Hello everyone and welcome back to Solid State Physics in a Nutshell,
brought to you by the physics department at the Colorado School of Mines.
Okay, so last time we took a look at
thermal conductivity and related it to the
heat capacity of your solid,
the velocity of the phonons and the mean
free path between collisions.
Today we're going to look a little bit more at this mean free path term
and in particular look at phonon phonon
scattering. The first thing to do is recall that
for phonons outside the first Brillouin
zone these phonons can be translated back
into the first brillouin zone
by some reciprocal lattice vector G with no loss
of information.
This is the whole Nyquist frequency argument.
So when it comes to phonons scattering
off of other phonons
we have to think about conservation loss.
Okay
so remember first of all, phonons are bosons
and quasi-particles.
And so we can create them, we can destroy
them no big deal.
Really that phonon creation and
destruction is at the heart of
phonon phonon scattering.
So for example, in terms of
conserving crystal momentum, I could have
a phonon one and a phonon two
plus some reciprocal lattice vector G.
Because of this whole quasi-momentum business
those two phonons could to come together and
form
phonon three. And not only do we conserve
crystal momentum in this way we also
conserve energy. So E1 plus E2 
equals E3. Likewise, on this lower
line here,
you can see how we could start with some
high energy phonon and that phonon
could then split into two other phonons
q2 and q3 plus some reciprocal lattice
vector G.
Looking to the right here we can see
this energy conservation law
through this phonon decaying into two
other phonons. If this whole business
seems really creepy, of phonons just
becoming other phonons,
we can think about the photon analogy
where we think about second harmonic generation..
If I have
two red photons and a material that's
active for second-harmonic generation
and I put those two photons into the crystal
there's a non-zero chance those two
photons are going to combine and
give me a green light.
Within phonon phonon scattering we have
two types of scattering.
First, we have normal scattering then we have this other type
called Umklapp scattering.
With normal scattering G equals 0 in the
crystal momentum conservation law,
which is to say that q1 plus q2 simply
equals q3. On the other hand, in
Umklapp scattering
G does not equal 0 in the crystal
momentum conservation law.
So lets take a look at an example to see how that
plays out.
So with normal phonon phonon
scattering we might have some one dimensional
dispersion like yea
and two acoustic phonons can combine
together to give me this
optical phonon.
So there's some notes though that we should
consider first.
As it doesn't change your crystal
momentum this is not going to be a scattering event
that actually reduces thermal
conductivity.
While it may not reduce thermal
conductivity through scattering,
what it does do is locally move the
system towards the Planck
distribution. As a final note,
this transition must involve allowed
states. You can't have two phonons
coming in and then try to create a phonon at some
point in the dispersion
that is forbidden.
Now let's take a look at Umklapp scattering. I'm going to show not only the first brillouin zone
but I'm also going to show that zone repeated to the right out to 3 pi over a.
Let's consider two phonons coming
in,
q1 and q2 plus some reciprocal lattice vector
G that's required to bring the phonon
back into the first brillouin zone
like so. That's going to give us a new
phonon
q3, where energy is conserved but crystal momentum, as you can tell, is not
conserved.
Some things to remember about Umklapp scattering.
The energy is still going to be conserved,
crystal momentum is not conserved and
and I have demonstrated this for the phonon
plus phonon
combining together to create a new phonon.
But the reverse process is identical in that you
can have
one phonon decay into two phonons
plus a reciprocal lattice
vector. Okay, now that we have some
understanding of normal and Umklapp scattering
processes
let's take a look at their impact on
thermal conductivity. We'll start
by illuminating a surface with light such that
you
locally heat it and basically create a phonon
source. We should also probably cool the
backside
so that it's going to act as a sink.
We can visually see this in terms of
a bunch of phonons
coming from this left side, all of which
have a positive vx velocity.
This gives us significant
phonon momentum in the positive x
direction.
If we imagine just normal scattering is happening
we may find a redistribution
of the phonon momentum, but there's not
going to be a net change in this q vector
to the right.
Therefore, there's effectively no
resistance to transmitting heat.
On the other hand, with Umklapp scattering we
find that transport becomes diffusive
because crystal momentum is not
conserved. And while we started with a
significant phonon momentum in the
positive x direction
after a few Umklapp scattering events occur
we now have a significant fraction of our
phonons
moving in the wrong direction. Okay,
this continues to be pretty hand wavey
but let's bring this Umklapp scattering
back to the phonon mean free path and
thermal conductivity.
Umklapp scattering is going to be the dominant
source of phonon scattering at
room temperature and above, with a scattering
rate proportional to population.
From this we obtain that the mean free path
for phonons is
inversely proportional to temperature.
Increase in temperature increases the
phonon population
and decreases the mean free path.
Okay, to recap what we have is a failure to
conserve crystal momentum
due to Umklapp scattering, which is enabled by an anharmonic lattice
which allows phonons to interact and
this leads to significantly lower
kappa than might otherwise be obtained than
if we were simply relying on
imperfections to scatter phonons.
Okay, that's all
for this video. Next time we're going to take a
look at the temperature dependence
of thermal conductivity.
See you then.
