Advanced Solid State Physics

elements) the phase factor cancels out.

=

∫

d^3 rΨ∗(r)HΨ(r)

〈Ψk|Ψk〉

Now to themnearest neighbors. They are all equivalent in the crystal (so there is one kind of atom,
and one atom per unit cell).ρis the distance to the nearest neighbor. The factor is because there is
a little bit of overlap between two nearest neighbors:

t=−

∫

d^3 rΨ∗(r−ρm)HΨ(r)

〈Ψk|Ψk〉

In the definition oftthere is a minus becausetis negative most of the time (but a negativetis not a
must!). If allmnearest neighbors are equivalent there aremNterms liketeikρm. All these terms have
a phase factor because there is a difference in the phase factor (the phase factor changes by moving
through the crystal, this is because there are differences when it moves for example in the y-direction
or in the x-direction)a 1 ,a 2 anda 3 are the primitive lattice vectors in real space.

If you have a wavefunction like in eqn. (56) and you calculate the energy like in (57) you get this
general formula to calculate the total energy with the sum over the phase factors. Equation (58) is
the basic formula to calculate the dispersion relationship.

E=−t

∑

m

eik~ρm (58)

~ρmis the distance to the nearest neighbors.

Doing this for a simple cubic crystal structure, all the atoms have 6 nearest neighbors (two in x-, two
in y- and two in z-direction). So we take the sum over the six nearest neighbors for the calculation
of the energy (eqn. (58)). The next equation is just for the nearest neighbors and the first term is
for example in plus x-direction withρ=a(which is the same in each direction in a simple cubic).
The second term goes in the minus x-direction withρ=aand so on for the y- and z-direction. And
then you see that you can put it together to a cosinus term, you get an energy versuskdispersion
relationship (for just the nearest neighbor terms). That looks like

E=−t(eikxa+e−ikxa+eikya+e−ikya+eikza+e−ikza) =− 2 t(cos(kxa) +cos(kya) +cos(kza)).(59)

For a bcc crystal there are one atom in the middle and eight nearest neighbors at the edges. In this
case the formula consists of 8 terms and looks like

E=−t(ei(kx

a 2 +kya 2 +kza 2 )

+ei(kx

a 2 +kya 2 −kza 2 )

+......=− 8 t

(

cos

(

kxa

2

)

+ cos

(

kya

2

)

+ cos

(

kza

2

))

.

This is the dispersion relationship for a bcc crystal in the tight binding model.

In an fcc crystal there are 12 nearest neighbors, so the formula has twelve terms and looks like:

E=−t(ei(kx

a 2 +kya 2 )

+ei(kx

a 2 −kya 2 )

+..) =

=− 4 t

(

cos

(

kxa

2

)

cos

(

kya

2

)

+ cos

(

kxa

2

)

cos

(

kza

2

)

+ cos

(

kya

2

)

cos

(

kza

2

))