@@ -270,43 +270,57 @@ $g(E) \propto E$ ⇒ 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
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. Check the [Fermi surface database](fermi_surfaces.md) in the attic. Explain why potassium and (most) other alkali metals can be described well with the Sommerfeld model.
2. Calculate the Fermi temperature, Fermi wave vector and Fermi velocity for potassium.
3. Why is the Fermi temperature much higher than room temperature?
4. Calculate the free electron density in Potassium.
5. 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?
1. Check the [Fermi surface database](fermi_surfaces.md) in the attic. Explain why potassium and (most) other alkali metals can be described well with the Sommerfeld model.
2. Calculate the Fermi temperature, Fermi wave vector and Fermi velocity for potassium.
3. Why is the Fermi temperature much higher than room temperature?
4. Calculate the free electron density in potassium.
5. 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, what is the distance between nearest-neighbour points in $\mathbf{k}$-space? What is the density of $\mathbf{k}$-points in n-dimensional $\mathbf{k}$-space?
2. The number of $\mathbf{k}$-points with a magnitude between $k$ and $k + dk$ is given by $g(k)dk$. Using the answer for (1), find $g(k)$ for 1D, 2D and 3D.
3. Now show that $g(k)$ for $n$ dimensions is given by
1. Assuming periodic boundary conditions, what is the distance between nearest-neighbour points in $\mathbf{k}$-space? What is the density of $\mathbf{k}$-points in n-dimensional $\mathbf{k}$-space?
2. The number of $\mathbf{k}$-points with a magnitude between $k$ and $k + dk$ is given by $g(k)dk$. Using the answer for 1, find $g(k)$ for 1D, 2D and 3D.
3. Now show that $g(k)$ for $n$ dimensions 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).
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 for $T = 0$.
#### Exercise 3: a hypothetical metal
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 $T = 1000$ $\mathrm{K}$.
??? 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 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 for $T = 0$.
#### Exercise 3: a hypothetical material
A hypothetical metal has a Fermi energy $\epsilon_F = 5.2 \,\mathrm{eV}$ and a DOS per unit volume $g(\epsilon) = 2 \times 10^{10} \,\mathrm{eV}^{-\frac{3}{2}} \sqrt{\epsilon}$.
1. Give an integral expression for the total energy of the electrons in this hypothetical material in terms of the DOS $g(\epsilon)$, the temperature $T$ and the chemical potential $\mu = \epsilon_F$.
2. Find the ground state energy at $T = 0$.
3. In order to obtain a good approximation of the integral for non-zero $T$, one can make use of the [Sommerfeld expansion](https://en.wikipedia.org/wiki/Sommerfeld_expansion). Using this expansion, find the difference between the total energy of the electrons for $T = 1000 \,\mathrm{K}$ with that of the ground state.
4. Now, find this difference in energy by calculating the integral found in 1 numerically. Compare your result with 3.
??? hint
You can do numerical integration in MATLAB with [`integral(fun,xmin,xmax)`](https://www.mathworks.com/help/matlab/ref/integral.html).
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).
3. Calculate the heat capacity of the metal in eV/K.
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.
5. Calculate the heat capacity for $T = 1000 \,\mathrm{K}$ in eV/K.
6. Numerically compute the heat capacity by approximating the derivative of energy difference found in 4 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 5.
#### 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:
$$ \epsilon(\mathbf{k}) = \pm c|\mathbf{k}|.$$ 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. 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 $T$.
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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, assume that 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 $T$.
