@@ -230,10 +230,10 @@ Suppose we have an $n$-dimensional hypercube with length $L$ for each side and c
4. Check that this equation 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 for the total number of electrons and for their total energy 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 $T = 0$.
7. Work out these integrals for $T = 0$.
#### 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 $T = 1000\mathrm{K}$.
A hypothetical metal has a Fermi energy $\epsilon_F = 5.2 \mathrm{eV}$, DOS per unit volume $g(\epsilon) = 2 \times 10^{10} \mathrm{eV}^{-\frac{3}{2}} \sqrt{\epsilon}$, and a temperature of $T = 1000$ $\mathrm{K}$.
1. Given these parameters, calculate the total energy of the electrons in the metal 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 found in 1. **Hint:** you can do numerical integration in MATLAB with [`integrate(fun,xmin,xmax)`](https://www.mathworks.com/help/matlab/ref/integral.html).
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@@ -241,10 +241,10 @@ A hypothetical metal (let's call it Akhmerovium) has a Fermi energy $\epsilon_F
4. Numerically compute the heat capacity by approximating the derivative of the total energy in 2 with respect to $T$. To this end, make use of the fact that $$\frac{dy}{dx}=\lim_{\Delta x \to 0} \frac{y(x + \Delta x) - y(x - \Delta x)}{2 \Delta x}.$$ Compare your result with 3.
#### Exercise 4: graphene
One of the most famous recently discovered materials is [graphene](https://en.wikipedia.org/wiki/Graphene, which consists of 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 by a linear relation:
One of the most famous recently discovered materials is [graphene](https://en.wikipedia.org/wiki/Graphene), which consists of 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 by 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 surface area $A$. Your result should be linear with $|\epsilon|$. Do not forget spin degeneracy, and take into account that graphene has an additional two-fold 'valley degeneracy'.
1. Make a sketch of the dispersion relation. What other well-known particles have a linear dispersion relation?
2. Using the dispersion relation and assuming periodic boundary conditions, derive an expression for the DOS of graphene. Your result should be linear with $|\epsilon|$. Do not forget spin degeneracy, and take into account that graphene has an additional two-fold 'valley degeneracy'.
3. At finite temperatures, electrons close to the Fermi level (i.e. not more than $k_B T$ below the Fermi level) will get thermally excited, thereby increasing their energy by $k_B T$. Calculate the difference between the energy of the thermally excited state and that of the ground state $E(T)-E_0$. To do so, 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 a function of the temperature $T$.
