 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 as you know the next three days I'll be giving a series of lectures. There will be about six of them and the outline you can see broadly. And we have a guest lecture by Professor Lundstrom which will be the next one for the Nano transistor. And I'll be doing it more or less the old fashioned way which is like it will be a blackboard lecture. So hopefully the pace will be you know comfortable and at the end of the day all three days. Actually we have a discussion session. It's not really not a lab session or anything because as you know there are no codes as such that we are trying to distribute. There's many codes that are already available on the Nano Hub. Many of you are familiar with it anyway. So that's not the purpose. Those sessions are really meant more as discussion sessions. And it would be helpful if you have some questions ahead of time that you either email or to Mr. Ganguly. I guess just introduce him or you can hand it over to him. Just you know if you want to hand write it on a piece of paper and hand it over to him. That's fine too if you have your own piece of paper or. And that will help me will look at the questions and try to structure the discussion a little bit. I mean not that you cannot ask questions then itself. That's fine too it will just help structure the thing a little. So that's the general purpose and so let me get started then. So as you know at the heart of this field of nano electronics or microelectronics is this device called this field effect transistor and which is basically a resistor. That is you have this channel with two contacts you call the source and the drain. And if you apply a voltage you get a current. And of course what makes it a transistor is that you can actually control the resistance through this third terminal called the gate. So that VG the gate voltage there shouldn't be any ideally there shouldn't be any current flowing through that terminal through the insulator. Although as it turns out as the insulators get thinner there is some current but ideally there shouldn't be any current. What that voltage should be doing is simply controlling the resistance. Now by the way in these set of lectures I really will try not to assume any specific background you know other than differential equations and matrices that's about it. So if you have any questions if you are not familiar with certain things then please do feel to raise them as you go along. I'm not really trying to not really assuming you know a lot about any of this necessarily. Now this relatively simple device and we'll talk about how it works and of course Professor Lundstrom's talk will essentially it will be about this nano transistor and how it works detail but this device is really at the heart of in all the wonderful things that computers do that every computer we have has about a billion of this in it. In fact the other day I was discussing with my colleague Professor Alam who was telling me that well you know actually in this world there's more transistors than there are ants because if you just look up on Google apparently they say there's about a million ants per human being for every human being that's about a million ants well but then if you look at the number of transistors per human being that's probably pretty no I have about a billion here and that's same with everyone else too so it's really a one of the I mean it's very numerous there's more of these than anything else you can think of if you think about it and now how did people get to pack a billion of these transistors into something that small and the reason of course is that over the years every transistor has gotten smaller and smaller and this picture then I guess one of the things I wanted to get clear to everyone is the length scales that in a millimeter is something we are all familiar with and if you go down by a factor of thousand that's the micrometer and another factor of thousand that's a nanometer and over the years what has happened is this transistor has gotten smaller and smaller and that is how people are able to fit so many of them into such a small space and you can do a little simple math actually that let's take something that's three centimeters by three centimeters now if each transistor can be fitted within say one micron by one micron then the number of transistors you could fit this way would be three times ten to the fourth and the number you'd go this way be three times ten to the fourth and so if you multiply that that comes out as nine times ten to the eight which is about a billion so that roughly gives you the scale of things I mean that's how each transistor is getting fitted within this one micron by one micron and that includes not just the channel which actually is a whole lot smaller but the contacts to in everything together that's about what it is okay and so today the commercial transistors if you look at the length of the channel that's really down to say tens of nanometers you know 50 nanometers which if you think about it is again a few hundred atoms that's a commercial transistors and if you look at research what people do in the research laboratories that's actually enough because they're measured even a hydrogen molecule they've measured the resistance of something as small as that see and this is all that has happened in the last 20 or 30 years and so back when I was a graduate student and that's like 20 30 years ago you know one of the questions that we often used to talk about is what's the resistance of something that's really small because you know the way you learn about resistance is ohm's law right you learn that R is equal to rho L over A and of course inverse of that you write conductance sigma A over L learn about this conductivity and resistivity and based on that you'd say as you keep making the length smaller and smaller of course the resistance should just get smaller now question is does that really hold on to atomic dimensions well no one really expects that because once you get smaller and smaller you get into a whole different regime of transport because with big devices you have what to call this diffusive transport that is electrons come in here and then they move around in random directions like this and kind of on the whole drift towards the drain that's what happens in big transistors whereas when you get down to small transistors you have more like what you call ballistic transport ballistic meaning like a bullet that it that's it and in today's devices it's somewhere in between yes number is pretty close to ballistic and so the whole nature of transports changing and one of the questions people used to ask is well what would happen to the resistance as you really cut down the length and back in the 80s of course the answers were not at all clear you know this made a very nice topic of discussion and it is almost like the weather you don't really know the answers something like when talk about it for endlessly but what happened over the last 20 years is over in the 80s and 90s as people actually started making these measurements by late 80s people had actually measured the resistance of ballistic conductors and the results now are actually pretty clear you see and although the results are clear the point I wanted to make is it hasn't yet broadly influenced the way we think about devices conduction and all that that just for historical reasons we still think the old way which is we first learn about big things and then we try to project that understanding down to small things that's just because of history of the subject you see but and of course it made a lot of sense 30 years ago because it's only at that end anyone knew the answers so you start where you know the answers and this and no one was sure of anything but now that we understand it more or less all the way across I'd say it makes more sense or at least it makes sense to complement that view with a different view the one that kind of starts from the bottom up and that's what this electronics from bottom up is about is the approach to try to think of conduction from this other end and as you know that's how you usually if you had a choice that's what you do normally see I mean how do you learn quantum mechanics you start from hydrogen atom and then you work up to big things you don't first learn about solids and then try to figure out the hydrogen atom I mean that would make the hydrogen atom very confusing and one of the points I wanted to get across is that of course when it comes to small things it's very confusing if you try to take your understanding from big things and try to project it down and so it's better to start at this end but actually even about big conductors you can get a lot of insight if you start from small things and see how it and try to go upwards and that's kind of the flavor of what we want to talk about in the next few lectures and as I saw from the outline I want to cover a quite a wide variety of topics see okay now one result that I'll try to get across in the next few minutes actually is that and I said that you know about this ohm's law and the question was what happens when the length gets really small and the answer actually is what's now believed is that you could just do this in other words it's as if when the length goes to zero that is some constant there so it the resistance doesn't really goes to zero but goes to some value which is kind of what you would expect if you think of that well you know there is a resistor and then I've got some contact resistance all this stuff that's connecting to it and you might think that well that's kind of the contact resistance what wasn't realized though before is that this contact resistance has a fundamental significance it has a special meaning and all that and is for small conductors it happens to be quantized as well so those are the issues that we'll be talking about anyway so this is then what I'd like to now the first step in understanding conduction always say is to draw as you know usually draw something they call a band diagram here it's kind of like that but what we have here is this axis is the energy axis this is the density of states so at different energies you have states available for conduction now if you have learned if you've taken any courses on devices often you start with something called a E.K. relation and later on I'll talk a little more about that but this concept is actually much more general in the sense that as you know E.K. relations only apply to periodic solids which where we have crystalline structure and all that whereas even it doesn't matter even if you're talking about a hydrogen atom or you're talking about an amorphous conductor anything anything there is a you can define a density of states that is at a given energy what states are available so this axis is energy this axis is the density of states and usually in many of these conductors in semiconductors there's a gap there's certain ranges where there are no states available and some ranges where the states are available so this is one of the first things you need to draw and you could treat that almost as an experimental input that is how do I know where these states are well this is what people any with new material that's the first thing they'll try to measure how do they measure these states well one of the common experiments is photoemission that is you hit it with light and you see what energy it takes to knock an electron out of the solid so usually you think of the vacuum level as being somewhere here that's usually about seven to eight volts down you hit it with four four tons with that energy and you can knock it out and by looking at what energy it takes you can actually map out this density of states so there's all kinds of experiments people have done over the years to map out this picture of what the states available are and then you have these contacts which are big regions and where we assume that let's say for starters at least that the density of states is more or less uniform lots of states available and the second concept that we need here is that of this Fermi function that is the question you ask is that these big states how are they occupied with electrons and the idea is that if you do at a low temperature then what you'd expect is everything up to some energy would be occupied and that's what's called the electrochemical potential of the Fermi energy that's the new one now if you had to draw this Fermi function then so this axis is energy this is the Fermi function everything above this energy would be empty so the Fermi function should be zero everything below this energy should be full and so that's one now that's at zero temperature when you raise the temperature it's spread out somewhat as you might expect and the spreading is over an energy scale of KT and KT at room temperature is about 25 millivolts so that's this Fermi function that I've written down that's the function that I've tried to draw here if you plot that this is energy that's what you get and when you apply a voltage across this structure any time there's a positive voltage it lowers all the energy levels positive makes it easier for electrons to get in there we are drawing electron energies so everything is lowered including the positive this Fermi energy or the electrochemical potential so everything up to here is filled so this mu one is different from mu two and that's what sort of makes it a non equilibrium problem that is at equilibrium there is a common electrochemical potential just like there's a common temperature for example whereas if you have two different temperatures then you can have heat flow similarly when you have two different chemical potentials you can have flow of electrons and so this is a non equilibrium problem and that what makes it non equilibrium of course is that voltage and because of the voltage you have these two different electrochemical potentials and you can draw the Fermi function here it looks something like this so what we want to write down is an expression for the conductance of this structure we like or the current through this structure now the first question you can ask is well why does current flow through this and this is where again if you look at the standard description people would say well current flows because there's an electric field and that I've always found confusing because of the following and that is that if it is an electric field you know you said that well you have applied a voltage so there's an electric field here and that's why electrons start flowing now if it's the electric field that drives the current then of course all these electrons should start moving including these which is not what happens at all so at that point then people tell you well you know of course a filled band cannot conduct and so nothing no current flows here and that's that kind of gives you the feeling that something very mysterious is happening which you don't quite understand and that is why current only flows up here and not down here but the point I want to make is that when you think about a small device it's actually pretty clear why the current flows only out here and not down here and the argument is very simple it's like this we have these states here's the electrochemical potential so what this contact would like to do is fill up all the states up to here because it likes to make bring this channel into equilibrium with itself equilibrium means you want to get the same electrochemical potential so it's really trying to fill up all these states and the other hand this contact of course has an electrochemical potential down here so it would really like to keep all these states empty and so what happens is this contact keeps filling them up and this contact keeps pulling them out and of course once it's out there then it flows out of this contact and goes back to the battery and the one that was left behind a new one comes in and that's why of course current keeps flowing forever that's basically it and from this point of view of course you immediately see why these electrons don't conduct it's nothing mysterious about a filled band doesn't conduct or anything it's just very simple it's this if you look at these states this contact wants to keep it filled why because you know the chemical potentials up here these states are all down here they just keep it filled that contact also wants to keep it filled you know because it's electrochemical potential is here well fine it just stays filled that's it so these are all filled but no current flows everyone's happy done so this is exactly why I feel I mean these don't conduct what conducts it up here and this is a very important point that when you're trying to understand conduction you don't really need to know necessarily everything about the density of states at all energies because as you know when you're looking at this density of states when I draw this picture here you're really looking at the tip of the iceberg in the sense that there are core electrons down here that if there's one S states and two S states two P states all kinds of energies way down there and you're just looking at this tip the valence electrons on top and the point is as far as current flow is concerned that's all you need to know all you need to know is about these states out here anything down here is relatively relevant unless you put a voltage big enough to start making it flow once you put a bigger voltage and this comes down here yes this will start conduct and that's what happens in graphene for example where you have states that are close by and with a reasonable voltage you might actually start conducting through those sure okay now given this picture then how do you write down the current and this is where it's a well we could do it this way first let's think of what the current would be if you just had one level here let's say we're a device with just one level okay so then though you could say that well the current would be like if an electron takes a time t to transfer from one contact to the other so the way you think about it is you have this entire reservoir full of electrons trying to get and there's this one little level and it has to squeeze through that and so every once in a while there's a time t typical time t that tells you how long it takes for an electron to get from here to there and in that case you would write your current as q over t that's the rate at which you get through but then you should multiply it by f1 minus f2 why well f1 sort of tells you that's this Fermi function that tells you whether electrons are available with that energy or not because if you're talking of things down here there's lots of electrons available trying to get through but then if you're talking of something up here there's no electrons there so that's this f1 that tells you rate at which things go this way and f2 tells you you'd have reverse flow once that are trying to get back this way and so it's f1 minus f2 and that has this essential physics that is why does current flow well because you have got these two contacts with two different Fermi functions two different agendas one trying to fill it up one trying to empty it etc all that physics is right here f1 minus f2 so that's what you get if you had one level now what we want to deal with is some general density of states okay so you look in a certain energy range D e how many states do I have in there well alright so that would be the current through that and I'll explain this in a minute I'll put a 2 there and explain that and then if you integrate this it will be that will be the current now why did I divide by 2 well in general and we'll talk more about this as we go on that when you look at these states I tend to think of these as you know like a highway that's connecting these two ends the thing is on that highway it's like there's some that are north bound lanes some that are southbound lanes and for going from one side to the other you have only half the lanes available basically because half go from left to right the other half are kind of going from right to left so that's how I justify this so this is it so the expression so what I'm what I tried to get to you get here is that first expression up there see so let me write that right so let me just write this up here now from this you can obtain an expression for the conductance the way you do it is that for small voltages what you can do is write this f1 minus f2 as dfde times the chemical potential difference mu1 minus mu2 now how did I do this this is what you could call sort of a Taylor series expansion the idea being that well you see f1 is this Fermi function with some mu1 so f1 looks like 1 divided by e minus mu1 over kt plus 1 and f2 is 1 divided by e to the power e minus mu2 over kt plus 1 so it's the same function but with a slightly different mu so if you are talking of a small voltage so mu1 and mu2 are different but by a small fraction of kt if it's a small voltage then you can write this difference f1 minus f2 as del f del mu that is how much does the function f change with mu and I'm using the equilibrium value just take that derivative and multiply it with mu1 minus mu2 so this would be like the first term in your Taylor series and then the point is that because of this nature of the first function it depends on e minus mu so the derivative of f with respect to mu is the same as the derivative of f with respect to e except with a minus sign that's all and so you get that this is it now so this mu1 minus mu2 that's what you can write as q times v that's the applied voltage because mu has the dimensions of energy it's this electrochemical potential and when I apply a voltage the amount by which the energy levels move up and down is q times v and so if v is 1 volt it changes by 1.6 times 10 to the minus 19th joules except that usually when we talk of energy energy we don't use m ks we just use electron volts I mean don't use joules so if you put 1 volt the energy will go down by 1 electron volt so that's this qv so if you put that in there is the integral d e minus del f del e and then q square d over 2t times v that's i and I can take the v out from here put it here so that's the conductance this is it now this quantity here it represents kind of an averaging and so you know it often I find it convenient to basically say well conductance is this and this is something that depends on energy and what you should do is if you actually want the conductance you should average it over energy according to that function because if you look at del f del e what does it look like say you see the Fermi function looks like this if you look at its derivative you see it's zero way up here then right around here there's a big change and then again it's zero so that function del f del e it has a peak right around here and that's what I tried to plot up there so what's on the right-hand side that's the Fermi function what's on the left-hand side that's like the derivative of that you just plot that out you'll see this function and the important thing to note is that this function firstly has a peak right here and has a width and the width is of the order of kt I mean around 4 kt or so the width because the energy range over which things change that's a few kt and the peak value is actually about 1 over 4 kt and the point is that the area under this curve is actually 1 so if this were really sharp you could think of it as a delta function you know with area 1 but basic point is that over a range of kt when you do something like this what you're doing is taking this function and averaging it over a range of the order of kt that's about it so you don't have to keep carrying this around you know this makes it look more complicated than physical it is physical it's really just that quantity averaged that's how you should think about now how do I show that the area is one well you know if you write down that function you'll see you'll have to take this take its derivative and all that but actually the fact that the area is one actually is very easy to show because you see it's integral da minus del f 0 del e so that's the function we're talking about and I'm integrating it from one end to the other so that's equal to of course f 0 one end to the other and of course one end it is one the other end it's zero so that's basically one then there's a minus sign that's it so the area is one that's very easy to see that's in a minute okay so the important concepts I wanted to make sure everyone's comfortable with first is this density of states what are they how many channels how many lanes do I have available for conduction that's of course central and in order to conduct what you really need is channels right around your chemical potential why because conduction is determined by the difference between the two Fermi functions for low voltages that basically amounts to looking at this narrow range of energies right around Katie because this tells you this is the conductance at a given energy but then actual conductance you get is averaged according to that that's how I think about it okay so that's the way that result is written up there that you have integral de let me put a 1 over q here so you make this q square so then this quantity is what I call the conductance as a function of energy g of e so this is the quantity that I wrote as g of e and then the idea is that any time you want the actual conductance you should average over df d in average according to this and so and what is the expression for this conductance that's what we just obtained which is that it's equal to q square d over 2t write this anymore so that's where we have come so far so what I'll do then in the next few minutes is we'll try to obtain another expression will go from here and get a expression that will get us to this idea that you know I said that when you get to small devices instead of the usual ohm's law rho l over a you actually get something like rho l plus lambda over a and that's the result I want to get and what I'll do is I'll start from here and we'll get there now when we want to apply this to something relatively big one of the things you expect is that this density of states would be proportional to the volume of the region you are considering why because if you think of stay in if you think of a region with say 10 states in it and another one another 10 and together this number of states would be like 20 in general you know we'll get into this thing about how you calculate the density of states how you model the density of state but the one basic point is that when you get down to something small of course there is no simple rule but once you get to big things usually the density of states is proportional to the volume so you take something that big and you put two of them you'll have twice as many states overall so D should be proportional to the area times the length so so this quantity here should be proportional to the volume of the solid and so and the question we are asking is how does the conductance or the resistance because this is conductance I could go resistance and write it the other way 2d over d and the thing is this is proportional to a times L now this would just tell you that the resistance should go down as a l but what we need to consider is how the transfer time changes the time it takes for an electron to get from left to right how does that depend on this area and length that is if you are considering a conductor whose length is l and whose cross section is this area a cross section or if it's a two dimensional thing you could just think of the width or if it's a three dimensional thing and the cross sectional area a that's this so what we need to talk about is what time does it take for an electron to get from left to right and this is where you'd say well you know if it's a ballistic conduction if electrons go straight through then the time is pretty clear it should be should just be t equals l over v right whatever velocity it is it has and it's kind of true if you want to be a little more accurate I'd say it is the velocity in the z direction z meaning I'll call this direction the z direction that's the direction of current flow and in general you have electrons going at various angles and of course what helps it get through is just the z component so you might say this and this since is different for different electrons you could say I'll just use the average bit now in this limit then you can see that t would be proportional to l and then your resistance would go as one over area because the length would cancel out and that's exactly what you see for ballistic conductors you see in ballistic conductors you know people have now made measurements on carbon nanotubes for example which are all which even for a fraction of a micron is ballistic and then you make it twice as long the resistance doesn't change really once you're in that limit so it's independent of length unlike ohm's law which tells you you make something twice resistance should be twice but actually in this ballistic regime it is independent of length but then you make the area bigger the resistance just does go down so it is inversely proportional to the area but independent of length and that comes out very nicely here now what happens in diffusive conductors is that as you know this diffusive motion means that electrons don't go straight through but instead they kind of have this random walk what you say is sort of like a drunken person trying to get through and you know he takes a few steps in one direction then another and so on and it of course takes him a whole lot longer now to get through it's not just L over V and this is this random walk problem that has been analyzed many different ways for the moment I'll just take the answer and if this you'd like to discuss this further we can talk in the discussion sessions about it but the basic result is that instead of being L over V it is more like L squared divided by two times something that people call the diffusion coefficient so it is Vz squared tau where tau is this mean free time that is the length of time that an electron that a length of time that an electron travels before it gets scattered before it turns around that's a mean free time and this is the quantity that you call the diffusion coefficient and the time it takes then is L squared divided by twice the diffusion coefficient this if you have not seen it before we can discuss further but let's go with this now the point is that once you accept this one then you can kind of see where ohm's law comes from because you see the density of states proportional to area times length this one's proportional to L square and then you can see you'll get L over a just like ohm's law and in general then we could say that the time it takes to cross a region has two parts to it for one that is proportional to length one that's proportional to L square again you can think of that as a Taylor series expansion also if you like that you know this time is proportional to I mean it depends on length the first term is proportional to L than this L square and when you go to big lengths of course this one will take over when you go to small lengths this one will not be one way to interpolate so if you write that then you see from here you can get 2 over q square d and then times the time and that's this so for the time I put in these two things that's it yes please Is there any way to attach to the two different terms in time because it's like those two terms are by two different mechanisms right? So the way I was viewing it is I guess the question was that these sound like two different mechanisms so very good question that is there a way to should there be a kind of waiting to this one and to that one I'd say the best way I justify this is by saying that let's say we write t as these two different lengths and then we get this from the long length limit and you get this one from the short length limit and then I say that this is the only combination that will work at both limits and hence I'm using it because anything else you put here or here would mean that it won't work in the either the long limit or the short limit that would be my justification now one could ask whether you know should there could there be an l cube term or so on and this is why I'd say this random walk problem has been analyzed very carefully in all over in many places many different ways and I think what they find is basically this okay so based on this then you'd write the resistance like this and now I think I can get this form that I had mentioned earlier that is let me just write the resistance as I think all I did was kind of took this out and put it here so I had a l over something plus l square over something I took l over this put it outside and then inside you get something like this and this is the quantity then you could we'll say we'll call the mean free path now roughly speaking you can see what it is vz square tau divided by vz so that's like vz times tau roughly speaking and that's basically what you think should be the mean free path the tau is the mean free time how far does it get in that time or its velocity times that time so you'll roughly see that now the part that is the needs more discussion and we can go into in the discussion section is what these averages give you because when you take those averages you get certain numerical factors that go with it so for example one of the things useful to know is that the average of vz square is actually equal to the magnitude of v square so we are assuming that let's say you have electrons with a certain velocity magnitude but any angle so this is the magnitude which is a fixed number and when you average over the angle you'll get this divided by the number of dimensions so in one dimension is just v square in two dimensions it will be half and three dimensions it will be one-third now the average of vz that one takes a little more discussion this one the way you can see why it is number of dimensions is something very simple let's say we have two dimensions then of course just you'd expect since everything is isotropic that v y square is equal to vz square and so if you look at v square that should be equal to like the sum of the two which should be equal to two times vz square hence vz square is half of v square so in two dimensions this would be in three dimensions you add the x to that story so you can actually see very quickly why the average of vz square is like half or one-third or one of the v square that's it whereas the average of vz it's a little takes a little more work but we can get into that if you like in the discussion session okay so this would then be the expression we have this is l plus lambda that's this mean free path and what's in front then is what we could call the resistivity this is then what you could call the resistivity row which is and so the conductivity oh actually I guess this I should call row over a so if I want resistivity I should really put an area and this then makes sense density of states per unit volume that's a number that's independent of we said the density of states should be proportional to the volume so when you divide it this would be density of states per unit volume so it's conductivity would be q square as density of states per unit volume times vz square in the simple model know oh the question was whether the tau depends on length usually and the picture we have here is that an electron goes for a length of time tau and then scatters and then scatters again and that should not that particular time should not depend on how long it has to go or not that's how you picture now if it's a very short device that's when of course it doesn't even have time to scatter but all that I think we have included by writing the total transfer time as l over vz plus l square over so the fact that when you get to really short lengths it doesn't depend on tau at all that physics is already included here and the time tau really comes in when you have long things and in long things you know how long it goes before it scatters that should not depend on how long it is that is that would be my argument so this expression we have here then for the conductive that's actually a standard expression you see this is conductivities just that it's not very familiar not many people not many people are familiar with this q square times the density of states per unit volume times the diffusion coefficient right and the reason is that usually it's not derived quite this simple you see the you the reason we could do it so simple is because we are talking about this very small resistor where we assume that electrons just go through this without changing energy what I assumed is electrons can scatter inside but it doesn't change energies and is right through and that is sort of what allowed me you know to say that well they have these two Fermi functions at the two ends and will calculate the resistance and allowed me to do this relatively simply whereas usually these are derived in much more with much more advanced approaches you know either going to the Boltzmann formalism or using cubo formalism but the final result is this that's the conductivities and now you take a standard textbook like say solid state physics like Ashcroft and Merman that formula is there it's just that it's in chapter 13 so that you often don't get that far and you don't often carry this in your head that's all like when you're thinking of things that's not how you're normally thinking you have some other you know like root formula other things that I'll relate to that you carry in your head usually so the result itself though is very yeah and the important thing is that when you think about it it's like this is something that depends on energy and if you actually want the conductivity you should average it using that Fermi function that D F D what I call the thermal broadening function the derivative that's the important thing so you have different things at this energy this energy etc and this is average that matters okay now one of the important results though that is not as widely appreciated which came out of this is that the resistance is not rho L over a but rho L plus lambda over a that we got by including this term because usually when you are analyzing these things you assume it's all in the diffusive limit but by putting that in you get this extra L plus lambda and and that also gets you this and in terms of conductivity what it means is that actual conductance is like sigma A over L plus lambda and resistance is rho L plus lambda that's the new part of it right now what it tells you if you think about it is almost like what you're saying is well if you made this shorter and shorter as it tends to zero it kind of behaves as if there is a length it has a length lambda that's this mean free path and after that when you get it really short it still looks like a conductor whose length is about a mean free path now that statement kind of bothers you a little bit because you see if you think about it you've got this something very short where there is no scattering so why should its mean free path be so important you know physically what does it mean to talk about the mean free path in a device that's really short and the point I want to make next is that you see that quantity if you look at the ballistic limit so if you take this R let's say we write it this way rho lambda over A times 1 plus L over lambda see it's exactly what I had here but I pulled the lambda out so this then is the resistance of something that is ballistic if your L goes to 0 that's this quantity and you might say well you know in a ballistic conductor what does it mean to talk about a mean free path anyway why should that matter and the answer is it really doesn't matter at all actually that row and the lambda is because of the way we did this this has no particular meaning in a ballistic conductor neither does that but the product actually has a meaning because you see if you look at our expressions that's row and that's lambda and if you multiply those two things you'll notice that tau just cancels out so mean free time is nowhere in the picture it's just because the way we got there it looks like there's a row and there's a lambda but none of them individually have any meaning so if you look at row lambda what is it if you're just multiplying those two things right so let's write that here so let's try to write row times lambda so lambda is whatever's up there so that's row lambda and then if I divide by a put that here so you'll notice what you get that's it no towels anywhere the mean free times out as you might expect so this quantity then as I say it's the ballistic resistance of a conductor and it's got nothing to know has no mean free times or anything in it and of course one of the important things that came out of the experiments in the late 80s and early 90s is that when they looked at small conductor they found that this conductance actually comes in integer multiples of this q squared over h so actually I'm writing resistance here so it'll be like h over q squared m and this this as you notice is a fundamental constant is this Planck's constant this is the charge on an electron squared and if you put in those numbers that comes to about 25 kilo ohms if you put in the Planck's constant put in the charge in an electron this would be about 25 kilo ohms 25.9 or so and in all the theory of small conductors of course usually this plays a very important role because what was discovered is that in these ballistic conductors the conductance that you the resistance that you get is like 25 kilo ohms or 25 kilo ohms divided by 2 25 kilo ohms divided by 4 etc. It comes in integer and the way you now think about it is that if you had just one channel then it would have been 25 kilo ohms so in a way this m is a measure of the number of channels in your conductor so any conductor you can think of as lots of channels and this sort of tells you the number of channels and in big conductors what happens is that's what becomes proportional to the area when you get down to small conductors of course it's not necessarily proportional you can say well there's two of them or ten of them twelve of them fifty sixteen of them etc but when you go to big conductors it gets proportional so so what is the resistivity of say copper from this point of view the way I would do it is you see if you wanted the resistivity of copper I'd say well it's like h over q squared m and then times guess this is what was that I guess we had rho lambda over a that's equal to h over q squared m so if you want rho you should do h over q squared m area and then 1 over lambda I suppose so from this point of view okay what is h over q square well that's this 25 kilo ohms how many channels do we have per unit area well in metals it's approximately equal to the number of atoms it's almost as if every atom in your cross-section gives you a channel in semi conductors it's a whole lot less you get like one channel for every like 100 nanometer by 100 nanometer or 10 nanometer by 10 nanometer but in a metal it's like every atom is giving you a channel actually and so for copper you could say well if I had say one channel for every one nanometer by one nanometer then the number of modes per unit area would be like 10 to the 18th per square meter for example so you'd have put like 10 to the power minus 18 meter square and then the mean free path in copper is about 50 nanometers or so I mean just so it cancels out maybe I'll put 25 nanometers and you can see what you get you'll get you'll get about the right order of magnitude oh meters for the resistivity the point I'm trying to make is that this is a whole different way of viewing it that what is it what determines the resistivity of copper well it's 25 kilo ohm per mode then you ask the question how many channels do I have how many highways how many lanes on my highway and there you say well I got about one for every say a third of a nanometer the third of a nanometer because that's the size of an atom you know three angstrom by three angstrom let's say you got one for every third of a nanometer so and you get some number so maybe 10 to the power minus 19th would be a better number here because three times three is about 10 for this discussion and then that tells you how many channels and then there's the mean free path that you put in that's then ways the resistivity you get okay so this is the viewpoint that I wanted to get across now what I want to do in the next few minutes then is connect a little bit to the standard expression for resistivity that you might be familiar with and that is where you see if I if you look at the results of there for conductivity I think we talked about this the first expression that's this density of states times diffusion coefficient for conductivity that I think you are we just discussed yeah this one this one we have seen this is something we have discussed and now I just want to tell you a little bit about this this expression where it comes from that's it okay have this expression so this is what I was writing as V z square tau and I'm writing it as V square tau divided by the number of dimensions as I mentioned the average of V z square is like V square divided by the number of dimensions now what I want to get at is this next expression so and this is kind of familiar in the sense that the expression that you may have seen because most discussions of transport usually start with this root formula and you calculate the conductivity and usually what you see is q square n tau over m this is an expression that this is the one that everyone carries in their head but what's the conductivity is q square and tau over m see and the problem with this one as I've said I mean the reason I want to stress this more than that is that this one's very general this is about the density of states and it also stresses this point that what really matters is the density of states right around the Fermi energy whereas this one kind of gives you an impression that conductivity depends on how many electrons you have and that's really not true because as I said you take the best insulator and the best conductor they have about the same total number of electrons so when you say that it depends on the electron density what you mean is let us look at only the electron density in this band not anything down there you of course if you include all that they would be wrong and so then they say well you know the filled bands don't conduct so you have to take only this one etc but the basic point of course is that conduction only depends on the density of states at this energy but what I want to show next is that the density of states here can be related to the total number of electrons here as long as it is just that much as long as I just include that if you adopt a certain model for the density of states you see so far I've avoided adopting any models I said only the density of states wherever it comes from so it means you can apply to molecules you could apply to amorphous conductors you could anything as long as you have the density of states you can go on from there but now I'm going to adopt a specific model which involves an ek relation of some sort right and so it would be of limited validity but the reason I'm kind of doing it is I want to connect here and the other important thing I'm describing it leaving more general is that I'm writing it as velocity divided by momentum whereas the usual expression has a mass and of course moment momentum is like mass times velocity it is just that that's only true in parabolic bands in a non parabolic band what I feel is that instead of this you should use really this and then what you get that effective the mass which is p over v might well be energy dependent and that is something one should take into account when trying to interpret things like mobility and what you're talking about so in graphene is a very good example again where p over v is actually energy dependent it's not just a number so how do you get from here to here that's the point I want to make now the way it works there that what we say is that one important insights in solid state physics back from the 40s and 50s was that inside a solid an electron behaves almost as if it has a free electron almost as if it's going through the vacuum so all these density of states you know how do you calculate them you solve Schrodinger equation you have to include all the nuclear potentials if you look inside a solid it's a very complicated object there's all these atomic potentials that the electron is going through but the great insight was that as long as you're talking about electrons within this conduction band it behaves almost as if it's going through vacuum almost as if we're talking of free electrons in vacuum but with a different mass right we have a certain ek relation so it's ek that I'll write it as ep relation so usually often you write it as p squared over 2m so p is momentum and usually people relate the p to k through this h bar k and this is the most common form of the ek relation or I guess there's often a constant added to this but graphene for example doesn't have a parabolic relation it doesn't depend on p square it depends actually linearly on p so that's one of the materials there's a lot of interest in so there it will be more like ep is equal to some constant times magnitude of p so when you plot this it might look something like this but when you plot that one it will look something like that so these are various k relations that you're used and then the way you calculate the density of states based on this is by saying that well given any box like this the only those moment are allowed I mean the way this is it's like you can choose any momentum and there's a corresponding energy level but then when you say that well if I have a box then only those moment are allowed which give you a integral number of wavelengths and how do you get the wavelength that's this de Broglie relation that you say that lambda is equal to h over p so the length of the box must be some integer times h over p and so accordingly the p because you the p will be spaced by units of h over L so you could write it as p is equal to integer and this is the simple picture that people use for counting the number of states for coming up with a model for this density of states because so far I've said you know density of states that we don't know where it comes from but given that let's calculate conductance current everything so now you're saying well we are going to adopt this model for it and based on this model the way you count states then you can define something which is what I'll call this n of p so let us define this n of p as the total number of states which have a total number of states who which have p less than some maximum value so total number of states that are contained in here so this is the momentum this is px py so how many do I have in here and in one dimension it would be something like okay how many do I have all the way from minus p to plus p for example and the point is that the allowed values would be spaced by h over l so it's a well the total number must be equal to 2p divided by h over l so in one dimension so this model then gives you a total number of states available that is proportional to the length of the solid make it twice you get twice as many total number of states up to a certain energy up to a certain momentum p now if you have to do this in two dimensions it would be sort of like p square p over you know h over l and then another h over w and ordinarily you might have thought 4 but then 4 would be if you are kind of finding how many states are within the square but actually you want the number of states within the circle so it's more like actually you know the pi and when you do it in three dimensions it is something like 4 pi over 3 p cube divided by h over l then you know h squared over area so this is how your norm count the total number of states up to a certain point so if all the states were filled up to a certain maximum momentum how many electrons would you have that would be the number now you want density of states how do you get density of states out of that you look at the derivative of that with respect to energy this gives a total number of states well if I increase it a little bit how many extra states do I pick up so that's how you usually do it so the bottom line is that when you calculate density of states it would be something like because what I just wrote is n of p but then e and p are really I'm assuming it's all isotropic so the energy can be related to the magnitude of the momentum that's this e p relation I have so given this I can always turn it into n of e e and p are related thanks so here for example I could easily have so it is density of states is the derivative of this n with respect to e but once I have this I could turn it into so you could have done it for example like dn dp and then dp dp that's how you could calculate this density of state now the important result that is useful here that's what I'll be showing in a minute is the following that is regardless of the ek relation no matter what it is what I can show is that as long as your counting states in this way what I can show is the following the density of states times velocity times momentum is equal to n of e times the dimensions this is a general result that I can show and I'll do that in a minute that's basically the last thing I'll do in this lecture so but if you accept this then you can kind of see how that relation becomes this why because density of states divided by the number of dimensions is like n divided by v times p and then so from here to here basically all I'm using is this particular relation that's it and the importance of this relation is you can see this a density of states that's a property at a given energy and that's like the total number of electrons up to that energy and the thing is it is relating the density of states that energy to the total number of electrons up to that energy if total number of states up to that energy so conduction depends only on the density of states right here but that relation will now relate it density of states to the total number of states in here everything that's that's the importance of this not parabolic any EP relation so so the power of this is that it works for any EP relation whatever it is any relation so it could be graphene it could be anything else it would this would still be true now and then you can see that the conductivity depends on this electron density could say well it depends on the electron density tau and then V over P and V over P is then what you could call mass and then this would be like the root formula okay so let me just take a couple of minutes to get this result so the basic idea is what we saw was that N of P that is the total number of states up to a certain momentum P is proportional to P to the power the number of dimensions and there's some constant up front but we don't need to worry about that one whatever it is because if it's one dimension is proportional to P two dimensions P square three dimensions P cubed so whatever it is it will depend on that and so when you take DNDP you will get K times D times P to the power D minus 1 and you do DND that's it okay now what we want is this density of states say D of E is equal to now by definition velocity is actually DEDP whenever you have any EP relation the way you define velocity this is group velocity it's actually derivative so this quantity is actually 1 over V therefore I take the V over to the side K D P to the power D minus 1 now I multiply by P on both sides so I multiply by another P so this becomes P to the power D and K times P to the power D that of course was N in the first place so that's basically the relation as you say this is what I said that density of states times V times P is equal to the total number of electrons times the number of dimensions and that is what I used in order to get from here to here and the reason I'm going through all this is as I said that this both these expressions for conductivity are in the literature this is the one you carry in your head sort of N q square tau over m except that instead of m the point I'm making is you should really take the ratio of velocity to momentum which may not be a number which may be actually energy dependent on the certain conditions but otherwise this is the one you carry in your head this one is also a standard literature except that it's not something you usually remember and the reason as I said you look at ashtrafter and merman this is in chapter one that's in chapter 13 I mean that's why you usually don't and that's but the way we did it this one came more natural and so what I wanted to do is connect this to this and philosophically there's an enormous difference because this one tells you conductivity depends on density of states at one energy this one tells you conductivity depends on all the electrons up to that energy there's a philosophical kind of two different things you see and this is the relation which of course is based on this particular model for density of states I mean the only assumption I made was this as long as you accept this this follow based on but this particular assumption of course may not hold in an amorphous conductor would not hold in a molecule for example etc. all kinds of places where you may not want to use this what that means is I would still be able to use this but may not be this that's all so let me stop here now and the next lecture did a professor Lundstrom on transistors and this afternoon I'll continue thank you very good question I guess the question was that this is transport which is a non-equilibrium process so why are we using an equilibrium for me function right now the point I want to make is that it's a non-equilibrium process but the non-equilibrium is because you've got this two different for me function that is the idea is at the ends of these two contacts which are big contacts so that they are essentially in local equilibrium that's the picture so this is an equilibrium that's an equilibrium by itself what is in between of course is can be badly out of equilibrium in general and so whatever f's appear in our equations like if you look at that top equation with the f1 minus f2 those are f's in the contacts so this afternoon I'll try to talk a little bit more about f inside the channel what does it look like and that one did not in general look like a Fermi function although often people kind of make that approximation to simplify calculations but there's a lot of discussion with it what that should but I'll talk a little more about it this afternoon. Yes please. So here we always talk about density of state in equations so what if there's no density of state at all like there's no emission theory that that can be done. So that would be like tunneling through something for example. Now there is still a density of local density of states it is so for example if this is a as you said so the question was what if this is a vacuum for example where there is no density of states so if this was like two things separated by vacuum and then I'd say the current really flows only when they're really close together and when they're close together means that if you carefully calculated the density of states you would find something here so usually in a vacuum let's say that all the density of states is way up there but when you get it close enough these states from the two contacts penetrate in so there is still something in there anyway so there's effectively some density of states but whether it would in that case whether it is useful to think of it this way I'm not completely sure what we'll talk about I guess tomorrow morning is when you talk about quantum transport is that from the quantum formalism you get a different expression for that conductance function up there so here I said conductance is q square d over t say from the quantum formalism you'll get a different expression for the conductance the overall though once you have that it's still the same expression for current it is just that instead of q square d over 2t you have a something more quantum mechanically defined function