nanoHUB U Fundamentals of Nanoelectronics: Basic Concepts/Lecture 2.2: E(p) or E(k:) Relation
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[Slide 1] Welcome back to Unit 2 of our course. This is Lecture 2 and we'll start from as we had discussed the energy momentum relation. 

[Slide 2] Now, what you may be familiar with is if you had say electrons in vacuum, not in a solid, but just in vacuum, then it could write the energy in the form of like something you could call the potential energy and something you call the kinetic energy, and the kinetic energy is proportional to the square of the momentum. This may or may not be familiar to you and we'll go over that in a minute. And if you plot energy versus momentum then it would look like parabola E is proportional to p squared, quadratic and the bottom of the band that is p equal to 0, energy is equal to U, so that's the bottom of the band here, the lowest energy. Now, what you may be more familiar with for the kinetic energy is not this p squared over 2m, but rather 1/2 mv squared and what I'd like to show you is that that actually becomes what I have written if you use the relations between momentum and velocity, that is, you see you could you write that just algebraically as mv whole square divided by m and then we know that momentum is m times v and so it becomes p square over 2m like I've written there, so those kind of look equivalent but one point I wanted to stress is that they're equivalent only in this special case where what you call a parabolic energy momentum relation; parabolic meaning this E is proportionate to p square, but in general, you could have all kinds of energy momentum relations and in that case this is not quite right in the sense that p is not always equal to mv, so the way we normally think is velocity is kind of the important thing. And then if you want momentum you multiply by m, but what I'd like to point out is that's not the general way of doing this, it's the momentum that comes first and from that you have to reduce the velocity in a certain way that we'll talk about. So, this of course is electron in a vacuum, so what is the energy momentum relation in a solid, well a solid is a pretty complicated object. You that I've tried to show here is this periodic arrangement of atoms inside, so from an electron's point of view it is seeing this array of, you know, this potential due to the background of atoms. So, this is a rather complicated thing and it's not immediately obvious that there should be an energy momentum relation because after all, you know, there's U is not quite a constant, it's changing periodically allover but if we assume a crystalline solid and this is true only of a perfect crystalline solid. What that means is all these atoms are arranged periodically and we have this unit cell then in an average sense over one unit cell, you can actually draw an energy momentum relation. Now this isn't meant to be obvious at all, alright, this as I said is one of the important insights of solid state physics, one of the important early insights of solid state physics and for a proper justification you read the quantum mechanics the quantum models which I am trying to avoid in this course, this is something we take up in the follow-up course where we get into quantum models, but in this course I'm generally not trying to, or rather trying not to get too deep into quantum mechanics. So, what I'm asking you to accept then is that in a solid electrons behave almost as if they're in vacuum, but then the energy momentum relation won't look anything as simple as that and in general could look like anything, and I've kind of drawn a typical sketch of in some certain solids, but in general you see the curve could be anything and there would be a bottom of the band which you'd call Ec, but then you might say well but I've been told that electrons in a solid have a certain effective mass so that they behave as if they're in vacuum with a different mass. Well, that's kind of true if you restrict you attention to a small region here. That is if you look right here, this kind of looks like a parabola and so you could kind of fit it to a parabola using a different mass, not the free electron mass, that's the free electron mass, this is a different mass and in different solids that mass will be different, so that's what you usually call the effective mass, but the point to remember is that that is really just an approximate description over a small range of energies. Why is that useful? Well, often you see if your chemical potential was here then conduction processes as we have discussed depend on the energy levels right around the chemical potential, so energies that are far away don't matter too much, so what matters is only the energies around here and as long as that is true, you could kind of pretend as if this describes the entire curve, of course you're not describing these parts correctly, possibly not even the upper parts, but that's okay, what matters is this part and as long as you fit your parabola well that's good enough. So, that's usually the justification for using this. Now, the point I want to make though is that as long as you have what you call parabolic bands it is okay 

[Slide 3] to say p equals mv like you are used to, on the other hand, the more general relation that will work in respect of the shape of the energy momentum relation is what I've written here. These will be the general form of the laws of mechanics that is the rate of changing of momentum is proportional to this derivative of energy with respect to distance, that is just Newton's Law, you know Newton's Law says that dp/dt rate of change of momentum is the force and force can be written as the gradient of energy, the potential energy that's this. It's this other one that may or may not be familiar to you and that says that the velocity which is dz/dt, the rate at which this electron's position changes, that's proportional to del E/del p, derivative of energy with respect to momentum. If you had a parabolic band, where energy is proportionate to pz squared then this gives you what you would be, what you are familiar with. Why? Because when I take the derivative of E with respect to pz, of course this is a constant so that pops out and I take derivative of pz square I get 2pz and the 2 cancels out so you get pz over m and that's your usual result, p equals mv. On the other hand if it is non-parabolic though, this wouldn't have worked out, I mean you wouldn't get p equals mv, so one common example is what sometimes people call a relativistic band that is actually even in vacuum you see when electrons are traveling very fast close to the speed of light then the energy momentum relation is not E equals p square over 2m, but rather it has this form, this is what you call the relativistic energy momentum relation where zero momentum means E equals mc square that's the most famous equation in physics and then as you increase the momentum it goes up, so this would be the general relativistic energy momentum relation. Now of course in a solid usually you don't have these relativistic speeds to worry to about, but the point is energy momentum relations could easily have this form. Now, this, it is only looks like this in form, but not in terms of the actual constants, that is, for the relativistic relation that c is the velocity of light, but when you have an energy momentum relation looking like this in a solid that c is not the velocity of light, it's usually like a couple of orders of magnitude smaller, it's only just mathematically it has that form. And the question we're asking is, if that's your energy momentum relation, then what is the relationship between momentum and velocity? Is it p equals mv, or is something else? And what you could check out easily is well take the derivative E square so I get 2E derivative of E with respect to pz and on the right-hand side, this is a constant derivative of 0, take the derivative of that we get 2c square pz; 2 is cancelled so what you have is this is the velocity vz is equal to c square pz divided by E. So, if you want the mass that's equal to p divided by v and that is like E divided by c square, so what the means is as the electron energy increases the mass gets bigger and bigger because mass is proportional to energy and with a little algebra but not quite straight forward, not like you would see right away, but if you sat down and did a little algebra you can show that this actually amounts to this m0 we just call the rest mass divided by 1 minus v square over c square and this is a form you might have seen in freshman physics, this relativistic enhancement of mass for example; that when a particle travels close to the velocity of light it's mass gets very large. You have heard that statement, and the point is that in general then when you have non-parabolic energy momentum relations the mass is not a constant it can easily depend with energy. And this is something to remember if you're using say the Drude formula for example because in Drude formula, as I said, the very important thing that is effective mass here and it's important to remember when using this that the mass itself could be energy dependent, so if your chemical potential happen to be higher up in the band you might need to use a different mass, etcetera. On the other hand, of course the way we are doing it in terms of density of states or number of modes or ballistic conductance the mass does not directly appear in there, it only appears indirectly through the density of states. 

[Slide 4] So, the important message though I want to convey is that in general in a solid you could have any shape of energy momentum relation, it doesn't have to be parabolic as it's in vacuum and when people use parabolic bands it's really an approximation, so it could have any shape and in general if you're describing the dynamics of an electron in a non-parabolic band, then the right way to write the laws of mechanics is in this form and this one is like what you have seen before dp/dt is force, but this one you may or may not be familiar with where velocity is del E/del p, so the point I made earlier usually you thing velocity comes first, you multiply by mass, get momentum, rather here the way you should think is momentum comes first, take that, look at the slope that's what gives you velocity. And the other important difference with electrons and vacuum, this is electron in a solid, that's electrons in vacuum, so one difference is as I said, you could have arbitrary shapes here this could be very non-parabolic. The other point is that you have multiple bands, that there's-- you could have a band of energy levels here another one here with a totally different shape, in fact usually the one here might even be curving downwards, you see which kind of corresponds to a negative mass actually. And that's quite possible, so in general then what I'm asking you to accept without going into the quantum mechanical basis for this, is that in a solid you can describe electrons in a solid in terms of this general energy momentum relations which are a little bit like that of electrons in vacuum; it's two important differences, one is this that the shape can be quite strange and the other is you could have multiple bands. 

[Slide 5] So with that now, we are ready to move on to the next step. So the next few lectures will be about how we get from that energy momentum relation, given energy momentum relation, to density of states, number of modes.