Before the start of this lecture, you should be able to:
- Apply perturbation theory to understand the band structure at crossings
- Derive the dispersion relation for a tight-binding model
- Recall the notion of the Fermi energy
- Calculate the Fermi surface, i.e. through the Fermi wavevector $k_F$, for two and three-dimensional systems
- Construct the unit cell of a crystal
!!! summary "Learning goals"
After this lecture you will be able to:
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@@ -16,19 +27,26 @@ _(based on chapter 16 of the book)_
- describe how the light absorption spectrum of a material relates to its band structure.
## Band structure
How are material properties related to the band structure?
In nature we see that some materials conduct electricity (conductors), some don't (insulators), and some do under specific circumstances (semiconductors).
The motion of electrons is described by the band structure (dispersion relation).
So far we have derived the dispersion relation for different type of models, such as the tight-binding model in which the electrons are bound by a strong potential, or when the electrons are perturbed by a small periodic potential as was the case in the NFEM.
We have not yet discussed how the band structure does affect the electrical properties of a material.
For a material to be a conductor, there should be available electron states at the Fermi level. Otherwise all the states are occupied, and all the currents cancel out.
A band structure of a 1D material may look similar to this:
Suppose we have a band structure of a 1D material similar to one below:
We see several **energy bands** that may be separated by a **band gap** or overlapping.
We see several energy bands that may either be separated by a **band gap** or overlap.
When the Fermi level lies in the band gap, the material is called a semiconductor (or dielectric or insulator). When the Fermi level lies within a band, it is a conductor (metal).
??? Question "Suppose the Fermi energy lies in the band gap. What feature about the band gap determines if a material is insulating or a semiconductor?"
Its size
### A simple requirement for insulators
In an insulator every single band is either completely filled or completely empty.
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@@ -47,18 +65,25 @@ We come to the important rule:
> Any material with an odd number of electrons per unit cell is a metal.
If the material has an even number of electrons per unit cell it may be a semiconductor, but only if the bands are not overlapping (see the figure above). For example: Si, Ge, Sn all have 4 valence electrons. Si (silicon, band gap 1.14 eV) and Ge (germanium, band gap 0.67 eV) are semiconductors, Sn (tin) is a metal. **Interesting feature: the heaviest material is a metal, why?**
If the material has an even number of electrons per unit cell it may be a semiconductor, but only if the bands are not overlapping (see the figure above), or it could be a metal. To illustrate this, let us consider the three elements Si (silicon), Ge (germanium) and Sn (tin). All three of these have 4 valence electrons. Both Si (band gap 1.14 eV) and Ge (band gap 0.67 eV) are semiconductors, while Sn (tin) is a metal.
??? Question "Why is Sn not a semiconductor but a metal?"
It has overlapping energy bands
## Fermi surface using a nearly free electron model
Sequence of steps (same procedure as in 1D, but harder because of the need to imagine a 2D dispersion relation):
In the end, we are also interested in studying the electrical properties, and thus the Fermi surface, in higher dimensions as well.
One could either do this through a tight-binding model approach (which is very similar to the 1D case), or through the NFEM (same procedure as in 1D, but harder because of the need to imagine a higher dispersion relation).
Let us illustarte the procedure in 2D.
To derive the corresponding Fermi surface using the NFEM we use the sequence of steps:
1. Compute $k_f$ using the free electron model (remember this is our starting point).
2. Plot the free electron model Fermi surface and the Brillouin zones.
3. Apply the perturbation where the Fermi surface crosses the Brillouin zone (due to avoided level crossings).
The resulting band structure looks like this (in the extended Brillouin zone scheme):
By doing so, the resulting band structure looks like the following (in the extended Brillouin zone scheme):
```python
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)
```
Observe that the top of the first band is above the bottom of the lowest band. Therefore if $V$ is sufficiently weak, the material can be conducting even with 2 electrons per unit cell!
Observe that the top of the first band is above the bottom of the second band. Therefore if $V$ is sufficiently weak, the material can be conducting even with 2 electrons per unit cell!
A larger $V$ makes the Fermi surface more distorted and eventually makes the material insulating.
Let's compare the almost parabolic dispersion of the nearly free electron model with a tight-binding model in 2D.
We now have a dispersion relation $E = E_0 - 2t(\cos k_x a + \cos k_y a)$, which looks like this:
For a tight-binding model we obtain the dispersion $E = E_0 - 2t(\cos k_x a + \cos k_y a)$:
```python
momenta=np.linspace(-pi,pi,100)
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```
### Light absorption
So far we have only discussed the effect of the Fermi energy and the band structure on the electrical properties of a material.
We know from nature that photons of external light sources can be reflected, transmitted, or absorbed by a material. Absorption, requires energy to transfer from the photon to electrons. In a filled band there are no available states where energy could be transferred (that's why insulators may be transparent).
Photons of external light can be reflected, transmitted, or absorbed by the material. Absorption, in turn requires energy transfer from the photon to electrons. In a filled band there are no available states where energy could be transferred (that's why insulators may be transparent).
When transition between two bands becomes possible due to photons having high energy, the absorption increases in a step-like fashion, see the sketch below for germanium.
When transition between two bands becomes possible due to photons having a high energy, the absorption increases in a step-like fashion, see the sketch below for germanium.
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### Exercise 1: 3D Fermi surfaces
Using the [periodic table](../fermi_surfaces) of the Fermi surfaces, answer the following questions:
#### Question 1.
Find 4 elements that are well described by the nearly-free electron model and 4 that are poorly described by it.
#### Question 2.
Is the Fermi surface of lithium or potassium better described by the free electron model? What about nearly-free electron model? Why?
#### Question 3.
Do you expect a crystal with a simple cubic lattice and monovalent atoms to be conducting?
#### Question 4.
What Fermi surface shape would you expect the [NaCl crystal](https://en.wikipedia.org/wiki/Sodium_chloride) to have? Explain your answer using both the atomic valences and the optical properties of this crystal.
1. Find 4 elements that are well described by the nearly-free electron model and 4 that are poorly described by it.
2. Is the Fermi surface of lithium better described by the free electron model or the NFEM? What about potassium? Why?
3. Do you expect a crystal with a simple cubic lattice and monovalent atoms to be conducting?
4. Na has 1 valence electron and Cl has 7. What Fermi surface shape would you expect the [NaCl crystal](https://en.wikipedia.org/wiki/Sodium_chloride) to have? Explain your answer using **a.** the atomic valences, and **b.** the optical properties of this crystal.
### Exercise 2: Tight-binding in 2D
Consider a rectangular lattice with lattice constants $a_x$ and $a_y$.
Suppose the hopping parameters in the two corresponding directions to be $-t_1$ and $-t_2$.
Consider a single orbital per atom and only nearest-neighbour interactions.
#### Question 1.
Write down a 2D tight-binding Schrödinger equation (expand to 2D the results of 1D).
#### Question 2.
Formulate the Bloch ansatz for the wave function.
#### Question 3.
Calculate the dispersion relation of this model.
#### Question 4.
What Fermi surface shape would this model have if the atoms are monovalent?
#### Question 5.
What Fermi surface shape would it have if the number of electrons per atom is much smaller than 1?
We limit ourselves to a single orbital per atom and only nearest-neighbour interactions.
1. Write down a 2D tight-binding Schrödinger equation (use your result of 1D).
2. Formulate the Bloch ansatz for the wave function.
3. Calculate the dispersion relation of this model.
4. The dispersion is plotted below.
What Fermi surface shape would this model have if the atoms are monovalent?
5. What Fermi surface shape would it have if the number of electrons per atom is much smaller than 1?
### Exercise 3: Nearly-free electron model in 2D
_(based on exercise 15.4 of the book)_
Suppose we have a square lattice with lattice constant $a$, with a periodic potential given by $V(x,y)=2V_{10}(\cos(2\pi x/a)+\cos(2\pi y/a))+4V_{11}\cos(2 \pi x/a)\cos(2 \pi y/a)$.
#### Question 1.
Use the nearly-free electron model to find the energy of state $\mathbf{q}=(\pi/a, 0)$.
1. Use the nearly-free electron model to find the energy of state $\mathbf{q}=(\pi/a, 0)$.
??? hint
This is analogous to the 1D case: the states that interact have $k$-vectors $(\pi/a,0)$ and $(-\pi/a,0)$; ($\psi_{+}\sim e^{i\pi x /a}$ ; $\psi_{-}\sim e^{-i\pi x /a}$).
??? hint
This is analogous to the 1D case: the states that interact have $k$-vectors $(\pi/a,0)$ and $(-\pi/a,0)$; ($\psi_{+}\sim e^{i\pi x /a}$ ; $\psi_{-}\sim e^{-i\pi x /a}$).
2. Let's now study the more complicated case of state $\mathbf{q}=(\pi/a,\pi/a)$. How many $k$-points have the same energy? Which ones?
3. Write down the nearly free electron model Hamiltonian near this point.
4. Find its eigenvalues.
#### Question 2.
Let's now study the more complicated case of state $\mathbf{q}=(\pi/a,\pi/a)$. How many $k$-points have the same energy? Which ones?
#### Question 3.
Write down the nearly free electron model Hamiltonian near this point.
#### Question 4.
Find its eigenvalues.
??? hint
Make use of the symmatries of the matrix Hamiltonian
