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nanoHUB-U Fundamentals of Nanoelectronics A L4.9: Heat & Electricity: Fuel Value of Information

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>>[Slide 1] Welcome back to
Unit 4 of our course. This is Lecture 9. [Slide 2] Now, as you might recall
in the last lecture, we talked about this expression
for entropy and how that kind of corresponds to information. That is if you have
a collection of spins and physically what we have
in mind is the whole bunch of magnetic impurities which
can either be up or down and they don’t move or anything, they’re just all localized spins
inside the channel of a device, and if the– all the spins
are up, there’s only one way that can happen because
they’re all up. That corresponds to
this entropy equal to zero. Whereas– And that corresponds to having information,
you know what it is. On the other hand, if it just
interacts with its surroundings, comes to equilibrium and goes
into this random collection, that corresponds to a
higher entropy state. And here, we are assuming
there’s no interaction between the spins. So that whether they are
up, all up or whether it is like up, down, doesn’t matter. The energies are all the same. So, there is no change in
energy going from this to this, just the change in entropy. And the point I’d like to
make, I think I mentioned this at the end of the last lecture, is that if these
localized spins were coupled to an appropriately designed
device, so we have this device where you have two contacts,
electrons can flow in and out of this channel. And let us say, these
spins were coupled to it in the sense these are
localized impurities which are probably
sitting inside the channel. And if I designed it
properly, then I could make use of this loss of information
to actually do useful work like charging up a battery
or lighting up a light bulb. And that’s the reason for
the title of this lecture. The fuel value of information
that is when using information, you’re converting
information into useful work. What the second law says is that
it should be possible to do so. Because ordinarily what the
second law would have said is, that there is no way you can
take heat from your surroundings and do something useful. Why? Because whenever you take
heat, the entropy goes down. And overall, second law requires
that entropy has to go up. But in this case, because
there’s a third terminal where the entropy is going up, it means that without
violating the second law, you could take some
amount of heat from the surroundings
and do useful work. Note that none of the energy is
coming from this terminal at all because whether these
spins are up– all up or random doesn’t
matter, energy stays the same. So, you’re not getting
any energy out of it. But the increase in entropy is
kind of letting you take heat or energy from elsewhere
and do useful work. So the second law says that
you can take a certain amount of heat out of your surrounding
and convert it into useful work as long as that amount is less
than T times the entropy cost that you have paid
here, this delta S, which means you should be
able to convert an amount of energy up to nkT log 2. What the second law doesn’t
tell you though is how to do it. It just says in principle,
it’s possible. But this is an inequality,
it’s not equality. So, if you don’t design
your device right, you won’t be able to. But if you design it right, you
should be able to, in principle. So in this lecture,
what I’ll describe to you is a concrete design
that will achieve just that. And the purpose is again, not that it’s a very important
device, the purpose is just to illustrate an important
conceptual point about entropy. [Slide 3] So, the device I have in mind
is what you might call an antiparallel spin valve. You see spin valves have, in
the last couple of decades, have got a lot of
attention as useful devices. What it is, is it’s
a normal device but with magnetic contacts. And here, what we have is
an antiparallel spin valve in the sense that
if one magnet is up, the other magnet
is antiparallel, pointing in the opposite
direction. And what we assume is that these
magnetic contacts are perfect, 100% efficiency. What that means is that if
you have an up contact magnet, it only talks to the up-spin
electrons in the channel. So, up-spin electrons can flow
in and out of this contact. Similarly, the down-spin
electrons only talk to the– this down contact. Now, in practice, real
magnetic contacts are– don’t have 100% efficiency. They might talk, say 80% to
that one and 20% to that one. But what we are assuming, because this is entirely
conceptual, it’s designed to illustrate a conceptual
point. So we are assuming these
magnets are perfect. So, this only talks to up-spin,
that only talks to down-spin. So, what that means is
that in this device, ordinarily no current can flow. Why? Because up-spin
electrons are connected to the left contact,
down-spins are connected to the right contact. But there is no path going from the right contact
to the left contact. You see, up-spins can come in
but then they can’t go anywhere. Down-spins can come
in from the drain but then they can’t go anywhere. Now, if we take this
channel now and couple it to these localized
spins, you know, these spins that I talked about, these are these just
localized impurities. Let’s say we couple them. Now, actually, current
flow becomes possible. Current flow becomes possible because we have this
interaction between them. There’s exchange
interaction which says that if you have an up-spin
electron in the channel, these are the moving electrons, and you have a down-spin
impurity, they can interact and flip. So, the up-spin becomes
down-spin and the down-spin
becomes up-spin. So, it would be something
like this. An up-spin electron in
the channel interacts with the localized down-spin. The localized down-spin
becomes an up-spin, and up-spin electron
becomes a down-spin. So, this is what– we would
call this exchange interaction. If both had up, nothing happens. If both had down,
nothing happens. But up down becomes down up
because of this interaction. Now, as a result,
what would happen? As you see, you have
all these up-spins here. They would let down-spins
converge to up-spins because they would interact. You see the down-spins
will interact with all the up-spin
localized ones and turn them into up-spins. And these up-spins would want
to come out into the contact. The key to the design of this
device is that up-spins has to come out from
the left contact, down-spins only talk
to the right contact. So, anytime you create an
up-spin, it goes to the source. And of course if it cannot
flow out because it’s like open circuited, then what
will happen is the source will become negative as
I have shown you. On the other hand, you lost the
down-spin so a new down-spin has to come in from here
making this end positive. So, the point is, if you take a
structure like this with all– interacting with this
collection of up-spins, up-spins in the channel, these
up-spin localized things, then it will be like a battery. There would be an
open-circuit voltage, one side will be negative,
one side will be positive. So, how do we estimate
this open-circuit voltage? Let us try to write the current. You could see that there is a
process which converts up-spins into down-spins,
up-spins into down-spins. And anytime an up flips into a
down, it is able to flow out. So, it gives you a
current from left to right, electron current
from left to right. So, that’s this f
up times 1 minus FD. This is the occupation
of an up-spin. This is the probability of a
down-spin state being empty so that the electron
can go into that. Similarly, there’s
the reverse process. Down turns into up, that
will depend on down– availability of a
down-spin electron and an empty up-spin state. And there’s another
factor we have to include. In order for an up-spin
to convert to a down-spin, you need a localized
down-spin impurity. That’s this n sub D.
Similarly, for this process, you need a localized
up-spin impurity. Now, what we can assume is that this up-spin channel is
essentially in equilibrium with the left contact. What that means is instead
of f up, I can write f1. But f1 is the Fermi
function in this contact. Similarly, instead of
f down, I can write f2, which is the Fermi
function in that contact. Now, you might say, well,
didn’t we always say that current is proportional
to f1 minus f2? So, how come you have such a
complicated expression though? And the answer is, yeah, this is
a good example why current need not always be proportional
to f1 minus f2. That only applies to
these elastic resistors and where no entropy driven
processes are occurring in that contact inside
the channel. They’re all in the contact. Now, here of course we have
actually have an entropy-driven process in the channel. Note however that if
nD is equal to nU, which is the usual equilibrium
state, that if all these spins where like 50, 50 because it’s
an equilibrium, then you see that nD would be equal to the nU and so the current
would be proportional to f1 times 1 minus f2
minus f2 times 1 minus f1. And now you’ll notice that these
cross terms f1, f2 cancel out, and so we get our old result that current is proportional
to f1 minus f2. But in general, the point is that because we now have
entropy driven processes being considered in the channel, current is not necessarily
proportional to f1 minus f2. What we have is this expression. So what we want to
do though is use that to find the
open-circuit voltage. [Slide 4] To find the open-circuit
voltage we say, well, current has to be zero. So, if current is zero, it
means the first term has to be equal to the second term. And so with a little
rearrangement, you could write it this way. Now, here there’s a little bit
of algebra that’s often comes in handy and that is if this
f is a Fermi function then, as you know, it has this form
of one divided by one plus E to the power x. And
so, 1 minus f over f, you can show it a
little algebra, is E to the power x. So, what
that means is I could write– instead of 1 minus f1 over f1 and I could write
exponential E minus mu1 over kT because that’s the x basically. E minus mu1 over kT. And here, you have
exponential E minus mu2 over kT. Again, just rearranging,
dividing this by this, you’ll get this exponential mu1
minus mu2 over kT is equal to n up divided by n down, which
you could write in terms of probabilities, that if
you have a total number of localized spins n, then n
up is n times the probability of an up-spin and down is
n times the probability of a down-spin. So, the ratio of n up
and n down is the same as this ratio of probabilities. And that immediately gives
you now an expression for this open-circuit
voltage that left to itself, this battery develops
this voltage of kT log P up over P down. And of course if it’s
an equilibrium, 50, 50 P up is equal P down, there
is no open-circuit voltage but if it is in the
state with all of them up then you do have an
open-circuit voltage. [Slide 5] So, how much energy can
you extract from it? So, the idea is you have this
open-circuit voltage and now if you put something here like a little light
bulb then you’d let– you’ll kind of draw
electrons out from here and they would flow
to your load. Now, the thing is that every
time you draw an electron out, of course a new one
has to be created, and in the process you need– this has to be flipped
into this. And in the process, one of these
localized spins will flip the other way. So, the energy you extract
is mu1 minus mu2 times dn, n being the number of electrons. But dn is actually
equal to minus dn up, because every time you
have to create an electron, the number of localized
up-spins goes down by 1. So, you could write
the energy then as dn up times mu1 minus
mu2 and if you saw, put in the mu1 minus mu2 in
terms of this P up and P down, what you’re doing is–
then you have this kT and log P up minus log P down. And instead of n up, I’m
writing n times P up. Now, with a little algebra,
what you can show is that what we have here could
be written in this form, it’s T times integral of
dS, S being the entropy. And that is of course the basic
result, that way I’m trying to prove that the maximum
energy that you could extract from this device
is actually equal to what second law allows you
to do which is T delta S. Now, in order to see that this is T
dS requires a bit of algebra, that is you first write S
using this second formula for the entropy that we
discussed in the last lecture, that you have these two
possibilities P up and P down so the entropy is P up log
P up plus P down log P down. And so when you find dS, when
you take the differential of this, you get
these two terms. And when you take the
differential of that, you get two terms again. But the point to note is because
the system is either up or down, it means the dP up plus
dP down, that must be zero so you can take that out. And here, instead of dP down,
you could put minus dP up. And this DdS, this expression
will basically become what you had here. So this was just in– designed
to show that the expression for energy becomes T times
integral dS, initial to final. And so, what it shows is that
you can extract this entropy. That what happens is,
you started with all ups, that was kind of
like information which gradually got
randomized in the process, the entropy increased,
information got lost. And you’ve extracted
an amount of energy, T delta S, to do useful work. [Slide 7] So this is what you could
call an info-battery, sort of, one that converts
information into useful work. And this battery consists of this antiparallel
spin valve, ideal. Two magnets pointing in opposite
direction with electrons that can– flowing in from one
and here you have down-spins that are connected
to this contact. Here, you have up-spins
connected to that contact. And they are interacting
with this background of localized spins which have
all been put in an up state. And that will given you
an open-circuit voltage. You could use that to
light up a light bulb. But of course, as you
draw energy from it, these localized spins
will get randomized. And eventually, it
will all be random and there won’t be
any voltage left. And it will run down
like a battery, of course, to the run down. And how much energy could
you extract from it? Well, the maximum is T times
the increase in entropy. Now, one point you might
say is that you should be– that’s interesting to note is
that S equals 0 doesn’t mean that all have to be up. It only means that you have to
know exactly what state it is. It’s this information. So, for example if you
knew they are all down, that will be fine too. You could build the battery
easily, that’s trivial. But let’s say that you knew
they are half up and half down. So let’s say you had
something like this. Half of them you
know is pointing up and other half is pointing down. Now, you might say,
well, now if I put up– try to make a battery
the way I described it, you don’t get anything,
don’t give you any voltage. Yes, that’s true, that
old design won’t work. You have to now figure
out a better design. For example, what you could do
is separate out the two parts, don’t keep them together. So, in that case, if you
now put two terminals here, this is negative and that’s
positive just like before because you have all
up-spin localized electrons. Here, you have down-spin. And so, here the polarity
would be reversed. So, you’d have plus
here and minus there. And what you could
do is, you know, you kind of have now
two batteries you see and the way you do, take two
batteries and put them end to end to get the series
combination so you could connect up this end and then take
your power out of this one, so you could build a battery. Now, remember again, of course,
if you had a very complicated, I guess, collection of spins
then designing a proper device could be pretty difficult and
building it could be impossible. But the real point here is
just the conceptual point, that if you knew exactly
what the distribution is, you could in principle
build an info-battery to extract this information,
I guess, in the form of useful energy. You could convert information
into energy in principle. In practice, building
the device, as I said, could be very hard. [Slide 7] So, basically as you know in
this unit, our whole purpose was to give you a little insight
into this very deep topic of what exactly this entropy
driven processes are about that, as we started in this
course, we say that, you know, if what makes it possible
to describe nanodevices in a relatively simple way and
what gives you a lot of insight into big devices as
well, is this separation between mechanics
and thermodynamics. And by and large so far,
we never talked much about the thermodynamics. So, we just said, well,
you have these contacts where they are held at with
certain Fermi functions. And the purpose of this unit
was to give you a better feeling for what these entropy
driven forces are– processes are about. And this last example in particular shows you a very
interesting example involving entropy driven processes
in the channel which actually do not
involve any energy exchange so it is kind of strictly
speaking still elastic but it’s entropy driven and how that could invalidate
this equation that we had been
using for example. So, I guess with that,
now it is time to sum up. And that’s what we’ll be
doing in the next lecture.

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