WEBVTT
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STUDENT: Hello, everyone.
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Today, I'm going to talk
about a very important model,
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analyze the energy
band of a crystal, that
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is the tight binding model.
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Before I talk about
tight binding model,
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let's now take a look
at free electron model.
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The potential energy
of free electron
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can be considered as
zero, so the Hamiltonian
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of the free electron system
is without the potential term.
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And the waveform function can be
simply written as a plane wave.
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By substituting
the wave function
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into Schrodinger
equation, we can
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gather energy
dispersion relation,
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which is a parabola
in one dimension case.
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And we usually form the parabola
into the first Brillouin zone.
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And this is the
reduced Brillouin zone.
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And in this figure, we can
see that the first band is
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at the bottom of the parabola.
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And the second band and third
band, fourth, and so on.
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Similar to one dimension case,
the energy of three dimension
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case is proportional to
kx squared plus ky squared
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plus kz squared.
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For a given energy,
the equal energy
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surface forms of a sphere.
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And if the energy is
equal to the Fermi energy,
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the states inside the sphere
will be occupied by electrons.
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Now let's look at the
two hydrogen atom system.
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When the two atoms are
very far from each other,
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the wave functions
don't overlap.
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As the two atoms going
towards each other,
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the electron of atom a starts
to build existence of atom b
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and the wave functions
start to overlap.
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We can see that when
the two atoms are close,
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two new orbital are created.
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One is the bonding
orbital and the other one
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is the antibonding orbital.
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For the bonding orbital,
we can see the probability
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to find the electron
between the two atoms
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is larger than before.
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And in the antibonding
orbital, we
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can see that in the
middle of the two atoms,
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the probability to find
the electron is zero.
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The overlap of the wave
function in bonding orbital,
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leading to a higher
banding energy, which
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leading to a more stable state.
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So the bonding orbital is the
ground state of the system.
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And antibonding orbital is
excited state of the system.
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Let us now talk about
the tight binding model.
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Assuming the potential
of a single atom is ur,
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then the Schrodinger equation
can be written like this.
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We now consider the
effect of other atoms
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as perturbation, as we have
seen in the last slide.
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Each energy level of
each atom split into two
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in a two hydrogen atom
system, the degeneracy
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of each energy level is two.
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For a system of n atoms, the
degeneracy of each energy level
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is n.
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From the perturbation
theory, the wave function
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for degenerative system can be
written as a linear combination
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of degenerate states.
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And if aj in the expression
is equal to the expression
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on the right, the function
satisfies the block theorem.
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We can also write the
first order correction
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of the energy, which is
the diagonal interest
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of the Hamiltonian matrix.
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Now we only consider the
integration of the atom itself
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and its nearest neighbors.
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So we can further
simplify the expression
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and the ways we set j 0 as the
integration of the atom itself.
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And j as the integration
of its nearest neighbors.
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Note that the integration
is determined by the overlap
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of the two wave functions.
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The more they overlap, the
wider the energy band is.
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And finally, we get our
final dispersion relation
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of tight binding model.
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Now let's see what has
happened after applying
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the tight binding model to
a one dimensional system,
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assuming that the distance
between atoms is a,
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each has only in a one
dimensional atom chain
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has two nearest neighbors.
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So the sum of the expression
has two components,
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where rho n equals
to a and the minus a.
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By using the Euler's formula,
the dispersion relation
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can be written as a
trigonometric form.
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The figures show the first two
bands of a one dimension system
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for free electron model
and tight binding model.
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We can say that because of the
inference of nearest neighbors,
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distortions appear at the
boundaries of the Brillouin
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zone, and leading to the
opening of a bandgap, which
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is quite different from
the free electron model.
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In three dimension cases,
the dispersion relations
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are more complicated.
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Let's now take a look at
the dispersion relations
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of these three kinds
of 3D lattices.
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For a simple cubic
lattice, there
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are six nearest
neighbors of one atom.
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So the sum has six components.
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And we can get the
dispersion relation like this
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after simplifying.
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While for body-centered
cubic, there
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are eight nearest
neighbors, which leads
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to an expression like this.
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For face-centered cubic, they
are 12 nearest neighbors.
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And we get a dispersion
relation like this.
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So what do the relations
on the last slide
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look like in the
first Brillouin zone?
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In 3D crystals, we can calculate
the reciprocal primitive
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vectors by using
the formula below.
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The reciprocal lattice of simple
cubic is also a simple cubic.
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And the reciprocal lattice
of a body-centered cubic
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is face-centered cubic,
while the reciprocal lattice
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of a face-centered cubic
is body-centered cubic.
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The first Brillouin
zone is defined
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as the region inside
the perpendicular
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bisectors of the segments
linking a point to its nearest
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neighbors.
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As we know, the
reciprocal primitive
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vectors we can plug
the first two Brillouin
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zones of this lattices.
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This graphs show the Bravais
lattice in real space
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and its energy dispersion
in the reciprocal space.
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For a simple cubic, the first
Brillouin zone is a cube.
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And when the energy is small,
the equal energy surface
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looks like a sphere, like
a free electron case.
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As we can see, when
the energy goes up,
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distortion begins to appear.
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And when the energy reaches the
boundary of the first Brillouin
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zone, openings of holes
appear at this boundaries.
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In this figures, we can see
that in the first Brillouin
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zone of BCC lattice, similar
to a single cubic lattice.
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Distortion gradually appear
as the energy goes up.
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And the surface is no
longer close to surface
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at the boundaries as the
energy goes higher and higher.
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And sodium is a BCC
crystal, because there
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is only one valence
electron per sodium atom.
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So the Fermi surface is pretty
far away from the boundaries
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and the shape of it is nearly
a sphere, which is like this.
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In FCC crystals, such
as copper and gold,
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their Fermi surface can be
well-described by tight bonding
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model.
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At the hexagon
faces, holes appear.
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Well because the distance
is further between the Fermi
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surface and the square faces.
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So this areas of the surface
won't extend and contact
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with the squares.
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Well, I hope that through
in this shot video
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you can understand what
tight bonding model is
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and how do we use it as a
tool to explain the phenomena
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in a real material.
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Thank you for your watching.