[Slide 1] Welcome back to Unit 1 of our course, The New Perspective.
Though this is the last lecture it's time to sum up what we did.
[Slide 2] Remember we started
with defining this concept of density of states.
That's very important as I said.
And if you want to describe current flow
through any device the first step is always
to draw this density of states diagram
and locate the Fermi energy.
Fermi energy or electrochemical potential,
which we just denote with this mu 1
and there should be a mu 2 on this side.
And they're separated by the amount of the applied voltage.
A positive side is lowered by the amount of this voltage,
V. Q times V. So that's the first picture
that you need to draw.
And then what we showed is that if you used this idea
of elastic transport, that is electrons go from one contact
to another without changing energy,
then you see you can write the current in this form.
That you can write the current at any energy
as some conductance function times the difference
between f1 and f2.
Because, as we discussed,
the essential point why current flows is
because f1 is different from f2.
So one contact tries to bring it into equilibrium with f1
another contact tries to bring it into equilibrium with f2.
And in the process electrons keep coming in and going out.
So that's the current through a given energy.
And since every energy channel is independent
because the way you're thinking is electrons,
whatever energy they have coming in,
they have exactly the same energy going out.
So you can take all these different energies
and just integrate it over energy to get the total current.
So that's the basic current formula that we obtained in,
I believe, lecture three.
We then went ahead and said let's assume the applied voltage
is small and obtain an expression for this current
versus voltage.
Rather current divided by voltage.
And that's the measured conductance and this is given
by an average of what we call the conductance function.
Something that depends on energy.
And if you have to average it over energy in order to compare
with measured values, and if you're at low temperatures,
than this function is very sharp.
And so the value of this only at the Fermi energy is what matters.
Fermi energy of this chemical potential.
So what matters is only the values right there.
Whereas if you're at relatively high temperature then you have
to average over energy window of the order of kT.
Now what's this expression for conductance?
It depends on density of states and the time it takes
for an electron to get from left to right.
We then talked about the ballistic conductance
where the time is given by length divided by velocity.
And so you have an expression for the ballistic conductance,
the Sharvin conductance
that we talked about.
And then we say that in general when it's not ballistic, see,
then you can write it
as the ballistic conductance times this mean free path
over L plus lambda.
And what's in the numerator here?
GB lambda is what you can identify
as conductivity times the area.
So this point of view then, kind of,
connects this ballistic conductance,
which is what you normally learn in a course
on mesoscopoic physics with the diffusive limit,
which is what you learn in a normal course on devices
or on solid state physics.
And this, kind of, connects the two.
You see it smoothly interpolates from one limit to the other.
So we don't have a separate story for ballistic
and another for diffusive.
It's just one continuous story.
If L is small you get the ballistic limit.
If the L is large compared
to lambda you get the diffusive limit.
And this lambda here we defined
as this diffusion coefficient two times the diffusion
coefficient divided by the average velocity.
And then we went ahead and-probably it was
[Slide 3] in lecture eight that we obtained an expression
for this lambda in terms of this velocity and time.
That is we showed that the average velocity would be this
v, which corresponds to a particular energy
that is electrons and a certain energy,
let's have a certain magnitude of the velocity, that's v.
And in different dimensions the averaging gives you is different
geometrical factors.
And similar diffusion coefficient is v square tau.
And then with these, different geometrical factors, again.
And if you look at lambda
from those you can deduce that's equal
to v times tau velocity times the mean free time,
which is normally what you'd have called a mean free path.
But then we have this extra factor in here that depends
on whether you are in 1D, 2D, or 3D.
Now why do you have these factors here?
Why not just define this as v tau if this has mean free path?
You know, v tau is a mean free path.
Well if you did that then what would happen is these equations
would look a little more complicated.
It would be like it's L plus - you'd have L plus some
geometrical factor times lambda,
which would depend on the dimensions.
But the way we have done it,
it's like this is a general expression
and then you redefine your mean free path depending
on the geometry.
[Slide 4] OK? So one point I stressed is that this point
of view not only gets you results for nanoelectronics
for very small devices, but also connects smoothly
to large devices.
And that's kind of important, you see, because lots of devices
that are of interest are really not quite
in the ballistic limit.
It's usually somewhere between ballistic and diffusive.
It may not be completely in the diffusive limit
but it's somewhere in there.
And so what you really want is a theory that goes smoothly.
That allows you to think in a continuous way
from one limit to the other.
Now you might say that, well, for large devices how is it
that we are able to get this expression for conductivity,
this general expression for conductivity that we talked
about that normally you need advanced formalisms to get?
You know, the one that is obtained from Boltzmann equation
or from what you call the Kubo formula.
We obtain that expression for the conductivity, actually.
Just from this elementary discussion.
And what made this possible is
because we were using this idea of a elastic resistor.
The idea that electrons go through the device elastically
and all the dissipation occurs in the contacts.
As I've mentioned before in the introduction
that these are two very different types of phenomena.
These are two branches of physics
that developed separately, you see?
And what usually makes transport such a difficult subject is
that this is all mixed up.
You see? So usually what you need is this Boltzmann equation
for semi-classical transport.
For quantum transport you'd need the quantum version of it.
Where it would be Schrodinger's equation plus
entropic forces.
Whereas when semi-classical transport what you use is the
Newton's Laws plus entropic forces,
which is what the Boltzmann equation is.
So in general your benchmark, your gold standard,
is this Boltzmann equation.
So all results that we are talking about here you can get
from the Boltzmann equation.
See? But the Boltzmann equation is takes some time
to get used to.
There are many things that you have to get familiar with
and you have to spend a semester or two, you see,
playing around with it to get there.
Whereas what we were able to do is get results very simply
because we made this assumption.
Because when you make this assumption then you see the
mechanics are separated from the thermodynamics.
And so you can take care of the thermodynamics in a way that's
so simple that you don't even realize what you're doing.
You see the way we took care of it is by saying, that you know,
we've got these two contacts, these big contacts,
and there's all kind of thermodynamic processes going
on there, and what they do is they maintain the contacts
in thermal equilibrium.
There's very complicated processes
that maintain a thermal equilibrium
and that's what is reflected in this f1 and f2.
So just by legislating that the electrons
and the contacts are distributed according
to the Fermi function, we took care of the thermodynamics.
But when an electron comes out in the contact,
you see, it is a hot electron.
It's got much more energy than it should have.
And then you see there's all kinds of complicated processes
that restore equilibrium.
But we never got into those details.
We just said that's OK.
We know that what all those thermodynamic processes will do
is they will maintain this reservoir at thermal equilibrium
with a Fermi function f2 or f1,
depending on which contact we are talking about.
So that's how, you see, we were able to get our results
in a relatively simple way
without explicitly involving the Boltzmann equation.
But whatever we are getting though is exactly what you'd
have got from the Boltzmann equation
and we'll talk a little bit about that
in the third unit of this course.
But you might say, well, but that assumption
that an electron does not lose energy in the middle that, well,
I can buy that for small devices but how can it justify
that for long devices?
[Slide 5] That's where, again, as I mentioned in the introduction,
the way we are treating long resistors is we are thinking
of it as if it's a series
of little nano-resistors.
Little elastic resistors.
Electron goes elastically for a while
and then loses some energy.
Again, goes elastically for a while, loses some energy.
Of course, in the real thing its continuous but here we, kind of,
approximated as if it's going discontinuously.
And then what we do is focus our attention on one
of these little units and deduce an expression
for conductivity and so on.
And the thing is the answers we are getting is exactly what you
get from a rigorous treatment of the Boltzmann equation.
So not that you're losing anything
by way of, you know, rigor here.
The actual answers come out exactly the same.
See? For low bias.
Now for if the bias is higher then, of course,
then this only gives you an approximately physical picture
and that's something we'll talk about a little in the next unit.
Now one subtle question that I wanted to stress then is
that when I go from this expression for current
to this expression
for conductance we use this Taylor series expansion.
And that expansion requires you to assume
that the applied voltage is much less than kT.
We say well, Taylor series expansion is something
that only holds for a small value of X. And the X
in this context is applied voltage over kT.
So you might say, well, but then are you saying
that you have this linear dependence of current
on voltage only for small voltages?
Because our flow kT's like 25 milli-electron volts.
So it means if you apply a voltage bigger
than 25 millivolts you'll think this expression won't work.
But we all know that when it comes
to big resistors they're often linear even
if you apply 10 volts across it.
Like if you got a little resistor
from Radio Shack it will probably be linear.
You could apply volts across it and it'd still be linear.
But this theory seems to say that anything more
than 25 millivolts and it won't be linear
because I won't be able to use Taylor series expansion.
And that is where, of course, what if you remember is
when you have a long resistor like the one you buy
from Radio Shack - see in order to apply this point
of view you have to break it up into little resistors.
And the length of each section should be, kind of,
much less than an inelastic scattering length,
in a length that an electron traverses before losing energy.
So you should use a relatively short section here.
See? And the point is
when you're applying this theory we're applying it to one
of this short sections.
And so in order to be linear the voltage drop across one
of these short sections should be less than kT.
So the idea is you shouldn't have more
than 25 millivolts applied
across a small section who's length is roughly how far an
electron travels before it relaxes energy.
[Slide 6] OK? So with that then I guess we are done
with this first unit of the course.
Note that in this discussion we kept things kind of general
in the sense that we expressed everything in terms of density
of state without talking too much
about what exactly the density of states is.
How do you model it?
Where do you get it from?
What it is?
Right? So the results are very general
and could apply to any solid.
Any solid meaning, you know, amorphous solids.
We could talk about molecules.
Could use this basic results that we talked about.
In the next unit what we'll do is we'll adopt something called
this energy band model.
Something that is widely used
for discussing crystalline semi-conductor.
Crystalline solids, in general.
And see how you can obtain expressions for the density
and apply it to the formulas, the expressions
that we developed in this unit.
Thank you.