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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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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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diagrams for a lot of things.
Last lecture,

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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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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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This might be a hydrogen atom,
for example.

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Here is an H two
molecular orbital diagram.

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And then we can consider a
system that might have three

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orbitals populated,
like this.

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And then, if we have a system
with four orbitals,

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it might be something like
that, four electrons and so on.

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We see that one of the features
is that the energy levels are

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starting to come closer together
as we get more and more of them.

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There are five.
Here is six.

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00:15:54 --> 00:16:00


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And then, onto seven.
And you start running out of

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

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

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

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So what do we do?
We draw them,

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

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What the idea is,
is that we have here this band

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

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

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

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them, an infinite number of
orbitals that are all

241
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interacting in some extended
solid material,

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we are going to be talking
today a little bit about silicon

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

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

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three-dimensions.
And so these molecular orbitals

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

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

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

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

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

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

252
00:17:37 --> 00:17:43
mind around going from both ends
to the same place in that type

253
00:17:43 --> 00:17:46
of continuum.
And the idea that these band

254
00:17:46 --> 00:17:52
structure diagrams that people
use to describe the electronic

255
00:17:52 --> 00:17:57
structure properties of extended
materials are really molecular

256
00:17:57 --> 00:18:04
orbital diagrams.
And so let's take a typical

257
00:18:04 --> 00:18:13
metal, where n equals three,
principle quantum number three.

258
00:18:13 --> 00:18:21
The atom has a 1s orbital.
It has a 2s and a set of 2p

259
00:18:21 --> 00:18:27
orbitals.
It has a 3s orbital and a set

260
00:18:27 --> 00:18:31
of 3p.
And so there is an atom,

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

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

263
00:18:38 --> 00:18:42
these orbitals,
that when you put all these

264
00:18:42 --> 00:18:46
atoms together into a piece of
solid sodium metal,

265
00:18:46 --> 00:18:49
we talked about that earlier in
the semester,

266
00:18:49 --> 00:18:53
these orbitals overlap,
spread out and form bands.

267
00:18:53 --> 00:18:58
And there is a band very low in
energy that is derived from the

268
00:18:58 --> 00:19:03
1s electrons in a metal like
sodium.

269
00:19:03 --> 00:19:07
And it is completely full.
And then there is a band from

270
00:19:07 --> 00:19:11
the 2s and there is a band,
accordingly,

271
00:19:11 --> 00:19:14
from the 2p.
And then there will be a band

272
00:19:14 --> 00:19:20
from the 3s, and a band from the
3p that I have run out of room

273
00:19:20 --> 00:19:23
to draw there.
And notice that the bands that

274
00:19:23 --> 00:19:28
originate from atomic orbitals
having the same principle

275
00:19:28 --> 00:19:34
quantum number here,
2s and 2p, are overlapping.

276
00:19:34 --> 00:19:40
In the case of a sodium atom,
this 2s band is completely full

277
00:19:40 --> 00:19:45
with electrons and the 2p band
is completely full.

278
00:19:45 --> 00:19:49
And this 3s band here,
in the case of sodium,

279
00:19:49 --> 00:19:52
is half full.
Furthermore,

280
00:19:52 --> 00:19:57
we are going to call these
filled bands that are at the

281
00:19:57 --> 00:20:02
highest energy.
This corresponds to our highest

282
00:20:02 --> 00:20:08
occupied molecular orbital.
That will be called the valance

283
00:20:08 --> 00:20:10
band.

284
00:20:10 --> 00:20:16


285
00:20:16 --> 00:20:22
And then up here,
the lowest unoccupied band is

286
00:20:22 --> 00:20:26
called the conduction band.

287
00:20:26 --> 00:20:34


288
00:20:34 --> 00:20:41
And so, in the case of sodium
metal, this 3s band is half

289
00:20:41 --> 00:20:42
full.

290
00:20:42 --> 00:20:47


291
00:20:47 --> 00:20:55
And, if we go over to
magnesium, this same 3s band is

292
00:20:55 --> 00:21:00
now full.
And if we go to aluminum,

293
00:21:00 --> 00:21:09
that 3s band is full,
and the 3p band is partly full.

294
00:21:09 --> 00:21:14


295
00:21:14 --> 00:21:17
And it is a consequence of the
fact that the electrons in the

296
00:21:17 --> 00:21:21
valance band are right here at
the same energy as the lowest

297
00:21:21 --> 00:21:25
part of the conduction band in a
metal that gives metals their

298
00:21:25 --> 00:21:27
luster.
It gives them their 100%

299
00:21:27 --> 00:21:30
optical reflectivity,
--

300
00:21:30 --> 00:21:34
-- these properties that we
very much associate with metals.

301
00:21:34 --> 00:21:37
And so from analyzing band
structure diagrams,

302
00:21:37 --> 00:21:40
even simplified ones,
like the ones you will find

303
00:21:40 --> 00:21:44
here and in your textbook today,
you can really say a lot about

304
00:21:44 --> 00:21:48
the properties of different
materials.

305
00:21:48 --> 00:21:58


306
00:21:58 --> 00:22:07
When you have a meeting of the
valance band and the conduction

307
00:22:07 --> 00:22:16
band, then your material is a
conductor and is metallic.

308
00:22:16 --> 00:22:22
And then, there are other
possibilities,

309
00:22:22 --> 00:22:28
of course.
You may have a valance band

310
00:22:28 --> 00:22:40
that is separated by some energy
gap from the conduction band.

311
00:22:40 --> 00:22:45
And if that is that is the
case, then you have a

312
00:22:45 --> 00:22:48
semi-conductor,
such as silicon.

313
00:22:48 --> 00:22:53
And then, finally,
you can have a large gap

314
00:22:53 --> 00:23:00
between your valance band and
your conduction band.

315
00:23:00 --> 00:23:03
And, in all cases,
like with an MO diagram,

316
00:23:03 --> 00:23:06
we are putting these things on
an energy axis.

317
00:23:06 --> 00:23:10
We are filling up electrons
from the bottom in this material

318
00:23:10 --> 00:23:15
from the standpoint of energy,
and so you have a large gap,

319
00:23:15 --> 00:23:16
here.

320
00:23:16 --> 00:23:22


321
00:23:22 --> 00:23:26
And you have a material that is
an insulator.

322
00:23:26 --> 00:23:30


323
00:23:30 --> 00:23:33
And I think you will appreciate
why that is in a moment,

324
00:23:33 --> 00:23:37
but I want to bring Boltzmann's
law to bear on the issue of

325
00:23:37 --> 00:23:41
electronic structure in extended
networks, like we are talking

326
00:23:41 --> 00:23:43
about today.

327
00:23:43 --> 00:23:50


328
00:23:50 --> 00:23:53
In materials like the ones I
have drawn over here,

329
00:23:53 --> 00:23:56
the ability to conduct
electricity is related to the

330
00:23:56 --> 00:24:01
probability of electrons being
in the conduction band.

331
00:24:01 --> 00:24:07


332
00:24:07 --> 00:24:13
So we need to know something
about electrons in the

333
00:24:13 --> 00:24:19
conduction band.
And, using a Boltzmann

334
00:24:19 --> 00:24:25
distribution,
we can write that probability

335
00:24:25 --> 00:24:33
as being related to one over (e
to the (delta E over RT)) plus

336
00:24:33 --> 00:24:40
one.

337
00:24:40 --> 00:24:44
And, with an expression like
this, this delta E here

338
00:24:44 --> 00:24:49
corresponds to our gap.
And so it is possible,

339
00:24:49 --> 00:24:54
then, to go ahead and estimate
the number of electrons that

340
00:24:54 --> 00:25:00
would be present in a cubic
centimeter of your material in

341
00:25:00 --> 00:25:06
the condition band as a function
of this energy gap.

342
00:25:06 --> 00:25:10
And so we can consider that for
materials like carbon or silicon

343
00:25:10 --> 00:25:14
or elemental germanium.
In the case of carbon,

344
00:25:14 --> 00:25:18
I am talking about diamond.
And you should definitely go to

345
00:25:18 --> 00:25:23
that specified website and look
at the diamond structure and try

346
00:25:23 --> 00:25:28
to get an appreciation for how
the carbon atoms in diamond pack

347
00:25:28 --> 00:25:32
in three dimensions.
From a hybridization

348
00:25:32 --> 00:25:37
standpoint, all the carbons in
graphite are sp two.

349
00:25:37 --> 00:25:42
Whereas, in diamond all the
carbons are sp three

350
00:25:42 --> 00:25:46
and tetrahedral.
And completing this table,

351
00:25:46 --> 00:25:47
--

352
00:25:47 --> 00:25:53


353
00:25:53 --> 00:26:00
-- we can write down delta E in
kilojoules per mole.

354
00:26:00 --> 00:26:06
The gap for diamond is
kilojoules per mole,

355
00:26:06 --> 00:26:10
for silicon,
117 kilojoules per mole,

356
00:26:10 --> 00:26:16
and for germanium,
66 kilojoules per mole.

357
00:26:16 --> 00:26:23
And here is the number of
electrons per centimeter cubed

358
00:26:23 --> 00:26:30
in the material in the
conduction band.

359
00:26:30 --> 00:26:37
And, based on this large energy
gap in the diamond structure,

360
00:26:37 --> 00:26:41
this value is on the order of
10^-27.

361
00:26:41 --> 00:26:45
Very small.
This is an insulator.

362
00:26:45 --> 00:26:49
Diamond is an insulator.

363
00:26:49 --> 00:26:54


364
00:26:54 --> 00:26:57
And, on the other hand,
silicon, the number of

365
00:26:57 --> 00:27:01
electrons per cubic centimeter
that are in the conduction band

366
00:27:01 --> 00:27:06
are on the order of 10^9.
This much smaller gap in the

367
00:27:06 --> 00:27:10
case of silicon,
despite the fact that the

368
00:27:10 --> 00:27:15
silicon atoms also are
tetrahedrally disposed with

369
00:27:15 --> 00:27:20
respect to their bonding,
just as in the diamond case,

370
00:27:20 --> 00:27:23
we have a much smaller gap,
10^9.

371
00:27:23 --> 00:27:27
And that makes silicon,
as you know,

372
00:27:27 --> 00:27:32
a semiconductor.
And then germanium,

373
00:27:32 --> 00:27:37
10^13, so even more.
It is getting closer and closer

374
00:27:37 --> 00:27:43
to being metallic as the gap
shrinks as we compare these

375
00:27:43 --> 00:27:45
materials.

376
00:27:45 --> 00:27:51


377
00:27:51 --> 00:27:56
And, having said that,
we need to talk about the

378
00:27:56 --> 00:28:01
different types of materials
that we can have.

379
00:28:01 --> 00:28:08
I want you to understand the
difference between intrinsic and

380
00:28:08 --> 00:28:12
extrinsic semiconductors.

381
00:28:12 --> 00:28:25


382
00:28:25 --> 00:28:28
If a semiconductor material is
an intrinsic semiconductor,

383
00:28:28 --> 00:28:33
that means that it is a
semiconductor in its pure state.

384
00:28:33 --> 00:28:50


385
00:28:50 --> 00:28:51
And why would we make that
reference?

386
00:28:51 --> 00:28:53
I mean normally,
we are always talking about

387
00:28:53 --> 00:28:55
pure things.
But, actually,

388
00:28:55 --> 00:28:57
you will see in a moment that
people do purposely make impure

389
00:28:57 --> 00:29:00
semiconductors for very good
reasons.

390
00:29:00 --> 00:29:04
And we will discuss
semiconductor when pure.

391
00:29:04 --> 00:29:11
And what that means is you have
a system like this with some

392
00:29:11 --> 00:29:16
kind of a small band gap,
as we have suggested.

393
00:29:16 --> 00:29:22
Here is our energy axis.
And what can happen is that,

394
00:29:22 --> 00:29:28
either thermally or upon
absorption of light energy,

395
00:29:28 --> 00:29:34
we can have promotion of an
electron from the valance band

396
00:29:34 --> 00:29:42
into the conduction band.
And so I will draw that new

397
00:29:42 --> 00:29:46
situation over here.
In other words,

398
00:29:46 --> 00:29:52
we may have thermal population
of our conduction band.

399
00:29:52 --> 00:29:55
And we generate,
accordingly,

400
00:29:55 --> 00:29:58
a hole.
In this intrinsic

401
00:29:58 --> 00:30:03
semiconductor,
for every electron that jumps

402
00:30:03 --> 00:30:09
up into the conduction band and
can then provide conductivity by

403
00:30:09 --> 00:30:12
electron transport --

404
00:30:12 --> 00:30:28


405
00:30:28 --> 00:30:32
Down here, in the valance band,
the missing electron generates

406
00:30:32 --> 00:30:34
a hole.
And that hole can move around

407
00:30:34 --> 00:30:37
freely in the valance band.
How does it do that?

408
00:30:37 --> 00:30:41
Well, it is a little bit like
the mechanism that we studied

409
00:30:41 --> 00:30:45
earlier for translocation of
protons in acidic water.

410
00:30:45 --> 00:30:49
If an electron that is nearby
the hole jumps into the hole,

411
00:30:49 --> 00:30:52
it creates another hole.
The hole thereby moves.

412
00:30:52 --> 00:30:55
And so here,
we can have hole transport in

413
00:30:55 --> 00:31:00
our valance band.
You can have conductivity

414
00:31:00 --> 00:31:05
occurring freely both in the
valance band and in the

415
00:31:05 --> 00:31:11
conduction band for an intrinsic
semiconductor of this type,

416
00:31:11 --> 00:31:15
where the number of holes is
equal to the number of

417
00:31:15 --> 00:31:19
electrons.
And you might begin to suspect

418
00:31:19 --> 00:31:23
that for an extrinsic
semiconductor,

419
00:31:23 --> 00:31:30
the number of holes does not
equal the number of electrons.

420
00:31:30 --> 00:31:52


421
00:31:52 --> 00:31:54
How does that work?
In these extrinsic

422
00:31:54 --> 00:31:59
semiconductors wherein you have
differing numbers of holes and

423
00:31:59 --> 00:32:03
electrons, you are purposely
adding a small percentage of

424
00:32:03 --> 00:32:10
impurities to your material.
Let me draw two pictures to

425
00:32:10 --> 00:32:17
represent this.
Here, I would like to draw just

426
00:32:17 --> 00:32:24
a simple tetrahedron.
We are looking at a very small

427
00:32:24 --> 00:32:32
part of the silicon structure.
Let me put the silicons in here

428
00:32:32 --> 00:32:35
in color.
Each silicon is coordinated to

429
00:32:35 --> 00:32:39
four other silicons in elemental
silicon.

430
00:32:39 --> 00:32:44
It is tetrahedral silicon all
through this three-dimensional

431
00:32:44 --> 00:32:48
material in which these bands
have been created,

432
00:32:48 --> 00:32:52
and in which we have valance
and conduction bands.

433
00:32:52 --> 00:32:57
But what if we have synthesized
our silicon with a little bit of

434
00:32:57 --> 00:33:01
boron impurity?
If we do that,

435
00:33:01 --> 00:33:05
then boron goes into a position
in the crystal lattice that is

436
00:33:05 --> 00:33:09
normally occupied by silicon,
so boron finds itself

437
00:33:09 --> 00:33:14
surrounded by four silicons,
each of which wishes to donate

438
00:33:14 --> 00:33:18
an electron to the boron to form
an electron pair bond.

439
00:33:18 --> 00:33:23
But boron only has three of the
needed four electrons to make

440
00:33:23 --> 00:33:29
those four two-electron bonds
and to generate the octet.

441
00:33:29 --> 00:33:33
And so what happens is it wants
to get an electron.

442
00:33:33 --> 00:33:37
And where can it get an
electron from?

443
00:33:37 --> 00:33:41
It can get an electron from the
valance band,

444
00:33:41 --> 00:33:46
from the HOMO of the system,
from what would be able to

445
00:33:46 --> 00:33:49
donate an electron in the
system.

446
00:33:49 --> 00:33:54
The way this then works is as
follows.

447
00:33:54 --> 00:33:58
You should think about the
structure of elemental silicon

448
00:33:58 --> 00:34:03
with a small number of boron
atoms dispersed throughout that

449
00:34:03 --> 00:34:08
structure as creating localized
negative charges that cannot

450
00:34:08 --> 00:34:12
move because they are localized
on these borons.

451
00:34:12 --> 00:34:16
And that generates for you a
band structure diagram like

452
00:34:16 --> 00:34:18
this.
You have bands,

453
00:34:18 --> 00:34:23
but then you have slipped in
there a little orbital from the

454
00:34:23 --> 00:34:28
boron, a little electronic state
here, an intermediate between

455
00:34:28 --> 00:34:33
the valance band and the
conduction band.

456
00:34:33 --> 00:34:36
So you have to choose your
impurity correctly,

457
00:34:36 --> 00:34:41
so that it has the right energy
with respect to the valance band

458
00:34:41 --> 00:34:46
and the conduction band in order
for the process that you want to

459
00:34:46 --> 00:34:48
occur.
Here is our boron-derived

460
00:34:48 --> 00:34:53
state, here, and it needs that
extra electron because it is

461
00:34:53 --> 00:34:56
electron deficient when you put
it in there.

462
00:34:56 --> 00:35:01
And so an electron jumps onto
the boron, we will represent

463
00:35:01 --> 00:35:04
that this way,
in the material like that.

464
00:35:04 --> 00:35:09
And this electron is fixed in
position --

465
00:35:09 --> 00:35:15


466
00:35:15 --> 00:35:20
-- because it is associated
with a negative charge that has

467
00:35:20 --> 00:35:25
formed on the boron.
And a corresponding hole is

468
00:35:25 --> 00:35:30
formed down in the conduction
band, and this hole transport

469
00:35:30 --> 00:35:34
can give rise to conductivity.

470
00:35:34 --> 00:35:42


471
00:35:42 --> 00:35:46
And so notice that in this type
of semiconductor,

472
00:35:46 --> 00:35:50
and this, by the way,
is a p-type semiconductor.

473
00:35:50 --> 00:35:54
P for positive.
You are putting in a hole on

474
00:35:54 --> 00:36:00
that boron, so this is a p-type
of semiconductor.

475
00:36:00 --> 00:36:03
This thing is stuck on the
boron, and it generates a hole

476
00:36:03 --> 00:36:07
that is free to move in the
valance band and give rise to

477
00:36:07 --> 00:36:10
conductivity.
And so in this extrinsic-type

478
00:36:10 --> 00:36:13
of semiconductor,
you are not necessarily getting

479
00:36:13 --> 00:36:17
any conductivity up here in the
normal conduction band but down

480
00:36:17 --> 00:36:21
in the valance band due to the
creation of the hole.

481
00:36:21 --> 00:36:25
And then the parallel situation
to that would be where you have

482
00:36:25 --> 00:36:29
something that you dope into the
structure that has one more

483
00:36:29 --> 00:36:34
electron than what is normally
in the structure.

484
00:36:34 --> 00:36:38
Normally, you are putting in
silicon atoms each of which has

485
00:36:38 --> 00:36:41
four valance electrons.
You put in a phosphorus atom,

486
00:36:41 --> 00:36:44
which isn't nearly the same
size as silicon,

487
00:36:44 --> 00:36:48
but has five valance electrons.
So you have this phosphorus in

488
00:36:48 --> 00:36:52
here, and it is tetrahedrally
coordinated to four silicons.

489
00:36:52 --> 00:36:56
And there is only a few
percentage of phosphorous atoms

490
00:36:56 --> 00:37:00
doped into this silicon
semiconductor.

491
00:37:00 --> 00:37:03
And the phosphorus,
what happens is it goes in

492
00:37:03 --> 00:37:05
there.
It has an extra electron.

493
00:37:05 --> 00:37:10
It wants to give it up so that
it can have just an octet and

494
00:37:10 --> 00:37:13
form these four bonds to the
four silicons.

495
00:37:13 --> 00:37:17
And when it gives up that
electron, the electron goes out

496
00:37:17 --> 00:37:22
and that forms a positive charge
that is localized and fixed on

497
00:37:22 --> 00:37:25
the phosphorous center in the
structure.

498
00:37:25 --> 00:37:30
And so, we can represent that
as follows.

499
00:37:30 --> 00:37:34
Where we have a phosphorus
state that we had chosen

500
00:37:34 --> 00:37:39
appropriately in energy to go
ahead and give up that electron,

501
00:37:39 --> 00:37:45
it gives up the electron to the
lowest unoccupied orbitals in

502
00:37:45 --> 00:37:50
the system, which is the bottom
part of your conduction band.

503
00:37:50 --> 00:37:54
This electron jumps up off the
phosphorus and into the

504
00:37:54 --> 00:38:00
conduction band,
and that leaves behind a hole.

505
00:38:00 --> 00:38:03
So you have a hole or a
positive charge,

506
00:38:03 --> 00:38:08
there, fixed in position.
And now you can have electron

507
00:38:08 --> 00:38:12
transport as your mechanism of
conductivity,

508
00:38:12 --> 00:38:15
up in the conduction band.
It is really,

509
00:38:15 --> 00:38:21
I think, quite fascinating to
think about the way in which the

510
00:38:21 --> 00:38:26
concept of the octet rule and
our understanding of bonding in

511
00:38:26 --> 00:38:31
tetrahedral centers can actually
lead us to understand the

512
00:38:31 --> 00:38:36
mechanism of conductivity in
solid materials that are either

513
00:38:36 --> 00:38:42
n-doped or p-doped.
Here, it is quite clear.

514
00:38:42 --> 00:38:47
And then, finally,
I just want to take you through

515
00:38:47 --> 00:38:54
the way in which you can put the
positive and negative doped

516
00:38:54 --> 00:39:00
materials together to create a
device like a light-emitting

517
00:39:00 --> 00:39:02
diode.

518
00:39:02 --> 00:39:14


519
00:39:14 --> 00:39:16
LED materials,
these are obviously great

520
00:39:16 --> 00:39:21
things because you can generate
light with a lot more efficiency

521
00:39:21 --> 00:39:24
in terms of energy than you can
with incandescent bulbs.

522
00:39:24 --> 00:39:27
You can get them in all
different colors.

523
00:39:27 --> 00:39:32
These are finding application
in so many different ways.

524
00:39:32 --> 00:39:35
One of the challenges that
chemists have taken on is the

525
00:39:35 --> 00:39:40
discovery of light-emitting
diode materials that are made of

526
00:39:40 --> 00:39:43
organic molecules,
actually, conducting organic

527
00:39:43 --> 00:39:47
molecules that have properties
correct for giving you very

528
00:39:47 --> 00:39:52
narrow emissions in the part of
the spectrum that you want to

529
00:39:52 --> 00:39:55
have coming out of your
light-emitting diode material.

530
00:39:55 --> 00:39:59
So chemistry is really very
strongly involved in making

531
00:39:59 --> 00:40:05
next-generation LED materials.
But I just want to tell you a

532
00:40:05 --> 00:40:09
little bit about how these
things work.

533
00:40:09 --> 00:40:14
The idea is that you have these
two types of semiconductors,

534
00:40:14 --> 00:40:18
and you juxtapose them at an
interface.

535
00:40:18 --> 00:40:21
Let me make the interface with
blue.

536
00:40:21 --> 00:40:25
You have a solid chunk of
material, here.

537
00:40:25 --> 00:40:30
This will be our N-type
semiconductor.

538
00:40:30 --> 00:40:32
And over here,
they have a continuous,

539
00:40:32 --> 00:40:35
possibly two dimensional
interface here in this

540
00:40:35 --> 00:40:38
three-dimensional chunk of
material.

541
00:40:38 --> 00:40:42
And you have a P-type doped
part of the system over here.

542
00:40:42 --> 00:40:45
And, in fact,
normally this would be the same

543
00:40:45 --> 00:40:49
basic semiconductor material on
both sides of the interface.

544
00:40:49 --> 00:40:54
And it would be just the doping
that changes on the left versus

545
00:40:54 --> 00:40:56
the right.
This material might be

546
00:40:56 --> 00:41:02
something like gallium nitride.
Now, notice that this is a 3/5

547
00:41:02 --> 00:41:06
type of material.
And this is gallium in Group 13

548
00:41:06 --> 00:41:10
and nitrogen in Group 15 of the
Periodic Table.

549
00:41:10 --> 00:41:15
You add three and five together
and you get eight,

550
00:41:15 --> 00:41:20
just the same number of valance
electrons you would if you had

551
00:41:20 --> 00:41:24
silicon and silicon.
But these materials have the

552
00:41:24 --> 00:41:30
property that they are a direct
band gap material.

553
00:41:30 --> 00:41:34
And, as direct band gap
materials, when the process that

554
00:41:34 --> 00:41:39
we are going to talk about here
takes place at the interface,

555
00:41:39 --> 00:41:43
then out of this interface
comes the light.

556
00:41:43 --> 00:41:47
And if your semiconductor
material is an indirect band gap

557
00:41:47 --> 00:41:52
material like silicone is,
then that is not the case.

558
00:41:52 --> 00:41:56
And that has to do with the
solid state physics of

559
00:41:56 --> 00:41:59
electronically,
where is the conduction band

560
00:41:59 --> 00:42:04
located relative to that valance
band?

561
00:42:04 --> 00:42:08
Has it shifted horizontally in
solid physics-speak relative to

562
00:42:08 --> 00:42:10
the valance band?
That's what makes a

563
00:42:10 --> 00:42:13
semiconductor an indirect band
gap material.

564
00:42:13 --> 00:42:17
If the conduction band is
vertically situated above the

565
00:42:17 --> 00:42:19
valance band,
then you get a direct band

566
00:42:19 --> 00:42:22
material.
And so solid state chemists are

567
00:42:22 --> 00:42:26
interested in designing new
materials that have a direct

568
00:42:26 --> 00:42:30
band gap and that can release
light when this process takes

569
00:42:30 --> 00:42:34
place at the interface.
And on both sides,

570
00:42:34 --> 00:42:39
I just want to sketch the band
structure for the materials.

571
00:42:39 --> 00:42:43
And I mean the gap,
actually, to be the same on

572
00:42:43 --> 00:42:47
both sides here.
And the difference is what we

573
00:42:47 --> 00:42:51
have doped it with.
In the case of the negative

574
00:42:51 --> 00:42:56
material, we have an electron up
here in the conduction band that

575
00:42:56 --> 00:42:58
came in with,
for example,

576
00:42:58 --> 00:43:03
our phosphorus atom and left
behind a hole there that is

577
00:43:03 --> 00:43:08
fixed in space.
And down here we have our

578
00:43:08 --> 00:43:11
valance band all full.
Over here similar,

579
00:43:11 --> 00:43:16
except our material starts out
with a hole down there because

580
00:43:16 --> 00:43:21
the electron has jumped up onto
the doped atom and become fixed

581
00:43:21 --> 00:43:26
in space as a negative charge.
And that left behind a hole in

582
00:43:26 --> 00:43:30
the valance band down there like
that.

583
00:43:30 --> 00:43:35
What you have is your N-doped
material here on the left,

584
00:43:35 --> 00:43:38
your P-doped material on the
right.

585
00:43:38 --> 00:43:42
And, of course,
what do you do then?

586
00:43:42 --> 00:43:47
You attach leads so that you
can connect it to a potential

587
00:43:47 --> 00:43:51
difference.
And, when you do that,

588
00:43:51 --> 00:43:57
you want your anode to be over
here, so that the electrons can

589
00:43:57 --> 00:44:03
flow that way up to the N part
of the device.

590
00:44:03 --> 00:44:07
And then electrons can flow
this way, which means,

591
00:44:07 --> 00:44:11
of course, that holes go that
way.

592
00:44:11 --> 00:44:16
And here is your cathode.
And so you can think of putting

593
00:44:16 --> 00:44:21
on your potential difference in
a system like this.

594
00:44:21 --> 00:44:28
And that has the effect of
ripping electrons out over here.

595
00:44:28 --> 00:44:31
And, if you rip an electron
out, let's say you take that

596
00:44:31 --> 00:44:35
negative charge back off of that
boron atom, you pull an electron

597
00:44:35 --> 00:44:39
out to go ahead and reduce
something down here in solution,

598
00:44:39 --> 00:44:42
if you are using a battery for
this sort of process,

599
00:44:42 --> 00:44:46
well, then another electron can
jump up here to take its place.

600
00:44:46 --> 00:44:49
But you are building up a
potential, here.

601
00:44:49 --> 00:44:52
And so, what happens?
The highest lying electron in

602
00:44:52 --> 00:44:56
the system over here in the
N-type semiconductor is sitting

603
00:44:56 --> 00:45:00
there right at the junction
right next to where electrons

604
00:45:00 --> 00:45:03
are needed.
It jumps across.

605
00:45:03 --> 00:45:05
Electrons flow downhill,
here.

606
00:45:05 --> 00:45:09
And you can look at it like
this, electrons flow down there,

607
00:45:09 --> 00:45:13
across the interface.
And when you hook this LED up

608
00:45:13 --> 00:45:17
to your potential difference
supply, the electron flow is

609
00:45:17 --> 00:45:21
unidirectional.
As the electrons jump across

610
00:45:21 --> 00:45:23
the interface,
they are going from the

611
00:45:23 --> 00:45:27
conduction band of the N-type
semiconductor,

612
00:45:27 --> 00:45:31
they are going down in here,
holes are being created,

613
00:45:31 --> 00:45:35
maybe electrons are being
pulled right off of that boron,

614
00:45:35 --> 00:45:39
and they are moving right
across here in a process that

615
00:45:39 --> 00:45:45
leads to the emission of light
right at the interface.

616
00:45:45 --> 00:45:49
It is like a waterfall of
electrons taking place all along

617
00:45:49 --> 00:45:52
this 2D interface.
Electrons are just moving

618
00:45:52 --> 00:45:56
across this interface and
dropping down in energy.

619
00:45:56 --> 00:46:01
As they drop down in energy,
a photon that is the energy of

620
00:46:01 --> 00:46:05
this energy difference between
the gap of this material,

621
00:46:05 --> 00:46:10
those photons are released for
each electron that transits this

622
00:46:10 --> 00:46:14
barrier.
And that is the principle of an

623
00:46:14 --> 00:46:16
LED.
And I hope you are enjoying

624
00:46:16 --> 00:46:21
seeing this connection between
molecular orbital theory for

625
00:46:21 --> 00:46:25
molecules being taken all the
way to solid state physics.

626
00:46:25 --> 46:28
Have a nice weekend and we will
see you on Monday.