@@ -208,49 +208,51 @@ $g(E) \propto E$ ⇒ total energy is $T \times T^2$ ⇒ $C_V \propto T^2$.
$g(E) \propto E^2$ ⇒ total energy is $T \times T^3$ ⇒ $C_V \propto T^3$.
## Exercises
### Exercise 1: potassium
### Exercises
#### Exercise 1: potassium
The Sommerfeld model provides a good description of free electrons in alkali metals such as potassium, which has a Fermi energy of 2.12 eV (data from Ashcroft, N. W. and Mermin, N. D., Solid State Physics, Saunders, 1976.).
1. Calculate the corresponding Fermi temperature, Fermi wave vector and Fermi velocity.
2. Why is the Fermi temperature much higher than room temperature?
3. Calculate the free electron density in Potassium.
4. Compare this with the actual electron density of Potassium, which can be calculated by using the density, atomic mass and atomic number of Potassium. What can you conclude from this?
### Exercise 2: the $n$-dimensional free electron model.
1. Calculate the corresponding Fermi temperature, Fermi wave vector and Fermi velocity.
2. Why is the Fermi temperature much higher than room temperature?
3. Calculate the free electron density in Potassium.
4. Compare this with the actual electron density of Potassium, which can be calculated by using the density, atomic mass and atomic number of Potassium. What can you conclude from this?
#### Exercise 2: the $n$-dimensional free electron model.
In the lecture, it has been explained that the density of states (DOS) of the free electron model is proportional to $1/\sqrt{\epsilon}$ in 1D, constant in 2D and proportional to $\sqrt{\epsilon}$ in 3D. In this exercise, we are going to derive the DOS of the free electron model for an arbitrary number of dimensions.
Suppose we have an $n$-dimensional hypercube with length $L$ for each side and contains free electrons.
1. Assuming periodic boundary conditions, how much volume in $\mathbf{k}$-space is there between $\mathbf{k}$-points?
2. Using the answer in 1, show that the density of $\mathbf{k}$-points with a magnitude between $k$ and $k + dk$ in $\mathbf{k}$-space $g(k)$ is given by
$$g(k) = \frac{1}{\Gamma(n/2)} \left( \frac{L }{ \sqrt{\pi}} \right)^n \left( \frac{k}{2} \right)^{n-1},$$ where $\Gamma(z)$ is the [gamma function](https://en.wikipedia.org/wiki/Gamma_function).
**Hint:** you will need the area of an $n$-dimensional sphere and this can be found on [Wikipedia](https://en.wikipedia.org/wiki/N-sphere#Volume_and_surface_area)(blue box on the right).
3. Using the expression above, calculate the DOS (do not forget the spin degeneracy).
4. Give an integral expression of the total number of electrons and total energy in the hypercube in terms of the DOS, the temperature $T$ and the chemical potential $\mu$ (_you do not have to work out these integrals_).
5. Work out these integrals in case of $T = 0$.
### Exercise 3: fermions in a box
1. Assuming periodic boundary conditions, how much volume in $\mathbf{k}$-space is there between $\mathbf{k}$-points?
2. Using the answer in 1, show that the density of $\mathbf{k}$-points with a magnitude between $k$ and $k + dk$ in $\mathbf{k}$-space $g(k)$ is given by
$$g(k) = \frac{1}{\Gamma(n/2)} \left( \frac{L }{ \sqrt{\pi}} \right)^n \left( \frac{k}{2} \right)^{n-1},$$ where $\Gamma(z)$ is the [gamma function](https://en.wikipedia.org/wiki/Gamma_function).
**Hint:** you will need the area of an $n$-dimensional sphere and this can be found on [Wikipedia](https://en.wikipedia.org/wiki/N-sphere#Volume_and_surface_area)(blue box on the right).
3. Using the expression above, calculate the DOS (do not forget the spin degeneracy).
4. Give an integral expression of the total number of electrons and total energy in the hypercube in terms of the DOS, the temperature $T$ and the chemical potential $\mu$ (_you do not have to work out these integrals_).
5. Work out these integrals in case of $T = 0$.
#### Exercise 3: fermions in a box
Given is a 3D box with free fermions with Fermi energy $\epsilon_F =100$, DOS $g(\epsilon) = 0.42\sqrt{\epsilon}$, and $k_B T = 2.5$.
1. Given these parameters, calculate the total energy of this fermion box using the triangle technique explained in the lecture notes. (Make a sketch!)
2. Numerically calculate the integral expression for the total energy and compare your result with that in a). **Hint:** you can do numerical integration in MATLAB with [`integrate(fun,xmin,xmax)`](https://www.mathworks.com/help/matlab/ref/integral.html).
3. Calculate the heat capacity of this fermion box in units of $k_B$.
4. Numerically compute the heat capacity by approximating the derivative of the total energy in 2 with respect to $T$. Compare your result with 3.
### Exercise 4: graphene
1. Given these parameters, calculate the total energy of this fermion box using the triangle technique explained in the lecture notes. (Make a sketch!)
2. Numerically calculate the integral expression for the total energy and compare your result with that in a). **Hint:** you can do numerical integration in MATLAB with [`integrate(fun,xmin,xmax)`](https://www.mathworks.com/help/matlab/ref/integral.html).
3. Calculate the heat capacity of this fermion box in units of $k_B$.
4. Numerically compute the heat capacity by approximating the derivative of the total energy in 2 with respect to $T$. Compare your result with 3.
#### Exercise 4: graphene
One of the most hyped materials in solid state physics is [graphene](https://en.wikipedia.org/wiki/Graphene), which are carbon atoms arranged in a 2D honeycomb structure. In this exercise, we will focus on the electrons in bulk graphene. Unlike in metals, electrons in graphene cannot be treated as 'free'. However, close to the Fermi level, the dispersion relation can be approximated as a linear relation:
$$ \epsilon(\mathbf{k}) = \pm c|\mathbf{k}|.$$ Note that the $\pm$ here means that there are two energy levels at a specified $\mathbf{k}$. The Fermi level is set at $\epsilon_F = 0$.
1. Make a sketch of the dispersion relation.
2. Using the dispersion relation, derive an expression for the DOS of graphene with an area surface of $A$. Your result should be linear to $|\epsilon|$.
At finite temperatures, electrons close to the Fermi level (i.e. not further below than $k_B T$ from the Fermi level) will get thermally excited and contribute $k_B T$ more energy with respect to the ground state.
3. Calculate the total energy of the exited state with respect to the ground state ($E-E_0$).
4. Calculate the heat capacity $C_V$ as function of the temperature $T$.
1. Make a sketch of the dispersion relation.
2. Using the dispersion relation, derive an expression for the DOS of graphene with an area surface of $A$. Your result should be linear to $|\epsilon|$.
3. At finite temperatures, electrons close to the Fermi level (i.e. not further below than $k_B T$ from the Fermi level) will get thermally excited and contribute $k_B T$ more energy with respect to the ground state. Calculate the total energy of the exited state with respect to the ground state ($E-E_0$).
4. Calculate the heat capacity $C_V$ as function of the temperature $T$.
### Exercise 5: Sommerfeld and Drude model
#### Exercise 5: Sommerfeld and Drude model
When an electrical field is applied, the center of the Fermi sphere will shift away from $\mathbf{k}=0$. In this exercise, we will calculate this shift $\Delta \mathbf{k}$ using both concepts from the Drude model and Sommerfeld model.
Suppose we have a 1D wire of length $L$ over which a bias voltage $V_0$ is applied. The wire has a resistivity $\rho$ and free electron density $n$.
1. Using the Drude model, give an expression of the average velocity of the electrons in the wire in terms of $L$, $V_0$, $\rho$, $n$ and natural constants/properties of electrons.
2. What is the corresponding average wave vector to this velocity?
3. Proof that this average wave vector is equal to the shift of the Fermi sphere in $\mathbf{k}$-space.
**Hints:**
- Find an integral expression of the _total_ wave vector of all electrons in the wire.
- Remember from the book that the total number of electrons in the wire is related to the Fermi wave vector.
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1. Using the Drude model, give an expression of the average velocity of the electrons in the wire in terms of $L$, $V_0$, $\rho$, $n$ and natural constants/properties of electrons.
2. What is the corresponding average wave vector to this velocity?
3. Proof that this average wave vector is equal to the shift of the Fermi sphere in $\mathbf{k}$-space.
**Hints:**
- Find an integral expression of the _total_ wave vector of all electrons in the wire.
- Remember from the book that the total number of electrons in the wire is related to the Fermi wave vector.
