| Total particle number | temperature-dependent | constant |
It is important to note the last difference in this table: warming a material up creates more thermally excited phonons.
The number of electrons, on the other hand, stays the same: the electrons may not appear out of nowhere.[^1]
!!! note "About $\hbar$"
Within quantum mechanics energy and frequency are related by Planck's constant: $\varepsilon = \hbar\omega$.
Similarly, $p = \hbar k$ relates a particle's momentum with its wave vector.
This relation is so unambiguous that you may encounter these terms used synonymously in scientific literature.
The last difference is important: warming a material up creates more thermally excited phonons.
The number of electrons, on the other hand, stays the same: the electrons may not appear out of nowhere.[^1]
## The Fermi sea
...
...
@@ -283,8 +285,8 @@ ax.legend()
draw_classic_axes(ax,xlabeloffset=.2)
```
## Relationship between the Fermi energy and the system parameters
We would like to relate the Fermi energy $\varepsilon_{F}$ to the system parameters the number of electrons in the system ($N$) and the volume of the box $V = L^3$.
## Relationship between the Fermi energy and the number of electrons
We would like to relate the Fermi energy $\varepsilon_{F}$ to the number of electrons in the system ($N$) and the volume of the box $V = L^3$.
To do so, we calculate the number of electrons in the system at $T = 0$ using the previously found density of states:
@@ -425,9 +427,9 @@ Thus the total energy is proportional to $T \times T^2$ and the heat capacity $C
## Conclusions
1. The Sommerfeld free electron model treats electrons as free particles with energy dispersion $\varepsilon = \frac{\hbar^2k^2}{2m}$.
1. The Sommerfeld free electron model treats electrons as free particles with energy dispersion $\varepsilon = \frac{\hbar^2|\mathbf{k}|^2}{2m}$.
2. The density of states $g(\varepsilon)$ follows from the dispersion relation by using a general procedure that is analogous to that for phonons.
3. The Fermi-Dirac distribution gives the probability of an electron state to be occupied.
3. The Fermi-Dirac distribution gives the probability of an electron state at energy $\varepsilon$ to be occupied.
4. The electron contribution to the heat capacity is proportional to $T$. It is much lower than phonon heat capacity at high temperatures, and much higher at low temperatures.
5. The scaling of heat capacity with $T$ can be quickly estimated by estimating the number of particles in an energy range $k_BT$ from the Fermi energy.
...
...
@@ -436,116 +438,52 @@ Thus the total energy is proportional to $T \times T^2$ and the heat capacity $C
### Warm-up questions
1. List the differences between electrons and phonons from your memory.
2. Write down the expression for the total energy of particles with the density of states $g(\varepsilon)$ and the occupation number $n_{F}(\beta(\varepsilon - \mu))$.
3. Explain what happens if a material is heated up to its Fermi temperature (assuming that material where this is possible exists).
4. Why can we not use the Sommerfeld expansion with a Fermi energy of the order of the thermal energy?
5. Is the heat capacity of a solid at temperatures near $T=0$ dominated by electrons or phonons?
### Exercise 1: potassium
The Sommerfeld model provides a good description of free electrons in alkali metals such as potassium (element K), which has a Fermi energy of $\varepsilon_{F} = 2.12$ eV (data from Ashcroft, N. W. and Mermin, N. D., Solid State Physics, Saunders, 1976.).
#### Question 1.
Check the [Fermi surface database](http://www.phys.ufl.edu/fermisurface/). Explain why potassium and (most) other alkali metals can be described well with the Sommerfeld model.
#### Question 2.
Calculate the Fermi temperature, Fermi wave vector and Fermi velocity for potassium.
#### Question 3.
Why is the Fermi temperature much higher than room temperature?
#### Question 4.
Calculate the free electron density $n$ in potassium.
#### Question 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 of the free electron model is proportional to $1/\sqrt{\varepsilon}$ in 1D, constant in 2D and proportional to $\sqrt{\varepsilon}$ in 3D. In this exercise, we are going to derive the density of states of the free electron model for an arbitrary number of dimensions.
Suppose we have a $n$-dimensional hypercube with length $L$ for each side that houses free electrons.
#### Question 1.
What is the distance between nearest-neighbour points in $\mathbf{k}$-space? Assume periodic boundary conditions.
What is the density of $\mathbf{k}$-points in n-dimensional $\mathbf{k}$-space?
#### Question 2.
The number of $\mathbf{k}$-points between $k$ and $k + dk$ is given by $g(k)dk$.
Using the answer for 1, find $g(k)$ for 1D, 2D and 3D.
#### Question 3.
Now show that $g(k)$ for $n$ dimensions is given by
3. Write down an integral expression for the total energy of particles with density of states $g(\varepsilon)$ and occupation number $n_{F}(\beta(\varepsilon - \mu))$.
4. Describe what is the Fermi surface. For the free-electron model, what does the Fermi surface look like in 1D, 2D, and 3D?
5. Argue if the heat capacity of a solid at temperatures near $T=0$ dominated by electrons or phonons.
where $\Gamma(z)$ is the [gamma function](https://en.wikipedia.org/wiki/Gamma_function).
### Exercise 1*: Deriving the density of states of the free-electron dispersion.
In this lecture, we found that the density of states $g(\varepsilon)$ in the free electron model is proportional to $1/\sqrt{\varepsilon}$ in 1D, constant in 2D, and proportional to $\sqrt{\varepsilon}$ in 3D. In this exercise, we will practice this important derivation again.
??? 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).
#### Question 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.
#### Question 5.
Using the expression in 3, calculate the density of states (do not forget the spin degeneracy).
#### Question 6.
Give an integral expression for the total number of electrons and for their total energy in terms of the density of states, the temperature $T$ and the chemical potential $\mu$ (_you do not have to work out these integrals_).
#### Question 7.
Work out these integrals for $T = 0$.
### Exercise 3: a hypothetical material
A hypothetical metal has a Fermi energy $\varepsilon_F = 5.2 \,\mathrm{eV}$ and a density of states $g(\varepsilon) = 2 \times 10^{10} \,\mathrm{eV}^{-\frac{3}{2}} \sqrt{\varepsilon}$.
#### Question 1.
Give an integral expression for the total energy of the electrons in this hypothetical material in terms of the density of states $g(\varepsilon)$, the temperature $T$ and the chemical potential $\mu = \varepsilon_F$.
#### Question 2.
Find the ground state energy at $T = 0$.
#### Question 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)(the first equation is all you need and you can neglect the $O\left(\frac{1}{\beta \mu}\right)^{4}$ term).
Using this expansion, find the difference between the total energy of the electrons for $T = 1000 \,\mathrm{K}$ with that of the ground state.
#### Question 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 python with [`scipy.integrate.quad(func, xmin, xmax)`](https://docs.scipy.org/doc/scipy/reference/generated/scipy.integrate.quad.html)
#### Question 5.
Calculate the heat capacity for $T = 1000 \,\mathrm{K}$ in eV/K.
#### Question 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
1. What is the distance between nearest-neighbour points in $\mathbf{k}$-space? Assume periodic boundary conditions.
What is the density of $\mathbf{k}$-points in 1, 2, and 3 dimensions?
2. Write down an integral expression for the number of states in k-space between $0 < k < k_{max}$. Do so for 1D, 2D and 3D. Do not forget the spin degeneracy.
3. Transform these integrals to integrals over energy for 1D, 2D and 3D. Indicate the integral boundaries. Extract the density of states. Are the integral boundaries important for the result?
### Exercise 2: Applying the free electron model to potassium
The Sommerfeld model provides a good description of free electrons in alkali metals such as potassium (element K), which has a Fermi energy of $\varepsilon_{F} = 2.12$ eV (data from Ashcroft, N. W. and Mermin, N. D., Solid State Physics, Saunders, 1976.).
Compare your result with 5.
1. Check the [Fermi surface database](http://www.phys.ufl.edu/fermisurface/). 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. Sketch the Fermi distribution at room temperature and indicate the role of the Fermi temperature.
3. Calculate the free electron density $n$ 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 4: graphene
### Exercise 3*: graphene
One of the most famous recently discovered materials is [graphene](https://en.wikipedia.org/wiki/Graphene). It 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:
$ \varepsilon(\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 $\varepsilon_F = 0$.
#### Question 1.
Make a sketch of the dispersion relation.
1. Make a sketch of the dispersion relation.
What other well-known particles have a linear dispersion relation?
#### Question 2.
Using the dispersion relation and assuming periodic boundary conditions, derive an expression for the density of states of graphene.
2. Using the dispersion relation and assuming periodic boundary conditions, derive an expression for the density of states of graphene.
Do not forget spin degeneracy, and take into account that graphene has an additional two-fold 'valley degeneracy' (hence there is a total of a fourfold degeneracy instead of two).
Your result should be linear with $|\varepsilon|$.
??? hint
It is convenient to first start by only considering the positive energy contributions $\varepsilon(\mathbf{k}) = + c|\mathbf{k}|$ and calculate the density of states for it. Then account for the negative energy contributions $\varepsilon(\mathbf{k}) = - c|\mathbf{k}|$ by adding it to the density of states for the positive energies. You can also make use of $\frac{\rm{d} |k|}{\rm{d}k} = \frac{k}{|k|}$.
#### Question 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
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
$$
n_{ex} = \frac{1}{2} g(-k_B T) k_B T.
$$
#### Question 4.
Calculate the heat capacity $C_e$ as a function of the temperature $T$.
4. Calculate the heat capacity $C_e$ as a function of the temperature $T$.
[^1]:This is not completely true, as we will see when learning about [semiconductors](13_semiconductors)
[^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.
[^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.
@@ -52,6 +52,57 @@ Consider now an ensemble of electrons in a metal. Include the Drude scattering t
Note that the differential equation now describes the *average* velocity of the electrons in the ensemble.
## Sommerfeld model
### Exercise 1: the $n$-dimensional free electron model.
In the Sommerfeld lecture, it has been explained that the density of states of the free electron model is proportional to $1/\sqrt{\varepsilon}$ in 1D, constant in 2D, and proportional to $\sqrt{\varepsilon}$ in 3D. In this exercise, we are going to derive the density of states of the free electron model for an arbitrary number of dimensions.
Suppose we have a $n$-dimensional hypercube with length $L$ for each side that houses free electrons.
1. What is the distance between nearest-neighbour points in $\mathbf{k}$-space? Assume periodic boundary conditions.
What is the density of $\mathbf{k}$-points in n-dimensional $\mathbf{k}$-space?
2. The number of $\mathbf{k}$-points 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
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 density of states (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 density of states, 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 2: A hypothetical material
A hypothetical metal has a Fermi energy $\varepsilon_F = 5.2 \,\mathrm{eV}$ and a density of states $g(\varepsilon) = 2 \times 10^{10} \,\mathrm{eV}^{-\frac{3}{2}} \sqrt{\varepsilon}$.
1. Give an integral expression for the total energy of the electrons in this hypothetical material in terms of the density of states $g(\varepsilon)$, the temperature $T$ and the chemical potential $\mu = \varepsilon_F$.
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)(the first equation is all you need and you can neglect the $O\left(\frac{1}{\beta \mu}\right)^{4}$ term).
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 python with [`scipy.integrate.quad(func, xmin, xmax)`](https://docs.scipy.org/doc/scipy/reference/generated/scipy.integrate.quad.html)
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
