@@ -367,7 +367,7 @@ We first find the band structure using the nearly free electron model.
To ease the calculating $\epsilon_0$ and $t$, calculate them for $| n = 0 \rangle $ and $ | n = 1 \rangle $.
You may also make use of the results obtained in [exercise 2 of lecture 5](/5_atoms_and_lcao/#exercise-2-application-of-the-lcao-model) or found on the [wikipedia](https://en.wikipedia.org/wiki/Delta_potential).
You may also make use of the results obtained in [exercise 2 of lecture 5](5_atoms_and_lcao.md#exercise-2-application-of-the-lcao-model) or found on the [wikipedia](https://en.wikipedia.org/wiki/Delta_potential).
4. Compare the bands obtained in exercise 1 and 2: what are the minima and bandwidths (difference between maximum and minimum) of those bands?
5. For what $a$ and $\lambda$ is the nearly free electron model more accurate? And for what $a$ and $\lambda$ is the tight binding model more accurate?
@@ -374,7 +374,7 @@ Furthermore, the chemical potential is set at $0 < \mu < E_G$.
6. What would the chemical potential $\mu$ be in case of an intrinsic semiconductor?
### Exercise 3*. Holes and electrons in a 1D tight-binding energy band
Recall that the dispersion relation of a [1D single orbital tight binding chain](/7_tight_binding/) is given by
Recall that the dispersion relation of a [1D single orbital tight binding chain](7_tight_binding.md) is given by
$$E(k)=\varepsilon + 2t \cos(ka),$$
...
...
@@ -386,7 +386,7 @@ where $a$ is the lattice constant and $\varepsilon$ and $t$ are tight binding pa
### Exercise 4*: The Hall coefficient when both electrons and holes are present in the system
1. Recall from the [Drude model](/3_drude_model/) that electrons give rise to a negative Hall coefficient. Explain why the Hall coefficient is positive if holes are the charge carriers in a material.
1. Recall from the [Drude model](3_drude_model.md) that electrons give rise to a negative Hall coefficient. Explain why the Hall coefficient is positive if holes are the charge carriers in a material.
2. Recall that the Hall coefficient $R_H$ describes the change of the transverse resistivity $\rho_{xy}$ due to a change in $B$: $\rho_{xy} = \frac{E_y}{J_x}=R_H B$. When both electrons and holes are present in a system, the Hall coefficient is given by
@@ -493,7 +493,7 @@ The dispersion of energy band 1 is $\varepsilon_1(\mathbf{k}) = \tfrac{\hbar^2k^
4. Express the number of electrons in the energy range $\varepsilon_a<\varepsilon<\infty$asanintegraloverenergy,assuming$T>0$.
5. Assuming $\varepsilon_a - E_F \gg k_B T$, explicitly calculate the integral of the previous subquestion.
[^1]:This is not completely true, as we will see when learning about [semiconductors](13_semiconductors)
[^1]:This is not completely true, as we will see when learning about [semiconductors](13_semiconductors.md)
[^2]:An[isotropic](https://en.wikipedia.org/wiki/Isotropic_solid) material means that the material is the same in all directions.
[^3]:The mean inter-particle distance is related to the electron density $n = N/V$ as $\langle r \rangle \propto n^{-1/3}$. The exact proportionality constant depends on the properties of the system. The Fermi wavelength sets the scale at which quantum interference effects of the electronic waves become important. In some materials (e.g. graphene) it can be on the 100 nm scale - accessible to nanofabrication techniques. Striking images of [electron interference at the atomic](https://en.wikipedia.org/wiki/Quantum_mirage) scale are visible with a scanning tunneling microscope.