Our Energy Sources, Electricity

>>[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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