So in the second part of this segment, I'm going to elaborate on the weak
localization phenomenon, and in particular, I will discuss in more detail
the self-crossing trajectories that are, are responsible for it.
And in particular, I will identify the physical circumstances under which they
are actually important. And this discussion is going to be under
the title I put here, Quantum corrections to diffusion, but before going back to
these quantum corrections that we already mentioned in the previous video let me
remind you what classical diffusion actually is.
Actually the phenomena of diffusion you can observe at home by performing a
rather, very simple experiment. If you put a droplet of dye in water and
watch it spread out with time. So this evolution of this droplet, time
evolution. Spreading out of this droplet is going to
be described by the diffusion equation presented here with some diffusion
coefficient D. So but the phenomenal diffusion the
diffusion equation itself are of course much more general than this particular
physical realization. And they happened whenever we have a large
number of particles experiencing a random walk.
So random is, if you can, you can think about it as particle which moves every
time step it moves in a certain direction it can go up, it can go right, it can go
up, it can go down, it can go left, up et cetera, et cetera.
And so, at each moment of time there is a probability, a equal probability to go in
any, in any direction. And so, if a lot of size particles are put
together. They, if, and if you look at those
particles from a large length scale. So, their density.
The evolution of their density is going to be described by this equation.
As with here rho is in general a, a function of a position and time.
So, how it is relevant to the previous discussion about electrons moving in the
metal? So there as we mentioned, so the disorders
the imperfections in the metal play the role of scatterers.
That's somehow randomized the electron trajectories.
And so, we can think of what electrons experiencing this random walk.
And since there are many, many such electrons in a piece of metal, so if we
want to extract their density the density is going to follow the diffusion equation.
So another interpretation of the diffusion equation is that this raw Is actually
probability density of finding a particular a random working particle in a
certain moment of time at a certain moment of time in a certain position in space.
So there's probability interpretation of this equation if you want.
Now another comment I would like to make about this equation is that actually in
some sense, it looks similar to the Schrodinger equation in quantum mechanics.
So if we were to put in the measuring constant i here, which we don't have to
put, but if we were imagine that some measuring time, so and so the coefficient
D we would have let's say minus h squared over 2 m.
Then this equation would have become a free [unknown] equation of a free particle
of the wave function. So this is a purely mathematical analogy I
should Could say, but this mathematical analogy in certain cases actually goes a
long way, and allows us to calculate things that otherwise very, would have
been very difficult to calculate. So, at this stage however, we don't want
necessarily to involve any, this analogy. Instead let me focus on the question of
relevance to the phenomenon of weak localization that I mentioned.
Namely on the self-crossing trajectory. So here, I'm talking about self-crossing
trajectories of some diffusive particle. So let's say, here actually, what I mean
is that there is some small length scale at which particle diffuses around, but as
a result, in a realization of this random walk, it may lead to this self-crossing
trajectory. And so from the perspective of this
crossing point, what it actually means is that.
I'm asking the question of what is the probability for a particle to return to
the same point where it started. So let's say I set an initial condition
for my density at t equals 0 to be a delta function in space.
A delta, very sharply localized density using a particular point.
Now it turns out one can actually solve for exactly the diffusion equation with
this initial condition is actually called the room function.
In this fundamental solution the general solution to the diffusion equation is
written here. So here again r denotes the coordinate in
space at T star, D is the diffusion coefficient and the small d denotes the
dimensionality of space. So it can be either one dimensional space
or three dimensional space, or two dimensional space.
And in order to, for us to find the probability to return to the origin.
Let's say we have, we are looking for the probability of a self rising trajectory
which starts from r equals zero. And goes back to r equals 0.
So it means that we have to set r to 0 in this ex, expression or set expon,
exponential to 1 and that's what we're going to get for the probability of
self-intersection. So again, what this guy implies is that if
we take random working particles, put it by hand in the origin and let it random
walk. So this guy gives us the probability of
its return to the vicinity of the probability density, more precisely the
return to the vicinity of this point in the time sheet.
So, now let us recall about what we were talking about this self crossing
trajectories. So the context in which they appeared was
the diffusion of electrons in the metal and we wanted to see again, we wanted to
find the probability of where an electron could diffuse from an initial point and at
some final point. And there are many different trajectories
of the q that so and we focused on the quantum terms.
The quantum interference terms were different trajectories where l1 not equal
to l2 would interfere with one another and the interference term was this cosine of
the difference in length between two given trajectories, times p fermi, the typical
momentum of an electron divided by h bar. And the statement was, unless l1 was
actually equal to l2 this interference term would kill each other.
Or every [unknown] to zero, because you will get just a collection of positive and
negative random numbers and on average you will get 0.
But, for the special kinds of terms where you can go clockwise or counterclockwise
around the certain trajectory, this delta l here would be 0 and loci of 0 is equal
to 1 so we'll get prescribed contribution from these guys.
And so what we're trying to figure out now is how probabilities for us to actually
get the self crossing trajectory[UNKNOWN] and this is, is determined by this
equation. But this guy tells us this probability
density to return to, to an original point in the particular time t.
While what're what we're interested in is any self-intersection, it doesn't have to
a-, occur necessarily in this particular time t.
It has to occur at some point. So in order to calculate the total
probability which I will denote as P sub total, I have to integrate this expression
over time, from some minimal, t to some maximal t.
So in here, I should mention that I'm dropping the overall coefficients.
I'm not keeping track of the coefficients. I'm just focusing on the most important
truth. Now the minimal time here is the minimal
time that the diffusion equation is able to resolve being an approximation it
cannot describe the motion shorter in time than the consecutive collision let's say
of the electron between two [unknown] so this guys of the order tau was actually
introduced. In the beginning of the previous video,
but I should mention that it's actually not important, what it actually is.
It, it is only important that there is some cutoff.
On the other hand, if there is no dephasing processes, if there is, you
know, the phase, quantum mechanical phase is perfectly preserved, then there is no
maximum cutoff, so this guy actually goes to infinity.
And therefore, the integral that we actually have is an integral from sum tau
to infinity of d, t over t to the power of d over 2.
And this integral we can now easily calculate.
And what is extremely important, what is crucial Is that this integral is finite
if, we are in 3-dimensional space in which we live, but it's infinite, it's equal to
infinity, if we are in one or two dimensions.
So physically it implies the following so that if we start let's say, imagine you
are in a spaceship and you start, you're sort of flying around in completely random
directions in 3 dimensions, so this result tells us that there are going, you're not
going to, you're going to be lost. You're never going to return to the point
where you started from. On the other hand, if you start walking
around in a city, in a town, in a city you don't know, and you, you walk randomly
long enough, you're actually guaranteed to return to the vicinity of the point where
you started. From the point of view of the electrons,
so this result implies I think we have a motion of electrons in a.
One and two dimensions. So there will be an infinite number of
self-rising trajectories and the probability of self rising is going to be
equal to 1. So this term is actually going to matter
enormously at low temperatures. And they're going to blow up and lead to
localization in, in low dimensions and this is actually what happens.
And so if we now actually think about the physical consequences, of course, again,
so this, in principle everything I'm talking about here is rather complicated
theory, so if we were to study it in full details.
But, I just basically want to, maybe I will leave this result here.
The totally probability of self-intersection is really the key.
The reason for this video is just to give you this result essentially.
What it tells us in conjunction with the path integral argument and some iterative
arguments. Is that in low dimensions the electrons
are going to get localized at lower temperatures.
They're going to tend to return to the same point.
And this localization is going to be consequence of the, wave nature of
particles. From the experimental point of view, what
it actually menas is that If you perform a measurement of, let's say, resistance,
which is 1 over conductivity, as a function of temperature.
So at high temperature we're, very with [unknown] disturb the phase.
So this is quantum interference phenomena. Basically unimportant whether or not the
trajectories cross or not doesn't, doesn't play any role.
So you will have some resistance, and it will flow actually decrease.
But then, at some point, it will actually shoot up and become an insulator.
Now, in this qualitative curve, so this part of the curve belongs to the weak
localization phenomenon that we have been discussing.
And this part is the classical due to conducitivity which goes back to more than
100 years ago. So, and the prediction is that in one and
two dimensions, per this argument, these quantum interference terms are going to
take over, and they're going to get an insulator, while in three dimensions,
actually, what's expected is they're going to remain metal, and the resistivity is
going to saturate in 3D at some finite value.
And this is indeed what happens. So, and there do exist quasi one
dimensional and quasi two dimensional [unknown] let's say nano wires and so
called semi conductor heterostructures, two dimensional fields where this is
observed Zero. So I should emphasize again that the full
theory of weak localization is an extremely complicated theory, and we just
got a flavor of it. So hopefully but it's not the full story
and interestingly enough, what I told you about here actually this weak
localization, is a precursor to a different phenomenon.
Which is actually called strong localization for which Phil Anderson, a
professor Emeritus at Princeton got his Nobel Prize in 1977.
So we obviously don't have time now to discuss in any detail this very
interesting and complicated theory of Phil Anderson, but let me just mention that
next week, we're going to go back to, to centered quantum mechanics and one problem
we're going to solve is a particle in a potential way out.
And in this problem we, we're going to see that the particle when put in a potential
well gets localized by the well with some discrete energy levels not continuum of
energy levels but discrete levels and so the answers of the theory of Anderson if
it can be summarized in a few seconds is that in a random landscape of such
potential wells, so this is let's say v of r.
So the energy levels that electrons are going to acquire are not going to match
each other. And so they're not going to be able to
move up from well to well because of the energy conservation.
And this would completely suppress the conductivity and this will lead to the
strong localization. So that's the only thing I'm going to say.
And next week we're going to study actually we're going to study simpler
phenomena using the Schoeringer formulation in particular the appearance
of discrete energy wells.