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Welcome, everybody,
on this snowy Friday.

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At the end of last hour,
and I just want to take a

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minute or two here to finish
this piece up,

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we were talking about ways of
writing kinetics expressions for

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a particular type of assumed
ligand substitution mechanism.

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In particular,
we were looking at dissociative

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substitution.
And now I would just like to

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take you through what happens if
you don't make the major

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assumption that we made last
time, which was that every time

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our intermediate five coordinate
complex ML five was

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formed, it would go onto
products.

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That was our assumption last
time, one of them.

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And this steady state
approximation is one that allows

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us to simplify the rate
expression.

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If we don't make that
assumption, what we say is that

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the change with time of the
concentration of the

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intermediate is approximately
zero.

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We know that at the very
beginning of the reaction,

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we have this ML five X
species.

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X is the ligand that
dissociates to give ML five,

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so at time zero there
is zero ML five

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concentration.
And let me just remind you of

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this plot of what happens in a
reaction like this.

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We have our initial ML five X
species that decays

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away, and we have our product ML
five Y that grows in.

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And if this reaction goes by
dissociative ligand

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substitution,
which would certainly be

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consistent with a d four
high spin electron count,

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as we saw last time,
then there may be some

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intermediate ML five
that at time zero has zero

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concentration.
And it may never really build

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up very much concentration.
And then at the end,

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it goes down to zero,
too, because when the reaction

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is over, all of our ML five has
a Y attached and is

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six-coordinate again.
If the concentration of the ML

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five intermediate never
really builds up very much,

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which is quite often the case,
it may not be observable,

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the amount of which is produced
during the reaction,

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then this approximation is
pretty valid and allows us to

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derive equations for the change
with time of the species that we

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can observe, namely the starting
material that is going away and

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the product that is coming in.
And so this is our

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concentration versus time plot
for the reaction.

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And with this approximation,
we can then set equal to zero

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the expression for the formation
of the intermediate,

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which is k1 times ML five X.

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This quantity k1,
if you remember back to the

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diagram we were using last time,
k1 is the rate constant

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associated with surmounting that
first energy barrier,

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k1 times ML five X.

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That produces ML five,
so this is producing

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it, but when you are in the
middle of the well,

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where the intermediate lies,
there are two ways to destroy

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ML five.
You can go minus k minus one ML

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five times X.

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This brings you,
then, back over to where you

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started.
And you can also lose it in a

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productive sense by k two times
ML five times our species Y,

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which brings us onto products.

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And we are still making the
assumption that once you get to

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products, the reaction is done
and you never go back.

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That assumption is built into
this analysis.

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And if we look at this,
we have set it equal to zero,

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meaning that ML five is
not building up.

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And so now, what we can do is
we can rearrange this expression

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and solve it for the
concentration of ML five.

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And you will see that you can
do that here.

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ML five becomes equal to k1
times ML five X --

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That is the expression for the
formation.

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These two brackets should be
done.

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ML five X over k minus one
times Y plus.

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Sorry.
That should be X,

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here.
Plus k2, times Y.

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The expression

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in the denominator here are
those two quantities that take

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us out of the well.
And on the top,

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we have the one quantity that
takes us into the well from the

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starting materials.
And so now, we have an

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expression for ML five
that we can plug into our

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expression for the rate.
The rate overall expressed as

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the appearance of the final
product is going to be equal to

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k2 times ML five,
this is going over the final

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barrier, times Y.
And so,

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if we take the expression here
for the rate,

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this is the value that we
expect for the appearance of

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products.
We take that substitute in this

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expression for the unobserved
intermediate,

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ML five,
into this.

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Then we have an expression for
the rate entirely in terms of

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things that we can either
observe, namely the starting

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material, the concentration we
would measure by some technique,

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like spectrophotometry as a
function of time,

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watching an absorbance band for
it decay away,

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for example.
And then also in terms of the

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concentration of Y which we were
talking about as the solvent,

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so it would have a large
invariant and known

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concentration.
And then, of course,

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since X is being produced in
the reaction whenever it

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dissociates from the starting
material, there is a

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relationship between the
concentration of X at any time

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and the concentration of our
starting material at any time.

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Now we get an expression that
we need to be able to integrate

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in order to actually produce,
given these parameters,

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k1, k minus 1,
and k2, to produce a predicted

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dataset to compare with the
experimental.

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And what one normally does is
you take an experimental

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dataset, you have the equations
that arise from your mechanistic

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hypothesis, and you do a least
squares fitting procedure to

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obtain the values of the
parameters, which are these

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phenomenological rate constants
associated with each step in the

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mechanism.
Extract the values of the

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parameters, and see what you
get, see if the mechanism that

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you have assumed can give you a
good fit to the data or whether

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it cannot.
That is the end of my

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discussion of an introduction to
kinetic analysis of chemical

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reactions, a really important
subject where differential

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equations really become very
important.

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Today, I want to get onto the
top of extended solids,

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and so I am going to talk about
dimensionality,

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here.

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And materials.

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And I want you to keep in mind
the framework with which we have

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been discussing chemical bonding
all throughout this semester

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because we are going to extend
that today to try to understand

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some of the properties of
systems that are not molecular,

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but extend infinity in some
number of dimensions.

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We talked about the H two
molecule.

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This is a very small molecular
system that we have described

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and talked about quite a bit.
This system has a radius,

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here, of about 0.64 angstroms,
or internuclear distance of

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about 0.64 angstroms.
And then, we can also consider

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molecules that become more
extended.

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There is an example of a
polyene.

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And my use of a polyene,
here, should lead you to think

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that when we consider polymer
chemistry, in some cases,

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we will be talking about
systems that are extended,

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maybe not infinitely,
but very greatly in one

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direction, or maybe more than
one direction.

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But polyenes are interesting
species because if this thing

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had all the double bonds,
trans, as I have drawn here,

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then what we have perpendicular
to the board would be a set of p

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orbitals, one p orbital
perpendicular to the board on

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each of these carbons.
And all of those p orbitals can

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overlap.
And then, we can start talking

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about the transport of electrons
down a chain like this in one

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dimension.
So I put H two up here

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as an example of what is
approximately a zero dimensional

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system.
Here is a system that extends

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somewhat in one dimension.
And then, as an example of a

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two dimensional system,
let me just draw up here a

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piece of one of the sheets of
the graphite structure.

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Graphite is one of the
allotropes of carbon.

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And graphite is a really nice
2d structure.

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I can draw in some of the
unsaturation here.

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And I don't mean to suggest
that this thing stops here.

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These carbons on the parameter
of the graphite system have

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bonds, and the structure looks
very much like what I have drawn

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here, if you repeat outward,
up, down, left,

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or right.
And what that leads to are

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two-dimensional planar arrays of
carbon atoms that,

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as with the polyene structure,
here this polyene that I drew

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with six bonds,
is about 13 angstroms long.

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And then the problem that you
get to with graphite is that

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these graphite sheets can have
dimensions of millimeters.

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You have gone from something
short to something very,

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very large.
And we are going infinitely.

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I am going to show you a little
bit more about this.

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I am going to let you look at
this website yourselves.

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You are going to find that I
did put into the notes for today

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the website that we are going to
look at.

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And it is for you to go ahead
and look at in three-dimensions

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at the structure of the
graphite.

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And also, in particular,
I want you to be able to look

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at the structure of the diamond
framework.

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One of the things that I want
you to be thinking about in

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association with today's lecture
are the way that atoms pack in

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three dimensions when you make
up a solid material.

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You will see why that is
important in just a moment.

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And so we are going to have to
go from bonds to bands in order

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to make this transition.

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And that means we are going to
have to talk about band

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structure today and where bands
come from.

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This, by the way,
the title for this panel from

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bonds to bands is actually the
inverse of a title that was

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penned for a beautiful article
written by Professor Roald

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Hoffmann.
And he was actually one of my

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teachers as an undergraduate at
Cornell University,

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and he also won the Nobel Prize
for his contributions to theory.

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And I refer you to his article
from bands to bonds if you want

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to learn more about this topic
because, although the concepts

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of solid state physics are often
discussed with very different

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terminology than the concepts of
electronic structure theory for

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molecules, there are a lot of
very important parallelisms.

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And one of the things that
Professor Hoffmann is very good

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at is in bridging the gap
between different branches of

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science that talk about the same
things but don't realize that

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they are talking about the same
things.

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00:14:33,000 --> 00:14:37,000
And so here we have a system
with a single orbital.

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We are going to look at the
number of orbitals.

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And here is a system with one.
And this is an energy level

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diagram.
We have used energy level

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00:14:49,000 --> 00:14:53,000
diagrams for a lot of things.
Last lecture,

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00:14:53,000 --> 00:14:57,000
we used them to discuss
potential energy surfaces of

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chemical reactions,
in addition to all these other

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properties.
Here is a system with one

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00:15:05,000 --> 00:15:07,000
orbital.
And then, as you know,

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when you have a system with two
orbitals, you can get bonding

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and antibonding.
This is the hydrogen problem.

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00:15:15,000 --> 00:15:17,000
This might be a hydrogen atom,
for example.

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00:15:17,000 --> 00:15:21,000
Here is an H two
molecular orbital diagram.

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00:15:21,000 --> 00:15:25,000
And then we can consider a
system that might have three

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00:15:25,000 --> 00:15:28,000
orbitals populated,
like this.

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00:15:28,000 --> 00:15:32,000
And then, if we have a system
with four orbitals,

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00:15:32,000 --> 00:15:38,000
it might be something like
that, four electrons and so on.

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00:15:38,000 --> 00:15:44,000
We see that one of the features
is that the energy levels are

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00:15:44,000 --> 00:15:50,000
starting to come closer together
as we get more and more of them.

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00:15:50,000 --> 00:15:54,000
There are five.
Here is six.

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00:16:00,000 --> 00:16:05,000
And then, onto seven.
And you start running out of

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00:16:05,000 --> 00:16:10,000
room to draw them.
And so, what people do then

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00:16:10,000 --> 00:16:16,000
with this problem,
we are only up to seven and we

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00:16:16,000 --> 00:16:22,000
have almost run out of space to
draw these things.

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00:16:22,000 --> 00:16:25,000
So what do we do?
We draw them,

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00:16:25,000 --> 00:16:32,000
when we get out to infinity
here, as a band.

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00:16:32,000 --> 00:16:35,000
What the idea is,
is that we have here this band

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00:16:35,000 --> 00:16:41,000
diagram, as we are going to call
them, a type of diagram in which

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00:16:41,000 --> 00:16:45,000
we are representing field
orbitals down here as some kind

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00:16:45,000 --> 00:16:49,000
of continuum.
Because there are so many of

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00:16:49,000 --> 00:16:53,000
them, an infinite number of
orbitals that are all

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00:16:53,000 --> 00:16:56,000
interacting in some extended
solid material,

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00:16:56,000 --> 00:17:01,000
we are going to be talking
today a little bit about silicon

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00:17:01,000 --> 00:17:05,000
and germanium and things like
gallium nitride,

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00:17:05,000 --> 00:17:10,000
in which you have a lattice
that extends periodically in

237
00:17:10,000 --> 00:17:15,000
three-dimensions.
And so these molecular orbitals

238
00:17:15,000 --> 00:17:18,000
spread out and cover the whole
material.

239
00:17:18,000 --> 00:17:23,000
Electrons can be anywhere at
once within this entire extended

240
00:17:23,000 --> 00:17:26,000
solid by virtue of these
delocalized orbitals.

241
00:17:26,000 --> 00:17:30,000
And then, just like in
molecules, there are empty

242
00:17:30,000 --> 00:17:33,000
orbitals.
And they occur,

243
00:17:33,000 --> 00:17:37,000
also, in a continuum.
I would like you to get your

244
00:17:37,000 --> 00:17:43,000
mind around going from both ends
to the same place in that type

245
00:17:43,000 --> 00:17:46,000
of continuum.
And the idea that these band

246
00:17:46,000 --> 00:17:52,000
structure diagrams that people
use to describe the electronic

247
00:17:52,000 --> 00:17:57,000
structure properties of extended
materials are really molecular

248
00:17:57,000 --> 00:18:04,000
orbital diagrams.
And so let's take a typical

249
00:18:04,000 --> 00:18:13,000
metal, where n equals three,
principle quantum number three.

250
00:18:13,000 --> 00:18:21,000
The atom has a 1s orbital.
It has a 2s and a set of 2p

251
00:18:21,000 --> 00:18:27,000
orbitals.
It has a 3s orbital and a set

252
00:18:27,000 --> 00:18:31,000
of 3p.
And so there is an atom,

253
00:18:31,000 --> 00:18:34,000
like a sodium atom,
for example.

254
00:18:34,000 --> 00:18:38,000
And here is our energy axis.
What happens is that each of

255
00:18:38,000 --> 00:18:42,000
these orbitals,
that when you put all these

256
00:18:42,000 --> 00:18:46,000
atoms together into a piece of
solid sodium metal,

257
00:18:46,000 --> 00:18:49,000
we talked about that earlier in
the semester,

258
00:18:49,000 --> 00:18:53,000
these orbitals overlap,
spread out and form bands.

259
00:18:53,000 --> 00:18:58,000
And there is a band very low in
energy that is derived from the

260
00:18:58,000 --> 00:19:03,000
1s electrons in a metal like
sodium.

261
00:19:03,000 --> 00:19:07,000
And it is completely full.
And then there is a band from

262
00:19:07,000 --> 00:19:11,000
the 2s and there is a band,
accordingly,

263
00:19:11,000 --> 00:19:14,000
from the 2p.
And then there will be a band

264
00:19:14,000 --> 00:19:20,000
from the 3s, and a band from the
3p that I have run out of room

265
00:19:20,000 --> 00:19:23,000
to draw there.
And notice that the bands that

266
00:19:23,000 --> 00:19:28,000
originate from atomic orbitals
having the same principle

267
00:19:28,000 --> 00:19:34,000
quantum number here,
2s and 2p, are overlapping.

268
00:19:34,000 --> 00:19:40,000
In the case of a sodium atom,
this 2s band is completely full

269
00:19:40,000 --> 00:19:45,000
with electrons and the 2p band
is completely full.

270
00:19:45,000 --> 00:19:49,000
And this 3s band here,
in the case of sodium,

271
00:19:49,000 --> 00:19:52,000
is half full.
Furthermore,

272
00:19:52,000 --> 00:19:57,000
we are going to call these
filled bands that are at the

273
00:19:57,000 --> 00:20:02,000
highest energy.
This corresponds to our highest

274
00:20:02,000 --> 00:20:08,000
occupied molecular orbital.
That will be called the valance

275
00:20:08,000 --> 00:20:10,000
band.

276
00:20:16,000 --> 00:20:22,000
And then up here,
the lowest unoccupied band is

277
00:20:22,000 --> 00:20:26,000
called the conduction band.

278
00:20:34,000 --> 00:20:41,000
And so, in the case of sodium
metal, this 3s band is half

279
00:20:41,000 --> 00:20:42,000
full.

280
00:20:47,000 --> 00:20:55,000
And, if we go over to
magnesium, this same 3s band is

281
00:20:55,000 --> 00:21:00,000
now full.
And if we go to aluminum,

282
00:21:00,000 --> 00:21:09,000
that 3s band is full,
and the 3p band is partly full.

283
00:21:14,000 --> 00:21:17,000
And it is a consequence of the
fact that the electrons in the

284
00:21:17,000 --> 00:21:21,000
valance band are right here at
the same energy as the lowest

285
00:21:21,000 --> 00:21:25,000
part of the conduction band in a
metal that gives metals their

286
00:21:25,000 --> 00:21:27,000
luster.
It gives them their 100%

287
00:21:27,000 --> 00:21:30,000
optical reflectivity,
--

288
00:21:30,000 --> 00:21:34,000
-- these properties that we
very much associate with metals.

289
00:21:34,000 --> 00:21:37,000
And so from analyzing band
structure diagrams,

290
00:21:37,000 --> 00:21:40,000
even simplified ones,
like the ones you will find

291
00:21:40,000 --> 00:21:44,000
here and in your textbook today,
you can really say a lot about

292
00:21:44,000 --> 00:21:48,000
the properties of different
materials.

293
00:21:58,000 --> 00:22:07,000
When you have a meeting of the
valance band and the conduction

294
00:22:07,000 --> 00:22:16,000
band, then your material is a
conductor and is metallic.

295
00:22:16,000 --> 00:22:22,000
And then, there are other
possibilities,

296
00:22:22,000 --> 00:22:28,000
of course.
You may have a valance band

297
00:22:28,000 --> 00:22:40,000
that is separated by some energy
gap from the conduction band.

298
00:22:40,000 --> 00:22:45,000
And if that is that is the
case, then you have a

299
00:22:45,000 --> 00:22:48,000
semi-conductor,
such as silicon.

300
00:22:48,000 --> 00:22:53,000
And then, finally,
you can have a large gap

301
00:22:53,000 --> 00:23:00,000
between your valance band and
your conduction band.

302
00:23:00,000 --> 00:23:03,000
And, in all cases,
like with an MO diagram,

303
00:23:03,000 --> 00:23:06,000
we are putting these things on
an energy axis.

304
00:23:06,000 --> 00:23:10,000
We are filling up electrons
from the bottom in this material

305
00:23:10,000 --> 00:23:15,000
from the standpoint of energy,
and so you have a large gap,

306
00:23:15,000 --> 00:23:16,000
here.

307
00:23:22,000 --> 00:23:26,000
And you have a material that is
an insulator.

308
00:23:30,000 --> 00:23:33,000
And I think you will appreciate
why that is in a moment,

309
00:23:33,000 --> 00:23:37,000
but I want to bring Boltzmann's
law to bear on the issue of

310
00:23:37,000 --> 00:23:41,000
electronic structure in extended
networks, like we are talking

311
00:23:41,000 --> 00:23:43,000
about today.

312
00:23:50,000 --> 00:23:53,000
In materials like the ones I
have drawn over here,

313
00:23:53,000 --> 00:23:56,000
the ability to conduct
electricity is related to the

314
00:23:56,000 --> 00:24:01,000
probability of electrons being
in the conduction band.

315
00:24:07,000 --> 00:24:13,000
So we need to know something
about electrons in the

316
00:24:13,000 --> 00:24:19,000
conduction band.
And, using a Boltzmann

317
00:24:19,000 --> 00:24:25,000
distribution,
we can write that probability

318
00:24:25,000 --> 00:24:33,000
as being related to one over (e
to the (delta E over RT)) plus

319
00:24:33,000 --> 00:24:40,000
one.

320
00:24:40,000 --> 00:24:44,000
And, with an expression like
this, this delta E here

321
00:24:44,000 --> 00:24:49,000
corresponds to our gap.
And so it is possible,

322
00:24:49,000 --> 00:24:54,000
then, to go ahead and estimate
the number of electrons that

323
00:24:54,000 --> 00:25:00,000
would be present in a cubic
centimeter of your material in

324
00:25:00,000 --> 00:25:06,000
the condition band as a function
of this energy gap.

325
00:25:06,000 --> 00:25:10,000
And so we can consider that for
materials like carbon or silicon

326
00:25:10,000 --> 00:25:14,000
or elemental germanium.
In the case of carbon,

327
00:25:14,000 --> 00:25:18,000
I am talking about diamond.
And you should definitely go to

328
00:25:18,000 --> 00:25:23,000
that specified website and look
at the diamond structure and try

329
00:25:23,000 --> 00:25:28,000
to get an appreciation for how
the carbon atoms in diamond pack

330
00:25:28,000 --> 00:25:32,000
in three dimensions.
From a hybridization

331
00:25:32,000 --> 00:25:37,000
standpoint, all the carbons in
graphite are sp two.

332
00:25:37,000 --> 00:25:42,000
Whereas, in diamond all the
carbons are sp three

333
00:25:42,000 --> 00:25:46,000
and tetrahedral.
And completing this table,

334
00:25:46,000 --> 00:25:47,000
--

335
00:25:53,000 --> 00:26:00,000
-- we can write down delta E in
kilojoules per mole.

336
00:26:00,000 --> 00:26:06,000
The gap for diamond is
kilojoules per mole,

337
00:26:06,000 --> 00:26:10,000
for silicon,
117 kilojoules per mole,

338
00:26:10,000 --> 00:26:16,000
and for germanium,
66 kilojoules per mole.

339
00:26:16,000 --> 00:26:23,000
And here is the number of
electrons per centimeter cubed

340
00:26:23,000 --> 00:26:30,000
in the material in the
conduction band.

341
00:26:30,000 --> 00:26:37,000
And, based on this large energy
gap in the diamond structure,

342
00:26:37,000 --> 00:26:41,000
this value is on the order of
10^-27.

343
00:26:41,000 --> 00:26:45,000
Very small.
This is an insulator.

344
00:26:45,000 --> 00:26:49,000
Diamond is an insulator.

345
00:26:54,000 --> 00:26:57,000
And, on the other hand,
silicon, the number of

346
00:26:57,000 --> 00:27:01,000
electrons per cubic centimeter
that are in the conduction band

347
00:27:01,000 --> 00:27:06,000
are on the order of 10^9.
This much smaller gap in the

348
00:27:06,000 --> 00:27:10,000
case of silicon,
despite the fact that the

349
00:27:10,000 --> 00:27:15,000
silicon atoms also are
tetrahedrally disposed with

350
00:27:15,000 --> 00:27:20,000
respect to their bonding,
just as in the diamond case,

351
00:27:20,000 --> 00:27:23,000
we have a much smaller gap,
10^9.

352
00:27:23,000 --> 00:27:27,000
And that makes silicon,
as you know,

353
00:27:27,000 --> 00:27:32,000
a semiconductor.
And then germanium,

354
00:27:32,000 --> 00:27:37,000
10^13, so even more.
It is getting closer and closer

355
00:27:37,000 --> 00:27:43,000
to being metallic as the gap
shrinks as we compare these

356
00:27:43,000 --> 00:27:45,000
materials.

357
00:27:51,000 --> 00:27:56,000
And, having said that,
we need to talk about the

358
00:27:56,000 --> 00:28:01,000
different types of materials
that we can have.

359
00:28:01,000 --> 00:28:08,000
I want you to understand the
difference between intrinsic and

360
00:28:08,000 --> 00:28:12,000
extrinsic semiconductors.

361
00:28:25,000 --> 00:28:28,000
If a semiconductor material is
an intrinsic semiconductor,

362
00:28:28,000 --> 00:28:33,000
that means that it is a
semiconductor in its pure state.

363
00:28:50,000 --> 00:28:51,000
And why would we make that
reference?

364
00:28:51,000 --> 00:28:53,000
I mean normally,
we are always talking about

365
00:28:53,000 --> 00:28:55,000
pure things.
But, actually,

366
00:28:55,000 --> 00:28:57,000
you will see in a moment that
people do purposely make impure

367
00:28:57,000 --> 00:29:00,000
semiconductors for very good
reasons.

368
00:29:00,000 --> 00:29:04,000
And we will discuss
semiconductor when pure.

369
00:29:04,000 --> 00:29:11,000
And what that means is you have
a system like this with some

370
00:29:11,000 --> 00:29:16,000
kind of a small band gap,
as we have suggested.

371
00:29:16,000 --> 00:29:22,000
Here is our energy axis.
And what can happen is that,

372
00:29:22,000 --> 00:29:28,000
either thermally or upon
absorption of light energy,

373
00:29:28,000 --> 00:29:34,000
we can have promotion of an
electron from the valance band

374
00:29:34,000 --> 00:29:42,000
into the conduction band.
And so I will draw that new

375
00:29:42,000 --> 00:29:46,000
situation over here.
In other words,

376
00:29:46,000 --> 00:29:52,000
we may have thermal population
of our conduction band.

377
00:29:52,000 --> 00:29:55,000
And we generate,
accordingly,

378
00:29:55,000 --> 00:29:58,000
a hole.
In this intrinsic

379
00:29:58,000 --> 00:30:03,000
semiconductor,
for every electron that jumps

380
00:30:03,000 --> 00:30:09,000
up into the conduction band and
can then provide conductivity by

381
00:30:09,000 --> 00:30:12,000
electron transport --

382
00:30:28,000 --> 00:30:32,000
Down here, in the valance band,
the missing electron generates

383
00:30:32,000 --> 00:30:34,000
a hole.
And that hole can move around

384
00:30:34,000 --> 00:30:37,000
freely in the valance band.
How does it do that?

385
00:30:37,000 --> 00:30:41,000
Well, it is a little bit like
the mechanism that we studied

386
00:30:41,000 --> 00:30:45,000
earlier for translocation of
protons in acidic water.

387
00:30:45,000 --> 00:30:49,000
If an electron that is nearby
the hole jumps into the hole,

388
00:30:49,000 --> 00:30:52,000
it creates another hole.
The hole thereby moves.

389
00:30:52,000 --> 00:30:55,000
And so here,
we can have hole transport in

390
00:30:55,000 --> 00:31:00,000
our valance band.
You can have conductivity

391
00:31:00,000 --> 00:31:05,000
occurring freely both in the
valance band and in the

392
00:31:05,000 --> 00:31:11,000
conduction band for an intrinsic
semiconductor of this type,

393
00:31:11,000 --> 00:31:15,000
where the number of holes is
equal to the number of

394
00:31:15,000 --> 00:31:19,000
electrons.
And you might begin to suspect

395
00:31:19,000 --> 00:31:23,000
that for an extrinsic
semiconductor,

396
00:31:23,000 --> 00:31:30,000
the number of holes does not
equal the number of electrons.

397
00:31:52,000 --> 00:31:54,000
How does that work?
In these extrinsic

398
00:31:54,000 --> 00:31:59,000
semiconductors wherein you have
differing numbers of holes and

399
00:31:59,000 --> 00:32:03,000
electrons, you are purposely
adding a small percentage of

400
00:32:03,000 --> 00:32:10,000
impurities to your material.
Let me draw two pictures to

401
00:32:10,000 --> 00:32:17,000
represent this.
Here, I would like to draw just

402
00:32:17,000 --> 00:32:24,000
a simple tetrahedron.
We are looking at a very small

403
00:32:24,000 --> 00:32:32,000
part of the silicon structure.
Let me put the silicons in here

404
00:32:32,000 --> 00:32:35,000
in color.
Each silicon is coordinated to

405
00:32:35,000 --> 00:32:39,000
four other silicons in elemental
silicon.

406
00:32:39,000 --> 00:32:44,000
It is tetrahedral silicon all
through this three-dimensional

407
00:32:44,000 --> 00:32:48,000
material in which these bands
have been created,

408
00:32:48,000 --> 00:32:52,000
and in which we have valance
and conduction bands.

409
00:32:52,000 --> 00:32:57,000
But what if we have synthesized
our silicon with a little bit of

410
00:32:57,000 --> 00:33:01,000
boron impurity?
If we do that,

411
00:33:01,000 --> 00:33:05,000
then boron goes into a position
in the crystal lattice that is

412
00:33:05,000 --> 00:33:09,000
normally occupied by silicon,
so boron finds itself

413
00:33:09,000 --> 00:33:14,000
surrounded by four silicons,
each of which wishes to donate

414
00:33:14,000 --> 00:33:18,000
an electron to the boron to form
an electron pair bond.

415
00:33:18,000 --> 00:33:23,000
But boron only has three of the
needed four electrons to make

416
00:33:23,000 --> 00:33:29,000
those four two-electron bonds
and to generate the octet.

417
00:33:29,000 --> 00:33:33,000
And so what happens is it wants
to get an electron.

418
00:33:33,000 --> 00:33:37,000
And where can it get an
electron from?

419
00:33:37,000 --> 00:33:41,000
It can get an electron from the
valance band,

420
00:33:41,000 --> 00:33:46,000
from the HOMO of the system,
from what would be able to

421
00:33:46,000 --> 00:33:49,000
donate an electron in the
system.

422
00:33:49,000 --> 00:33:54,000
The way this then works is as
follows.

423
00:33:54,000 --> 00:33:58,000
You should think about the
structure of elemental silicon

424
00:33:58,000 --> 00:34:03,000
with a small number of boron
atoms dispersed throughout that

425
00:34:03,000 --> 00:34:08,000
structure as creating localized
negative charges that cannot

426
00:34:08,000 --> 00:34:12,000
move because they are localized
on these borons.

427
00:34:12,000 --> 00:34:16,000
And that generates for you a
band structure diagram like

428
00:34:16,000 --> 00:34:18,000
this.
You have bands,

429
00:34:18,000 --> 00:34:23,000
but then you have slipped in
there a little orbital from the

430
00:34:23,000 --> 00:34:28,000
boron, a little electronic state
here, an intermediate between

431
00:34:28,000 --> 00:34:33,000
the valance band and the
conduction band.

432
00:34:33,000 --> 00:34:36,000
So you have to choose your
impurity correctly,

433
00:34:36,000 --> 00:34:41,000
so that it has the right energy
with respect to the valance band

434
00:34:41,000 --> 00:34:46,000
and the conduction band in order
for the process that you want to

435
00:34:46,000 --> 00:34:48,000
occur.
Here is our boron-derived

436
00:34:48,000 --> 00:34:53,000
state, here, and it needs that
extra electron because it is

437
00:34:53,000 --> 00:34:56,000
electron deficient when you put
it in there.

438
00:34:56,000 --> 00:35:01,000
And so an electron jumps onto
the boron, we will represent

439
00:35:01,000 --> 00:35:04,000
that this way,
in the material like that.

440
00:35:04,000 --> 00:35:09,000
And this electron is fixed in
position --

441
00:35:15,000 --> 00:35:20,000
-- because it is associated
with a negative charge that has

442
00:35:20,000 --> 00:35:25,000
formed on the boron.
And a corresponding hole is

443
00:35:25,000 --> 00:35:30,000
formed down in the conduction
band, and this hole transport

444
00:35:30,000 --> 00:35:34,000
can give rise to conductivity.

445
00:35:42,000 --> 00:35:46,000
And so notice that in this type
of semiconductor,

446
00:35:46,000 --> 00:35:50,000
and this, by the way,
is a p-type semiconductor.

447
00:35:50,000 --> 00:35:54,000
P for positive.
You are putting in a hole on

448
00:35:54,000 --> 00:36:00,000
that boron, so this is a p-type
of semiconductor.

449
00:36:00,000 --> 00:36:03,000
This thing is stuck on the
boron, and it generates a hole

450
00:36:03,000 --> 00:36:07,000
that is free to move in the
valance band and give rise to

451
00:36:07,000 --> 00:36:10,000
conductivity.
And so in this extrinsic-type

452
00:36:10,000 --> 00:36:13,000
of semiconductor,
you are not necessarily getting

453
00:36:13,000 --> 00:36:17,000
any conductivity up here in the
normal conduction band but down

454
00:36:17,000 --> 00:36:21,000
in the valance band due to the
creation of the hole.

455
00:36:21,000 --> 00:36:25,000
And then the parallel situation
to that would be where you have

456
00:36:25,000 --> 00:36:29,000
something that you dope into the
structure that has one more

457
00:36:29,000 --> 00:36:34,000
electron than what is normally
in the structure.

458
00:36:34,000 --> 00:36:38,000
Normally, you are putting in
silicon atoms each of which has

459
00:36:38,000 --> 00:36:41,000
four valance electrons.
You put in a phosphorus atom,

460
00:36:41,000 --> 00:36:44,000
which isn't nearly the same
size as silicon,

461
00:36:44,000 --> 00:36:48,000
but has five valance electrons.
So you have this phosphorus in

462
00:36:48,000 --> 00:36:52,000
here, and it is tetrahedrally
coordinated to four silicons.

463
00:36:52,000 --> 00:36:56,000
And there is only a few
percentage of phosphorous atoms

464
00:36:56,000 --> 00:37:00,000
doped into this silicon
semiconductor.

465
00:37:00,000 --> 00:37:03,000
And the phosphorus,
what happens is it goes in

466
00:37:03,000 --> 00:37:05,000
there.
It has an extra electron.

467
00:37:05,000 --> 00:37:10,000
It wants to give it up so that
it can have just an octet and

468
00:37:10,000 --> 00:37:13,000
form these four bonds to the
four silicons.

469
00:37:13,000 --> 00:37:17,000
And when it gives up that
electron, the electron goes out

470
00:37:17,000 --> 00:37:22,000
and that forms a positive charge
that is localized and fixed on

471
00:37:22,000 --> 00:37:25,000
the phosphorous center in the
structure.

472
00:37:25,000 --> 00:37:30,000
And so, we can represent that
as follows.

473
00:37:30,000 --> 00:37:34,000
Where we have a phosphorus
state that we had chosen

474
00:37:34,000 --> 00:37:39,000
appropriately in energy to go
ahead and give up that electron,

475
00:37:39,000 --> 00:37:45,000
it gives up the electron to the
lowest unoccupied orbitals in

476
00:37:45,000 --> 00:37:50,000
the system, which is the bottom
part of your conduction band.

477
00:37:50,000 --> 00:37:54,000
This electron jumps up off the
phosphorus and into the

478
00:37:54,000 --> 00:38:00,000
conduction band,
and that leaves behind a hole.

479
00:38:00,000 --> 00:38:03,000
So you have a hole or a
positive charge,

480
00:38:03,000 --> 00:38:08,000
there, fixed in position.
And now you can have electron

481
00:38:08,000 --> 00:38:12,000
transport as your mechanism of
conductivity,

482
00:38:12,000 --> 00:38:15,000
up in the conduction band.
It is really,

483
00:38:15,000 --> 00:38:21,000
I think, quite fascinating to
think about the way in which the

484
00:38:21,000 --> 00:38:26,000
concept of the octet rule and
our understanding of bonding in

485
00:38:26,000 --> 00:38:31,000
tetrahedral centers can actually
lead us to understand the

486
00:38:31,000 --> 00:38:36,000
mechanism of conductivity in
solid materials that are either

487
00:38:36,000 --> 00:38:42,000
n-doped or p-doped.
Here, it is quite clear.

488
00:38:42,000 --> 00:38:47,000
And then, finally,
I just want to take you through

489
00:38:47,000 --> 00:38:54,000
the way in which you can put the
positive and negative doped

490
00:38:54,000 --> 00:39:00,000
materials together to create a
device like a light-emitting

491
00:39:00,000 --> 00:39:02,000
diode.

492
00:39:14,000 --> 00:39:16,000
LED materials,
these are obviously great

493
00:39:16,000 --> 00:39:21,000
things because you can generate
light with a lot more efficiency

494
00:39:21,000 --> 00:39:24,000
in terms of energy than you can
with incandescent bulbs.

495
00:39:24,000 --> 00:39:27,000
You can get them in all
different colors.

496
00:39:27,000 --> 00:39:32,000
These are finding application
in so many different ways.

497
00:39:32,000 --> 00:39:35,000
One of the challenges that
chemists have taken on is the

498
00:39:35,000 --> 00:39:40,000
discovery of light-emitting
diode materials that are made of

499
00:39:40,000 --> 00:39:43,000
organic molecules,
actually, conducting organic

500
00:39:43,000 --> 00:39:47,000
molecules that have properties
correct for giving you very

501
00:39:47,000 --> 00:39:52,000
narrow emissions in the part of
the spectrum that you want to

502
00:39:52,000 --> 00:39:55,000
have coming out of your
light-emitting diode material.

503
00:39:55,000 --> 00:39:59,000
So chemistry is really very
strongly involved in making

504
00:39:59,000 --> 00:40:05,000
next-generation LED materials.
But I just want to tell you a

505
00:40:05,000 --> 00:40:09,000
little bit about how these
things work.

506
00:40:09,000 --> 00:40:14,000
The idea is that you have these
two types of semiconductors,

507
00:40:14,000 --> 00:40:18,000
and you juxtapose them at an
interface.

508
00:40:18,000 --> 00:40:21,000
Let me make the interface with
blue.

509
00:40:21,000 --> 00:40:25,000
You have a solid chunk of
material, here.

510
00:40:25,000 --> 00:40:30,000
This will be our N-type
semiconductor.

511
00:40:30,000 --> 00:40:32,000
And over here,
they have a continuous,

512
00:40:32,000 --> 00:40:35,000
possibly two dimensional
interface here in this

513
00:40:35,000 --> 00:40:38,000
three-dimensional chunk of
material.

514
00:40:38,000 --> 00:40:42,000
And you have a P-type doped
part of the system over here.

515
00:40:42,000 --> 00:40:45,000
And, in fact,
normally this would be the same

516
00:40:45,000 --> 00:40:49,000
basic semiconductor material on
both sides of the interface.

517
00:40:49,000 --> 00:40:54,000
And it would be just the doping
that changes on the left versus

518
00:40:54,000 --> 00:40:56,000
the right.
This material might be

519
00:40:56,000 --> 00:41:02,000
something like gallium nitride.
Now, notice that this is a 3/5

520
00:41:02,000 --> 00:41:06,000
type of material.
And this is gallium in Group 13

521
00:41:06,000 --> 00:41:10,000
and nitrogen in Group 15 of the
Periodic Table.

522
00:41:10,000 --> 00:41:15,000
You add three and five together
and you get eight,

523
00:41:15,000 --> 00:41:20,000
just the same number of valance
electrons you would if you had

524
00:41:20,000 --> 00:41:24,000
silicon and silicon.
But these materials have the

525
00:41:24,000 --> 00:41:30,000
property that they are a direct
band gap material.

526
00:41:30,000 --> 00:41:34,000
And, as direct band gap
materials, when the process that

527
00:41:34,000 --> 00:41:39,000
we are going to talk about here
takes place at the interface,

528
00:41:39,000 --> 00:41:43,000
then out of this interface
comes the light.

529
00:41:43,000 --> 00:41:47,000
And if your semiconductor
material is an indirect band gap

530
00:41:47,000 --> 00:41:52,000
material like silicone is,
then that is not the case.

531
00:41:52,000 --> 00:41:56,000
And that has to do with the
solid state physics of

532
00:41:56,000 --> 00:41:59,000
electronically,
where is the conduction band

533
00:41:59,000 --> 00:42:04,000
located relative to that valance
band?

534
00:42:04,000 --> 00:42:08,000
Has it shifted horizontally in
solid physics-speak relative to

535
00:42:08,000 --> 00:42:10,000
the valance band?
That's what makes a

536
00:42:10,000 --> 00:42:13,000
semiconductor an indirect band
gap material.

537
00:42:13,000 --> 00:42:17,000
If the conduction band is
vertically situated above the

538
00:42:17,000 --> 00:42:19,000
valance band,
then you get a direct band

539
00:42:19,000 --> 00:42:22,000
material.
And so solid state chemists are

540
00:42:22,000 --> 00:42:26,000
interested in designing new
materials that have a direct

541
00:42:26,000 --> 00:42:30,000
band gap and that can release
light when this process takes

542
00:42:30,000 --> 00:42:34,000
place at the interface.
And on both sides,

543
00:42:34,000 --> 00:42:39,000
I just want to sketch the band
structure for the materials.

544
00:42:39,000 --> 00:42:43,000
And I mean the gap,
actually, to be the same on

545
00:42:43,000 --> 00:42:47,000
both sides here.
And the difference is what we

546
00:42:47,000 --> 00:42:51,000
have doped it with.
In the case of the negative

547
00:42:51,000 --> 00:42:56,000
material, we have an electron up
here in the conduction band that

548
00:42:56,000 --> 00:42:58,000
came in with,
for example,

549
00:42:58,000 --> 00:43:03,000
our phosphorus atom and left
behind a hole there that is

550
00:43:03,000 --> 00:43:08,000
fixed in space.
And down here we have our

551
00:43:08,000 --> 00:43:11,000
valance band all full.
Over here similar,

552
00:43:11,000 --> 00:43:16,000
except our material starts out
with a hole down there because

553
00:43:16,000 --> 00:43:21,000
the electron has jumped up onto
the doped atom and become fixed

554
00:43:21,000 --> 00:43:26,000
in space as a negative charge.
And that left behind a hole in

555
00:43:26,000 --> 00:43:30,000
the valance band down there like
that.

556
00:43:30,000 --> 00:43:35,000
What you have is your N-doped
material here on the left,

557
00:43:35,000 --> 00:43:38,000
your P-doped material on the
right.

558
00:43:38,000 --> 00:43:42,000
And, of course,
what do you do then?

559
00:43:42,000 --> 00:43:47,000
You attach leads so that you
can connect it to a potential

560
00:43:47,000 --> 00:43:51,000
difference.
And, when you do that,

561
00:43:51,000 --> 00:43:57,000
you want your anode to be over
here, so that the electrons can

562
00:43:57,000 --> 00:44:03,000
flow that way up to the N part
of the device.

563
00:44:03,000 --> 00:44:07,000
And then electrons can flow
this way, which means,

564
00:44:07,000 --> 00:44:11,000
of course, that holes go that
way.

565
00:44:11,000 --> 00:44:16,000
And here is your cathode.
And so you can think of putting

566
00:44:16,000 --> 00:44:21,000
on your potential difference in
a system like this.

567
00:44:21,000 --> 00:44:28,000
And that has the effect of
ripping electrons out over here.

568
00:44:28,000 --> 00:44:31,000
And, if you rip an electron
out, let's say you take that

569
00:44:31,000 --> 00:44:35,000
negative charge back off of that
boron atom, you pull an electron

570
00:44:35,000 --> 00:44:39,000
out to go ahead and reduce
something down here in solution,

571
00:44:39,000 --> 00:44:42,000
if you are using a battery for
this sort of process,

572
00:44:42,000 --> 00:44:46,000
well, then another electron can
jump up here to take its place.

573
00:44:46,000 --> 00:44:49,000
But you are building up a
potential, here.

574
00:44:49,000 --> 00:44:52,000
And so, what happens?
The highest lying electron in

575
00:44:52,000 --> 00:44:56,000
the system over here in the
N-type semiconductor is sitting

576
00:44:56,000 --> 00:45:00,000
there right at the junction
right next to where electrons

577
00:45:00,000 --> 00:45:03,000
are needed.
It jumps across.

578
00:45:03,000 --> 00:45:05,000
Electrons flow downhill,
here.

579
00:45:05,000 --> 00:45:09,000
And you can look at it like
this, electrons flow down there,

580
00:45:09,000 --> 00:45:13,000
across the interface.
And when you hook this LED up

581
00:45:13,000 --> 00:45:17,000
to your potential difference
supply, the electron flow is

582
00:45:17,000 --> 00:45:21,000
unidirectional.
As the electrons jump across

583
00:45:21,000 --> 00:45:23,000
the interface,
they are going from the

584
00:45:23,000 --> 00:45:27,000
conduction band of the N-type
semiconductor,

585
00:45:27,000 --> 00:45:31,000
they are going down in here,
holes are being created,

586
00:45:31,000 --> 00:45:35,000
maybe electrons are being
pulled right off of that boron,

587
00:45:35,000 --> 00:45:39,000
and they are moving right
across here in a process that

588
00:45:39,000 --> 00:45:45,000
leads to the emission of light
right at the interface.

589
00:45:45,000 --> 00:45:49,000
It is like a waterfall of
electrons taking place all along

590
00:45:49,000 --> 00:45:52,000
this 2D interface.
Electrons are just moving

591
00:45:52,000 --> 00:45:56,000
across this interface and
dropping down in energy.

592
00:45:56,000 --> 00:46:01,000
As they drop down in energy,
a photon that is the energy of

593
00:46:01,000 --> 00:46:05,000
this energy difference between
the gap of this material,

594
00:46:05,000 --> 00:46:10,000
those photons are released for
each electron that transits this

595
00:46:10,000 --> 00:46:14,000
barrier.
And that is the principle of an

596
00:46:14,000 --> 00:46:16,000
LED.
And I hope you are enjoying

597
00:46:16,000 --> 00:46:21,000
seeing this connection between
molecular orbital theory for

598
00:46:21,000 --> 00:46:25,000
molecules being taken all the
way to solid state physics.

599
00:46:25,000 --> 00:46:28,000
Have a nice weekend and we will
see you on Monday.