@@ -29,23 +29,37 @@ _(based on chapter 4 of the book)_
- calculate the electron contribution to the specific heat of a solid.
- describe central 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?"
## Quantization of k-space and density of states in the free electron model
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:
Atoms in a metal provide conduction electrons from their outer shells (often s-shells). These electrons can be described as waves, analogous to phonons. The Hamiltonian of a free electron is:
| | Phonons | Electrons |
| - | - | - |
| Governed by | Wave equation | Schrödinger equation |
The difference with phonons is that electrons are _fermions_, implying that there are only 2 electron states (due to spin) per $k$-value.
At $T=0$ the states with zero energy get completely filled up because the electrons cannot disappear.
Our first goal is to compute the density of states. We start by expressing the total number of states $N$ as an integral over k-space. Assuming three dimensions and spherical symmetry (the dispersion in the free electron model is isotropic), we find
## Density of states
Our first goal is to compute the density of states, which we will do now using a more explicit algorithm.
By definition density of states is the number of states per energy interval.
Therefore if we compute the *total* number of states $N(ε)$ with energy lower than $ε$, then $g(ε) = dN(ε)/dε$.
Let us apply this to the 3D case for a start.
Assuming three dimensions and spherical symmetry (the dispersion in the free electron model is isotropic), we find
Given the number of electrons in a system, we can now fill up these states starting from the lowest energy until we run out of electrons, at which point we reach the _Fermi energy_.
Note that $N$ now denotes the total number of electrons in the system.
Alternatively, we can express $N$ as an integral over k-space up to the _Fermi wavenumber_, which is the wavenumber associated with the Fermi energy $k_{\rm F}=\sqrt{2m\varepsilon_{\rm F}}/\hbar$
Alternatively, we can express $N$ as an integral over k-space up to the _Fermi wavenumber_, which is the wavenumber associated with the Fermi energy $k_{\rm F}=\sqrt{2mε_{\rm F}}/\hbar$
From the last equation it follows that the _Fermi wavelength_ $\lambda_{\rm F}\equiv 2\pi/k_{\rm F}$ is on the order of the atomic spacing for typical free electron densities in metals.
The bold line represents all filled states at $T=0$. This is called the _Fermi sea_.
New concept: _Fermi surface_ = all points in k-space with $\varepsilon=\varepsilon_{\rm F}$. For free electrons in 3D, the Fermi surface is the surface of a sphere.
New concept: _Fermi surface_ = all points in k-space with $ε=ε_{\rm F}$. For free electrons in 3D, the Fermi surface is the surface of a sphere.
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@@ -185,16 +202,16 @@ The orange circle represents the Fermi surface at finite current $\rightarrow$ t
## Finite temperature, heat capacity
We now extend our discussion to $T>0$ by including a temperature dependent occupation function $n_F(\varepsilon,T)$ into our expression for the total number of electrons:
We now extend our discussion to $T>0$ by including a temperature dependent occupation function $n_F(ε,T)$ into our expression for the total number of electrons:
where the chemical potential $\mu=\varepsilon_{\rm F}$ if $T=0$. Typically $\varepsilon_{\rm F}/k_{\rm B}$~70 000 K (~7 eV), whereas room temperature is only 300 K (~30 meV). Therefore, thermal smearing occurs only very close to the Fermi energy.
where the chemical potential $\mu=ε_{\rm F}$ if $T=0$. Typically $ε_{\rm F}/k_{\rm B}$~70 000 K (~7 eV), whereas room temperature is only 300 K (~30 meV). Therefore, thermal smearing occurs only very close to the Fermi energy.
Having included the temperature dependence, we can now calculate the electronic contribution to the heat capacity $C_{\rm V,e}$.
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@@ -225,7 +242,7 @@ Having included the temperature dependence, we can now calculate the electronic
We will estimate $C_{\rm V,e}$ and derive its scaling with temperature using the triangle method depicted in the figure. A finite temperature causes electrons in the top triangle to be excited to the bottom triangle. Because the base of this triangle scales with $k_{\rm B}T$ and its height with $ g(\varepsilon_{\rm F})$, it follows that the number of excited electrons $N_{\rm exc} \approx g(\varepsilon_{\rm F})k_{\rm B}T$ (neglecting pre-factors of order 1).
We will estimate $C_{\rm V,e}$ and derive its scaling with temperature using the triangle method depicted in the figure. A finite temperature causes electrons in the top triangle to be excited to the bottom triangle. Because the base of this triangle scales with $k_{\rm B}T$ and its height with $ g(ε_{\rm F})$, it follows that the number of excited electrons $N_{\rm exc} \approx g(ε_{\rm F})k_{\rm B}T$ (neglecting pre-factors of order 1).
These electrons gain $k_{\rm B}T$ of energy, so the total extra energy is
where we used $N=\frac{2}{3}\varepsilon_{\rm F}g(\varepsilon_{\rm F})$ and we defined the _Fermi temperature_ $T_{\rm F}=\frac{\varepsilon_{\rm F}}{k_{\rm B}}$.
where we used $N=\frac{2}{3}ε_{\rm F}g(ε_{\rm F})$ and we defined the _Fermi temperature_ $T_{\rm F}=\frac{ε_{\rm F}}{k_{\rm B}}$.
How does $C_{\rm V,e}$ relate to the phonon contribution $C_{\rm V,p}$?
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@@ -275,19 +292,19 @@ Behavior of $C_{\rm V}$ can be very quickly memorized or understood using the fo
#### Example 1: electrons
$g(\varepsilon_{\rm F})$ roughly constant ⇒ total energy in the thermal state is $T \times [T\times g(\varepsilon_{\rm F})]$ ⇒ $C_{\rm V} \propto T$.
$g(ε_{\rm F})$ roughly constant ⇒ total energy in the thermal state is $T \times [T\times g(ε_{\rm F})]$ ⇒ $C_{\rm V} \propto T$.
#### Example 2: graphene with $E_F=0$ (midterm 2018)
$g(\varepsilon) \propto \varepsilon$ ⇒ total energy is $T \times T^2$ ⇒ $C_{\rm V} \propto T^2$.
$g(ε) \propto ε$ ⇒ total energy is $T \times T^2$ ⇒ $C_{\rm V} \propto T^2$.
#### Example 3: phonons in 3D at low temperatures.
$g(\varepsilon) \propto \varepsilon^2$ ⇒ total energy is $T \times T^3$ ⇒ $C_{\rm V} \propto T^3$.
$g(ε) \propto ε^2$ ⇒ total energy is $T \times T^3$ ⇒ $C_{\rm V} \propto T^3$.
## Conclusions
1. The Sommerfeld free electron model treats electrons as waves with dispersion $\varepsilon=\frac{\hbar^2k^2}{2m}$.
1. The Sommerfeld free electron model treats electrons as waves with dispersion $ε=\frac{\hbar^2k^2}{2m}$.
2. The density of states (DOS) can be derived from the dispersion relation. This procedure is general, and analogous to e.g. that for phonons (see lecture 2 - Debye model).
3. The Fermi-Dirac distribution describes the probability an electron state is occupied.
4. The free-electron heat capacity is linear with $T$.
