PROFESSOR: --called case studies.
That is, we really look at how some of the tools
that we have learned during this class--
some of the energy tools, like density functional theory,
some of the statistical tools, like thermodynamic
integration-- come together and are actually
used to predict the properties of materials and structures
of real interest.
This is-- I called it lecture 20,
but I think I have been slightly inconsistent with the numbering
of my lectures.
We count also the labs.
Anyhow, everything is on the web.
So what will you learn today is actually
how to become an expert in two of these jobs
for the 21st century.
One is the virtual alchemist, and the one that is fancier,
I think, is the nanotechnologist.
There was probably an ad for GE a few months
ago on TV in which there was a nanotechnologist marrying
a supermodel.
It was sort of fairly intriguing for all of us
doing the initial calculations.
But more seriously, what we are really going to focus on
is how can we break the n-cube barrier of electronic structure
calculation.
You have learned that, say, when we study a system using
density functional theory, the computational cost scales
as the cube of the number of atoms.
And that basically scales very rapidly
the size of the system that you can do.
I mean, even on a supercomputer nowadays,
we can't really do more than 500 atoms, say.
And on a regular cluster, we are more likely close to 100 atoms.
And the reason of the cubic scaling in DFT
is ultimately due to the orthonormality constraints.
Every single particle orbital needs
to be orthogonal to every other single particle orbital.
So if you double the size of your system,
you double the number of electrons.
That is, you have twice as many orbitals
that need to be orthogonal with twice as many other orbitals.
So you have four times more matrix elements
that you need to calculate.
But also, those matrix elements are
calculated in a unit cell that is twice as large.
So there is another factor of two,
and that gives us a factor of eight.
And of course, there is a lot of effort in the community
nowadays to go towards linear scaling algorithm, algorithm
that just double in computational cost
when you double the size of your unit cell.
And they are all based on the sort of physical principle
that, in many ways, nature is local.
So your electron needs only to know what's happening nearby
and doesn't need to be really orthogonal to electrons
that are very, very far away.
And you see an example of sort of where this linear scaling
direction are heading.
Other methods scale even worse than cubic.
We have sort of discussed that sort of standard Hartree-Fock
scales as the full power, and highly correlated
quantum chemistry methods scale as the fifth, sixth, seventh
power.
An interesting exception, I think,
among the methods that try to go beyond DFT in accuracy
is what people call quantum Monte Carlo approaches.
They tend to be approaches that are very accurate but based
on a stochastic sampling of the wave function,
so really, a stochastic solution for the wave
function in the Schrodinger equation.
And those in principle scale as the cube.
And there have been sort of recent progress
in making them truly linear scaling.
So I think they are a very promising avenue
for the future.
But in this class, we'll sort of confine ourselves to DFT.
And in particular, we'll look at two examples in which we
use electronic structure DFT calculation to actually obtain
the relevant parameters for a Hamiltonian
that basically models the important degrees of freedom
for our system.
So we'll actually look at semiconductor alloys, where
you'll see, very naturally, one can sort of
solve the problem of substitutional disorder
on that alloy using an Ising-like Hamiltonian, so
something that you have seen in the previous classes.
And then we'll look at another example
to functionalize nanotubes, that is
carbon nanotubes, to which organic ligands have
been covalently attached.
And you'll see how we sort of develop a linear scaling
tight binding-like procedure.
We'll explain this.
That allows us actually to study,
with full DFT precision, systems that have tens of thousands
of atoms in the units cell.
So let's start with the first example, that
of semiconductor alloys.
Semiconductor alloys have been of central interest
since, I would say, the late 80s, when
people started to realize that there wasn't only silicon.
And in particular, it was very important to tune
out the physical properties of your semiconductor,
and alloying is a very simple way of doing that.
If you, say, mix together something like gallium arsenide
and aluminum arsenide, depending on the concentration,
you have a substitutional alloy with a bandgap
that is variable.
So it can be used very efficiently, say,
to tune the emission frequency of your laser light,
if this semiconductor is your active layer.
And this is something which, in our department,
in the material science course, there
is a lot of ongoing effort.
In particular, Professor Fitzgerald
works a lot on some of these alloys
to actually improve mobility of the carriers.
And I had here some kind of beautiful picture
of actually an array of vertical KVT lasers
in which sort of a number of vertical pillars
were grown on a surface with a composition
gradient for that active layer.
So what you achieve on a single breadboard
is a very high density of emitting lasers.
And each one emits at a slightly different wavelength,
because the bandgap is tuned by the compositional gradient
in that array.
And another set of reasons that sort of intrigued and prompted
the study of the semiconductor alloys
was actually what is called the ordering,
multilayer ordering, superlattice
ordering in those alloys.
That is, when doing X-ray diffraction
of a standard semiconductor, you would get a set of broad pixel.
And what people note is that under certain growth
circumstances, they would find naturally in the diffraction
figures some intermediate peaks.
So in this case, some peaks are closer to the origin,
or in this case, sort of the green dots.
So yeah, midway between the two red dots.
What it means is that there is actually
some kind of ordering that has periodicity that
is in that particular crystallography direction,
has a wavelength that is twice as large
or a wave vector that is one half.
And so it sort of hints at some kind of ordering.
So there were a number of reasons
why these were interesting.
And I'll show you sort of one of the ways one could actually
tackle the problem of studying these systems.
I had another figure here.
They just didn't come-- they didn't come out--
in which, again, I was sort of showing you
basically the whole set of semiconductors
and plotting that as a function of lattice parameter
and bandgap.
And basically showing that, if you look at all, say,
the III-V semiconductor, they sit in a certain region
of this diagram.
But in reality, there isn't anything
that is really useful that-- say, as the bandgap of silicon.
But it meets at the wavelength for which conventional fiber
optics cables are transparent.
And so a lot of effort in trying to engineer alloys
or new materials that go towards that soft spot in which you
can grow things that seem more thick on silicon.
So you can use all your silicon technology,
but they have the right direct bandgap
that is appropriate to optical fibers, 1.5 microns.
OK, so from the conceptual or computational point of view,
our problem is really trying to figure out
a way to calculate efficiently the energetics and the phase
ability of a semiconductor alloy.
And I sort of shown you here an example.
Remember, semiconductors have the diamond or the zinc blende
structure.
So they are made by two, say, interpenetrating lattices.
And what I'm showing here is just one
of the two FCC sub-lattices of a typical zinc blende structure.
Say, suppose that you are studying gallium aluminum
arsenide.
Arsenic is the group V semiconductor.
Gallium and aluminum are group III.
So you would have a sub-lattice where
gallium and aluminum sit, and this
is what you are looking at.
It's an FCC sub-lattice-- and where there
can be substitutional disorder.
On each side, there is a certain probability
of finding either a gallium or an aluminum that
is basically proportional to the concentration.
So if you have a 50/50 alloy, you
will have 50% probability of finding gallium
in a certain side.
And then there is a second sub-lattice
that is almost invisible that would
be given by the anion or the arsenic side.
Of course, one could sort of study this system
with a sort of brute force approach.
We could really look at large supercell,
and perform many calculations, and find out
which arrangement of atoms give us lower energy configuration,
figure out what, say, would be the bandgap
or what would be the lattice parameter
of certain configuration, and try
to figure out what are the macroscopic properties.
But statistics at this stage will always be limited.
If you want to study a supercell that is large enough,
you probably need, again, a cell in which
the correlation between faraway atoms is lost,
and you might easily need something like 100 atoms.
But then in particular, you need a lot
of statistics that are true to the 100 ways of arranging atoms
on that lattice.
And you can only calculate a few of these.
And so you become very limited in analyzing the thermodynamics
of your problem.
At the end, what you really care about
is understanding what is the energy of a configuration.
That is, what is the energy as a function of-- you have seen
this before--
configuration of variable sigma.
We use the nomenclature in which sigma is equal to 1,
if, say, we have a gallium atom here.
This example specifically is actually
for gallium indium phosphide, so we'll have,
again, group III indium.
And sigma equal to minus 1.
That means that there is indium on one side.
This will make actually the connection with the Ising model
very, very apparent.
And so what you want to do is calculate the energetics
of a lot of configuration and then extract
the thermodynamic properties.
And you have seen already in Professor
[? Seder's ?] lectures, and you'll
see again partially in this lecture and the next lecture
how you can use actually Monte Carlo approaches
to extract basically what are the average ensemble
properties.
But our first goal for us is actually
finding out an inexpensive way to calculate
the energy of an arbitrary configuration of atoms,
without having to do the explicit first principle
calculation.
And this is where the fundamental concept
that you'll see in this lecture comes about.
That is, how we actually devise from the initial calculation,
how we extract from the initial calculation, the parameters
for the model Hamiltonian that describes
the energetic of my system for any arbitrary configuration.
And as usual, you need to put ingenuity
in solving a difficult problem.
And this is one of the possible solution.
You'll actually see some of different alternative
approaches in one of the next lecture.
But what we are doing here is, we are actually
sort of exploiting the fact that, say, if you are looking
at a substitutional alloy, you have substitutions
on a sub-lattice between atoms that are really chemically very
similar.
If you are studying, say, gallium indium phosphide,
gallium and indium are going to be chemically very similar.
They are both group III.
They are just sort of in a slightly different in shape.
And of course, their orbitals will be sort of slightly
different, and they will correspond
to a different period in the periodic table.
But their chemical properties are still fairly similar.
And so for a moment, we could actually think--
and I'll show you how it works mathematically--
that a real configuration of a crystal--
and remember, from the point of view of density function
theory, having a certain arrangement
of red and blue atoms means that there
are very specific pseudo potentials
sitting on each one of the sides.
But one of these arrangements could actually
be sort of considered as a perturbation on what we
call a virtual crystal, a crystal that has
full translational symmetry.
So the sort of idea that I'm trying to give you--
and we'll see it mathematically in a way--
is that we could think at the real configuration of a system
as a perturbation in which we take a pseudo atom that
doesn't exist but is somehow very
similar to both gallium and indium.
And then we turn that with a small change
in the pseudo potential in either gallium or indium.
And if we can do that, well, then we
can study this problem with density functional theory,
because this is actually a system that
has two atoms per unit cell.
And then we can try and use perturbation theory
to figure out what would be the energy of our configuration
over there.
And this is actually fairly trivial
to sort of see it in mathematical form,
when you actually think again at the fact
that in density functional theory,
the potential acting on the electron
is really given by this array of pseudo potential
sitting on lattice sides.
And so if we have a real configuration-- remember,
the red and blue crystal on the left--
well, to that, it will correspond
an external potential that is an array of red and blue pseudo
potential, that can actually be rewritten
in terms of the semi-sum and the semi-difference of pseudo
potential.
So this would be the exact expression.
We have the true pseudo potential
on each sort of FCC sub-lattice site.
But we write, say, gallium pseudo potential
as being given by the sum of the average between the gallium
and the indium pseudo potential, say, plus one
half the semi-difference.
You see, if I'm sitting on a site,
and I'm telling you that the pseudo potential acting
on that side is going to be the sum between the average
between gallium and indium plus one half the semi-difference,
you see that the indium cancels out, and the gallium sums up.
And what you have is the gallium pseudo potential there.
And so a sigma variable equal to plus 1 at the site R
really gives you a gallium pseudo potential in that site.
And a sigma variable equal to minus 1
gives you an indium pseudo potential in that site.
But in this form, and in the limit in which the two pseudo
potential are fairly similar, you
can see that this term is actually small.
The more similar gallium and indium
are, the smaller this system is.
But this term here overall is the only one
where the configurational variables are present.
There are no configurational variable here.
And so this represents, really, the green average virtual
crystal to which at every site, we
add or subtract depending on the sign of sigma
R, the semi-difference of the two pseudo potential.
And this term really turns a green atom
in a red or blue atom.
And for the specific case of semiconductor alloys,
this term is smaller, because disorder
takes place between atoms that are very small.
And so what we are going to do is use density function theory
to study this crystal then use perturbation theory
to understand what is going on when we do this perturbation.
And so this is also sort of a chance
for me to at least hint at how we actually
perform perturbation theory in density functional theory
itself.
And that's a very useful tool.
Say, it's used, and you'll see it,
and you have seen it before, to calculate things
like phonon dispersions.
That is, what is the force or what
is the frequency of perturbation in the atomic position
with a certain wavelength that is really
the normal mode of the crystal.
Or it can be used in a variety of problems,
say, to study what is the response of a system
to an electric field and then calculate
Raman or infrared spectra.
It can be used to calculate the response of a system
to a magnetic field and then calculate the NMR spectra,
or it can even be used to calculate higher order
important physical quantities.
If we go to sort of higher order response,
we can start to calculate what is the third order
Hamiltonian in the phonons, so what are the--
in the phonons, what are the lifetime of phonons.
Or we can calculate how an electronic state couples
with a phonon state.
And so we can actually get out the parameters,
say, for superconductivity.
That is, when electrons travel through a crystal paired
because of the interaction with the phonon degrees of freedom.
Or we could calculate, say, resonant Raman scattering
that, again, is sort of a strong electron-phonon component.
But in general, the problem is just this.
Perturbation theory looks at what
happens to a certain system that has an external potential v0
when you add a perturbation.
And the perturbation is sort of parameterized
by a certain strength lambda.
So for small lambda, the perturbation is weak.
So we are sort of thinking at the limit of what happens
when the perturbation is weak.
So you add a lambda delta v term to your external potential.
And in this specific case, that delta v term
is going to be that semi-difference
between the pseudo potential.
So it's the term that really transform a virtual atom
in a real gallium and in a real indium.
What happens when you apply this small perturbation is that,
of course, your new self-consistent charge density
will change.
So if, for a given external potential v0,
you add a ground state charge density n0 for a perturbation
in which you add the term lambda delta v,
your new ground state charge density
is going to be a certain charge density that
will define us n lambda.
But we can think of this n lambda
as being sort of expanded in a power series of power lambdas.
So the new n lambda is going to be given by the original ground
state charge density plus a term that we
call the linear response that is a first order in lambda,
plus a term that is a second-order response,
plus a term that is third order and fourth order
and so on and so forth.
And the smaller the lambda is, the less relevant
will be the higher order terms.
And what is really important is just this first order response
in the charge density.
And this actually sort of gives us a lot of power.
That is, just knowing the first order response in the charge
density allows us to calculate the energy of a system
in the presence of this perturbation
up to the second order.
And this basically can be seen by using
Hellmann-Feynman theorem here.
This is-- you have seen it sort of when we discussed forces,
and it's really a sort of direct consequence of the ground state
properties of the orbitals in the Schrodinger equation.
But basically, what does Hellmann-Feynman theorem
tells us?
Well, it tells us that if we need
to look at the change in energy with respect
to a certain perturbation, the only thing
that we need to calculate is actually
the expectation value of the orbitals with respect
to the perturbation.
So this is really Hellmann-Feynman theorem.
You might remember this.
We have discussed it.
We don't need to take into account the derivative
of, say, lambda--
sorry, I'm using too many lambdas here.
But let's call it lambda again here.
We don't need to take into account the changes in the wave
functions themselves.
We just calculate the expectation value.
So for any given arbitrary lambda,
the derivative of the energy, the change
in the energy with respect to lambda,
is just given by this simple integral.
And so you see that if we then say that we actually
expand the energy up to the first-order term, what we have
is that the energy is equal to e0 plus lambda d
e over d lambda.
And then there would be a one half lambda square term.
But then in itself, we can expand--
it's a little bit father, so I'm sort
of skipping some of the passages-- we can expand this n
lambda in powers of lambda.
And so what you see is that up to the second order
in lambda, the only terms that are
relevant in the expansion of your energy--
and this is what is important--
are, for the first order term, just the integral
of the perturbation with respect to the ground state's charge
density, and for the second-order term,
the integral of the perturbation with the first order
change in the charge density.
So perturbation induces a change.
And if we are able to calculate the linear response to that,
this n1, we actually get correction
to the energy up to the second-order term.
And so there is a whole branch of density functional theory
that's called density functional perturbation theory that
really deals on how to calculate this linear order perturbation.
And I really won't go into the detail.
The sort of other difficulty with respect
to a standard set of quantum mechanical perturbation theory
is that in density functional theory,
the Hamiltonian is in itself self-consistent.
And so you have an added layer of complexity,
because it depends from the charge density itself.
And so when you have got a perturbation
in your external potential, the perturbation
in your Kohn-Sham potential contains also term
in the Hartree and in the exchange correlation term.
So we won't really go into these details,
but there is, again, an iterative self-consistent
procedure to obtain the self-consistent linear order
response to an external perturbation.
And all of this is actually implemented in the PWSCF code.
So some of you might at certain point
need to use that, again, to calculate quantities
like the phonon dispersions.
And there are actually examples in the distribution
that tell you how to do that.
And so if we go back to our original problem, all of this
was actually to convince you that if we
are able to calculate this linear response,
the first order change in the charge density induced
by a certain perturbation, we actually
have the energy of a system up to the second order
in the perturbation, where the first order term, again,
is just given by the integral of the ground state charge
density in the perturbation.
And the second order term is given
by the integral of the linear response
times the perturbation.
But you see, what is now the fundamental step
is that these quantities don't depend
on the configurational variables.
These are quantities that are calculated on the ground
state-- that is, the green crystal--
something that doesn't depend on configuration.
So this can be calculated once and for all just
on the system that contains two atoms per unit cell.
And once you have calculated this,
this will give you the energy up to the second order
in the configurational variables.
So doing this calculation will take some time,
but it's really on a two-atom unit cell.
But once you have this, what you have
is really a analytical and extremely inexpensive
expression to calculate the energy up to the second order.
So as long as your disorder is weak,
you will get a very good approximation of the energy
just by doing this set of sums that a computer will
take probably a nanosecond or something like this.
And again, this really looks like an Ising model
Hamiltonian.
If you remember what you had in the Ising model--
is that you had a sort of coupling between nearby spins
that was parameterized by a certain exchange parameter
here.
The only difference is that, being these
are semiconductor alloys, interactions
tend to be long range.
And so it's not only atoms or sigma variables sitting nearby
that sort of interact, but the interaction sort of
goes on farther and farther.
And so it's a sort of long range Ising Hamiltonian.
But once you have this, you can do all the thermodynamics
that you want to your problem and very inexpensively.
I'll sort of go very quickly through this.
But again, what we have seen up to now is really this sort
of fundamental concept that the electronic structure
calculation or our energy calculation allows us
to extract parameters-- in this case,
the J interaction constants--
that gives us the energy of our system.
And as usual, when you really need
to find out what are the parameters to extract,
you need to think at what are your relevant degrees
of freedom.
So you need to think at the physics of your problem.
And if you have a semiconductor alloy,
the first degree of freedom that comes to mind correctly
is the configurational degree of freedom.
That is, we can have on a certain site one atom, gallium,
or another atom, indium.
But actually, it's not the only degrees of freedom that is
relevant, because depending on which atom you put on a certain
side, what you have is that, depending on also what is
the surrounding environment-- if it's a gallium between gallium
atoms or if it's a gallium between indium atoms--
that atom will see a broken symmetry.
And so it will want to relax from the ideal sort
of periodic Bravais lattice position
and will likely like to move around,
either to make space for larger atoms that
are nearby or sort of to make space for itself if there
are smaller atoms.
And so this is sort of where intuition becomes important.
And so you need to decide what are the important degrees
of freedom.
And in this case, what we are saying
is that there are not only configurational variables that
are important degrees of freedom,
but there are also displacements of the atoms in response
to a certain configuration away from equilibrium.
And so once you have identified that there
is this second set of degrees of freedom,
well, what you are really saying is that we really
need to consider, in the expansion of the energy
over all my degrees of freedom, not
only the configurational variables but also
the displacement of an atom, say,
from its equilibrium position.
And so we need to expand their energy up
to the second order, not only in configuration but also
in displacements.
And because we are expanding up to the second order
in configuration and in displacements,
there is going to be also a mixed term that
is half configuration and half displacement.
If you want this term, think of F
as being the second derivative of the energy with respect
to u and with respect towards sigma.
So this term here is telling us what
is the force that is induced on the atom R
when, on the side R prime, there is a certain atom given
by the configurational variable sigma.
So we really need to expand in all our degrees of freedom.
But then we can do something more.
And we can even sort of apply an equilibrium condition.
Because at the end, what we want to do for the system is not
really the dynamics, but what we really want to do
is the thermodynamics of the different configurations.
So what we are saying is that, well,
suppose that we have chosen a certain configuration of atoms.
Now, what we really need to take into account
is not only how the energy changes but also
how the atom moves around to accommodate the configuration
as best as possible.
But that really means that the atoms will move in order
to minimize the energy, or if you want,
in order to have really zero force at a certain point.
If you start from the periodic lattice
and you sort of decide a certain configuration, on those sites,
there will be forces.
Atoms move a little bit away from that site,
so that their force is 0.
So what we have is here the sort of physical embodiment
of the degrees of freedom.
The energy is expanded up to the second order.
But then we also apply an equilibrium condition.
That is, the force on the ions for a given configuration sigma
must be 0.
So the derivative, the first order derivative
of the energy with displacement, that is the force, must be 0.
And if you apply that at this and you work the algebra,
you really obtain, for a given set
of configurational variable, what are the displacements away
from equilibrium that are induced
by this configurational variable.
And of course, they are mediated by this response function
of our system, the force on an atom induced
by our configurational variable, and really,
the second-order derivative, the stiffness
of our potential energy surface with respect
to atomic displacements.
And so we use this equilibrium condition.
That is, for a given configuration,
the displacement are really given.
And we take it, and we stick it back inside.
So we substitute, for the green u of R, this expression here.
Those are all sort of tensorial quantities
which I sort of removed the sums.
But so we are back to our spin Ising Hamiltonian
in which the only thing that is relevant
is the configurational variables,
because at the end, that's the only thing that can happen.
We can have a certain atom in a certain place,
but we take into account the fact that the ions sort of
go back, move to equilibrium.
And we take that into account by a renormalization
of the interaction constant j by, if you want,
the displacement term or what we would call the elastic terms.
OK, so we have all the tools to calculate this regional j, this
that would be nothing else than the dynamical force
constant matrix that you diagonalize
in your final calculation or this mixed term that
in the literature are called Kanzaki forces.
We do this on the virtual crystal.
And then we really have our spin Hamiltonian,
including the energetics effects of atomic displacements.
And so we have really the Hamiltonian of our system,
and we can start doing the thermodynamics of it,
with a small caveat that what we have done up to now
is really this step.
We have been able to calculate, if you
want, the energy of a certain configuration,
starting from a virtual crystal.
But we haven't really-- and this is a subtlety that I won't go
into the detail--
we haven't really taken into account
that our sort of end original products
have actually different lattice parameters.
So in itself, at the end, if we need
to understand phased ability, we need
to understand what is the lower Gibbs
free energy for our problem, that is,
if entropy drives this configuration
to be stable, or, if you want, if enthalpy and elastic
interaction drive segregation in two
compounds with different lattice parameters
to be the stable state.
And of course, the closer the lattice parameters
are, the more irrelevant this term is going to be
and the more relevant term in which
the disorder atoms will be.
And temperature is always going to favor this.
But this might become very expensive
if the original compounds have very different lattice
parameters, because we really need
to stretch things to start having
the possibility of mixing them together.
And all sort of-- the phase diagonal stability
of this system is really driven by these two competing sort
of forces, the temperature that want
to mix up things around but chemistry
that, in order to mix things around,
you need to have atoms that are chemically very similar.
And you need to have lattice parameters that
are fairly similar, so that you don't build up a lot of strain.
And so, again, without going into detail,
it is actually very easy to calculate
some of those elastic terms.
And so when we do that, what we really have is a way
to calculate how much energy it takes to stretch things so they
have the same sort of unit cell and then how much--
and that was the tricky part--
it cost to mix things around, it cost
to have a certain configuration.
And the very nice thing is that now, we
can sort of test our perturbation theory approach
and compare it with full blown density functional theory
calculations for cells that are small
enough so that you can actually do a density functional theory
calculation.
These are older results.
Now we could do much larger cells.
But the concept stands.
We can do a full self-consistent calculation, say,
for a super lattice of gallium planes and indium
planes in the 111 direction.
And so we can calculate what is the lattice
parameter of this structure and what
is actually the sort of Gibbs free energy
of mixing for this system.
And we can do that with a full explicit calculation,
and we can do it with a set of perturbation approach.
And you see, we can compare the equilibrium lattice parameters
in both cases.
And basically, the difference-- this is atomic units.
This is PWSCF-- is basically negligible.
And the energies are also very good.
This sort of error that the perturbation theory makes
is actually here, just 1 millielectron volt per atom.
And as you sort of move around, you
see that there are lot of cases in which the error is larger,
4 millielectron volt. And 4 millielectron
volt is very small, but it's still
10% of this mixing free energy.
So it's sort of somehow relevant.
But we, I would say, still do very well.
And calculating this energy of mixing
is absolutely inexpensive.
It takes a nanosecond on your computer.
You just basically sum with the appropriate configurational
variables the term with your interaction constants.
This requires a full self-consistent calculation.
So with this approach, you can calculate the energies
of millions of structures that are
very large, while with this, you can only
calculate tens of structures that are reasonably small.
So it's very appropriate to use the linear response term
to then do-- to use one of these statistical sampling methods
that you have seen with Professor [? Seder. ?]
And again, I won't go into detail, because you
have seen them partially, and you'll
see them over and over again.
But just to remind you sort of the concepts,
if you want to understand what is
the stability of a certain structure, what you really
need to compare are their Gibbs free energy.
So you want to operate into the thermodynamic ensemble in which
you have constant pressure and constant temperature,
because that's really what your experiment is.
You're at a certain temperature, and you let the system relax.
And as usual, you can't calculate--
explicitly, there is not a formula
that gives you free energy.
It's a logarithm of a partitioned function.
And so there are sort of statistical ways
by which that can be calculated.
And one of the most powerful approaches
is that given by thermodynamic integration.
That really is based on the fact that while we can't calculate
free energy, we can actually calculate
the derivative of free energy.
So the derivative of the free energy with a certain
parameter, like the derivative of the free energy,
the Gibbs free energy with respect to the number
of particles, is actually going to be given by the ensemble
average-- that's when we use the sort of brackets,
the bracket sort of symbol--
by the ensemble average of the derivative of the energy
with respect to the number of particles.
So that allows us to calculate derivatives of free energy
in the appropriate thermodynamic ensemble.
And so what we can do is calculate free energy
differences by integrating the derivative of free energy
from state A and state B. And obviously, this integral
will be done by step-wise calculating these derivatives
at different points.
But these derivatives, that in the case of the Gibbs
free energy with respect to the number of particles, nothing
less than the chemical potential as a function
of the concentration, needs to be evaluated
for every concentration.
Say, in going from concentration A to concentration B,
what we really want to figure out
is what is the free energy difference between these two
concentration.
So each of these points in this integral
requires an entire statistical sampling and then
an entire statistical simulation.
And so this is what you would actually do.
You choose a temperature, say, 1,000 kelvin,
and you choose a pressure, say, 0 pressure or 1 atmosphere,
that for all practical purposes is 0.
And then under these conditions, you
have your total energy expressed in configurational variables.
And you decide what is your chemical potential, that is,
what is sort of the additional cost of switching
an atom from gallium to indium.
And then you start with your Monte Carlo sampling.
You keep switching atoms from gallium to indium
or from indium to gallium.
Each of these switches will have a certain energy
cost that is weighed with the Metropolis factor.
And in this simulation, at a given pressure
and at a given chemical potential,
after of millions of sweeps and millions of trials in which you
try to change a spin variable from plus 1 or minus 1,
you do this over and over again, and you end up with a point
there that tells you what is the concentration of, say,
gallium 0.9 for a given chemical potential.
And you repeat this over and over again.
That is, you sweep all your possible chemical potential.
And for each choice of chemical potential,
you get a certain average concentration.
And so you do this over and over again.
And so you see, what you get is a curve
that tells you what is the concentration as a function
of the chemical potential.
And you get one of these curves for each temperature
at which you do the simulation.
So you get concentration as a function of chemical potential.
And if we go back for a moment to the previous slide,
this is really the inverse of what you need.
You need the thermodynamic integration
to calculate the free energy difference between two
concentration, the integral of the average chemical potential
for a given concentration.
So the only thing that you have to do
is, really, look at this curve, turn it around if you want to,
so that you have x as a function of mu.
And say, for the 1,000 kelvin case,
you'll have something like this.
So you have sort of inverted--
your simulations have given you concentration as a function
of chemical potential.
You invert that, and you integrate it,
and you get a free energy curve.
And you see, things get interesting
when you start being below a critical temperature.
So you see, we got 820 kelvin.
And you really see that the curve is still continuous,
but it sort of starts to have a slope in the middle that
is going towards infinity.
And if we go below, at 650 kelvin, what happens
is that we have opened a miscibility gap.
That is, there is a region of concentrations from 0.1 to 0.8
that never appear.
So at 650 kelvin, this system doesn't
want to mix in the concentration regime between 0.8 and 0.1.
And this is typical of model alloys.
Miscibility gap has opened.
The two things don't mix.
And if you think at sort of how your curve could look like,
we could sort of somehow interpolate
in some arbitrary way this curve and then invert it.
And when you integrate the data--
it seems that the computer has gone mad.
And let's see.
OK, we'll keep the green forever.
It's now in the realm of the computer.
But this is good, because when you invert this
and you integrate, what you see is
that for your free energy as a function of the concentration--
is that what you would really obtain
is real free energies only from 0 to 0.1 or from between 1
and something like 0.8, 0.85.
And in the middle, you have this famous miscibility
gap in which, really, the free energy of the solid solution
is higher than the free energy of the n compounds.
And so you have the max error and the tangent.
And so what you have is the composition of your system.
And so you can calculate your phase diagram.
And at the end, for this system, it would really look like this.
Below 820 kelvin, you open a miscibility gap
in which, really, the two systems
don't want to mix together.
And so in many ways, semiconductor alloys
are really model solutions.
And below the critical temperature,
they start to segregate.
And of course, you can do a lot of other intriguing things.
That is, in your simulation, you figure out, say,
what is the distributions of, say,
gallium-phosphorus distance or indium-phosphorus distances
at any given concentrations.
And what you actually discover is
that there is a sort of very bimodal distribution,
that there are short distances with the phosphorus,
and there are long distance with the phosphorus-- that
are, of course, due to the fact that these
are gallium-phosphorus distances and these are
indium-phosphorus distances.
But they sort of depends with the concentration.
But those are things that can be measured.
And so in this sort of graph, I've
put an example of what are the numbers that we
get from the Monte Carlo simulations
on the Ising Hamiltonian and what are actually the EXAES
results that people measure.
And you see, there is really a very good agreement.
Say, these are the short gallium phosphide distances
as a function of concentration between what
the experiment measure and what your thermodynamical DFT is
actually predicting.
And I think I'll skip this.
There were some other details about epitaxial system.
But otherwise, I'm worried I won't
be able to do the second part.
So this was the first example.
Again, what is important there is that we
have a physical problem.
We have identified what are the important degrees of freedom,
and we have calculated those degrees of freedom
from calculations that are actually manageable.
But those degrees of freedom in those calculation
have given us the parameter of a Hamiltonian that, in this case,
was a Ising-like long range Hamiltonian in which we
can do sort of much more complex statistical mechanics analysis.
And so we can actually obtain phase diagrams.
That was one example.
The other example that I want to show you
here is instead how we sort of repeat
the same process in order instead
to calculate Hamiltonians that have a linear scaling cost.
That is, that allow us to calculate
the electronic structure and energies
of systems that are really large,
that are sort of thousands or tens of thousands of atoms.
And you know, this is because this comes from some research
example.
We were interested in sort of studying
what is the effect on a carbon nanotube
of organic functionalization.
That is, what happens if we take a ligand--
this would be-- it's a benzene ring.
So again, let's kind of call it phenyl moiety.
And this sort of ring is attached with a covalent bond
to an infinitely longer carbon nanotube.
And people have started doing this.
And of course, when it happens, what you
have is a nanotube that has a random array
of these funny rings.
So again, we have the thermodynamical problem
of finding out what is the effect of this configuration
of this order on some of the intriguing properties
of the carbon nanotube.
Like, there are sort of ballistic electrical
conductivity, that is electrical conductivity, that
is independent of length, so very different
from an ohmic conductor--
or what is their thermal conductivity.
But again, we really don't have the computational resources,
and no one has the computational resources
to study, with density functional theory, a system
with tens of thousands of atoms.
And you'll see this in sort of the last part of the class,
really.
It's not just a matter of waiting a few more years.
I mean, the scaling cost kills us in many ways.
So we really need to use sort of ingenuity.
And what we are going to use here is this idea.
We want to break away from this orthonormalization
constraint that is really what drives
the cubic scaling of a system.
That is, what we want to do is using a system
as sort of picture in which we really
look at Kohn-Sham orbitals that are localized,
so we don't have to worry about orthonormality with orbitals
that are far away.
Because if two orbitals are far away, and both of them
are localized, their overlap is going to be 0.
And so they are orthogonal by default,
without us having to sort of impose that.
And the way we are going to do this
is, again, a mapping from the full density functional theory
calculation.
So we start--
I'm giving you here the example of silicon.
If you look at the band structure of silicon,
again, what you have are very localized
core orbitals down there that don't really have
any effect on the chemistry.
They corresponds to flat bands.
But what is interesting is what the valence electrons do,
and this is sort of the very celebrated band
structure of silicon.
This in particular is the gamma point, so at 0, 0, 0.
And so because silicon has two atoms
per unit cell, four valence electrons per atom,
we have eight valence electrons.
Spin the generator.
So we have bands that contain two electrons each.
Sorry.
So what we have is four valence bands.
And you see, 1, 2, 3, 4.
As we move around the Brillouin zone,
they can become the generator.
And those that are studying semiconductor physics,
they will know that this is the top of the valence band,
and we have heavy holes and light holes
depending on the curvature up here.
But so what is the electronic structure of our problem?
Well, remember, the band index and the quasimomentum k
across the Brillouin zone are really
the quantum numbers for the crystal, for my problem.
If I study a molecule, and I have
a molecule with eight electrons, I will have only four--
in the case of spin degeneracy--
eigenstates that are occupied.
If, instead of studying a molecule with eight electrons,
I study a crystal with eight electrons per unit cell,
instead of having four states, I have four bands.
And the energy of each eigenstate,
thanks to Bloch theorem, can be cataloged
depending on this band index and depending on the quasimomentum.
And the quasimomentum really sort of tells me
how the phase of things changes when we go from one unit cell
to the other.
The same concept takes place also not in--
for the electronic eigenstates but for the normal modes,
for the phonon modes of a crystal.
We classify them according to wavelength
and according to band index.
And so your infinite number of electrons in the crystal
can all sort of be summarized in this picture, in which we
have a continuum of states.
So if you want, we have mapped the fact
that we had eight electrons repeated infinitely
into a problem in which we have eight electrons, that is four
bands, with a sort of continuum variation,
depending on which wavelength we are looking at.
And so this picture sort of represents
all the possible occupied states in your system
that are infinite but are classified in a group of four
with a continuum index k.
Now, the interesting things about quantum mechanics
is that you can change your representation
and still look at the same problem.
It's the same thing in classical mechanics.
If you are sort of studying something in three dimension,
you can rotate your coordinate system.
And so your equation of motion will change.
Your trajectory will change.
But really, you are just moving everything accordingly,
and you are looking at the same problem
in a different coordinate system.
That is even more true in quantum mechanics, in which you
can perform a unitary transformation between all
your eigenstates and still look at the same problem.
Usually, what we sort of choose to be
is into a representation in which
we are looking at eigenstates of the Hamiltonian.
So each of these states here is exactly
the characteristic of being an eigenstate of the Hamiltonian.
You apply the Hamiltonian to that state.
What you obtain is a constant that
is the energy eigenvalue times the state itself.
But we can mix this around, and we can still
look at the same problem, just in a different representation.
Exactly as in classical mechanics,
we can do an orthogonal transformation
of our coordinate system and look at the same problem.
So what we can do is actually do a unitary transformation,
a unitary rotation, of our orbitals
and still look at the same problem.
And again, without really going into the details of what it is
and what quantum mechanics is, it's
something that would look like this.
That is, if we start from our eigenstates, at each k point--
that is, at each point in the Brillouin zone--
if we have those four bands, we can mix them together
with a unitary transformation.
But not only that, we can sort of mix those mixed bands
with all the other bands at different k points
with the appropriate unitary factor.
And so what we can obtain is a new set of orbitals that
are, again, infinite and that really represent
the same quantum mechanical problem,
just sort of turned it around.
And our quantum numbers that before,
where the band index and the quasimomentum
become some new quantum numbers, that will be-- again,
one of them will be related to the bands.
So for silicon, we'll go from 1 to 4.
And the other one will count all the sort
of infinite number of electrons that we
have when we go from one isolated unit cell to one unit
cell periodically repeated.
And before, what we had was a quasicontinuum,
basically, of wavelengths.
And in this new representation, what we
have is a quasicontinuum of localization vectors.
So we go from extended Bloch orbitals
that are eigenstates of the Hamiltonian
to localized orbitals in which the electron is sitting in each
of the different unit cells.
And really, this is an arbitrary transformation.
And these matrices are sort of our parameters,
the sort of quantum mechanical equivalent of the three Euler
angles that rotate the Cartesian system from one
representation to another.
And so we can choose, say, this set of transformation,
so that what we obtain is the same quantum mechanical picture
but in a representation that is as localized as possible.
And how we do that sort of requires a lot of algebra.
But this is the sort of very intriguing result
that takes place when we actually make it in practice.
We take-- again, all these states and each of them
is delocalized.
It's Bloch orbital energy eigenstates
that is extended over the whole crystal.
We mix all these things together with a criterion
that those mixing transformation are
chosen so that the new states are as localized as possible.
And what we obtain at the end is fairly beautiful.
If we look at, say, something like silicon, again,
sort of a perennial diamond structure,
one silicon atom fourfold coordinated
with the other silicon atoms--
and it's a bit difficult to see, but what we would have
is that one silicon atom, in red,
is connected to four silicon atoms.
And this blue over blue silicon atom
is connected to other four silicon atoms.
And the localized orbital that you obtain
is really what the chemist would have told us probably in 1930.
It's really a covalent bond between the two orbitals.
So what we have is that these two atoms here
are connected by an orbital that has,
really, this pile up of covalent charge between those two atoms.
And there is a certain amount of sort of back-bonding
between each of those two atoms and the remaining three
neighbors.
So this is how the localized orbital looks like.
And that's been just obtained by a unitary transformation.
But when we have this player here,
it becomes very easy now to construct
the electronic structure of a system that
might have some amount of configurational disorder.
Because, if instead of having somewhere,
say, silicon-silicon in an infinite silicon matrix,
we have maybe gallium and arsenic for a moment locally.
Well, maybe what we can do, and I'll show you how we'll do it,
maybe we can put there the exact orbital corresponding
to the gallium arsenic bond.
And if you look at it, it's actually going to be-- again,
because things are similar, it's going
to be very similar to the silicon orbital.
The gallium arsenic covalent bonds
is just, because of the difference
of electronic activity, sort of tilted more
towards the electronegative arsenic in this problem.
Or say, again, to give you the feeling,
it's nothing less than the sort of standard chemical picture.
If we have a system that is still silicon,
but it's amorphous silicon--
and I've got here atoms around.
But amorphous silicon is something
in which a tetrahedral short range holder is still
preserved on average but long range holder is lost.
Well, atoms that are tetrahedrally coordinated
won't have a perfect 109 degrees angles,
but their bonds still look very much like those.
And sometimes, you have atoms that are five-fold coordinated.
And so they have their own bond.
And if you want, these are really
what we call the LEGO bricks of electronic structure.
That is, in a very complex system,
the electronic structure of the system
is going to be made up locally by the appropriate orbitals
that corresponds to the local environment.
And the nature and physics is inherently local,
so there is a large degree of transferability.
And again, to give you a sort of more feeling for this,
suppose that instead of looking at a crystal,
we look at a molecule.
We look at something like a benzene ring.
And you go to a representation in which states, instead
of being eigenstates of the Hamiltonian,
are localized states.
What you see are electronic states that are really
carbon-hydrogen single bonds, or what
you see are carbon-carbon single bonds, or what you see
are carbon-carbon double bonds, in which
you have both this orbital and this reflection
across the symmetry plane of the benzene molecule.
So really, what you are seeing is very intuitive chemical
picture.
But the way it's been obtained is actually
from an exact transformation of our quantum mechanical problem,
solving density functional theory, that is,
with a certain degree of accuracy.
And we'll see now how we apply this
to the case of carbon nanotubes, because what we really
want to do is study chemical disorder on these carbon
nanotubes.
And carbon nanotube is nothing else than a sheet of graphite--
that is what is called the graphene sheet--
that has been rolled over together to create a tube.
So if you take this hexagonal lattice,
there are sort of discrete infinite way of rolling things
together.
And the way the carbon nanotube community
has agreed to sort of use a nomenclature
is by defining what is called a chiral vector, that,
in this case here, would be this CH vector.
So if, on a graphene sheet, you define a vector,
that is really just an integer number
of times the two a1 and a2 primitive lattice vectors--
that is, once you have defined this green CH,
you have defined your nanotube, by saying
that you are cutting the graphene sheet
perpendicular to this vector and rolling what you have cut,
so that the top of the vector touches the bottom.
And so you have sort of a perfect cylinder.
And then there is a nomenclature.
And you can sort of-- a generic nanotube
will be called armchair, zig-zag, or chiral nanotube,
depending on how you have rolled things around.
And so if your chiral vector is an n number of a1s
and an n number of a2, equal, your chiral vector
will be this blue vector here.
And so what you are going to do is,
you are going to roll the nanotube so
that perpendicular to the direction of the tube,
you can clearly see this sort of zig-zag pattern.
And so an armchair--
sorry, a zig-zag-- oh, sorry.
A zig-zag nanotube is the one in which
you have a chiral vector that is made
by an integer number of times a1 and zero
number a2 or vice versa.
So in the form n, 0 or 0, m.
And so you see, this blue vector for the zig-zag
is going to be 1, 2, 3, 4, 5 times a1.
So this will be the zig-zag nanotube,
and you see this zig-zag pattern.
If, instead, you choose n for a1 equal to the n for a2, what
you have is a vector that goes in this direction,
like the red vector here.
And so what you are doing is actually
sort of cutting the nanotube like this
and rolling it together.
And again, now perpendicular to the axis of the nanotube,
you see this pattern.
That is called an armchair pattern.
And so an armchair nanotube is identified by these two quantum
numbers.
And then identical zig-zag is n, 0 or 0, n.
And everything else has a sort of more ordered chirality
and is called, generically speaking, a chiral nanotube.
OK, so this is how they look like.
You see, a zig-zag, very long in this direction.
We cut it, we see the zig-zag pattern.
An armchair, we cut, and we see that sort of hexagonal pattern
that, for some reason, is called an armchair.
And now the interesting bit comes
when we look at the band structure of this system.
We could have started, say, from the band structure of graphite
or graphene, in which, again, the Brillouin zone--
the extended Brillouin zone is this hexagon.
And what happens is a bit difficult to visualize.
But actually, what happens is that there
are only six points--
so the black dots there--
at which they are unoccupied states
touch the occupied states.
So usually-- and we have seen it over and over again--
in something like a metal, we will
have maybe a band like this.
And then we'll have Fermi energy that cuts the occupied states
from the unoccupied states.
In graphite, the band dispersion is much more exotic.
It actually looks like this.
In certain points of the Brillouin zone,
it's actually sort of straight lines.
You'll see it in a moment.
And the Fermi surface is exactly here.
So there is only-- the Fermi energy is here.
The Fermi surface is on a point.
And seen from above, if you want,
these are where these combs are that I've sort of cut there
in a profile.
What happens when you go from graphene to a nanotube?
Well, instead of having a system that
is infinitely B-dimensional, you get a system in which,
by rolling one of the dimensions, what
you do actually is you quantize your quantum
numbers in one direction.
So you still have a Brillouin zone
that is a continuum in the direction of the tube axis.
But because you have rolled it, maybe you have only 10 hexagons
instead of an infinite number of hexagons in the direction
perpendicular to the axis.
Now, what you are effectively doing
is really looking at the band structure only along
a certain number of cuts of this Brillouin zone.
And so depending on how you roll your system, what you can have
is that this subset of lines that corresponds
to having a continuum index in one direction
become a discrete index, because in one direction,
we go from infinite hexagon to a finite number of hexagons.
And so depending on how many and what
is the orientation of the cut, we
can have that these lines cross or do not cross these points.
And if they cross those points, we'll have a metallic nanotube.
If they don't, we'll have a semiconducting nanotube.
So just a very slight change in orientation
in the chirality vector might allow us to miss this point
and make something that was metallic into a semiconductor.
And the rule to remember this is trivial.
If the difference between the indices
is 0 or a multiple of three, your nanotube
is going to be metallic.
Otherwise, it's semiconducting.
So say, all armchair nanotubes are metallic,
because m is equal to n, so the difference is 0.
So they are metallic.
Zig-zag nanotubes, well, only 1 out of 3
is going to be metallic.
6, 0 is going to be metallic.
7, 0, 8, 0 are semiconducting.
9, 0 is metallic, and so on and so forth.
And this sort of only breaks down
when we have very narrow nanotubes
in which this sort of model picture of what is called band
folding starts to break apart, because the way we have thought
of the nanotube is as being identical to a perfect sheet
of graphene that has been quantized in one direction.
But really, there is curvature.
And if the nanotube is narrow, there is a lot of curvature.
And so that sort of might change things around a little bit.
But so here is an example.
You see, if you have a metallic nanotube,
like in this case, when you quantize it in one direction,
you are rolling it, you actually end up
with lines that go through the black points.
And so what you have is that your band structure there
sort of sees these cones of the graphite system.
If you have, say, in this case a semiconducting a0 system, when
you quantize anything in that perpendicular direction,
your lines actually do not go through the black points.
And so you never encounter a corner,
and you never find that gap.
And again, a very slight change in orientation
can give you a very variable gap and can bring you
to be metallic very quickly.
And so we start really going again into the details.
We really need to take all our occupied states--
that is, the one below here that I'm not showing,
everything below the Fermi energy
that is here for this metallic nanotube--
and we actually want to take a little bit of the states
above, because those are also important.
And with all these states, the red, the blue, and the black--
so all of the occupied states and a bit of the unoccupied
one--
we want to sort of mix them together
and find what are the localized orbitals coming from this.
And I won't go into the details on how we sort of extracted
those when you have some of the discontinuity that comes about
in a metal.
But so basically, what's happening
is that we start from our electronic structure
calculation, maybe of the carbon nanotube
with a functionalized ligand.
And in a new quantum mechanical representation,
what we have obtained are sort of the building blocks
of the pristine nanotube, and the building
blocks of the ligand, and things like what is exactly
the orbital that connects a carbon
atom in the ligand to a carbon atom in the sidewall,
with sort of full first principle precision.
And so with these blocks now, we can actually
construct not only the band structure
and the electronic structure of this system periodically
repeated, but also a system in which the ligand is first here,
here, and then here, with all the chemical disorder
that we want.
And because the orbitals do not overlap with each other,
the computational cost of doing this is very, very reduced.
Because basically, in our new representation,
orbitals that are far away are orthogonal with each other,
when we actually represent the Hamiltonian--
remember, in a lot of your PWSCF calculation,
you have represented the Hamiltonian
in a plane wave basis set.
And so in a plane wave basis set,
your Hamiltonian has the diagonally dominant part,
that is g squared over 2.
But then it also terms all over the place
that corresponds to the Fourier transform of the potential.
When we actually represent the Hamiltonian
on this basis of localized orbitals, what you have
is that this is truly diagonally dominant.
And as soon as we consider orbitals
that are far away from each other,
there is no Hamiltonian matrix between them.
If we calculate the matrix element between this,
the Hamiltonian, and this green one,
it's going to be 0, because they really do not overlap.
So this is what is called a sparse diagonally dominant
Hamiltonian.
And diagonalizing this is something
that actually doesn't require cubic scaling
but only requires linear scaling.
So one can really sort of study very large systems.
And to show you an enlargement of a cutout
from a perfect nanotube, you see, this is what you get.
You get orbitals that look either as beautiful sort
of bonding combination of sp2 orbitals--
if you think, we know from chemistry
that the carbon is sp2 hybridized.
So it means that the graphitic backbone
is given by a bonding combination
of an sp2 orbital sitting here and an sp2 sitting here.
And the same happens here.
And then there is a set of pz orbitals sitting on each tube.
And when those mix together, they
give rise to the metallic graphitic-like bonds.
And again, this looks a lot like a pz orbital,
but it is nowhere similar to the pz orbital of a carbon atom.
It's the pz orbital specific to a carbon nanotube.
So with this, we can-- as previously
in the case of the alloys, we can first validate our system.
That is, we can start constructing
the electronic structure of something
that we can also fully calculate from first principle.
And we can compare them and these continuous lines
from the models and dots from the full calculation.
And you see, the model works perfectly
and doesn't describe the higher energy
bonds that have not been included
in our sort of orbital construction procedure.
But for all the bonds that we wanted to consider,
this is perfect.
And once we sort of feel confident that this works,
we can actually use it to calculate the--
let me actually go farther--
we can actually use it to calculate
the electronic structure of systems
that are too large to attack directly, something like this.
This has just a conducting region of 3,000 atoms
and a very long sort of semi-infinite leads
on the side.
And we can figure out a lot of what happens,
say, to the ballistic conductivity of this system.
And I'm not really going into the details.
I wanted to go before concluding to a previous slide
here, in which I was actually showing how,
in practice, the matrix element between a localized
orbital and another localized orbital
changes as a function of the distance,
say, as we sit on a carbon atom, and we
look at the overlap between the pz orbital here
and we move away from the nanotube.
And you see, in a semi-logarithmic scale,
we have very good exponential decay.
Certain point goes into the numerical noise.
And we can really look at--
validate what are the effects of our truncation by cutting
shorter and shorter Hamiltonians, by neglecting
a large number of terms that are far away.
And you see, there is almost an invisible effect
on the band structure.
So that really sort of tells us how much localized things are,
how much they overlap, and if you want,
tells us how long range or short range chemistry is.
That is very important when we consider
one of this configuration.
So with this, I'll conclude.
Again, the two important messages
that you want to remember from this class
is how you take a physical problem,
and you can use density functional calculation
to find out what are the important parameters
and construct a model for your energy.
The first case where configurational variables
inducing relaxation and gives us Ising-like Hamiltonian,
in this case, we have localized orbitals.
And once we determine those, from a small calculation
or from a fragment, we can actually
calculate the very large system.