JURGIS RUZA: In this demonstration,
I'm going to show you how to construct 2D Brillouin zones
and hopefully achieve a better understanding of them
and what they are.
So a Brillouin zone is an important concept
in material science and solid state physics
alike because it is used to describe
the behavior of an electron in a perfect crystal system.
So what is a Brillouin zone?
A Brillouin zone is a particular choice of the unit
cell of the reciprocal lattice.
It is defined as the Wigner-Seitz cell
of the reciprocal lattice.
It is constructed as the set of points
enclosed by the Bragg planes--
the planes perpendicular to a connection line
from the origin to each lattice point passing
through the midpoint.
Alternatively, it is defined as the set
of points closer to the origin than to any other reciprocal
lattice point.
The whole reciprocal space may be covered without overlap
with copies of such Brillouin zone.
OK.
So that was a rather convoluted definition.
Let's do it again.
Let's brush up and be sure that we understand this definition.
We're going to define the reciprocal space and the Bragg
planes as well.
But to define both of these, we'll
also do a quick revision of what is the erect lattice so
that we can go from there to the reciprocal lattice.
The microscopic perfect crystal is
formed by adding identical building blocks.
So to say unit cells consisting of atoms and groups of atoms.
A unit cell is the smallest component
of a crystal that, once tacked together
with pure translational repetition,
reproduces the whole crystal, which essentially
means that you can take the same thing
over and over and over again and get the whole system done.
So the groups of atoms, these unit cells that
form the microscopic crystal by infinite repetition,
is called the basis.
OK.
That seems quite clear.
And the basis is formed in such a way
that it forms the lattice, more commonly known
as the Bravais lattice.
Every point of a Bravais lattice is
equivalent to every other point, which
means that the arrangement atoms in a crystal
is the same when viewed from different lattice points.
OK.
That also seems quite understandable,
and you should probably know that by now.
So any fundamental lattice must be
definable by three primitive translational vectors--
a1, a2, and a3.
The combination of these vectors is usually
to find the crystal translational vector r,
such that r is equal to a1 n1 plus a2 n2 plus a3 n3,
where n are just arbitrary integers to show
the size of our lattice.
The crystal lattice is repeated an infinite amount of times
to create the perfect crystal structure,
and each of those lattice are translationally symmetric.
Another way to look at it is that one cannot tell
their position in the crystal structure because every lattice
looks the same.
OK.
So that seems to make sense.
So now, let's go through reciprocal space.
So every lattice has a reciprocal lattice
associated to it.
In crystallography terms, the reciprocal lattice
is the fraction prior of a crystal,
or in quantum mechanics it's describe as k space,
with k being for k wave vectors.
In 3D lattice, the vectors would be b1, b2, and b3.
And they can be denoted as-- we'll look at just b1.
b1 is equal to two parts of the cross-products of the vectors
a2 and a3 from our direct lattice divided
by the triple cross scalar product of a1, a2,
and a3, in which case this cross-product of a2 and a3
is the area of our vector of our two vectors,
and the triple scalar product is the volume of our system.
By simplifying it, we can just get
2 pi over the height of our unit cell,
or we can put it this way.
The larger our direct lattice, the smaller in comparison
our reciprocal lattice becomes.
Another observation that could actually
be made by the reciprocal lattice
is that the reciprocal lattice of the reciprocal lattice
is the direct lattice.
But OK, for simplicity's sake, let's look
at a transformation from 2D lattice
to a reciprocal lattice.
So we have a visualization here where
we can change the length of our x vector in direct space
and the length of our y vector in direct space.
And we can change whether or not we're
seeing this as a direct lattice or whether we're
seeing this pattern as a reciprocal lattice.
So as we can see, by increasing the length
of our direct vector, we change the sizes
of our reciprocal lattice vectors, and the other way
around.
So now we had a definition of the reciprocal space
and direct space.
Let's go back to our definition.
So the first Brillouin zone can be
defined as a set of points in reciprocal space
that can be reached from a specific point of origin
without crossing any Bragg planes.
So what are Bragg planes?
A Bragg plane, or in this case, a Bragg line,
is a Bragg line which perpendicularly bisects
a reciprocal lattice vector--
a vector which connects two lattice points.
And the closest Bragg planes are essentially
crossing the Brillouin zone.
Now, we can show the Bragg planes
with the closest neighbors, with this being our original lattice
point.
And these are the four closest neighbors in a simple,
I'd say cubic, but it's actually just the square lattice
because it's in 2D.
And if we add the second largest vectors,
for the second closest neighbors you can see these ones.
And essentially, it conveys the same information.
When we go to a higher order of closest neighbors,
we can see that the system gets a lot more complex.
So now we've seen what are Bragg planes,
we can go towards Brillouin zones.
So this is the first Brillouin zone.
It is what it seems it is.
As you can imagine, the Bragg planes just go here.
And the Brillouin zone shows us the area
in reciprocal space that is closer to our lattice point
than any other lattice point, which are essentially the Bragg
planes as well.
As we can see, our reciprocal lattice origin point
is in here.
The square is closer to this point than to any other point.
And after these lines, it gets the other way around.
So we can move on forward to a higher
order of Brillouin zones.
And this is the Brillouin zone for the second closest
neighbor.
As you can see, it is rather similar.
It's just takes the second closest neighbors
and essentially draws another square.
But it's a bit tilted to the edge.
So now let's look at the third one.
For the third Brillouin zone, it gets a bit more complex
because if we scroll a bit backwards,
we can see that the Bragg planes for the third closest neighbors
are these ones.
So we might think that this whole thing would
be the third Brillouin zone, but it's actually not,
because with every next system it gets a bit more complex.
And it's actually taking account both the third Bragg
planes and the first ones.
So now let's do another thing.
Let's turn on that we can see all the Bragg planes
and turn up another notch.
So here we can see that the fourth Brillouin
zone gets a lot more complex.
And we can see all of the Bragg planes
for the closest neighbors.
And these lines get quite difficult to understand
by drawing themselves, but we can help ourselves out
with this visualization.
So yeah, we can see that they start
to interact with each other, and thus make
a more difficult structure.
And if we go to the fifth one and show
that we can see all the Brillouin zones--
so here we can see that it becomes quite a nice drawing.
If we were to keep adding them--
I mean, let's just do it.
Let's add until the ninth one.
So here we can see a high order Brillouin zone.
It actually looks kind of nice, which is also interesting
because it's also an art form drawing high order Brillouin
zones.
The higher you go, the more complex and more discrete
the system gets.
And it essentially looks nicer.
So after looking at this, we can get
a short definition of how to construct 2D Brillouin zones.
So n-th Brillouin zone can be defined as the area, or volume
if we look in 3D, in reciprocal space that
can be reached from the origin by crossing
exactly n minus 1 Bragg planes.
So we can look also at a Brillouin zone for a system
where the atoms are not perfectly
in a perfect square lattice but are offset a bit, making, so
to say, a triangular lattice.
And as we can see here, the first Brillouin zone
is a hexagon.
And the second one already gets a bit more difficult
by forming a star shape.
And the third one is, again, so to say, a hexagon.
And it goes on like that.
And then, so what information does the Brillouin zone hold
and what does it give us?
In short, vectors in the Brillouin zone
or on its boundary characterize states in the system
with lattice periodicity.
For example, phonon or electron states.
But for that, a whole other video.
This code for this demonstration was taken and edited
from mathematical demonstrations made by Jaroslaw Klos.