check it out now yo nanohub online instruction welcome everyone to lecture
for of week two n atoms to materials today we're going to build on what we've
been working on to talk about the electronic structure of crystals okay so
instead of talking about just maybe two atoms or a few atoms we're going to talk
about a large collection of atoms and we're going to see that eventually we'll
be able to use the fact that it's a very large collection to our advantage so
we're going to start by considering a 1d one-dimensional chain of atoms that you
see at the bottom they're arranged in a pattern they're separated by a bond
distance a that's called the lattice parameter in this 1d model and as before
we're going to think about the wave function of the crystal in terms of a
local combination of atomic orbitals okay they were going to try to create
orbitals for the entire crystal in terms of atomic orbitals that we know from the
previous week so it microscopic crystal would have 10 to the 23 22 to 24 atoms
okay kind of Avogadro's number type of atoms so if you look at the linear
dimension of a microscopic object you might have been to the eight atoms or so
of course I'm not going to be able to draw 10 to 10 to the 8 atoms so my
linear chain is significantly shorter than that project imagine the dc's have
lots of atoms okay all right so that's what we have let's consider one s type
orbitals as the atomic orbitals and let's try to build crystalline orbitals
that are let's start with the lowest possible
energy okay let's create out of 1s orbitals the lowest possible energy for
the crystal so that's the 1s orbital for the first item okay that's how it looks
and I know that for the crystal for the ground state structure the lowest energy
wave function it will have no notes okay so building on what we know for the
diatomic molecule we know that we're going to have to combine the individual
atomic orbitals in a combination that's all symmetric okay it's plus 1 plus plus
plus so let me sketch what this wave function would look like
that's the last item there so all all symmetric combinations I'm combining all
the 1s orbitals and I have no nodes in the entire wave function that's the
lowest possible wave function that I can create now let's think about the worst
possible case the highest energy configuration that will be a
configuration where we combine the 1s orbitals all in anti-bonding
configurations at the bottom they're all bonding configuration so so let's do
that it's going to look plus minus the wave function will change sign at every
bond okay so that's the that's the lowest highest energy possible
configuration all anti-bonding configurations I combine the wave
functions splicing the first time I'm gonna change the sign for the second
item and so on and so forth so for the crystals we also have the same idea of
bonding and anti-bonding combination of orbitals and what I'm showing here at
the top and at the bottom are they extreme anti bonding and extreme bonding
possibilities but in the crystal we have all sorts of other options I start if I
start with n atomic orbitals I need to end up with n with the same number of
crystalline orbitals okay so I'm going to have as many possibilities of
crystalline orbitals as I have atoms or atomic orbitals so so let's sketch some
of the intermediate ones so the first excited state based on what we know so
far now it's good to have one note okay the ground state has 0 notes the first
excited state will have a single note so let's sketch what a wave function like
that would oak at the middle of the crystal it's
going to have change sign oops add one extra spike there so so that'd be a
first excited configuration with a single dog now I can imagine that
there's many others where I have two up two down two up two down and so on and
so forth okay so what happens in the crystal is that between the extreme
bonding and extreme anti-bonding I have a lot of possibilities lots of possible
wave functions and energies associated with them and there are so many that
it's actually they form a continuum in energies okay so if i have a molecule if
i have two atoms when i bring them together as we've done before i'm going
to combine the wave function and i'm going to combine into a bonding and
anti-bonding configuration right what I'm plotting here in this plot is the
energy as a function of inverse separations between the arms okay so I
start off with the atoms far away and I bring them together and as I do that
they want the atomic orbitals mix and their energies change because of poly
and they split ok into bonding and anti-bonding now if you still bringing
two molecules I bring lots of molecules what happens is that between the extreme
bonding that's at the bottom and the extreme anti-bonding at the top all of
those states intermediate states are available all those energies between
extreme die bonding and extreme bonding are
available because I'm going to have this family of wave functions as we described
in the previous slide that have 0 nodes that they at the very bottom and then
one know that the two notes and the three notes so essentially they across
this energy band what I'm going to have is approximately 10 to the 8 States okay
because we we said that we have about 10 to the 8 atoms along a linear dimension
of a gopher crystal so those are so many that we can see there any of those
energies to be available and instead of talking about discrete energy states we
talk about energy bands okay given a lattice parameter okay for a given
separation there's going to be a region of energies okay that's going to be
allowed where I'm going to have electronic States available for me okay
so instead of having bonding and anti-bonding I'm going to have a band
and it is the origin of the electronic bands in in crystals okay all right so
let's go back to an old friend okay we've looked at this graph before what
happens when we bring to lithium row atoms together and we worked out in a
previous lecture what happens with the S orbitals and the P orbitals as we bring
them together and how symmetries lead to this interesting behavior so now we're
going to worry about silicon let's pick siliconic we could also do it with
carbon so silicon or carbon if I bring two of those atoms I'm going to have the
S orbitals occupied the three or the 2's orbitals occupied
and i'm going to have one third one
third of the p-orbitals occupied okay only one third so this what happens if I
bring two atoms together now if I bring lots of items together as we said before
what happens is first of all I have these bands i don't have individual
states but all the intermediate states are available so as the s-orbital and as
the p orbital open up all those energy states are available and but also what
happens is that as the estate's open up into bonding and anti-bonding and the P
States open up into bonding and anti-bonding they overlap okay and that
happens in the crystals and that least a very interesting process so what we
discussed is that one-third of the peace states are going to be occupied so let
me mark that line okay this line here represents the one-third of those
orbitals that are occupied so i'm going to use blue for energy levels that are
occupied okay all of my s orbitals are occupied one third the lower third of
the peep and is going to be occupied okay now all those states are all those
states exists okay so I'm going to use red for the empty States okay now
something very interesting happens when the highest occupied state meets the
anti-bonding state of the P of the S orbitals at this point there's a gap
that forms a range of energies where there's no
states available whatsoever okay and so these are the empty States and at the
bottom I have the occupied States okay so this is very interesting what happens
is that if we look at large bond distances okay to the left of the graph
what we have is that the highest occupied state okay the highest blue
state they blue state with highest energy and the lowest unoccupied state
that's take the red state with the lowest energy there next to one another
okay so one of the bands are partially occupied that's the characteristic of a
metal now if I go to the right what happens is that there is a gap of energy
between the highest occupied state this be the top of this barrier here and the
lowest unoccupied state that's the lowest red state that's a semiconductor
or a metal where there is a finite amount of energy that i need to give to
the system in order to promote an electron from the occupied state to the
next empty statement that of course leads to electrical insulators and
electrical resistivity so and this graph is very interesting so we were talking
about silicon or carbon okay and we're going to discuss that column four of the
periodic table and it starts with carbon okay carbon is at the top of that column
is the smallest item the shortest bond distance so in this graph carbon would
be to the right okay let me use a third color now and so carbon would be
somewhere here and what we would predict based on this
description is that the band gap for carbon to be very very large okay and we
go to the lab and carbon in the diamond structure okay has a band gap of about
5.5 electron volts now one step below carbon in the in the periodic table is
silicon okay so silicon would have a slightly larger lattice parameter that
means moving to the left in our our diagram and we'd predict it to have a
smaller band gap and the band gap of silicon is about 1.1 1.2 EV so again
that's consistent with our picture of the gap reducing as we move to the left
of the diagram then below silicone is germanium band gap 1.6 electron volts
and that's also consistent as we move to the left so this be silicon here
germanium would be around here and just below germanium okay it's teen until
it's in a very interesting position tin has zero band gap okay so Dean is right
here at at the other point where the gap opens and teen is right there at the
edge and actually teen exhibits a phase transition with temperature so the low
temperature face of team of teen your low temperature thermal expansion it
being smaller bonds so the low temperature phase is also the diamond
structure like carbon silicon and germanium now if you increase the
temperature the atoms move a little bit and they expand it moves to the right in
this picture and teen undergoes a transformation from this covalent
covalently bonded the diamond structure you're very open structure to a more
clothes packed tetragonal structure that's called beta team
and so so did is at that very interesting position at the transit at
the transition between covalency and semiconductors and metallic behavior
right below teen is led and of course lead is an FCC metals were on the left
of this diagram ok so again you're looking at these four column for you can
see this interesting transition between an insulator carbon to a metal in this
picture of the interplay between bonding and anti-bonding and an SP splitting
explains the trends okay so to sum up what happens is that if they bonding
anti-bonding splitting okay the opening up of each of these bands if that
splitting is larger than the S P splitting ok this is the S P split so if
the haunting and bonding anti-bonding splitting dominates then you have
semiconductors and our insulators and covalent bonding dominates okay diamond
structures these are open structures where each autumn makes only four bonds
however if the SP splitting dominates that means moving to the left in this
diagram then metallic bonding dominates metallic character dominates there's no
band gap okay all the atoms are the highest unoccupied state is right next
to the I'm sorry the lowest unoccupied state is right next to the highest
occupied state and you have a metal more close-packed bonding less covalent or
directional bonding that leads to open structures like diamond okay alright so
so in this lecture we developed an intuitive understanding
of crystals and under band structures their electronic structures and next
lecture we're going to do is put these in a slightly more mathematical term and
start thinking about how can we quantify and how can we account for this very
large number of wave functions ten to the eighth wave functions that we have
in each of these bands okay but that's for next lecture thank you very much