@@ -223,12 +223,14 @@ In the lecture, it has been explained that the density of states (DOS) of the fr
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
2. Using the answer in 1, find the density of $\mathbf{k}$-points with a magnitude between $k$ and $k + dk$ in $\mathbf{k}$-space $g(k)$ for the 1D, 2D and 3D free electron model.
3. Show now that $g(k)$ for any number of dimensions $n$ 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 $T = 0$.
**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).
4. Check that this equation in 3, is consistent with your answers in 2. **Hint:** check [Wikipedia](https://en.wikipedia.org/wiki/Particular_values_of_the_gamma_function) to find out how to deal with half-integer values in the gamma function.
5. Using the expression in 3, calculate the DOS (do not forget the spin degeneracy).
6. 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_).
7. Work out these integrals in case of $T = 0$.
#### Exercise 3: a hypothetical metal
A hypothetical metal (let's call it Akhmerovium) has a Fermi energy $\epsilon_F = 5.2 \mathrm{eV}$, DOS $g(\epsilon) = 2 \times 10^{10} \mathrm{eV}^{-\frac{3}{2}} \sqrt{\epsilon}$, and a temperature of $T = 1000 \mathrm{K}$.
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@@ -243,6 +245,6 @@ One of the most hyped materials in solid state physics is [graphene](https://en.
$$ \epsilon(\mathbf{k}) = \pm c|\mathbf{k}|.$$ 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 $|\epsilon|$.
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 ($E-E_0$).
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 $|\epsilon|$. Furthermore, it should be noted that graphene also has a so-called valley degeneracy (on top of the spin degeneracy) which gives an additional factor of 2.
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 ($E-E_0$). To this end, show first that the number of electrons that will get excited is given by $$n_{ex} = \frac{1}{2} g(-k_B T) k_B T.$$
4. Calculate the heat capacity $C_V$ as function of the temperature $T$.
