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JEFFREY GROSSMAN: So we're here, but don't forget about this.
And so the quiz is next Thursday.
I'm going to send an email out with office
hours for extra time.
But if anyone wants to meet anytime,
just send me an email, please.
This PSET 2, we decided is going to be due
the day before on May 9.
And that's when your PSET 3 will be posted as well.
And PSET 3 will have to do with the things we've been talking
about the last two lectures and today, which
is what you can do with solids.
Any questions about the plan, the schedule?
OK.
Right.
Now, there are other practice problems.
I'll throw one out from last year's quiz.
So that's a practice quiz that I gave out
maybe a year ago or two.
And then there's last year's quiz,
and I'll pull a problem or two out from that,
and we'll do one today, and then maybe we'll
look at another one on Tuesday.
What I want to cover in terms of the lecture material today
is, I want to briefly start with that band structure
and caspase, and make sure we're feeling good about it,
and then tell you a few of the things you can do.
We talked about the band structure,
we've talked about how it has so much information.
Well, what other kinds of information is there?
What can you do with these sorts of properties of solids?
I love this cartoon.
"At some point, his theory becomes so abstract,
it can only be conveyed using interpretive dance."
And mostly it's because of the visuals
that I get on my colleagues here at MIT just dancing.
But we're not going to get this complicated, right?
So the point isn't for me to do a lecture on transport theory,
and tell you all about phonons and go into great detail
about these different things.
That is, again, not been our point here.
The point is to keep relevant and keep pretty applied
in this class.
So I'm going to give you just a little hint as to what
you can do with the DOS and the band structure.
But we're not going to go into great detail.
But you can find those details in many places online
and in books.
And if you come to me, I'm happy to talk about these concepts
that we'll talk about.
So the plan is, we're going to do
a little bit of review, caspase, feeling good
about oneness, all that.
And then I want to show you that problem.
We'll talk about just one band structure
problem that I happened to give last year on the exam.
And then we'll talk a little bit about some properties,
electron transport a couple of minutes.
And then we'll talk about magnetization,
and then we'll do some calculations together again,
because I think that's been really enjoyable.
Actually, seriously, I always enjoy calculating together.
But how many of you feel like that's useful,
to do some simulations here in class?
One, two, more?
OK.
Because please ask questions.
This is the time.
Because this is where we can, if you have any questions, please
stop.
And say, I got no idea what you're talking about.
That's OK.
I love those.
That's what I want to hear.
[LAUGHS] Well, that's actually not what I want to hear.
But if I hear it, it's fine.
That helps me to know where we need
to have a little discussion.
Now, this is the concept that we have
to be really good in our understanding.
But this is really the key of this second part
of the second part, if you want, of the class.
Which is, we did quantum, we did quantum for atoms,
we did quantum for molecules, and now it's all about solids.
And the big difference is that when you have a solid,
you have over and over and over again,
the same atom, or same couple of atoms in your basis.
And therefore, the same potential that the electron
feels.
And over and over again, how many times?
AUDIENCE: [INAUDIBLE] 2?
JEFFREY GROSSMAN: If I had to give a number.
AUDIENCE: Oh my god, there's a couple thousand.
JEFFREY GROSSMAN: Well, I'd say more like a couple of 10
to the 23rd thousands.
AUDIENCE: [LAUGHTER]
JEFFREY GROSSMAN: 10 to the 23rds.
If I take a piece of material, this big, this big, even
this big.
Even really tiny.
You got like, 10 23rd repeats of this potential.
That's a lot.
So it's almost infinite.
And when you have that periodically repeating
well that the electron feels, something happens.
And the something that happens was
this mathematical thing that, who
can remember what it's called?
It was a theorem, I believe.
AUDIENCE: Bloch's theorem?
JEFFREY GROSSMAN: Bloch's theorem.
And we don't need to know much about Bloch's theorem
except the endgame.
The key is what it means.
And it's a mathematical consequence.
It's a consequence of having an electron, and therefore
its solution being a wave function that solves
the Schrodinger equation.
It's a consequence of having that
in the context of a periodic potential.
You can't get around it.
It's a theorem.
So because of Bloch's theorem, you
got to have this requirement, if you
want to think about it as a requirement,
that when the wave function repeats,
the properties just repeat.
The density's the same here as it is here
as it is here, 10 to the 23rd out.
But when the wave function repeats,
it picks up this phase.
And that phase is a number that lives in k-space, that
lives in the inverse space.
And that's why we spent all that time
talking about what inverse space is, what reciprocal space is,
what k-space is.
And that phase is, you can think about it as another quantum
number.
So the result of solving the Schrodinger equation
and getting psi in a periodic potential
gives you a new number.
But it's very different than these numbers.
How is it different?
Yeah.
AUDIENCE: The vector.
JEFFREY GROSSMAN: OK, yeah.
Right.
These are like 1 or 2 or 1, 2, 0, minus 1,
whereas this is a place in space.
And what else is really interesting about that vector?
AUDIENCE: It's in reciprocal space.
JEFFREY GROSSMAN: It's a reciprocal space.
Absolutely.
Nice job.
You guys did that together, totally.
Team effort, man.
I love it.
And what else?
It's a vector in reciprocal space.
What else?
AUDIENCE: Kind of in the Brillouin zone.
JEFFREY GROSSMAN: OK, well, it doesn't have to be.
But we said it doesn't really matter if you go outside of it,
because you just loop back.
But what about its variation in the Brillouin zone?
How can it vary?
AUDIENCE: Anything.
JEFFREY GROSSMAN: Yeah.
This is a vector that can be anything.
This thing can be anything.
These things, can they be anything?
Can l be pi?
AUDIENCE: No.
JEFFREY GROSSMAN: Or square root of 2, right?
So it's a different kind of quantum number.
But you see, it has the same impact.
Because, what do these things do?
What does this do, anyway?
What's one of the things that this does?
Or this?
What does l do?
When I change l from 0 to 1, what's
one of the things that happens?
AUDIENCE: Moving from, like, s to p?
JEFFREY GROSSMAN: Yeah, yeah, OK.
So let's write that down.
Well, let's change n up to here to s, 1s, 2s.
And now you just said I'm moving to p.
Now, what is this that I'm always just assuming you know,
is what that vertical axis is?
What is it?
AUDIENCE: Fill energy from lowest to highest.
JEFFREY GROSSMAN: Yeah.
So this is the energy going from lowest to highest, exactly.
And so what did changing the quantum number, l, from 0 to 1
do?
AUDIENCE: It moved from s to p.
JEFFREY GROSSMAN: Yeah, it moves it around.
It moves it around.
OK, so you can change the energy,
and actually, you can also not change the energy.
m doesn't change the energy.
It gives degeneracy.
But certainly you can change the energy
by changing these quantum numbers.
That's what you can do with k.
That's what k does.
k, which is a result of living in a periodically repeating
world, k changes the levels.
It's the same idea.
It's very different, but it has the same impact.
It changes these levels around.
And it's very different because it can do it continuously,
as opposed to these big leaps that we have in the atom.
Or in the molecule.
In the molecule, there are also discrete levels.
And so we got this picture that we came to last time,
where you see, k can be a vector.
And it can live anywhere in this inverse space.
And anytime you change it, you may change the energy levels.
You may change them.
So it's a really important thing when you have a solid.
Now, again, if I have a periodically repeating atom,
but the distance between them is like, 100 angstroms.
What's the impact of k going to be?
We did it in class.
AUDIENCE: It's going to be relatively small.
JEFFREY GROSSMAN: Relatively small.
Like, almost nothing.
Because the whole effect comes from the fact
that you have an electron in a lattice of periodic potentials,
where they're sort of interacting,
where they're close enough that there's some interaction.
If there's no interaction, and your atoms are really,
really far apart, then it's basically
like you just have isolated atoms,
sitting in their potentials.
And a whole lot of empty space in between.
And then the levels will just look flat
as a function of k-space.
Doesn't mean you can't vary k.
You could still vary k.
But the bands that you get will be completely flat.
They'll go back to your molecular level description.
So does everybody see that?
All right.
And we talked about that.
And so then the question was, well, what playground of k
matters?
And this is something we talked about,
but it's really important.
So I want to review it here.
So we just said that k is this new index that
is a continuous variable, that changes the energy levels.
It moves them around.
OK, so what does that mean for a calculation, right?
Well, one thing that it means is that when
I do a calculation, if an energy level can now do this,
as a function of k, then that means
I need to actually do my calculation
with a mesh of k-points.
How fine a mesh do I need?
What's the answer to that?
AUDIENCE: Depends.
JEFFREY GROSSMAN: Thank you.
On what?
AUDIENCE: How curvy the bands are.
JEFFREY GROSSMAN: Yeah.
How curvy are my bands?
AUDIENCE: Pretty curvy.
JEFFREY GROSSMAN: This is pretty curvy.
So maybe I need a lot of k-points to capture this.
You got a grid.
I'm solving for this problem on a grid of k-points, basically.
And if I only have a k-point here and here, well,
let's say I have it actually even worse, I have it here
and here, I'm not going to really capture
the right physics here.
So the density of k-points has to be such
that you capture the features, in terms of how
much these energy levels vary.
So let's go back to these two atoms that are very far apart.
How many k-points do I need for those?
AUDIENCE: Two.
JEFFREY GROSSMAN: What if they're really far apart?
AUDIENCE: One.
JEFFREY GROSSMAN: One.
Whoa.
All the calculations you've been doing, until we did the solids,
everything before that was actually 1 k-point.
It was at 000.
We never varied it.
We never worried about it.
It was never an index.
OK, so now, if the bands are flat,
you could just have one k-point, and you've
got the variation of the bands nailed, since they don't vary.
So you just need the one.
But if they vary a lot, like if I bring atoms together,
and they vary a lot, and there's a lot of interaction,
then I just need more and more k-points to represent that
correctly.
OK, so the k-point is something that's
now an essential part of your calculation.
Now, that's talking about what you
need to get the density right, which
is to get the wave function right, which is
to get the energy levels right.
That's a new component of a quantum mechanical simulation.
But then there was this other thing
that's actually a little bit separate in a way, which
is to visualize that variation.
Now, visualizing this variation leads to these plots,
which are called
AUDIENCE: Band structures.
JEFFREY GROSSMAN: Band structures.
This is the band structure of the material.
Ooh, it's another kind of phase map,
for those of you who felt the excitement of phase diagrams
in 301-2.
This is like a phase map for that electron.
It's a really important description
of how electrons behave in materials.
Everything going OK over there?
AUDIENCE: Yeah.
JEFFREY GROSSMAN: It looked like you got hurt.
No, you're good.
OK.
All right, I got a little concerned, but we're over it.
[LAUGHS] Sorry.
The phase map is a walkthrough k-space.
Now, I just mentioned that you have a grid that you just
have to put on here to do the calculation
to get the wiggles that may or may not be there.
Well, you should take a guess it whether you
think they're there, and then you should see,
and then you should converge it.
But now I'm taking a walk around in that zone.
And the question is, where am I going to go?
Well, I'm using 4 by 4 by 4 k-points.
So I'm using a grid of 64 points.
And I'm plopping it down into this Brillouin zone.
Should I just look at the energies
at each of those points?
Well, you could do that, right?
Ideally, you'd like to be able to visualize
this variation all over k-space, at every single point.
But why don't we do that?
Tell me why we don't usually do that.
AUDIENCE: Would we be more interested in isosurfaces?
JEFFREY GROSSMAN: Well, that is true,
but that's more of a density thing.
So that's where you'd want to know the isosurfaces
of the charged density.
Yeah?
AUDIENCE: Once you know a certain amount,
it's repetitive throughout this direction.
JEFFREY GROSSMAN: OK, that's definitely true.
But let's say you didn't know that.
I'd like to do the most accurate calculation possible.
Let's say I put a grid over this k-space of a million
by a million by a million.
Don't do that on the nanoHUB.
We'll get calls.
[LAUGHS] But let's say I did that.
Now, I've got a million by a million
by a million points in here, where
each point has a new set of energy eigenvalues.
Why can't I just visualize that?
Yeah.
AUDIENCE: Would it just take too long to simulate?
JEFFREY GROSSMAN: Well, it would take a whole lot of time
to simulate.
What about visualizing it, though?
How would you visualize that?
AUDIENCE: [INAUDIBLE] 3D [INAUDIBLE],, right?
JEFFREY GROSSMAN: But see, it's not even really just 3D, right?
Because at each one of these points,
I've got a spectrum of energies.
So it would be making your eigenvalues,
each one into a 3D curve.
And I want to look at them all.
You could do that.
But you got to remember, if I go down to here,
you can't just do it within this volume element, right?
My eigenvalues might shift up, down in different ways,
depending on where I am.
How am I actually going to see that?
So in a way, what we have is actually a strong visualization
challenge.
There's sort of a good story here,
because luckily, we can just look
at a 2D map of this variation and get all the information we
need, for almost all materials.
And so, how do you get it a 2D map?
Well then, instead of doing the whole of volume,
you just walk through the volume along lines.
You literally just walk along this line, and then this line,
and then down, and then back, and then up.
And as you're doing this walk through k-space,
you plot the energy bands, and you connect them together.
And you draw lines between those connections.
This is the same energy level just
moving around, along that walk.
So that seems like an oversimplification,
because I'm not going through the whole Brillouin zone
by any means.
I'm just going through very particular lines in there.
And yet it's all we need.
And the reason is that most of the things
I want to know about a material, most of the things
I want to know, are going to come
from the variation in these bands between high symmetry
points in k-space.
So I can go to other points in the Brillouin zone,
and I'll get other levels.
I'll get a different set of levels.
But it's probably not going to add
to my knowledge of the material and the properties
that I can get.
So for example, I'm very interested in this
where this has a maximum, the valence band, and where
the conduction band has a minimum, right?
And you'll notice, the minimum for silicon
is actually not at this high symmetry point x.
But it does come along the path to x from another high symmetry
point, gamma.
It came along that path.
And there's no place in k-space that gives me a lower point,
you see.
So I nailed the thing I needed I nailed it
by going from one high symmetry point to another.
And that's what a band diagram is.
It's a walk through k-space along lines
between high symmetry points.
And that gives you the map that you need.
And these sometimes look really complicated, but just think,
it's saving you a whole lot of trouble
compared to all the data you actually could have.
But it's giving you what you need.
Are there any questions about that?
That's sort of the key point here from this band structure.
And we've talked about this, how you get your bands,
just like in an atom or a molecule.
You get your energy levels from solving the Schrodinger
equation.
And then you fill them up, using the Pauli exclusion rule,
right?
Add a little bit of Hund's rule to it,
although that can be violated.
Don't tell anyone.
And you fill them up and then you're done.
And when you're done, you've got your Fermi energy.
Now, it's the same exact thing here in the band structure.
You fill your bands up, until you've filled them
up with the number of electrons that you
had in your simulation.
And then you've got your Fermi energy,
and then if there's a small gap there, and no bands cross it,
it cannot be a what?
It cannot be a--
AUDIENCE: Conductor.
JEFFREY GROSSMAN: A metal.
It can be a conductor.
AUDIENCE: I don't know.
JEFFREY GROSSMAN: It can be a conductor, a semiconductor,
or a really bad insulating conductor, also known
as an insulator.
But it can't be a metal.
Because if no bands cross, if there's a gap here
between where you filled to and the next place,
then it's not a metal.
So we talked about that.
And that's, certainly if you have
an odd number of electrons, it has to be a metal.
And there's a calculation, I showed this last time
of an insulator.
And we also talked about how this density of states
is another really important thing here.
And since this is in your homework as well,
in your current PSET, I'll get to transport in a sec.
I want to make sure we know what density of states is.
Somebody tell me what the density of states of the carbon
atom would look like.
AUDIENCE: [INAUDIBLE]
JEFFREY GROSSMAN: OK, so let's draw.
So we had the carbon atom would have 1s and 2s and 2p.
And how many would I put here?
AUDIENCE: 2.
JEFFREY GROSSMAN: OK, and what about the next one?
AUDIENCE: 2.
JEFFREY GROSSMAN: OK, I'm liking it.
And here?
AUDIENCE: 2.
JEFFREY GROSSMAN: 2?
OK, like that.
AUDIENCE: Aw.
JEFFREY GROSSMAN: Oh, wait, hold on.
OK, thank you, like this.
Oh, sorry.
All right.
That would be an open shell singlet.
Thank you very much.
OK, now.
OK, so what's my density of states over this?
How do I do it?
OK, yeah.
So I'm seeing people go like this.
So this was energy here.
And now this is going to be energy here.
OK, what's my next step?
AUDIENCE: Drawing a curve.
JEFFREY GROSSMAN: Drawing a curve.
OK, I'm going to draw a curve.
What is that?
AUDIENCE: 1s.
JEFFREY GROSSMAN: It's 1s.
Why not?
I've just broadened it.
We'll make that 1s.
And what's my next curve I want to draw?
OK, sure.
And it should be spaced such that it
should be the right energy spacing
that I get from my quantum mechanical simulations.
OK.
And what's next?
2p.
Let's make this go out further.
OK, now 2p.
Now, how should I do this?
Should I make it the same?
AUDIENCE: Or shift it down [INAUDIBLE] energy.
JEFFREY GROSSMAN: Shift it down.
AUDIENCE: Same.
JEFFREY GROSSMAN: Same.
Is it different than 2s?
AUDIENCE: [INAUDIBLE]
JEFFREY GROSSMAN: Well, let's see.
I had two electrons in each of these.
How many electrons could go into 2p?
AUDIENCE: 6.
JEFFREY GROSSMAN: 6.
So I actually have more states there.
I have 3 times as many as here.
So in a way, if you're thinking about these
as sort of a vertical line, that is spread out
by some smoothing function, just to make it
into a nice, pretty curve, if it's an atom, if it's really
an atom, then it's really just a line.
It's a delta function.
But then this should have three lines all on top of each other,
right?
And so if I look at that, it's going
to lead to a bigger peak, that has more area under it.
Three times as much, yeah.
AUDIENCE: [INAUDIBLE]
JEFFREY GROSSMAN: OK.
So you can fit 6 electrons into this peak.
This is what the density of states is, right?
It is the density of states.
[LAUGHS] It is these levels.
It's these levels.
OK.
Does everybody see that?
Now, where's my Fermi energy for the carbon atom?
AUDIENCE: [INAUDIBLE]
JEFFREY GROSSMAN: Is it over here?
AUDIENCE: No.
[INAUDIBLE]
JEFFREY GROSSMAN: Here?
AUDIENCE: No.
JEFFREY GROSSMAN: It feels like it should be here, right?
We like Fermi energies between peaks,
but that's just because we like semiconductors.
No, we like metals too, and insulators.
They're all good.
So where should it be?
Does this sound good?
AUDIENCE: No.
JEFFREY GROSSMAN: Over here?
Hmm?
Somebody tell me when to stop.
Somebody say stop.
AUDIENCE: Stop.
JEFFREY GROSSMAN: Yeah.
OK, now why am I stopping here?
So that's my Fermi energy for the carbon atom.
Why is it there?
AUDIENCE: [INAUDIBLE]
JEFFREY GROSSMAN: Well, it's an atom.
So let's not think about it necessarily as conducting.
AUDIENCE: It's where you stop filling.
JEFFREY GROSSMAN: That's the definition, right?
Here I filled and I stopped.
AUDIENCE: [INAUDIBLE]
JEFFREY GROSSMAN: Yeah.
So here I filled and I stopped, and here I filled
and I stopped.
And what the gentleman over there said is,
it's where you stop filling, right?
And so here is where I stop filling,
because I had this peak of states, right?
It's a peak of states, the density of states, that is only
filled by 2 electrons, right?
Does everybody see that?
So if I had something else, like say, neon,
would my Fermi energy be inside the peak?
AUDIENCE: No.
JEFFREY GROSSMAN: No.
AUDIENCE: It would be just out.
JEFFREY GROSSMAN: Just outside of it, because I
would have filled it, right?
Yeah.
AUDIENCE: Besides the change in the location of the Fermi
energy, would the density states look the same for nitrogen?
JEFFREY GROSSMAN: Very good question.
It would, actually.
In the sense that if you're looking at atoms,
a density of states is a little bit
of an odd thing for an atom.
But I'm trying to explain it in the context of atoms,
because I really want you guys to see the connection,
and really what it is.
But if you look at atoms, they're
nothing more than levels.
And those discrete levels become little shapes
that you broaden a little.
So it would look the same as nitrogen,
because in nitrogen, you would have that.
Now, what does that do?
Well, it'll change these energy level positions.
It'll shift them around.
This may even change.
The 1s may go down or up, this may change a little.
And then here, you're going to change not the peak,
because you still got 6 states, and you still
got 6 electrons you can put in there.
But you will change the Fermi energy within the peak,
because you can fill it by one more.
And just like going all the way back to our first
or second lecture, where we had the orbitals of the hydrogen
atom.
So we had the nucleus and we had this electron
that does not orbit.
But there it is orbiting.
And then we had the 1s, and then we had the 2s.
And remember how we talked about,
it can actually make transitions between the two
without actually existing between the two.
There are some times it has to cross paths,
which are totally forbidden.
It's sort of exciting, it's like some sort
of interesting fiction.
That this electron is going on an adventure.
And well, that's what this is, right?
There is no state in here, in this energy.
So the DOS tells you that too.
That is what's in the DOS.
That information is contained in the DOS.
There cannot be an electron with an energy where there's no DOS,
there's no states to support it there.
So for an atom, like the hydrogen atom or the carbon
atom as well, that's fairly obvious,
because you had your levels and you knew it.
Couldn't be in between those levels.
That's the quantization stuff we talked about.
But now when you get to solids, and it's a more messy picture.
But it's the same understanding that you get out of it.
It's the same exact understanding.
So what you do for atom, molecule, or solid
is you take your levels and you turn them over,
and you count up the number of states,
and that has to be the same.
And you get shapes to the DOS.
If it was an atom, I just had these little lines
here from the levels, and I broaden them.
Now, the levels are no longer flat.
If it's a solid, they can vary.
And when you turn that on its side,
it can be really different, right?
But it's telling you the same thing.
It's telling you exactly the same thing.
You're filling your electrons up through here.
And you're stopping, well, I don't know,
is that the Fermi energy?
I didn't tell you.
But if it is, then you're stopping there.
And those are your unoccupied bands, and so forth.
Does everybody see that?
It's the same idea.
But it's more complicated when you have these curvinesses.
Yeah.
Right?
We good?
Sort of?
Any questions?
Thoughts?
Yeah.
AUDIENCE: So when you're looking at allotropic systems--
so in this case, you have done--
I'm assuming that the graphite, this
would be completely different.
JEFFREY GROSSMAN: Completely different.
AUDIENCE: And so how does that tie
to solving the density of states?
JEFFREY GROSSMAN: You can write the density
of states for any material.
And the same concept always holds, right?
That you have these accessible places where
the electron can be, with some probability
of them being there, right?
In other words, it's all about how many
possible energy levels you have as a function of energy.
That's really what it is.
And so, whatever material you have, the density of states
simply has the same meaning, but it will look very different.
And in diamond, you're going to get a big bandgap,
and in graphite, well, it's a metal.
In graphene, you have a crossing at one
part in the band structure, goes right through what
would have been the gap.
OK?
And that's why when you do your homework, when you do PSET 5,
there are two ways I want you to think about how
these molecules absorb light.
One way is that they just have some cutoff,
below which in energy or above which in wavelength,
they cannot absorb light.
So that's just a single cutoff.
But the other way is to actually use
the features in this plot of levels,
of accessible states for the electrons,
to use this information and weight the absorption
of the sun by this.
Because that's actually what's really going on, is that
the molecule cannot absorb the same at all the wavelengths
of light.
Because in some places it can't absorb any light.
In some parts of the DOS, there's no probability.
So that turns out to have a really big impact.
OK.
Questions, thoughts?
Now, let me, before I go on, it's OK.
I thought maybe we'd have time to look at Intro to Solar,
but that would have been ahead anyway.
Oh wait, no.
Where is the view?
Huh.
That's not looking very good.
I had what I thought was a nice example of a problem.
No.
Hang on one second.
Oh boy.
Never show anyone your desktop in a presentation.
That's a bad idea.
But anyway, oh, final quiz.
AUDIENCE: That's last year.
JEFFREY GROSSMAN: Yeah, that's from last year.
Yeah, don't worry.
It's in a very safe location for this year.
AUDIENCE: [LAUGHS]
JEFFREY GROSSMAN: Actually, it's very unsafe, it's up here.
Anyway, wait.
Where did it go?
Ah.
Why can't I see it?
AUDIENCE: [LAUGHS]
JEFFREY GROSSMAN: Here we go.
OK, that's the problem I wanted to talk about.
OK, so this was a problem from last year's quiz.
AUDIENCE: [LAUGHS]
JEFFREY GROSSMAN: OK, so your advisor has just spilled coffee
on the culmination of your entire summer UROP work.
This is not a true story.
The prediction of the band structure
and of an amazing new material for solar cells.
Unfortunately, in your excitement to show your work,
you forgot to back up your data.
You dropped your computer in a hot tub
and you forgot everything you did.
AUDIENCE: [LAUGHS]
JEFFREY GROSSMAN: So you only have that one printout
now stained with coffee.
Can you believe it?
This is what you got.
Nothing else.
OK?
Was this material-- you want to put it back together,
because you know-- it was 4:00 in the morning,
you had had a lot of NoDoz.
But still, you're pretty sure you
remember it was going to change the whole field of solar, OK?
Now, from this, was this material metal,
a semiconductor, or an insulator?
AUDIENCE: Semiconductor.
JEFFREY GROSSMAN: Tell me why.
Really?
Seriously?
Yeah.
AUDIENCE: Doesn't look like The bands are crossing.
Looks like the tallest bandgap is just under the 0.
I can't really tell.
And then it doesn't seem like the bandgap
for the bands is crossing, either.
JEFFREY GROSSMAN: It doesn't look like it?
Can you be certain?
AUDIENCE: No.
It's also a fuller cell.
JEFFREY GROSSMAN: Well, you do need metal,
and maybe it was a revolution in the metal contact.
That wouldn't revolutionize solar PV, but still.
Again, see, we're inclined to want
to connect these things up.
We're sort of used to staring at silicon,
and we think that's going to do that.
But I don't know that you have enough information here.
I'm not sure you know enough about this
to know that this band didn't actually split, and one of them
crossed up and went this way.
How do you know?
There is one thing you know it's not.
There's one of those three you know it's not.
AUDIENCE: It's not an insulator.
JEFFREY GROSSMAN: Not an insulator.
Why not?
AUDIENCE: [INAUDIBLE]
JEFFREY GROSSMAN: Because you can make out
at least that much, that these bands,
they're about an EV or two apart.
They may be closer, they may overlap.
But it is not going to be an insulator, right?
But you can't tell.
So this a little bit of a tricky question.
You seem to remember that the material would absorb light
very efficiently.
Reconstruct the band structure plot
that would correspond to such a material.
What are you supposed to do here?
Yeah.
AUDIENCE: [INAUDIBLE] associated with some wavelength
or a region of wavelengths alike.
JEFFREY GROSSMAN: OK, you could certainly do that.
And in a way, any band structure above the bandgap
is associated with the region of light.
Because except for the bandgap, below which you cannot absorb
light, as soon as you start having states,
you can absorb light.
But the key is here.
It absorbs light very efficiently.
So what do I need to do in here?
AUDIENCE: Put a direct bandgap.
JEFFREY GROSSMAN: Yeah.
You got to put a direct bandgap in there somehow.
That's the key.
So you got to reconstruct this.
And it didn't matter to me where.
And there were all kinds of really creative curves
people drew.
But somewhere in here, maybe this
is going to come down a little and then go down here.
And then maybe come in like here, where you can't see.
But somehow, you got to try to make this a direct bandgap
material.
It's actually pretty hard, because the stain
doesn't give you a lot of room.
But it's actually, that mirrors how hard it
is to do that for silicon in real life.
The stain and silicon are deeply connected.
One minute later, you realize that actually, it
didn't absorb light efficiently at all,
but instead it was a revolutionary metal electrode.
Now what would you draw?
AUDIENCE: Overlapping.
JEFFREY GROSSMAN: Right.
You'd cross it up.
Maybe you'd try to get a lot of wiggles in here
if you could, or just certainly several crossings,
so that the electrons don't really see any gap.
You can't see a gap.
So that's kind of a nice problem.
And you can see, there's not a very long problem.
But you've got to know a few things to answer the question
about these sorts of topics.
OK, so we'll do another one of those next Tuesday, or two.
Wait, where am I here?
Let's talk about a few other properties
that you can calculate.
Again, I'm not going to go into the details of transport.
But this is sort of basic double E or basic physics stuff.
If you put an electron in a field,
it's going to push it with a force.
And so the F equals ma of that.
Yeah?
AUDIENCE: I have a question.
[INAUDIBLE]
JEFFREY GROSSMAN: Please.
AUDIENCE: [INAUDIBLE] absorb things that are greater here.
It absorbs light that has energy greater than the gap.
JEFFREY GROSSMAN: Greater than the gap.
AUDIENCE: But don't you want a very small gap
for a semiconductor?
JEFFREY GROSSMAN: For what application?
For solar cells?
AUDIENCE: Oh yeah, sure.
JEFFREY GROSSMAN: So that's a very good point.
And who can tell me why you can't have too low of a gap?
And I'll talk about this as we move into solar PV.
AUDIENCE: Because I think the barrier's not high enough,
and they keep hopping over.
[INAUDIBLE]
JEFFREY GROSSMAN: That wouldn't necessarily be a problem.
Yeah.
AUDIENCE: It would probably get saturated with lower energy.
JEFFREY GROSSMAN: And what's wrong with that, though?
AUDIENCE: You're not going to be able to use
the full potential of the cell.
JEFFREY GROSSMAN: Well, that's what it is, is it
actually is what sets the potential of the cell.
So this goes back to, and it's a very good question, right?
Why not just use the lowest gap possible?
And then you'll absorb all the light, or most of the light.
Well, the reason is that it goes back to this picture
that when you're taking photons and making them into electrons,
what you're really doing is it's again,
it's that pumping water analogy.
You're giving the electrons energy
and pushing them up a hill.
Now, in the solar fuel's case, we're
using that push to change the molecule's configuration
to store energy.
In the solar cell case, we're using that push
to get just energy, voltage out, right?
So I'd like to not get just a teeny, teeny bit of voltage.
I'd like to get it up a hill.
Now, if I make my band gap such that I can only push it really,
really high, you might think you want that.
But then what happens?
AUDIENCE: Doesn't absorb any light.
JEFFREY GROSSMAN: Doesn't absorb any light.
So now you have these two competing effects.
AUDIENCE: Small bandgap.
JEFFREY GROSSMAN: To absorb the light.
Larger bandgap to give you more potential for yourself.
AUDIENCE: And large bandgap, more potential, small bandgap,
small potential.
JEFFREY GROSSMAN: But more light.
And there's a sweet spot.
And I'll talk about that.
I'll talk about that a little bit on Tuesday.
That that is two competing attributes of solar PV
that limit how much energy you can get from the sun.
So if you put an electron in a field,
it's going to move until you hit this equilibrium point where
the acceleration is 0, because the force it feels
is balanced by the scattering.
That's sort of basic transport, electron transport equations.
And so you can get an equation for the drift velocity
of the electron.
Or the thing that we care about a lot
is this electrical conductivity, something that you can measure,
which is nothing more than the drift velocity times the number
density.
That's actually a hard thing to separate out,
but it can be done in experiment.
And so the current is that, and it's
equal to the electrical connectivity times the field.
Now, here's the thing.
That equation, then, if you just look at these very simple F
equals ma for an electron in a field,
that very simply comes down to this equation.
And here's the thing.
That mass, what we want now is, well,
what's the electron conductivity in the solid?
What's the electron conductivity in the solid?
And it turns out that that mass that's
on the previous page, that m, it becomes
a kind of effective mass of the electron that
can be different than just the mass
of the electron, the standard mass of the electron.
The reason is that it's in this periodic crystal.
Remember, the electron is going to feel
this very strange environment when
it's in a periodic crystal.
And it's going to feel differences in its energy band.
And what that does is it changes,
actually the effective mass of the electron,
as it's in different parts of k-space.
Well, it turns out that that effective mass is directly
related to this curvature.
I'm just rewriting the previous equation,
but now I'm putting in the band structure here.
These little v's are nothing more than the slope of E
in the band structure, the slope of the energy
in the band structure versus k.
So this is called effective mass approximation.
It's an approximation, but it's a pretty good one.
The electron feels a different weight to it.
It's heavy or light.
Think about it that way.
Depending on whether it's in a flat part of the band, where
it's really heavy, or whether it's
in a curvy part of the band, where it's
feeling more light and bouncy.
And that changes the connectivity
that can be measured.
So just from the curvature of these bands,
it's a really nice way of getting
a sense of how mobile an electron or hole is
in a material.
All you have to do is you just take
an integral over the derivative of the slope,
basically, as you go along k-space there.
So does anybody know what a Fermi function is?
It's just a wait.
But-- yeah?
How many of you don't know what a Fermi function is?
OK.
So this comes from Fermi-Dirac statistics.
It's not in any way critical here.
But it's an occupation function.
It's saying that at a given temperature,
you're likely to have electrons or not at a certain energy.
And so the Fermi function at 0 temperature,
F is going to look like this.
Where this would be 1, and this would be 0.
And the Fermi function is 1 over e
to the E minus EF divided by KT, plus 1.
Where e is where you are in the band
structure. e is where you are in the band structure,
and you're looking at this as you go above the Fermi energy.
Can electrons be there?
Well, at 0 temperature, no.
But actually, when you go to some finite temperature,
just temperature alone can kick electrons up.
OK, that's what the Fermi function is in that equation
for.
Because at some temperature, you can actually
have a distribution that looks more like this.
T equals 0, T equals something.
Now, what's important here?
Well, what's important is that when
you think about conductivity, you want to know, you see.
The difference here between drift velocity and current
or connectivity is that you got to know something about how
many carriers you have.
And that's going to be temperature-dependent.
Does everybody see that?
That's why this is important.
Now, but what if my Fermi function is like this?
And OK, let's say I have silicon,
and I'm going to now plot on top of this,
I'm going to use a different color, and plot on top of this
the density of states.
And let's say my density of states was like this.
OK, I don't know if it is or not.
And then here's the bandgap.
This is my DOS.
That doesn't look like a different color at all.
AUDIENCE: [LAUGHS]
JEFFREY GROSSMAN: It's not even close.
It's like they painted it yellow on the outside.
And seriously, it's totally white.
All right, anyway.
So here's my Fermi function.
I'm at room temperature, and this is
my distribution of electrons.
This is the gap.
Tell me what's going on here.
Somebody explain this to me.
Yeah, Sam?
AUDIENCE: Even though there's probability
that you can have electrons in that space, because there's
no space for them to go, you still [INAUDIBLE]..
JEFFREY GROSSMAN: I love it.
That's absolutely right.
You cannot have electrons there.
The Fermi function says I can.
In your face, Fermi function, says the DOS.
I didn't say that.
I would never say that to a Fermi function.
But the DOS said it.
The DOS said, uh-uh!
AUDIENCE: [LAUGHS]
JEFFREY GROSSMAN: We got zero probability here, man!
You can't be here.
So the DOS said, no states, no deal.
Go away.
So the Fermi function tried.
It said, hey, I got enough energy from temperature
to give you a few electrons.
And the DOS says, I don't want them.
Because they're not at the energy where I've got
any states.
Now, what can I do, then, to populate
the conduction band of silicon?
What can I do?
AUDIENCE: More thermal energy.
JEFFREY GROSSMAN: More thermal energy.
I'm going to run my solar cell at 10,000 degrees.
Well, you'll get more and more.
And you will start to get electrons in there.
But actually, that's not always practical.
AUDIENCE: [LAUGHS]
JEFFREY GROSSMAN: So how do we get electrons into there?
Then there are other reasons electrons
come into those states, not just from thermal.
But what's one way that we can get electrons in there?
AUDIENCE: Doping.
JEFFREY GROSSMAN: Yeah.
What does doping do?
AUDIENCE: [INAUDIBLE] changes it from energy.
JEFFREY GROSSMAN: Well, it can change it from the energy.
But what might I dope silicon with?
AUDIENCE: Phosphorus.
JEFFREY GROSSMAN: Phosphorus.
Why?
AUDIENCE: Excess electron.
JEFFREY GROSSMAN: Excess electron,
but compared to silicon.
Now, if I have an excess electron sitting in my crystal,
but really the crystal is still kind of silicony,
because I don't have many of these excess electrons.
I just have one phosphorus atom every 1000.
So it's really still silicon.
But I filled all my electrons up to here.
And now I've got one extra.
Where does it go?
AUDIENCE: It goes in there.
JEFFREY GROSSMAN: Goes in there.
And now I'm populating it without thermal energy, right?
I don't need to work with my Fermi function,
I did it chemically.
I put an atom in there that had a spare electron, right?
And that is actually, in fact, exactly what
happens in real materials.
Who knows what it's called when you have an extra electron,
or an excess of electrons?
AUDIENCE: N-type?
JEFFREY GROSSMAN: N-type.
Now, what if I wanted holes?
What if I wanted extra holes?
AUDIENCE: B-type.
JEFFREY GROSSMAN: B-type.
And what would I put in there?
AUDIENCE: Boron.
JEFFREY GROSSMAN: Boron.
Because?
AUDIENCE: It's an electron.
JEFFREY GROSSMAN: It's an electron.
Now, you could dope it with many other things.
You don't have to dope silicon with just phosphorus or boron.
It turns out that those are easy and there
are some advantages in terms of their processing,
and chemically, they can be happy.
But there are many ways of doping materials.
Many, many ways of doping materials.
And one of the things you do with doping,
there are many things you do with doping.
But one of the things you do is you get electrons or holes
into these bands.
You populate them, right?
But this expression is ignoring that.
This is just saying, well, we're going
to see if there's some thermal energy there,
and we'll use that.
OK?
Yeah.
Any questions about that?
Now, we talked about this a lot.
So I'm not going to go over this again,
that the direct transitions are easy for photons.
The indirect ones where you shift where you are in k-space
are hard, OK?
And that is why silicon is an expensive solar cell.
So we talked about this, and I don't
want to belabor the point, because I
want to try to get to some simulations, if possible.
Any questions about that point, though, the optical
transitions?
All right.
So the kinds of things you can get.
Well, you can get optical properties, right?
And you can also get magnetic properties
from these densities of states.
So we talked about the optical properties.
And the optical properties is really the context of your PSET
5, but it's for a molecule.
We haven't talked about megnetism,
and I want to come back to it, because I did mention
that electrons have spin.
And therefore, they also have a magnetic moment.
And we call this, just think of it as some constant
that we call the Bohr magneton.
It has a value.
Well, and if you think about an atom or a molecule,
you could just get that by knowing how many electrons
are spinning in one direction more than the other, right?
So you can just count how many spin up
electrons minus how many spin down electrons you have,
and multiply that by the Bohr magneton.
And that will give you the magnetic moment
of the atom or molecule.
But what would I do for a solid?
Do I still have it?
Yeah.
See, there's the atomic picture where I could just count.
And in this case, the difference is 3.
Right?
But now, I've got a solid.
So what do I want to do?
What do I want to look at for a solid?
AUDIENCE: Density of states.
JEFFREY GROSSMAN: Density of states.
And it turns out that you can plot
the density of states for spin up and spin down electrons
separately.
And they can be different.
They can occupy the two spins differently
in a more complicated way, but very much same
idea as what can happen in and out of them.
But it's more complicated as you get
these curvy bands and all kinds of other things
that can happen in a solid.
So now, you look at the separate density of states for electrons
with spin up versus spin down.
And that is something that I didn't talk about yet.
It's an input parameter in SIESTA
that is checked yes by default. And it's
called spin polarized, question mark,
I think is what it's called.
Spin polarized?
I should give the right intonation.
And you say yes or no.
And you say yes, because that's the default.
But what are you doing when you say yes?
Well, what you're doing is you're
telling the calculation to allow the up and down electrons
to have freedom from one another.
That's actually critical.
So if you didn't do that, you would
be simulating the material with the constraint
that the up and down electrons are exactly
at the same energies.
So spin polarized, the reason you
do calculations that are not spin polarized
is because many systems don't have
a difference between up and down electrons, right?
And they are actually all the same, right?
What do I have if I fill this up?
What atom is that?
AUDIENCE: Neon.
JEFFREY GROSSMAN: It's neon.
And here, every up electron is the same as every down electron
for a given state.
Everything is the same here.
So it's not like this up electron of this p orbital
wanted to be there, and the down one wanted to be there.
That doesn't happen for neon.
And it doesn't happen for many systems and many materials.
But certainly in some cases, and certainly
for magnetic materials, that's what causes magnetism.
It's that you have a difference between spin up and spin down.
And that as you occupy the density of states,
literally you get a difference in the number, right?
So that's what causes the magnetism.
So for many materials, there is no spin polarization.
Or in other words, there is no difference.
Spin up and spin down is the same here,
same here, same here.
But if it's a magnetic material, what you'll find
is that if you plot the density of states
separately for spin up and spin down
electrons, like in this case for iron, there's a difference.
And in fact, the magnetic moment of that material
is nothing more than the integral of that difference.
Again, you're integrating the DOS.
And you're finding something really cool about the material.
So that's the same thing as just counting
number of up a number of down, but now you're
counting it in this very complex filling of states
that happens in a material.
Any questions?
So if I did a calculation of iron with spin polarized
checked no, then you see, these would be identical.
And I'd get no magnetism.
And you can do that.
You can actually do the calculation of iron
with spin polarized checked no, and it's completely wrong.
It's not the ground state of iron.
And if you do it with spin polarized on,
then it allows those up and down electrons
to move relative to each other.
And the outputs, you'll see that in the plots,
there's something called the projected density of states.
And it's projected for one spin and another spin.
And that would allow you to calculate
the magnetic properties of materials
using the nanoHUB tool.
And I think last year we might have done a homework on that.
But this year, we won't.
But I want you to know about it, because it's kind of cool.
OK?
Any questions?
Mm-hm.
A couple other things you can do,
and first principles, quantum molecular dynamics,
you can do it.
You can do MD, Just like you did MD in the first part.
You're just getting f from the Hellmann-Feynman theorem, which
I referred to a couple lectures ago.
But it's basically just a way of getting the forces
using quantum mechanics.
And I don't really want to go into it in detail,
but I just want to tell you, you can
get the force on these atoms from these quantum mechanical
simulations.
And you can do dynamics.
Here's one I did a long time ago.
And here the problem was that, how do nanotubes grow,
which was a big question, OK?
And so what you have is, you have a piece of iron here.
And this is what happens when you
grow nanotubes, is you have these seed particles,
like iron.
It can be other things, nickel, other things will work.
Not gold.
And then you flow some kind of gas over it
that dumps carbon onto the catalyst.
Now, the problem is that understanding how that then
leads to the growth of a tube is still an open question,
by the way.
Still an open question.
Nobody actually really knows the full answer to that.
But certainly, people were just debating,
does the carbon go inside the catalyst or not?
Does it stay on the surface, does it saturate the catalyst
first, and then come out from the bulk, from the particle?
And that's a perfect problem for quantum,
because there's so many different bonding environments,
and so many possibilities here, that classical potentials will
fail.
They're not going to capture all of the different complexities
in how carbon needs to bond to iron, inside and out.
And so it's a great problem for quantum mechanics.
And we studied this, and we fed the simulation carbon,
and we just watched how the tube winds up making its way out
of the system.
And we showed that it doesn't saturate, actually,
inside the iron, but rather stays on the surface.
And it makes for cool movies.
Somebody tell me something you notice about this.
Besides just, it's kind of a fun movie.
But do you notice anything?
There's the initial nanotube growth.
And that's where it ended.
So I notice two things.
Yeah.
AUDIENCE: Some rings are more than 6 carbons.
JEFFREY GROSSMAN: Oh yeah.
Actually, that's a really good point.
So that was a very interesting part of the analysis is,
what kind of carbon you get.
When you catalyze carbon on a surface, what does it do?
Does it form nice graphene and then curve it?
Or does it form the defects so that you can curve it?
Where are the penalties in energy coming from?
But actually, I was going to say some negative things, two
negative things.
They're not negatives, but they're limitations.
Tell me something obvious about this
compared to, say, the protein simulations
we did in the first part.
AUDIENCE: Very small.
JEFFREY GROSSMAN: Yeah.
This is really tiny.
And how long do you think this went?
AUDIENCE: [INAUDIBLE]
JEFFREY GROSSMAN: What is that?
AUDIENCE: When was it found?
What year?
JEFFREY GROSSMAN: I'm not going to tell you, because that
would make me feel old.
A while ago.
[LAUGHS] But anyway, how long can you
run a molecular dynamics simulation having to do
quantum mechanics at each step?
AUDIENCE: Very short period.
JEFFREY GROSSMAN: Yeah.
AUDIENCE: [INAUDIBLE]
JEFFREY GROSSMAN: Well, how long could you
do proteins with classical MD?
Huge systems.
Could you do them for seconds?
No.
But how long?
AUDIENCE: Milliseconds?
JEFFREY GROSSMAN: Yeah, actually even longer.
With classical MD, you can get up to microseconds now.
Certainly nanoseconds, just 10, 100 nanoseconds, no problem.
Here, we're at 10 picoseconds, right?
Even 2 picoseconds.
Nowadays, you can push 50, 100 picoseconds,
but you wouldn't go further than that.
It's really hard.
So the time and size are really limited.
And that's why with quantum MD, you
study small systems for short timescales.
This was now cluster growth.
I think quantum, these are just other things you can do.
This is supposed to be just making
sure you see that it's not just the band structure in DOS.
There are other things.
You can do a lot of things with quantum simulations.
Water is a really cool problem.
And I love Cavendish.
I love reading about Cavendish.
Has anybody read about Cavendish?
He's such a fascinating guy.
He's a university dropout.
He compared the connectivities of electrolytes
and expressed a version of Ohm's law way ahead of time.
And he measured the gravitational constant.
And he came up with really water, which at that point
was phlogiston and dephlogisticated air, which
I love those names.
Which is hydrogen and oxygen.
But anyway, water has these really interesting properties.
This is a wonderful website to read about water.
I'll give you the answer.
It actually looks like this.
Water sees water like this, not like this.
So it really sees it much more spherical
than we usually draw it.
And yet, it does have these very definite bondings
that it likes to do.
And the point I want to make is, when you go into an MD code,
and you put water in there, did you use water
in your protein simulations?
Yeah.
Which water did you pick?
You must have picked a potential,
or the code picked a potential.
Right?
Tip 3P to 4P.
Well, there's hundreds of them.
These are just a few water potentials.
Why are there so many?
Well, because some of them reproduce some things better
than others, right?
And that gets to be a really serious problem
if you want to study some of the fundamental problems of water.
And there's so many mysteries that we still don't know,
in terms of what happens when water evaporates, freezes.
And so, which one is best?
Well, this is a perfect problem for quantum mechanics.
And I won't go through the details.
But you can do water with quantum.
And this is a good way to validate
which potential you use in your classical simulations.
You can also calculate phonons.
You can just displace atoms and calculate energy differences,
and get frequencies.
So you can calculate phonons.
And when you calculate photons in a crystal,
that's different than in a molecule.
When you calculate phonons in a crystal,
you're getting the, waves the sound waves.
You're going to get the acoustic waves in the material, right?
You're just using water.
In a water molecule, how many phonon modes do I have?
In a water molecule?
Who can tell me?
Take a guess.
AUDIENCE: 6.
JEFFREY GROSSMAN: It's a good guess.
Let's get rid of translational and rotational.
How many waves can the water molecules stretch?
AUDIENCE: Two.
JEFFREY GROSSMAN: Yeah, there's actually three, right?
Because the OHs can stretch, and the bond can stretch.
But then there's two different ways that it can go, right?
And so those are your phonon modes for the molecule.
It's just stretch frequencies, right?
And they can be measured.
In a solid, you calculate your phonon modes
by moving atoms around in the lattice.
And that actually gives you the behavior
of the sound waves in the material, which tells you
something about how sound travels in the solid,
and you can calculate the sound velocity,
you can calculate thermal transport.
You can calculate the heat capacity of the material.
All of this are things that we do with quantum mechanics
that we're not going to do in any of our problems,
or anymore.
But I just wanted you to be aware of some of the things
that people do in research, and that
are really a big part of quantum mechanical simulations.
So those are some of the properties.
So that was sort of the point here.
You can do a wide calculator, a wide range of properties.
This has been our focus, these two.
And we're going to stay with that focus.
But there's a very large range of things you can do.
Now, I want to just end.
I got 5 minutes, OK?
And we'll start solar with this on Tuesday.
But for five minutes, I want to do a simulation.
So we'll start solar on Tuesday.
But here we are.
You know which tool it is.
It's our favorite.
AUDIENCE: Are we [INAUDIBLE] Tuesday, or no?
JEFFREY GROSSMAN: And we'll do another problem or two
in class.
And I can do a little review.
I sort of did that on Tuesday of this week.
Does anybody want to do more review next week?
What do you guys think?
A little bit?
Yeah?
OK.
What I'll do next Tuesday is we'll
do another quiz problem from last year like we did today.
I'll ask if you have any questions about this practice
quiz but, I'll also post the solutions.
And I'll go over some key things to keep in mind for the quiz.
So we'll spend the first half of class on that, then.
But let me just spend the last few minutes here.
Let's do a simulation.
AUDIENCE: It's open notes, right?
JEFFREY GROSSMAN: It's open notes.
Open notes, yeah.
AUDIENCE: Does that mean computer is open?
JEFFREY GROSSMAN: I think no.
Because that's like, everything, right?
What did you guys do in the first part?
AUDIENCE: We had the computers.
JEFFREY GROSSMAN: You had the computers?
AUDIENCE: We had the computers, but we weren't
allowed to use the internet.
So I don't know how they were checking that.
But I [INAUDIBLE] my computer, so I don't know.
JEFFREY GROSSMAN: OK, let's put it this way.
If we do open computers, then I'll make the problems harder.
AUDIENCE: No.
JEFFREY GROSSMAN: OK.
[LAUGHS] All right.
Because Wikipedia's pretty powerful, right?
And Google.
And so if you have that, you can find a lot of close answers
with a quick search.
So I'd have to be a little more creative.
But you won't need your computers.
And it's full open notes, right?
OK.
Now here's what I wanted to do.
OK.
Let's go to our favorite thing here, SIESTA.
There it is.
So I wanted to show you first of all, this feature here,
which is that spin polarized.
Do you see?
Yes?
No.
OK?
Now, when it's yes, and I calculate silicon,
here's the calculation of the density of states of silicon,
is what I want to show you.
What you are going to find.
Oh, I didn't fill out the structure.
Is that, OK, so this is just rounding out
our SIESTA understanding of what the tool does.
Because there were just these few things that
we hadn't talked about.
This is the density of states.
And you now really feel your oneness with this.
You know what these are.
You look at this, and you get emotional, which is OK.
But look at this.
I also have here, because I did, let's see
if it does this right.
Total, and there.
Look at that, OK.
You see, it's automatically doing
what's called the projected density of states.
And what a projected density of states is, is it's
the density of states, but only for a certain part
of the system.
Which turns out to be really interesting.
Like sometimes you just want to know the density
of states for one type of atom.
Where is that atom contributing to the states?
Or in this case, because it has been polarized,
it's giving me the density of states for spin down and spin
up separately.
And look at this.
They are, while spin down, spin down, you can't see it.
There's some spin up in there.
They're exactly on top of each other.
What does that mean?
AUDIENCE: Same energy.
JEFFREY GROSSMAN: All the spin up
electrons when they're paired with spin down,
they all have the same energy.
They have the exact same position on this.
And they lead to this total DOS here.
They just add up.
So if I did silicon with spin polarized off,
then the answer would be exactly the same.
So there was really no need to do it with spin polarized,
but it's the default. And now if I do iron, you see?
And I press simulate.
Then what you'll find is that you should see a difference.
Let's see if we do.
Here is iron.
And now you have the total density of states.
Oh, what is this?
I forget, is this an insulator?
AUDIENCE: No, it's a [INAUDIBLE]..
JEFFREY GROSSMAN: How do I know?
AUDIENCE: [INAUDIBLE]
JEFFREY GROSSMAN: Because I got some DOS here, right?
At the Fermi energy.
Oh, we got to do it for iron.
And now look at this.
This is beautiful.
OK, so this is the total, the green.
That's the total.
And look at this.
The red is the spin down.
And the blue is the spin up.
And look at those differences.
What does it mean?
Whoa!
I mean, you've got to integrate it to know.
But there's a difference.
There's a big difference in the spin up
and spin down electrons and where
they sit in energy space in the magnetic material.
There can be.
That's what makes it a magnet.
And so if you integrate that difference up to here,
well then, you're going to get this magnetization.
If I simulated iron with spin polarized turned off,
it would force those to be exactly the same,
and they would lie exactly on top of each other,
just like they just did in silicon.
But that would be not iron.
So you got to be aware of what this spin thing is,
and when you need it, when it's important.
Keeping it on is safe.
But I wanted you to know what it means.
So OK.
Very good.
Have a good weekend.