 You can follow along with this presentation by going to nanohub.org and downloading the corresponding slides. Enjoy the show. Okay, welcome back and thanks for hanging in there. One more lecture. So what I'm going to be talking about now is the general model for conduction that Professor Dada discussed in the first half of his talk. I'm going to go through it one more time. We'll do it slowly and I'll do one or two things just a little bit differently. I'll do it in a slightly different notation but it's the form of the model that I'm going to use for the next eight lectures. So it's worth seeing it again and getting comfortable with it. So it's a general way to think about transport. Okay, so here are the things we'll go through and I'm going to begin by talking about a simple model device for a nano device and we'll begin like this. So we'll think of the device. So the device is some material. This is like the channel of a field effect transistor or something and it's characterized by a density of states, D. So that might come from a band structure of silicon but it might come from a set of energy levels from a molecule. It can be any material that conducts current. Okay, we haven't got a device yet because we have to put contacts on the device. So we'll attach two contacts and on the left and the right I have two contacts. The assumption is that these are large contacts with lots of inelastic scattering that maintain thermodynamic equilibrium. Of course things can't be exactly a thermodynamic equilibrium or there wouldn't be any current to flow but it's very, very close, close enough. So F1 is the Fermi function of the left contact. F2 is the Fermi function of the right contact. The left contact is grounded. The voltage is zero. The right contact has a voltage V applied to it. Later on it'll have a different temperature applied to it as well. Okay, so we describe both of those by a Fermi function because they're an equilibrium and I'll write the Fermi level as EF not as mu because I'm going to be following more standard electrical engineering terminology where mu is always the mobility. So you have to keep that straight. So EF1 is the Fermi level of the left contact. EF2 is the Fermi level of the right contact and it's different because it applied a positive voltage lowers the Fermi level. So EF2 is EF1 minus QV. That's what the voltage does. Okay and we might have a gate. We might be trying to make a nano transistor. We'd apply a voltage to the gate and the gate voltage would change the potential inside the device. So it would change the self-consistent potential, that U, and that would just move the states up and down. That's how we'd make a transistor. We'd just move those energy states up and down with an external gate voltage. I'm actually not going to be talking about transistors in this talk, in fact in any of the lectures. So you can make a nice little model for a nano transistor with this model but that's not what we're going to be talking about. And then finally we have to talk about the contacts. So the contacts are always important so we need to specify the contacts somehow. The easiest way to specify contacts is in terms of these times. So if that's a small molecule and we put an electron there, it might take a time tau 1 to escape into contact 1. It might take a time tau 2 to escape into contact 2. So we'll have to talk about exactly what those times are a little bit later. But we could also express them in terms of energy. It looks like an uncertainty relation. So gamma is the broadening and it's h bar over tau. That just expresses those times in units of 1 over energy or h bar over energy. If this were a molecule, then what gamma would correspond to is that if a molecule were isolated its energy levels would be discrete. If a molecule is attached to the outside world there'd be some finite lifetime of the electrons in those states and the energy levels would broaden. That's why we call it the broadening. So current will flow in contact 1 and out or in the right contact out the left contact so we define it to be positive when it flows in the right contact. And then we begin by asking two questions. How is the number of electrons in the device related to the Fermi levels of those two contacts? And obviously it's going to also be related to the density of states of that channel material as well and to those characteristic times. And the second question is how is the current related to those same quantities? Those are really those are the two questions that we're asking today. So here's our device again. And now first of all there are a few assumptions here that I just need to mention. So one is I'm going to be describing this density of states or device by a band structure and E of K. And for most of the lectures I'm going to assume a parabolic band. Energy is h bar squared k squared over 2m. That's actually not necessary. All I need is a density of states. This might be a set of molecular levels that might be an amorphous material or something else. We don't need a band structure. But for these lectures we're going to assume we have a nanotube with a band structure. We have a piece of silicon with a band structure. We're going to make that assumption and I'll refer you to Professor Dada's books and lectures that talk about a little more generally about how you do that without a band structure. The contacts that I said have strong inelastic scattering. They maintain thermal equilibrium. We assume that's the only place that inelastic scattering occurs. So that's where the electrons exchange energy with the phonons and everything comes into thermal equilibrium. U is this self-consistent potential. So an electron sees the average potential from all of the other electrons. That's a mean field theory. That's what we normally do in semiconductor device analysis. There are some conditions under which that's not good enough. And some examples are things like single electron charging. Where that you have to take into account the discreteness of the electron and the interactions between the electrons much more carefully and not in this average sense. That's beyond the scope of what we're going to be talking about. And again I'll refer you to some of Professor Dada's lectures where he discusses those effects. And finally as I mentioned the electrons travel through the device elastically. They might scatter but they don't change energy. They can only change energy when they get into the contacts and dissipate energy. So we have independent energy channels. We can ask ourselves what happens at each energy and then to get the total current we can integrate or add up all of the energy channels. That's the assumption. Oh and one more thing. We're going to assume these contacts have a special nature. They're perfectly absorbing or reflecting. So if an electron exits the device and goes into contact to it gets absorbed and thermalizes it doesn't reflect and come back into the device. That helps us with the bookkeeping. We know inside the device where the electrons come from. That's actually in most cases that's not a bad description of real contacts. They are very often very close to an absorbing contact so even though this model might look simplified the experimental evidence is that it's actually quite good. So that's the model device and we're just going to try to find out how many electrons are in it how much current flows through it. So the first step is to formulate the mathematical model. So we can think of it this way. Let's assume that we only have the left contact attached. No current is going to flow. Electrons might flow out of the contact into the device everything will settle down and will be in thermodynamic equilibrium. So if we do that we can write some little equations. The number of electrons in the device when it's only connected to contact one it tells me that I'm at equilibrium. We're just going to fill up the states in the device according to the Fermi level in the contact. It'll just come to equilibrium. There'll be one Fermi level it'll describe how all of the states are occupied whether they're in the contact or the device. And if I want to know how long that will take if I attach it it'll take a little bit for the charge to flow around and for that to happen. The simplest equation that I could think of writing would be something like this. The rate of change of the electron number in the device is proportional to the difference between the equilibrium number that the contact wants to be there and the number that is actually there and then divided by this characteristic time. So if I don't have enough electrons in the device this rate will be positive and electrons will flow from the contact into the device. If I have somehow started the process with too many electrons in the device this rate will be negative and electrons will flow out of the device into the contact and after a time tau one everything will settle down and be in equilibrium. Now one of the things that Professor Fischer mentioned is that you always have to keep the notation straight and different people use different types of notation. So I need to just point out when you're reading books and papers and things you have to be careful about various factors. How did the author do this or that? I'm going to always include the factor of two spin in the density of states because that's typically the way we do it in semiconductor device textbooks and things. So we'll have to be careful about where these factors do come in. Alright now it's just as easy if I had the device just connected to the second contact then the device would want to come into equilibrium with the second contact. Again no current would flow it's only connected to one contact. So I can write the same thing the number of electrons in the device would be the density of states times the Fermi function of contact two and the flow in or out to achieve that equilibrium would be described by this simple rate equation. So now in general we've got both contacts attached and each one one contact has a voltage applied to it and the other one doesn't so the Fermi levels are in different locations so each contact wants to fill up the device in different ways. So I can simply add the rate that they flow in or out from contact one to the rate that they flow in or out from contact two and I can add the two up. Now initially there will be a transient but if I wait long enough everything will settle down to steady state and there will be no change with time. So I have an equation there that I can solve for the number of electrons in the device that will result. And the answer is this, it's some weighted average between what the two contacts would like to see in their equilibrium. N1 superscript zero is the number of electrons in the device if it's in equilibrium with contact one and N2 superscript zero would be the number of electrons in the device if it were in equilibrium with contact two. At two different Fermi levels it can't be in equilibrium with both contacts at the same time. So it's just some weighted average between the two and the weights depend on how strong the connection is to the two contacts. If it's weakly connected to one contact it will mostly be determined by the other contact. If it's equally weighted both of them will have equal weights. N1 zero is just the density of states in the channel times the Fermi function of the contact. This is a little nano device the channel is not in thermodynamic equilibrium there is no Fermi function in the channel the Fermi function always refers to the contacts. And to keep things simple for the rest of the lecture and all of the subsequent lectures two contacts are identical these two times are the same. So we get a very simple answer surprisingly simple. The number of electrons half the states are filled by contact one or half the states are available to be filled by contact one half the states are available to be filled by contact two their contacts are equal the states are just going to divide up split in half but the weights are determined by the Fermi functions of contact one and contact two. So these are all independent energy channels so if I want the total number of electrons I just integrate over all of the channels. Now this is really if you remember one of the first things you learn in a semiconductor device course we compute the equilibrium number of electrons and the conduction band of silicon or whatever it's just an integral of the density of states times the equilibrium Fermi function. Now we're out of equilibrium we have two different contacts with two different biases but the result is very similar it's an integral of half of the density of states times the equilibrium Fermi function of the first contact plus the other half of the density of states times the equilibrium Fermi function of the second contact so you can describe this out of equilibrium situation very much like we describe the equilibrium situation we just have to remember that there are two different Fermi levels there are two different sets of states that are being filled by those Fermi levels. One more thing in terms of notation that we'll have to keep straight so the density of states if this is a three-dimensional channel then if its volume is twice as big there'll be twice as many states so the density of states will be proportional to the volume if the channel is a two-dimensional sheet like a MOSFET inversion layer of the sheet and if it's a nanowire like a carbon nanotube then the density of states will be proportional to the length of the nanotube normally in semiconductor courses we express density of states as per unit volume or per unit area or per unit length and per energy as well but right now I'm not doing it I'm doing it as total number of electrons. One of the advantages is I can do 1d, 2d or 3d it's all the same it's only at the end when I decide how I want to divide things up and if I were doing it in 1d I would ask what's the density of electrons but it would be the density per unit length 2d we usually ask what's the sheet carrier density the density per square centimeter the density per cubic centimeter it's just a number divided by volume so we'll keep that so we were posing these two questions and we've answered the first so the second question then is how much current is flowing and that we can answer too so we'll go back to our model device and we describe these two fluxes 2dT and dn 2dT the contacts are either putting electrons into the device F1 would be positive or they're pulling it out F1 would be negative so both contacts are trying to fill up the device according to their Fermi level and in steady state the sum of the two has to be zero what comes in one contact just goes out the other contact so if I want the steady state current which is the only question I'm going to ask we just solve that little equation and now the current then will just be q times the flux that's coming in the first contact let's say that's positive then that same flux in steady state is going out the other contact so it would be a negative flux so it's minus q times F2 if you're in physics q is the electron charge and it has a sign if you're in electrical engineering q is the magnitude of the charge so q is the magnitude of the charge I think physicists would write this as E is the magnitude of the charge people can't agree so you just have to keep track of it ok so we have these expressions for the current we have already formulated what the fluxes are the equilibrium carrier densities are just products of density of states times Fermi function so we'll just insert those quantities in those expressions let's see I really have two equations there I'm relating it in terms of the flux in contact 1 which is equal and opposite to the flux in contact 2 so if I add those two equations 2 times the current and then divide by 2 I get the second expression then I just plug these quantities in and I expand it a little bit and we have an expression for the current just straight forward algebra so we've answered our second question the current is 2q over h times gamma pi d over 2 F1 minus F2 this is an equation we want to get familiar with this we're going to use this throughout the next 8 lectures I'm going to express this in slightly different form but it will be the same equation the important points here to remember there have to be a difference in Fermi levels or current won't flow if both contacts have the same Fermi energy there's no reason for current to flow the density of states remember I included the factor 2 in the density of states because that's the way most semiconductor people do it so density of states divided by 2 is the density of states per spin you might ask why do I have a 2 out front and divided by 2 because this is the way people like to write it the 2 out front we think of as accounting for spin the d divided by 2 is my density of states per spin and then pi is just pi and gamma is this broadening ok so it's all it's all very simple all we have to do now for the rest of the lectures is just think about these terms a little bit so the next thing I want to talk about is this concept of the number of channels for conduction people call that modes talk about why we call that modes so this is the expression we've derived just a few simple straight forward arguments following the work that Professor Dada has done over the past few years now we have this expression now let's look at this and see if we can make some more sense of it so we have this term gamma pi d over 2 you know what is this well gamma has the units of energy it's the broadening density of states has units of 1 over energy pi doesn't have any units so this is some number dimensionless you know what is it well we're going to show that it is the number of channels for conduction at the energy e and we'll have to see if we can see why that is so you know everything we're doing here we can do in 1d, 2d and 3d as I told you in my introduction I'm mostly going to work out examples in 2d just because you get bored of seeing it done in all three dimensions and lots of problems involved 2d so let's think of a sheet like maybe the inversion layer of a MOSFET so those are our two contacts dimensional density of states so now just a little bit more notation when I write d sub 2d I'm going to mean the density of states that you find in the semiconductor textbooks the number of states per square centimeter per electron volt so the total density of states which is d of e is that quantity times the area of the channel quantity if you have parabolic bands and you've had an introductory semiconductor course you've probably worked out the density of states in 2d is effective mass over pi h bar squared that's something we know so let's see if we can figure out what this gamma or tau is I think professor Dada already discussed it so we know the answer but we'll see it one more time so these are our two expressions that we've developed for the current and for the number of electrons in the device now let me divide the two so what's on top is the total charge in the device q times n is the total amount of charge in the device and i is the current so I'll just divide those two now let me apply a large enough voltage at contact 2 such that it lowers the Fermi level in contact 2 such that it has no probability of occupying any states in the channel its Fermi level is pulled down so low so now f2 is basically 0 and all of the electrons that come into the device come in from contact 1 so I can simplify that expression and I get an f1 divided by an f1 so what we find is that the total charge in the device divided by the current is just this time tau dimensionally you can see that that's correct current is charged divided by time so we're dividing charge by charge divided by time this is what electrical engineers would call a charge control time commonly used to analyze devices so that's what this tau is and we can figure out what it is for our little nano device so let's say if I look along the sheet in the x direction from contact 1 to contact 2 and I ask what is the density it's going to be the sheet carrier one thing I should point out this is a small nano device I'm only asking for the number of electrons in the device or if it's a long one that's under low bias and the concentration of electrons is uniform if I want to spatially resolve things inside the device I have to get much more sophisticated I could do a non-equilibrium Green's function simulation I could solve a Boltzmann transport equation in space but we're assuming that the density of electrons is uniform inside the device it's a short small device so we have a density of electrons we have a current I can always write current as the product of charge times velocity that's pretty fundamental the charge is just the sheet carrier density times w times q the velocity is the average velocity in the direction of current flow that's what this average vx plus is the average velocity flowing in the plus x direction that's almost a definition so if I just divide those two quantities I'll divide stored charge by the current we find that this transit time is L divided by the length of the device by the average velocity that they're traveling at that's the transit time and you saw that from Professor Dada probably guessed that so let's see what we have here we have to be a little bit more careful Professor Dada also mentioned that you might have to do some averaging over angles if this were a 1D nanowire and everything was just moving in the x direction we'd be done we're coming in from contact 1 and they can have any angle between plus and minus pi over 2 and we want the average velocity the x directed velocity of those electrons that are moving in the xy plane so vx is just v cosine theta where v is the magnitude of the velocity so I have to take the average value of cosine theta that's easy to do distributed between minus pi over 2 and plus pi over 2 so all of the incoming electrons are distributed over a range of angles of pi so the average cosine theta is just 2 over pi so this average velocity in the x direction is just 2 over pi times the magnitude of the velocity and again I'm using parabolic bands over and over again and the danger that you shouldn't let yourself fall into is thinking that any of this applies just to parabolic bands that's why we have lecture 10 when we'll do graphene so what's the magnitude of the velocity in a parabolic band 1 half mv squared is equal to energy so the velocity is just proportional to the square root of kinetic energy okay so now we can go back to this question remember the question is what is this quantity m the product of gamma pi d over 2 well the broadening is just h bar over tau and tau is l over the average velocity the density of states we know also so we find that the number of channels is just width proportional to the width so that's kind of intuitive the wider the channel is in the direction coming out of the page the more channels there are it's proportional to average velocity times density of states so this is actually very general if I want to find the number of channels this is the way that I can do it it's still not giving me a physical sense as to what these are but this is a prescription that works all the time it's just h bar over 4 times the average velocity times the 1d density of states in 2d it's proportional to the width and it's width times h bar over 4 average velocity times 2d density of states and in 3d it's a similar thing but it's proportional to the cross sectional area but the question still is what does this mean why do we call them modes okay so we know what's in here we know what the average velocity is we know how it's related to energy we know what the 2d density of states is for parabolic bands so we can just plug everything in and we can work out an expression for what m is and this is the expression then the question is what does that mean well if I look down here for parabolic bands I can solve for k k is the wave vector of the electron waves 2pi over the Broglie wavelength 2pi over lambda so I can solve that bottom equation on the right for k put it back into my expression for m and we find that m is wk over pi okay it's getting simpler k the wave vector the wave vector is 2pi over lambda so m is the width of the channel divided by the half wavelength of the electron at that energy okay so what does that mean m is so now it's easy to remember it's just a number of half wavelengths that fit into w so if I go back to my 2d conductor if I assume that there's only one subband occupied in the z direction then the question is how I have some finite width so the wave function has to go to zero at the two edges of the channel and how many channels are there associated with that finite width well it's just w divided by half a wavelength so look if there's one if m is equal to one if there's one channel that the half wavelength of the electron is just equal to the width that means that the wave function has to go to zero at both sides and the wave function fits in what if there are two conducting channels that means that the wavelength of the electron wave is w that means something like this is the mode that fits in but I also have a mode like this that has a smaller wave vector in the y direction that also fits in so now I've got two different k y's that fit in so each one of those is a separate channel for conduction and that's why I have two channels for conduction so this is why we call them modes this looks like an electron wave guide so people saw an analogy to wave guides you establish the boundary conditions for electrons at the two boundary conditions and it looks like a wave guide like one for electromagnetics so that's why we call these modes the number of channels is equal to the number of modes so we have a simple physical interpretation m is the number of electron half wavelengths that fit into w that's what gamma gamma pi density of states over 2 is so let me just point out the density of states is probably something you've all seen you've worked it out for parabolic bands you can work it out in 1D, 2D or 3D and you have these characteristic shapes in 1D it goes as 1 over the square root of energy so it goes to infinity at the bottom of the band in 2D the density of states is constant in parabolic bands in 3D the density of states goes as the square root of energy but now we have this other quantity m and m is just velocity times density of states so if I work out m in 1D, 2D and 3D m is just constant in 1D if I have one subband occupied m is just one in 2D it's proportional to the square root of energy because as the width gets wider more and more of these modes can fit into the width and in 3D it turns out to go linearly with energy so just to summarize the density of states is probably something that you're all familiar with when we want the number of electrons we integrate the density of states times the Fermi function if we divide this number of modes or channels we need to understand what it is because we integrate m when we want to find the current and it's worth remembering that the two are related and that the number of modes is proportional to the average velocity in the direction of transport times the density of states and it's also worth remembering but you shouldn't forget that it also depends on the dispersion of the band structure because I assume parabolic bands in all three cases if we go to graphene you'll say graphene is a 2D conductor but its number of modes won't be what I just showed you because graphene doesn't have a parabolic band so it depends on both so continuing on what we're talking about is transmission and I remember Professor Fischer mentioned this I think Professor Dada might have too so let's talk about transmission so back to our channel ballistic transport electrons just come in one contact and shoot across the other to the other and exit without any reflections it's a perfectly absorbing contact but they might come in at an angle and that's why we did that average over angle to get the average velocity in the x direction that's ballistic transport if there's any scattering from the boundaries we assume that the boundaries are atomically flat and it's specular scattering so angle of incidence equals angle of reflection and it doesn't have any effect on the current flow and that'll be a scattering mechanism but we're ballistic here now what's diffusive transport I think Professor Dada discussed this a little bit too so in diffusive transport an electron comes in ballistic for a while but then it encounters something a charged impurity, a phonon maybe a roughness at the edge of the sample and it scatters in some random direction and it undergoes a random walk gets a little ways and then it scatters again and there's no assurance that it will come out contact two but some of them do, some of them turn around and go back out contact one if I have applied a positive voltage on contact two slightly more of them will come out of contact two than contact one but this is what we mean by diffusive transport so electrons undergo a random walk some terminate at contact one some at contact two the average distance between collisions is the main free path this is lambda, this is what Professor Fischer labeled with a capital lambda I'm going to use a small lambda of course I also use lambda for wavelength but then I have a lambda b for de Broglie wavelength so I don't know any nice solution it's actually helpful because you have to be sure you know what the meaning of these equations is because you're never sure what the symbol means Diffusive transport just means the device is a lot longer than the mean free path now since this broadening is gamma over transit time it's going to take a lot longer for the electrons to get across the device when it's diffusive it's much easier in ballistic it just shoots across so our transit time is going to be affected and we have to figure out that so we know what the average velocity is in the x direction for ballistic transport that's easy to deduce what is the average velocity in the x direction when it's diffusive and Professor Dada also mentioned this so the way I'm going to do this is I'm going to assume that we have a channel that is much longer than the mean free path so I can go back and do classical things like fixed law diffusion I know what the answer is because it's a diffusive sample so I know that the current is just proportional it's a diffusion coefficient times a gradient in the carrier concentration just fixed law and the minus sign is just because I define current to be positive when it flows in that second contact so if you solve a problem like this it's basically solving a diffusion equation and you inject electrons in one contact and we apply a positive voltage on the other so that the Fermi function is pulled way down and we don't inject any electrons from that contact the Fermi level is too low then the profile will look something like this and you'll get this if you solve fixed law in the continuity equation this is what you'll get you'll get a linear profile you might remember from semiconductor courses if carriers recombine inside the region this will be exponential but if there's no recombination it's just linear we had some concentration at the beginning so if you see mean free paths long we have a contact at the other end that just lets carriers go out but doesn't let any come in so that concentration is very low and the current we can just say current flows down the concentration gradient so it's just given by fixed law so what would this time tau be? you know because well I'll just say the stored charge by current the stored charge is the area under that rectangle the current is just given by the slope of that line times the diffusion coefficient if you divide those two you get length squared over 2 times diffusion coefficient okay no tesh thing I also have a problem with d's some d's are densities of states some d's are diffusion coefficients alright I write it d sub n I think of it as the electron diffusion coefficient but of course this works in a p-type sample too that's my notation alright so that's actually that's a very useful thing to remember many of you probably know that answer how long does it take for particles to diffuse across a region length of the region squared divided by 2 times the diffusion coefficient alright so now we can go back we have this parameter gamma in our current equation to evaluate and when we did it ballistically it was just h-bar divided by the ballistic transit time when we do it diffusively it's h-bar divided by the diffusive transit time that's the only thing that will change the ballistic transit time is just L divided by the average ballistic velocity the diffusive transit time is L squared divided by 2 times the diffusion coefficient now we'll see this a little bit later I'll I'm going to plop it in here now the diffusion coefficient is related to the mean free path and actually you can show and we'll come back to this later in some other lectures that the diffusion coefficient is the average velocity in the direction of transport times the mean free path divided by 2 alright that isn't intuitively obvious but we'll see where it comes from but at least you can see it's dimensionally correct velocity is centimeters per second mean free path is centimeters so it has units of centimeters squared per second so we're trying to evaluate this broadening gamma sub D for diffusive transport so let me write it as the ballistic quantity h-bar over ballistic transit time times the ratio of ballistic transit time over diffusive transit time ok so gamma for the diffusive case is just gamma for the ballistic case times this ratio of transit times ok the ratio of transit times we have right here we can just compute and if you do that you'll find that the diffusive gamma is mean free path divided by length times the ballistic gamma and since we've assumed diffusive transport the mean free path is much smaller than the length so the diffusive transit time or the diffusive broadening is much less than the ballistic broadening diffusive transit time is much greater than the ballistic transit time ok so when we were computing current we had this number the number of channels was important if we go back to our current expression we have to replace gamma sub B the ballistic quantity with the diffusive quantity so all we have to do and you can see that the ballistic quantity is just a diffusive quantity times this racial mean free path divided by length so we can simply replace M in the ballistic case by the product of M times a number right gamma our mean free path divided by L is just a number that number will call T transmission and it's very much less than one for ballistic transit transport it's just the ratio of the mean free path divided by the length Professor Fischer uses a similar expression you had some Greek letter squiggle or something for this ok I'm using T you'll notice earlier I used T sub L for temperature that's the temperature of the lattice and since we're near equilibrium the temperature of the lattice is also the temperature of the electrons because I wanted to use T for transmission so that's another one of these notation things I have to worry about ok so we can take this expression that we could derive very simply current is 2q over h times an integral of gamma pi d over 2 times f1 minus f2 and we can write it in an equivalent way and this is the way I'm mostly going to use for these lectures they're just algebraically the same 2q over h times transmission times number of channels times f1 minus f2 two different ways of saying the same thing ok and in the diffusive case that transmission we just saw was the ratio of lambda mean free path divided by length in the ballistic case T was just one it wasn't there now a lot of problems these days are in this quasi ballistic regime 10, 20 years ago we were almost always in the diffusive regime and people did experiments in mesoscopic physics where they were in the ballistic regime these days at room temperature in practical electronic devices we're frequently in between those two and sometimes we're actually quite close to ballistic here in this regime where you're not quite sure which is the best assumption and as professor Dada mentioned that this is almost considered two different topics there are people that work on ballistic transport and use Landauer approach for that there are people that work on diffusive transport and use drift diffusion equations or Boltzmann equations for that but there's really no reason I mean it's just transport there's no reason to artificially divide these you have to use two separate approaches for them and in fact this will seem ad hoc right now but it's not if I write T as mean free path over mean free path plus length you can see that it has the proper answer in both limits when L is much longer than the mean free path it gives what it needs to give for the diffusive case when the length is much shorter than the mean free path which is the answer in the ballistic case now it's actually not an ad hoc thing to give me the correct answer between the two endpoints but professor Fischer had a reference to professor Dada's was it 1995 book on electronic conduction you can derive this and we'll do that in lecture 6 it actually works continuously from diffusive to ballistic and anywhere in between so it gives you a very nice technique that's one of the reasons that we like this method is we can we can treat transport from both limits and in between with one way of doing that ok so just a couple more things and we'll be ready to call it a day so we have this expression for the current I can write it in a couple of different ways of doing it and there are many problems in which we use the expression this way but that's not the subject of the next 8 lectures that I'm going to talk about the next 8 lectures are about near equilibrium transport so near equilibrium transport means that we apply a small voltage or a small temperature difference and f1 and f2 are not very much different that's what we mean by near equilibrium transport so since f1 and f2 are not very different we can linearize it let's see how you do that we know what f1 and f2 are for the time being we're going to assume same temperatures for the both of the contacts when we let the temperatures be different we get thermoelectric effects so I can expand f2 as a Taylor series expansion it's f1 plus some small additional amount so that's the derivative of f1 with respect to the Fermi level times the difference in the Fermi levels because the two temperatures are the same the only thing that's different between the two contacts is they have different Fermi energies that's all now energy is minus q times voltage for electrons so the difference in Fermi energy is coming because I've applied a positive voltage to the right side of q times v now also if you look at these Fermi functions you can see e and ef energy in the Fermi energy kind of come in similar ways the only difference is one has a negative sign so I could just as easily write df by ef or minus df by energy same thing so the result of doing all of that is that when I linearize this difference f1 minus f2 I find it's just proportional to the voltage this is what's going to give me near equilibrium transport and the bottom line then is that we can simplify that expression general expression we had for current into an expression for small differences in the two Fermi functions or small applied voltages we get this expression so the key factor now is this minus fde this has become an f0 because we're near equilibrium so f1 is close to the equilibrium Fermi function f0 f2 is close to the equilibrium just a little bit different okay so that's linear transport so I want to mention here just briefly again this is a topic Professor Dada touched on this so we can we should mention it again here most transport if you look at traditional books from the 50's, 60's, 70's probably 80's too near equilibrium transport is all about transport in the bulk it means a structure that is many mean free paths long and you don't even think about the context ballistic transport you have to think about context everything is determined by the context so let's see how we connect to traditional transport theory we have this expression you know we know what the broadening is for diffusive transport the transit time is L squared over 2D the density of states in this expression is the total density of states per energy so I can write it as the density of states per energy per square centimeter times the area of the sheet now the current this is one of these little careful doing these little devices we apply a positive voltage to the second contact the current if the current flows in we call that a positive voltage that's the way we people call that the circuit convention but if my x axis points in the other direction then that's a current flowing in the minus x direction so if I want a current equation I have to worry about that minus sign and then we'll expand this for small differences in the Fermi level so I'll get this and we'll lump things together here so now I have the current is proportional to a diffusion coefficient that diffusion coefficient came in here from the transit time which was L squared over 2 diffusion coefficient times the density of states per unit energy per square centimeter times the minus the FDE and then when we expand F1 minus F2 we ended up with a delta Fn over L remember F sub n is the difference between the Fermi levels of the two contacts and now in a bulk sample I'm going to think about it a little differently and we're going to think about it as a position dependent quasi Fermi level so delta Fn over L that's like a derivative so it's a small step now to lump everything in the curly brackets I'll call that conductivity to take delta Fn over L and call that a gradient of the quasi Fermi energy and I get this expression for current density J is equal to conductivity times the gradient of the quasi Fermi energy divided by Q so it has units of both so that's a classic textbook description of low field transport current is proportional to the gradient of the electrochemical potential or quasi Fermi level whatever we call it and we've got that we've got that just by taking our small nano device and making it very very long so there was many mean free paths long and what comes out is this and we have this expression for conductivity and remember Professor Dada showed this also so this is actually a little better than NQ mu or mu is Q tau over M and you see this in standard textbooks too the conductivity is an integral of the diffusion coefficient at energy E and the density of states at energy E weighted by this minus Df to E so in a bulk sample like this we'll apply a voltage the voltage will just tilt the conduction band so it will have a constant slope the quasi Fermi level will be parallel if there are no diffusion currents there and that's our picture now how do we think about what we're doing here and this is the thing like Professor Dada said we have to be a little bit careful because conceptually the way we think about this is we have this long bulk device it's near equilibrium everywhere and instead of thinking about inelastic scattering occurring only in the contacts and elastic scattering occurring in between we just go into a chunk of this infinitely long material and we'll say okay all of the inelastic scattering occurs in these two shaded regions those we're going to think of as our contacts and in between we can have the elastic scattering on average if it's a long device everything will average out just fine it's not actually separated this way but we can conceptually think of it that way we have to be careful about those contacts because as we'll discuss tomorrow morning in the first lecture and as Professor Dada mentioned there is this quantum contact resistance but in any device there are only two real contacts and you can't add up those quantum contact resistances at each little dx of the infinite device they only come at the end but that's an easy so if we just take our normal little expression for this nano device make it many mean-free paths long we get the classic expression that everybody has known for probably more than 50 years about what the current flow is so that's a very fundamental description now I can do one more thing if you're an electrical engineer you know about drift diffusion equations so sometimes we do this backwards sometimes we start with a drift diffusion equation and show that the equation on the top left can be derived from it but that's not the way to do it the equation on the top left is really a very fundamental description of current flow but if I look at that so let me think about a 2D semiconductor let me assume that it's non-degenerate so the carrier density can be expressed as an effective density of states capital N sub 2D times an exponential of the difference between the quasi Fermi level and the bottom of the conduction band the effective density of states is given by that expression so I can solve the top equation then for the quasi Fermi energy F sub N or electrochemical potential and then I can take its gradient and plug it in for the gradient on the first expression and I can write conductivity as NQ mu and if you do all of that and put it together this is the current the equation that you'll get things out the first term is what we call the drift term the second term is what we call the diffusion term the diffusion coefficient is a thermal velocity in the x direction times a mean free path divided by 2 the mobility is given by an Einstein relation it's just a classic drift diffusion equation so drift diffusion equations are not fundamental but I just wanted to connect them to something we've probably seen before they do assume that you're near equilibrium we can't get around that they also assume that we're at constant temperature you know later on in tomorrow or Wednesday morning we'll talk about what a drift diffusion equation looks like when there's a temperature gradient as well I did it for Maxwell Boltzmann statistics here but there was no reason I had to do that I could have used Fermi Dirac statistics and I just would have gotten the more complicated Einstein relation alright so that's the drift diffusion equation now another thing we should ask about is what about holes in thinking about this I've been sort of thinking about electrons flowing through the conduction band and I've written a quasi-Fermi level for the conduction band now you know in many semiconductors we have a band gap for silicon E B and we usually have two different electrochemical potentials or two different quasi-Fermi levels now F sub N is the quasi-Fermi level for electrons in the conduction band and the assumption is that the electrons are distributed in energy according to a near equilibrium distribution a Fermi Dirac distribution it's just that instead of the equilibrium Fermi energy we put in the quasi-Fermi level now the holes are not necessarily in equilibrium with the electrons in the conduction band we might be shining light on this and we might have vastly increased the number of both electrons and holes so we need two separate quasi-Fermi levels one to describe the population of the valence band and one to describe the population of the conduction band now you might ask how do we compute the conductivity in the valence band and you could get yourself turned around easily you could start thinking well I shouldn't be talking about F the Fermi function I should be talking about 1-F and I should be thinking about holes but it turns out you don't have to these are the expressions we got and what happens to these expressions when the conduction is occurring through the valence band we just replace the subscript n with the subscript p and we don't change anything else right that F there is the Fermi function and if you think of that in the model when we talked about it in the model that Professor Dada talked about this morning you had a Fermi level and one contact you had a Fermi level and another contact if there were states in between you had current flow because electrons were flowing through those states if the electrons are flowing through the conduction band states we call that n-type if the electrons are flowing through the valence band states we call that p-type we don't have to think about holes it's electrons flowing in both cases so all of the expressions that we have work for either electrons or for holes or in some cases we have bipolar conduction we have both occurring at the same time we just integrate over both bands so from time to time in these lectures I'll stop and there are things like c-back coefficients are positive for n-type and negative for p-type we'll stop and talk about that momentarily but I won't be saying a whole lot about holes because everything I'm saying about electrons you can evaluate parameters for holes in just the same way ok so let's summarize and then see if there are any questions so the key idea to keep in mind is that current flows whenever there's a difference in Fermi levels or if it's a bulk material wherever there's a gradient an electrochemical potential we develop these really simple expressions for how we compute the current flow that are remarkably useful they're very very simple but in practice they turn out to apply to a wide class of problems very simply and these are equivalent so I can go back and forth whichever one is most convenient for the problem at hand I'm primarily going to use the one on the bottom for the rest of the lectures now for small applied biases we just expand f1 it's just a little bit different from f2 we just expand it and then we get current is proportional to voltage if we only have a temperature difference we'll get current is proportional to the temperature difference that it will be linear and we'll get expressions like this if we want to go to the bulk then we just let the conductor be many mean free paths long and we naturally recover the conventional expressions for current flow in a bulk material okay and I'll just point out that I've tried to convey this as simply and as clearly as I could this is the conceptual basis and we're going to work out the consequences of all of this in the next several lectures for those of you that are interested in going more deeply into this there's much more that can be said and Professor Dada's notes are the best references I think those of you that have a flash drive have two or three chapters from the volume that's coming out I believe that talk much more about the fundamentals about these concepts if you want even more information they can refer you to several lectures of Professor Dada I can also refer you to my course on the nano hub if you want to see how to work out densities of states and things like that as discussed there a little bit but that's pretty much it okay so thanks for hanging in there it's been a long day thank you so if there are any questions we can take them now yes I wanted to ask a question from Mr. Gama at the beginning of the talk we presented as a covering part of the interface later on we discussed the diffusion channel you related to the gamma parameter with the diffusion coefficient in the length of the conductor so my question is if the interpretation of gamma has changed so that's a good question you're paying close attention to what I'm saying I hoped I had done that quickly enough but it's a very good question and I can ask Professor Dada to comment if he likes to if you're thinking about if you're trying to compute the IV characteristics of a molecule I think you would think of these tau 1 and tau 2 as the escape time you put an electron on a molecule this is how long it takes to get out if you're trying to think about the current flow through a carbon nanotube or something then the longest time is probably the time it takes to travel across the device and the relevant time becomes a transit time so I think the physical interpretation of that particular time probably depends on which problem you're thinking about is that the way you would think about it usually part of it is the time it takes to get from one contact to another and the major part of it could be the interfaces or it could be connected so in all these things you're assuming you have great contact real time it depends yes first 5-10 minutes you said that we are looking at the steady state so everything we made was for the steady state and you said we connect the channel to the toaster and the channel to the second and then you sum up that you said that it is zero then you turn to the bias banded case but I don't understand why you apply the steady state results to the biased state ok so the question is under bias why do we make the steady state assumption the question is after is zero for example in the slide number 10 slide number what 10 let's take a look this one is this the right one so your question is why is this zero that's just a out of equilibrium we can still be in steady state that just mean when I applied a bias and waited long enough everything will settle down there will be some constant current and there will be some constant number of carriers in there it doesn't mean that things aren't happening in time the electrons are continually coming in one contact and continually going out the other but the electron density has settled down the system will settle itself down until it reaches a steady state value so this would occur in equilibrium but it also occurs if I just have a problem where nothing is changing in time and I'm out of equilibrium so these are all sometimes people ask well what about you have an electronic device and you're switching it so these are these are all steady state they're assuming after the switching event you wait long enough until everything settles down and the current is constant and these are the problems that we're trying to understand in most devices if you operate very very quickly you may be able to see frequency limitations in transistors because the time these transit times the time it takes for an electron to go from the source to the drain starts to become an appreciable time if you have a rapidly varying AC signal going on and if you go to extremely high frequencies you can sometimes see some of these effects yes so we interpret M as the number of units but when we actually calculate it do we always get an integer or does it really mean when we don't get it we think of M as the number of modes but do we get an integer if the structure is very small then you can just count the number and sometimes if you have a nano wire or you have a very narrow nano ribbon you might have 4, 5, 10, 12 modes and you can see its discreteness and you can just count them now as the channel gets wider and wider and more and more half wavelengths can fit into it you just get more and more and it starts to look like a continuous quantity so if you have a wide structure then you don't attempt to count them but in principle there are a finite number there there's so closely spaced you can't count them this interpretation applied to diffusive transport as well yeah yeah so the picture we have is that we have the same channels for conduction diffusive transport it's just that the probability that an electron that comes into that channel the probability that it goes out the other side is now much smaller but the number of channels is the same yeah I have a question on this slide so you made an assumption that tau 1 is equal to tau 2 then you got E over 2 terms in your expressions so if you don't make that assumption then you get some terms in your expressions but what the argument that for Shraddatta gave in the morning that why we have dE divided by 2 term dE divided by 2 term in the expressions he said that it's because the plus k states have like half of the states are plus k states and half of them are minus k states so how do we interpret these two yeah I mean you can just see from the math here that if tau 1 and tau 2 are equal they're going to split in half and then you say physically what's going on and you can see in the ballistic case you can see why it happens electrons come in from contact 1 and they're flowing in a positive direction they can only occupy the plus k states and only half of the states are that way and in the other way if they come in from the other contact half of the states that they can occupy but even when there's diffusive scattering and it's all mixed up you're still going to if the contacts are the same they're still going to divide that way equally but in the ballistic case if tau 1 is not equal to tau 2 then we'll get some we'll not get the e over 2 that's right and I think people that worried about molecular conductors a few years ago it was very difficult for them to make the two contacts identical and they would see asymmetries in the I.B. characteristics that they frequently attributed to the fact that one contact was much worse than the other so they could see effects that are attributed to that mm-hmm looks like 18 18 this one maybe explain what I'm using the broadening occurs and then why does it have to do with tau so the question you're asking is why does the broadening occur yeah and how is it related to tau alright so there are two answers I can give to this you know one is that on that previous slide 10 that we were looking at when I just went through the description you know and we characterize our device by these two times tau 1 and tau 2 which told us something about how fast charge could move in and out I could have done everything in terms of tau and never even brought up gamma right I just introduced it at that stage as an equivalent way I mean I said gamma is equal to h bar over tau but that's not quite it actually does have a physical significance right and if you think back again to if this little device is a molecule there's always this uncertainty relation delta E delta T is greater than h bar so if you have a molecule it has discrete energy levels if you attach it to two contacts and it's now attached to the outside world the energy levels broaden if you want to estimate how much the energy levels broaden you use this uncertainty relation delta E is h bar over the average time that an electron spends in one of those levels so you can give that physical interpretation to it yeah you didn't say too much about how tough it was how tough it was yeah I'm giving you some incentive to stay for the rest of the lectures yeah no I did so one of the questions is what is the mean free path and how do you calculate the mean free path that's going to be important I'm going to do two or three lectures here where I'm just going to assume that some smart person has calculated the mean free path and given it to us it's just a given quantity but in lecture six we'll talk about scattering I'll talk about where that comes from and how you compute it how do you put it this way so we'll talk about MOSFET how do you how do you model so are you concerned about pushing it in terms of applied voltages that you yeah right so in the next lecture we'll analyze some MOSFET characteristics where this really applies is down near VD is near zero you know the initial part of a MOSFET characteristic it operates like a voltage controlled resistor that's where this applies to and we can get some estimate of what the mean free path is of electrons in a channel you know by analyzing that region now if you go up to higher biases you can still develop a model you might know this little model that I published it's just that the f1-f2 you can't use a Taylor series expansion for it because now f2 is much much different than f1 but you can use these concepts and develop a Landauer theory of the MOSFET that works under higher bias so if you look at I think the first summer school on our electronics in the bottom web page you'll see I had a set of lectures there on that approach I think I might do it again next summer you should come back next summer okay any other questions alright thanks for hanging in that well we have one more one more question about the poles we changed the prescriptions if you change the prescription for the thermal energy for the thermal energy for poles no see this is the point you have to be very careful so you can get very confused about holes the point I was trying to make is those expressions that I developed they describe electron flow whether the electrons are flowing through the conduction band or through the valence band now if you're going to say I don't want to think about electrons flowing in the valence band I prefer to think about holes in the valence band then you have to go through and you have to redo these things and you have to be very very careful that you don't make any mistakes but my point is you don't have to do any of that it applies for electrons flowing to the conduction band or electrons flowing to the valence band if you want the conductivity of a p-type material use the same expressions the same Fermi function not a 1 minus F and the Fermi level would be the same well if I'm out of equilibrium it would be the quasi Fermi level for the valence band or quasi Fermi level for holes if you want when you're out of equilibrium in a semiconductor device you have two different electrochemical potentials so if you're talking about electrons flowing in the conduction band use the electron quasi Fermi level if you're talking about electrons flowing in the valence band you use the whole quasi Fermi level