So in this final part of the lecture we'll solve the Cooper problem, which is
a problem of weakly attractive electrons near the thermos surface that form pairs,
so-called Cooper pairs bound states of 2 electrons.
And as we discussed these Cooper pairs, the, at low temperatures, they both
condense into [INAUDIBLE] condensate should become superfluid, and this is
this amazing state that we know as superconductor.
I should mention that we will see actually the solution to the Cooper
problem will involve essentially the same method we used in the first lecture this
week in the context of single particles quantum mechanics, so it's going to be
actually very simple. Technically, a very simple solution for a
Nobel prize winning work anyway. So, it's not trivial but, you know, this
is a very important work. So, here I'm actually showing the the
title of the original paper by Leon Cooper.
So and this work of Cooper eventually has progressed, has developed into a much
more sophisticated theory, which is the Bardeen-Cooper-Schrieffer theory of
superconductivity, also called the BCS theory.
So, and this BCS theory has been the cornerstone of the theory of
superconductivity for many, many years. And frankly, there is nothing better on
the market in some sense. So, this BCS theory describes very well
most conventional superconductors. And it provides some understanding also,
in the physics of unconventional superconductors.
Although they're sensuously a high temperatures super conductors for example
although their understanding is not complete.
now as I well I shown many times already excitations for various Nobel prizes and
and that just caused briefly a number of Nobel prize winning works and they should
mid, admit that. In most of these cases I had to sort of
simplify things enormously when I was providing explanations, sort of dumb it
down to some degree. So here, well I'm not going to discuss
the BCS-theory, this is a very complicated theory, but the paper of
Cooper is actually very, very simple. So, the original paper which started it
all is very simple. And we will be able to understand
essentially its main message and the derivation.
Now but before going into this derivation let me discuss the origin of this
attractive interaction. So I basically, w,hat I, what I want to
solve again is a problem of electrons interacting via an attractive potential.
But where is it exactly coming from? So this is actually very complicated and
intriguing question and well, it sometimes the mystery of let's say high
temperature of super conductivity is in that, we don't really know where it is
sort glue this attractive interaction might be coming from.
So the origin of electron interaction and conventional super conductors is also far
from obvious actually. So, but there was a breakthrough paper
back in the beginning of the 50s. so I'm showing you here the title of this
paper, Superconductivity in Iso, of Isotopes of Mercury.
And to appreciate the importance of this paper, let me just remind you of the
basic picture of, what a metal is. So, well, if you have a metallic system,
well, assorted system, so the reason it's assorted is because, the ions form a
crystal lattice. So they are positioned in space in a
regular fashion. So I'm just drawing here for the sake of
simplicity, as a square a lot as 2D but in relatives, it could be more
complicated. Three crystal lattice And, you should
think about having positive ions sitting here in these slides, and these line are
slides. And these lattice is sort of elastic,
well it's a recent object but it can have a oscillations, waves running through it,
and we're going to discuss a little bit in the few, in a few weeks, and so these
oscillations of the lattice are called phonons.
Now electrons in the metal are moving moving around, they're free to move
around, but they move around in the on the background of this lattice.
So this red guys here are electrons. Okay?
So this is pretty much what, well it's very [UNKNOWN] model, simplified, over
simplified model of, metallic soil. Now, so this paper, what this paper has
achieved is that it looked at different isotopes of mercury which is a
superconductor. And let me remind you that the isotopes
are essentially different versions of the same chemical elements.
and basically everything is the same for different isotopes apart from the number
of neutrons in each ion sitting on the lattice size, but basically.
So different isotopes means that everything is the same, but the only
difference is the mass of these objects on the lattice size.
Which well now really shouldn't affect too much superconductivity because
clearly superconductivity's coming from whatever electron, electrons are doing
and it shouldn't have much to do with the lattice.
Right? So because these guys are actually, well
they can oscillate a little bit near their equilibrium positions, but they
cannot conduct electricity. So as it turns out, and this is really
the main message of the paper, that the transition temperature, in the
superconductor, the temperature at which the resistivity drops exactly to zero,
depended very strongly on the mass of these guys.
So and the behavior the ions the lower was the transition temperatures.
So here maybe it is hard to see so there are a few points so this has average mass
number and this has transition temperature.
So so this is basically let me write a tc and this is the mass of this ions on
lattice size. Okay and so this was a very clear-cut
trend and it was a smoking gun of that superconductivity had something to do
With interactions between the electrons, and decrease the lattice.
So a little later, people realized that what actually happens is that this these
electrons, moving around, they whenever the electron passes through a region in a
lattice, essentially because electron's negatively charged.
And the ions sitting on lattice size are positively charged, so this electron
polarizes the lattice, okay, so locally. And it takes some time for the lattice
sort of to relax back. And then if a second electron comes in,
so, let's say I have here now this guy goes away, and there's a second electron
which comes in into the same, region. So, this, electron, this second electron
is attracted in some sense to, to this region and, this, results in an effective
correlation, effective, phonon-mediated attraction between the electrons.
It's a very tricky business. So, I, if you don't understand it, well,
don't be surprised. I don't understand it either.
Attraction but that's how it works so, we'll have to you know just accept this
fact. So that what happens is that electrons in
some sense they exchange waves elastic waves which are called phonons and this
leads to effective attraction. Well, and by the way for the model for
the actual solution, I'm going to present the origin of the attraction does matter
that much it will be just some attraction, some constant minus v not
but, you know, it's good to know the physics behind it.
For the actual compilation we'll need a specific mathematical model of this
electron, electron attraction. And the true attraction the true
interaction is actually quite complicated.
But here I will present what I call a spherical cow model of the phonon
mediated attraction between electrons. which I should say actually is used
commonly even in research papers. So it works perfectly well.
And even though we know how to write the true model which would involve all the
complications of the theory. So the results, the outcomes, of the
simplified model and the I'm not going to give a little more, or pretty close to
each other, so there is really no reason for us to complicate things.
In any case, so what we were talking about is that again, electrons moving
around in a presence of a crystal lattice, they exchange, waves drawing
through the lattice are these phonons. And so this diagram is an example of a
fine metallic like diagram which shows this kind of a change.
And in any process like this we must satisfy basic conservation laws
particularly when must ensure that both energy and momentum are conserved.
So and therefore what's important here in understanding the structure of this
interaction is the typical energies involved in such processes.
So what we know, what we've already discussed is the the typical energy of
the electrons in the in the middle. So and this typical energy, I denoted as
E sub F, the Fermi energy divided by the Boseman constant, sort of to convert into
a more, familiar temperature units so the corresponding typical sort of temperature
of electrons is going to be about 10,000 Kelvin.
So this guys which are actually the main players.
Of the theory have the energy, which you shall see with this temperature scale.
Now it turns out that the phonons, it's not obvious from the previous discussion.
It's not something we can get at, I'm just giving you an experimental fact, is
that if the typical energy of the phonons, which I will call s h times I
mean divide, so called divide frequency divided by the Boseman constant.
So when convert it into temperatures, it's about 400 kelvin.
A few hundred kelvin, so, which is two orders of magnitude smaller than the
energy of the electron. So what we should think about is that,
this phonon actually being exchanged by the electrons have energy which is much,
much more than the electron energy itself.
And, this, puts certain constrains on the possible, momentum and energies of the
electrons that may experience sort of a process.
And the simplest, sort of, version of the interaction which takes this into account
is, presented here. So this is an equation for an
interaction, momentum independent interaction between the two electrons V
of p and what it tells us that the two electrons attract each other in this
language if they are located in the narrow shell near the Fermi surface and
the width of the shell is of the order of this phonon energy.
So basically if, two electrons can exchange phonons, they do so and this
results in the attraction which is manifested here through this negative
coefficient. And V naught is some constant.
And if two electrons are located well, beyond this narrow region So then they
don't have a mechanism to attract each other.
And their interaction, therefore, is 0. So if we look just at the upper part.
So v of p is equal to minus v, 0. So this would correspond to, essentially,
a local attraction real speed. So this is something that we actually
saw. In, lecture number five for the delta
potential. So this is the kind of problem we,
discussed already. And, but, however, then, we were talking
about the, one particle in a, in an quantum well.
So now we're talking about two particles two excitations interacting with each
other by this potential. But per, well put the results of the
previous segment in this lecture we know that the two particle problem and quantum
mechanics is to a large degree equivalent to a single particle quantumic angle
problems and as we discussed in the previous video the only difference
between the. single particle problem in potential v of
r and 2 particle problem of 2 particles interacting by the same potential is that
in the lecture we will get the reduced mass instead of the mass of eac, of each
individual particle. And in the case of two identical
electrons the reduced mass is equal to m r 2.
So we, we have this simply this Schrodinger equation.
And here I write it first in real space sort of but keeping in mind that there
are in fact additional constraints on the interaction.
So if we do if we follow the same route that we did in lecture number five and do
Fourier transform. So, we can write this equation in this
form, so delta of r emptying on psi of r picks up psi of of 0.
And so here I'm going to have this integral over all momentum.
But basically a way to introduce constraints on the potential which we
just discussed would be to limit the integration or momentum here by only
momenta that appear in the vicinity of the Fermi surface, you know, so that the
possibility of phonon exchange exists. And so this equation essentially, these 2
equations with the appropriate sort of caveats.
About where this reaction is possible is exactly the Cooper pairing problem, and
as you can see it's essentially identical mathematically to the kinds of problems
we saw in lecture number five in the single particle of quantum mechanics.
And so, well we can we can follow the same route but at at this stage we may be
a little bit surprised by this resemblance.
Because, I mentioned that this, eventually the result of this calculation
is going to be the appearance of a bound state between the two electrons.
But, as we discussed, weak attraction in, three dimensions, and this is a
3-dimensional problem, does not result in a bound state.
So delta function potentially 3D does not have a bound state.
Uh,however, an interesting thing that happens here is that because electrons
really play a role in, in this, pairing and this interaction, with for-nodes,
they exist in the vicinity of this firmer surface.
And this surface is two dimensional. This, gives rise, in a sense, to
reduction of dimensionality. Although, in real space, we started with
a three dimensional, system, so this effectively what you are dealing with,
with here has used dimension. And this gives rise to the appearance of
a bond state in a very unexpected way. So, to see this we essentially have to
repeat exactly the same steps as we did in parts 3 and 4 of lecture number 5.
So, so the only difference here is due to the fact that this energy in the left
hand, in the right-hand side, if we want to find, really consists of two parts.
So this is the energy of the electrons that already have a finite energy at the
at the Fermi surface. So remember is that we're talking here
about these excitations that have the energy of two E-Fermi, and then there is
an energy due to the interaction which we want to negative.
So if this guy delta is negative, it means that we can have former bonds
stayed, and by doing so lowering, lower the energy.
So and well this fact actually makes all the difference.
Actually this is how mathematically the fact that we're dealing with this
manifest itself in this equation. Since the remaining calculation follows
almost 1 to 1, that in lecture number 5, and I also present this particular
calculation, all the details in additional materials, so you can read
through this. So I'm not going to repeat this again in
this video. So what I'm going to do, I'm going to
just present the final result for this Our key parameter delta which again is
the energy of the bounced state of two electrons that form sort of a large
molecule, if you want, which is called Gutterberg here.
So and this delta is is going to be of the water of minus h of energy so this is
phonon energy, typical phonon energy that we discussed in the previous slide.
But the key thing here is that there is multiplied by an.
Exponential of minus one, some constant, I'll call it n naught, and define it in a
second, times v naught. So where n naught is equal to m, p fermi,
so this is this threshold momentum, divided by 4 pi squared, hq.
So this is just constant, but what I want to emphasize, well, first of all, there
is a solution, which is great. So the electrons indeed,can find a way to
lower their energy by forming these [UNKNOWN] that eventually form the
super-conductor. but what is not so great is that these
small interaction energy appears in the exponential in, minus 1 over d naught.
So if you notice, small then, well the, energy of the bound state, so
called,[UNKNOWN], is exponentially small and this is exactly what happens.
And those of you followed, all segments in lecture number five can recognize.
This result is a result that we have seen, actually, in the context of
two-dimensional, single particle problem. And this reflects this reduction in
dimensionality that I mentioned before. So, the last two things I'm going to
mention in the end is. First of all, this result.
For delta is a function of not is a very unusual function which we've already seen
but, this function is special because it doesn't have a Taylor expansion.
So there is no way one can approach this result by doing so called perturbation
theory. And this may be one of the reasons why it
took so long for people to figure out the heat of super conductivity, because there
was no way to approach, it sort of in a conservative way.
And, the last thing I am going to mention, is that well, if we did not have
this exponential, if we were to image that this delta was the order of this
phonon energy. This would have actually implied that we
can get super conductivity at very high temperatures up to room temperatures.
So, this would have been great we would have the ability to transport electricity
with no losses. So, this is really the grand challenge in
the field of how to go beyond this coupling this so called weak coupling.
Bearing model, so how to first of all describe this situation where the
interaction is so strong and how to get rid of this exponentially this is a sort
of practical problem that there's no solution at the moment.