[Slide 1] Welcome back to our, the
first unit of our course.
This is The New Perspective,
so this is the third lecture
where we'll talk about
this current flow
and obtain an expression for
the current through a nanodevice.
[Slide 2] So as we discussed
in the last lecture,
the first thing you need
is this density of states
and the electrochemical
potential
which tells you how
far these states are filled.
At equilibrium, both
contacts have the same mu,
same electrochemical potential.
Out of equilibrium, you have
to 2 different electrochemical
potentials,
the positive side is
lowered with respect
to the negative side
by the amount qV.
Now why does current flow
when you apply a bias?
Well, the idea is
that if you look
at these states this
contact wants to fill them
up because it wants to bring it
into equilibrium with itself,
but this contact wants to
empty it because again,
it wants to bring it into
equilibrium with itself see,
so every time the state is
empty, the source fills it up
and the drain takes it out
and they electron then goes
out of the drain into the
battery and a new one comes in
and so this process
goes on forever.
Now why do states down
here don't contribute
to current flow?
Well, because as far as these
states are concerned both
contacts want to keep them
filled, so it just stays filled,
end of story, nothing else,
no current flows, okay?
And this is the part that
is hard to understand.
If you think of electric
fields as driving electrons,
because in that case electrons
down here also you might say
should start moving, okay?
But, what is well-known,
widely accepted is
that current flows only
in the small energy window
around the Fermi energy or
the electrochemical potential.
States deep down here play
no role in current flow.
[Slide 3] Now, let's try to make
this quantitative,
let's try to get an
expression for the current
and let's first do
this approximately
and the next slide I'll
do it a little better.
So first let me assume that
over this entire energy window
of where we have applied the
volt, you know, which is opened
up by this applied voltage.
Let's assume the density of
states is almost constant,
so then how many states
do we have in this window?
Well, it will be D times qV.
So, how many electrons
will we have in here?
Well, it's like half of that.
That's because if you think
about it you have these states,
the left contact wants to fill
them up, the right contact wants
to empty it, so what
you'd expect is
that these states would be
kind of filled half the time,
so we'd get filled up, taken
out, okay, that's filled up,
taken out and as long as
it's equally connected
to both contacts, you would
think these electrons would be,
would fill up the states
like half the time.
And so the number of electrons
in the channel would be this.
Now the flow of electrons,
that's this electrons
per second,
that's this current divided by
q. That is current is the amount
of charge that flows per
second and if you divide it
by the charge on 1 electron
then what you get is how many
electrons flow per second, and
this is the amount of electrons
in the channel, this is
the flow of electrons,
and they are related by, so
number of electrons is equal
to the flow of electrons
per second times the time
that an electron
spends in the channel.
And I'll try to explain this
point a little better shortly,
but if you accept this then
you see you immediately get an
expression for the
conductance; current divided
by voltage just simple
arithmetic and you get algebra
and you get q squared D over 2t,
so that's the density of states
and that's this time it
takes for an electron
to get from left to right.
So, where did this
equation come from?
Well, one analogy I've often
used is supposing we have a
graduate program, let's say
where say a 100 students
graduate with a PhD
every year and let's say
every student take the 4 years
to get a PhD.
Then what would be the
steady state number
of graduate students
in your program?
So, say a 100 per year
times 4 years per student,
so you would have 400
students in your program.
If you think about that,
but this idea is, you know,
is something that you would use
to estimate the steady
state number of PhD
students in a graduate program.
You see new students continually
come in and leave and graduate
with a degree, so
same here you see,
new electrons continually come
in and leave, but steady state
if you want to know how many
electrons are in channel that's
like your graduate
program, then that number
of electrons would be
the flow times the time
that each electron
spends in the channel.
Okay? So, this is an
equation, this express,
relation is often quite useful.
You see sometimes it may be
easier to estimate I you know I
and then you can estimate t,
or if you know t then you
can estimate I etcetera,
so you have occasion to
come back to this later.
Okay. So, but if you
accept this as you see,
you can get the conductance
right away, but I assume here is
that the density of
states is constant
over this energy window, so
let's now do it a little better,
that is let us assume
that it is not constant,
[Slide 4] so we consider a
small energy range dE.
Okay? So, let's us consider the
small energy in d so the number
of states in that energy window
is density of states times dE.
And so the number of
electrons is that divided by 2.
So, previously we done D times
qV over 2, now we are D(E)dE
over 2 and then there's
the current
and then there's the time spent
and in principle the times spent
can also depend on the energy
because at certain energies the
electrons might have a bigger
high of velocity and
zip right through,
other energies it
might take longer,
so we could write it this way
and then you can rearrange it
and write the current
as dE qD over 2t.
Now this is then the current
that you'd have if the states
at this were completely full
one and this end were 0.
On the other hand, in general,
these states may
be partially full,
this states may be partially
empty, so you might have like,
so the occupation on
this side is described
by the Fermi function here,
occupation on this
side is described
by the Fermi function
over here and so
in general you would have
this factor here f1 minus f2.
So for example, if both
sides have the same f then
as we discussed there
would be no current flow.
Current flow is because, you
know, one wants to fill it up,
the other one wants to empty
it, but energy is down here
for example, f1 is equal
to f2 both are equal to 1
and so there is no current flow,
so this factor here
automatically takes care
of this part; the current
is driven by the difference
in this Fermi functions, that
is, and the two Fermi functions
in the two contacts are different
and that's what causes current
to flow, so this is the
current at a single energy
and the thing is that in this
viewpoint we are assuming
that an electron goes from left
to right at the same energy
without changing energies and
that makes it very simple,
you see, you can think of
different energy levels
as all being independent, so
this is elastic resistor idea;
that current is carried
by different energy
channels all independently.
And so if you want the total
current, you just add it up,
and add it up means
you integrate it.
So once you have done that,
you have got this expression
for current which you
could rewrite in this form
that current is equal to this f1
minus f2 times the quantity here
that we could call the
conductance function
and that conductance would be
written as q squared D over 2t.
See, so this would
be the general form
of the current equation and what
we had done earlier was done a
simpler version where we assume
that this conductance
is independent of energy
in that energy window and
so we were kind of able
to pull it out,
that's kind of what we did
in last slide, but this is
the more general expression.
[Slide 5] So with that then we're
now ready to move on
and obtain an expression for
the conductance in general
and that's what we'll
do in the next lecture.
Thank you.
