are doing collinear spin products when everything is in one direction just call that Z and you are done of course for homework problems you could call that X and see whether you get the same answer because you could have called that any direction but in terms of thinking just call that but you need all this whenever it is non-collinear spin products where multiple directions are involved and how can multiple directions be involved well if you had a magnet that was neither parallel or antiparallel to this but in some other direction or if you had some internal things like a magnetic field which actually rotate your spins as it go along so you injected it this way but by the time it got there it is in some other direction that is this handly effect that people have observed in many devices that with a magnetic field you can rotate the spin or this Rashba based on this Rashba field that is the spin orbit coupling that is what is now established in this high spin orbit materials like indium arsenide and so on if you put an electric field in this direction and an electron is moving in this direction it effectively sees a magnetic field in this direction and this is a relativistic effect this spin orbit interaction basically is a relativistic correction to the Schrodinger equation usually relativity doesn't play much of a role in most things but this is where people say well because these nuclear fields are enormous actually there is a relativistic effect and this has been known for a long time in all materials and the bigger the atom the larger it is so usually in silicon it is relatively small compared to gallium arsenide it is relatively even smaller but indium arsenide is a material where with a gate field people have found that is a relatively sizable effect and which people have measured the effects of this on transport and what it does is effectively have a magnetic field this way so what that means is you inject a spin like this but because of the spin orbit coupling by the time it comes here it could be in that direction then what you measure of course will depend on the angle between this and that and that of course you wouldn't get out of a simple up down diffusion model you need something else so it's important to be clear on where you need this part and where you don't okay let me stop here and continue