Before the start of this lecture, you should be able to:
- Write down the Fermi-Dirac distribution function.
- Write down the Schrödinger equation and solve it in free space.
- Write down the solutions to the Schrödinger Equation for a free particle.
- Apply periodic boundary conditions, and compute the density of states
given a sufficiently simple dispersion relation.
@@ -23,39 +23,63 @@ _(based on chapter 4 of the book)_
After this lecture you will be able to:
- calculate the electron density of states in 1D, 2D, and 3D using the Sommerfeld free-electron model.
- express the number and energy of electrons in a system in terms of integrals over k-space.
- use the Fermi distribution to extend the previous learning goal to finite T.
- calculate the electron contribution to the specific heat of a solid.
- describe central terms such as the Fermi energy, Fermi temperature, and Fermi wavevector.
- Calculate the electron density of states in 1D, 2D, and 3D using the Sommerfeld free-electron model.
- Express the number and energy of electrons in a system in terms of integrals over k-space for $T = 0$.
- Use the Fermi-Dirac distribution to extend the previous learning goal to $T>0$.
- Calculate the electron contribution to the specific heat of a solid.
- Describe terms such as the Fermi energy, Fermi temperature, and Fermi wavevector.
## Electrons vs phonons
> Two electrons are sitting on a bench. Another one approaches and asks: "May I join you guys?"
> The first two immediately reply: "Who do you think we are? Bosons?"
Having learned the statistical properties of phonons and the [Debye model](2_debye_model.md), let us use these as a starting point for comparing with the electrons.
Here is a table comparing the most important properties of electrons and phonons:
Having learned the statistical properties of [phonons in the Debye model](2_debye_model.md), let us use these as a starting point for comparing with the electrons.
We consider the electrons as _free particles_ in a cubic box of size $L^3$ with periodic boundary conditions.
In that case, the solutions to the Schrödinger equation will be a plane waves
$$
\psi ∝ \exp(i\mathbf{k} \cdot \mathbf{r})
$$
where $\mathbf{k}$ is the electron's wavevector.
As a result of the periodic boundary conditions, $\mathbf{k}$ must take values $\frac{2\pi}{L} (n_x, n_y, n_z)$.
The plane wave's corresponding eigenenergies are given by the following dispersion relation
where $\beta = \frac{1}{k_{\rm B}T}$, \epsilon is the energy of the electrons, described by the previously mentioned dispersion relations, and $\mu$ is the electron's _chemical potential_.
Using the Fermi-Dirac distribution, we can write the number of electrons in the system as
$$
\begin{align}
N &= 2_s \sum_{\mathbf{k}} n_{FD}(\beta(\epsilon-\mu))\\
&\approx 2_s \left( \frac{L}{2 \pi} \right)^3 \int d \mathbf{k} n_{FD}(\beta(\epsilon-\mu))
\end{align}
$$
The factor $2_s$ accounts for the spin degeneracy.
In similar fashion the expression for the total energy is given by
$$
E = 2_s \left( \frac{L}{2 \pi} \right)^3 \int d \mathbf{k} \epsilon(\mathbf{k}) n_{FD}(\beta(\epsilon-\mu))
$$
Note that this expression is extremely similar to the expression of the total energy of phonons in the Debye model.
The only differences are that we used $n_{FD}$ instead of $n_{BE}$ and a factor 2 instead of 3.
In the table below we summarize the properties of both phonons and electrons
<!---
Move this table downwards and also remove the schrodinger and wave equation.
-->
| | Phonons | Electrons |
| - | - | - |
| Governed by | Wave equation | Schrödinger equation |