@@ -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 $k_{\rmF}=\sqrt{2m\varepsilon_{\rmF}}/\hbar$
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 $k_\mathrm{F}=\sqrt{2mε_\mathrm{F}}/\hbar$
$$
N \overset{\mathrm{3D}}{=} 2\frac{L^3}{(2\pi)^3}\int_0^{k_{\rmF}} 4\pi k^2{\rmd}k= 2\frac{V}{(2\pi)^3} \frac{4}{3}\pi k_{\rmF}^3
N \overset{\mathrm{3D}}{=} 2\frac{L^3}{(2\pi)^3}\int_0^{k_\mathrm{F}} 4\pi k^2\mathrm{d}k= 2\frac{V}{(2\pi)^3} \frac{4}{3}\pi k_\mathrm{F}^3
$$
These equations allow us to relate $\varepsilon_{\rmF}$ and $k_{\rmF}$ to the electron density $N/V$:
These equations allow us to relate $ε_\mathrm{F}$ and $k_\mathrm{F}$ to the electron density $N/V$:
From the last equation it follows that the _Fermi wavelength_ $\lambda_{\rmF}\equiv 2\pi/k_{\rmF}$ is on the order of the atomic spacing for typical free electron densities in metals.
From the last equation it follows that the _Fermi wavelength_ $\lambda_\mathrm{F}\equiv 2\pi/k_\mathrm{F}$ is on the order of the atomic spacing for typical free electron densities in metals.
Example: For copper, the Fermi energy is ~7 eV. It would take a temperature of $\sim 70 000$K for electrons to gain such energy through a thermal excitation! The _Fermi velocity_ $v_{\rmF}=\frac{\hbar k_{\rmF}}{m}\approx$ 1750 km/s $\rightarrow$ electrons run with a significant fraction of the speed of light, only because lower energy states are already filled by other electrons.
Example: For copper, the Fermi energy is ~7 eV. It would take a temperature of $\sim 70 000$K for electrons to gain such energy through a thermal excitation! The _Fermi velocity_ $v_\mathrm{F}=\frac{\hbar k_\mathrm{F}}{m}\approx$ 1750 km/s $\rightarrow$ electrons run with a significant fraction of the speed of light, only because lower energy states are already filled by other electrons.
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 $ε=ε_\mathrm{F}$. For free electrons in 3D, the Fermi surface is the surface of a sphere.
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@@ -185,16 +201,14 @@ 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 $\varepsilon_{\rmF}/k_{\rmB}$~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 $\mu=ε_\mathrm{F}$ if $T=0$. Typically $ε_\mathrm{F}/k_\mathrm{B}$~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 $C_{\rmV,e}$.
Having included the temperature dependence, we can now calculate the electronic contribution to the heat capacity $C_\mathrm{V,e}$.
We will estimate $C_{\rmV,e}$ 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 $k_{\rmB}T$ and its height with $ g(\varepsilon_{\rm F})$, it follows that the number of excited electrons $N_{\rmexc} \approx g(\varepsilon_{\rm F})k_{\rmB}T$ (neglecting pre-factors of order 1).
We will estimate $C_\mathrm{V,e}$ 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 $k_\mathrm{B}T$ and its height with $ g(ε_\mathrm{F})$, it follows that the number of excited electrons $N_\mathrm{exc} \approx g(ε_\mathrm{F})k_\mathrm{B}T$ (neglecting pre-factors of order 1).
These electrons gain $k_{\rmB}T$ of energy, so the total extra energy is
These electrons gain $k_\mathrm{B}T$ of energy, so the total extra energy is
where we used $N=\frac{2}{3}\varepsilon_{\rmF}g(\varepsilon_{\rm F})$ and we defined the _Fermi temperature_ $T_{\rmF}=\frac{\varepsilon_{\rm F}}{k_{\rmB}}$.
where we used $N=\frac{2}{3}ε_\mathrm{F}g(ε_\mathrm{F})$ and we defined the _Fermi temperature_ $T_\mathrm{F}=\frac{ε_\mathrm{F}}{k_\mathrm{B}}$.
How does $C_{\rmV,e}$ relate to the phonon contribution $C_{\rmV,p}$?
How does $C_\mathrm{V,e}$ relate to the phonon contribution $C_\mathrm{V,p}$?
- At room temperature, $C_{\rmV,p}=3Nk_{\rmB}\gg C_{\rmV,e}$
- Near $T=0$, $C_{\rmV,p}\propto T^3$ and $C_{\rmV,e}\propto T$ $\rightarrow$ competition.
- At room temperature, $C_\mathrm{V,p}=3Nk_\mathrm{B}\gg C_\mathrm{V,e}$
- Near $T=0$, $C_\mathrm{V,p}\propto T^3$ and $C_\mathrm{V,e}\propto T$ $\rightarrow$ competition.
## Useful trick: scaling of $C_V$
Behavior of $C_{\rmV}$ can be very quickly memorized or understood using the following mnemonic rule
Behavior of $C_\mathrm{V}$ can be very quickly memorized or understood using the following mnemonic rule
> Particles in within an energy range ~$kT$ become thermally excited, and each carries extra energy $kT$.
#### Example 1: electrons
$g(\varepsilon_{\rm F})$ roughly constant ⇒ total energy in the thermal state is $T \times [T\times g(\varepsilon_{\rmF})]$ ⇒ $C_{\rmV} \propto T$.
$g(ε_\mathrm{F})$ roughly constant ⇒ total energy in the thermal state is $T \times [T\times g(ε_\mathrm{F})]$ ⇒ $C_\mathrm{V} \propto T$.
#### Example 2: graphene with $E_F=0$ (midterm 2018)
$g(\varepsilon) \propto \varepsilon$ ⇒ total energy is $T \times T^2$ ⇒ $C_{\rmV} \propto T^2$.
$g(ε) \propto ε$ ⇒ total energy is $T \times T^2$ ⇒ $C_\mathrm{V} \propto T^2$.
#### Example 3: phonons in 3D at low temperatures.
$g(\varepsilon) \propto \varepsilon^2$ ⇒ total energy is $T \times T^3$ ⇒ $C_{\rmV} \propto T^3$.
$g(ε) \propto ε^2$ ⇒ total energy is $T \times T^3$ ⇒ $C_\mathrm{V} \propto T^3$.
## 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 $T$.
5. The scaling of heat capacity with $T$ can be derived quickly by estimating the nr of particles in an energy range $k_{\rmB}T$, using the DOS.
5. The scaling of heat capacity with $T$ can be derived quickly by estimating the nr of particles in an energy range $k_\mathrm{B}T$, using the DOS.
## Exercises
#### Exercise 1: potassium
### 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 $g(E)$ and the occupation number $n(E, T)$.
3. Explain what happens if a material is heated up to its Fermi temperature (assuming that materials where it is possible exist).
### 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.
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.
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.
### 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.
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@@ -324,7 +345,7 @@ Suppose we have an $n$-dimensional hypercube with length $L$ for each side and c
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
### 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 $\mu = \epsilon_F$.
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@@ -338,7 +359,7 @@ A hypothetical metal has a Fermi energy $\epsilon_F = 5.2 \, \mathrm{eV}$ and a
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
### 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 $\epsilon_F = 0$.
