Sommerfeld Free Electron Gas Theory: The Quantum Approach
The Drude model provided a good start for understanding metals, but it failed catastrophically when predicting thermal properties like specific heat. Arnold Sommerfeld solved these issues not by adding new forces, but by treating electrons as quantum mechanical particles (fermions) that must obey the Pauli Exclusion Principle. The result is the Free Electron Gas model, which forms the foundation of modern solid-state physics.
1. Motivation and Context
The Drude Failure: Drude treated electrons as a classical gas obeying Maxwell-Boltzmann statistics. This predicted that every electron should contribute \(\frac{3}{2}k_B\) to the heat capacity. Experimentally, the electronic heat capacity is about 100 times smaller at room temperature.
The Sommerfeld Fix:
• Keep: The idea that electrons are "free" (no interaction with ions or each other).
• Change: Electrons are indistinguishable fermions with spin-1/2. They obey Fermi-Dirac statistics, not classical statistics.
2. Fundamental Assumptions
- Free Electrons: The potential \(V(\mathbf{r})\) inside the metal is assumed to be constant (usually zero). The periodic lattice is ignored.
- Pauli Exclusion: No two electrons can occupy the same quantum state.
- Spin Degeneracy: Each spatial quantum state can hold exactly two electrons (one spin up, one spin down).
- Relaxation Time: Scattering still exists (phenomenological \(\tau\)), but the dynamics between collisions are quantum mechanical.
3. Quantum Mechanical Description
We solve the time-independent Schrödinger equation for a free particle in a box of volume \(V = L^3\).
The solution is a plane wave: \[ \psi_{\mathbf{k}}(\mathbf{r}) = \frac{1}{\sqrt{V}} e^{i\mathbf{k}\cdot\mathbf{r}} \]
The energy eigenvalues depend on the wavevector \(\mathbf{k}\): \[ E(\mathbf{k}) = \frac{\hbar^2 k^2}{2m} \] The momentum is \(\mathbf{p} = \hbar \mathbf{k}\).
4. Boundary Conditions and Quantization
To avoid surface effects, we use Born-von Karman (Periodic) Boundary Conditions: \[ \psi(x+L, y, z) = \psi(x, y, z) \]
This restricts the allowed values of the wavevector components \(k_x, k_y, k_z\) to discrete values: \[ k_x = \frac{2\pi n_x}{L}, \quad k_y = \frac{2\pi n_y}{L}, \quad k_z = \frac{2\pi n_z}{L} \] where \(n_x, n_y, n_z\) are integers.
Volume per state in k-space: Since the separation between allowed \(k\) values is \(2\pi/L\), the volume occupied by a single state in k-space is: \[ (\Delta k)^3 = \left(\frac{2\pi}{L}\right)^3 = \frac{8\pi^3}{V} \]
5. Density of States (DOS) Derivation
The Density of States, \(g(E)\), tells us how many electron states exist at a specific energy level. We derive this in two steps.
Step A: Counting States in k-space
Consider a spherical shell in k-space with radius \(k\) and thickness \(dk\). The volume of this shell is \(4\pi k^2 dk\).
The number of allowed states \(dN\) in this shell is:
(The factor of 2 accounts for spin degeneracy.)
\[ dN = 2 \times \frac{4\pi k^2 dk}{(2\pi/L)^3} = 2 \frac{4\pi k^2 dk}{8\pi^3/V} \] \[ dN = \frac{V}{\pi^2} k^2 dk \]Thus, the density of states in k-space is \(g(k) = \frac{V}{\pi^2} k^2\).
Step B: Converting to Energy Space
We need \(g(E)\), defined such that \(dN = g(E)dE\).
From the dispersion relation \(E = \frac{\hbar^2 k^2}{2m}\), we solve for \(k\):
\[ k = \sqrt{\frac{2mE}{\hbar^2}} = \frac{\sqrt{2m}}{\hbar} E^{1/2} \]
Differentiating with respect to \(E\): \[ \frac{dk}{dE} = \frac{\sqrt{2m}}{\hbar} \cdot \frac{1}{2} E^{-1/2} = \frac{1}{\hbar} \sqrt{\frac{m}{2E}} \]
Now substitute \(k\) and \(dk/dE\) into \(g(E) = g(k) \frac{dk}{dE}\):
\[ g(E) = \left( \frac{V}{\pi^2} k^2 \right) \left( \frac{dk}{dE} \right) \] \[ g(E) = \frac{V}{\pi^2} \left( \frac{2mE}{\hbar^2} \right) \left( \frac{1}{\hbar}\sqrt{\frac{m}{2E}} \right) \]Simplifying leads to the famous result:
6. The Fermi Energy (\(E_F\))
At absolute zero (\(T=0\)), electrons fill the lowest energy states first up to a maximum energy called the Fermi Energy (\(E_F\)).
The total number of electrons \(N\) is the integral of the DOS up to \(E_F\):
Substituting our expression for \(g(E)\):
\[ N = \frac{V}{2\pi^2} \left( \frac{2m}{\hbar^2} \right)^{3/2} \int_0^{E_F} E^{1/2} dE \]Perform the integration \(\int E^{1/2} dE = \frac{2}{3} E^{3/2}\):
\[ N = \frac{V}{2\pi^2} \left( \frac{2m}{\hbar^2} \right)^{3/2} \left( \frac{2}{3} E_F^{3/2} \right) \] \[ N = \frac{V}{3\pi^2} \left( \frac{2m}{\hbar^2} \right)^{3/2} E_F^{3/2} \]Now, we define the electron density \(n = N/V\) and solve for \(E_F\):
Associated with \(E_F\) are the Fermi Wavevector \(k_F\) and Fermi Temperature \(T_F\): \[ k_F = (3\pi^2 n)^{1/3} \] \[ T_F = \frac{E_F}{k_B} \]
For typical metals, \(T_F \approx 10,000 \text{ K}\) to \(50,000 \text{ K}\). Since room temperature \(T \ll T_F\), the electron gas is highly degenerate.
7. Fermi-Dirac Statistics
At non-zero temperatures, the probability of a state with energy \(E\) being occupied is given by the Fermi-Dirac distribution:
where \(\mu\) is the chemical potential.
At T=0: \(f(E)\) is a step function (1 below \(E_F\), 0 above).
At T>0: The step softens. Only electrons within a range of \(\sim k_B T\) near \(E_F\) can be thermally excited.
8. Physical Consequences
(a) Electronic Specific Heat
Classically, all electrons gain thermal energy. In the Sommerfeld model, only the tiny fraction of electrons near \(E_F\) (roughly \(T/T_F\)) can absorb thermal energy.
This leads to a specific heat that is linear in temperature:
\[ C_e = \gamma T \]
This matches experimental data perfectly at low temperatures and explains why the electronic contribution is negligible at room temperature.
(b) Pauli Paramagnetism
External magnetic fields affect electron spins. Only those near the Fermi surface can flip their spins to align with the field, leading to a weak, temperature-independent paramagnetic susceptibility.
9. Limitations
While successful, the Free Electron Model has limits:
- It assumes a constant potential, ignoring the periodic lattice. Therefore, it cannot explain Band Gaps or the difference between insulators and semiconductors.
- It fails for transition metals where d-electrons are not "free".
- It ignores electron-electron interactions (which are treated in Fermi Liquid Theory).